Applied Well Test Interpretation - John Lee

March 19, 2018 | Author: Ali Khaleel Farouk | Category: Petroleum Reservoir, Permeability (Earth Sciences), Nature, Mathematics, Science

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Applied Well Test Interpretation

Applied Well Test Interpretation

John P. Spivey Phoenix Reservoir Engineering W. John Lee University of Houston

Society of Petroleum Engineers

© Copyright 2013 Society of Petroleum Engineers All rights reserved. No portion of this book may be reproduced in any form or by any means, including electronic storage and retrieval systems, except by explicit, prior written permission of the publisher except for brief passages excerpted for review and critical purposes.

Disclaimer This book was prepared by members of the Society of Petroleum Engineers and their well-qualified colleagues from material published in the recognized technical literature and from their own individual experience and expertise. While the material presented is believed to be based on sound technical knowledge, neither the Society of Petroleum Engineers nor any of the authors or editors herein provide a warranty either expressed or implied in its application. Correspondingly, the discussion of materials, methods, or techniques that may be covered by letters patents implies no freedom to use such materials, methods, or techniques without permission through appropriate licensing. Nothing described within this book should be construed to lessen the need to apply sound engineering judgment nor to carefully apply accepted engineering practices in the design, implementation, or application of the techniques described herein.

ISBN 978-1-61399-307-1

Society of Petroleum Engineers 222 Palisades Creek Drive Richardson, TX 75080-2040 USA http://www.spe.org/store [email protected] 1.972.952.9393

Introduction The objective of this textbook is to introduce the reader to the fundamentals of applied well test interpretation. The text focuses on the most basic well testing scenario: a single-well test on a well producing a single-phase fluid, from a single-layer, homogeneous reservoir. Although simple, this scenario illustrates most of the elements required for interpretation in more complex scenarios. Chapter 1—Introduction to Applied Well Test Analysis opens with an overview of different types of well tests, common applications and objectives in well testing, and alternatives to conventional testing. The chapter continues with a review of reservoir rock and fluid properties and ends with a brief discussion of the effects of graphical scales on data presentation. Chapter 2—Fluid Flow in Porous Media covers the assumptions on which the diffusivity equation is based, then introduces the concepts of superposition in space, superposition in time, and radius of investigation. The remainder of the chapter focuses on the applied topics of wellbore damage and stimulation, pseudosteady-state flow, and wellbore storage. Chapter 3—Radial Flow Semilog Analysis introduces semilog methods for estimating permeability and skin factor from data in infinite-acting radial flow for both drawdown and buildup tests. The chapter also discusses classical methods of estimating average reservoir pressure using the semilog plot. Chapter 4—Log-Log Type Curve Analysis discusses the Gringarten-Bourdet pressure and pressure derivative type curves and the log-log field data plot. The chapter discusses use of the log-log plot to qualitatively evaluate whether or not a well is damaged or stimulated, and to identify wellbore storage and the infinite-acting radial flow period. The chapter covers estimation of permeability, skin factor, and wellbore storage coefficient from the loglog field data plot without using type curves. The chapter closes with a discussion of some common methods used to calculate the logarithmic derivative from field data. Chapter 5—Pressure Transient Testing for Gas Wells introduces the real-gas pseudopressure and pseudotime transforms and their normalized counterparts, adjusted pressure and adjusted time, to allow the use of methods developed for slightly compressible liquids to be used for analysis of gas well test data. Chapter 6—Flow Regimes and the Diagnostic Plot introduces the common flow regimes and the use of the standard log-log and flow-regime specific diagnostic plots for flow-regime identification. For each flow regime, examples of one or more reservoir models that exhibit the flow regime are given. Chapter 7—Bounded Reservoir Behavior covers the most common models of single-layer reservoir behavior. For each model, the flow regimes that may be exhibited and the order in which they occur are discussed. Chapter 8—Variable Flow Rate History discusses various methods for treating a variable flow-rate history, from ignoring prior history for short buildups following long drawdowns to deconvolution. The chapter discusses the effects of some common types of boundaries on the shape of the log-log buildup test for different ways of plotting the data. The chapter then provides a spatial interpretation of a variable rate history as a pressure profile in the reservoir. The effects of rate history on a subsequent buildup are discussed, as are the differences between the drawdown and buildup responses for a well in a closed reservoir, a reservoir with a constant-pressure boundary, and a radial composite reservoir. A method for graphing the rate history preceding a buildup along with the pressure response during the buildup is introduced to help distinguish rate-history-induced features in the derivative from those caused by boundaries. Chapter 9—Wellbore Phenomena addresses an issue that impacts any well test to one degree or another, yet has received only sporadic attention in the well testing literature. A number of different wellbore phenomena that

may affect the shape of the pressure response, such as changing wellbore storage, a rising or falling fluid interface, and completion cleanup, are discussed. In addition, other phenomena that affect the pressure response but have no impact on well productivity (such as pressure fluctuations from earth tides or daily changes in wellhead temperature, gauge problems, or data processing artifacts) are also addressed. Chapter 10—Near-Wellbore Phenomena covers phenomena present in the near-wellbore area that do impact the well performance, including geometric skin factor for a perforated completion, a limited-entry or partial penetration completion, or a deviated well, and non-Darcy skin factor for both drawdown and buildup. Chapter 11—Well Test Interpretation Workflow presents a recommended workflow (more accurately, a workflow framework or checklist) for well test interpretation. The major steps are the same for virtually any well test interpretation: collect the data, QC the data, identify flow regimes, select a reservoir model, estimate model parameters, and validate the results. Chapter 12—Well Test Design Workflow presents a recommended workflow for well test design. As with well test interpretation, the major steps in well test design are the same for most situations: define the test objectives, collect data, estimate unknown reservoir properties, estimate test duration, estimate test flow rate, and determine flow rate sequence Disclaimer. The phrases “recommended procedure,” “recommended practice,” or other similar phrases refer to procedures or practices recommended by the authors and do not imply endorsement by the Society of Petroleum Engineers.

Acknowledgments I’d like to thank all of the representatives of the various providers of transducers and gauges who helped educate me on the operating principles of the transducers used in today’s gauges. Two in particular replied very patiently to my many questions and multiple e-mails: Jason Blackburn with QuartzDyne, Inc. and Sheldon Nadeau with DataCan. I’d also like to thank those who have taken the PetroSkills Well Test Design and Analysis short course during the last 7 years, as well as my fellow instructors, Dr. Iskander Diyashev and Dr. Rosalind Archer, for their valuable feedback on early drafts of the text. John Spivey Thank you to the SPE staff who worked on the book, most notably Mattie Tanner and Jennifer Wegman. John Lee

To all of the teachers, professors, and mentors who have encouraged, challenged, and inspired me throughout my life. – John Spivey To my former and present students at Mississippi State University, Texas A&M University, and the University of Houston. – John Lee

Contents Introduction�� v Acknowledgments�� vii 1. Introduction to Applied Well Test Interpretation�� 1 1.1 Introduction�� 1 1.2 Applications of Well Testing�� 2 1.3 Alternatives to Conventional Well Testing�� 3 1.4 Forward and Inverse Problems�� 5 1.5 Well Test Interpretation Methods�� 6 1.6 Rock and Fluid Properties�� 6 1.7 Graph Scales�� 15 1.8 Summary�� 19 Nomenclature�� 19 2. Fluid Flow in Porous Media�� 21 2.1 Introduction�� 21 2.2 Steady-State Flow�� 22 2.3 Development of the Diffusivity Equation�� 24 2.4 Infinite-Acting Radial Flow—Ei-Function Solution�� 26 2.5 Principle of Superposition�� 26 2.6 Radius of Investigation�� 30 2.7 Damage and Stimulation�� 34 2.8 Pseudosteady-State Flow�� 36 2.9 Wellbore Storage�� 44 2.10 Summary�� 47 Nomenclature�� 48 3. Radial Flow Semilog Analysis�� 49 3.1 Introduction�� 49 3.2 Drawdown Tests�� 50 3.3 Buildup Test Following Constant-Rate Production�� 57 3.4 Estimating Average Reservoir Pressure�� 65 3.5 Flow Rate Variations Before Shut-In�� 72 3.6 Summary�� 74 Nomenclature�� 75 4. Log-Log Type Curve Analysis�� 77 4.1 Introduction�� 77 4.2 Dimensionless Variables�� 78 4.3 Gringarten-Bourdet Type Curves�� 79 4.4 Manual Parameter Estimation Using the Log-Log Plot�� 90 4.5 Calculating the Logarithmic Derivative From Field Data�� 93 4.6 Summary�� 96 Nomenclature�� 97 5. Pressure Transient Testing for Gas Wells�� 99 5.1 Introduction�� 99 5.2 Gas Flow Equation�� 99 5.3 Gas-Well Drawdown Tests�� 103 5.4 Gas-Well Buildup Tests�� 110 5.5 Summary�� 120 Nomenclature�� 120

6. Flow Regimes and the Diagnostic Plot�� 123 6.1 Introduction�� 123 6.2 Power-Law Function Pressure Response�� 124 6.3 Radial Flow�� 126 6.4 Linear Flow�� 131 6.5 Volumetric Behavior�� 140 6.6 Spherical Flow�� 145 6.7 Bilinear Flow�� 152 6.8 Other Flow Regimes�� 155 6.9 Practical Aspects of Flow Regime Identification�� 156 6.10 Summary�� 158 Nomenclature�� 159 7. Bounded Reservoir Behavior�� 161 7.1 Introduction�� 161 7.2 Types of Boundaries�� 162 7.3 Linear Boundaries�� 165 7.4 Circular Reservoir Boundaries�� 169 7.5 Multiple Linear Boundaries�� 172 7.6 Composite Reservoir Models�� 179 7.7 Summary�� 184 Nomenclature�� 184 8. Variable Flow Rate History�� 187 8.1 Introduction�� 187 8.2 Methods for Variable Rate/Variable Pressure Problems�� 188 8.3 Effect of Boundaries on a Subsequent Buildup�� 197 8.4 Spatial Interpretation of Flow Rate History�� 204 8.5 Effect of a Change in Flow Rate on a Subsequent Buildup Test�� 208 8.6 Convolution and Deconvolution�� 219 8.7 Rate Normalization�� 220 8.8 Deconvolution�� 222 8.9 Summary�� 227 Nomenclature�� 228 9. Wellbore Phenomena�� 231 9.1 Introduction�� 231 9.2 Variable Wellbore Storage�� 232 9.3 Cleanup�� 239 9.4 Movement of Gas-Liquid Interface in Wellbore�� 240 9.5 Activity During Operations�� 244 9.6 Pressure Oscillations�� 248 9.7 Gauge Problems�� 249 9.8 Data Processing Errors and Artifacts�� 250 9.9 Distinguishing Wellbore Phenomena From Reservoir Phenomena�� 253 9.10 Summary�� 257 Nomenclature�� 258 10. Near-Wellbore Phenomena�� 259 10.1 Introduction�� 259 10.2 Finite-Thickness Skin Factor�� 259 10.3 Perforated Completion�� 260 10.4 Partial Penetration/Limited Entry Completion�� 261 10.5 Deviated Wellbore�� 262 10.6 Rate-Dependent Skin Factor�� 265 10.7 Rate-Dependent Skin Factor and Wellbore Storage�� 268 10.8 Estimating Non-Darcy Coefficient D From Multiple Transient Tests�� 270 10.9 Summary�� 270 Nomenclature�� 271

11. Well Test Interpretation Workflow�� 273 11.1 Introduction�� 273 11.2 Collect Data�� 273 11.3 Review and Quality Control Data�� 279 11.4 Deconvolve Data�� 284 11.5 Identify Flow Regimes�� 284 11.6 Select Reservoir Model�� 285 11.7 Estimate Model Parameters�� 287 11.8 Simulate or History-Match Pressure Response�� 289 11.9 Calculate Confidence Intervals�� 295 11.10 Interpret Model Parameters�� 301 11.11 Validate Results�� 303 11.12 Field Example�� 306 11.13 Summary�� 316 Nomenclature�� 317 12. Well Test Design Workflow�� 319 12.1 Introduction�� 319 12.2 Typical Test Design Scenarios�� 319 12.3 Define Test Objectives�� 323 12.4 Consider Alternatives to Conventional Well Testing�� 325 12.5 Collect Data�� 327 12.6 Estimate Reservoir Properties�� 327 12.7 Permeability Estimates�� 328 12.8 Estimate Test Duration To Reach Desired Flow Regime�� 331 12.9 Estimate Test Duration Based on Economics�� 336 12.10 Estimate Test Rate and Determine Flow Rate Sequence�� 340 12.11 Estimate Magnitude of Pressure Response�� 344 12.12 Select Gauges�� 345 12.13 Simulate Test�� 349 12.14 Design Example�� 350 12.15 Summary�� 356 Nomenclature�� 356 References�� 359 Author Index�� 367 Subject Index�� 369

Chapter 1

Introduction to Applied Well Test Interpretation “‘Begin at the beginning,’ the King said gravely, ‘and go on till you come to the end: then stop.’” —Lewis Carroll, Alice’s Adventures in Wonderland (1865) “The last thing one settles in writing a book is what one should put in first.” —Blaise Pascal, Pensées 1.1 Introduction Well test interpretation is the process of obtaining information about a reservoir through examining and analyzing the pressure-transient response caused by a change in production rate. This information is used to make reservoirmanagement decisions. It is important to note that the information obtained from well test interpretation may be qualitative as well as quantitative. Identification of the presence and nature of a no-flow boundary or a down-dip aquifer is just as important as, if not more important than, estimating the distance to the boundary. This textbook focuses on well test interpretation rather than well test analysis. Much of the literature on well test analysis focuses on finding solutions to the transient flow equation for new boundary conditions, using advanced tools such as Bessel functions, Laplace and Fourier transforms, and Green functions. Even applied well test analysis is often thought of as simply estimating reservoir properties by drawing straight lines or matching curves. Interpretation, in contrast, seeks first to understand the nature of the reservoir that produced the pressure response, and only then to quantify the physical properties that describe the reservoir. Section 1.2 provides an overview of the types of well tests and applications and objectives of well test interpretation. Section 1.3 gives a brief description of several alternatives to conventional well testing, including wireline formation testing, permanent monitoring, rate-transient analysis, log analysis, and core analysis. Section 1.4 discusses well test interpretation as an inverse problem, where the system input (flow rate) and output (pressure response) are known, and the system (reservoir model) must be determined. Section 1.5 introduces the three categories of well test interpretation methods, straight-line analysis, type-curve matching, and simulation/history matching. Section 1.6 defines and discusses the major variables used in well test interpretation, including rock properties such as porosity, saturation, pore-volume compressibility, permeability, and net-pay or net sand thickness, fluid properties such as formation volume factor, viscosity, and compressibility, and other properties such as wellbore radius and total compressibility. Section 1.7 discusses some of the properties of the more common graph scales used in well test interpretation. After completing this chapter, you should be able to 1. List the major types of well tests and give the principal goal for each type. 2. List five alternatives to conventional well testing, and describe the advantages, disadvantages, and applications of each. 3. List the two major tasks of well test interpretation. 4. List the three classes of analysis methods used in well test interpretation. 5. Define and give typical values for each of the primary variables used in well test interpretation. 6. Describe the effect of graphing data on Cartesian, semilog, and log-log scales.

2 Applied Well Test Interpretation

1.2 Applications of Well Testing 1.2.1 Test Types. There are a wide range of types of well tests, as well as a number of different ways of categorizing well tests. Tests may be classified by the way in which they are carried out, whether they involve one or multiple wells, whether the test is run by opening up a well for production or shutting in a producing well, whether the well is used for production or injection, and whether the well is an exploration well or a well in a mature field. Deliverability Tests vs. Transient Tests. The primary objectives of deliverability tests are to obtain fluid samples, to determine well deliverability, to determine the well potential, or to develop inflow performance curves for system analysis. Primary objectives of transient tests may include to estimate in-situ permeability to oil or gas, to estimate average drainage area pressure, to evaluate stimulation treatment effectiveness, and to estimate drainage area and/or fluids in place. Multiwell Tests vs. Single-Well Tests. In a multiwell test, the rate is changed at one well (the active well), and the pressure response is measured at one or more offset, or observation, wells. Multiwell tests are often more complex than single-well tests. Multiwell tests are usually run to quantify the degree of communication between wells or to estimate directional permeability. In a single-well test, the rate is changed, and the pressure response is measured at the well being tested. Objectives of single-well tests may include to quantify the degree of damage or stimulation, to estimate the in-situ permeability, to estimate the drainage area or fluids in place, and to estimate the average drainage area pressure. Buildup Tests vs. Drawdown Tests. In a drawdown test, the reservoir is initially at uniform pressure, and the well to be tested is shut in. The well is opened for production, ideally at constant flow rate, and the resulting pressure response is measured as the pressure draws down. In a buildup test, the well has been producing (again, ideally at constant rate) for a period of time, thus creating a pressure gradient in the reservoir. The well is shut in, and the resulting pressure response is measured as the pressure builds up. Exploration-Well Tests vs. Development-Well Tests. For a well test on an exploration well, the focus is on the entire reservoir. Objectives of an exploration-well test usually include fluid sampling, estimating initial reservoir pressure, evaluating well productivity, estimating distances to boundaries, and estimating fluids in place. Exploration-well tests often must be conducted while the rig is on location, making such tests extremely expensive. The expense is justified by the major investment decisions that will be made based on the information obtained from the test. In contrast, in development-well testing, the focus is on the individual well and the near-wellbore area. Objectives of a well test on a development well might include estimating average drainage area pressure, evaluating stimulation treatment effectiveness, quantifying wellbore damage, and estimating reservoir permeability. Compared to exploration-well tests, development-well tests are relatively inexpensive. The results of development-well tests are used in minor investment decisions such as whether or not to do a stimulation treatment. 1.2.2 Applications and Objectives of Well Test Interpretation. Well test interpretation plays a role in many stages in the life of a well including exploration, appraisal, development, reservoir characterization, and production engineering. Exploration and Appraisal. Well testing plays an extremely important role during exploration and appraisal. The information gained through well testing will be instrumental in making major investment decisions such as whether to set a platform, build a pipeline, develop a field, or sign a long-term production contract. The two most important questions that well testing in exploration and appraisal wells can address are “How much oil or gas does this reservoir contain?” and “At what rate can wells in this reservoir produce?” Flow tests are often used during exploration testing; these tests are designed to investigate enough of the reservoir to assess the economic viability of the prospect. The deliverability is obtained from the flow rates achieved during the test. Distances to boundaries, reservoir pore volume, and estimates of oil or gas in place may be obtained from the pressure-transient response. Reservoir Engineering. In reservoir engineering, well testing may address questions such as “What is the insitu permeability?,” “What are the nature of and distances to reservoir boundaries?,” and “What is the average reservoir pressure?” A knowledge of the in-situ permeability is important in reservoir simulation and production forecasting. The nature of and distance to reservoir boundaries is important in reservoir simulation and in well-spacing decisions. The average reservoir pressure is used in reservoir simulation, well optimization, and material-balance calculations. Production Engineering. The focus in production engineering is on the individual well. The questions addressed by well testing in production engineering reflect this focus: “Is the well damaged?” and “How effective was the stimulation treatment?”

Introduction to Applied Well Test Interpretation 3

Poor well performance may be caused by reservoir characteristics, such as low permeability or low reservoir pressure; by near-wellbore conditions, such as damage from mud-filtrate invasion, non-Darcy flow, or plugged perforations; or by some combination of these factors. Well test interpretation can help identify the cause of poor well performance, so that the appropriate remedial action may be taken. 1.3 Alternatives to Conventional Well Testing There are several alternatives to pressure-transient testing that may also be used to estimate permeability and other reservoir properties. Alternatives include wireline formation testing, data from permanent gauges, rate-transient analysis, log-derived permeability estimates, and core analysis. In general, these methods supplement or complement, rather than replace, conventional pressure-transient testing. 1.3.1 Wireline-Formation Testing. In wireline formation testing, a relatively small amount of fluid (a few liters) is withdrawn from a short interval (1 m or less). The pressure response during both the withdrawal period and the subsequent shut-in period is recorded. Formation tests may be conducted open hole, with a small probe forced through the mudcake or with a short interval (typically 1 m) isolated by dual packers. Formation tests may even be conducted in cased hole, with the tool drilling through the casing and cement to provide hydraulic communication with the formation, collecting a fluid sample and pressure data, and plugging the hole in the casing in a single trip (Burgess et al. 2002). Unlike conventional pressure-transient testing, formation testing can provide multiple small-scale measurements of in-situ permeability and pressure across the reservoir interval. Multiple downhole fluid samples may also be collected at different depths. Because of the small volume of fluid withdrawn from the formation, wireline formation testing has a rather limited depth of investigation and provides little or no information about lateral reservoir boundaries. In highpermeability formations, the limited production volume may not create a large enough pressure response to allow formation permeability to be estimated. Normally, the small-scale permeability estimates from formation testing must be scaled up before use in forecasting well performance. Finally, skin factors obtained from wireline formation testing are of limited use for predicting well performance as completed for production (Elshahawi et al. 2008). Applications of wireline formation testing include collection of downhole fluid samples, estimation of formation pressure and permeability as a function of depth, identification of depleted zones, and estimation of vertical permeability. Graphing pressure vs. depth allows the reservoir pressure gradient to be determined, thus enabling identification of the formation fluid and the location of fluid contact. Vertical interference testing provides estimates of vertical permeability, which is not usually available from conventional pressure-transient testing. Permeability estimates from wireline formation testing may also be used to calibrate correlations for estimating permeability from wireline logs. Although wireline formation testing is considered an alternative to conventional testing, it relies on the same theoretical foundations as conventional pressure-transient testing. As a result, many of the methods discussed in this test find also direct application in wireline formation testing. 1.3.2 Permanent Monitoring. Permanently installed pressure gauges have become common in recent years. Permanent gauges offer a number of advantages over conventional testing but also have some disadvantages as well. Perhaps the two primary advantages of permanent gauges are (1) they allow shut-in periods that occur as a result of normal production operations to be analyzed as buildup tests, thereby eliminating the expense of loss or deferral of production, and (2) they provide frequent tests allowing continual monitoring of skin factor, reservoir permeability, and movement of fluid contacts, allowing timely remedial action. On the other hand, data quality from permanent gauges obtained during routine production operations is not as high as that from conventional tests run under controlled conditions. Permanent monitoring must include both rate and pressure measurements to be useful. Because of the large quantity of data that is acquired, automated preprocessing and analysis are essential to realize their full potential. Like wireline formation testing, interpretation of permanent gauge data is based on the same theoretical foundation as conventional pressure-transient testing. 1.3.3 Rate-Transient Analysis. Rate-transient analysis uses production and pressure data obtained during normal production operations to estimate reservoir properties. Methods used for rate-transient analysis are very similar to those used for pressure-transient analysis. Under the right conditions, rate-transient analysis can provide estimates of initial hydrocarbons in place, reservoir permeability, and skin factor (for vertical wells) or fracture half length (for hydraulically fractured wells).

4 Applied Well Test Interpretation

Advantages of rate-transient analysis over pressure-transient analysis include lower cost of data collection (no special equipment, no need to shut in the well, no need to interrupt normal operations), and the ability to reach a much larger radius of investigation than could be obtained by pressure-transient testing. This latter advantage has made rate-transient analysis one of the most important reservoir characterization tools for low permeability formations. While pressure-transient data is collected under controlled conditions, specifically for the purpose of estimating reservoir properties, data used for rate-transient analysis is collected during normal production operations. Thus, rate-transient data are typically noisier and are more likely to be subject to changes in operating conditions than are pressure-transient data. In addition, early-time data are often inadequate to reliably distinguish between damaged wells and undamaged wells. With rate-transient analysis, it is also difficult to track changes in skin factor, effective permeability, and drainage pressure over time. The primary application of rate-transient analysis is estimation of permeability, stimulation effectiveness, and drainage area or original hydrocarbons in place, for moderate to low permeability wells that are produced at capacity. Rate-transient analysis is particularly important for estimating original hydrocarbons in place and reserves in low permeability reservoirs where it is difficult (if not impossible) to estimate average reservoir pressures for material balance analysis. Rate-transient analysis is based on the same foundations as conventional well testing, wireline formation testing, and analysis of permanent gauge data. 1.3.4 Log-Derived Permeability Estimates. In many reservoirs, it may be possible to develop correlations to estimate permeability from a standard logging suite. Although not applicable to all reservoirs, log-derived permeability estimates may be quite useful. Typically, core data will be used to identify two or more distinct rock types. Separate correlations are then developed for each rock type. The correlations are calibrated with permeability estimates from wireline-formation testing or core analysis. Unlike fluid property correlations, log permeability correlations have to be custom built for each rock type in each formation. In some cases, it may not be possible to develop correlations that are accurate enough to be useful. In other cases, the correlations may serve as permeability indicators rather than reliable permeability estimates. In older fields, core or formation test data to calibrate correlations may not be available. As with permeability estimates from wireline formation testing or from core analysis, log-derived permeabilities must be scaled up to use in preformation forecasting or comparison with permeability estimates from conventional pressure-transient tests. If feasible, log-derived permeability estimates allow a limited amount of formation test or core data to be extrapolated to all wells that have a standard suite of logs. 1.3.5 Core Analysis. Conventional core analysis provides measurements of fluid saturations, porosity, and absolute permeability. Depending on the specific tests requested, special core analysis may provide measurements of capillary pressure, relative permeability, and stress-dependent porosity and/or permeability. Although core analysis is an essential component of a complete reservoir study, it is often difficult to relate laboratory-derived permeabilities to those obtained from pressure-transient test interpretation or required for performance forecasting. Some of the reasons for this difficulty include • Differences in scale—Core plug sizes are typically measured in centimeters or inches; pressure-transient testing measures permeabilities over distances of tens of feet to several hundred feet. • Differences in conditions—Conventional core analysis permeability measurements are conducted at room temperature and low confining pressures. Pressure-transient testing provides estimates of permeability at in-situ temperature and stress conditions. • Differences in fluids—Conventional core analysis permeability measurements are typically conducted using air under single-phase conditions. Pressure-transient testing provides estimates of permeability to reservoir fluids with reservoir fluid saturations. The combined effects of stress and connate water may cause the in-situ permeability for low permeability or stress-sensitive formations may be two or three orders of magnitude lower than that measured on unstressed, dry cores (Jones and Owens 1980). • Incomplete core recovery—Incomplete core recovery may be particularly significant in unconsolidated sands or in thick formations. • Fracture permeability not captured—Virtually all of the bulk permeability in naturally fractured reservoirs is provided by the fracture network. If the fractures are widely spaced, a core may not intersect a representative sample of the fractures. Even when a whole core cuts a natural fracture, it may not be possible to determine the in-situ fracture aperture and thus, the permeability.

Introduction to Applied Well Test Interpretation 5

• Improper lab technique—The use of inappropriate lab technique may cause irreversible changes in the core during cleaning and drying. • Biased core plug sampling—If lab personnel are inexperienced or lack adequate training, cores plugs may be preferentially selected from the more permeable portions of the whole core. 1.4 Forward and Inverse Problems In the forward (or direct) problem, Fig. 1.1, the input flow rate history and the system reservoir model are both known, and the resulting pressure response must be determined. Any given reservoir model will always give the same output pressure response for a given input flow rate history. In other words, the solution to the forward problem is unique. In contrast, in the inverse problem, Fig. 1.2, the input flow rate history and output pressure response are known, while the unknown reservoir model must be found. Many different reservoir models may give essentially the same pressure response for a given input flow rate history. The solution to the inverse problem is not unique. A forward (or direct) problem has a unique solution. For any given input and model, the output will always be the same. An inverse problem does not have a unique solution; many different models can produce the same output for a given input.

q

t Known input Known model

p

System

? t Unknown output

Fig. 1.1—Forward problem.

q

t Known input

? Unknown model System

p

t Known output Fig. 1.2—Inverse problem.

6 Applied Well Test Interpretation

Well test interpretation is an inverse problem. The two central problems of well testing are (1) to identify the unknown reservoir model, and (2) to estimate the unknown parameters of that model. Although we often focus on parameter estimation, correct model identification is an essential component of any interpretation. If the model is wrong, the resulting parameter estimates are useless. Correct model identification is an essential component of well test interpretation. If the model is wrong, the resulting parameter estimates are useless. 1.5 Well Test Interpretation Methods Virtually, all quantitative well test interpretation methods fall into one of three categories: straight-line analysis, type-curve analysis, and simulation/history matching. Each of these categories has certain strengths and weaknesses. In our preferred workflow, type-curve and straight-line methods are used to identify the reservoir model and obtain preliminary estimates of reservoir properties, while history matching with an analytical or numerical model is used to verify and, if necessary, fine-tune the preliminary interpretation. 1.5.1 Straight-Line Methods. In straight-line analysis methods, the analysis must first identify data that exhibit a specific type of flow or flow regime, such as infinite-acting radial flow. The data are then graphed, perhaps after transforming pressure and time variables to account for non-ideal phenomena or flow conditions. Finally, a straight line is drawn through the data exhibiting the desired flow regime. From the slope and intercept of the straight line, the analyst then calculates reservoir properties, such as permeability, and/or well properties, such as skin factor. Straight-line methods were historically the first methods to be used in the petroleum industry. They are easy to perform by hand or implement in a spreadsheet. When the desired flow regimes are present and can be correctly identified, straight-line analysis methods give excellent results. Unfortunately, straight-line methods often require the pressure and time data be transformed. Another, more significant disadvantage of straight-line methods is that specific flow regimes must be present to determine certain reservoir properties. Finally, because straight-line methods use only data within a single flow regime, there is no guarantee that the resulting parameter estimates will be consistent with the rest of the pressure data. 1.5.2 Type-Curve Methods. In type-curve analysis methods, the analyst graphs the pressure and time data, perhaps after transforming to account for non-ideal phenomena, on a log-log scale. This field data graph is then matched to the pressure response (or type curve) from an ideal reservoir model. The shape and position of the field data relative to the type curve are used to estimate formation and/or well properties such as permeability, skin factor, and wellbore storage. Type-curve methods honor more of the data than straight-line methods and are thus less likely to give results that are inconsistent with the rest of the data. Type-curve methods may be used manually with paper type curves. However, the procedure is tedious and time-consuming, and paper type curves are available for only a limited number of different reservoir models. On-screen matching with computer-generated type curves is fast and flexible. However, some commercial well test software packages have limited type-curve matching capability. 1.5.3 Simulation and History Matching. In the simulation/history-matching approach, an analytical or numerical model is used to calculate the pressure response for known reservoir properties and assumed values for unknown reservoir properties. The values of the unknown properties are then adjusted until the calculated pressure response matches the observed field data. Matching may be conducted manually, or automatically using a nonlinear regression algorithm. The biggest advantage of history matching with a simulator is that more phenomena can be incorporated in the analysis, allowing most if not all of the data to be used in an integrated analysis, thereby ensuring consistency of the interpretation with the entire data set. However, history matching may be very time-consuming, unless good initial estimates of the unknown reservoir properties are available. In practice, we have found that the best workflow is to begin with type-curve and straight-line methods to get initial estimates of reservoir properties, then use simulation and history matching to verify or to fine-tune those initial estimates. 1.6 Rock and Fluid Properties A variety of rock and fluid property data are used as input data in well test interpretation. These data can be categorized as (1) rock property data, usually obtained from openhole logs, (2) fluid property data, usually obtained from correlations or lab measurements, and (3) other data.

Introduction to Applied Well Test Interpretation 7 Pore space

Grains

Fig. 1.3—Porosity is a measure of the capacity of the reservoir rock to store fluids.

1.6.1 Rock Properties. In this section, we will discuss the porosity, fluid saturations, permeability, pore volume compressibility, and net-pay thickness. Porosity. Porosity, f, is a measure of the capacity of the reservoir rock to store fluids (see Fig. 1.3), defined as the ratio of the volume of the pore space to the total or bulk volume of the reservoir rock, Eq. 1.1:

φ=

pore volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.1) bulk volume

When used in equations in well test interpretation, porosity is always expressed as a fraction. In reports, graphs, and tables, porosity is often given in %. Porosity may vary over a wide range, depending on the type of reservoir. Coalbed methane reservoirs may have porosity as low as 0.5%, while a diatomite may have a porosity as high as 60%. Table 1.1 gives typical ranges of porosity for a number of common rock types. Porosities estimated from openhole logs typically have an uncertainty of 5% to 15% (Spivey and Pursell 1998). Saturation. Fluid saturation is defined as the fraction of pore space occupied by a particular fluid (see Fig. 1.4), Eqs. 1.2, 1.3, and 1.4: So =

oil volume , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.2) pore volume

Sg =

gas volume , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.3) pore volume

Sw =

water volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.4) pore volume

TABLE 1.1—TYPICAL POROSITY VALUES FOR DIFFERENT LITHOLOGIES Rock Type

Porosity Range

Diatomite

50–60%

Chalk

50–60%

Clean sandstone

25–30%

Typical sandstone

15–25%

Shaly sandstone

5–15%

Tight gas sand

5–12%

Limestone

2–15%

Fractured shale

0.5–8%

Coalbed methane

0.5–4%

8 Applied Well Test Interpretation Oil

Grains

Water

Fig. 1.4—Saturation is the fraction of the pore space occupied by a particular fluid.

In well test interpretation, when used in equations, saturation is always expressed as a fraction. In reports, graphs, and tables, saturation is often given in %. Almost all reservoir rocks have a nonzero water saturation. In well test interpretation, we usually assume that the water is immobile and that a single fluid, which may be either oil or gas, occupies the remaining pore space. In this case, the water saturation is referred to as the connate or irreducible water saturation. Generally speaking, the lower the porosity, the higher the irreducible water saturation. Table 1.2 gives typical values for water saturation for various common lithologies. Water saturations estimated from logs may have uncertainties of 10 to 40% (Spivey and Pursell 1998). Permeability. Permeability, k, is a measure of the capacity of a porous medium to transmit fluid. Eq. 1.5 is a statement of Darcy’s law for linear steady-state flow of a single-phase fluid, as shown in Fig. 1.5. Eq. 1.5 may be used as a definition of permeability: k=

qµ L , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.5) A ( p1 − p2 )

where q is the flow rate in cm3/s, A is the cross-sectional area open to flow in cm2, L is the length of the flow path in cm, p1 and p2 are the inlet and outlet pressures, respectively, in atmospheres (atm), μ is the viscosity of the fluid in centipoise (cp), and k is the permeability in Darcys.

TABLE 1.2—TYPICAL WATER SATURATION VALUES FOR DIFFERENT LITHOLOGIES Rock Type

Water Saturation Range

Clean sandstone

15–20%

Typical sandstone

20–30%

Shaly sandstone

30–40%

Tight gas sand

35–70%

Limestone

15–40%

Coalbed methane

100%

p1

p2

A q

L Fig. 1.5—Permeability is a measure of the capacity of the rock to allow flow of fluid.

Introduction to Applied Well Test Interpretation 9 TABLE 1.3—TYPICAL PERMEABILITY VALUES FOR DIFFERENT LITHOLOGIES Rock Type

Permeability Range

Clean sandstone

10–1 000+ md

Typical sandstone

10–100 md

Shaly sandstone

0.1–10 md

Tight gas sand

1–100 µd

Limestone

1 µd –10 000+ md

Coalbed methane

1–100 md

Fractured shale

0.1–100 md

Permeability varies over an extremely wide range. Matrix permeability for a naturally fractured shale may be as low as 2 × 10-8 md, while permeability for a clean, well-sorted sandstone may exceed 1 Darcy. Table 1.3 gives typical ranges for permeability for common reservoir rocks. A “tight gas” reservoir is defined by the US Federal Energy Regulatory Commission as a formation having an average in-situ permeability to gas of 0.1 md or lower. The permeability measured in the laboratory under single-phase conditions is called the absolute permeability. The permeability estimated from well test data is the in-situ, effective permeability to the flowing fluid. The in-situ permeability may be one to three orders of magnitude lower than the absolute permeability at zero net overburden pressure. Pore-Volume Compressibility. Pore-volume compressibility, cf , sometimes called the formation compressibility or rock compressibility, is defined as the fractional increase in pore volume corresponding to a unit increase in pore pressure, Eq. 1.6. 1 dVp Vp dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.6) 1 dφ = φ dp

cf =

Pore-volume compressibility has units of reciprocal pressure. In oilfield units, compressibility is in psi-1, sometimes written as square inches per pound (sip). A particularly convenient unit for compressibility is the microsip (msip), where 1 msip = 10-6 psi-1. As with the other rock properties, pore-volume compressibility varies over a broad range, from as low as 2 × 10-6 psi-1 to 100 × 10-6 psi-1 or even higher. For limestones and consolidated sandstones, the pore-volume compressibility increases significantly with decreasing initial sample porosity, but for friable and unconsolidated sandstones, the pore-volume compressibility bears little relationship to porosity (Newman 1973). Table 1.4 gives typical ranges of pore-volume compressibility for common reservoir rock types. It should be noted that errors in pore-volume compressibility may have a dramatic impact (as much as a factor of 10) on estimates of original oil in place obtained from pressure-transient tests. An error in pore-volume compressibility can have a dramatic effect (a factor of 10) on the estimate of original oil in place obtained from pressure-transient analysis.

TABLE 1.4—TYPICAL PORE VOLUME COMPRESSIBILITY VALUES FOR DIFFERENT LITHOLOGIES Rock Type

Pore Volume Compressibility Range

Consolidated sandstone

3–20 × 10–6 psi-1

Typical sandstone

3–40 × 10–6 psi-1

Unconsolidated sandstone

3–80 × 10–6 psi-1

Limestone

3–80 × 10–6 psi-1

10 Applied Well Test Interpretation

Pore-volume compressibility can be obtained from special core analysis measurements of stress-dependent porosity from the slope of a graph of ln φ vs. net overburden pressure. Net Sand Thickness. Net sand thickness, h, is the thickness of the reservoir actually contributing to flow of reservoir fluids. Gross thickness, on the other hand, includes the entire reservoir section, which may include nonreservoir rock such as interbedded shale, etc. For purposes of well test interpretation, net sand thickness is measured in the direction perpendicular to the plane of the upper and lower bed boundaries. In a vertical well drilled through a horizontal reservoir (Fig. 1.6a), the net sand thickness may be obtained directly from the logs. For a deviated well producing from a horizontal reservoir (Fig. 1.6b), measured depths on the logs must be converted to true vertical depths to estimate net pay. For a vertical well (Fig. 1.6c) or deviated well (Fig. 1.6d), intersecting a dipping reservoir, neither the measured depth nor the true vertical depth represents the net sand thickness. In a horizontal well (Fig. 1.6e), the wellbore may not penetrate the full net sand interval. Even without having to make depth corrections for a deviated well or a dipping pay zone, net sand thickness is one of the more difficult parameters to estimate accurately. Thus, the results of well test interpretation are often presented as a permeability-thickness product, kh, instead of a permeability, k. Net sand thickness is normally estimated from openhole logs by using porosity, clay content, and water saturation cutoffs. Log-derived net sand estimates may have uncertainties from 15 to 50% (Spivey and Pursell 1998). 1.6.2 Fluid Properties. Fluid properties may be estimated from correlations or obtained from laboratory measurements on reservoir fluid samples. The better fluid property correlations are quite accurate, rivaling the accuracy of routine laboratory measurements. Formation Volume Factor. The formation volume factor, B, defined as the volume of fluid at reservoir conditions necessary to produce one unit of the same fluid at surface conditions, is a measure of the change in volume of fluid as it is produced from the reservoir to the stock tank or pipeline. The formation volume factor must account for (1) decrease in volume because of the decrease in temperature from the reservoir to the surface, (2) increase in volume because of the decrease in pressure from the reservoir to the surface, and, for oil and water, (3) decrease in volume because of the loss of dissolved gas. For oil and water, the formation volume factor in oilfield units is bbl/STB. For gas, the formation-volume factor, as used in this text, is in bbl/Mscf. Fig. 1.7 shows the pressure dependence of the formation-volume factor for the three oilfield fluids oil, water, and gas. The oil formation volume factor, Bo, is the volume of oil (plus dissolved gas) at reservoir conditions necessary to give one barrel of oil at stock tank conditions. For oil at pressures above the bubblepoint pressure (undersaturated oil), the formation volume factor increases with decreasing pressure. For oil at pressures below the original bubblepoint pressure, where the oil is in equilibrium with a free gas phase (saturated oil), the formation-volume factor decreases with decreasing pressure. This effect is most pronounced at low pressures. The gas formation volume factor, Bg, is the volume of gas at reservoir conditions necessary to give 1 Mscf of gas from the separator and is approximately inversely proportional to pressure. The water formation volume factor, Bw, is the volume of water (plus dissolved gas) at reservoir conditions necessary to give one barrel of water at stock tank conditions. The water formation-volume factor is a monotonically

h

h

(a)

h

(b)

h

(c)

h

(d)

(e)

Fig. 1.6—Net sand thickness is measured along a direction perpendicular to the bedding planes.

Introduction to Applied Well Test Interpretation 11 1.1 Formation Volume Factor, bbl/STB

Formation Volume Factor, bbl/STB

1.4 1.3 1.2 1.1 1 0.9 0.8

0

1000

2000 3000 Pressure, psi

4000

1.08 1.06 1.04 1.02 1

5000

0

1000

2000 3000 Pressure, psi

4000

5000

Water

Oil

Formation Volume Factor, bbl/Mscf

25 20 15 10 5 0

0

1000

2000 3000 Pressure, psi

4000

5000

Gas Fig. 1.7—Pressure dependence of formation-volume factor for oilfield fluids.

decreasing function of pressure. The slope is greater for undersaturated water (water at a pressure above its bubblepoint pressure) than for saturated water (water in equilibrium with a free gas saturation). Table 1.5 gives typical ranges for formation volume factor for various reservoir fluids. Fluid Compressibility. The coefficient of isothermal compressibility, c, usually called simply the compressibility, is the fractional decrease in fluid volume corresponding to a unit increase in pore pressure (see Fig. 1.8). For gases, the compressibility may be defined as cg = −

1 dBg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.7) Bg dp

At pressures above the bubblepoint pressure, the oil and water compressibilities are defined as co = −

1 dBo , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.8) Bo dp

TABLE 1.5—TYPICAL FORMATION VOLUME VALUES FOR RESERVOIR FLUIDS Rock Type

Formation-Volume Factor Range

Low GOR black oil

1–1.3 bbl/STB

High GOR black oil

1.3–2 bbl/STB

Volatile oil

2–4 bbl/STB

Water

1–1.1 bbl/STB

Gas

0.5–25 bbl/Mscf

12 Applied Well Test Interpretation

1E-01

1E-01

Compressibility, psi −1

1E+00

Compressibility, psi −1

1E+00

1E-02 1E-03 1E-04 1E-05 1E-06

0

1000

2000 3000 Pressure, psi

4000

1E-02 1E-03 1E-04 1E-05 1E-06

5000

0

1000

2000 3000 Pressure, psi

4000

5000

Water

Oil

Compressibility, psi −1

1E+00 1E-01 1E-02 1E-03 1E-04 1E-05 1E-06

0

1000

2000 3000 Pressure, psi

4000

5000

Gas Fig. 1.8—Pressure dependence of compressibility for oilfield fluids.

and cw = −

1 dBw , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.9) Bw dp

respectively. For undersaturated oil (oil above its bubblepoint pressure), the oil compressibility is typically only one-tenth that of the same oil at pressures below its original bubblepoint pressure. For oil (or water) containing dissolved gas, the compressibility includes the effects of gas coming out of solution as well as expansion of the oil itself. At pressures above the bubblepoint pressure, no gas comes out of solution as pressure is reduced. At pressures below the original bubblepoint pressure, the oil volume actually decreases as gas comes out of solution. At pressures below the original bubblepoint pressure, the oil and water compressibilities are defined as co = −

1 dBo 1 Bg dRso + , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.10) Bo dp 1, 000 Bo dp

and cw = −

1 dBw 1 Bg dRsw + , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.11) Bw dp 1, 000 Bw dp

respectively. The gas compressibility is approximately equal to 1/p at low pressures (p < 1,000 psi). At higher pressures, this approximation is not very good, but it still provides a useful rule of thumb. For an ideal gas, the compressibility is exactly 1/p.

Introduction to Applied Well Test Interpretation 13

The oil compressibility increases by a factor of ten as the pressure drops below the bubblepoint pressure. As we will see later in the course, this is a good reason to test oil wells as early as possible before a free gas saturation is allowed to develop in the reservoir. A similar effect causes the water compressibility to increase by a factor of perhaps two as the pressure drops below the bubblepoint pressure. As may be seen from Eqs. 1.7 through 1.11, the fluid compressibility is not independent of the formation volume factor B and the solution gas-oil or gas-water ratio, Rs. For consistency, if correlations are used to estimate B and Rs, the same correlations should be used to calculate the compressibility using Eqs. 1.7 through 1.11 (Spivey et al. 2007). Table 1.6 gives typical values for fluid compressibility for various reservoir fluids. Viscosity.Viscosity is a measure of the ability of a fluid to flow under an applied shear stress (see Fig. 1.9). Technically, viscosity is defined as the rate of shear strain to the applied shear stress. For flow in porous media, the viscosity determines how easily the fluid flows through the porous media in the presence of a pressure gradient.

TABLE 1.6—TYPICAL COMPRESSIBILITY VALUES FOR RESERVOIR FLUIDS Rock Type

Compressibility Range

Undersaturated oil

4–20 × 10–6 psi–1

Saturated oil

20–200 × 10–6 psi–1

Undersaturated water

2–3.3 × 10–6 psi–1

Saturated water

2.5–10 × 10–6 psi–1

Gas

20 × 10–6–10 × 10–3 psi–1

8

1

7 0.98 Viscosity, cp

5 4 3 3 2

0.96 0.94 0.92

1 0

0

1000

2000 3000 Pressure, psi

4000

0.9

5000

0

1000

2000 3000 Pressure, psi

Water

Oil 0.05 0.04 Viscosity, cp

Viscosity, cp

6

0.03 0.02 0.01 0

0

1000

2000 3000 Pressure, psi

4000

5000

Gas Fig. 1.9—Pressure dependence of viscosity for oilfield fluids.

4000

5000

14 Applied Well Test Interpretation TABLE 1.7—TYPICAL VISCOSITY RANGES FOR RESERVOIR FLUIDS Rock Type

Viscosity Range

Oil

0.3–5000 cp

Gas

0.01–0.045 cp

Water

0.25–1.5 cp

Oil viscosity may vary over an extremely wide range. In contrast, water and gas viscosities lie within rather narrow ranges. For gas, the viscosity is only a weak function of pressure. At low pressures, where the gas behavior is similar to that of an ideal gas, the viscosity is roughly independent of pressure. As the pressure increases, the gas viscosity increases somewhat. The water viscosity is a monotonically increasing function of pressure. For undersaturated oil (oil at a pressure above its bubblepoint pressure), the viscosity decreases with decreasing pressure. For saturated oil (oil in equilibrium with a free gas phase), the viscosity increases with decreasing pressure as gas comes out of solution in the oil. This effect is most pronounced at low pressures. Table 1.7 gives typical values for viscosity for various reservoir fluids. 1.6.3 Other Properties. Two other variables used in well test interpretation are neither rock nor fluid properties: the total compressibility and the wellbore radius. Total Compressibility. The total compressibility, defined in Eq. 1.12, reflects change of both pore volume and fluid volume with change in pressure: ct = c f + Soco + Sgcg + Swcw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.12) The fluid saturations in Eq. 1.12 may, but do not have to be, mobile. Thus, an immobile connate water saturation or a subcritical gas saturation will still contribute to the total compressibility, although there is only a single phase flowing. Typically, for undersaturated oil reservoirs, the total compressibility will be close to that of the oil alone. This is not always the case; for a heavy oil, in a low porosity carbonate, for example, the pore-volume compressibility may be as high as 100 × 10-6 psi-1, while the oil compressibility may be no more than 2 × 10-6 psi-1. For gas reservoirs, the gas compressibility usually dominates (especially for low-pressure gas reservoirs), but occasionally, the pore-volume compressibility can be significant. Wellbore Radius. The wellbore radius appears in many well testing equations. It is basically a reference value to allow us to compare performance of the well to some ideal value (see Fig. 1.10). Because of this, we recommend that the wellbore radius be calculated from the nominal bit size. This allows the performance of the well to be compared to that of a well drilled in gauge through the formation, completed open hole, with neither damage nor stimulation. Errors in wellbore radius have only a minor impact on well test results. An error of a factor of two, as might occur because of using the diameter instead of the radius, will only cause an error of ln(2), or 0.69, in the skin factor. An error of 0.69 in skin factor is likely to be important only for highly stimulated wells. If bit size is not readily available, a “universal value” of 0.3 ft or 0.1 m may be used, as proposed by Brown and Hawkes (2005).

Washout

Damage

Realistic wellbore

In gauge, no damage, openhole completion

Ideal wellbore

Fig. 1.10—Actual wellbore compared with ideal wellbore.

Introduction to Applied Well Test Interpretation 15

The wellbore radius serves only as a reference for the behavior of an ideal well. Using the nominal bit size to calculate the wellbore radius allows the well performance to be compared to that of a well drilled in gauge, completed open hole, with neither damage nor stimulation. 1.7 Graph Scales Several different graph scales are used in well test interpretation. A specific scale may be chosen to emphasize a particular part of the data, or to make data in a particular flow regime follow a straight line. The log-log scale is often used because the shape of the curve may reveal some important characteristic of the data set, such as damage or stimulation, independently of the numeric values involved. The three scales most frequently encountered in well test interpretation are the Cartesian scale, the semilog scale, and the log-log scale. These scales have several characteristics that are important for the well test interpreter to remember. 1.7.1 Properties of Semilog and Log-Log Scales. In well testing, we often refer to the number of log cycles. On a logarithmic scale, one log cycle is equivalent to a change in value by a factor of 10. The distance from 1 to 10, from 25 to 250, and from 0.0137 to 0.137 are all one log cycle in length. One-half of a log cycle is one-half the distance between adjacent powers of ten, equivalent to a change in value by a factor of 10 , or 3.1622…. Similarly, one-third of a log cycle is equivalent to a change in value by a factor of 3 10 , or 2.1544…. To quickly estimate distances of one-half or one-third of a log cycle on a graph, one-half of a log cycle is approximately a factor of 3, and one-third of log cycle is approximately a factor of 2. The logarithmic time axis of the semilog and log-log scales expands the early-time data. In Fig. 1.11, the pressure change has a range from 10 to 180 psi, but the data for about half of the pressure range is obscured by the y-axis. In contrast, expansion of the time scale makes it very easy to read the early pressure data in Fig. 1.12. Similarly, the logarithmic time axis of the semilog and log-log scales compresses the late-time data. For a test with data recorded at uniform time intervals, on a Cartesian scale, 90% of the data will occupy 90% of the scale (Fig. 1.11). In contrast, on a logarithmic time scale, the last log cycle of the time axis will contain 90% of the data regardless of the number of log cycles on the graph (Fig. 1.12). For transient radial flow, the last log cycle also represents 90% of the reservoir volume investigated during the test. Another interesting (and useful) characteristic of the log scale is that a constant multiple of a value will always lie the same distance from the original value, independently of that value. For example, the distance between any two of the pairs of values (2, 6), (9, 27), (125, 375), or, in general, (x, 3x) will be the same when plotted on a logarithmic scale. Because of this characteristic of equal multiples lying the same distance apart on a logarithmic scale, the graphs of the functions y = f ( x ) and y = g( x ) = f (bx ) will have exactly the same shape on a semilog scale. The graph of

200 180

Pressure Change, psi

160 140 120 100 80 60 40

90 %

20 0

0

20

40

60

80

100

120

140

160

180

Time, hr Fig. 1.11—On a Cartesian scale, 90% of the data takes up 90% of the scale.

16 Applied Well Test Interpretation

200 180

Pressure Change, psi

160 140 120 100 80 60 40

90 %

20 0 0.01

0.1

1

10

100

1000

Time, hr Fig. 1.12—On a logarithmic scale, the last log cycle contains 90% of the data.

g(x) will simply be shifted to the right (for b > 1) or left (for b < 1). Similarly, the graphs of the functions y = f ( x ) and y = h( x ) = af (bx ) will have the same shape on a log-log scale, translated both horizontally and vertically. This last characteristic is the principle on which log-log type-curve analysis is based. In Fig. 1.13, the x- and y-values for each point of Curve A were multiplied by 20 and 0.3, respectively, and then graphed as Curve B. Similarly, Curve C was obtained by multiplying the x- and y-values for Curve A by factors of 2 and 10, respectively. On a Cartesian scale, 90% of the data will occupy 90% of the scale. On a logarithmic scale, the last log cycle will contain 90% of the data. 1.7.2 “Hidden” Behavior. When a specific graph scale is chosen to emphasize a particular part of the data, another part of the data is simultaneously de-emphasized. This can cause important features or behavior of the data to be overlooked unless the interpreter looks at several different scales as routine practice. This point may be illustrated by the following examples.

10000

Pressure Change, psi

C 1000

A 100

B 10

1 0.01

0.1

1

10

100

1000

10000

Time, hr Fig. 1.13—Shape is preserved on a log-log scale when the abscissa and ordinate of each point are multiplied by constant factors.

Introduction to Applied Well Test Interpretation 17

Consider the match of a synthetic data set shown in Figs. 1.14 and 1.15. Looking only at the Cartesian scale in Fig. 1.14, the match appears to be almost perfect. However, the semilog scale in Fig. 1.15 shows that the match is unacceptable (in this case, the synthetic data set had significant damage, while the pressure response for the match was calculated with the same permeability, but no damage. In this case, the early-time data that reveal the presence of damage are compressed so much on the Cartesian scale, that the difference between the curves cannot be detected. Switching to the semilog scale emphasizes the early-time data, revealing the poor match between the synthetic data set and the match. Figs. 1.16 and 1.17 are from a field data set that illustrates a situation that frequently occurs in gas wells. Fig. 1.16 shows a log-log graph of the field data set, showing two curves: the pressure-change curve (labeled “Delt Pres”) and the logarithmic-derivative curve (labeled “Deriv”). The pressure-change curve and the 3100

Obs Fit

Pressure, psi

3050

3000

2950

2900

2850 0

20

40

80

60

Time, hr Fig. 1.14—Perfect fit of field data—or is it?

3100

Obs Fit

Pressure, psi

3050

3000

2950

2900

2850 0.001

0.01

0.1

1

10

100

Time, hr Fig. 1.15—Perfect fit of field data is not so perfect when viewed on a different scale.

18 Applied Well Test Interpretation 100000

Delt Pres Deriv

∆pa , psi

10000

1000

100

10 0.001

0.01

0.1

1

10

100

∆taeq , hr Fig. 1.16—Why does the logarithmic derivative curve show spikes when the pressure change curve is smooth?

6000 5000

Delt Pres Deriv

∆pa , psi

4000 3000 2000 1000 0 −1000 0.001

0.01

0.1

1

10

100

∆taeq , hr Fig. 1.17—On the Cartesian pressure scale, the pressure data does not look so smooth. The spikes in the logarithmic derivative curve are caused by abrupt shifts in the pressure data.

logarithmic-derivative curve are both calculated from the field pressure data (do not worry about exactly what the plotting functions are; we’ll get to that in a later chapter). But if both curves are calculated from the same field data, why does the logarithmic-derivative curve show a pair of spikes in opposite directions, an upward spike between 2 and 3 hours, and a downward spike between 6 and 7 hours, while the pressure-change curve is perfectly smooth? Redrawing the same data on the semilog graph in Fig. 1.17 shows that the pressure-change curve is not smooth. In fact, the two spikes in the logarithmic-derivative curve correspond to a pair of abrupt shifts, in opposite directions, in the pressure-change curve. Any behavior in the logarithmic derivative can be found in the pressure data, but it may take a bit more work to find it. Notice that the logarithmic derivative in Fig. 1.17 becomes negative during the second spike, between 6 and 7 hours. The negative derivative cannot be graphed on the logarithmic pressure scale in Fig. 1.16. Different scales emphasize different parts of the data, while de-emphasizing the remaining data. The analyst must always use multiple types of graphs to ensure that unusual behavior is not overlooked.

Introduction to Applied Well Test Interpretation 19

1.8 Summary This chapter may be summarized by the following key points: 1. Well testing is an inverse problem. That is, the system input (production rate history) and system output (pressure response) are known, and the properties of the system (reservoir) are to be determined. 2. Alternatives to conventional well testing include wireline formation testing, continuous monitoring with permanent gauges, rate-transient analysis, log-derived permeability estimates, and core analysis. 3. The two major tasks of well test interpretation are model identification and parameter estimation. Parameter estimates are meaningless if an inappropriate model is selected for the interpretation. 4. Well test parameter estimation methods may be grouped into one of three classes: straight-line methods, type-curve methods, and simulation/history matching. 5. The logarithmic scale compresses late-time data such that 90% of the data fall in the last log cycle of the graph. For data in infinite-acting radial flow, the last log cycle also represents 90% of the reservoir volume investigated during the test. 6. Relying on a single graph scale for interpretation may cause important behavior to be overlooked, leading to invalid results. Nomenclature A = area, ft2 B = formation-volume factor, bbl/STB (oil, water), bbl/Mscf (gas) c = compressibility, psi-1 cf = formation or pore-volume compressibility, psi-1 ct = total compressibility, psi-1 h = net-pay thickness, ft k = permeability, md L = distance, ft p = pressure, psia q = flow rate, STB/D (oil, water), Mscf/D (gas) Rs = solution gas-liquid ratio, scf/STB rw = wellbore radius, ft S = saturation, fraction Vp = pore volume, ft3 μ = viscosity, cp f = porosity, fraction Subscripts g = gas o = oil w = water

Chapter 2

Fluid Flow in Porous Media “If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” –Roger Bacon “First master the fundamentals.” –Larry Bird 2.1 Introduction The theory of well testing begins with an understanding of fluid flow in porous media. Section 2.2 gives a brief review of Darcy’s law for steady-state flow in linear and radial systems. In Section 2.3, the continuity equation, Darcy’s law, and the equation of state for a slightly compressible liquid are used to develop the diffusivity equation, describing single-phase flow of a slightly compressible liquid. The Ei-function solution to the diffusivity equation, for radial flow to a line-source well producing at constant rate from an infinite-acting reservoir, is introduced in Section 2.4, along with the logarithmic approximation to the Ei function. Section 2.5 introduces the principle of superposition as a tool for calculating the pressure response for complex boundary conditions. The principle of superposition in space is used to calculate the pressure response at an arbitrary point in the reservoir caused by production from two wells. The principle of superposition in time is then used to calculate the pressure response for a well with a variable-rate history. The radius of investigation concept is introduced in Section 2.6. In spite of its simplicity, the radius of investigation concept is a very powerful tool for developing an intuitive understanding of the movement of pressure transients through the reservoir. Section 2.7 introduces the concept of skin factor as a means of characterizing damage or stimulation in a well. The skin factor may be used to calculate the additional pressure drop caused by skin, or vice versa. The skin factor may also be expressed as an apparent wellbore radius, the radius that an undamaged, unstimulated well would have to have to exhibit the same pressure response as the well in question. Section 2.8 discusses the pressure response for pseudosteady-state flow (PSSF) from a closed reservoir. For constant-rate production from a closed reservoir, PSSF will occur after the pressure transient has reached all of the reservoir boundaries. The flowing wellbore pressure is a linear function of time, with the slope inversely proportional to the pore volume of the reservoir. Wellbore storage (WBS) often distorts the shape of the pressure response, as discussed in Section 2.9. Perhaps the two most common WBS situations are a fluid-filled wellbore and a wellbore with a rising or falling liquid level. Both these situations exhibit a constant WBS coefficient and may be described by the same solution to the diffusivity equation. After completing this chapter, you should be able to 1. Calculate the pressure drop for steady-state flow in both linear and radial systems. 2. List the three physical principles used to develop the diffusivity equation. 3. Calculate the pressure response for a well producing at constant rate, in an infinite-acting reservoir using the Ei-function solution. 4. Calculate the pressure response for two or more wells producing at constant rate from an infinite-acting reservoir using the principle of superposition in space.

22 Applied Well Test Interpretation

5. Calculate the pressure response for a well producing with a varying rate history from an infinite-acting reservoir using the principle of superposition in time. 6. Calculate the radius of investigation that will be achieved at a given time, and the time required to achieve a desired radius of investigation. 7. Calculate the pressure drop caused by skin factor. 8. Calculate the apparent wellbore radius for a given skin factor. 9. Calculate the pressure drop during pseudosteady-state flow for constant-rate production from a closed reservoir. 10. Estimate reservoir pore volume, original oil in place, drainage area, and productivity index from data during pseudosteady-state flow. 11. Calculate the wellbore-storage coefficient for a fluid-filled wellbore or for a rising liquid level. 2.2 Steady-State Flow The focus of this text is the interpretation of well test data under transient flow conditions, that is, flow during which pressure changes with time. Before discussing transient flow, it may be useful to review steady-state flow, in which pressure is constant, independent of time. In the following sections, we apply Darcy’s law to develop equations for the pressure drop for steady-state flow for both linear and radial systems. In a later chapter, we will discuss transient flow in linear and radial systems. In a well test interpretation context, the terms linear flow and radial flow are understood to refer to transient flow unless explicitly identified as steady-state or PSSF. 2.2.1 Darcy’s Law. Darcy’s law describes the movement of fluid through a porous medium under the influence of a pressure gradient. For constant flow rate q across a core of length L and cross-sectional area A, as illustrated in Fig. 2.1, the flow rate and pressure are related by q=

kA ∆p , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.1) µ L

where the flow rate q is in cm3/s, the area A is in cm2, the length L is in cm, the pressures p1 and p2 are in atm, the permeability k is in Darcys, and the viscosity μ is in cp. We may write Darcy’s law in differential form as ux = −

k dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.2) µ dx

In differential form, Darcy’s law states that the superficial velocity is proportional to the pressure gradient and opposite in sign, with the constant of proportionality given by the permeability (a property of the porous medium) divided by the viscosity (a property of the fluid). Darcy’s law is an empirical law that holds at low flow velocities, typical of those encountered throughout most of a reservoir. Darcy’s law in differential form, as defined in Eq. 2.2, applies to gases as well as liquids for most reservoir conditions. 2.2.2 Steady-State Linear Flow. If we solve Eq. 2.1 for the pressure drop and write the result in field units (flow rate q in STB/D, area A in ft2, length L in ft, pressure drop Δp in psi, permeability k in md, and viscosity µ in cp), we obtain ∆p =

1 qBµ L , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.3) 0.001127 kA p1

p2

A q

L Fig. 2.1—Linear flow through a core sample.

Fluid Flow in Porous Media 23 1500

Pressure, psi

1400

1300

1200

1100

1000

0

200

400 600 Distance From Outlet, ft

800

1000

Fig. 2.2—Pressure profile during steady-state linear flow.

where the formation volume factor B is included to convert the flow rate q from stock tank conditions to reservoir conditions. If we assume the area A is rectangular, with uniform thickness h and width w, we obtain 1 qBµ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.4) 0.001127 khw

∆p =

For steady-state linear flow, if the outlet pressure po is known, the pressure at any distance x upstream from the outlet may be calculated from p ( x ) = po +

1 qBµ x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.5) 0.001127 khw

Thus, for steady-state linear flow, a Cartesian graph of pressure p vs. distance x will be a straight line, as shown in Fig. 2.2. 2.2.3 Steady-State Radial Flow. To obtain an equation for steady-state radial flow through a formation with thickness h, we write Darcy’s law in differential form for radial flow as ur = −

k dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.6) µ dr

At any distance r, the Darcy velocity ur is related to the volumetric flow rate q by ur =

q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.7) 2π rh

Substituting and rearranging, we have q k dp =− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.8) 2π rh µ dr Separating variables, we obtain dp =

qµ dr , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.9) 2π kh r

24 Applied Well Test Interpretation 1500

Pressure, psi

1400

1300

1200

1100

1000

0

200 400 600 800 Distance From Axis of Wellbore, ft

1000

Fig. 2.3—Pressure profile during steady-state radial flow, Cartesian graph.

which is easily integrated to obtain the pressure drop for steady-state radial flow from radius re to radius rw as ∆p =

r  qµ ln  e  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.10) 2π kh  rw 

In field units, with flow rate q in STB/D, Eq. 2.10 becomes ∆p =

1 qBµ  re  ln   0.001127 2π kh  rw 

=

r  qBµ ln  e  0.00708kh  rw 

=

141.2qBµ  re  ln  .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.11) kh  rw 

For steady-state radial flow, if the wellbore pressure pw is known, the pressure at any other radius r may be calculated from p (r ) = pw +

141.2qBµ  r  ln   , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.12) kh  rw 

where the flow rate q is positive for production and negative for injection. For steady-state radial flow, a Cartesian graph of pressure p vs. radius r shows a logarithmic dependence on radius, as shown in Fig. 2.3. A semilog graph of pressure p vs. radius r will be a straight line, as shown in Fig. 2.4. 2.3 Development of the Diffusivity Equation The diffusivity equation describes transient flow of a slightly compressible liquid through a porous medium. The diffusivity equation is derived by combining three physical principles: (1) Darcy’s law, discussed previously, (2) the continuity equation, and (3) the equation of state for a slightly compressible liquid. For convenience, we will use a consistent system of units throughout this section. 2.3.1 Continuity Equation. The continuity equation is a statement of the law of conservation of matter relating the net rate of mass influx into an infinitesimal volume to the rate of accumulation of mass within the volume as illustrated in Fig. 2.5. For 1D linear flow, the continuity equation may be written (in consistent units) as: ∂ ∂ ( ρux ) = − ( ρφ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.13) ∂t ∂x

Fluid Flow in Porous Media 25 1500

Pressure, psi

1400

1300

1200

1100

1000 0.1

1

10

100

1000

Distance From Axis of Wellbore, ft Fig. 2.4—Pressure profile during steady-state radial flow, semilog graph.

ρux (x)

ρux (x + ∆x)

Fig. 2.5—The continuity equation relates the net rate of mass influx of fluid into an infinitesimal volume to the rate of accumulation of mass within the volume.

The velocity ux in the continuity equation is the superficial velocity, defined as the volumetric flow rate per unit cross-sectional area, ux = q A. The average velocity of the fluid is given by v x = ux φ . 2.3.2 Equation of State for a Slightly Compressible Liquid. The equation of state for a slightly compressible liquid states that the compressibility, defined as the fractional change in volume per unit change of pressure, is independent of pressure: c=

1  dρ  1  dV    = −   = const. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.14) V  dp T ρ  dp T

Eq. 2.14 is a good approximation for the behavior of oil or water under single-phase conditions. It may also be used for gas wells if the drawdown is small, so that all pressures lie within a fairly narrow range such that the gas properties remain approximately constant. Integrating Eq. 2.14 gives the density and volume as functions of pressure:

ρ ( p) = ρ 0ec ( p − p0 ), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.15) and V ( p) = V0e−c ( p − p0 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.16) For a slightly compressible liquid, we may write

ρ ( p) = ρ 0ec ( p − p0 ) ≅ ρ 0 [1 + c ( p − p0 )], and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.17) V ( p) = V0e−c ( p − p0 ) ≅ V0 [1 − c ( p − p0 )] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.18) What exactly is slightly compressible? One useful rule of thumb is that the compressibility can be considered small if the first neglected term in the Taylor series approximations in Eqs. 2.17 and 2.18 is much less than 1. When c∆pmax ≤ 0.1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.19)

26 Applied Well Test Interpretation

the first neglected term will be approximately 0.005 or less. For single-phase oil and water wells, Eq. 2.19 will always be satisfied. For a gas well, we can replace the compressibility in Eq. 2.19 with the ideal gas approximation to obtain c∆pmax ≈

∆pmax ≤ 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.20) p

Thus, if the maximum drawdown during a test is less than 10% of the initial pressure, we may use the liquid equations for a gas well, regardless of the reservoir pressure. 2.3.3 Diffusivity Equation. Combining Eqs. 2.2, 2.13, and 2.14 gives the diffusivity equation for 1D, linear flow of a slightly compressible liquid (in consistent units): ∂2 p φµct ∂p = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.21) ∂x 2 k ∂t or, in radial coordinates, 1 ∂  ∂p  φµct ∂p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.22) r  = r ∂r  ∂r  k ∂t The diffusivity equation is a linear partial differential equation, first order in time and second order in space. Because the diffusivity equation is linear, we can apply techniques such as the principle of superposition in space and time, Laplace and Fourier transforms, and Green’s functions to obtain solutions to the diffusivity equation for a wide variety of different boundary conditions. 2.4 Infinite-Acting Radial Flow—Ei-Function Solution One of the most important solutions to the diffusivity equation is the Ei-function solution, which describes infinite-acting radial flow (IARF). For an infinite, homogeneous, horizontal reservoir, having uniform initial pressure, with an infinitesimal line-source well producing at constant rate q beginning at time zero, we may write the solution to Eq. 2.22, in oilfield units, as p (r , t ) = pi +

70.6qBµ  948φµct r 2  Ei −  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.23) kh kt  

where the exponential integral function Ei is defined by Ei ( − x ) = ∫

∞ x

e− y dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.24) y

Eq. 2.23 gives the pressure at time t and distance r from the center of the wellbore. Fig. 2.6 shows a graph of the Ei function. For large values of the argument x, the Ei function is approximately 0: −Ei (− x ) ≅ 0, x > 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.25) For small values of x, the Ei function may be approximated by a logarithmic function: −Ei (− x ) ≅ − ln (1.781x ) , x < 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.26) For a given time t, we can use Eq. 2.23 to calculate the pressure distribution in the reservoir as a function of position, as shown in Fig. 2.7. Or, for a given distance r, we can use Eq. 2.23 to calculate the pressure as a function of time, as shown in Fig. 2.8. 2.5 Principle of Superposition Because the diffusivity equation is linear, the principle of superposition may be used to compute solutions for complex boundary conditions using only linear combinations of solutions for relatively simple boundary conditions. In this section, we will apply the principle of superposition to obtain solutions for the pressure response for

Fluid Flow in Porous Media 27

10 9

–E i (–x) –ln(1.781x)

8

–E i (–x)

7 6 5 4 3 2 1 –E i(–x) ≅ –ln(1.781x) 0 0.0001

0.001

0.01

–Ei(–x) ≅ 0

0.1 x

1

10

100

Fig. 2.6—Graph of the Ei-function solution to the diffusivity equation for a line-source well.

8000 7000 ri = 400 ft @ t = 10 hr

Pressure, psi

6000 5000 4000 t = 10 hr

3000 2000 1000 0

1

10 100 1000 Distance From Axis of Wellbore, ft

10000

Fig. 2.7—Pressure distribution in a hypothetical reservoir at a time of 10 hours.

8000 7500

Pressure, psi

7000

@ t = 10 hr, ri = 400 ft

6500 6000

r = 400 ft

5500 5000 4500 4000 0.1

1

10 Time, hr

100

1000

Fig. 2.8—Pressure response in a hypothetical reservoir at a distance of 400 ft.

28 Applied Well Test Interpretation

multiple wells (superposition in space) in an infinite-acting reservoir and for the pressure response for a single well with a piecewise constant flow rate profile (superposition in time). 2.5.1 Superposition in Space. Perhaps the simplest application of the principle of superposition is the calculation of the pressure response at any point in an infinite-acting reservoir caused by production from two different wells. Consider the situation shown in Fig. 2.9. Two wells, Well A and Well B, produce at rates qA and qB, respectively, beginning at time t = 0, from an infinite-acting reservoir. We want to calculate the pressure response at Point C caused by production from Wells A and B. We can calculate the pressure response at Point C caused by production from Well A using the Ei-function solution as

( pi − p) A,C = −

70.6q A Bµ  948φµct rAC 2  Ei −  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.27) kh kt  

where rAC is the distance from Well A to Point C. Similarly, the pressure response at Point C caused by production from Well B may be written as

( pi − p) B,C = −

70.6qB Bµ  948φµct rBC 2  Ei −  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.28) kh kt  

where rBC is the distance from Well B to Point C. Because the diffusivity equation is linear, we can simply add Eqs. 2.27 and 2.28 to obtain the pressure response at Point C:

( pi − p)C = ( pi − p) A,C + ( pi − p) B,C =−

70.6q A Bµ  948φµct rAC 2  70.6qB Bµ  948φµct rBC 2  Ei − Ei −  . . . . . . . . . . . . . . . . . . . . . . . (2.29) − kh kt kh kt    

Thus, we can write the pressure at Point C as pC = pi − ( pi − p) A,C − ( pi − p) B,C = pi +

70.6q A Bµ  948φµct rAC 2  70.6qB Bµ  948φµct rBC 2  Ei − Ei −  . . . . . . . . . . . . . . . . . . . . . . . . . . (2.30) + kh kt kh kt    

The principle of superposition allows us to extend this calculation to obtain the pressure response at any point in an infinite-acting reservoir, for N wells producing at different rates, using the Ei-function solution: N

pC ( t ) = pi + ∑ j =1

70.6q j Bµ kh

 948φµct rjC 2  Ei −  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.31) kt   Well B

Well A rBC rAC

Point C Fig. 2.9—Superposition in space.

Fluid Flow in Porous Media 29

q2 q1

t0

t1 Fig. 2.10—Superposition in time.

2.5.2 Superposition in Time. The principle of superposition also allows us to calculate the pressure response for a well with a variable-rate history by adding the pressure responses caused by each rate change in the rate history. For example, consider the rate history shown in Fig. 2.10. The wellbore pressure response for constant-rate production at rate q1 beginning at time t0 = 0 may be written as

( pi − p)q = − 1

=−

70.6q1 Bµ  948φµct rw 2  Ei −  kh kt   70.6 ( q1 − q0 ) Bµ  948φµct rw 2  Ei −  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.32) kh  k ( t − t0 ) 

where q0 and t0 are both defined to be zero, and where the pressure is evaluated at the sandface r = rw. Similarly, the pressure response for a rate change q2 − q1 at time t1 may be written as

( pi − p)q

2

− q1

=−

70.6 ( q2 − q1 ) Bµ  948φµct rw 2  Ei −  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.33) kh  k ( t − t1 ) 

Adding the pressure responses in Eqs. 2.32 and 2.33 gives the wellbore pressure at any time t > t1: pwf ( t ) = pi − ( pi − p)q1 − ( pi − p)(q2 − q1 ) = pi +

70.6 ( q1 − q0 ) Bµ  948φµct rw 2  70.6 ( q2 − q1 ) Bµ  948φµct rw 2  Ei − Ei −  +  . . . . . . . . . . . . . . . (2.34) kh kh  k ( t − t0 )   k ( t − t1 ) 

As with superposition in space, we may extend the calculation to obtain the pressure at any time t following n rate changes, as shown in Fig. 2.11: pwf ( t ) = pi +

n

∑

j =1

70.6 ( q j − q j −1 ) Bµ kh

 948φµc r 2  t w  Ei −  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.35)  k (t − t j − 1 ) 

q2 q1 q3

qn−1 qn

t0

t1

t2

tn−1

Fig. 2.11—Superposition in time, multiple rate changes.

30 Applied Well Test Interpretation

2.5.3 Pressure Response for a General Rate History. If we add a factor t j − t j − 1 in both the numerator and denominator of Eq. 2.35, we obtain 70.6 Bµ ( q j − q j − 1 )  948φµ µct rw 2   ( t j − t j − 1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.36) Ei −  kh ( t j − t j − 1 ) j =1  k (t − t j − 1 )  n

pwf ( t ) = pi + ∑

We can approximate any variable-rate history to an arbitrary accuracy by using smaller and smaller time intervals. In the limit as the duration of the time intervals approaches zero, the sum becomes an integral:

()

pwf t = pi + ∫

t

τ =0



( ) 70.6khBµ Ei  −

q′ τ



948φµ ct rw 2   d τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.37) k t −τ 

(

)

Although we have developed Eq. 2.37 specifically for a line-source well in an infinite-acting reservoir, we may write in general

()

pwf t = pi − ∫

t

τ =0

() (

)

q ′ τ pu t − τ d τ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.38)

where pu ( t ) is the unit pressure response for a well producing at constant unit rate. Eq. 2.38 is one form of the convolution equation, which we will consider at greater length in Chapter 8. 2.6 Radius of Investigation The radius of investigation is one of the most useful, yet least understood, concepts in well test interpretation. 2.6.1 Radius of Investigation—Drawdown. If we set the argument x of the Ei function to 1, then solve for t as a function of distance from the wellbore ri, we have t=

948φµct ri 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.39) k

Similarly, solving for ri, we have ri =

kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.40) 948φµct

Here we have used ri to denote the distance at which the argument of the Ei function is 1 for a given time t. This distance is referred to as the radius of investigation. The radius of investigation is a simple, yet powerful, tool for understanding the movement of a pressure transient through the reservoir. In Fig. 2.12, the pressure distribution is shown at several different points in time. As time increases, the pressure transient moves further into the reservoir. The radius of investigation provides a convenient dividing line

8000 7000 0.01 hr 0.1 hr

Pressure

6000

1 hr

10 hr

100 hr 1000 hr

5000 4000 3000 2000 1000 0

1

10 100 1000 Distance From Axis of Wellbore

10000

Fig. 2.12—Radius of investigation and movement of a pressure transient through the reservoir, drawdown.

Fluid Flow in Porous Media 31

for distinguishing the region distant from the well, where the pressure is essentially unaffected by the pressure transient, from the near-well region influenced by the pressure transient. 2.6.2 Radius of Investigation—Buildup. Every change in rate at the wellbore creates a new pressure transient that propagates into the reservoir independently of other pressure transients. Fig. 2.13 shows the pressure profile in the reservoir at several points in time during a buildup, with the radius of investigation for the buildup transient calculated using the shut-in time Δt. Meanwhile, the drawdown transient continues to move into the reservoir. For example, at 1000 hours into the buildup, the drawdown transient has moved to the point labeled “ri @ 2000 hour.” In spite of the name, the effects of a pressure transient do not end abruptly at the radius of investigation, as can be seen from Fig. 2.7 and Fig. 2.8. In Fig. 2.7, at t = 10 hours, the pressure has fallen slightly below the initial pressure at distances shortly beyond the radius of investigation of 400 ft; in Fig. 2.8, the pressure falls below the initial pressure shortly before the radius of investigation reaches 400 ft at time t = 10 hours. Every change in rate at the wellbore creates a new pressure transient that propagates into the reservoir independently of any other pressure transients. 2.6.3 Discussion. Although the radius of investigation was developed for single-phase flow of a slightly compressible liquid, we may use the radius of investigation for gas wells by evaluating the viscosity and compressibility at the initial pressure for a new well, or at the current average drainage area pressure for a producing well. One common misconception is that a higher flow rate will cause the pressure transient to move further into the reservoir in a given time. However, as Fig. 2.14 shows, changing the flow rate has no effect on the radius of investigation. 8000 1000 hr

7000

100 hr

Pressure

6000

10 hr

5000

ri @ 2000hr

1 hr

4000

0.1 hr

3000

0.01 hr

2000 1000 0

1

10

100

1000

10000

Distance From Axis of Wellbore Fig. 2.13—Radius of investigation and movement of a pressure transient through the reservoir, buildup. 8000 7000

q/2

Pressure

6000 q

5000 4000 3000

2q

2000 1000 0

1

10 100 1000 Distance From Axis of Wellbore

Fig. 2.14—Effect of flow rate on radius of investigation.

10000

32 Applied Well Test Interpretation

It is impossible to increase the radius of investigation at a given time by producing at a higher flow rate.

As may be seen from Eq. 2.39, the only parameters that affect the radius of investigation are the formation permeability, k, the fluid viscosity, μ, the total compressibility, ct, and the porosity, f. Except for the porosity, each of these parameters may vary over several orders of magnitude. As a result, for a given time t, the radius of investigation may be thousands of feet for a high permeability light oil reservoir above its bubblepoint, while that for a moderate permeability heavy oil reservoir below its original bubblepoint pressure may be only a few dozen feet. This fact will have a major influence on how pressure transient tests are conducted and how much information can be obtained from a test of a specified duration in different types of reservoirs. Table 2.1 shows the radius of investigation after 24 hours and the time required to reach a radius of investigation of 880 ft, for an oil reservoir with typical rock and fluid properties and different permeabilities. 2.6.4 Derivation of Radius of Investigation. Why did we define the radius of investigation by setting the argument of the Ei function to 1? Other choices are certainly possible. For example, we could have chosen the radius of investigation, rî , to be the distance at which the logarithmic approximation to the Ei function is zero. The logarithmic approximation to the Ei function is zero when the argument of the logarithm is 1, or 1.781x = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.41) Thus, we have x=

948φµct rî 2 1 = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.42) kt 1.781

which leads to t=

1688φµct rî 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.43) k

So, exactly what is special about setting the argument of the Ei function to 1? Consider the following. Taking the time derivative of Eq. 2.23, we have  948φµct r 2  ∂p 70.6qBµ exp − =−  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.44) ∂t kht kt  

TABLE 2.1—RADIUS OF INVESTIGATION EXAMPLE, OIL RESERVOIR Reservoir and Fluid Properties

m

1.7 cp

co 12.0 × 10–6 psi–1

f 18%

cw 3.0 × 10–6 psi–1

Sw 30%

cf 4.0 × 10–6 psi–1 ct 13.3 × 10–6 psi–1

k

ri @ t = 24 hour

Time required to reach ri = 880 ft

md

ft

1000

2,494

3

100

789

29.8

1.25

10

249

299.0

12.5

1

79

2,990.0

125

hr

days

Fluid Flow in Porous Media 33 18 ri = 400 ft at t = 10 hr

|dp/dt| at r = 400 ft, psi/hr

16 14 12 10 8 6 4 2 0 0.1

1

10

100

1000

10000

t, hr Fig. 2.15—Magnitude of time rate of change of pressure as a function of time, at a distance r = 400 ft from the axis of the wellbore.

At any fixed distance r, the magnitude of the rate of change of pressure initially increases, reaches a maximum, then decreases asymptotically toward 0. For the example drawdown test given above, the time derivative of pressure at a radius of 400 ft is shown in Fig. 2.15. To determine the time at which the maximum occurs at given distance, we set the second derivative of the pressure to zero and solve for t as a function of r:  948φµct r 2   948φµct r 2  ∂2 p 70.6qBµ − 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.45) exp − =−  2 2 ∂t kht kt kt    or −

 948φµct r 2   948φµct r 2  70.6qBµ exp − − 1 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.46)  2 kht kt kt   

For Eq. 2.46 to be satisfied, one of the three terms on the left side must be zero. It is easy to show that the first term decreases monotonically, approaching zero at large times: lim

t →∞

70.6qBµ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.47) kht 2

The second term increases monotonically as t decreases, approaching zero as t approaches zero:  948φµct r 2  lim exp −  = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.48) t →0 kt   Neither Eq. 2.47 nor Eq. 2.48 describes an extremum in the derivative. On the other hand, setting the third term to zero, and replacing r by ri, gives  948φµct ri 2  − 1 = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.49)  kt   which leads to the desired extremum. Thus, at a given distance ri, the magnitude of the rate of change of pressure will reach a maximum at a time t given by t=

948φµct ri 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.50) k

which is the same as Eq. 2.39.

34 Applied Well Test Interpretation

Thus, we may define the radius of investigation ri for a given time t as that distance at which the magnitude of the time derivative of pressure reaches a maximum value (in time) at the given time t. We will revisit the radius of investigation in Section 2.8.5, after we have introduced the pressure response for PSSF from a closed circular reservoir. 2.7 Damage and Stimulation In developing Eq. 2.23, we assumed that the well is vertical, has been completed open hole over the entire productive interval, and has been neither damaged nor stimulated. In many cases, we can account for damage or stimulation, partial penetration, cased and perforated or gravel-pack completion, and deviation from the vertical by a single number called the skin factor (Van Everdingen 1953). The primary effect of each of these situations is to change the pressure at the wellbore from that predicted by Eq. 2.23. 2.7.1 Skin Modeled as Annular Region of Altered Permeability. In one common model, near-wellbore damage or stimulation is assumed to be caused by an annulus of altered permeability near the well, while the permeability in the rest of the reservoir is unchanged (Van Everdingen 1953; Hawkins 1956). If the permeability in the altered zone is lower than the reservoir permeability, the well is damaged; if it is higher than the reservoir permeability, the well is stimulated. Fig. 2.16 shows a realistic sketch of near-wellbore damage along with a sketch of the simplified model. The annular region affected by damage or stimulation has inner radius rw, outer radius ra, and uniform permeability ka. Fig. 2.17a shows the near-wellbore pressure gradient for a well in which the permeability in the altered zone is the same as the original reservoir permeability. Since there is no damage or stimulation, the pressure drop across the altered zone annulus is given by ∆pu =

141.2qBµ ra ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.51) kh rw

With the altered zone present, as shown in Fig. 2.17b, the pressure drop is given by ∆pd =

141.2qBµ ra ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.52) ka h rw

Realistic wellbore damage

ka

k

ra

Damage model Fig. 2.16—Realistic and idealized models for near-wellbore damage.

∆pu ∆pd k

k ra

rw (a)

ka rw

k

ra (b)

Fig. 2.17—Calculating the additional pressure drop across the altered zone.

Fluid Flow in Porous Media 35

The additional pressure drop caused by damage or stimulation is obtained by subtracting Eq. 2.51 from Eq. 2.52: ∆ps = ∆pd − ∆pu =

141.2qBµ kh

 k  ra  − 1  ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.53)  ka  rw

Following Hawkins (1956), we define the skin factor s as  r k s =  − 1 ln a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.54)  rw  ka allowing us to write the equation for the additional pressure drop simply as ∆ps =

141.2qBµ s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.55) kh

If the permeability in the altered zone ka is lower than the formation permeability k, we say that the well is damaged. The skin factor as calculated using Eq. 2.54 will be positive for a damaged well. On the other hand, if the permeability in the altered zone is higher than the formation permeability, the well is stimulated, and the skin factor will be negative. Fig. 2.18 compares the pressure drop profile across the altered zone for a damaged well and for a stimulated well. From Fig. 2.18, it is apparent that there is a lower bound to the value that Δps can have for a given radius of the altered zone, achieved when the pressure drop across the altered zone is zero. In the limit as the permeability in the altered zone increases without bound (equivalent to removing the reservoir rock within the altered zone), Eq. 2.54 becomes s = − ln

rwa , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.56) rw

where rwa is the apparent wellbore radius, defined as the wellbore radius of a well with neither damage nor stimulation that would have the same productivity as the original well. The apparent wellbore radius is more physically meaningful for stimulated wells than for damaged wells, as will be seen in the exercises at the end of the chapter. Eq. 2.56 may be solved for the apparent wellbore radius in terms of the skin factor: rwa = rwe− s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.57) To obtain a given negative skin factor, a stimulation treatment must contact the formation to a depth at least equal to the apparent wellbore radius. To achieve a given negative skin factor with a hydraulic fracture treatment, the fracture half-length must be at least 2rwa. For a horizontal well, the lateral length must be at least 4rwa to achieve the desired negative skin factor.

ka > k

∆ps < 0 ∆ps > 0 ka < k

1

k

10 100 1000 Distance From Axis of Wellbore

10000

Fig. 2.18—Pressure profile across altered zone, damage, and stimulation.

36 Applied Well Test Interpretation

To obtain a given negative skin factor s, a stimulation treatment must contact the formation to a depth at least rwa = rwexp(–s). To achieve a skin factor s with a hydraulic fracture, the fracture half-length Lf must be at least 2rwa. To achieve a skin factor s with a horizontal well, the lateral length Lh must be at least 4rwa. 2.7.2 Wellbore Pressure Equation. We may calculate the pressure at the wellbore by evaluating Eq. 2.23 at the wellbore and adding the additional pressure drop caused by skin from Eq. 2.55: pwf ( t ) = pi +

70.6qBµ  948φµct rw 2  141.2qBµ Ei − s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.58) − kh kt kh  

Note that Eq. 2.58 applies only at the wellbore, while Eq. 2.23 is valid only at points beyond the annular altered zone. At the wellbore, where rw is small, the logarithmic approximation to the Ei function becomes valid almost immediately, allowing us to write pwf ( t ) = pi −

  kt 70.6qBµ   ln   + 2s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.59) 2 kh   1, 688φµct rw  

Rewriting Eq. 2.59 using common logarithms and rearranging gives pwf ( t ) = pi −

  k  162.6qBµ  log t + log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.60) 2 kh  φµct rw   

Eq. 2.60 is the basis for conventional semilog analysis of drawdown and buildup tests. 2.8 Pseudosteady-State Flow PSSF will occur for constant-rate production from any closed reservoir after the pressure transient has reached all of the reservoir boundaries. Once PSSF has been established, the pressure decreases at a constant rate that is independent of position in the reservoir. The rate of change of pressure with time is inversely proportional to the reservoir pore volume. 2.8.1 PSSF Equation. Fig. 2.19 shows the pressure profile in a hypothetical closed circular reservoir being produced at constant rate, during the IARF and PSSF periods. During the IARF period, the profiles are shown at times forming a geometric series (equally spaced on a log scale), while during the PSSF period, the profiles are shown at times forming an arithmetic series (equally spaced on a linear scale). 8000 7000

IARF

6000

PSSF

Pressure

5000 4000 3000 2000 1000 0

1

10

100

1000

Distance From Axis of Wellbore Fig. 2.19—Reservoir pressure profiles for IARF and PSSF.

10000

Fluid Flow in Porous Media 37

During IARF, a graph of pressure vs. time will form a straight line on a semilog scale, Fig. 2.20. During PSSF, a graph of pressure vs. time will form a straight line on a Cartesian scale, Fig. 2.21. The equation for PSSF for a closed circular reservoir may be written as pwf ( t ) = pi −

0.0744 qBt 141.2qBµ   re  3  − ln   − + s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.61) re 2 hφct kh   rw  4 

For a closed reservoir of arbitrary geometry, we may write pwf ( t ) = pi −

0.234 qBt 141.2qBµ  1  10.06 A  3  −  ln   − + s, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.62) 2 Ahφct kh  2  C Arw  4 

where CA is a shape factor that depends on the shape of the reservoir and the position of the well in the reservoir. Eq. 2.62 is the basis for reservoir limits testing, often used in exploration wells to estimate or place a lower bound on hydrocarbons in place.

Pressure, psi

7000

6500

6000

IARF 5500 10

PSSF

100

1000

10000

Time, hr Fig. 2.20—A graph of pressure vs. time forms a straight line on a semilog scale during IARF.

Pressure, psi

7000

6500

6000

IARF 5500

0

PSSF

1000

2000

3000

4000

5000

Time, hr Fig. 2.21—A graph of pressure vs. time forms a straight line on a Cartesian scale during PSSF flow.

38 Applied Well Test Interpretation

Eqs. 2.61 and 2.62 may be written in terms of the average pressure, thereby eliminating the time dependence, as p ( t ) − pwf ( t ) =

141.2qBµ   re  3  ln   − + s , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.63) kh   rw  4 

and p ( t ) − pwf ( t ) =

141.2qBµ  1  10.06 A  3   ln   − + s , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.64) 2 kh  2  C Arw  4 

respectively. Eq. 2.63 gives additional insight into the meaning of the skin factor s. The skin factor inside the brackets in Eq. 2.63 describes the pressure drop across the damaged zone, so the remaining terms inside the brackets must represent the pressure drop in the reservoir. For typical combinations of well spacing and wellbore radius, the term ln (re rw ) − 3 4 is approximately 7 (Brown 1984). This means that, for a well with a skin factor of 7, the pressure drop caused by the damage is about the same magnitude as the entire pressure drop in the reservoir beyond the damaged zone. Further, since the total pressure drop must be positive, we have r  3 ln  e  − + s > 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.65)  rw  4 or r  3 s > − ln  e  + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.66)  rw  4 Eq. 2.66 puts a strict lower bound on the most negative skin factor that can be achieved for a given combination of wellbore radius and drainage area. 2.8.2 Drainage Area Shape Factor and Shape Factor Skin. Letting re be the radius of a circle of area A, Eq. 2.64 may be rearranged to obtain p ( t ) − pwf ( t ) =

141.2qBµ   re  3 1  10.06π  ln   − + s + ln   kh 2  C A    rw  4 =

 141.2qBµ   re  3 ln   − + s + sCA , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.67) kh    rw  4

where sCA is a shape factor skin defined by Fetkovich and Vienot (1985) as sCA =

1  10.06π  ln   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.68) 2  CA 

Table 2.2 gives the shape factor CA and the shape factor skin sCA, for several simple reservoir geometries. Table 2.2 also gives the dimensionless time to end of IARF, te IARF AD, and the dimensionless time to the beginning of PSSF, tb PSSF AD. These last two columns are defined in terms of a dimensionless time based on the drainage area A in square feet: t AD =

0.0002637 kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.69) φµct A

Fluid Flow in Porous Media 39 TABLE 2.2—SHAPE FACTORS FOR SIMPLE DRAINAGE AREA GEOMETRIES Reservoir Shape, Well Location

Shape Factor CA

Shape Factor Skin sCA

Dimensionless Time to End of IARF* te IARF AD

Dimensionless Time to Start of PSSF† tb PSSF AD

0.10

0.06

Circle

31.62

0 Square

30.8828

0.0115

0.09

0.05

12.9851

0.445

0.03

0.25

4.5132

0.973

0.01

0.25

2 × 1 Rectangle

21.8369

0.185

0.025

0.15

10.8374

0.535

0.025

0.15

4.5141

0.973

0.06

0.5

2.0769

1.361

0.02

0.5

*Dimensionless pressure from logarithmic straight-line equation within 1% of exact solution. †

Dimensionless pressure from pseudosteady-state equation within 1% of exact solution.

The time to end of IARF is defined as the time at which the pressure response Δp = pi – pwf for the bounded reservoir deviates by 1% from that given by the logarithmic approximation to the Ei-function solution for IARF, Eq. 2.60. Similarly, the time to the beginning of PSSF is defined as the time at which the pressure response for the bounded reservoir is within 1% of that given by the solution for PSSF, Eq. 2.67. Interestingly, for a well in the center of a closed circular reservoir, by these definitions, PSSF begins before IARF ends; the equations for both IARF and PSSF are within 1% of the full bounded-reservoir solution during the period from tAD = 0.06 to tAD = 0.1. More comprehensive shape factor tables may be found in Earlougher (1977), Lee (1982), Fetkovich and Vienot (1985), and Lee et al. (2003). 2.8.3 Productivity Index and Completion Efficiency. The single-phase productivity index is defined as J=

q , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.70) p − pwf

40 Applied Well Test Interpretation

where q is the flow rate in STB/D, p is the average drainage area pressure in psi, pwf is the stabilized flowing bottomhole pressure, and J is the single-phase productivity index in STB/D/psi. The PSSF equation, Eq. 2.67, may be rearranged to give the productivity index as J=

=

kh  r  3 1  10.06π   141.2 Bµ  ln  e  − + s + ln   2  C A     rw  4 kh  r  3  141.2 Bµ  ln  e  − + s + sCA    rw  4 

The completion efficiency is defined as the ratio of the actual productivity index to the productivity index that the well would have had if it were neither damaged nor stimulated (Lee 1982): E=

Jactual , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.71) J ideal

where E is the completion efficiency, expressed as a fraction. A damaged well will have a completion efficiency less than 100%, while a stimulated well will have a completion efficiency greater than 100%. The completion efficiency may also be written in terms of the pressure drop as E=

p − pwf − ∆ ps p − pwf

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.72)

or in terms of the reservoir size, skin factor, and shape factor skin as

E=

r ln  e  rw r ln  e  rw

 3  − + sCA  4

 3  − + s + sCA  4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.73)

Note that the shape factor skin sCA appears in both numerator and denominator of Eq. 2.73, while the damage/ stimulation skin factor s appears only in the denominator. For a well in the center of a closed circular drainage area with typical wellbore radius and well spacing, we can write E≅

7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.74) 7+s

Thus, a well with a damage skin of 7 will have a completion efficiency of approximately 50%, while a well with a damage skin of 63 will have a completion efficiency of roughly 10%. A similar rule of thumb has been used for many years to estimate the theoretical increase in production for a given decrease in skin factor, assuming fixed pressure drawdown (Brown and Hawkes 2005): qnew 8 + sold ≅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.75) qold 8 + snew This equation assumes that the flowing bottomhole pressure is not affected by flow rate. To determine whether a particular well is a candidate for stimulation, the engineer must consider all pressure losses from the reservoir to the surface. The skin factor alone must not be used for stimulation candidate selection. The skin factor alone must not be used for stimulation candidate selection; the engineer must consider all pressure losses in the system to determine whether a given well is a good stimulation candidate.

Fluid Flow in Porous Media 41

2.8.4 Reservoir Limits Testing. If we compare Eq. 2.62 to the equation of a straight line, y = mx + b, we find that, during PSSF, a graph of flowing bottomhole pressure pwf(t) vs. t will have a slope mpss given by mpss = −

0.234qB , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.76) Ahφ ct

and an intercept b given by bpss = pi −

141.2qBµ kh

 1  10.06 A  3  − + s  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.77)  ln  2   2  C A rw  4 

Reservoir Limits Test Analysis—Recommended Procedure. The following procedure may be used to analyze data in PSSF for a well producing at constant rate: 1. Graph the flowing bottomhole pressure, pwf, vs. the test time, t, on a Cartesian scale. 2. Draw a straight line through the data in PSSF. 3. Find the slope mpss and the intercept bpss of the straight line. 4. Calculate the reservoir pore volume, Vp, and the oil in place, N, from the slope mpss as Vp = Aφ h =

N=

0.234 qB , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.78) m pss ct

Vp (1 − Sw ) 5.615 B

=

q (1 − Sw ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.79) 24 m pss ct

5. If the formation net pay and porosity are known, calculate the drainage area from the pore volume: A=

Vp

φh

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.80)

6. If the initial reservoir pressure pi is known, calculate the productivity index, J, of the well from the intercept bpss as J=

q q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.81) = p − pwf pi − b pss

Example 2.1—Reservoir Limits Test Analysis. The following example illustrates the recommended procedure for analyzing a reservoir limits test. Problem. Given the rock and fluid property data in Table 2.3, analyze the reservoir limits test data given in Table 2.4. Solution. We follow the recommended procedure, as follows. 1. Graph the flowing bottomhole pressure, pwf, vs. the test time, t, on a Cartesian scale, as shown in Fig. 2.22. 2. Draw a straight line through the data in PSSF, Fig. 2.22.

TABLE 2.3—ROCK AND FLUID PROPERTY DATA FOR RESERVOIR LIMITS ANALYSIS EXAMPLE Reservoir and Fluid Properties q

250 STB/D

SW

25%

φ

21%

B

1.328 bbl/STB

h

15 ft

µ

0.61 cp

rW

0.32 ft

ct

16.1 × 10–6 psi–1

42 Applied Well Test Interpretation TABLE 2.4—TEST DATA FOR RESERVOIR LIMITS EXAMPLE t hr

pws psia

t hr

pws psia

t hr

pws psia

t hr

pws psia

0 0.5 0.75 1 1.25 1.5 2 2.5 3 4

4419.0 4240.8 4225.4 4219.5 4217.0 4215.4 4213.3 4212.2 4211.4 4210.2

5 6 7 8 9 10 11 12 14 16

4209.6 4208.3 4207.4 4207.3 4207.0 4206.3 4206.2 4205. 7 4204.9 4203.7

18 20 22 24 27 30 33 36 39 42

4203.4 4202.5 4202.1 4201.2 4199.7 4198.4 4197.7 4196.8 4195.4 4194.5

45 48 51 54 57 60 63 66 69 72

4193.4 4192.5 4190.8 4189.9 4188.5 4187.8 4186.9 4185.2 4184.1 4183.8

3. Find the slope mpss and the intercept bpss of the straight line. We read the intercept bpss from the graph as 4,210.0 psi. The slope mpss is given by m pss =

4, 210.0 − 4,180.3 = 0.371 psi/hr. 0 − 80

4. We calculate the reservoir pore volume, Vp, and the oil in place, N, from the slope mpss as Vp = =

0.234 qB m pss ct

( 0.234 )( 250 )(1.328) ( 0.371) (16.1 × 10 ) −6

= 13 × 10 6 ft 3 , and N= =

q (1 − Sw ) 24 m pss ct

( 250 ) (1 − 0.25) ( 24 )( 0.371) (16.1 × 10 ) −6

= 1.31 × 10 6 STB. 4250 4240

Pressure, psi

4230 4220 4210 4200

4,210.0 psi

4190

4,180.3 psi

4180 4170

0

10

20

30

40 Time, hr

50

60

Fig. 2.22—Reservoir limits test analysis example.

70

80

Fluid Flow in Porous Media 43

5. Since the formation net pay and porosity are known, we calculate the drainage area from the pore volume: A= =

Vp

φh 13 × 10 6 ( 0.21)(15)

= 4.13 × 10 6 ft 2 = 94.7 acree . 6. Since the initial reservoir pressure pi is known, calculate the productivity index, J, of the well from the intercept bpss as J=

q pi − b pss

250 4419 − 4210 = 1.20 STB/D/psi. =

2.8.5 Radius of Investigation Revisited. In Section 2.6, we showed that setting the argument of the Ei function to 1 leads to the radius of investigation equation, Eq. 2.39. In this section, we develop the radius of investigation equation from a different perspective. During IARF, the pressure response is given by Eq. 2.59. On the other hand, during PSSF to a well in the center of a closed circular reservoir, the pressure response is given by Eq. 2.61. The derivative of Eq. 2.59 with respect to time is given by dpwf dt

=

70.6qBµ 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.82) kh t

showing that the derivative decreases monotonically as time increases. Taking the derivative of Eq. 2.61, we have dpwf dt

=

0.0744 qB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.83) re 2 hφct

If we set Eqs. 2.82 and 2.83 equal and solve for the time at which the slope of the pressure response for IARF is equal to that for PSSF from a closed circular reservoir of radius re, we find that t=

948φµct re 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.84) k

which is Eq. 2.39 with the drainage radius re substituted in place of the radius of investigation, ri. For constant-rate production of a slightly compressible liquid from a closed reservoir, the radius of investigation at the end of the IARF period gives a lower bound on the drainage radius, and thus a lower bound on hydrocarbons in place. For constant-rate production of a slightly compressible liquid from a closed reservoir, the radius of investigation at the end of the IARF period gives a lower bound on the drainage radius, and thus a lower bound on hydrocarbons in place.

Mattar and Zoral (1992) call the derivative with respect to time, dpw dt , the primary pressure derivative. They argue that any reservoir pressure response will always give a monotonically decreasing primary pressure derivative. PSSF will always be the last flow regime encountered for constant-rate production of a slightly compressible liquid from a closed reservoir. Therefore, for a closed reservoir, any value of the primary pressure

44 Applied Well Test Interpretation

derivative observed before PSSF is established represents an upper bound on the value of the primary pressure derivative during PSSF and thus, provides a lower bound on the size of the reservoir. For a closed reservoir, any value of the primary pressure derivative observed before PSSF is established represents an upper bound on the value of the primary pressure derivative during PSSF and thus, provides a lower bound on the size of the reservoir. 2.9 Wellbore Storage The Ei-function solution assumes that the sandface flow rate rises instantly from zero to the final constant rate q at time t = 0 at the beginning of a drawdown test. In using superposition to calculate the buildup pressure response using the Ei-function solution, we assume the sandface flow rate drops instantly from q to zero at time Δt = 0 at the beginning of the buildup test. However, because of the finite volume of the wellbore between the sandface and the valve used to control the flow rate (Fig. 2.23), the sandface rate does not change immediately upon opening or closing the valve. Instead, the first fluid produced upon opening the valve comes from the wellbore. As the pressure in the wellbore decreases because of withdrawal of fluid, the sandface flow rate rises rapidly at first, asymptotically approaching a constant value equal to the surface flow rate q, as shown in Fig. 2.24. Similarly, when the well is shut in at the surface following a period of production at constant rate, fluid continues to flow into the wellbore, asymptotically approaching zero, as shown in Fig. 2.25. During buildup, this phenomenon is often referred to as “afterflow.”

Fig. 2.23—Wellbore storage, fluid-filled wellbore.

Surface flow rate

Sandface flow rate

Fig. 2.24—Wellbore storage, drawdown test, constant surface flow rate. Sandface flow rate rises slowly, asymptotically approaching surface flow rate.

Fluid Flow in Porous Media 45

Sandface flow rate

Surface flow rate Fig. 2.25—Wellbore storage, buildup test, illustrating afterflow. Sandface flow rate decreases slowly, asymptotically approaching zero.

In both drawdown and buildup, WBS distorts the shape of the pressure response. The effect of constant WBS on the pressure response will be discussed in Chapter 4. The effect of variable WBS will be discussed in Chapter 9. Two common situations can cause a constant WBS: a well filled with a single-phase fluid, and a well with a rising or falling liquid level. 2.9.1 Fluid-Filled Wellbore. Fig. 2.23 shows a well filled with a single-phase fluid. This situation is very common in gas wells, where there is either no liquid in the wellbore or the liquid level is constant with time. Oil wells with surface shut-in usually do not exhibit a liquid-filled wellbore because the pressure at the wellhead will almost always be below the bubblepoint pressure of the oil. A fluid-filled wellbore may occur in a water-injection well when the pump is shut in for a falloff test, before the well goes on vacuum. We can write a material-balance equation for the relationship between the surface and sandface flow rates for a wellbore filled with a single-phase fluid as 24cwbVwb

dpw = − ( q ( t ) − qsf ( t )) Bwb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.85) dt

where cwb is the compressibility of the wellbore fluid in psi–1, Bwb is the formation volume factor of the fluid in the wellbore, Vwb is the wellbore volume in bbl, pw is the wellbore pressure, and qsf is the sandface flow rate. By introducing a WBS coefficient C, defined as C = cwbVwb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.86) we can rewrite Eq. 2.85 as 24C

dpw = − ( q ( t ) − qsf ( t )) Bwb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.87) dt

Eq. 2.87 is general, applying to variable flow rate production as well as drawdown and buildup. Just before the beginning of a drawdown, the sandface flow rate and surface flow rate are both zero. Upon opening the valve, the surface flow rate rises instantly to a constant value q. The rate of pressure change immediately after the valve is opened, when the sandface flow rate is approximately zero, is then given by dp dpw − q − 0 B 24 24C C w≅ ≅ − ( − 0 ) wb wb dt =− − qBwb = wb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.88) The flowing wellbore pressure pwf during this WBS-dominated period is then given by pi − pwf ≅

qBwb t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.89) 24C

46 Applied Well Test Interpretation

Similarly, just before shutting in for a buildup, the sandface and surface flow rates are both q. Immediately after shutting in the well before the sandface flow rate has begun to drop, the rate of pressure change in the wellbore is given by 24C

dpw ≅ − ( 0 − qsf ) Bwb, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.90) dt = qBwb

while the shut-in wellbore pressure pws is given by pws − pwf ≅

qBwb ∆t, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.91) 24C

where pwf is the flowing bottomhole pressure at the moment of shut-in. For a well producing through casing from a normally pressured reservoir, filled with a single-phase gas, shut in at the surface, the WBS coefficient is approximately 0.05 psi, independent of depth. The WBS coefficient for a well filled with a single-phase liquid is typically one to two orders of magnitude smaller than for a well filled with a single-phase gas. The analytical solution for the pressure response for production at constant rate q, from a well with constant WBS and skin, in an infinite-acting reservoir is rather complex, with no simple closed form. However, the solution can be presented graphically as a set of drawdown type curves, as we will see in Chapter 4. 2.9.2 Rising or Falling Liquid Level. Many wells have a rising or falling liquid level instead of a wellbore filled with a single-phase fluid, as shown in Fig. 2.26. This situation is particularly common in oil wells produced by rod pump, where the well is pumped off (i.e., the liquid level is maintained at a constant height a short distance above the pump as long as the well is producing), and in water-injection wells, where the well goes on vacuum a short time after the pump is turned off. A wellbore material-balance equation for a well with a rising or falling liquid level may be written as 24

144 Awb d ( pw − pt ) = − ( q − qsf ) Bwb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.92) dt 5.615 ρ wb

where pt is the pressure at the top of the liquid column, rwb is the density of the liquid in lbm/ft3, and Awb is the cross-sectional area of the wellbore in square feet. If pt is constant, we have 24

144 Awb dpw = − ( q − qsf ) Bwb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.93) 5.615 ρ wb dt

Fig. 2.26—Wellbore storage, rising liquid level.

Fluid Flow in Porous Media 47

For the changing liquid level case, we define the WBS coefficient as C=

A 144 Awb = 25.65 wb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.94) ρ wb 5.615 ρ wb

allowing us to simplify Eq. 2.93 as 24C

dpw = − ( q − qsf ) Bwb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.95) dt

Note that Eq. 2.95 has exactly the same form as Eq. 2.87. This allows us to use solutions for the pressure response for a well with constant WBS coefficient and constant skin factor, producing at constant rate, to describe both a well filled with a single-phase fluid and a well with a changing liquid level. The constant WBS and skin model is thus a very general and widely applicable model, a fact that we will take advantage of in Chapter 4 when we introduce type curves for constant WBS and skin factor. If the wellbore area Awb is given as tubing or casing capacity in bbl/ft and the density of the liquid rwb is given as liquid gradient in psi/ft, the WBS coefficient for a changing liquid level may be written simply as C=

Awb ( in bbl ft ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.96) ρ wb ( in psi ft )

The WBS coefficient for a well with a changing liquid level is approximately 0.05 psi, depending on the size of the casing and the density of the liquid. The WBS coefficient for a well with either a gas-filled wellbore or a changing liquid level (two of the most common situations) is roughly 0.05 psi. 2.10 Summary The following points summarize this chapter: 1. The diffusivity equation describing the flow of a slightly compressible liquid in a porous medium is derived by combining the continuity equation, Darcy’s law, and the equation of state for a slightly compressible liquid. 2. The Ei-function solution to the diffusivity equation may be used to calculate the pressure response at any point in an infinite-acting homogeneous reservoir for constant-rate production from a vertical well. 3. The principle of superposition in space may be used to calculate the pressure response for multiple wells, each producing at constant rate. 4. The principle of superposition in time may be used to calculate the pressure response for an arbitrary rate history. 5. The radius of investigation is a measure of the distance a transient has moved into the reservoir at a given time. 6. Each rate change creates a new pressure transient that propagates into the reservoir at the same speed. 7. The radius of investigation is independent of the flow rate. 8. Near-wellbore damage caused by mud filtrate invasion or other factors may be described by a skin factor. 9. During infinite-acting radial flow, the flowing bottomhole pressure is a linear function of the logarithm of time. 10. During pseudosteady-state flow, the flowing bottomhole pressure is a linear function of time. 11. The reservoir pore volume, original oil in place, drainage area, and productivity index may be estimated from data in pseudosteady-state flow. 12. Wellbore storage causes the sandface flow rate to be different from the surface flow rate, thereby distorting the shape of the resulting pressure response. 13. Wellbore storage may be caused by a well filled with a single-phase fluid of constant compressibility or by a well with a rising or falling liquid level.

48 Applied Well Test Interpretation

Nomenclature A = drainage area of well, ft2 Awb = cross-sectional area of wellbore, ft2 B = formation volume factor, bbl/STB c = compressibility, psi–1 C = WBS coefficient, bbl/psi cf = formation or pore volume compressibility, psi–1 ct = total compressibility, psi–1 h = net pay thickness, ft j = summation index J = productivity index, STB/D/psi k = permeability, md n = number of rate changes N = number of wells N = original oil in place, STB p = pressure, psi pi = initial pressure, psi q = flow rate, STB/D r = distance from the axis of the wellbore, ft ri = radius of investigation, ft rw = wellbore radius, ft rwa = apparent wellbore radius, ft s = skin factor, dimensionless S = saturation, fraction t = time, hour Vp = reservoir pore volume, ft3 Vwb = wellbore volume, bbl x = argument of Ei function, dimensionless y = integration variable, dimensionless rwb = density of fluid in wellbore, lbm/ft3 μ = viscosity, cp f = porosity, fraction t = time integration variable, hour Subscripts a = property of altered zone g = gas o = oil pss = pseudosteady state sf = sandface conditions w = water wb = wellbore conditions wf = flowing wellbore conditions ws = shut-in wellbore conditions x = spatial coordinate direction

Chapter 3

Radial Flow Semilog Analysis “Seeing there is nothing (right well beloved Students of Mathematics) that is so trublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.” —John Napier, Canon of Logarithms, 1614 3.1 Introduction Infinite-acting radial flow (IARF) is arguably the most common flow regime encountered in pressure transient analysis. This chapter presents straight-line analysis methods for analyzing data in IARF. The pressure response during IARF is a linear function of the logarithm of time. Thus, straight-line analysis methods for IARF use either logarithmic time scales or time plotting functions involving logarithms or sums of logarithms. Section 3.2 addresses the topic of drawdown tests, in which the well is initially shut in, and the reservoir is at uniform pressure. The well is opened to flow, and the resulting pressure response is measured as a function of time. The section opens with a discussion of analysis of drawdown tests conducted at constant rate. Unfortunately, it is difficult to maintain a perfectly constant flow rate, and even small changes in flow rate over the course of a test may radically distort the shape of the pressure response. If the rate is changing slowly and smoothly, and if the reservoir is infinite acting for all times of interest, rate normalization may be used to account for changing flow rate. In many cases, there is the need to test wells that have been producing for an extended period of time, where the assumption of uniform reservoir pressure is violated. Section 3.3 introduces the buildup test as a means of eliminating the assumption of uniform reservoir pressure at the beginning of the test. The section then discusses the Miller-Dyes-Hutchinson (MDH) and Horner methods for analyzing data from pressure-buildup tests following constant rate production. In Section 3.4, we address the issue of estimating average drainage area pressure, using methods based on extrapolation of the semilog straight line. These methods include the Horner method, for reservoirs that are infinite acting from the beginning of production to the end of the buildup; the Dietz and Ramey-Cobb methods, which assume the reservoir has produced at constant rate long enough to reach pseudosteady-state flow, and the Matthews-Brons-Hazebroek (MBH) method, which was the most general method available before widespread use of personal computers. Section 3.5 first discusses the effect of flow rate variations on a subsequent buildup. The section then introduces the Horner pseudoproducing-time approximation for analyzing buildup tests following a multirate flow period in which the last constant-rate flow period before shut-in is much longer in duration than the buildup test. The section closes by showing how to adjust the flowing bottomhole pressure and time of shut-in to account for a momentary change in flow rate immediately before shut-in. After completing this chapter, you should be able to 1. Estimate permeability and skin factor from a constant-rate drawdown test. 2. Estimate permeability and skin factor for a test in which the rate is changing slowly and smoothly. 3. Estimate permeability and skin factor from a pressure-buildup test using the MDH method, the Horner method, and the Agarwal equivalent-time method.

50 Applied Well Test Interpretation

4. Explain the advantages and disadvantages of the Miller-Dyes-Hutchinson, Horner, and equivalent-time methods. 5. Estimate average drainage-area pressure using the Dietz, Ramey-Cobb, and Matthews-Brons-Hazebroek methods. 6. Calculate the Horner pseudoproducing time. 7. List the conditions under which the Horner pseudoproducing time may be used. 8. Adjust the flowing bottomhole pressure and time of shut-in to account for a brief change in flow rate immediately before shutting in for a buildup test. 3.2 Drawdown Tests The constant-rate drawdown test is the easiest test protocol to treat theoretically because it most nearly approximates the “constant terminal rate” boundary condition under which most analytical solutions have been obtained. In this section, we will first present the analysis procedure for a constant-rate drawdown test. We will then present an approximate method known as rate normalization to allow us to analyze drawdown tests in infinite-acting reservoirs in which the rate varies slowly and smoothly with time. 3.2.1 Drawdown—Constant-Rate Production. In a constant-rate drawdown test, the well is initially shut in, and the reservoir is at uniform pressure. The well is produced at constant rate q, and the flowing bottomhole pressure pwf is measured as a function of time as the pressure draws down. As we saw in Chapter 2, the pressure response for a well producing at constant rate from an infinite-acting reservoir may be written as pwf ( t ) = pi −

  k  162.6qBµ  log t + log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.1) 2 kh  φµct rw   

Comparison of Eq. 3.1 with the equation of a straight line, y = mx + b, suggests that a graph of pwf (t) vs. log(t) for drawdown data exhibiting IARF will be a straight line with slope m given by m=−

162.6qBµ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.2) kh

and y-intercept b given by b = p1hr = pi −

 162.6qBµ   k  log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.3) 2 kh    φµct rw 

Because the independent variable is log(t), the y-intercept is read from the y-axis, where log(t) = 0, corresponding to a time of 1 hour. Thus, the intercept b is usually written as p1hr for a drawdown test. Fig. 3.1 shows a graph of pressure vs. time for a drawdown test, showing the straight line through the data in IARF, the slope m, and the intercept p1hr. Drawdown Semilog Analysis—Recommended Procedure. The following procedure may be used to analyze a constant-rate drawdown test exhibiting IARF: 1. Graph the flowing bottomhole pressure, pwf , vs. the test time, t, on a semilog scale. 2. Identify data in IARF. (At this point, simply find the data that fall on a straight line. We will learn how to identify radial flow and other flow regimes unambiguously using the diagnostic plot in Chapter 6). 3. Draw a straight line through the selected data, and find the slope m and intercept p1hr. Note that the intercept p1hr must be read from the straight line or its extrapolation, not from the measured field pressure data. 4. Calculate the permeability k from the slope m from 162.6qBµ mh 162.6qBµ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.4) mh

k =−

Radial Flow Semilog Analysis 51 2800 2700 2600

pwf , psia

2500 2400 2300 Slope m

2200 2100 2000

Intercept p1hr

1900 1800 0.001

0.01

0.1

1

10

100

t, hr Fig. 3.1—Semilog analysis for drawdown test.

Note that the permeability is conventionally written with the absolute value of the slope. For a drawdown test, the slope m must always be negative. 5. Calculate the skin factor s from the slope m and the intercept p1hr as  p − p   k  s = 1.151  i 1hr − log   + 3.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.5) 2  φµct rw   m  6. Calculate the radius of investigation at the beginning and end of the apparent semilog straight line: ri =

kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.6) 948φµct

In practice, wellbore storage almost always distorts the shape of the pressure response at early times. If the test lasts long enough, boundaries will also affect the pressure response at late times. The most difficult part of semilog analysis is correctly identifying the data that are in IARF, so that the correct semilog straight line can be drawn. For semilog analysis of a drawdown test, the slope m of the semilog straight line gives the permeability and the intercept p1hr gives the skin factor.

Example 3.1—Drawdown Semilog Analysis. The following example illustrates the procedure for analyzing a constant-rate drawdown test using semilog analysis. Problem. Given the rock and fluid properties in Table 3.1, analyze the pressure drawdown test data in Table 3.2 and Fig. 3.2.

TABLE 3.1—ROCK AND FLUID PROPERTY DATA FOR DRAWDOWN ANALYSIS EXAMPLE Reservoir and Fluid Properties q

125 STB/D

pi

2,750 psi

f

22%

B

1.152 bbl/STB

h

32 ft

m

2.122 cp

rw

0.25 ft

ct

10.9 × 10–6 psi–1

52 Applied Well Test Interpretation TABLE 3.2—TEST DATA FOR DRAWDOWN SEMILOG ANALYSIS EXAMPLE t hr

pwf psia

t hr

pwf psia

t hr

pwf psia

t hr

pwf psia

0.0010

2748.95

0.0869

2655.67

0.993

2196.92

10.545

1995.75

0.0021

2745.62

0.0988

2642.29

1.118

2170.70

11.865

1991.15

0.0034

2744.63

0.1121

2627.50

1.259

2148.33

13.349

1988.67

0.0048

2745.49

0.1271

2614.76

1.417

2126.44

15.018

1984.74

0.0064

2741.70

0.1440

2598.79

1.595

2108.50

16.897

1979.34

0.0082

2742.00

0.1630

2582.16

1.795

2090.87

19.010

1981.14

0.0102

2736.69

0.1844

2564.54

2.021

2080.73

21.387

1973.78

0.0125

2737.26

0.208

2545.27

2.275

2066.59

24.061

1970.58

0.0151

2733.72

0.236

2523.21

2.560

2054.29

27.070

1967.59

0.0180

2729.13

0.266

2501.07

2.881

2048.25

30.455

1965.50

0.0212

2724.23

0.300

2475.93

3.242

2039.49

34.262

1961.64

0.0249

2720.57

0.339

2451.83

3.648

2035.32

38.546

1957.61

0.0290

2715.83

0.382

2422.80

4.105

2029.91

43.366

1955.90

0.0336

2710.70

0.431

2397.61

4.619

2025.01

48.787

1951.21

0.0388

2706.63

0.486

2367.50

5.198

2018.87

54.787

1949.05

0.0447

2698.17

0.547

2338.18

5.848

2016.40

60.787

1945.70

0.0512

2692.75

0.617

2309.21

6.580

2011.11

66.787

1942.51

0.0587

2684.56

0.695

2277.84

7.404

2007.46

72.000

1941.14

0.0670

2676.82

0.783

2251.46

8.331

2003.24

0.0764

2665.33

0.882

2222.09

9.373

2000.53

2800 2700 2600

pwf , psia

2500 2400 2300

p1 = 2,250 psia

2200 2100 2000

p1hr = 2,060 psia

1900

p2 = 1,930 psia

1800 0.001 t1 = 0.001 hr

0.01

0.1

1

10

t, hr

100 t2 = 100 hr

Fig. 3.2—Drawdown semilog analysis example.

Solution. We follow the recommended procedure for the analysis. 1. Graph the flowing bottomhole pressure, pwf , vs. the test time, t, on a semilog scale, Fig. 3.2. 2. Identify data in IARF. The early data appear to be distorted by wellbore storage. The data toward the end of the test appear to fall on a straight line, so we select this data for analysis. 3. Draw a straight line through the selected data, and find the slope m and intercept p1hr. Reading two points on the straight line (as far apart as possible for best accuracy), we find the slope m to be

Radial Flow Semilog Analysis 53

m=

2, 250 − 1, 930 log ( 0.001) − log (100 )

320 −3 − 2 = − 64 psi/cycle. =

We read the intercept p1hr from the extrapolation of the straight line at a time of 1 hour as 2,060 psia. 4. Calculate the permeability k from the slope m using Eq. 3.4: k= =

162.6qBµ mh

(162.6)(125)(1.152)( 2.122) (64 ) (32)

= 24.3 md. Note that the permeability is conventionally written with the absolute value of the slope. For a drawdown test, the slope m must always be negative. 5. Calculate the skin factor s from the slope m and the intercept p1hr using Eq. 3.5 as  p − p   k  1hr 3 23 s = 1.151  i − log  . +    φµc r 2    m t w    ( 2, 750 ) − ( 2, 060 ) = 1.151  (64 )   ( 24.26) − log   0.22 2.122 10.9 × 10 −6 0.25 2 )( ) ( ) ( = 7.06.

(

)

   + 3.23     

6. Calculate the radius of investigation at the beginning and end of the apparent semilog straight line. The semilog straight line begins around t = 6 hours, and lasts until the end of the test at 72 hours. At the beginning of the straight line at t = 6 hours, the radius of investigation is given by ri = =

kt 948φµct

( 24.3)( 72) (948)( 0.22)( 2.122) (10.9 × 10 ) −6

= 602 ft. In the same way, we calculate the radius investigation at the end of the test as 602 ft. 3.2.2 Drawdown—Smoothly Varying Rate. In practice, it is often difficult to maintain a strictly constant rate during a drawdown test. How constant does the rate have to be for the constant-rate assumption to be valid? To illustrate how sensitive the drawdown analysis is to even small variations in flow rate, consider the following example. Fig. 3.3 shows the rate profile for a test with a 10% decrease in rate during the test. The resulting pressure response is shown in Fig. 3.4. Even with a change in flow rate as small as 10%, the pressure response is increasing by the end of the test. If the last few points of the test data were used to estimate permeability under the assumption of constant flow rate, and the algebraic sign of the slope were carried through the calculation, the calculated permeability would be negative.

54 Applied Well Test Interpretation 25

q, STB/D

20 10% decrease in rate during test

15

10

5

0

0

1

2

3

4

5

6

7

t, hr Fig. 3.3—Slowly decreasing rate—10% decrease during test. 2000 1900 1800

pwf , psia

1700 1600 1500 1400 1300

Even with only 10% change in rate, pressure starts increasing by end of test

1200 1100 1000 0.10

0.1

1

10

t, hr Fig. 3.4—Pressure response for test with 10% decrease in rate during test.

Winestock and Colpitts (1965) showed that, if the rate is changing slowly and smoothly, the pressure response for a varying flow rate in an infinite-acting reservoir may be modeled as pi − pwf ( t ) q (t )

=

  k  162.6 Bµ  log t + log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.7) 2 kh   φµct rw  

This technique is commonly known as rate normalization. To use rate normalization, we graph the rate-normalized pressure change ∆p q = [ pi − pwf (t )] q(t ) vs. the test time t on a semilog scale, as shown in Fig. 3.5. Drawdown Semilog Analysis, Smoothly Varying Rate—Recommended Procedure. The following procedure may be used to analyze a variable-rate drawdown test in an infinite-acting reservoir, when the rates are changing slowly and smoothly: 1. Graph the rate-normalized pressure change, ∆p q = [ pi − pwf (t )] q(t ), vs. the test time, t, on a semilog scale. 2. Identify data in IARF. 3. Draw a straight line through the selected data, and find the slope m′ and intercept (Δp/q)1hr. Note that the intercept (Δp/q)1hr must read from the straight line or its extrapolation, not from the measured field pressure data.

Radial Flow Semilog Analysis 55 100

(pi –pwf ) /q, psia

90

80

m′ (∆p/q)1hr

70

60

50

40 0.01

0.1

1

10

t, hr Fig. 3.5—Drawdown analysis, smoothly varying rate, with rate normalization.

4. Calculate the permeability k from the slope m′ from k=

162.6 Bµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.8) m′h

The slope m′ must always be positive. 5. Calculate the skin factor s from the slope m′ and the intercept (Δp/q)1hr as  ( ∆p q )   k  1hr − log  s = 1.151   + 3.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.9) 2  φµct rw   m′  6. Calculate the radius of investigation at the start and end of the semilog straight line: ri =

kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.10) 948φµct

The rate-normalized pressure change may be used to analyze data for a flow test in which the rate is changing slowly and smoothly.

Example 3.2—Drawdown Semilog Analysis, Smoothly Varying Rate. The following example illustrates the procedure for using rate normalization to analyze a test with a smoothly varying rate. Problem. Analyze the test show in Figs. 3.3 through 3.5. Rock and fluid property data are given in Table 3.3; pressure and rate data are given in Table 3.4. TABLE 3.3—ROCK AND FLUID PROPERTY DATA FOR SEMILOG ANALYSIS EXAMPLE, DRAWDOWN WITH SMOOTHLY VARYING RATE Reservoir and Fluid Properties pi

2,750 psi

B

f

18.5%

m

6.5 cp

h

35 ft

ct

8.42 × 10–6 psi–1

rw

0.25 ft

1.15 bbl/STB

56 Applied Well Test Interpretation TABLE 3.4—TEST DATA FOR SEMILOG ANALYSIS EXAMPLE, DRAWDOWN WITH SMOOTHLY VARYING RATE t hr

pwf psia

q STB/D

Δp/q psi/STB/D

t hr

pwf psia

q STB/D

Δp/q psi/STB/D

0.00600

1759.42

19.998

49.53

0.219

1494.16

19.927

63.02

0.00693

1748.83

19.998

50.06

0.253

1484.03

19.916

63.57

0.00800

1738.21

19.997

50.60

0.292

1474.01

19.903

64.11

0.00924

1727.57

19.997

51.13

0.337

1464.13

19.888

64.66

0.01067

1716.90

19.996

51.67

0.390

1454.42

19.870

65.20

0.01232

1706.22

19.996

52.20

0.450

1444.88

19.850

65.75

0.01423

1695.52

19.995

52.74

0.520

1435.57

19.827

66.29

0.01643

1684.82

19.995

53.27

0.600

1426.50

19.800

66.84

0.01897

1674.10

19.994

53.81

0.693

1417.73

19.769

67.39

0.0219

1663.38

19.993

54.35

0.800

1409.29

19.733

67.94

0.0253

1652.66

19.992

54.89

0.924

1401.25

19.692

68.49

0.0292

1641.94

19.990

55.43

1.067

1393.68

19.644

69.05

0.0337

1631.22

19.989

55.97

1.232

1386.64

19.589

69.60

0.0390

1620.51

19.987

56.51

1.423

1380.23

19.526

70.15

0.0450

1609.81

19.985

57.05

1.643

1374.54

19.452

70.71

0.0520

1599.12

19.983

57.59

1.897

1369.69

19.368

71.27

0.0600

1588.46

19.980

58.14

2.191

1365.83

19.270

71.83

0.0693

1577.81

19.977

58.68

2.530

1363.12

19.157

72.40

0.0800

1567.19

19.973

59.22

2.922

1361.73

19.026

72.97

0.0924

1556.60

19.969

59.76

3.374

1361.89

18.875

73.54

0.1067

1546.05

19.964

60.31

3.896

1363.85

18.701

74.12

0.1232

1535.54

19.959

60.85

4.499

1367.90

18.500

74.71

0.1423

1525.09

19.953

61.39

5.195

1374.38

18.268

75.30

0.1643

1514.70

19.945

61.94

5.999

1383.70

18.000

75.91

0.1897

1504.39

19.937

62.48

Solution. We follow the recommended procedure for using rate normalization to analyze a drawdown test where the rate varies slowly and smoothly. 1. Graph the rate-normalized pressure change, ∆p q = [ pi − pwf (t )] q(t ), vs. the test time, t, on a semilog scale. The graph is shown in Fig. 3.6. 2. Identify data in IARF. All of the test data appear to fall on a single straight line, so we assume all the data are in IARF. 100

(pi –pwf )/q, psia

90

80

70

(∆p/q)2 = 78 psi/(STB/D) (∆p/q)1hr = 69 psi/(STB/D)

60

50

(∆p/q)1 = 51.5 psi/(STB/D)

40 0.01 t1 = 0.01 hr

0.1

1 t, hr

10 t2 = 10 hr

Fig. 3.6—Example semilog analysis, drawdown with smoothly changing rates.

Radial Flow Semilog Analysis 57

3. Draw a straight line through the selected data, and find the slope m′ and intercept (Δp/q)1hr. We calculate the slope as m′ = =

78 − 51.5 log (10 ) − log ( 0.01) 26.5 1 − ( −2 )

= 8.83 psi/(STB/D)/cycle. 4. Calculate the permeability k from the slope m′ from k= =

162.6 Bµ m′h (162.6)(1.15)(6.5)

(8.83)(35)

= 3.93 md. 5. Calculate the skin factor s from the slope m′ and the intercept (Δp/q)1hr as  k  ( ∆p q )1hr − log  s = 1.151   φµc r 2 m′ t w  

   + 3.23  

  (3.93)  69 = 1.151  − log   0.185 6.5 8.42 × 10 −6 0.25 2  8.83 )( ) ( ) (  = 4.25.

(

)

   + 3.23     

6. Calculate the radius of investigation at the start and end of the semilog straight line. All of the data fall on a single straight line, so we calculate the radius of investigation at the beginning of the test, t = 0.006 hours, as 1.6 ft, and at the end of the test, t = 6 hours, as 50 ft. 3.3 Buildup Test Following Constant-Rate Production One of the underlying assumptions for Eqs. 3.4 and 3.5 is that the flow rate q is constant during the drawdown. Even slight variations in flow rate may have a significant effect on the slope of the pressure response. In practice, it is difficult to maintain a strictly constant flow rate. Another assumption is that the well is shut in and that the pressure is uniform throughout the reservoir before the test begins. However, we often need to test wells that have been producing for some time, creating a pressure gradient in the vicinity of the wellbore. These two considerations led to the development of the buildup test. In an ideal buildup test, the well has been producing at constant rate q for a time tp. The well is then shut in, and the shut-in bottomhole pressure pws is measured as a function of elapsed test time Dt as the pressure builds up. The resulting pressure response may be analyzed a variety of different ways. We will consider the classic MDH (Miller et al. 1950a) and Horner methods (Horner 1951), as well as a method based on the Agarwal equivalent time (Agarwal 1980). We can model a buildup as a series of two separate rate changes, the first from zero to rate q, beginning at time t = 0, the second from q to 0, beginning at time t = tp, as shown in Fig. 3.7. Because the diffusivity equation is linear, we can calculate the buildup pressure response using the principal of superposition.

∆t

tp tp + ∆t

Fig. 3.7—Ideal buildup rate sequence.

58 Applied Well Test Interpretation

The pressure response caused by production at rate q beginning at time t = 0 is given by 162.6qBµ kh

∆p+ q = −

  k log t + log  2   φµct rw

   − 3.23 + 0.869s , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.11)  

while the pressure response caused by a rate change –q beginning at time t = tp is given by  k 162.6qBµ  log ( ∆t ) + log  2 kh   φµct rw

∆p− q = +

   − 3.23 + 0.869s  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.12)  

where Dt is defined as ∆t = t − t p. The shut-in bottomhole pressure during buildup is then calculated from pws ( ∆t ) = pi + ∆p+ q + ∆p− q    k  3 . 23 0 869 − + s . log t p + ∆t + log    2     φµct rw    k  162.6qBµ  + . s 3 . 23 − + 0 869 log ( ∆t ) + log   , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.13)  2  kh    φµct rw 

= pi −

162.6qBµ kh

(

)

where we have replaced t by t p + ∆t . 3.3.1 MDH Method. If the shut-in time Δt is much less than the producing time tp, the second term in Eq. 3.43 may be approximated by    k  3 23 0 869 − + . . s log t p + ∆t + log    2     φµct rw    k  162.6qBµ  s 3 . 23 0 . 869 − + ≅− log t p + log    2  kh    φµct rw  = −  pi − pwf t p  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.14)

∆p+ q = −

162.6qBµ kh

(

)

( )

( )

Thus, we may approximate Eq. 3.13 as pws ( ∆t ) = pi + ∆p+ q + ∆p− q

(

≅ pi − pi − pwf

)

   k  s 3 . 23 0 . 869 − +  log ( ∆t ) + log   2    φµct rw     k  162.6qBµ  = pwf + . . s − + 3 23 0 869 log ( ∆t ) + log   . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.15)  2  kh    φµct rw  +

162.6qBµ kh

where pwf is defined as the flowing bottomhole pressure at the instant of shut-in. Comparing Eq. 3.15 with the equation of a straight line, y = mx + b, suggests that a graph of pws(t) vs. log(Δt) will exhibit a straight line with a slope m given by m=

162.6qBµ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.16) kh

and an intercept b given by b = p1hr = pwf +

 162.6qBµ   k  log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3.17) 2 kh    φµct rw 

Eqs. 3.15, 3.16, and 3.17 provide the basis for the MDH method (Miller et al. 1950).

Radial Flow Semilog Analysis 59

Buildup Analysis, MDH Method—Recommended Procedure. The following modification of the MDH method may be used to analyze pressure-buildup data to obtain permeability and skin factor. If the reservoir is infinite acting throughout the flow and buildup periods, the MDH method may also be used to estimate the initial reservoir pressure. 1. Graph the shut-in bottomhole pressure, pws, vs. the shut-in time Δt, on a semilog scale. 2. Identify the data falling on a single straight line, exhibiting IARF. In this method, only data where Δt 100. 11. The mobility and storativity ratios used to describe the composite reservoir models are determined by the fluid and rock systems. The analyst must ensure that the ratios used in an analysis represent realistic values for the reservoir system. Nomenclature A = drainage area, ft2 B = formation volume factor, bbl/STB Bw = water formation volume factor, bbl/STB ct = total compressibility, psi–1 D = diffusivity ratio for composite reservoir, D = ( k φµct )2 ( k φµct )1, dimensionless h = net pay thickness, ft

Bounded Reservoir Behavior 185

k = permeability, md L = distance to boundary, ft L = distance to apex of wedge reservoir, ft L1 = distance to closer boundary of a channel or wedge reservoir, ft L2 = distance to further boundary of a channel or wedge reservoir, ft M = mobility ratio for composite reservoir, M = ( k µ )2 ( k µ )1, dimensionless M1 = mobility for Region 1 of composite reservoir, M1 = ( k µ )1, md/cp M2 = mobility for Region 2 of composite reservoir, M 2 = ( k µ )2, md/cp p = pressure, psia pi = initial pressure, psi pwf = flowing bottomhole pressure, psi q = flow rate, STB/D ra = outer radius of inner zone annulus in radial-composite model, ft re = drainage radius, ft rw = wellbore radius, ft s = skin factor, dimensionless Sw = water saturation, fraction S = storativity ratio for composite reservoir, S = (φct )2 (φct )1, dimensionless S1 = storativity ratio for Region 1 of composite reservoir, S1 = (φct )1, psi–1 S2 = storativity ratio for Region 1 of composite reservoir, S2 = (φct )2, psi–1 t = time, hour tAD = dimensionless time based on drainage area w = channel width, ft Wi = volume of water injected, STB q = angle between two no-flow boundaries, deg m = viscosity, cp f = porosity, fraction Subscripts 1 = properties of inner zone in a radial-composite reservoir 2 = properties of outer zone in a radial-composite reservoir b = beginning of flow regime D = dimensionless quantity e = end of flow regime x,y = Cartesian coordinate directions

Chapter 8

Variable Flow Rate History “Nothing endures but change.” —Heraclitus, as quoted by Diogenes Laertius in Lives of the Philosophers “You could not step twice into the same rivers; for other waters are ever flowing on to you.” —Heraclitus, On the Universe 8.1 Introduction In Chapter 7 we discussed the drawdown pressure responses for several common reservoir models. In this chapter, we will look first at the effect of a change in flow rate on the subsequent buildup. We will then present and discuss a variety of methods for taking a variable flow rate history into account during pressure transient analysis. Section 8.2 reviews the classic methods for accounting for rate changes for buildup tests following a variable flow rate history. The section discusses the Miller-Dyes-Hutchinson (MDH) method, in which a buildup is treated like a drawdown, as well as superposition time functions and equivalent-time functions. Section 8.3 shows the effects and limitations of using simple methods for graphing the data for a buildup following constant-rate production, for an infinite-acting reservoir, a reservoir with a single no-flow boundary, an infinite channel reservoir, and a closed circular reservoir. Section 8.4 interprets the flow rate history for radial flow in terms of the pressure gradients created in the reservoir by the flow history. Section 8.5 shows the effect of a change in flow rate on the shape of a subsequent buildup test. The section begins by illustrating the effect of a single change in flow rate for a well in an infinite-acting reservoir. The differences between drawdown and buildup in a closed reservoir are discussed, showing that the pressure buildup (PBU) response for a closed reservoir looks more like the drawdown response for a well with a circular constant pressure boundary than it does like the drawdown pressure response for a closed reservoir. Rate history can be incorporated into the diagnostic plot to help identify portions of the pressure response that are likely to be affected by the rate history. The section continues with a discussion of the effect of the flow regime on the length of the flow period prior to shut-in for the buildup pressure response to follow the drawdown pressure response. Of the flow regimes considered (radial flow, linear flow, spherical flow, and bilinear flow), linear flow requires the longest flow period prior to shut-in (100× duration of the shut-in) for the buildup pressure response to match the drawdown response, while spherical flow requires the shortest flow period (4× duration of the shut-in). The section closes by recommending that analysis should include detailed rate history for a period before shut-in the least of the following times: (1) time to pseudosteady-state or steady-state flow; (2) 10× duration of the shut-in (for radial flow), and (3) the complete production history. Section 8.6 discusses the convolution equation and the assumptions under which it is valid. The processes of convolution and deconvolution are contrasted. The use of analysis of pressure-transient data by history matching using an analytical solution and the superposition equation are briefly discussed. Section 8.7 presents rate normalization as an approximate method for deconvolution for wells in which the rate is changing slowly and smoothly. The Winestock-Colpitts method for rate normalization of drawdown tests and the Gladfelter method for buildup tests are discussed. The section closes by discussing the application of rate normalization plus material balance time for analysis of production data for both oil and gas wells producing under depletion drive.

188 Applied Well Test Interpretation

Section 8.8 provides an introduction to the use of deconvolution. The primary applications of deconvolution are introduced. The conditions necessary for deconvolution to be valid are discussed. Suggestions for practical application of deconvolution are provided. Finally, history matching using a generic reservoir model is suggested as a approximate deconvolution method. After completing this chapter, you should be able to 1. Recognize and sketch the drawdown and buildup pressure responses for a well in a closed circular reservoir, for a well in a reservoir with a circular constant-pressure boundary, and for a well in a radial composite reservoir with a low mobility outer zone. 2. Draw a diagnostic plot incorporating the flow rate history. 3. Determine how much detailed rate history before shut-in should be used for analysis. 4. Explain the difference between convolution and deconvolution. 5. Give the conditions under which rate normalization is valid. 6. List the major applications of deconvolution. 7. Give the recommended guidelines for practical application of deconvolution. 8.2 Methods for Variable Rate/Variable Pressure Problems Through the years, many different approaches have been used to account for changing flow rates. Each of these approaches has its advantages and disadvantages. 1. Treat the buildup pressure change as if it were a drawdown test. This approach works only if the flow period before the buildup lasts long enough to have negligible effect on the buildup. For radial flow, a good rule of thumb is that the flow period preceding the buildup should be at least 10 times the duration of the buildup. This approach may be used for diagnostics or preliminary interpretation with straight-line analysis or manual type curve matching. However, if the rate history is available, this method should not be used for the final interpretation. 2. Use a superposition time function to make multirate pressure data during a specific flow regime fall on a single straight line. This approach assumes a single flow regime dominates the pressure response throughout the flow rate history. When applicable, superposition time function methods are particularly useful for straight-line methods but should not be used for a final interpretation. 3. Use an equivalent-time plotting function to make the pressure response during buildup look like the constant-rate drawdown pressure response. This approach is a special case of a superposition-time function and has the same limitations. When an equivalent-time function is applicable, it may be used to analyze buildups as if they were drawdowns. The equivalent-time functions, as with all superposition-time functions, should not be used for a final interpretation. 4. If the flow regime existing at the time of shut-in is known, and if there is enough pressure data during the flow period to identify the slope and intercept of the appropriate straight-line plot, desuperposition may be used (Slider 1971). In desuperposition, the drawdown pressure response is extrapolated forward in time to estimate the flowing pressure that would have occurred if the well had not been shut in, then subtracting the estimated flowing pressure from the measured buildup pressure to reconstruct the constant-rate pressure response. The reconstructed constant-rate pressure response may then be analyzed as a constantrate drawdown test. 5. Select an appropriate reservoir model; then use superposition or convolution to calculate the pressure response for the given rate history, and tune the parameters of the model to history match the observed pressure response. This approach assumes that an appropriate reservoir model is available. 6. Use rate normalization to account for a changing flow rate during a test period (Gladfelter et al. 1955; Winestock and Colpitts 1965). This approach requires that the rate be changing slowly and smoothly. It only accounts for changes in pressure resulting from simultaneous changes in flow rate during a single test period; it cannot account for the effects of changes in flow during before flow rate history. 7. Use deconvolution to solve for the constant-rate or unit-rate pressure response, given the flow rate history and the resulting variable rate pressure response. The deconvolved pressure response can then be analyzed as if it were a constant-rate drawdown test. Because of the inherent instability in deconvolution, this approach requires higher quality data and more attention to certain details than other methods require.

Variable Flow Rate History 189

In this section, we review several methods based treating a buildup as a drawdown, superposition time functions, and equivalent time. 8.2.1 MDH and Related Methods. The MDH method for semilog analysis of PBU data relies on the fact that the pressure response for a short buildup following a long, constant-rate flow period is relatively unaffected by any earlier changes in flow rate. Analysis of buildup pressure data by graphing the shut-in pressure change vs. the shut-in time, then type curve matching with drawdown type curves is essentially the same as the MDH method. In both situations, the rate history is essentially ignored, and the buildup pressure response is treated as a drawdown. Use of the MDH and related methods should be restricted to the following two situations. (1) The flow rate at the moment of shut-in is known, but no other rate history is available. In the worst case scenario, it may be necessary to calculate the flow rate at shut-in from the estimated wellbore-storage coefficient and the pressure response during the wellbore-storage-dominated period. (2) The flow rate before shut-in was constant for a period of time at least 10 times the duration of the buildup, and the purpose of the analysis is only to get preliminary results for permeability and skin factor. If linear flow or bilinear flow is present, the flow rate before buildup must be constant for a period 100 or 20 times the duration of the buildup, respectively. If detailed flow rate history is available, the MDH and similar methods should never be used for the final analysis. 8.2.2 Superposition Time Functions. The Horner method for semilog analysis of PBU data is an example of a superposition time function. Superposition time functions are derived for data in a specific flow regime, using superposition along with the functional form of the pressure dependence for the specific flow regime to develop a method for graphing variable flow rate pressure data that will give a straight line for data within the desired flow regime. In addition to radial flow, superposition time functions have also been developed for linear flow, bilinear flow, and spherical flow. Superposition time functions are derived for data in a specific flow regime. Superposition time functions should never be used for a final interpretation when other flow regimes occur during the test. Bourdet Superposition Time Function. One of the most popular superposition time functions in current use is the Bourdet superposition time function, which may be developed as follows. Assuming infinite-acting radial flow, we may write the pressure response for a variable flow rate history in terms of the logarithmic approximation to the Ei function as n

p ( t ) = pi − ∑

j =1

  k  162.6 ( q j − q j − 1 ) Bµ  log ( t − t j − 1 ) + log   − 3.23 + 0.869s . . . . . . . . . . . . . . . . . . . (8.1) 2 kh  φµct rw   

We can use Eq. 8.1 to calculate the bottomhole pressure at the end of flow period n–1 as n −1

p ( tn − 1 ) = pi − ∑

j =1

 k  162.6 ( q j − q j − 1 ) Bµ   log ( tn − 1 − t j − 1 ) + log   − 3.23 + 0.869s . . . . . . . . . . . . . . . (8.2) 2 kh    φµct rw 

Subtracting Eq. 8.1 from Eq. 8.2 and rearranging, we have p ( tn − 1 ) − p ( t ) qn − qn − 1

=

 q j − q j − 1     k   log ( t − t j − 1 ) + log   − 3.23 + 0.869s  2  φµct rw    n − qn − 1  

162.6 Bµ kh

∑  q

162.6 Bµ kh

∑  q

−

n

j =1

n −1

j =1

 qj − qj −1    n − qn − 1 

   k  log ( tn − 1 − t j − 1 ) + log   − 3.23 + 0.869s . . . . . . . . . . . (8.3) 2    φµct rw 

Simplifying, we have p ( tn − 1 ) − p ( t ) qn − qn − 1

=

162.6 Bµ kh

n −1  log ( t − tn − 1 ) + ∑  j =1

  qj − qj −1   t − t j −1   − 3.23 + 0.869s . . . . . . . . . . . (8.4)   log   q n − qn − 1   tn − 1 − t j − 1  

190 Applied Well Test Interpretation

In terms of natural logarithms, p ( tn − 1 ) − p ( t ) qn − qn − 1

=

70.6 Bµ kh

n −1  ln ( t − tn − 1 ) + ∑  j =1

  qj − qj −1   t − t j −1   − 7.43 + 2s . . . . . . . . . . . . . . . . . . (8.5)   ln    qn − qn − 1   tn − 1 − t j − 1 

Either Eq. 8.4 or 8.5 may be used to develop a superposition time function method. We may define a superposition time function based on Eq. 8.4 as n −1   t − t j −1  q − qj −1   log   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.6) X10 = log ( t − tn − 1 ) + ∑  j  tn − 1 − t j − 1  j = 1  qn − qn − 1 

Bourdet et al. (1989) defined their superposition time function based on Eq. 8.5: n −1  q − qj −1   t − t j −1   ln   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.7) X e = ln ( t − tn − 1 ) + ∑  j  tn − 1 − t j − 1  j = 1  qn − qn − 1 

As shown here, superposition time functions may be defined in terms of either common or natural logarithms. When using pressure-transient software, the technical reference manual should be consulted to determine how the superposition time function is defined for that particular software package. Bourdet Superposition Time Function Method—Procedure. The following procedure may be used to analyze multirate flow test data to obtain permeability and skin factor using the Bourdet superposition time function, Eq. 8.7. p ( tn − 1 ) − p ( t )

vs. the superposition time function Xe, Eq. 8.7, on a qn − qn − 1 Cartesian scale. Data from different flow periods may be drawn on a single graph. All data should fall on a single straight line if the assumptions underlying the method are satisfied. 2. Identify the data falling on the straight line, exhibiting infinite-acting radial flow. Data may deviate from the straight line at early times, representing wellbore storage. Deviation of the data from the straight line at late times suggests the presence of boundaries. The method will still give good estimates of permeability and skin factor if the early data affected by wellbore storage and the late data affected by boundaries are identified and ignored during the analysis. 3. Draw a straight line through the selected data. Find the slope m″ from the semilog straight line, and read the intercept b″ from the straight line (or its extrapolation) at Xe = 0. 4. Calculate the permeability from the slope m″ as 1. Graph the pressure-plotting function

k=

70.6 Bµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.8) m″h

5. Calculate the skin factor s from the slope m″ and the intercept b″:  b″  k s = 0.5  − ln  2 ″ m  φµ ct rw 

   + 7.43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.9)  

The Bourdet method allows data from both flow and shut-in periods to be included in the analysis. The method may be used even when boundaries are encountered during the test, provided the data influenced by boundaries are identified and ignored. Finally, for the case of constant wellbore storage and skin factor, wellbore-storage distorted data from different flow periods will all follow a single curve, facilitating identification of data distorted by wellbore storage. Although the Bourdet superposition time function was derived assuming infinite-acting radial flow, it is often used as a general-purpose plotting function for type curve and history matching. 8.2.3 Equivalent-Time Functions. An equivalent-time function is a specific type of superposition time function developed so that the superposition equation for the pressure change during a flow or buildup period following a variable rate flow history will have the same form as the equation for the pressure change during a constant-rate drawdown test, with the flow time in the constant-rate drawdown equation replaced by the equivalent-time function.

Variable Flow Rate History 191

Properties of Equivalent-Time Functions. Equivalent-time functions have been developed for radial flow, linear flow, bilinear flow, and spherical flow. The radial equivalent time for buildup following constant-rate production is defined as ∆te =

t p ∆t t p + ∆t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.10)

The linear equivalent time for buildup following constant-rate production is defined as 2

∆teL =  t p + ∆t − t p + ∆t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.11) The bilinear equivalent time is 4

∆teB =  4 t p + 4 ∆t − 4 t p + ∆t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.12) The spherical equivalent time is ∆teS =

1  1  1 1   + − ∆t t p + ∆t   t p

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.13)

All of the above definitions of equivalent time have the following properties. The equivalent time ∆teX is always less than the shut-in time ∆t: ∆teX < ∆t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.14) In other words, the pressure change at any given elapsed time during a buildup will always be smaller than the pressure change at the same elapsed time during a drawdown. At early times, the equivalent time ∆teX is approximately equal to the shut-in time ∆t: ∆teX ≅ ∆t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.15) At early times, the pressure change during a buildup is approximately equal to the pressure change during drawdown at the same elapsed test time. The equivalent time ∆teX is always less than the producing time tp: ∆teX < tp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.16) The pressure change during buildup, no matter how long the buildup lasts, is always smaller than the pressure change at the end of the drawdown period. At large shut-in times, the equivalent time ∆teX approaches the producing time tp: lim ∆teX = t p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.17)

∆t → ∞

During buildup in an infinite system, the pressure during buildup will approach the original pressure at infinite shut-in time. From Eq. 8.17, it is obvious that the equivalent-time function compresses the time scale, especially for buildups longer than the duration of the preceding flow period. This has caused significant confusion in calculating the radius of investigation. One common misconception is that the radius of investigation during buildup is limited to the radius of investigation achieved during the flow period preceding buildup, thus, the equivalent time should be used to calculate the radius of investigation. However, the pressure response from an impulse test, conducted with a very short flow period, shows that the pressure transient during shut-in moves the same distance into the reservoir, independent of the duration

192 Applied Well Test Interpretation

∫ (pi -pws)dt, ∆t (pi -pws), psi-hr

10000

1000

100

10 0.01

0.1

1

∆t, hr

10

100

1000

Fig. 8.1—Pressure response for an impulse test with a 15-minute flow period, for a well with a single no-flow boundary 500 ft away, showing that the pressure transient continues to move through the reservoir during shut in.

of the flow period. Fig. 8.1 shows the pressure response for an impulse test with a 15-minute flow period on a well near a single no-flow boundary. The “pressure change” plotting function in Fig 8.1 is actually the integral given by ∆t

f∆p =

∫ ( p − p (τ )) dτ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.18) i

ws

τ =0

while the “derivative” plotting function is the product of the shut-in time and the change in pressure, fderiv = ( pi − pws ( ∆t )) ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.19) As shown in Fig 8.1, the pressure transient during the buildup period continues to move through the reservoir, clearly demonstrating the effect of the no-flow boundary on the pressure response at shut-in times between 20 and 200 hours, times much longer than the 15-minute duration of the flow period prior to shut-in. Agarwal Multirate Equivalent-Time Method. Noting that n −1

10

X10

= exp ( X e ) = ( t − tn − 1 ) ∏ j =1

q −q



 t − t j − 1   q j − q j − 1    n n − 1  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.20)   tn − 1 − t j − 1 

has the dimensions of time, Agarwal (1980) defined the multirate equivalent time, Δte, as n −1

∆te = ( t − tn − 1 ) ∏ j =1

q −q



 t − t j − 1   q j − q j − 1     n n − 1  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.21) t t −  n −1 j −1 

allowing Eqs. 8.4 and 8.5 to be written as p ( tn − 1 ) − p ( t ) q n − qn − 1

=

  k  162.6 Bµ  log ( ∆te ) + log   − 3.23 + 0.869s , . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.22) 2 kh   φµct rw  

=

  k  70.6 Bµ  ln ( ∆te ) + ln   − 7.43 + 2s, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.23) 2 kh   φµct rw  

and p ( tn − 1 ) − p ( t ) qn − qn − 1

Variable Flow Rate History 193

respectively. Eq. 8.7 can be used to calculate the Agarwal multirate equivalent time from ∆te = exp ( X e ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.24) Computationally, Eqs. 8.7 and 8.24 are preferred to Eq. 8.21, because the exponents in Eq. 8.21 may easily exceed the range allowed by typical computer systems. Agarwal Multirate Equivalent-Time Method—Procedure. The following procedure may be used to analyze multirate flow test data to obtain permeability and skin factor using the Agarwal multirate equivalent-time method. p ( tn − 1 ) − p ( t )

, Eq. 8.22, vs. the time-plotting function Δte, defined qn − 1 − qn by Eq. 8.21, on a semilog scale. As with the Bourdet superposition-time function method, data from different flow periods may be drawn on the same graph. All data should fall on a single straight line if the assumptions underlying the method are satisfied. 2. Identify the data falling on the straight line, exhibiting infinite-acting radial flow. Data may deviate from the straight line at early times, representing wellbore storage. Deviation of the data from the straight line at late times suggests the presence of boundaries. The method will still give good estimates of permeability and skin factor if the early data affected by wellbore storage and the late data affected by boundaries are identified and ignored during the analysis. 3. Draw a straight line through the selected data. Find the slope m′ from the semilog straight line, and read the intercept b′ from the straight line (or its extrapolation) at an equivalent time of 1 hour. 4. Calculate the permeability from the slope m′ as

1. Graph the pressure-plotting function

k=

162.6 Bµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.25) m′h

5. Calculate the skin factor s from the slope m′ and the intercept b′:  b′  k s = 1.151  − log  2 m ′  φµ ct rw 

   + 3.23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.26)  

Mathematically, the Agarwal multirate equivalent-time method based on Eq. 8.21 is identical to the Bourdet superposition time function method. We prefer the equivalent-time method to the superposition time function method because the x-axis of the equivalent time graph is in hours. This allows the analyst to more easily correlate events observed on the semilog graph with events in real time and to calculate the corresponding radius of investigation. Example 8.1—Multirate Analysis. In this section, we will analyze a flow-after-flow test on an oil well using the Bourdet superposition time function method and the Agarwal multirate equivalent-time method. Problem. A multirate flow-after-flow test has been run on a new oil well. The rock and fluid property data are given in Table 8.1, and rate and pressure data are given in Table 8.2. The reservoir is believed to be large enough that boundaries will not be observed during the 24-hour test. Solution—Bourdet Superposition Time Function Method. We follow the procedure for analyzing multirate flow test data using the Bourdet superposition time function defined by Eq. 8.7. 1. Graph the pressure-plotting function, Eq. 8.22, vs. the time-plotting function Xe, defined by Eq. 8.7, on a Cartesian scale. For t = 0.25 hour in the first flow period (n = 1), the time-plotting function is n −1  q − qj −1   t − t j −1   ln   X e = ln ( t − tn − 1 ) + ∑  j  tn − 1 − t j − 1  j = 1  qn − qn − 1 

= ln ( t − t0 ) = ln ( 0.25 − 0 ) = − 1.3863.

194 Applied Well Test Interpretation TABLE 8.1—ROCK AND FLUID PROPERTY DATA FOR MULTIRATE ANALYSIS EXAMPLE Reservoir and Fluid Properties

f h rw

18% 43 ft 0.33 ft

B m ct

1.115 bbl/STB 5.47 cp 9.77 × 10–6 psi–1

TABLE 8.2—TEST DATA FOR MULTIRATE ANALYSIS EXAMPLE Field Data t hr

pwf psia

0

1927.03

Field Data Bourdet STF

Agarwal Δte hr

Field Data PPF psi/STB/D

t hr

Flow period 1–q = 10.7 STB/D 0.25 0.50 0.75 1 2 3 4 5 6

1908.65 1907.87 1907.50 1907.20 1906.59 1906.22 1905.97 1905.82 1905.62

–1.3863 –0.6931 –0.2877 0 0.6931 1.0986 1.3863 1.6094 1.7918

0.25 0.5 0.75 1 2 3 4 5 6

1888.85 1888.03 1887.65 1887.37 1886.70 1886.25 1885.93 1885.65 1885.43

–1.3417 –0.6058 –0.1591 0.1683 1.0072 1.5413 1.9440 2.2712 2.5486

0.2614 0.5457 0.8529 1.1833 2.7381 4.6707 6.9869 9.6914 12.7887

pwf psia

Bourdet STF

1.7178 1.7907 1.8252 1.8533 1.9103 1.9449 1.9682 1.9822 2.0009

12.25 12.50 12.75 13 14 15 16 17 18

1869.45 1868.70 1868.37 1868.07 1867.38 1866.85 1866.46 1866.22 1865.96

–1.3196 –0.5618 –0.0938 0.2545 1.1737 1.7826 2.2556 2.6489 2.9887

qn − qn − 1

=

1.7112 1.7949 1.8337 1.8622 1.9306 1.9765 2.0092 2.0378 2.0602

18.25 18.50 18.75 19 20 21 22 23 24

1850.66 1849.95 1849.56 1849.29 1848.52 1848.09 1847.7 1847.34 1847.07

–1.3044 –0.5316 –0.0488 0.3142 1.2902 1.9533 2.4781 2.9210 3.3084

p ( t0 ) − p ( t ) q1 − q0

1, 927.03 − 1, 908.65 10.7 − 0 = 1.7178 psi/STB/D.

=

For t = 9 hr in the second flow period (n = 2), the time-plotting function is n − 1 q − qj −1   t − t j −1   ln   X e = ln ( t − tn − 1 ) + ∑  j  tn − 1 − t j − 1  j = 1  qn − qn − 1 

 q − q0   t − t0  = ln ( t − t1 ) +  1   ln   q2 − q1   t1 − t0   10.7 − 0   9 − 0  = ln ( 9 − 6) +    ln   20.5 − 10.7   6 − 0  = 1.5413.

PPF psi/STB/D

0.2673 0.5702 0.9105 1.2899 3.2338 5.9454 9.5408 14.1385 19.8594

1.7183 1.7989 1.8344 1.8667 1.9409 1.9978 2.0398 2.0656 2.0935

Flow period 4–38.7 STB/D

and the pressure-plotting function is p ( tn − 1 ) − p ( t )

Agarwal Δte hr

Flow period 3–29.8 STB/D

Flow period 2–20.5 STB/D 6.25 6.50 6.75 7 8 9 10 11 12

Plotting Functions

0.2713 0.5877 0.9524 1.3692 3.6334 7.0522 11.9187 18.5606 27.3411

1.7191 1.7989 1.8427 1.8730 1.9596 2.0079 2.0517 2.0921 2.1225

Variable Flow Rate History 195

and the pressure-plotting function is p ( tn − 1 ) − p ( t ) qn − qn − 1

=

p ( t1 ) − p ( t ) q2 − q1

=

1, 905.62 − 1, 886.25 20.5 − 10.7

= 1.9765 psi/STB/D.

Values of the time and pressure-plotting functions are given in Table 8.2. Fig. 8.2 shows the superposition time function graph. 2. Identify the data falling on the straight line, exhibiting infinite-acting radial flow. From Fig. 8.2, data for having Xe greater than 0 appear to fall on a single straight line. Earlier data deviate from the straight line, probably because of wellbore storage. 3. Draw a straight line through the selected data. Find the slope m″ from the semilog straight line, and read the intercept b″ from the straight line (or its extrapolation) at Xe = 0. The slope is obtained from 2.180 − 1.687 = 0.08217 psi/STB the two points where the straight line intersects the left and right sides of the graph m″ = 4 − (−2) 2.180 − 1.687 m″ = = 0.08217 psi/STB/D, and the intercept b″ is read from the intersection of the straight line and the y-axis 4 − (−2) as b″ = 1.850 psi/STB/D. 4. Calculate the permeability from the slope m″ as k=

70.6 Bµ ( 70.6) (1.115) ( 5.47) = 121.9 md. = m″h ( 0.08217) ( 43)

5. Calculate the skin factor s from the slope m″ and the intercept b″:  k   b″  s = 0.5  − ln   + 7.43 2  φµct rw   m″      1.850 121.9  + − ln  = 0.5  7 43 .  2 −6  ( 0.18) ( 5.47) ( 9.77 × 10 ) ( 0.33)   0.08217 

Rate-Normalized Pressure Change, psi/STB/D

= 5.69.

2.2 2.180 2.1 2.0 1.9

1.850

1.8 1.7 1.6

Flow 1 Flow 2 Flow 3 Flow 4

1.687 -2

-1

0

1

2

3

4

Superposition Time Function Fig. 8.2—Multirate example, Bourdet superposition-time function graph.

196 Applied Well Test Interpretation

Solution—Agarwal Multirate Equivalent Time Method. We follow the recommended procedure for analyzing multirate flow test data using the Agarwal multirate equivalent time method. p ( tn − 1 ) − p ( t )

vs. the time-plotting function Δte, defined by Eq. 8.21, qn − qn − 1 on a semilog scale. The pressure-plotting function for this method is the same as that for the superposition time function method just presented. The time-plotting function is given by ∆te = exp ( X e ). For t = 0.25 hours in the first flow period,

1. Graph the pressure-plotting function

∆te = exp ( X e ) = exp (− 1.3863) = 0.25 hr For t = 9 hours in the second flow period, ∆te = exp ( X e ) = exp (1.5413) = 4.671 hr . Values of the time and pressure-plotting functions are given in Table 8.2. Fig. 8.3 shows the Agarwal multirate equivalent time graph. 2. Identify the data falling on the straight line, exhibiting infinite-acting radial flow. From Fig. 8.3, data for having Δte greater than 1 hour appear to fall on a single straight line. Earlier data deviate from the straight line, probably because of wellbore storage. 3. Draw a straight line through the selected data. Find the slope m′ from the semilog straight line, and read the intercept b′ from the straight line (or its extrapolation) at an equivalent time of 1 hour. The slope is obtained from the two points where the straight line intersects the left and right sides of the graph m′ =

2.232 − 1.659 = 0.1910 psi/ST TB/D/cycle, log (100 ) − log ( 0.1)

and the intercept b′ is read from the intersection of the straight line and the y-axis as b′ = 1.850 psi/STB/D. 4. Calculate the permeability from the slope m′ as 162.6 Bµ (162.6) (1.115) ( 5.47) = 120.7 md. = m′h ( 0.1910 ) ( 43) Rate-Normalized Pressure Change, psi/STB/D

k=

2.3 2.2

2.232

2.1 2.0 1.9

1.850

1.8

Flow 1 Flow 2 Flow 3 Flow 4

1.7 1.659 1.6 0.1

1

10

100

Multirate Equivalent Time, hr Fig. 8.3—Multirate example, Agarwal multirate equivalent-time graph.

Variable Flow Rate History 197

5. Calculate the skin factor s from the slope m′ and the intercept b′:  b′  k s = 1.151  − log  2  φµ ct rw  m′

   + 3.23  

  120.7  1.850 = 1.151  − log   ( 0.18 ) ( 5.47 ) ( 9.77 × 10−6 ) ( 0.33)2 0.1910  

   + 3.23   

= 5.59 8.2.4 Field Example 1. This example was presented in Bourdet et al. (1983) and in Bourdet et al. (1989). The test comprised a 15.33 hour flow period followed by a 30 hour buildup. Table 8.3 gives the rock and fluid property data for this data set; Table 8.4 shows the results for the example. In this field example, the shut-in time is almost twice the duration of the flow period, so the rate history must be taken into account. The data set was analyzed using radial equivalent time and superposition. Fig. 8.4 shows a log-log plot of pressure change and pressure derivative for an interpretation using nonlinear regression to match the buildup pressure response as a drawdown using the radial equivalent time. The match is not satisfactory, as shown by the poor agreement between the field data logarithmic derivative and the simulated derivative at the end of the test. Fig. 8.5 shows a log-log plot of pressure change and pressure derivative for an interpretation using nonlinear regression to match the buildup pressure response with full superposition. The match is much better than that from the analysis using radial equivalent time. 8.3 Effect of Boundaries on a Subsequent Buildup In this section, we look at the effect of reservoir boundaries on the succeeding buildup test. We consider four different reservoir models: (1) infinite-acting reservoir, (2) reservoir with a single no-flow boundary, (3) infinite channel reservoir, and (4) closed circular reservoir. For each reservoir model, we will discuss the effect of three time-plotting functions: (1) shut-in time, with derivative calculated with respect to shut-in time; (2) equivalent time, with derivative calculated with respect to equivalent time; and (3) shut-in time, with the derivative calculated with respect to equivalent time. 8.3.1 Infinite-Acting Reservoir. Fig. 8.6 shows the pressure response for both drawdown and buildup for a well with no wellbore storage, in an infinite-acting reservoir. The drawdown derivative is a horizontal line, indicating infinite-acting radial flow. The buildup derivative, calculated with respect to the shut-in time, is horizontal at shut-in times much smaller than the producing time (in this case, tpD = 106). For shut-in times much larger than the producing time, the buildup pressure change approaches a constant, while the derivative approaches a negative

TABLE 8.3—ROCK AND FLUID PROPERTY DATA FOR FIELD EXAMPLE 1 Reservoir and Fluid Properties

f

25%

B

h

107 ft

m

2.5 cp

rw

0.29 ft

ct

4.2 × 10–6 psi–1

tp

15.33 hour

q

174.0 STB/D

1.06 bbl/STB

TABLE 8.4—RESULTS FOR FIELD EXAMPLE 1

Method

pi psia

k md

s

C bbl/psi

Equivalent time

3,871.4

18.4

17.2

0.0089

Superposition

3,877.1

11.7

8.7

0.0091

198 Applied Well Test Interpretation 1000 Obs Fit ∆p, t(dp/dt), psi

1000

100

10

1 0.001

0.01

0.1

1

10

100

1000

∆teq, hr Fig. 8.4—Field Example 1, equivalent time analysis.

1000 Obs Fit ∆p, t(dp/dt), psi

1000

100

10

1 0.001

0.01

0.1

1

10

100

1000

∆t, hr Fig. 8.5—Field Example 1, superposition analysis.

1E+02

pD

1E+01

1E+00

Drawdown

1E-01

1E-02 1E+03

Buildup (∆t)

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.6—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup graphed vs. shut-in time.

Variable Flow Rate History 199 1E+02

pD

1E+01

1E+00

Drawdown Buildup (∆te)

1E-01

1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.7—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup graphed vs. equivalent time.

unit-slope line. The buildup pressure response is very similar to, but not identical, to the drawdown pressure response for a well near a single constant pressure boundary (Fig. 7.8). The radial equivalent time was developed assuming infinite-acting radial flow. Thus, it should be no surprise that the radial equivalent time does indeed make a buildup for a well in an infinite-acting reservoir look exactly like the drawdown pressure response, as shown in Fig. 8.7. Note, however, that the equivalent time cannot be larger than the producing time, regardless of the length of the preceding flow period. This causes the time scale to be compressed; in this case, the pressure response was calculated for a producing time tpD = 106 with a maximum shut-in time of 109. To avoid the time scale compression, Bourdet recommends calculating the derivative with respect to the equivalent time, but graphing the derivative vs. the shut-in time (Bourdet et al. 1983; Bourdet et al. 1989). Fig. 8.8 shows the same data, redrawn per Bourdet’s recommendation. As in Fig. 8.7, the derivative is horizontal and overlays the drawdown pressure response. Although Bourdet graphs the derivative vs. the shut-in time, his approach is inconsistent in that he graphs the pressure change curve vs. the equivalent time. 8.3.2 Reservoir With a Single No-Flow Boundary. Figs. 8.9, 8.10, and 8.11 show drawdown and buildup pressure responses for a well near a single no-flow boundary. The buildup response is shown for producing times tpD of 104, 106, and 108. For the tpD = 104 case, the flow period ends long before the radius of investigation reaches the boundary; for the tpD = 106 case, the flow period ends about the same time the radius of investigation reaches the boundary; for the tpD = 108 case, the flow period ends long after the radius of investigation reaches the 1E+02

pD

1E+01

1E+00

Drawdown Buildup (∆te×dp/d∆te plotted vs. ∆t)

1E-01

1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.8—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup pressure change graphed vs. equivalent time, buildup derivative calculated with respect to equivalent time, graphed vs. shut-in time.

200 Applied Well Test Interpretation 1E+02

pD

1E+01 Drawdown

1E+00

1E-01

tpD = 1×108

Buildup (∆t) 1E-02 1E+03

tpD = 1×104 1E+04

1E+05

tpD = 1×106 1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.9—Drawdown and buildup pressure responses for a well in a reservoir with a single no-flow boundary, drawdown graphed vs. elapsed test time, buildup graphed vs. shut-in time.

1E+02

pD

1E+01 tpD = 1×104

1E+00

tpD = 1×108

tpD = 1×106

Drawdown Buildup (∆te) 1E-01

1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.10—Drawdown and buildup pressure responses for a well in a reservoir with a single no-flow boundary, drawdown graphed vs. elapsed test time, buildup graphed vs. equivalent time.

1E+02

pD

1E+01

Drawdown

1E+00

tpD = 1×108 1E-01

Buildup (∆te×dp/d∆te plotted vs. ∆t)

1E-02 1E+03

1E+04

1E+05

1E+06 tD

tpD = 1×106 tpD = 1×104

1E+07

1E+08

1E+09

Fig. 8.11—Drawdown and buildup pressure responses for a well in a reservoir with a single no-flow boundary, drawdown graphed vs. elapsed test time, buildup pressure change graphed vs. equivalent time, buildup derivative calculated with respect to equivalent time, graphed vs. shut-in time.

Variable Flow Rate History 201

boundary. In Fig. 8.9, The buildup pressure response is graphed vs. the shut-in time. As with the infinite-reservoir case, the derivative approaches a negative unit-slope line at late times. In Fig. 8.10, the drawdown pressure response is graphed vs. the elapsed test time, while the buildup pressure responses are graphed vs. the radial equivalent time. For the tpD = 108 case, the buildup derivative follows the drawdown derivative almost perfectly. In this case, both drawdown and buildup transients are in hemiradial flow by the time the shut-in time approaches the producing time. Derivation of a “hemiradial equivalent time” would result in the same equation as for the radial equivalent time. For the tpD = 104 case, time scale compression causes the derivative to curve abruptly upward as the equivalent time approaches the producing time. A similar phenomenon occurs for the tpD = 106 case, although the derivative does not increase as sharply as in the tpD = 104 case. The buildup pressure curves agree very well with the drawdown curve for all three cases. Fig. 8.11 shows the same datasets graphed per Bourdet’s recommendation. Although the buildup derivative curves do not overlay the drawdown curve exactly, the shapes are very similar, and could be used to guide model identification. 8.3.3 Infinite Channel Reservoir. Figs. 8.12 through 8.15 show drawdown and buildup pressure responses for a well in an infinite channel reservoir; buildup curves are shown for tpD = 104, 106, and 108. At late times, the buildup curves in Fig. 8.12 all approach negative half-slope lines. The buildup derivative for the tpD = 108 case is similar to the drawdown derivative for a well in a channel with a constant pressure boundary at one end, Fig. 7.27. Graphing the derivative vs. the equivalent time for a well in an infinite channel, as shown in Fig. 8.13, exhibits even more extreme behavior because of time compression than for the single no-flow boundary 1E+02

1E+01

pD

Drawdown 1E+00

tpD = 1×108

1E-01 Buildup (∆t) tpD = 1×106

tpD = 1×104 1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.12—Drawdown and buildup pressure responses for a well in an infinite channel reservoir, drawdown graphed vs. elapsed test time, buildup graphed vs. shut-in time.

1E+02 tpD = 1×106

tpD = 1×104 1E+01

pD

Drawdown tpD = 1×108

1E+00 Buildup (∆te) 1E-01

1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.13—Drawdown and buildup pressure responses for a well in an infinite channel reservoir, drawdown graphed vs. elapsed test time, buildup graphed vs. equivalent time.

202 Applied Well Test Interpretation 1E+02

1E+01

pD

Drawdown tpD = 1×108

1E+00

tpD = 1×106 1E-01

1E-02 1E+03

Buildup (∆te×dp/d∆te plotted vs. ∆t)

1E+04

1E+05

1E+06 tD

1E+07

tpD = 1×104

1E+08

1E+09

Fig. 8.14—Drawdown and buildup pressure responses for a well in an infinite channel reservoir, drawdown graphed vs. elapsed test time, buildup pressure change graphed vs. equivalent time, buildup derivative calculated with respect to equivalent time, graphed vs. shut-in time. 1E+02

1E+01 tpD = 1×108

pD

Drawdown 1E+00 tpD = 1×106 1E-01

Buildup (∆te)

tpD = 1×104 1E-02 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.15—Drawdown and buildup pressure responses for a well in an infinite channel reservoir, drawdown graphed vs. elapsed test time, buildup pressure change graphed vs. linear equivalent time, buildup derivative calculated with respect to linear equivalent time, graphed vs. shut-in time.

case in Fig. 8.10. Furthermore, the buildup derivative does not follow the drawdown derivative, even for the tpD = 108 case. Using Bourdet’s recommendation, Fig. 8.14, results in the correct half-slope for the buildup derivative, allowing use for model identification, but the derivative is distorted and shifted to the right relative to the drawdown curve. 2 An equivalent time for linear flow may be defined as ∆teL = t p + ∆t − t p + ∆t . Using the linear equivalent time, as shown in Fig. 8.15, makes the buildup derivative follow the drawdown derivative for the tpD = 108 case but fails spectacularly for the tpD = 104 case.

(

)

8.3.4 Closed Circular Reservoir. The final reservoir model discussed in this series is the well in the center of a closed circular reservoir (Fig. 8.16). Buildup curves are shown for tpD = 104, 106, and 108. For tpD = 104, the reservoir is still infinite-acting at the end of the production period. For the second case, tpD = 106, the production period ends just after pseudosteady state begins; for the final case, tpD = 108, the flow period lasts 100 times longer than the time to reach pseudosteady-state flow. Fig. 8.16 shows the drawdown and buildup pressure responses for a well in the center of a closed circular reservoir, with the buildup pressure response graphed vs. shut-in time. The pressure change and derivative curves for the tpD = 106 and 108 cases are virtually identical and appear as a single set of curves. Once boundary effects become apparent, the buildup curves look nothing like the drawdown curves. Indeed, the buildup curves for the tpD = 106 and 108 cases looks more like the drawdown pressure response for a well in the center of a reservoir with a circular constant pressure boundary, Fig. 7.12.

Variable Flow Rate History 203 1E+02 Drawdown

pD

1E+01

1E+00

Buildup (∆t)

1E-01 tpD = 104 1E-02 1E+03

1E+04

1E+05

tpD =106, 108 1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.16—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup graphed vs. shut-in time.

Neither use of equivalent time nor the Bourdet method, Figs. 8.17 and 8.18, respectively, succeed in making the buildup pressure response reproduce the drawdown pressure response. If anything, if the reservoir is still infiniteacting at the end of the production period, as in the tpD = 104 case, the Bourdet method makes the buildup pressure response resemble the constant pressure boundary case even more closely (cf. Fig. 7.12). 8.3.5 Comments. No simple graphing method, such as the shut-in time method, the equivalent time method, or Bourdet’s method discussed in this section, is able to make a buildup pressure response for an arbitrary reservoir model look like the corresponding drawdown pressure response. It should be noted that the buildup cases discussed in this section are extreme. Normally, these methods would be used only when the duration of the shut-in is no more than two or three times the duration of the flow period. Slug tests and impulse tests both have a flow period that is extremely short compared to the duration of the shut-in period, but their interpretation requires specialized methods not discussed in this volume. Graphing the buildup pressure response vs. the shut-in time (MDH method) works very well for a well exhibiting infinite-acting radial flow provided the shut-in time is less than 10% of the producing time. However, a system with linear flow may require significantly longer producing time before shut-in. Fig. 8.12 shows the buildup derivative diverging from the drawdown derivative as early as 1% of the producing time for the tpD = 108 case. Use of equivalent time both for calculating the derivative and graphing the pressure change and derivative works almost perfectly for cases in which some form of radial flow is present during both flow and shut-in periods. Its biggest disadvantage in these cases is the extreme time scale compression that occurs when the shut-in time is much larger than the producing time. One situation that is quite common occurs when only the rate history for the time the pressure gauges are in the well; previous production history is ignored. In this situation, the shape of 1E+02 Drawdown

pD

1E+01

1E+00 Buildup (∆te)

1E-01

tpD = 104 1E-02 1E+03

1E+04

tpD = 106 1E+05

tpD = 108 1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.17—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup graphed vs. equivalent time.

204 Applied Well Test Interpretation 1E+02 Drawdown 1E+01

tpD = 106

tpD = 104

pD

tpD = 108 1E+00 tpD = 104 1E-01

tpD = 106 tpD = 108

1E-02 1E+03

1E+04

1E+05

Buildup (∆te×dp/d∆te plotted vs. ∆t) 1E+06 tD

1E+07

1E+08

1E+09

Fig. 8.18—Drawdown and buildup pressure responses for infinite-acting radial flow, drawdown graphed vs. elapsed test time, buildup pressure change graphed vs. equivalent time, buildup derivative graphed vs. shut-in time.

derivative curve calculated using an equivalent time based on the incorrect producing time will be similar to that shown in Figs. 8.10 and 8.13 for the tpD = 104 cases. This phenomenon will be illustrated in “Field Example 2” in Section 8.5.6. The Bourdet method is widely used. It is exact for infinite-acting radial flow and avoids the time compression effect of the use of equivalent time alone. For the single no-flow boundary and infinite channel cases considered here, it makes the buildup pressure response resemble the drawdown response, at least qualitatively, allowing it to be used in model identification. However, it is not suitable for parameter estimation. Perhaps the most serious drawback to the Bourdet method is that it works well, except when it does not, as in the closed circular reservoir case. This may lull the interpreter into a false sense of security, leading to gross interpretation errors when the Bourdet method is not applicable. In general, these methods are useful for diagnostic and model identification, as well as preliminary estimates of reservoir properties, but should never be used for the final interpretation. We strongly recommend that the final parameter estimates be obtained or confirmed through simulating the complete rate history and comparing the simulated pressure response with the original field data. No simple graphing procedure will make the buildup pressure response in an arbitrary reservoir model look like the corresponding drawdown pressure response. 8.4 Spatial Interpretation of Flow Rate History It may be helpful to understand the effects of rate history on a subsequent buildup by visualizing the rate history as a pressure distribution in the reservoir. In essence, the rate history is “stored” in the pressure profile in the reservoir at the moment of shut-in. 8.4.1 Constant Rate Flow/Buildup Sequence. Fig. 8.19 shows the reservoir pressure profile for a well produced at constant rate of 100 STB/D for 10,000 hour, from an infinite-acting reservoir, calculated using the Ei-function solution. Near the sandface, the pressure profile is linear function of the logarithm of the distance from the axis of the wellbore. This is the same behavior as steady-state radial flow noted in Chapter 2. Consider a cylindrical annulus of inner radius r, and outer radius r + dr. If the flow rate q (r ) across the annulus is known, the pressure drop across the annulus may be calculated by modifying Eq. 2.12 as p (r + dr ) − p (r ) =

141.2qBµ  r + dr  ln   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.27)  r  kh

Rearranging, and taking the limit as dr → 0, we find dp dp 141.2qBµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.28) =r = d ln (r ) dr kh

Variable Flow Rate History 205 5000

p, psia

4800

4600

4400

4200

1

10

100

1000

10000

100000

r/rw Fig. 8.19—Reservoir pressure profile for a vertical well produced at constant rate from an infinite-acting reservoir.

Solving for flow rate, we have q (r ) =

kh dp r , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.29) 141.2 Bµ dr

Eq. 8.29 gives the flow rate q (r ) at distance r from the wellbore in terms of the pressure gradient dp dr . Fig. 8.20 shows the flow rate in the reservoir, as a function of distance from the axis of the wellbore, calculated from Eq. 8.29 for the pressure distribution in Fig. 8.19. Fig. 8.20 also shows the rate history for the well, with the beginning of production coinciding with the radius of investigation since the beginning of production. Fig. 8.21 shows the pressure profile at time ∆t for a buildup following a constant-rate flow period of duration tp; Fig. 8.22 shows the corresponding reservoir flow rate along with the rate history. The flow profile in the reservoir is a smoothed version of the flow rate history. 8.4.2 Two-Rate Flow Period Followed by Buildup. Fig. 8.23 shows the pressure profile at the end of a two-rate production history, comprising an initial 10,000 hour flow period at 100 STB/D followed by a final 3.16 hour flow period at 50 STB/D. Fig. 8.24 shows the corresponding reservoir flow rate along with the rate history. The flow profile in the reservoir is a smoothed version of the flow rate history. Fig. 8.25 shows the pressure profiles at two different times, ∆t1 = 0.00316 hour, and ∆t2 = 31.6 hour, for a buildup following the two-rate flow rate history in Figs. 8.23 and 8.24. The elapsed time ∆t1 is much shorter than the duration of the final 50 STB/D flow period, while ∆t2 is much longer than the duration of the final flow period. Fig. 8.26 shows the corresponding reservoir flow rate along with the rate history. As can be seen from Fig. 8.26, 120 100

q, STB/D

80 60 40 20 0

1

10

100

1000 r/rw

10000 r 100000 iD

Fig. 8.20—Reservoir flow rate profile for a vertical well produced at constant rate from an infinite-acting reservoir.

206 Applied Well Test Interpretation 5000

p, psia

4800

4600

4400

4200

1

10

100

1000

10000

100000

r /rw Fig. 8.21—Reservoir pressure profile for a buildup test on a vertical well produced at constant rate from an infiniteacting reservoir.

120 100

q, STB/D

80 60 40 20 riD (∆t) 0

1

10

100

1000

riD (tp+∆t) 10000

100000

r/rw Fig. 8.22—Reservoir flow rate profile for a buildup test on vertical well produced at constant rate from an infinite-acting reservoir.

5000

p, psia

4800

4600

4400

4200

1

10

100

1000

10000

100000

r/rw Fig. 8.23—Reservoir pressure profile for a vertical well produced at two different rates from an infinite-acting reservoir.

Variable Flow Rate History 207 120 100

q, STB/D

80 60 riD (tp2)

40 20

riD (tp1+tp2) 0

1

10

100

1000

10000

100000

r/rw Fig. 8.24—Reservoir flow rate profile for a vertical well produced at two different rates from an infinite-acting reservoir.

5000

p, psia

4800

∆t2

4600

∆t1

4400

4200

1

10

100

1000

10000

100000

r/rw Fig. 8.25—Reservoir pressure profile for a buildup test on a vertical well produced at two different rates from an infiniteacting reservoir.

120 100

q, STB/D

80 60 40 20 ∆t1 0

1

10

∆t2 100

1000

10000

100000

r/rw Fig. 8.26—Reservoir flow rate profile for a buildup test on a vertical well produced at two different rates from an infiniteacting reservoir.

208 Applied Well Test Interpretation

the buildup “sees” the 50 STB/D flow rate for the final flow period at time ∆t1, and the 100 STB/D flow rate for the initial flow period at time ∆t2. 8.5 Effect of a Change in Flow Rate on a Subsequent Buildup Test As shown in Figs. 8.23 and 8.24, a decrease in flow rate shortly before a buildup distorts the shape of the pressure profile in the reservoir at the moment of shut-in. Near the wellbore, the pressure gradient is small, reflecting the lower flow rate immediately before the shut-in. Further away from the wellbore, the pressure gradient is large, reflecting the earlier, higher flow rate present during the long initial flow period. Figs. 8.25 and 8.26 show the pressure and rate profiles at two different times during the buildup. At time ∆t1, early in the shut-in period, the buildup pressure “sees” only the near-wellbore pressure gradient caused by the final flow rate. Later, after the shut-in pressure transient has moved deeper into the reservoir at time ∆t2, the pressure response “sees” the pressure gradient caused by the flow rate during the long initial flow period. In general, each time the flow rate at the wellbore changes, a new pressure transient begins moving through the reservoir, causing the reservoir pressure profile to vary radially away from the wellbore. The pressure profile at the moment of shut-in contains all the information that the reservoir “knows” about the rate history of the well before the buildup. As the radius of investigation moves through the reservoir, the wellbore pressure response sees the pressure profile established by successively earlier flow periods, effectively “reading” the flow rate history in reverse. As may be seen in Fig. 8.24, the flow rate profile in the reservoir is not an exact replica of the flow rate history. Rather, it is has been smoothed rather drastically. (The first flow period in Fig. 8.24 is more than 3000 times longer than the second flow period. Even so, the reservoir flow rate profile does not show a plateau at 100 STB/D.) All information about the past rate history of a well is “stored” in the pressure profile in the reservoir at the moment of shut-in. During a buildup test, as the radius of investigation moves away from the wellbore, the well “reads” a smoothed version of the flow rate history in reverse. This phenomenon has several major implications for our understanding of buildup pressure behavior. First, changes in the pressure gradient caused by flow rate changes preceding the buildup will affect the resulting buildup pressure response. If the flow rate history is ignored during buildup analysis, radial variation in the reservoir pressure gradient will appear to be caused by radial variation in reservoir properties. Depending on the specific rate history, the pressure response may appear to be that caused by a no-flow or constant pressure boundary, or simply a variation in reservoir quality. Second, pressure gradients caused by the flow rates much earlier than the duration of the buildup are too far away from the wellbore to affect the buildup pressure response. Thus, only recent flow rates have to be taken into account while analyzing a PBU test. The most recent flow rates have the most effect on the buildup response, and the earliest flow rates the least effect. Third, the drastic smoothing that occurs means that rate changes of short duration relative to the shut-in time cannot be seen as individual events. Finally, whenever the pressure gradient caused by a particular flow rate period disappears from the reservoir, the information about that flow rate period is effectively erased from the reservoir’s memory and cannot have any effect on the PBU response. For a closed reservoir, once a particular flow period has lasted long enough for pseudosteady-state flow to be established at that flow rate, all previous flow rate periods are erased and have no effect on the subsequent buildup pressure response. Similarly, for a well near a constant pressure boundary, after steadystate flow is established, the previous flow rate history has no effect on the pressure response of a buildup test. 8.5.1 Effect of a Single Rate Change. Fig. 8.27 shows the pressure response for a 120 hour buildup following a 1024 hour, constant-rate flow period. The pressure derivative is calculated with respect to the shut-in time, and the pressure change and pressure derivative are graphed as functions of the shut-in time. Because of the long flow period before buildup, the buildup pressure response is almost identical to the drawdown pressure response. Fig. 8.28 shows the pressure response for a 120 hour buildup following a 1024 hour flow period with a decrease in flow rate. The well produced 200 STB/D for the first 1016 hours, then 100 STB/D for the last 8 hours. If the flow rate history is not taken into account during the analysis, the pressure response could easily be mistaken for that of a well near a single linear no-flow boundary. Fig. 8.29 shows the pressure response for a 120 hour buildup following an 8 hour flow period at a constant rate of 100 STB/D. If the short duration of the flow period is not taken into account during the analysis, the pressure response could easily be mistaken for that of a well near a single linear constant pressure boundary.

Variable Flow Rate History 209 10000

∆p, tdp/dt, psi

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.27—Buildup pressure response for a well in an infinite reservoir, 1024 hour, constant rate flow period before shut-in.

10000

∆p, tdp/dt, psi

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.28—Buildup pressure response for a well in an infinite reservoir with a decrease in flow rate prior to shut-in may look like that for a well near a no-flow boundary if the rate history is ignored.

10000

∆p, tdp/dt, psi

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.29—Buildup pressure response for a well in an infinite reservoir with a short flow period prior to shut-in may look like that for a well near a constant pressure boundary if the rate history is ignored.

210 Applied Well Test Interpretation 10000

∆p, tdp/dt, psi

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.30—A well in an infinite reservoir with a short period of reduced rate flow prior to shut-in may appear to have a larger wellbore-storage coefficient and lower skin factor if the reduced rate period is ignored.

Rate changes before buildup may look like reservoir boundaries if not taken into account during analysis. The buildup pressure response in Fig. 8.30 appears to be undistorted. However, the flow rate was reduced from 200 STB/D to 100 STB for a very brief period (0.012 hours) before shut-in. Ignoring the final flow period, analyzing the test assuming a constant rate of 200 STB/D gives the correct permeability. However, the skin factor is too low, and the wellbore storage coefficient is too high. 8.5.2 Drawdown and Buildup in Bounded Reservoirs. Fig. 8.31 compares the pressure responses for constant-rate drawdown and for buildup following constant-rate production for three different reservoir models. Figs. 8.31a and 8.31b show the drawdown and buildup pressure responses for a well in a closed circular reservoir. As discussed in Chapter 7, the drawdown pressure response in Fig. 8.31a exhibits a unit-slope line after the beginning of pseudosteady-state flow. The buildup pressure response in the same reservoir, Fig. 8.31b, looks nothing like the drawdown pressure response. During buildup, after the radius of investigation has reached all the reservoir boundaries, the pressure begins to stabilize. As the pressure stabilizes, the pressure change approaches a constant value ∆p = p − pwf , and the pressure derivative decreases exponentially. The drawdown pressure response for a well in a reservoir with a circular constant pressure boundary is shown in Fig. 8.31c. If a reservoir model for analysis of the buildup in Fig. 8.31b is chosen based only on its similarity to the drawdown pressure response, the circular constant pressure boundary model would be an obvious choice. An excellent fit of the data would be likely, even if the rate history is taken into account during the analysis, because the buildup pressure response for the circular constant pressure boundary, Fig. 8.31d, is identical to the drawdown response, Fig. 8.31c. The buildup pressure response for a closed circular reservoir looks almost identical to the drawdown pressure response for a reservoir with a circular constant pressure boundary. On the other hand, if a buildup pressure response shows a unit-slope line at the end of the test, the buildup response cannot be caused by a closed reservoir. Instead, there must be some source of energy supplying fluid to repressurize that portion of the reservoir in which the well is located. This situation can occur, for example, for a well in a radial composite reservoir with a much lower mobility in the outer region, as shown in Figs. 8.31e and 8.31f. A unit-slope derivative at late times in a drawdown test suggests a closed reservoir model. A unit-slope derivative at late times in a buildup test guarantees that the reservoir is not closed, since there must be some source of energy that is repressurizing or recharging the reservoir.

Variable Flow Rate History 211 (d) 1000

∆p, tdp/dt, psi

∆p, tdp/dt, psi

(a) 1000

100

10

1 0.01

0.1

1 ∆t, hr

10

∆p, tdp/dt, psi

∆p, tdp/dt, psi

10

0.1

1 ∆t, hr

10

0.1

1 ∆t, hr

10

1000

10

10

1000

(f) 1000

∆p, tdp/dt, psi

∆p, tdp/dt, psi

1 ∆t, hr

100

1 0.01

1000

(c) 1000

100

10

1 0.01

0.1

(e) 1000

100

1 0.01

10

1 0.01

1000

(b) 1000

100

0.1

1 ∆t, hr

10

1000

100

10

1 0.01

0.1

1 ∆t, hr

10

1000

Fig. 8.31—Buildup pressure response for a well in a closed reservoir looks like the drawdown pressure response for a well in a reservoir with a circular constant pressure boundary. (a) Drawdown test in reservoir with closed circular boundary; (b) buildup test in reservoir with closed circular boundary; (c) drawdown test in reservoir with circular constant pressure boundary; (d) buildup test in reservoir with circular constant pressure boundary; (e) drawdown test in a radial composite reservoir with a low mobility outer zone; and (f) buildup test in a radial composite reservoir with a low mobility outer zone.

8.5.3 Incorporating Rate History in the Diagnostic Plot. One option for visually evaluating the effect of rate history on the buildup pressure response is to include the rate history on the diagnostic plot. The rate history during the flow period preceding the buildup, q ( t ), is graphed vs. the time until the buildup begins, ( t p − t ), where tp is the elapsed time at the start of the buildup, and t is the elapsed time at any point during the flow period. Thus, the rate history is plotted in reverse, with the final rate on the left of the graph and the initial rate on the right. Fig. 8.32 shows the pressure response for a well in an infinite-acting reservoir. The well was produced at a constant rate of 200 STB/D for 1012 hours. The rate was then produced at 100 STB/D for the final 12 hours

212 Applied Well Test Interpretation

before the well was shut in at tp = 1024 hours. The rate change from 200 STB/D to 100 STB/D is graphed at 1024 – 1012 = 12 hours. The resulting pressure response clearly shows the increase in the derivative corresponding to the rate change but smoothed out over ½ to ²⁄³ log cycle in each direction. Fig. 8.33 shows the pressure response for a similar case, where the rate decreased from 200 STB/D to 100 STB/D during the last 0.012 hour before shut-in. The rate history diagnostic shows that the effect of the final reduced-rate flow period will be almost completely hidden in wellbore storage. However, if the test is analyzed using a constant flow rate of 200 STB/D, and the pressure at the end of the short flow is used as pwf , the resulting analysis gives a skin factor that is too low and a wellbore-storage coefficient that is too high. Before a buildup can be conducted on a producing well, the pressure gauge must be run into the wellbore. Often, especially for high rate wells, this requires that the well be shut in while the gauge is being lowered into the well. This shut-in period can distort the shape of the pressure response if it is not taken into account during analysis. One option is to bring the well back onto production at the same rate it was producing before the gauge was run in the hole and to continue producing long enough that the temporary shut-in will not affect the buildup pressure response. Fig. 8.34 shows the rate history diagnostic plot for a well in an infinite-acting reservoir, in which the well produced at 100 STB/D for 1008 hours. The well was then shut in for 8 hours, opened back up to flow at 100 STB/D for 8 more hours, then shut in for the buildup test. Although the resulting distortion of the buildup pressure response shown in Fig. 8.34 is small, it is not negligible. For this case, the duration of the 8 hour flow period should be taken into account in the analysis of the buildup test. If the well is shut in for a shorter period of time, or if the well is produced for a longer period following the shut-in, the distortion of the buildup pressure response can be reduced. Fig. 8.35 shows the rate history diagnostic

∆p, tdp/dt, psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.32—Including the rate history on the diagnostic plot helps identify the effect of changes in rate on the resulting pressure response, rate decrease 12 hours before shut-in. 10000

∆p, tdp/dt, psi

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.33—Rate history diagnostic plot, rate decrease 0.012 hours before shut-in.

Variable Flow Rate History 213

∆p, tdp/dt, psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.34—Rate history diagnostic plot, well shut in for 8 hours, beginning 16 hours before buildup.

∆p, tdp/dt, psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.35—Rate history diagnostic plot, well shut in for 2 hours, beginning 12 hours before buildup.

plot for a buildup test in which the temporary shut-in lasted only 2 hours, and the well was then produced for 10 hours before being shut in for the buildup test. For this case, the short shut-in period appears to have no effect on the buildup pressure response. One recommendation is that the well be brought back on production for a period 10 times as long as the period the well was shut in while running in with the gauges. Figs. 8.36 through 8.40 show the effects of three different series of random flow rates, each lasting 276 hours, with random changes in flow rate every 12 hours. In each case, the flow rates during the random flow period were normalized to give an average rate of 100 STB/D. In Figs. 8.36 through 8.38, the random flow rate period was followed by a 12-hour flow period at a constant rate of 100 STB/D. In Fig. 8.36, the last few rates during the random flow rate period included rates that were both lower and higher than the average. The resulting effect on the buildup pressure response is surprisingly small. In Fig. 8.37, five of the last six rates during the random flow rate period were higher than the average. The buildup pressure response shows an increase in the pressure derivative during the time period influenced by the higher flow rates. In Fig. 8.38, three of the last four rates during the random flow rate period were lower than the average. The buildup pressure response shows a decrease in the pressure derivative during the time period influenced by the lower flow rates. Fig. 8.39 shows the same sequence of random flow rates as in Fig. 8.37, but with the random flow rate period ending 120 hours before shut-in. The buildup pressure response in Fig. 8.39 is much less affected by the random flow rate sequence ending 120 hours before the buildup than in Fig. 8.37, where the random rate sequence ends 12 hours before the buildup.

214 Applied Well Test Interpretation

∆p, tdp/dt, psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.36—Rate variations above and below the average rate before a final 12-hour flow period at constant rate have little effect on the pressure response during buildup.

∆p, tdp/dt, psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆t, hr

10

100

1000

Fig. 8.37—Rate history with higher than average rates before a final 12-hour constant rate flow period causes the pressure derivative to deviate upward from the horizontal line during buildup.

∆p, t(dp/dt ), psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆ t, hr

10

100

1000

Fig. 8.38—Rate history with lower than average rates before a final 12-hour constant rate flow period causes the pressure derivative to deviate downward from the horizontal line during buildup.

Variable Flow Rate History 215

∆p, t (dp/dt ), psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆ t, hr

10

100

1000

Fig. 8.39—Rate history with higher than average rates prior to a final 120-hour constant rate flow period causes a much smaller deviation in the pressure derivative than for the 12-hour constant flow period case.

∆p, t (dp/dt ), psi; q, STB/D

10000

1000

100

10

1 0.001

0.01

0.1

1 ∆ t, hr

10

100

1000

Fig. 8.40—Rate history with lower than average rates prior to a final 120-hour constant rate flow period causes a much smaller deviation in the pressure derivative than for the 12-hour constant flow period case.

Fig. 8.40 shows the same sequence of random flow rates as in Fig. 8.38, but with the random flow rates ending 120 hours before shut-in. Again, the buildup pressure response in Fig. 8.40 is much less affected by the random flow rate sequence ending 120 hours before the buildup than in Fig. 8.38, where the random rate sequence ends 12 hours before shut-in. 8.5.4 Effect of Flow Regime. In the previous section, we looked at the effect of flow rate history on reservoirs with infinite-acting radial flow. How do other flow regimes interact with the flow rate history? Fig. 8.41 shows drawdown and buildup pressure responses for radial flow, linear flow, spherical flow, and bilinear flow. In each case, the buildup pressure change and derivative are calculated with respect to the shut-in time. The time at which the buildup derivative deviates from the drawdown derivative depends on the flow regime. For radial flow, the buildup derivative deviates from the drawdown derivative by 10% at a shut-in time ∆t t p = 0.111. The corresponding deviation in the buildup and drawdown derivatives occurs at ∆t t p = 0.010 for linear flow, ∆t t p = 0.0487 for bilinear flow, and ∆t t p = 0.275 for spherical flow. 8.5.5 Variable Flow Rate History—How Much Is Enough? “How much of the variable flow rate history should I include in my analysis?” This question has received surprisingly little attention in the literature.

216 Applied Well Test Interpretation (a)

(c)

100

100

∆pD, tD (dpD /dtD)

∆pD, tD (dpD /dtD)

Drawdown 10

1

Drawdown Buildup

1

0.1

Buildup

0.1

10

10% Difference

10% Difference 0.01 0.0001

0.001

0.01

0.1 ∆tD /tpD

1

10

0.01 0.0001 0.001

100

(b) 100

0.01

10

100

Drawdown

10

Buildup

∆pD, tD (dpD /dtD)

∆pD, tD (dpD /dtD)

1

100

(d) Drawdown

1

0.1 ∆tD /tpD

10

0.1

0.1

Buildup

1 10% Difference

10% Difference 0.01 0.0001 0.001

0.01

0.1 ∆tD /tpD

1

10

100

0.01 0.0001 0.001

0.01

0.1 ∆tD /tpD

1

10

100

Fig. 8.41—The time at which the buildup pressure derivative deviates from the drawdown derivative depends on the type of flow regime: (a) radial flow; (b) linear flow; (c) spherical flow; and (d) bilinear flow.

Daungkaew et al. (2000) concluded that an accurate rate history should be used for the last 40% of the cumulative production history, while the first 60% of the cumulative production history may be approximated with a Horner pseudoproducing time. However, their theoretical analysis considered only infinite-acting radial flow. Further, in their investigation, every buildup lasted 2½ times the duration of the flow rate history. Taking into account the reservoir model and the duration of the buildup can significantly reduce the amount of detailed rate history that must be included in the analysis. We recommend including a detailed flow rate history immediately preceding the buildup equal to (or greater than) the shortest of the following times: (1) the time to reach pseudosteady-state flow (for a closed reservoir) or steady-state flow (for a reservoir with a constant pressure boundary); (2) a time equal to 10 times the duration of the buildup (100 times if linear flow is present, or 20 times if bilinear flow is present); or (3) the entire flow rate history. Reservoirs with significant heterogeneity (layered reservoir, radial composite reservoir with lower mobility outer region, etc.) may require much more of the production history to be included. Time to Pseudosteady-State Flow. Pinson (1972) and Kazemi (1974) both showed that average reservoir pressures estimated using the Matthews-Brons-Hazebroek (MBH) (Matthews et al. 1954) method could be estimated accurately for reservoirs in which pseudosteady state had been achieved by replacing the actual producing time tp by the time to pseudosteady state, tpss, in both the Horner analysis to obtain the false pressure p* and in the MBH analysis to calculate p from p*. Two points should be noted about this conclusion. First, the MBH method for estimating average reservoir pressure is based on extrapolating the infinite-acting radial flow semilog straight line to infinite shut-in time. This implies that the tpss approximation must predict the correct pressure response from infinite-acting radial flow until the reservoir pressure has stabilized. Second, the conclusion is independent of the reservoir model used in the analysis. Thus, the Pinson/Kazemi tpss approximation is valid for any reservoir model, for any time from infinite-acting radial flow through reservoir pressure stabilization. The conclusion that including the rate history only for times less than tpss before the buildup is consistent with the concept that all information about the past rate history of a well is contained in the reservoir

Variable Flow Rate History 217

Dimensionless Pressure

100

10

1

0.1

0.01

1

10

100

1000

10000

100000 1000000

Dimensionless Equivalent Time Fig. 8.42—Dimensionless pressure change and derivative, vs. dimensionless equivalent time, log-log graph. Equivalent time calculated using last 10 days of production history.

pressure profile at shut-in. For a reservoir in which pseudosteady-state flow (or steady-state flow) has been achieved, the pressure profile is independent of any earlier flow rate history. Thus, it is impossible for flow rate history earlier than a time tpss before shut-in to have any effect on the shape of the subsequent buildup pressure response. Ten Times (10×) Duration of the Buildup. Gringarten et al. (1979) noted that one rule of thumb for determining when buildup data could be analyzed with drawdown type curves was the producing time was 10 times greater than the producing time. They confirmed this rule of thumb for slightly damaged and stimulated wells, reporting that the dimensionless buildup pressure was within 1% of the dimensionless drawdown pressure as long as the ratio ∆t t p was less than 0.1. The rule of thumb that buildups may be analyzed using drawdown type curves provided tp ≥ 10∆t, also appears to be adequate for estimating how much detailed flow rate history must be included during analysis, at least for radial flow. Flow rate history at times earlier than 10 times the duration of the buildup test should be approximated using the Horner pseudoproducing time*. For linear flow, it may be necessary to include much more of the flow rate history. Using the results from the previous section on when the buildup derivative deviates from the drawdown derivative, we recommend including detailed flow rate history for a minimum of 100 times the duration of the shut-in if linear flow is present, or 20 times the shut-in duration if bilinear flow is present. On the other hand, for spherical flow, detailed flow rate history may be necessary only for a period 4 times the shut-in duration. The above guidelines appear to be reasonable for single-layer, homogeneous reservoirs. Much longer flow rate history may be required for layered reservoirs, compartmentalized reservoirs, or composite reservoirs with low mobility outer region. Entire Flow Rate History. If neither of the first two criteria are met, the entire rate history should be taken into account during analysis. This will normally be the case during analysis of exploration well tests, where the flow period is part of the test sequence, and thus the necessary rate data are readily available. Buildup test analysis should incorporate detailed flow rate data for a period of time before the buildup at least the smallest of (1) the time to pseudosteady-state or steady-state flow, (2) 10 times the duration of the buildup test (for radial flow), and (3) the entire flow rate history. 8.5.6 Field Example 2. Gas Well A was located in a small fault block in a mature field. It was suspected that one or more of the faults allowed some degree of communication with other reservoir compartments. The well had been shut in for approximately a year. The well was produced for a little more than three months, a test string was run in the hole, the well was produced for 10 more days, then was shut in for a three month buildup test. When the data were first loaded into the software, only the last 10 days of production (during which the test string was in the hole) were included in the production history. Figs. 8.42 and 8.43 show the log-log plot and semilog plot, respectively, for the data set with the 10-day production history. The extremely steep derivative on This practice assumes that the flow period used for qlast is long enough to justify using the tp approximation (i.e., don’t just take the last month of production as qlast if rates are fluctuating). Otherwise, some sort of averaging is required.

*

218 Applied Well Test Interpretation

Dimensionless Pressure

16 14 12 10 8 6 4 2 0

1

10

100 1000 10000 100000 1000000 Dimensionless Equivalent Time

Fig. 8.43—Dimensionless pressure change vs. dimensionless equivalent time, semilog graph. Equivalent time calculated using last 10 days of production history.

the log-log plot and the abrupt hook at the end of the test on the semilog plot are caused by using the truncated rate history. The rising pressure curve on the semilog plot shows that there is significant pressure support throughout the shut-in period. It was thought that a yearlong shut-in should be enough for the reservoir pressure to stabilize. So, the three months of production after the yearlong shut-in were added to the production history table. Fig. 8.44 shows the resulting log-log graph. Even with the longer production history, the derivative curve is still steepening upward at the end of the buildup, although it is not as severe as in Fig. 8.42. Because the steeper-than-unit-slope derivative could not be easily explained any other way, five years of production data before the yearlong shut-in were added to the production history. Fig. 8.45 shows the log-log graph, where the time-plotting function has been changed to the dimensionless shut-in time, which is independent of the rate history. Also shown in Fig. 8.45 as a dashed derivative curve is an approximate manual history match using an infinite radial composite reservoir with pseudosteady-state dual porosity behavior. The match was reasonable, but the history match derivative curve did not follow the unit-slope line of the field data derivative. The simulated derivative slope is becoming shallower, suggesting that the flow rate into the model is decreasing with time, while the field data derivative remains a straight, unit-slope line. As a last step, the entire production history was entered, using the Horner pseudoproducing time approximation for the period before the yearlong shut-in. Fig. 8.46 shows a log-log graph of the final match, using the same reservoir model as in Fig. 8.45, but with the entire flow rate history. The calculated pressure response for the full rate history, Fig. 8.46, was different from that for Fig. 8.45, in spite of the fact that Fig. 8.45 incorporated flow rate history equal to 24 times the duration of the buildup test. Without additional external data, it is impossible to verify that the final model is the best model for the interpretation. However, the fact that the simulated pressure response continued to change as additional production history were included indicates that, for this test, the entire production history must be used for the interpretation.

Dimensionless Pressure

100

10

1

0.1

0.01

1

10

100

1000

10000

100000 1000000

Dimensionless Equivalent Time Fig. 8.44—Dimensionless pressure change vs. dimensionless equivalent time, log-log graph. Equivalent time calculated using last 110 days of production history.

Variable Flow Rate History 219

Dimensionless Pressure

100

10

1

0.1

0.01 1

10

100 1000 10000 100000 1000000 Dimensionless Shut-in Time

Fig. 8.45—Dimensionless pressure change vs. dimensionless shut-in time, log-log graph. Manual history match using 6 years of production history.

Dimensionless Pressure

100

10

1

0.1

0.01

1

10

100 1000 10000 100000 1000000 Dimensionless Shut-in Time

Fig. 8.46—Dimensionless pressure change vs. dimensionless shut-in time, log-log graph. Manual history match using entire production history.

8.6 Convolution and Deconvolution The most general method for taking into account a variable rate history is the convolution equation. The convolution equation is based on applying the principle of superposition in time to an arbitrary rate history. Convolution is the process of calculating the variable rate pressure response for a given flow rate history, when the constant-rate pressure response is known. Convolution is a forward problem. Mathematically, convolution is a stable operation with a unique solution. Deconvolution is the process of calculating the constant-rate pressure response for a given flow rate history and the resulting variable rate pressure response. Deconvolution is an inverse problem. Unlike convolution, deconvolution is mathematically unstable and has no unique solution. 8.6.1 Convolution. For any reservoir for which the diffusivity equation is valid, the pressure response for a well with a rate history comprising a series of N constant-rate flow periods can be obtained using superposition in time as N

pw ( t ) = pi − ∑ ( qi − qi − 1 ) ∆pu ( t − ti − 1 ), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.30) i =1

where ∆pu ( t ) is the unit pressure response for a well producing at constant unit rate. There are several assumptions implicit in Eq. 8.30. (1) The flow equation is linear. In practice, this condition is met for single-phase flow of oil or water, provided rock properties are independent of pressure. The condition is also met for gas wells if the total drawdown during the test is small. (2) The reservoir is at uniform initial pressure pi. In particular, Eq. 8.30 will not hold for a well producing commingled from a multilayer reservoir where the layers have different initial pressures. (3) There is no interference from offset wells. (4) Boundary conditions are linear. Examples of nonlinear boundary conditions include phase segregation, non-Darcy flow, and changing skin factor.

220 Applied Well Test Interpretation

Approximating an arbitrary rate history as a series of constant-rate flow periods, then taking the limit as the duration of each flow period approaches zero, we obtain the convolution equation for the pressure response for an arbitrary rate history: ∆p ( t ) = pi − pw ( t ) = ∫

t

τ =0

q ′ (τ ) ∆pu ( t − τ ) dτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.31)

By integrating by parts and interchanging the arguments of the rate and pressure terms inside the integrand, the convolution equation can be written a number of different equivalent forms: ∆p ( t ) = ∫

t

=∫

t

=∫

t

=∫

t

τ =0

τ =0

τ =0

τ =0

q ′ (τ ) ∆pu ( t − τ ) dτ q ′ ( t − τ ) ∆ pu (τ ) dτ q (τ ) ∆ pu′ ( t − τ ) dτ q ( t − τ ) ∆pu′ (τ ) dτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.32)

Convolution is nothing more than the principle of superposition in time, extended to an arbitrary variable rate history. Convolution is a stable mathematical operation with a unique solution. Convolution is the principle of superposition in time extended to an arbitrary variable rate history. Convolution is a forward problem. Convolution is a stable mathematical operation with a unique solution. 8.6.2 Deconvolution. Deconvolution, the inverse problem, is the process of solving Eq. 8.31 for the unit pressure response ∆pu ( t ) when the rate history q ( t ) and the resulting pressure response ∆p ( t ) are known. Unlike convolution, in general, deconvolution is an unstable mathematic operation, with no guarantee that a unique solution exists. Deconvolution is the calculation of the unit pressure response Dpu(t) when the rate history q(t) and the resulting pressure response Dp(t) are known. Deconvolution is an inverse problem. The process of deconvolution is numerically unstable and has no unique solution. Although Eq. 8.31 is derived from the principle of superposition, which in turn relies on the linearity of the diffusivity equation, there is no guarantee that a solution to Eq. 8.31 is also a solution to the diffusivity equation. Thus, successful application of deconvolution must impose conditions on ∆pu ( t ) to ensure that it is a solution to the diffusivity equation. The process of imposing constraints on the solution ∆pu ( t ) is called regularization. 8.6.3 Superposition and History Matching. If the reservoir model is known, then the nature of the unit pressure response ∆pu ( t ) is also known, although the specific parameters such as permeability, skin factor, and distances to boundaries that describe the model may not be known. Estimating the model parameters by history matching the observed pressure response, either manually or using nonlinear regression, relies on the use of superposition (Eq. 8.30) to calculate the model pressure response ∆p ( t ) from the known rate history q ( t ) and the unit pressure response ∆pu ( t ). 8.7 Rate Normalization Rate normalization is a simple, approximate deconvolution method. In spite of its simplicity, it is widely applicable; because of its simplicity, it is widely applied. The primary limitation of rate normalization is that it requires that the flow rates (and pressures) be changing slowly and smoothly. Another major limitation of rate normalization is that it only accounts for changes in pressure response caused by simultaneous changes in rate. Rate normalization cannot account for changes in pressure response during buildup caused by rate changes during the preceding flow period, as in Figs. 8.32, 8.34, 8.37, and 8.38. Rate normalization is an approximate deconvolution method that can be used when the flow rate is changing slowly and smoothly. Rate normalization only accounts for changes in the pressure response caused by simultaneous changes in the flow rate; it cannot account for pressure changes during buildup caused by rate changes during the preceding flow period.

Variable Flow Rate History 221

8.7.1 Drawdown Tests—Winestock and Colpitts Method. Winestock and Colpitts (1965) conducted a study of the use of rate normalization to analyze flow tests in gas wells. Their study included numerical simulation, theoretical analysis, and field tests. They wrote the normalized flow equation for slightly compressible liquids as pi − pwf ( t ) q (t )

= 141.2

  Bµ  1   0.0002637 kt   ln  0 809 + .  + s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.33)  kh  2   φµct rw 2   

The also presented a similar equation for gases, including the effects of non-Darcy flow, in terms of pressure squared. 8.7.2 Buildup Tests—Wellbore Storage and Gladfelter Deconvolution. Gladfelter et al. (1955) applied the concept of rate normalization to analysis of buildup tests in wells producing by rod pump. Gladfelter used a buildup pressure-plotting function defined as ∆pn = { pws ( t ) − pwf }

q , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.34) q − qaf ( t )

where ∆pn is the rate-normalized pressure change, q is the stabilized flow rate before the buildup test, and qaf is the measured or estimated afterflow rate. The Gladfelter study, like the Winestock and Colpitts study, was limited to systems exhibiting infinite-acting radial flow. Fetkovich and Vienot (1984) extended the use of rate normalization to drawdown and buildup tests on hydraulically fractured wells and to wells experiencing multiphase flow. 8.7.3 Production Data Analysis—Material Balance Time. Blasingame and Lee (1986) studied variable rate reservoir limits testing for oil wells. They introduced the material balance time for oil wells as tmb =

24 N p qo

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.35)

They showed that, for production of a slightly compressible liquid from a closed reservoir of arbitrary shape, a Cartesian graph of the normalized pressure change ∆p q, defined as ∆p pi − pwf ( t ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.36) = q qo ( t ) vs. the material-balance time tmb from Eq. 8.35, gave a straight line with slope m pss =

0.234 Bo , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.37) Ahφct

and intercept b pss =

141.2 Boµ o  1  10.06 A  3   ln   − + s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.38) 2 kh  2  C Arw  4 

Blasingame and Lee (1988) extended their earlier work to reservoir limits testing of gas wells, defining the adjusted material balance time as tamb =

24G pa qg

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.39)

where Gpa is the adjusted cumulative production, defined as G pa = µ gi cti

t

qg dt

0

g t

∫ µ c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.40)

222 Applied Well Test Interpretation

Blasingame and Lee showed that a graph of the normalized adjusted pressure change, ∆pa qg, defined as ∆pa pai − pawf ( t ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.41) = qg qg ( t ) vs. the adjusted material balance time tamb gave a straight line with slope m pss =

0.234 Bgi Ahφcti

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.42)

and intercept b pss =

141.2 Bgi µ gi  1  10.06 A  3   ln   − + s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.43) 2 kh  2  C Arw  4 

A number of other authors (Blasingame et al. 1991; Palacio and Blasingame 1993; Agarwal et al. 1999) extended the application of the rate-normalized pressure change/material balance time to transient flow for both vertical and hydraulically fractured wells. 8.8 Deconvolution In spite of the difficulty of performing deconvolution, it offers so many advantages and applications that the potential benefits outweigh the difficulties involved. In the following sections, we describe and discuss (1) the major applications for deconvolution, (2) the conditions necessary for deconvolution to be valid, (3) some practical aspects involved in applying deconvolution, (4) some factors to consider during test design to facilitate deconvolution, and (5) a method for approximate deconvolution by history matching using a generic reservoir model. 8.8.1 Applications of Deconvolution. In this section, we discuss three major applications of deconvolution in pressure-transient test interpretation: (1) reducing the effect of wellbore storage to reveal near-wellbore reservoir characteristics, (2) determining the unit-rate pressure response for the total duration of a test with several different flow and shut-in periods; and (3) finding the unit-rate pressure response for flow regime identification independent of flow history. Other applications of deconvolution include estimation of individual layer properties for multilayer reservoirs using measured pressures and downhole flow rates, analysis of rate and pressure data from permanent gauges, and analysis of long-term production data. Reduce Effects of Wellbore Storage. One important application of deconvolution is to reduce the effects of wellbore storage, allowing evaluation of reservoir characteristics from shorter-term test data (Gladfelter et al. 1955; Fetkovich and Vienot 1984). Deconvolution can be performed using either measured or estimated afterflow rates. If a downhole flow meter is used to measure the sandface flow rate, the deconvolved pressure response will still include the effects of wellbore storage for the volume between the sandface and the flowmeter. If the wellbore-storage coefficient is known, the sandface flow rate may be calculated from the derivative of the measured shut-in pressure as a function of time: qsf =

24C dpws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.44) B dt

Eq. 8.44 is valid even when the wellbore-storage coefficient is time or pressure dependent, as long as it is known. For oil wells producing by rod pump, acoustic fluid-level measurement data may be used to obtain both shut-in bottomhole pressure and afterflow rate. Fig. 8.47 shows the pressure response for a well near a single no-flow boundary. Wellbore storage lasts too long for the characteristic no-flow boundary pressure response to be visible. Using the Gladfelter method to deconvolve the pressure data, calculating the afterflow rate from the known wellbore-storage coefficient reveals the presence of the no-flow boundary, as shown in Fig. 8.48. Note that Gladfelter deconvolution does not exactly reproduce the shape of the pressure response caused by the single no-flow boundary.

Variable Flow Rate History 223

∆p, t(dp/dt ), psi

1000

100

10

1 0.01

0.1

1

10

100

1000

∆t, hr Fig. 8.47—Well near a single no-flow boundary, boundary pressure response hidden in wellbore storage.

∆p, t(dp/dt ), psi

1000

100

10

1 0.01

0.1

1

10

100

1000

∆t, hr Fig. 8.48—Well near a single no-flow boundary, Gladfelter deconvolution shows presence of boundary.

Determine Long-Term Reservoir Pressure Response. A second important application of deconvolution is extract the unit pressure response from a long pressure-transient test with several different flow and shut-in periods. In principle, the pressure response for the entire test duration could be determined, even though the individual flow and shut-in periods are much shorter in duration than the total length of the test. Fig. 8.49 shows the test rates and resulting pressure response for a multirate test on a well in a reservoir with a single no-flow boundary 500 ft from the wellbore. The test comprises a series of four flow and buildup periods, each 48 hours long. The radius of investigation achieved during a single flow or buildup period is 645 ft, not quite enough to see the effects of the boundary. The radius of investigation for the entire 384 hour test sequence is 1820 ft, large enough to define the boundary with some confidence. Fig. 8.50 shows the reservoir pressure response for constant-rate production at 100 STB/D for a 48-hour flow period (solid line), along with the pressure response for constant-rate production at 100 STB/D for a 384-hour flow period (dashed line). In principle, deconvolution can provided the constant-rate pressure response the 384-hour duration of the test, even though no single flow period lasts longer than 48 hours. Determine Unit-Rate Pressure Response for Flow Regime Identification Independent of Flow History. Another application of convolution is to obtain the constant-rate pressure response for identification of flow regimes independent of flow history. This allows pressure responses caused by physical reservoir heterogeneities to be distinguished from similar pressure responses caused by changes in flow rate, as in Fig. 8.32. 8.8.2 Nonlinear Phenomena. The convolution equation, Eq. 8.31, is valid only if the flow equation and boundary conditions are all linear. Because deconvolution is based on the convolution equation, it also depends on the linearity of the flow equation and boundary conditions. Linearity is particularly important for deconvolution, since the process of deconvolution is ill posed and inherently unstable. Nonlinearity may arise in either the reservoir flow equation or in the boundary conditions. Nonlinearities in the flow equation include pressure-dependent rock and fluid properties and multiphase flow. Reservoir nonlinearities

3200

1200

3000

1000

2800

80

2600

600

2400

40

2200

20

2000

0

50

100

150 200 250 Elapsed Time, hr

300

350

Flow Rate, STB/D

Pressure, psi

224 Applied Well Test Interpretation

0 40

Fig. 8.49—Multirate test on well in reservoir with single no-flow boundary.

∆p, t(dp/dt ), psi

1000

100

10

48hr 1 0.01

0.1

1

∆t, hr

10

384 hr 100

1000

Fig. 8.50—Comparison of a constant rate reservoir pressure response seen at the end of a single flow period with that seen at end of the entire test.

can be particularly severe for retrograde gas reservoirs producing below the dew point pressure, or oil reservoirs producing below the bubble point pressure, or water-drive reservoirs with changing fluid saturations. Nonlinearities in the boundary conditions include variable wellbore storage, such as phase segregation or pressure-dependent wellbore storage; changing skin factor, such as non-Darcy flow, cleanup, stimulation, or degradation in skin factor over time, and changing boundary conditions caused by production from offset wells or water encroachment from a down-dip aquifer. 8.8.3 Practical Aspects of Deconvolution. Because deconvolution is an unstable operation with no unique solution, some degree of care is necessary for successful application. Levitan et al. (2006) present several guidelines to improve results from deconvolution. Use Buildup Data for Deconvolution. Buildup data provide much better deconvolution than flow period data because of the higher quality of the data. Flow data are inherently noisy. Further, small fluctuations in flow rate may occur that are not measured with the accuracy necessary for successful deconvolution. Synchronize Rate and Pressure Data. It is necessary to synchronize rate data with the pressure data much more carefully when doing deconvolution than when doing conventional analysis. Levitan et al. (2006) found that minor mismatches in synchronization, which would have only a localized effect on the results from conventional analysis, could distort the shape of the deconvolved pressure response for the entire test period. Remove Data Distorted by Nonlinear Phenomena. Identify and either remove or ignore data distorted by nonlinear phenomena such as phase segregation before attempting deconvolution. Only data that meet the criteria for validity of the convolution equation, Eq. 8.31, should be used during deconvolution.

Variable Flow Rate History 225

Response Function (MMPSI2/CP/M) 10–4 10–3

Use Complete Rate History. Levitan et al. (2006) caution that all of the rate history should be used, since the convolution equation assumes that the reservoir is at uniform pressure at the beginning of flow. They suggested that rate history preceding a buildup by a period at least twice the duration of the shut-in could be incorporated as an average rate, honoring the actual producing time. This recommendation is less restrictive than the rule of thumb that flow rate history at least 10 times the duration of the shut-in be included in analysis. It may be better to include all of the rate history for an exploration well test, particularly in the case of linear flow, than to rely on the validity of Levitan’s “twice the duration of the shut-in” recommendation. Deconvolve Each Buildup Separately. Independent deconvolution of each buildup should give the same deconvolved pressure response. Any discrepancies between the deconvolved pressure responses must be reconciled before the results of deconvolution can be accepted as valid. Adjust Initial Pressure for Consistency Between Buildups. Levitan et al. (2006) found that deconvolution was very sensitive to the initial pressure. Figs. 8.51, 8.52, and 8.53 show the effect of changing the initial pressure had on the deconvolution of two different PBU periods in the same test sequence.

from PBU 2 .11 .05

.11

from PBU 1

10–5

.05

10–3

10–2

10–1

∆t, hr

100

101

from PBU 2 from PBU 1

.11

10–3

.05

.11

10–4

.05

10–5

Response Function (MMPSI2/CP/M)

Fig. 8.51—Deconvolved pressure responses for two different buildups, assumed initial pressure 6,310 psi (Levitan et al. 2006).

10–3

10–2

10–1

∆t, hr

100

101

Fig. 8.52—Deconvolved pressure responses for two different buildups, assumed initial pressure 6,320 psi (Levitan et al. 2006).

from PBU 2 from PBU 1 10–3

.11 .05

.11

10–4

.05

10–5

Response Function (MMPSI2/CP/M)

226 Applied Well Test Interpretation

10–3

10–2

10–1

100

101

∆t, hr Fig. 8.53—Deconvolved pressure responses for two different buildups, assumed initial pressure 6,314.3 psi (Levitan et al. 2006).

In Fig. 8.51, the initial pressure was assumed to be 6,310 psi, lower than the actual pressure. Deconvolution of the PBU 1 showed the pressure derivative decreasing between 3 and 10 hours, whereas the deconvolved pressure response for PBU 2 showed a monotonically increasing derivative during the same time period. In Fig. 8.52, the initial pressure was assumed to be 6,320 psi, higher than the actual pressure. Again, the deconvolved pressure responses for the two buildups are different. In this case, PBU 1 shows a higher derivative between 1 and 10 hours than does PBU 2. An initial pressure of 6,314.3 psi, Fig. 8.53, gave excellent agreement between the deconvolved pressure derivatives for the two buildups. The deconvolved pressure change curves for the two buildups show that the skin factor was higher during the first buildup than during the second buildup. 8.8.4 Planning for Deconvolution During Test Design. Pressure-transient test design for a new well, especially for an exploration well, should take into account the likelihood that deconvolution will be used during test interpretation. Use Downhole Shut-In. The use of a downhole shut-in tool will minimize the likelihood of phase segregation effects as well as minimizing the duration of wellbore-storage effects. The shorter duration of wellbore storage effects will allow more of the data to be used during deconvolution, possibly revealing near-wellbore features that could not be resolved with surface shut-in. Hang Gauge at the Top of the Test Interval. The gauge assembly should be hung as close as possible to the top of the test interval to prevent problems caused by changing fluid density between the test interval and the gauge. Changes in fluid density during a buildup test, such as a moving gas-liquid interface between the test zone and the gauge, will cause a time-dependent offset in the measured pressure that cannot be accounted for during deconvolution. Test Zones Independently. If there are multiple target zones in the well that are not in hydrostatic equilibrium, the zones should be tested independently. If multiple zones are tested commingled, wellbore crossflow will prevent use of deconvolution during analysis. Multiple Buildups for Each Zone. The test sequence for each tested zone should include at least two extended buildup periods. Initial flow should be long enough to clean up the well. Ideally, after cleanup the well should be produced at a constant rate for a time period equal to the duration of the subsequent buildup. At a minimum, there should be at least one more flow period at constant rate, followed by a second buildup. Deconvolution of the two buildups separately allows the deconvolved pressure response to be validated. Complete All Pressure-Transient Testing Before Conducting Any Maximum Flow Rate Testing. For each zone, complete all pressure-transient testing before conducting any maximum flow rate testing. Attempting to produce a well at the highest flow rate possible will almost always cause the pressure to fall below the bubble point or dew point pressure. For an oil well, the pressure dropping below the dew point pressure near the wellbore will cause an extreme change in the total compressibility, along with reduction in relative permeability to oil caused by the presence of a free gas saturation. If the pressure rises through the bubblepoint during buildup,

Variable Flow Rate History 227

∆p, t(dp/dt ), psi

1000

100

10

48 hr 1 0.01

0.1

1

10

384 hr 100

1000

∆ t, hr Fig. 8.54—Approximate deconvolution using history matching with a generic reservoir model.

forcing gas back into solution may cause a decreasing total compressibility that cannot be accounted for during deconvolution. For a retrograde gas well, although the pressure falling below the dew point will not cause a severe change in compressibility as it does for an oil well, it may cause a reduction in near-wellbore permeability and a time-dependent skin factor as a condensate ring accumulates near the wellbore. During shut-in, the rising near-wellbore pressure may revaporize some of the condensate, again creating nonlinear fluid behavior that cannot be accounted for during deconvolution. 8.8.5 Approximate Deconvolution by History Matching. An approximate deconvolution may be obtained using a generic reservoir model to match the pressure response for a variable rate/variable pressure history. Suitable generic reservoir models might include the well in a rectangular reservoir model and the radial composite reservoir model. The particular reservoir model is not important, as long as the model is flexible enough to match the observed pressure response. If the matched pressure response matches the observed pressure response for all flow and shut-in periods, the drawdown pressure response for the model used for the match is an approximate deconvolution of the variable rate/pressure history, even if the reservoir model used for the match does not provide an accurate description of the reservoir. In this case, the parameter estimates for the matched model should be ignored (they are meaningless, since the model is wrong). The drawdown pressure response for the matched model is the desired deconvolved pressure response. Fig. 8.54 shows an approximate deconvolution obtained by history matching the last buildup period of the test sequence shown in Fig. 8.49, using a radial composite model. Although only the test data for the final buildup were used in the history match, the history match honored the initial pressure, and the approximate deconvolved pressure response (narrow solid line) agrees very well with the true constant-rate pressure response (open circles). 8.9 Summary Key points discussed in this chapter include 1. All information about the past rate history of a well is stored in the pressure profile in the reservoir at the moment of shut-in. During a buildup test, as the radius of investigation moves away from the wellbore, the well reads the flow rate history in reverse. 2. Rate changes before buildup may look like reservoir boundaries if not taken into account during analysis. 3. The pressure response for a buildup in a closed reservoir looks nothing like the drawdown pressure response. In fact, it can easily be confused with the pressure response for a well in a reservoir with a circular constant pressure boundary. 4. A drawdown test in a closed reservoir will exhibit a unit-slope derivative after pseudosteady-state flow has been achieved. However, a buildup test that exhibits a unit-slope line at the end of the test cannot be from a closed reservoir.

228 Applied Well Test Interpretation

5. The rate history may be incorporated into the diagnostic plot to help identify portions of the buildup pressure response that are affect by the preceding flow rate history. 6. Buildup test analysis for a well in a homogeneous single-layer reservoir should incorporate detailed flow rate data for a period of time before the buildup equal to the smallest of (1) the time to pseudosteady-state or steady-state flow, (2) 10 times the duration of the buildup test (for radial flow), and (3) the entire flow rate history. 7. Convolution is the principle of superposition in time extended to an arbitrary variable rate history. Convolution is a forward problem. Convolution is a stable mathematical operation with a unique solution. 8. Deconvolution is the calculation of the unit pressure response Dpu(t) when the rate history q(t) and the resulting pressure response Dp(t) are known. Deconvolution is an inverse problem. The process of deconvolution is numerically unstable and has no unique solution. 9. Rate normalization is an approximate deconvolution method that may be used when the flow rate is changing slowly and smoothly. Rate normalization may be used for reducing the effect of wellbore storage during buildup, correcting for slowly changing rates during flow tests. In conjunction with the material balance time, rate normalization may also be used for analyzing production data for both oil and gas wells producing by depletion drive. 10. Deconvolution may be used to (1) reduce or eliminate the effects of wellbore storage, (2) determine the unit-rate pressure response for a time period equal to the total duration of a series of flow and shut-in periods, and (3) determine the unit-rate pressure response for identification of flow regimes independent of the flow history. 11. Suggestions for applying deconvolution include (1) use only buildup data for deconvolution, (2) synchronize rate and pressure data and (3) remove data distorted by nonlinear phenomena before attempting deconvolution, (4) use the complete rate history, (5) deconvolve each buildup period separately, and (6) reconcile deconvolved pressure response for different buildup periods by adjusting the initial pressure used for the deconvolution. Nomenclature A = drainage area, ft2 b = intercept B = formation volume factor, bbl/STB C = wellbore-storage coefficient, bbl/psi CA = Deitz drainage area shape factor, dimensionless ct = total compressibility, psi–1 f = plotting function h = formation thickness, ft k = permeability, cp m = slope (units vary) p = pressure, psi p* = Horner false pressure, psi p = average pressure, psi Dpu = unit pressure response, psi q = production rate, STB/D rw = wellbore radius, ft s = skin factor t = time, hour tp = producing time, hour Dt = shut-in time, hour Dte = equivalent time, hour f = porosity, fraction m = viscosity, cp t = time integration variable, hour Subscripts a = adjusted af = afterflow amb = adjusted material balance B = bilinear flow

Variable Flow Rate History 229

deriv = pressure derivative curve g = gas i = index i = initial value L = linear flow mb = material balance n = rate normalized N = number of flow periods Np = cumulative oil production, STB o = oil Dp = pressure change curve pss = pseudosteady state S = spherical flow sf = sandface w = wellbore conditions wf = flowing wellbore conditions ws = shut-in wellbore conditions X = generic flow regime

Chapter 9

Wellbore Phenomena “In theory there is no difference between theory and practice. In practice there is.” —variously attributed to Yogi Berra, Jan van de Snepsheut, and Albert Einstein 9.1 Introduction Wellbore phenomena, the topic of this chapter, and near-wellbore phenomena, to be discussed in Chapter 10, both have significant influence on the pressure response during a pressure-transient test. However, unlike near-wellbore phenomena, wellbore phenomena are rarely important during production. The task of the analyst is to identify the cause of wellbore phenomena to determine whether to ignore, and if not, how to account for, the effect of the phenomena on the pressure response. Section 9.2 discusses a variety of situations that can cause a changing wellbore-storage (WBS) coefficient. These include an abrupt change in WBS coefficient, phase segregation, thermal effects, and compression of a gas column above a rising liquid level. Section 9.3 describes a simple wellbore cleanup model proposed by Larsen and Kviljo (1990). The cleanup model predicts pressure responses during flow periods that are very similar to those predicted by variable WBS caused by temperature changes. Section 9.4 discusses the effect of a gas-liquid interface moving between the completion and the gauge on the pressure response recorded by the gauge and gives suggestions on how the effect may be minimized or eliminated. Section 9.5 gives three examples of phenomena that may occur because of activities during testing operations, including a leak or opening a valve, setting or unsetting a bridge plug, and raising or lowering the gauge during a test. Section 9.6 describes three situations that may cause oscillations in pressure during a buildup test, including inertial effects that arise when shutting in a high-rate oil well, regular oscillations caused by earth tides, and irregular oscillations caused by diurnal variations in wellhead temperature. Section 9.7 describes some of the problems that may occur with modern electronic gauges. Section 9.8 shows how to determine the correct flowing bottomhole pressure and the time of shut-in for a buildup test, and the effect on the resulting pressure and pressure derivative if the incorrect pressure or time is chosen. The section also describes the banding effect on the pressure derivative caused by low gauge precision. Section 9.9 gives several criteria that can be used to distinguish wellbore phenomena from reservoir phenomena. Reservoir phenomena are always smooth; any abrupt disturbance in the pressure must be caused by wellbore effects. Wellbore phenomena may be smooth if they occur early in a test. After completing this chapter, you should be able to 1. Recognize the presence of, and identify the cause of, a changing wellbore-storage coefficient. 2. Analyze a test exhibiting changing wellbore-storage. 3. Identify effects caused by movement of a gas-liquid interface between the completion and the gauge. 4. Identify effects caused by activities during testing operations. 5. Identify the cause of pressure oscillations from momentum effects, earth tides, and diurnal temperature changes. 6. Distinguish apparent effects caused by gauge malfunction from actual changes in wellbore pressure. 7. Identify the correct flowing bottomhole pressure, and the time of shut-in, for analysis of a pressure buildup test. 8. Identify artifacts in the derivative caused by low gauge precision. 9. Distinguish wellbore phenomena from reservoir phenomena. 10. Determine whether or not changes in pressure caused by a particular wellbore effect must be taken into account when calculating skin factor and additional pressure drop because of skin.

232 Applied Well Test Interpretation

9.2 Variable Wellbore Storage In Chapter 2, we introduced the concept of WBS, showing that the storage coefficient is a constant for a wellbore filled with a single-phase fluid and for a well with a rising liquid level. However, there are many other common situations in which the WBS coefficient varies with time during a test. The following sections discuss several situations in which the WBS coefficient varies with time, including an abrupt increase or decrease in WBS coefficient, an apparent increase in WBS coefficient caused by phase segregation or thermal effects, an apparent increase or decrease in WBS coefficient because of thermal effects, and an apparent increase in WBS coefficient because of composite wellbore phenomena. 9.2.1 Abrupt Change in WBS Coefficient. Consider a water injection well, in which the flowing wellhead pressure during injection is higher than atmospheric pressure, yet the static reservoir pressure is too low to support a column of water reaching to the surface. In this situation, a falloff test will exhibit an abrupt increase in WBS coefficient when the shut-in wellhead pressure falls below atmospheric pressure and the well goes on vacuum. Fig. 9.1 shows the pressure response for a well in an infinite-acting reservoir with constant skin factor, and an abrupt increase in WBS coefficient. Immediately after shut-in, the well is filled with single-phase water. Thus, the WBS coefficient is initially very low. When the well goes on vacuum, the WBS coefficient abruptly increases to that for a well with a falling liquid level (at tD = 100 in Fig. 9.1). Because of the abrupt decrease in WBS coefficient, the pressure derivative is discontinuous, shifting instantaneously to the unit-slope line corresponding to the new, higher WBS coefficient. The pressure change curve flattens out, then approaches the unit-slope line corresponding to the new WBS coefficient. Similarly, Fig. 9.2 shows the pressure response for a well exhibiting an abrupt decrease in WBS coefficient. This situation may occur, for example, during an injection test on a well where the reservoir pressure cannot 100

pwD

10

1

0.1

0.01

10

100

1000

10000 tD

100000

1000000 10000000

Fig. 9.1—Pressure response for a well with an abrupt increase in WBS coefficient. Adapted from Earlougher et al. (1973). 100

pwD

10

1

0.1

0.01 10

100

1000

10000 tD

100000

1000000 10000000

Fig. 9.2—Pressure response for an abrupt decrease in WBS coefficient. Adapted from Earlougher et al. (1973).

Wellbore Phenomena 233

pT ′

pT ρ = VL ρL + VG ρG

pT ′ = pG′ = pG

≅ VL ρL

= pT +

1 VL ρL gh 2

pB ′= pT′ + ρLgVLh 3 ρL VL gh 2 1 = pB + ρL VL gh 2 = pT + pB

pB = pT + ρLVL gh

pB ′

Fig. 9.3—Redistribution of phases causes an increase in pressure at both the top and the bottom of the wellbore.

support a column of fluid to the surface, or during a buildup test on a pumping well with a packer, when the rising liquid level in the annulus reaches the packer (Earlougher et al. 1973; Earlougher 1977). 9.2.2 Phase Segregation. Stegemeier and Matthews (1958) reported that approximately 75% of the wells in a oil field in South Texas exhibited an unusual pressure response during pressure buildup, where the pressure would rise to a maximum, then decline before reaching a stabilized reservoir pressure. They noted that this humping effect appeared to be inversely related to the productivity index of the well. The pressure hump appeared earlier and was larger in magnitude for wells in which the annulus was packed off, than for wells without a packer. Stegemeier and Matthews attributed this pressure response to segregation of the oil and gas phases following shut-in. Consider a column with a mixture of liquid and gas flowing upward as shown in Fig. 9.3a. Neglecting frictional forces, the pressure at the bottom of the column is given by the pressure at the top of the column plus the hydrostatic head of the mixture. Assuming the gas density is much lower than the liquid density, the pressure at the bottom of the column is given by pB = pT + ρ LVL gh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.1) If valves at the top and bottom of the column are closed, the gas and liquid separate as shown in Fig. 9.3b. Assuming the liquid is incompressible and the volume of the column is constant, the volume occupied by the gas after the phases separate is the same as the total volume occupied by the gas bubbles before the valves were closed. If the temperature is also constant, the average pressure in the gas after closing the valves must also be the same as the average pressure in the gas bubbles before closing the valves. But the average pressure in the gas bubbles, assuming the bubbles are evenly distributed throughout the column, is given by pB = pT +

1 ρ LVL gh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.2) 2

Thus, the pressure in the gas at the top of the column after phase segregation is higher than the pressure at the top of the column before phase segregation, by an amount ∆pB =

1 ρ LVL gh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.3) 2

Adding the hydrostatic head of the liquid column, the same pressure increase is seen to occur at the base of the column after segregation*. 9.2.3 Fair Variable-WBS/Phase Segregation Model. Fair (1981) proposed an empirical model for the pressure change accompanying phase segregation, based on several reasonable assumptions about the form of the pressure change function pf: (1) The phase segregation pressure change is zero at the beginning of the buildup, *

Drilling engineers will recognize this phenomenon as the same principle that causes the wellbore pressure increase during a gas kick.

234 Applied Well Test Interpretation

(2) the pressure change approaches a constant (positive) value at large times, (3) the pressure change increases monotonically to its maximum value, and (4) the derivative of the pressure change approaches zero at large times. These four conditions can be expressed mathematically as lim pφ = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.4) t →0

lim pφ = Cφ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.5)

t →∞

dpφ dt

> 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.6)

and lim

dpφ

t →∞

dt

= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.7)

To satisfy the criteria given in Eqs. 9.4 through 9.7, Fair chose an exponential function of the form pφ = Cφ [1 − exp (− t α )], . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.8) where pf is the time-dependent pressure change caused by phase segregation, Cf is the maximum pressure change, and a is the characteristic bubble rise time. Fair chose Eq. 9.8 because it had a simple Laplace domain representation and agreed with the limited measured lab data available on phase segregation. Fair showed that the pressure change function given in Eq. 9.8 caused the early pressure response to exhibit an apparent WBS coefficient lower than the true WBS coefficient, given in dimensionless form by 1 1 Cφ D , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.9) = + CaD C D α D where Cf D is the dimensionless maximum pressure change, defined as Cφ D =

kh Cφ, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.10) 141.2qBµ

aD is the dimensionless characteristic rise time, defined as

αD =

0.0002637 k α , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.11) φµct rw 2

and CaD is the dimensionless apparent WBS coefficient, defined as CaD =

0.894 Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.12) φ hct rw 2

In Fair’s phase segregation model, the pressure response will initially follow a unit-slope line, given by pwD =

tD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.13) CaD

If phase segregation effects end quickly enough (i.e., the bubble rise time is much shorter than the duration of conventional WBS), the pressure response will show a shift to a larger WBS coefficient, following a second unitslope line given by the conventional WBS coefficient as pwD =

tD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.14) CD

Wellbore Phenomena 235

Once both phase segregation and conventional WBS effects have ended, the pressure response will approach that for a well with no WBS. Fig. 9.4 shows several pressure responses predicted by the Fair phase segregation/variable-WBS model, along with the corresponding constant-WBS pressure response. In Fig. 9.4a, phase segregation effects end very early and dissipate before the end of the WBS-dominated period. The pressure response shows two unit-slope lines, the first corresponding to the apparent WBS coefficient Ca from Fair’s model, the second corresponding to the true WBS coefficient C calculated from the wellbore geometry. In Fig. 9.4b, phase segregation effects end during the transition out of WBS. The pressure response again shows a unit-slope line corresponding to the apparent WBS coefficient Ca. Figs. 9.4c and 9.4d show the effect of phase segregation lasting long enough and having a large enough change in pressure, to cause the WBS to rise above the pressure trend that would have occurred in the absence of phase segregation and conventional WBS. This pressure hump causes a temporary reversal of flow into the formation. Eventually, the phase segregation effects dissipate, and the pressure response returns to the pressure trend for the constant WBS case. If the test lasts long enough for the horizontal portion of the derivative to appear after WBS and phase segregation effects have ended, the pressure data may be analyzed by ignoring that portion of the data affected by wellbore phenomena. If the horizontal derivative is not present, it may be possible to history match the pressure data using a variable-WBS model, as in Fig. 9.5. However, it will be more difficult to get a unique answer, and the results will be subject to a greater degree of uncertainty. 9.2.4 Hegeman, Hallford, Joseph Variable-WBS Model. Hegeman et al. (1993) introduced a variable-WBS model satisfying the same criteria as Fair’s model, replacing Fair’s exponential function pressure function with an error function,

10

10

1

1

pwD

(c) 100

pwD

(a) 100

0.1

0.01 0.001

0.1

0.01

0.1

1

10

100

1000

tD /CD

0.01 0.001

10

10 pwD

(d) 100

pwD

(b) 100

1

0.1

1 tD /CD

10

100

1000

0.01

0.1

1 tD /CD

10

100

1000

1

0.1

0.1

0.01 0.001

0.01

0.01

0.1

1 tD /CD

10

100

1000

0.01 0.001

Fig. 9.4—Pressure responses predicted by Fair’s phase segregation/variable wellbore-storage model, increasing WBS coefficient (a) Fair’s phase segregation model, CDe2s = 1000, CaD /CD = 0.1, aD /CD = 0.01; (b) Fair’s phase segregation model, CDe2s = 1000, CaD /CD = 0.1, aD /CD = 0.1; (c) Fair’s phase segregation model, CDe2s = 1000, CaD /CD = 0.1, aD /CD = 1; (d) Fair’s phase segregation model, CDe2s = 1000, CaD /CD = 0.1, aD /CD = 10.

236 Applied Well Test Interpretation

∆p, ∆t (dp/dt), psi

1000

100

10

1 0.01

0.1

10

1

100

1000

∆te , hr Fig. 9.5—Example data set analyzed using Fair’s variable WBS/phase segregation model, showing pressure hump.

pφ D = Cφ D erf ( t D α D ) ��(9.15) Eq. 9.15 gives a sharper, more abrupt transition out of WBS than that predicted by Fair’s model, Eq. 9.8. Fig. 9.6 shows several pressure responses predicted by the Hegeman-Hallford-Joseph phase segregation/‌variableWBS model, along with the corresponding constant WBS pressure response. 9.2.5 Thermal Effects. Fair (1996) showed that thermal effects could cause either a pressure increase or decrease after shut-in. A pressure increase could be modeled as variable WBS with an apparent increase in the WBS coefficient, using either Fair’s exponential variable-WBS model, or the Hegeman et al. (1993) error function

10

10 pwD

(c) 100

pwD

(a) 100

1

0.1

0.1

0.01 0.001

1

0.01

0.1

1

10

100

0.01 0.01

1000

0.1

1

10 tD /CD

100

1000

10000

1

10 tD /CD

100

1000

10000

tD /CD

10

10

pwD

(d) 100

pwD

(b) 100

1

0.1

0.01 0.001

1

0.1

0.01

0.1

1 tD /CD

10

100

1000

0.01 0.01

0.1

Fig. 9.6—Pressure responses predicted by the Hegeman-Hallford-Joseph phase segregation/variable WBS model, increasing wellbore storage coefficient (a) CDe2s = 1000, CaD /CD = 0.1, aD /CD = 0.01; (b) CDe2s = 1000, CaD /CD = 0.1, aD /CD = 0.1; (c) CDe2s = 1000, CaD /CD = 0.1,aD /CD = 1; (d) CDe2s = 1000, CaD /CD = 0.1, aD /CD = 10.

Wellbore Phenomena 237

variable-WBS model. A pressure decrease caused by thermal effects could be modeled as variable WBS with an apparent decrease in the WBS coefficient. 9.2.6 Spivey Variable-WBS Models. Spivey and Lee (1999) proposed two dual-compartment WBS models. These two models both give similar pressure responses to those predicted by the Fair (1981) and Hegeman et al. (1993) variable-WBS models, although the neither of the Spivey and Lee models exhibit the classic pressure hump caused by phase segregation. 9.2.7 Well With a Gas Column Above a Rising Gas-Liquid Interface. Consider a pumping well with a rising liquid level and a packer as shown in Fig. 9.7. As the liquid level rises, it compresses the gas. The effective WBS coefficient can be calculated as follows. The WBS coefficient C in bbl/psi can be interpreted as the reservoir volume of fluid in barrels that must be added to the wellbore to cause the wellbore pressure to increase by 1 psi. Thus, the pressure change in psi caused by the addition of 1 barrel of fluid is 1 C. As we saw in Chapter 2, the WBS coefficient for a well filled with a single-phase gas is given by C1φ g = cwbVwb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.16) Influx of one barrel of liquid will cause the pressure in the gas column to increase by an amount ∆pgas =

1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.17) = C1φ g cgwbVgwb

Similarly, the WBS coefficient for a changing liquid level is given by Ccll = 25.65

Awb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.18) ρ wb

Influx of one barrel of liquid will cause the pressure exerted by the hydrostatic head of the liquid column to increase by an amount

Fig. 9.7—Well with a gas column above a rising gas-liquid interface.

238 Applied Well Test Interpretation

∆phydrostatic =

1 ρlwb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.19) = Ccll 25.65 Alwb

The total pressure change caused by influx of one barrel of liquid for the well shown in Fig. 9.7 is thus ∆ptot = ∆phydrostatic + ∆pgas =

1 1 + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.20) Ccll C1φ g

The WBS coefficient caused by the combined effect of the rising liquid level and compression of the gas column is therefore Ctot =

1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.21) = 1 1 ∆ptot + Ccll C1φ g

As the liquid level rises, the volume of the gas column decreases, the gas pressure increases, and the gas compressibility decreases. Thus, the WBS coefficient given by Eq. 9.21 is pressure dependent. Example [adapted from Example 11.1 in Earlougher (1977)]. Consider a well with 2-in. tubing with a gas column above a rising water level. Initially the gas column is 116 ft high, and the pressure in the gas is 457 psi. The capacity of the 2-in. tubing is 0.004 bbl/ft. Assuming the ideal gas law is valid, the storage coefficient for the single-phase gas alone would be C1φ g = cwbVwb  1  =  (116) ( 0.004 )  457  = 1.015 × 10 −3 bbl psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.22) The storage coefficient for the rising water level alone would be Ccll = =

Awb ρ wb 0.004 bbl ft 0.433 psi ft

= 9.24 × 10 −3 bbbl psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.23) Thus, the combined WBS coefficient would be Ctot =

1 1 = = 9.15 × 10 −4 bbl psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.24) 1 1 1 1 + + Ccll C1φ g 1.015 × 10 −3 9.24 × 10 −3

When the water level has risen 25 ft, reducing the height of the gas column to 91 ft, the pressure in the gas would be p2 = p1

V1 (116) = 582.5 psi, = ( 457) V2 ( 91)

the storage coefficient for the gas zone would be C1φ g = cwbVwb  1  =  ( 91) ( 0.004 )  582.5  = 6.25 × 10 −4 bbl psi,

Wellbore Phenomena 239

and the combined WBS coefficient would be Ctot =

1 1 = = 5.885 × 10 −4 bbl psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.25) 1 1 1 1 + + Ccll C1φ g 6.25 × 10 −4 9.24 × 10 −3

Note that a 22% decrease in the height of the gas column causes a 36% decrease in the WBS coefficient. 9.3 Cleanup A new well often cleans up during early production, as completion fluids and/or mud filtrate is produced from the near-wellbore region. This cleanup phenomenon may result in a decreasing, time-dependent skin factor. 9.3.1 Larsen and Kviljo Cleanup Model. Larsen and Kviljo (1990) proposed a simple model to describe a decreasing time-dependent skin factor during cleanup. Their cleanup model assumed a hyperbolic equation for skin factor, of the form s=

a + c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.26) b+t

The skin factor calculated from Eq. 9.26 decreases with time, asymptotically approaching a constant value c. Fig. 9.8 shows a well exhibiting time-dependent skin factor during cleanup.

1× 106 0

Gas Rate, Sm3/ D

2× 106

9.3.2 Variable WBS Caused by Thermal Effects. Although the pressure response shown in Fig. 9.9 is very similar to that in Fig. 9.8, the cause of the pressure response appears to be variable WBS caused by changing temperature during drawdown, rather than decreasing skin factor caused by cleanup. In Fig. 9.8, the cleanup effect is smaller during the second flow period than that during the first flow period, as would be expected for cleanup. The corresponding effect in Fig. 9.9 increases with each successive flow period.

10

20

30

40

50

40

50

Bottomhole Pressure, KPa 35000 40000 45000 50000

Time, hr

10

20

30 Time, hr

Fig. 9.8—Pressure response for Well A, exhibiting time-dependent skin factor during cleanup (Larsen and Kviljo 1990).

240 Applied Well Test Interpretation 11000

Pressure, psi

10500

10000

9500

9000

8500

0

10

20

30 Time, hr

40

50

60

Fig. 9.9—Pressure response for well with variable WBS caused by thermal effects during drawdown.

Figs. 9.10 and 9.11 show a match of the final flow period from Fig. 9.9 using Fair’s variable-WBS model, on log-log and Cartesian scales, respectively. Although the pressure response looks like the pressure hump caused by phase segregation (and is described by the same mathematical model), this is for a flow period, whereas phase redistribution occurs only for buildup periods. 9.4 Movement of Gas-Liquid Interface in Wellbore Movement of a gas-liquid interface in the wellbore can cause a time-dependent pressure offset between the pressure at the perforations (which we are trying to measure) and the pressure actually recorded by the gauges (Mattar and Santo 1992; Mattar 1996). Although this is a wellbore phenomenon, it may occur at any time during a buildup test. Mattar (1996) estimates that approximately three-fourths of tests on wells flowing a mixture of gas and liquid exhibit this behavior. 9.4.1 Movement of a Gas-Liquid Interface. Fig. 9.12 shows a schematic of movement of a gas-liquid interface past the perforations and the gauges. In Fig. 9.12a, the gas-liquid interface is below the perforations. The gauges record the pressure at the perforations pw (the gas density is assumed to be negligible in this illustration).

10000

∆p, t(dp/dt), psi

1000

100

10

1

0.1 0.0001

0.001

0.01

0.1 ∆te , hr

1

10

100

Fig. 9.10—Pressure response for final flow period, well with variable WBS caused by thermal effects during drawdown, matched with Fair’s VWBS model.

Wellbore Phenomena 241 11000

Obs Fit

Pressure, psi

10500

10000

9500

9000

8500 48

50

52

54

56

58

Time, hr Fig. 9.11—Pressure response for final flow period, well with variable WBS caused by thermal effects during drawdown, matched with Fair’s VWBS model.

pg 2 = pw

pg 2 = pw – ρgh(t)

pg 2 = pw – ρgh(t)

pg 2 = pw – ρgh2 h(t) h()t hh2

pg 1 = pw pw (t)

A

pg 1 = pw – ρgh(t) h()t h(t)

B

h()t h(t)

pg 1 = pw – ρgh1

pg 1 = pw – ρgh1

pw (t)

pw (t)

hh11

pw (t)

C

D

Fig. 9.12—Movement of a gas-liquid interface past the perforations and the gauges.

pg1 = pg 2 = pw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.27) In Fig. 9.12b, the liquid interface has risen above the perforations. As the interface continues to rise, both pressure gauges now record the pressure at the perforations, minus a time-dependent static head of liquid, pg1 = pg 2 = pw − ρ gh ( t ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.28) In Fig. 9.12c, the liquid interface has risen above the lower gauge. The lower gauge records the pressure at the perforations, minus a time-independent static head of liquid. The liquid interface is still moving between the perforations and the upper gauge, so the upper records the pressure at the perforations minus a time-dependent static head of liquid. pg1 = pw − ρ gh1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.29) pg 2 = pw − ρ gh ( t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.30)

242 Applied Well Test Interpretation

Finally, in Fig. 9.12d, the liquid interface has risen above both gauges. Although the interface continues to rise, both pressure gauges now record the pressure at the perforations, minus a time-independent static head of liquid. pg1 = pw − ρ gh1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.31) pg 2 = pw − ρ gh2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.32) Fig. 9.13 shows the resulting pressure response for a liquid level rising below the completion (A), between the completion and the lower gauge (B), between the lower and upper gauges (C) and finally above the upper gauge (D). The liquid level may move downward as well as upward, as show in Fig. 9.14. Often, both effects will occur during a single test, as shown in the following field example. 9.4.2 Field Example. Fig. 9.15 shows a field case with probable movement of a gas-liquid interface downward past the gauge and perforations. The liquid interface appears to move faster as it approaches the perforations, as suggested by the increase in slope for a couple of hours just before the interface passes the perforations around 48 hours.

pw (t)

pg 1(t) pg 2(t) p

A

B

C

D

t Fig. 9.13—Pressure response recorded during upward movement of a gas-liquid interface past the perforations and the gauges.

p pw (t) pg 1(t) pg 2(t)

D

C

B

A

t Fig. 9.14—Pressure response recorded during downward movement of a gas-liquid interface past the gauges and the perforations. Adapted from Mattar (1996).

Wellbore Phenomena 243 2700

p, psia

2690

2680

2670

2660

2650 20

30

40

50

60

70

∆t, hr Fig. 9.15—Field example showing probable movement of a gas-liquid interface downward past the gauge and perforations.

The movement of a gas-liquid interface between the gauge and the perforations distorts the shape of the pressure derivative curve, as shown in Figs. 9.16 and 9.17. The derivative curve shown in Fig. 9.16 shows two periods where a gas-liquid interface distorts the pressure response. During the first period, A, the interface appears to be rising between the perforations and the gauge. The derivative calculated from the recorded pressure data shows a downward spike during this period. During the second period, B, (also shown in Fig. 9.15) the interface is falling between the gauge and the perforations. The derivative shows an upward spike during this period. Gas well buildup tests will often show pairs of spikes in the derivative, with one downward spike and one upward spike, as in Fig. 9.16. When you encounter such a test, look closely at the pressure data to identify possible liquid movement between the perforations and the gauge. It may be helpful to set the smoothing parameter to zero (or its lowest setting), as shown in Fig. 9.17, to identify points where the derivative is discontinuous. Although it may be tempting to try to “smooth out” the derivative by increasing the value of the smoothing parameter L, this is a bad practice. Smoothing should only be used for noisy data, that is, data subject to random fluctuations having both higher frequency and smaller amplitude than the reservoir pressure response. Movement of a gas-liquid interface between the perforations and the gauge causes a distortion of the pressure response that is not random, is not high frequency data, and may be comparable in magnitude to, or even larger than, the reservoir effects being measured.

∆pa , ∆ta (dpa /dta ), psi

1000

100

B A

10

1

0.1 0.001

0.01

0.1

1 ∆ta , hr

10

100

1000

Fig. 9.16—Field example of gas well with liquid level moving upward between perforations and gauge (a), then back downward (b), smoothing parameter L = 0.1.

244 Applied Well Test Interpretation

∆pa , ∆ta (dpa /dta ), psi

1000

100

10

1

0.1 0.001

0.01

0.1

1 ∆ta , hr

10

100

1000

Fig. 9.17—Field example of gas well with liquid level moving upward between perforations and gauge, then back downward, smoothing parameter L = 0.

Smoothing should be used only to reduce the effects of random, high frequency noise in the derivative. Do not attempt to smooth away effects that are not random! 9.4.3 Prevention and Remediation. Perhaps the only feasible way to prevent (or at least minimize) liquid movement between the gauge and the perforations is to hang the gauge as close as possible to the perforations. If a production log has been run, it might be a good idea to place the gauge assembly at the depth where the maximum gas influx rate was observed. If liquid movement is observed, it may be possible to subtract out the pressure offset before test interpretation. If accurate estimates of the wellbore fluid densities are available, and the distance between the perforations and the gauges are known, it is possible to calculate the magnitude of the pressure offset caused by liquid movement. The pressure derivative curve should not be affected by the liquid movement when the gas-liquid interface is below the perforations or above the gauge. During the period in which the gas-liquid interface is moving between the perforations and the gauge, the derivative curve does not represent reservoir behavior and should be ignored. 9.5 Activity During Operations Many activities during test operations may cause changes in wellbore pressure recorded by the pressure gauges during a test. This section gives examples of just a few of the many activities that may affect the pressure response during a test. 9.5.1 Abrupt Leak. Fig. 9.18 shows the pressure response for an abrupt leak into a lower pressure volume. Almost the same effect may be caused by a valve being opened inadvertently during a buildup test. At the onset of the leak, the pressure drops quickly as fluid flows flow from the wellbore into the lower pressure volume. Eventually, the pressure reaches equilibrium, and the pressure and derivative response return to the original trend. The time required for the pressure response to return to the original trend depends on the volume of the lower pressure region. Wellhead casing and tubing pressure data may help to confirm the presence of a leak if a test shows a pressure response similar to Fig. 9.18. Stegemeier and Matthews (1958) discuss the pressure response for a well producing from two isolated zones, where the isolation tool failed, allowing pressure communication between the lower tested zone and the upper high pressure zone as shown in Fig. 9.19. 9.5.2 Valve Opened/Choke Changed. Often, a valve may be inadvertently opened or closed momentarily. Or, the choke size may be changed momentarily with no record in the testing report. These events are easily to identify on the pressure vs. time graph (see Figs. 9.20 through 9.22 for examples). 9.5.3 Setting and Unsetting a Bridge Plug. Fig. 9.23 shows the pressure responses recorded by three gauges in a well with multiple zones producing commingled (Frantz et al. 2001). The well was produced from all three zones

Wellbore Phenomena 245

pD , tD (dpD /dtD )

100

Leak begins

10

1

0.1 0.1

1

100

10

10000

1000

tD /CD

Fig. 9.18—Abrupt leak into a lower pressure volume distorts the buildup pressure response. 3450 Upper zone pressure 3400

Pressure, psi

3350 Tool failure 3300

Lower zone buildup

3250 3200 3150 3100 10000

1000

100

10

1

(tp + ∆t)/∆t

Fig. 9.19—Abrupt leak into a higher pressure volume distorts the buildup pressure response (after Stegemeier and Matthews 1958).

11000

Pressure, psi

10000

Choke change?

9000

8000 Valve opened? 7000

0

10

20

30 40 Time, hr

50

60

70

Fig. 9.20—Example of suspected choke change and temporary leak or opened valve.

246 Applied Well Test Interpretation 9800

Pressure, psi

9700

9600

9500

9400

9300

6

6.5

7

7.5 Time, hr

8

8.5

9

Fig. 9.21—Example of suspected choke change, detail. 11000

Pressure, psi

10000

9000

8000

7000 43.7

43.8

43.9

44

Time, hr Fig. 9.22—Example of suspected temporary leak or opened valve, detail.

simultaneously for several hours, then retrievable bridge plugs were set to isolate each zone. Memory gauges were used to record the pressure buildup for each zone independently. For this particular well, the cement bond was inadequate to provide total zonal isolation. As the bridge plugs above the upper two zones were set or unset, the pressures recorded for the lower zones showed communication with the upper zones. Note the similarity of the shape of the pressure response measured in the Cubero and Paquate zones when the Two Wells bridge plug is unset at 3.3 days in Fig. 9.23, and the pressure response caused by the leak in Fig. 9.18. 9.5.4 Moving the Gauge. Occasionally, the pressure gauge may be lowered or raised during a test. The resulting change in pressure may be quite puzzling if the analyst is not aware of the fact that the gauge was moved. * Fig. 9.18 shows the pressure response for two flow and buildup periods on an oil well in a high permeability reservoir. There is an abrupt decrease in pressure during the second flow period between 80 and 90 hours, immediately before the second buildup. The pressure drops from approximately 3813 psia to 3588 psia, a drop of 225 psi. The pressure change occurred when the gauge was raised 200 meters to obtain data needed to calculate the flowing pressure gradient (Fig. 9.24). * One engineer, who shall remain nameless, admitted that he had, on occasion, gotten bored while conducting a well test, and had raised and lowered the gauge more or less randomly.

Wellbore Phenomena 247 3,500 Cubero Paquate Two Wells

Bottomhole Pressure, psia

3,000

2,500

2,000

1,500 Communication observed when setting and unsetting retrievable bridge plugs

1,000

500

0 0.0

0.5

1.0

1.5

2.0 2.5 Time, days

3.0

3.5

4.0

4.5

Fig. 9.23—Pressure responses for three zones tested simultaneously with inadequate zonal isolation (Frantz et al. 2001).

3950.0 CASE B-1 3925.0 3900.0 3875.0 3850.0

Pressure, psig

3825.0 3800.0 3775.0 3750.0 3725.0 3700.0 3675.0 3650.0 3625.0 3500.0 3575.0

0.0

20.0

40.0

60.0

80.0

100.0

120.0

Fig. 9.24—Decrease in pressure caused by raising the gauge 200 meters during the flow period immediately before the second buildup (Cinco-Ley et al. 1985).

248 Applied Well Test Interpretation

9.6 Pressure Oscillations There are a number of phenomena that may cause pressure oscillations during a pressure-transient test. Such phenomena include inertial effects, earth tides, and diurnal fluctuations in wellhead temperature. 9.6.1 Inertial Effects. In high-rate oil wells, the momentum of the fluid moving up the wellbore can cause pressure oscillations when the well is shut in (Fair 1996), a phenomenon commonly referred to as fluid hammer. Pressure oscillations caused by inertial effects have a very short period of oscillation and die out rapidly from frictional dampening. Cinco-Ley et al. (1985) reported several examples of pressure oscillations in high-rate oil wells. In one case, Well B-1 produced an average flow rate of 23,900 STB/D from a high permeability fractured carbonate. The pressure response from the second buildup period on Well B-1 is shown in Fig. 9.25. In this field example, the period of oscillation is approximately 1 minute. 9.6.2 Earth Tides. The gravitational pull of the moon, and to a lesser extent, the sun, causes a slight distortion in the shape of the earth. As the earth rotates, the resulting changes in stress, or earth tides, cause small, regular oscillations in the reservoir pressure (Arditty et al. 1978). The oscillations have a period of 12:25, with an amplitude that varies over a 14-day cycle as the smaller amplitude solar tides alternately reinforce or counteract the effects of the lunar tides. Fig. 9.26 shows earth-tide induced pressure oscillations for a long-term interference test in a heavy oil reservoir, illustrating the variation in amplitude. Although the forces that cause earth tides are the same

CASE B-1

pws , psia

3738

3736

3734

3732 10–3

10–2

10–1

1

∆t tp+ ∆t Fig. 9.25—Pressure oscillations caused by inertial effects in high-rate oil well (Cinco-Ley et al. 1985).

0.15

∆ Pressure, psi

0.1 0.05 0 –0.05 –0.1 –0.15 1000

1100

1200 1300 Time, hr

1400

1500

Fig. 9.26—Earth-tide induced pressure oscillations during long-term interference test.

Wellbore Phenomena 249

that create ocean tides, earth tides occur in both offshore and onshore reservoirs. Earth-tide induced oscillations are very regular, almost sinusoidal, with amplitude from 0.01 to 1 psi. Tidal oscillations can be removed either by filtering in the frequency domain via Fourier transform analysis, or by directly calculating and subtracting the tidal component (Mattar 1996). 9.6.3 Diurnal Temperature Changes. In areas where the wellhead is exposed to daily temperature extremes, wellhead temperature fluctuations may cause irregular pressure oscillations, even with downhole pressure measurement. Fig. 9.27 shows downhole pressure oscillations observed in a buildup test in an oil well in the Sahara desert, where wellhead temperatures varied from 70°F at night to as high as 145°F in the daytime. Although the amplitude of these oscillations is similar to those of earth-tide induced oscillations, they are readily distinguished by their 24-hour periodicity and irregular appearance. 9.7 Gauge Problems There are many ways in which gauge problems can affect the recorded pressure response during a test. This section discusses several gauge-related problems that have been reported with modern electronic gauges. Earlougher (1977) gives a catalog of a number of problems that may be encountered when using mechanical gauges. 9.7.1 Gauge Plugging. Fig. 9.28 shows the pressure responses for two different gauges at the same depth, one inside gauge and one outside gauge (Mattar 1996). The inside gauge pressure response during the final flow period was distorted by hydrates plugging the perforations of the gauge carrier, causing an increasing pressure to be recorded during the flow period, while the outside gauge recorded a decreasing pressure. Two other inside gauges exhibited the same increasing pressure behavior (Mattar and Santo 1992), showing that one cannot rely on “majority rules” to select the best data set to use for analysis. 9.7.2 Thermal Sensitivity. Fig. 9.29 shows a Horner plot for two different gauges in a high permeability gas reservoir (Veneruso et al. 1991). The lower sensitivity (±2 psi) strain gauge used as a backup recorded a noisy, but interpretable, signal, while the data from the higher sensitivity (±0.1 psi) quartz primary gauge was unusable because of distortion of the pressure response from thermal transients during the test. 9.7.3 Gauge Sampling Artifacts. Mattar (1996) reported pressure sampling artifacts caused by a change in sampling frequency, Fig. 9.30, and from alternating temperature/pressure sampling points, Fig. 9.31. The cause of the pressure shift coincident with the change in sampling frequency is not known. The “sawtooth” appearance of the pressure response in Fig. 9.31 might caused by the way pressure is calculated from the raw transducer data. Since the pressure transducers used in electronic gauges are sensitive to temperature as well as pressure, converting the raw data (frequency in the case of a quartz gauge) from the transducer to pressure must take into account the changing temperature as well. If the temperature from the most recent temperature sampling point is used to calculate the pressure, the calculated pressure would drift from

0.3

∆ Pressure, psi

0.2 0.1 0 –0.1 –0.2 –0.3

0

20

40

60

80 100 Time, hr

120

140

160

Fig. 9.27—Irregular oscillations in bottomhole pressure caused by diurnal fluctuations in wellhead temperature.

250 Applied Well Test Interpretation (a) 10400

9600

8800

8000 Note increasing pressure 7200

6400 0

(b)

10

20

30

40

50

60

10400

Pressure, kPa

9600 8800 8000 Note decreasing pressure

7200 6400

0

10

20

30 Time, hr

40

50

60

Fig. 9.28—Different gauges show different pressure responses when one is plugged by hydrates (Mattar 1996).

the true pressure as the temperature changes. The result would be a jump in pressure each time the temperature is sampled. 9.7.4 Unidentified Pressure Phenomena. It is not always possible to identify the source of changes in wellbore pressure. If data from multiple gauges are available, the possibility of gauge problems can be eliminated if the same pressure response is observed in multiple gauges, especially if the gauges are of different types. Fig. 9.32 shows an unidentified pressure response recorded during a test, where the same phenomenon was observed in both strain and quartz gauges. 9.8 Data Processing Errors and Artifacts Although not strictly speaking wellbore phenomena, the choice of time zero and the corresponding final flowing bottomhole pressure before shut-in have effects similar to those caused by true wellbore phenomena. Because of compression of the time and pressure scales on the log-log plot, an error in estimating either pwf or time zero will cause the greatest distortion of the pressure change and pressure derivative curves at early times. 9.8.1 Error in Identifying Final Flowing Bottomhole Pressure. An error in identifying the correct flowing bottomhole pressure will have little or no effect on the derivative curve. (Theoretically, there should be no effect on the derivative. However, different numerical differentiation schemes may cause small differences in the calculated derivative.) If the value of pwf is too low, the pressure change will be too large at the beginning of the test. Instead of falling on a unit-slope line during the wellbore-storage-dominated period, the pressure change curve will asymptotically approach a unit-slope line, as in Fig. 9.33a.

Wellbore Phenomena 251 Horner Plot

39000

p, psi

38850

3870.0

3855.0

3840.0

3825.0

Pressure, SG Pressure, OG 100

101

102 (tp+ ∆t) /∆t

103

Fig. 9.29—Lower resolution gauge gives noisy, but interpretable signal, while higher resolution gauge gives uninterpretable signal because of thermal transients (Veneruso et al. 1991).

Pressure, kPag

20900

Change in sampling frequency

20780 9.8

10.2

10.6

11

11.4

Time, hr Fig. 9.30—Apparent shift in pressure caused by a change in sampling frequency (Mattar 1996).

On the other hand, if pwf is too high, the pressure change will be too small at the beginning of the test. The pressure curve will initially have a slope greater than one and approach the unit-slope line from below, as in Fig. 9.33b. If the selected value for pwf is higher than the first few pressure data points in the buildup, the corresponding data points for the pressure change curve will be missing from the log-log plot. Special care should be used in determining the final flowing bottomhole pressure pwf . An error in estimating pwf will cause a corresponding error in the additional pressure drop because of skin Dps as well as in the skin factor s. 9.8.2 Error in Identifying Shut-in Time. The effects of incorrectly identifying the shut-in time are similar to those from incorrect identification of the final flowing pressure before shut-in but distinguishable by the fact that both the pressure change and the pressure derivative curve are affected, as shown in Fig. 9.34. The effect caused by time zero being too early, Fig. 9.34a, is very similar to the effect of pwf being too high, Fig. 9.33b. Similarly, the effect of time zero being too late, Fig. 9.34b, is very similar to that caused by pwf being too low, Fig. 9.33a.

252 Applied Well Test Interpretation

Pressure, kPa

8760

Temperature sampling points 8660 3.2

3.3

3.4 3.5 Time, hr

3.6

3.7

Fig. 9.31—Pressure artifacts caused by temperature sampling (Mattar 1996).

Pressure, kPa

8320

Quartz gauge 8280

Strain gauge

8240

8200

0

100

200

300 Time, hr

400

500

600

Fig. 9.32—Unidentified pressure phenomenon confirmed by two different types of gauges (Mattar 1996).

9.8.3 Determining Time and Flowing Pressure at Beginning of Buildup. Fig. 9.35 shows a field example of a buildup test on a deep, high pressure gas well. The well was shut in at the surface. Pressure data were recorded downhole, with a 5-second sampling interval. It took approximately 25 seconds for the valve to close fully for the buildup. In this situation, where should you choose the time and pressure for the beginning of shut-in? Point A in Fig. 9.35 shows the last measured pressure before the valve began to close; Point B shows the first measured pressure after the gauge is fully closed. Point C is the point of intersection between two straight lines one drawn through the last few data points when the valve was fully open and the other drawn through the first few data points after the valve was fully closed. Which of these three points, A, B, or C, should be chosen for analysis? If Point A is chosen as the moment of shut-in (Fig. 9.36a), the early time data do not follow a unit-slope line. Instead, the early data show a slope steeper than a unit-slope line, similar to the pressure response caused by a decreasing wellbore-storage coefficient. If Point B is chosen as the moment of shut-in (Fig. 9.36b), the early data follow a unit-slope line almost perfectly. However, note that the pressure rises roughly 350 psi between Points A and B. If the pressure at Point B is taken as pwf, the calculated pressure drop because of skin will be too low by an amount equal to the change in pressure between Point A and Point B. Analysis of the resulting data would give the correct WBS coefficient, but the skin factor would be too low. We recommend using Point C as the beginning of shut-in (Fig. 9.36c). Although the early data do not fall on a unit-slope line until the third point, the data are not distorted as much as in Fig. 9.36a, and the

Wellbore Phenomena 253 (a)

10000

∆p, t (dp/dt ), psi

1000

100

10

1 0.001

(b)

0.01

0.1

1 ∆t, hr

10

100

1000

0.01

0.1

1 ∆t, hr

10

100

1000

10000

∆p, t (dp/dt ), psi

1000

100

10

1 0.001

Fig. 9.33—Effect of using the wrong final flowing bottomhole pressure before shut in on the log-log plot: (a) pwf too low; (b) pwf too high.

calculated skin factor and additional pressure drop caused by skin will be more accurate than if Point B is used to determine pwf. 9.8.4 Incorrect Producing Time. One common artifact that shares some of the characteristics of wellbore phenomena is the use of an incorrect producing time when calculating the equivalent-time function. Because the equivalent-time function compresses the time scale of the buildup, a short producing time (often the time the gauge was on bottom before the well was brought back on production) causes the pressure and derivative to be distorted. Figs. 8.42 and 8.43 show a field example where the data were initially input using a short production period. 9.8.5 Gauge Precision Artifacts. Often, the combination of low gauge precision and small pressure changes may cause the calculated pressure derivative to exhibit a banded or striped appearance, as in Fig. 9.37. This effect may be avoided by using high resolution gauges when designing a test. If observed during a test, it may be possible to remove the banding by changing the method used to calculate the derivative or by increasing the value of the smoothing parameter. 9.9 Distinguishing Wellbore Phenomena From Reservoir Phenomena One of the more challenging tasks the engineer faces is distinguishing wellbore phenomena from reservoir phenomena. Often, wellbore phenomena have caused significant distortion of the shape of the pressure-transient response observed during a test, yet have negligible effect on well performance. However, these phenomena must be identified to determine whether and how to correct for them during analysis. 9.9.1 Abrupt Change in Pressure, Pressure Derivative, or Slope of Pressure Derivative. Any abrupt change in the wellbore pressure, the derivative of the pressure, or the slope of the pressure derivative must be caused by

254 Applied Well Test Interpretation (a)

10000

∆p, t (dp/dt ), psi

1000

100

10

1 0.001

(b)

0.01

0.1

1 ∆t, hr

10

100

1000

0.01

0.1

1 ∆t, hr

10

100

1000

10000

∆p, t (dp/dt ), psi

1000

100

10

1 0.001

Fig. 9.34—Effect of using the wrong time zero on the log-log plot: (a) t0 too early; (b) t0 too late.

10500 10000

Pressure, psi

9500 9000 8500 8000

A

7500 B

7000 6500

C 0

0.01

0.02 0.03 Elapsed Time, hr

0.04

0.05

Fig. 9.35—What is the right “moment of shut-in” when valve closes slowly?

some wellbore phenomenon. Because of the nature of the diffusivity equation, the pressure transient caused by a sudden disturbance in pressure at some point in the reservoir will spread out (diffuse) as the transient moves away from its source. This characteristic appears to be true for fluid flow in porous media in general, even when rock and fluid properties are pressure dependent and multiphase flow is occurring.

Wellbore Phenomena 255 10000

∆pa , ∆ta (dpa /d∆ta ), psi

(a)

1000

100

10

1 0.0001

0.001

0.01

0.1

1

10

0.1

1

10

0.1

1

10

∆ta , hr

10000

∆pa , ∆ta (dpa /d∆ta ), psi

(b)

1000

100

10

1 0.0001

0.001

0.01 ∆ta , hr

10000

∆pa , ∆ta (dpa /d∆ta ), psi

(c)

1000

100

10

1 0.0001

0.001

0.01 ∆ta , hr

Fig. 9.36—Choice of “moment of shut-in” affects the shape of the early pressure response: (a) Point A chosen as moment of shut-in; (b) Point B chosen as moment of shut-in; (c) Point C chosen as moment of shut-in.

256 Applied Well Test Interpretation

∆p, ∆t(dp/d∆t), psi

1000

100

10

1 0.0001

0.001

0.01

0.1

1

10

∆t, hr Fig. 9.37—Banding in the pressure derivative is an artifact caused by low gauge precision.

Any abrupt change in the wellbore pressure, the pressure derivative, or the slope of the pressure derivative must be caused by some wellbore phenomenon. 9.9.2 Timing and Duration of Pressure Disturbance. Any pressure response caused by reservoir boundaries, by rate changes before buildup, or by production or injection from offset wells will be smooth and have smooth derivatives. However, wellbore phenomena may or may not be smooth. As a rule of thumb, wellbore phenomena occur on a much shorter time scale, seconds to a few hours, than reservoir phenomena, which may occur over a time scale of hours to days. A smoothly changing pressure response occurring at least one-half log cycle after the end of obvious WBS effects is probably a reservoir phenomenon. Because of compression of the time scale on a log-log graph, the short duration for wellbore effects will appear as an abrupt change in pressure late in a test, whereas the same phenomenon occurring near the beginning of the test might be quite smooth. 9.9.3 Primary Pressure Derivative. According to Mattar and Zaoral (1992), the primary pressure derivative will always be monotonically non-increasing. This is consistent with work by Coats et al. (1964) showing that derivatives of solutions to the diffusivity equation alternate in sign. For constant rate production from a reservoir with uniform initial pressure and any combination of no-flow and constant pressure outer boundaries, the following criteria must hold: pwD > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.33) d 2 k − 1 pwD ≥ 0, k = 1,2,3, …. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.34) dt D 2 k − 1 d 2 k pwD ≤ 0, k = 1,2,3, … . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9.35) dt D 2 k However, Coats’ conclusions are based on the properties of the diffusivity equation, which assumes constant rock and fluid properties. Thus, Eqs. 9.33 through 9.35 may not hold if rock and/or fluid properties are functions of pressure. For example, consider a buildup on an oil well with a free gas saturation in an annular region around the wellbore, but the bulk of the reservoir is above the bubble point pressure. When the wellbore pressure rises above the bubble point pressure, gas will be forced back into solution within the annular region, causing a significant reduction in total compressibility. This reduction in compressibility would cause the derivative to be steeper than unit-slope, just as a decreasing wellbore compressibility would cause the derivative to be steeper than unit-slope during WBS. Fig. 9.38 shows the pressure response for a well in a volatile oil reservoir where the pressure passed through the original bubblepoint pressure during the buildup.

Wellbore Phenomena 257 10000

∆p, t dp /dt, psi

1000

100 Steeper than unit slope caused by decreasing total compressibility

10

1 0.001

0.01

0.1

1 ∆te, hr

10

100

1000

Fig. 9.38—Derivative showing greater-than-unit slope caused by decreasing reservoir compressibility as gas goes back into solution.

9.9.4 Duration of Transition. Because of the nature of the diffusivity equation, no reservoir phenomenon will exhibit an instantaneous transition from one type of flow to another. Among the sharpest transitions between flow regimes are (1) infinite-acting radial flow to pseudosteady-state flow for a well in a closed circular reservoir, (2) infinite-acting radial flow to linear flow for a well centered in an infinite channel, and (3) linear flow to pseudosteady-state flow for a well centered in a long rectangular reservoir. As discussed in Chapter 7, using a 5% deviation criterion, the transition from infinite-acting radial flow to pseudosteady-state flow takes slightly less than one-quarter log cycle. Similarly, the transition from infinite-acting radial flow to channel linear flow takes slightly more than one-eighth log cycle. 9.10 Summary One of the more common mistakes in pressure-transient interpretation is neglecting to determine the correct flowing bottomhole pressure for buildup analysis. The skin factor will be wrong if the flowing bottomhole pressure is not correct. The analyst should always consider whether any of the pressure changes observed during a test will cause a difference in pressure between flow and shut-in. Most of the phenomena discussed in this chapter cause only a temporary distortion in the pressure response during the test. If the test lasts long enough, the pressure response will eventually return to the trend that it would have following if the phenomenon had not occurred. However, two situations can cause a permanent shift in the pressure between the flow period and the buildup: a change in fluid density between the tested zone and the gauge (e.g., movement of a liquid interface) or a change in position of the gauge. If the difference in pressure caused by either of these two situations is not taken into account during analysis, the calculated additional pressure drop because of skin and the resulting skin factor will incorrectly include the permanent change in pressure between flow and buildup. Key points discussed in this chapter include the following: 1. Wellbore phenomena almost always have a negligible effect on well performance but must be identified and accounted for during pressure-transient analysis. 2. A variety of phenomena can cause varying wellbore storage, including phase segregation and temperature changes during a test. 3. Activities during testing operations may affect the pressure response observed during a test. 4. Movement of a gas-liquid interface between the perforations, and the gauge will cause a time-dependent pressure difference between the pressure at the sandface and the pressure recorded by the gauge. 5. Oscillations in pressure may occur because of momentum effects, earth tides, and diurnal changes in wellhead temperature. Momentum effects have a very short period (a minute or less), occur only in high-rate wells, and decay very quickly. Earth tides cause regular, small amplitude oscillations with a 12:25 period. Diurnal temperature changes cause irregular oscillations with a 24:00 period. 6. A variety of gauge-related problems may occur during a test. These must be identified before attempting further analysis. Multiple gauges of different types are recommended to improve the chance of getting some interpretable data.

258 Applied Well Test Interpretation

7. An error in determining the flowing bottomhole pressure before shut-in will cause a corresponding error in the additional pressure drop because of skin, and thus in the skin factor itself. 8. The pressure responses caused by reservoir phenomena are always smooth. Any abrupt change in the pressure, pressure derivative, or slope of the pressure derivative must be caused by wellbore phenomena. 9. Wellbore phenomena may affect the pressure response are not limited to the wellbore storage period. They may occur at any time during a test. Nomenclature Awb = wellbore area, ft2 c = compressibility, psi–1 C = WBS coefficient, bbl/psi h = formation thickness, ft h = height of column, ft k = index k = permeability, cp L = smoothing parameter, dimensionless p = pressure, psi q = production rate, STB/D rw = wellbore radius, ft s = skin factor t = time, hour V = volume fraction, dimensionless Vwb = wellbore volume, bbl a = characteristic bubble rise time, hour f = porosity, fraction r = density, lbm/ft3 m = viscosity, cp Subscripts 1fg = single-phase gas a = apparent B = bottom cll = changing liquid level D = dimensionless g = gas g = gauge G = gas l = liquid L = liquid t = total T = top w = wellbore conditions wb = wellbore conditions f = phase segregation

Chapter 10

Near-Wellbore Phenomena “It ain’t the things you don’t know that hurt you; it’s the things you do know that ain’t so.” —Anonymous (variously attributed to Mark Twain, Satchel Paige, and Will Rogers) 10.1 Introduction In contrast to wellbore phenomena, which rarely affect the long-term performance of a well, near-wellbore phenomena almost always affect a well’s long-term performance. Each of the phenomena discussed in this chapter causes a transient pressure response at early times. This transient period is frequently obscured by wellbore storage, allowing the pressure response to be modeled as a constant skin factor. However, under certain conditions, the transient response may be visible during the transition from wellbore storage to infinite-acting radial flow. Section 10.2 gives a brief review of the finite-thickness skin factor model originally proposed by Hawkins (1956). Sections 10.3 through 10.5 introduce the concept of geometric skin factor and present geometric skin factor models for three common types of completion, cased-hole/perforated completion (Section 10.3), partial penetration/ limited entry completion (Section 10.4), and deviated well completion (Section 10.5). Section 10.6 discusses the factors that contribute to steady-state non-Darcy pressure drop near the wellbore for radial flow and for spherical flow. In both cases, 90% of the non-Darcy pressure drop is shown to occur within a few feet of the wellbore. Section 10.7 presents type curves for a vertical well in an infinite-acting reservoir with constant wellbore storage and non-Darcy skin factor, for both buildup and drawdown. For a buildup test, the derivative is much steeper for non-Darcy skin than for constant skin during the transition from wellbore storage to infinite-acting radial flow. For a drawdown test, the end of wellbore storage is delayed relative to the constant skin case. Section 10.8 shows how to estimate the non-Darcy flow coefficient D from the skin factor estimated from two or more buildups following flow periods at different rates. After completing this chapter, you should be able to 1. Calculate the geometric skin factor for (1) a cased-hole/perforated completion, (2) a partially penetrating/ limited entry completion, and (3) a deviated well completion. 2. Calculate the rate-dependent skin factor for (1) radial flow and (2) spherical flow, for both oil and gas wells. 3. Identify the presence of non-Darcy skin factor in a single buildup test. 4. Estimate the non-Darcy flow coefficient from multiple transient tests run at different rates. 10.2 Finite-Thickness Skin Factor Van Everdingen (1953) introduced the dimensionless skin effect, s, to account for the additional pressure drop caused by near-wellbore damage to the formation. In the van Everdingen model, the additional pressure drop occurs across an infinitesimally thin zone at the sandface. Hawkins (1956) introduced a finite-thickness skin factor model, with a damaged annulus of permeability ka and outer radius ra surrounding the well, as shown in Fig. 2.17. Hawkins showed that the skin factor for this model could be calculated from  r k s =  − 1 ln a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.1)  rw  ka

260 Applied Well Test Interpretation

Unfortunately, the data necessary to calculate skin factor from Eq. 10.1 for a well damaged by mud filtrate invasion or stimulated by an acid treatment are not readily available. The Hawkins damage model is basically the same as the radial composite model discussed in Chapter 7. If the damaged zone is large enough, and wellbore storage small enough, the pressure transient response may show infinite-acting radial flow in the inner zone, followed by a long transition into infinite-acting radial flow in the outer zone, as shown in Figs. 7.30 through 7.33. Chu provides an excellent field example of this type of behavior in Example 4 in his paper (Chu 1996). 10.3 Perforated Completion In solving the diffusivity equation for a vertical well, it was implicitly assumed that the well was completed open hole, and was open to the formation over the entire pay interval. Any deviation from the vertical, fully penetrating well with openhole completion will cause the well to have a different flowing wellbore pressure from that predicted by the ideal model. The additional pressure drop caused by the wellbore and completion geometry is often taken into account by a geometric skin factor term, sometimes referred to as a pseudoskin. In this section, we discuss a model for the geometric skin factor for a perforated completion. The following sections will discuss models for a limited entry or partial penetration completion, and for a deviated well. Many papers have been written on the performance of perforated completions, most of them in the field of production systems, or nodal, analysis. McLeod (1983) proposed a model including damage from drilling, geometric skin from converging flow into the perforations, and damage from perforating, as shown in Fig. 10.1. McLeod wrote the total skin factor as s = s p + sd + sdp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.2) where s is the total skin factor, sp is the geometric skin from converging flow, sd is the damage skin factor from drilling damage, and sdp is the damage from perforating. The geometric skin factor, sp, depends on the perforation tunnel radius and length as well as perforation spacing and phasing. Hong (1975) and Locke (1981) give nomographs for estimating sp. The damage skin factor sd , is given by  r k sd =  − 1 ln d , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.3)  rw  kd where rd is the depth of the drilling damage, rw is the wellbore radius (measured from the center of the wellbore to the outside of the cement), k is the formation permeability, and kd is the permeability of the zone damaged by drilling. The perforation damage skin factor, sdp, is given by  kh   1 1 r   sdp =  −  ln dp , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.4)  nL p   kdp kd  rp rw

CL kdp

rp

Lp

rdp

kR

kd rd Fig. 10.1—Well with a perforated completion (McLeod 1983).

Near-Wellbore Phenomena 261

where kdp is the permeability of the damaged zone around the perforation, n is the total number of perforations, Lp is the length of the perforation tunnel (measured from the outside of the cement to the tip), rdp is the radius of damage around the perforation, and rp is the radius of the perforation tunnel. 10.4 Partial Penetration/Limited Entry Completion Often, for any of a number of reasons, a well may not be completed over the entire interval, as shown in Fig. 10.2. If there is a bottom water drive, the well may be completed at the top of the sand interval to delay water coning. Similarly, the well may be completed near the bottom of the interval to delay gas coning. 10.4.1 Geometric Skin During Infinite-Acting Radial Flow. The pressure response for a well with a partial penetration or limited entry completion may be estimated from the following equation: st =

h s + s p, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.5) hp

where st is the total skin factor, s is the skin factor for a fully penetrating completion, h is the net sand thickness, hp is the height of the completed interval, and sp is a positive geometric skin factor caused by converging flow to the completed interval. The skin factor s may include the effects of drilling damage (Eq. 10.1), perforated completion (Eq. 10.2) or rate-dependent skin factor (Eq. 10.19). The skin factor s may also include a negative skin factor from stimulation, as long as the apparent wellbore radius rwa meets the following condition: rwa = rwe− s < 0.2h p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.6) Papatzacos (1987) presented an approximate equation for estimating the geometric skin factor component sp as follows. First, calculate the dimensionless wellbore radius, rD, the fraction of the total interval completed, hpD, and the fraction of the distance from the top of the completion to the top of the interval, h1D from: 1

r k  2 rD = w  v  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.7) h  kh  h pD = h p h, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.8) and h1D = h1 h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.9) Next, calculate intermediate results A and B: A=

1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.10) h1D + h pD 4

h1

hp h

Fig. 10.2—Well with limited entry completion.

262 Applied Well Test Interpretation

B=

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.11) h1D + 3h pD 4

Finally, calculate the geometric skin factor, sp, caused by partial penetration: 1  h  1   A − 1 2  1 π pD  ln s p =  − 1 ln +   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.12)  h r h h B 2 2 + − 1     pD  D pD pD 

Eq. 10.12 is valid only for those cases where the wellbore radius is much less than the height of the completed interval; according to Papatzacos, the ratio rD h pD should be less than 0.2. Fig. 10.3 shows the geometric skin factor sp for a well completed at the top (or bottom) of the interval for various values of rD and hpD. Note that the geometric skin increases significantly for small values of hpD (i.e., small fraction of the total interval completed) and for small values of rD (i.e., small wellbore radius, large sand thickness, or low vertical/horizontal permeability ratio). The geometric skin factor for a partially penetrating well is always positive. It increases in magnitude as the penetration ratio gets smaller or the formation gets thicker. 10.4.2 Transient Pressure Response for a Partial-Penetration/Limited-Entry Completion. Fig. 10.4 shows the pressure response for a well completed at the top (or bottom) of the interval, for rD = 0.0001. At very early times, the pressure response exhibits radial flow over the height of the completed interval hp, followed by hemispherical flow, then by infinite-acting radial flow through the entire interval, h. Note that fully developed hemispherical flow, identified by the logarithmic derivative having a slope of –1 for at least ¹⁄³ to ½ log cycle, occurs only for hpD ≤ 0.1. 10.5 Deviated Wellbore Cinco et al. (1975) studied the transient pressure behavior for a fully penetrating deviated wellbore in an infinite-acting homogeneous reservoir (Fig. 10.5). They found that the pressure response exhibited three distinct flow periods: (1) an early radial flow period, (2) a transition period, and (3) a pseudoradial flow period. The early radial and pseudoradial flow periods gave a straight line on a semilog graph of pressure vs. time. The line for pseudoradial flow period had the same slope, m, as that for a vertical well, 162.6qBµ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.13) kh

45 40

= rD

35

25

rD

15 10 5

1

20

00

30

0.

sp

m=

=0

.00

3

rD = 0

.01 rD = 0.0 3

0 0.1

1 hpD

rD = 0.1

Fig. 10.3—Geometric skin factor for a partially penetrating well, h1D = 0.

Near-Wellbore Phenomena 263 1E+03 hpD = 0.01 0.025 0.05 0.1

1E+02

0.25

pD

0.01 00 5 0.025 1E+01

0.5 1

0.05 0.1 0.25 0.5

1E+00

hpD = 1

1E-01 1E+03

1E+04

1E+05

1E+06 tD

1E+07

1E+08

1E+09

Fig. 10.4—Pressure response for a partially penetrating well, rD = 0.0001, h1D = 0.

h

qw

Fig. 10.5—Well with deviated wellbore.

while the slope m′ for the early radial flow period was smaller by a factor cosθ w′ : m′ = m cosθ w′ =

162.6qBµ cosθ w′ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.14) kh

where θ w′ is given by   k θ w′ = tan −1  v tan θ w  , degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.15)   kh 10.5.1 Geometric Skin Factor During Infinite-Acting Radial Flow for a Deviated Well. During the pseudoradial flow period, the skin factor is given by* st = s cosθ w′ + sθ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.16) Cinco et al. (1975) inadvertently omit the factor cos q ′w. If the infinitesimal skin factor model is valid, the asymptotic late-time pressure response for a well with skin may be easily obtained from that for a well without skin by replacing rw with rwa. Using the rwa substitution in the drawdown pressure equation for infinite-acting radial flow (Eq. 3.1) and in the definition of hD, noting that, for 1.865 1  θ w′  ≅ 1 − cos θ w′ , the result shown in Eq. 10.16 is obtained. 0 ≤ q ′w ≤ 75°,   ln(10)  56  *

264 Applied Well Test Interpretation

where st is the total skin factor that would be observed during a test, s is the skin factor for a vertical well with the same amount of damage or stimulation, and sq is the negative geometric skin factor caused by the deviated well completion. The geometric skin factor sq can be estimated from θ ′  sθ = −  w   41 

2.06

θ′  − w   56 

1.865

h  log10  D  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.17)  100 

where the dimensionless thickness hD is given by hD =

h rw

kh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.18) kv

Eq. 10.17 is valid for angles q ′w from 0 to 75°. Cinco, Miller, and Ramey do not give a lower limit on hD. The smallest value of hD in their study was 100, but the results appear to be reasonable for hD ≥ 10. Fig. 10.6 shows the geometric skin factor sq for several values of hD, for angles q ′w from 0 to 75°. Note that the geometric skin factor is always negative. It increases in magnitude with increasing hD and increasing angle q ′w. While decreasing kv/kh increases hD (which would give a larger negative skin for fixed q ′w ), decreasing kv/kh also decreases the angle q ′w, so the net effect is a smaller negative skin factor. The geometric skin factor for a deviated well is always negative. It increases in magnitude with the degree of deviation from the vertical and the thickness of the formation. Fig. 10.7 shows the effect of the vertical/horizontal permeability ratio kv/kh on the relationship between q ′w and qw. For small kv/kh, the apparent angle q ′w is much smaller than the actual angle q ′w. This means that in spite of the fact that Eq. 10.17 is valid only for apparent angles q ′w less than 75°, it may be valid for larger actual angles. For example, for kv/kh, equal to 0.1, an actual angle of 85° has an equivalent angle of 75°. 10.5.2 Transient Flow for a Deviated Well. For a highly deviated well with small wellbore-storage coefficient, the shape of the transition from wellbore storage to infinite-acting radial flow may be affected by the geometry of the completion. In this case, the transient pressure response must be considered. Fig. 10.8 shows the pressure and pressure derivative for a fully penetrating deviated well on a log-log scale. Fig. 10.9 shows the pressure response on a semilog scale. At very early times, the pressure response exhibits radial flow over the length of the completed interval h cosθ w′ , followed by a long transition into infinite-acting radial flow over the net sand interval, h.

0 −1 −2

sq

−3 −4 −5 −6 −7

0

10

20

30

40

50

60

70

q ′w, degrees Fig. 10.6—Geometric skin factor for a well with a deviated wellbore.

80

Near-Wellbore Phenomena 265 90

75

q′w, degrees

60

45

30

15

0

0

15

30

45 60 q w, degrees

75

90

Fig. 10.7—Effect of kv /kh on equivalent angle q ′w .

1E+02 q′w = 0°

30°

45°

1E+01

pD

60°

75°

85°

1E+00 60° 1E-01

75° q′w = 85°

1E-02 1E+04

1E+05

1E+06

1E+07 tD

1E+08

1E+09

1E+10

Fig. 10.8—Pressure response for a deviated well, hD = 5000, log-log scale.

10.6 Rate-Dependent Skin Factor Rate-dependent skin factor, often called non-Darcy skin, is commonly assumed to occur only for gas wells (Winestock and Colpitts 1965). However, non-Darcy skin can also occur in oil and water wells when high velocity flow is present (Fetkovich 1973; Jones et al. 1976). The rate-dependent skin factor may be written as s′ = s + Dq , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.19) where s′ is the total skin factor observed at flow rate q, s is the skin factor from Darcy pressure drop, including geometric, damage, and stimulation components, and D is the non-Darcy flow coefficient in D/STB or D/Mscf. 10.6.1 Steady-State Radial Flow. For spherical and radial flow, non-Darcy skin factor is inherently a near-wellbore phenomenon. For steady-state radial flow, if we integrate the pressure drop from the sandface for radial flow, we find pe − pw = − ∫

re

rw

dp dr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.20) dr

266 Applied Well Test Interpretation 14 12 10

pD

8 6 4 2 0 1E+04

1E+05

1E+06

1E+07 tD

1E+08

1E+09

1E+10

Fig. 10.9—Pressure response for a deviated well, hD = 5000, semilog scale. Adapted from Cinco et al. (1975).

For non-Darcy flow, the pressure gradient is proportional to the square of the velocity. The velocity, in turn, is inversely proportional to the radius r. Thus, we have C dr r2 1 1 =C  −   rw re 

pe − pw = − ∫

re

rw

=

C  rw  1 −  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.21) rw  re 

Thus, 90% of the total pressure drop caused by radial non-Darcy flow occurs within a distance re = 10rw. Thus, for a typical vertical well with wellbore radius of 0.25 to 0.5 ft, 90% of the total non-Darcy skin factor occurs within a distance of 2.5 to 5 ft of the well, a depth which is easily reached with even a matrix acid treatment. For radial flow to a vertical well with an openhole completion, 90% of the pressure drop caused by non-Darcy flow occurs within a distance 10rw, or 2½ to 5 ft, from the center of the wellbore. A more detailed treatment shows that the total non-Darcy skin factor for a well producing single-phase oil from an openhole completion is given by D=

1.635 × 10 −16 βρ o, wf Bo, wf kh

µ o, wf h p2 rw

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.22)

The coefficient b is the inertial flow coefficient, a property of the reservoir rock. A number of correlations have been developed to predict b. One popular correlation (Jones 1987) is

β≈

1.88 × 1010 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.23) k 1.47φ 0.53

where b has the units of 1/ft, k is in md, and f is a fraction. Aminian et al. (2008) suggest constructing formation-specific correlations for b, using a relationship of the form a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.24) kb to provide improved estimates of non-Darcy skin from single-rate gas well tests.

β=

Near-Wellbore Phenomena 267

For a gas well producing from an openhole completion, the non-Darcy skin factor may be estimated from D= =

2.7136 × 10 −15 β Mpsc kh µ g. wf Tsc hp2rw 7.86 × 10 −14 βγ g psc kh

µ g. wf Tsc h p2rw

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.25)

Eqs. 10.22 and 10.25 give only rough estimates of the actual non-Darcy skin component. According to Ramey (1965), “comparison of values of non-Darcy flow constants estimated from Eq. 31 [Eq. 10.25] with those obtained from well-test data has indicated estimated values may be in error by as much as 100 percent.” There are at least three factors that may contribute to this discrepancy. First, the velocity coefficient b is rarely known and must be estimated from correlations; the available correlations give only order-of-magnitude estimates for b for reservoir rock. Second, b is heavily influenced by reservoir heterogeneity (Jones 1987). Finally, Eqs. 10.22 and 10.25 are specifically for openhole completions and thus, should not be expected to apply to cased and perforated completions. As Saleh and Stewart (1996) note, “the use of a radial (openhole) formula for the non-Darcy flow contribution in a perforated well is completely erroneous.” 10.6.2 Steady-State Spherical Flow. For steady-state non-Darcy flow in a spherical system, the pressure gradient is proportional to the square of the velocity. The velocity, in turn, is inversely proportional to the square of the radius r. Thus, we have C dr r4 1 C1 =  3 − 3 3  rs re 

pe − ps = − ∫

re

rs

=

C 3rs 3

 rs 3  1 − 3  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.26)  re 

where rs is the inner radius of the spherical system. Thus, 90% of the total pressure drop caused by spherical nonDarcy flow occurs within a distance re = 3 10rs ≅ 2.15rs. For a cased and perforated completion, it is convenient to define the equivalent spherical radius of the perforation tunnel, rs, as rs = rpl p , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.27) where rp is the radius of a perforation tunnel, and lp is the length of the perforation tunnel in contact with the formation. Thus, for a well with a cased and perforated completion, 90% of the total non-Darcy skin factor occurs within a few inches of the perforations. For spherical flow to a cased and perforated completion, 90% of the pressure drop caused by non-Darcy flow occurs within a distance approximately 2.15rs, or less than 1 ft, from the perforations. For single-phase flow of oil from a well with a cased-hole/perforated completion (Saleh and Stewart 1996), D = 1.3625 × 10 −17

GND βρ o, wf Bo, wf kh

µ o, wf h p2 ns2 rs3

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.28)

where GND is a dimensionless non-Darcy geometric factor of order unity, ns is the shot density in shots per foot, hp is the height of the perforated interval in feet, and rs is the equivalent spherical radius in feet. The non-Darcy geometric factor GND depends on the permeability anisotropy, as well as perforation density, phasing, and size. For spiral shot patterns, Saleh and Stewart suggest setting GND to 1.3 (Saleh and Stewart 1996).

268 Applied Well Test Interpretation

The corresponding equation for a single-phase flow of gas from a well with a cased-hole/perforated completion is (Saleh and Stewart 1996) D= =

2.2613 × 10 −16 GND β Mpsc kh µ g, wf Tsc h p2 ns2 rs3 6.55 × 10 −15 GND βγ g psc kh

µ g, wf Tsc h p2 ns2 rs3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10.29)

For a cased-hole/perforated completion, the non-Darcy flow coefficient is inversely proportional to the square of the height of the perforated interval, the square of the shot density, and the cube of the equivalent spherical perforation radius. 10.7 Rate-Dependent Skin Factor and Wellbore Storage In Chapter 4, we discussed the Gringarten-Bourdet type curves for constant rate production from a vertical well in a homogeneous, infinite-acting reservoir with constant wellbore storage and skin factor. Although buildup tests on many wells fit the type curves very well, in many other cases the pressure response deviates from the type curve either during the early unit-slope line or during the transition from the unit-slope line to infinite-acting radial flow. Deviation of the pressure response from the type curve means that one or more of the assumptions on which the type curve was based is not valid for the test under consideration. In Chapter 9 we discussed the Fair (1981) and Hegeman et al. (1993) models for variable wellbore storage. What happens when the skin factor changes instead of the wellbore-storage coefficient? Spivey et al. (2004) presented type curves for a well with constant wellbore storage and a combination of constant and rate-dependent skin factor in an infinite-acting reservoir. During a drawdown test, the flow rate across the completion is initially zero, rises rapidly when production begins, then asymptotically approaching the sandface flow rate as wellbore-storage effects end. Thus, rate-dependent skin factor for the drawdown case is a case of increasing skin factor. On the other hand, during a buildup test, the flow rate across the completion is initially equal to the sandface flow rate, decreases rapidly when the well is shut in, then asymptotically approaches zero as wellbore storage ends. This produces a decreasing skin factor during the test. Thus, the cases of drawdown and buildup must be treated separately. 10.7.1 Buildup. Fig. 10.10 shows the sandface flow rate for a buildup test in a well with various combinations of Darcy and non-Darcy skin factor. Compared to the constant skin factor case (aND = 0), the sandface flow rate in 1.0 CDe2s′=1060 CD=100

0.9 aND = 0

0.8

aND = 0.5

0.7

aND = 1

qsfD

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

tD /CD Fig. 10.10—Sandface flow rate, buildup with rate-dependent skin factor (Spivey et al. 2004).

Near-Wellbore Phenomena 269 1E+2

pwD, tD(dpwD/dtD)

CDe2s′=1060

1E+1

aND = 1 aND = 0.75 aND = 0.5

aND = 0.25 aND = 0

1E+0

1E-1 1E+0

1E+1

1E+2 tD /CD

1E+3

1E+4

Fig. 10.11—Pressure response, buildup with rate-dependent skin factor (Spivey et al. 2004).

1.0

CDe2s′=1060 CD=100

0.9 0.8 0.7

qsfD

0.6 0.5

aND = 1

0.4

aND = 0.5

0.3

aND = 0

0.2 0.1 0.0 1E+1

1E+2

1E+3

1E+4

1E+5

1E+6

tD Fig. 10.12—Sandface flow rate, drawdown with rate-dependent skin factor (Spivey et al. 2004).

the non-Darcy skin case (aND = 1) initially declines more slowly, but then declines much more rapidly. This rapid decline in the flow rate causes the pressure derivative during the transition out of wellbore storage in Fig. 10.11 to drop almost vertically as wellbore-storage effects end. As a result, wellbore-storage effects for non-Darcy skin factor end much sooner than for a test with an equivalent Darcy skin factor. The pressure derivative for a buildup test on a well with high non-Darcy skin will be much steeper during the transition from wellbore storage to infinite-acting radial flow than that for a well with constant skin. 10.7.2 Drawdown. Fig. 10.12 shows the sandface flow rate during drawdown for Darcy and non-Darcy skin factor. During drawdown, the sandface flow rate for the non-Darcy skin case increases much more rapidly at first than does the Darcy skin case. The pressure response, Fig. 10.13, shows that the transition from wellbore storage to infinite-acting radial flow has a shape similar to that for the constant skin case, but shifted to the right.

270 Applied Well Test Interpretation 1E+2 CDe2s′=1060

pwD, tD(dpwD/dtD)

aND = 1

aND = 0.75

1E+1

aND = 0.5 aND = 0.25 aND = 0

1E+0

1E-1 1E+0

1E+1

1E+2 tD /CD

1E+3

1E+4

Fig. 10.13—Pressure response, drawdown with rate-dependent skin factor (Spivey et al. 2004).

60 y = 2.15x + 9.17

50

s′ = s + Dq

40 30 20 10 0

0

5

10 15 q, MMscf/D

20

25

Fig. 10.14—Estimating s and D for rate-dependent skin factor.

10.8 Estimating Non-Darcy Coefficient D From Multiple Transient Tests If a series of transient tests have been run at different flow rates, as is often the case with gas wells, the Darcy and non-Darcy skin components can be determined as follows. The total skin seen during each test is given by Eq. 10.19. Thus, a graph of s′ vs. q may be used to estimate the Darcy skin factor, s, from the y-intercept, and the nonDarcy flow coefficient, D, from the slope. Fig. 10.14 illustrates the procedure. A gas well deliverability test, comprising a series of four flow and shutin periods, was analyzed to obtain permeability and skin factor for each of the four buildups. The skin factor observed during each buildup was graphed vs. the flow rate for the preceding flow period. The y-intercept of the best fit straight line gives the Darcy skin component, s, as 9.17, while the slope gives the non-Darcy flow coefficient, D, as 2.15 D/MMscf. 10.9 Summary Key points in this chapter include the following: 1. The geometric skin factor for a partially penetrating well is always positive. It increases in magnitude as the penetration ratio gets smaller or the formation gets thicker. 2. A partially penetrating well with damage will magnify the damage skin by the ratio of the thickness of the formation to the height of the completed interval, in addition to the geometric skin.

Near-Wellbore Phenomena 271

3. The geometric skin factor for a deviated well is always negative. It increases in magnitude with the degree of deviation from the vertical and the thickness of the formation. 4. For radial flow to a vertical well with an openhole completion, 90% of the pressure drop caused by non-Darcy flow occurs within a distance 10rw, or 2½ to 5 ft, from the center of the wellbore. 5. For spherical flow to a cased and perforated completion, 90% of the pressure drop caused by non-Darcy flow occurs within a distance approximately 2.15rs, typically less than 1 ft, from the perforations. 6. For a cased-hole/perforated completion, the non-Darcy flow coefficient is inversely proportional to the square of the height of the perforated interval, the square of the shot density, and the cube of the equivalent spherical perforation radius. 7. The pressure derivative for a buildup test on a well with high non-Darcy skin will be much steeper during the transition from wellbore storage to infinite-acting radial flow than that for a well with constant skin. Nomenclature A = intermediate result, dimensionless B = intermediate result, dimensionless B = formation volume factor, bbl/STB C = coefficient, units vary depending on equation D = non-Darcy flow coefficient, D/STB or D/Mscf G = non-Darcy geometric factor, dimensionless h = formation thickness, ft h1 = distance from top of zone to top of completed interval, ft hp = height of completed interval, ft k = permeability, md lp = length of perforation tunnel, ft Lp = length of perforation tunnel, ft m = slope of straight line, psi/~ m′ = slope of straight line during early radial flow for deviated well, psi/~ n = number of perforations ns = perforation density, shots/ft M = molecular weight, lbm/lbm-mole p = pressure, psia q = flow rate, STB/D r = radial coordinate, ft rp = perforation radius, ft rs = equivalent spherical radius, ft rw = wellbore radius, ft rwa = apparent wellbore radius, ft s = skin factor, dimensionless s′ = total skin factor including both Darcy and rate-dependent skin, dimensionless T = temperature, °R b = inertial flow coefficient, ft–1 gg = gas specific gravity, air = 1 m = viscosity, cp r = density, lbm/ft3 qw = angle from vertical, degrees qw′ = angle from vertical in equivalent isotropic system, degrees f = porosity, fraction Subscripts a = altered zone d = damage D = dimensionless dp = damage from perforating e = exterior g = gas h = horizontal

272 Applied Well Test Interpretation

o = oil p = partial penetration p = perforation s = spherical sc = standard conditions t = total v = vertical w = wellbore wf = flowing wellbore conditions q = wellbore deviation

Chapter 11

Well Test Interpretation Workflow “Never accept the proposition that just because a solution satisfies a problem, that it must be the only solution.” —Raymond E. Feist “We consider it a good principle to explain the phenomena by the simplest hypothesis possible.” —Ptolemy 11.1 Introduction In previous chapters, we have discussed a wide variety of wellbore and near-wellbore phenomena, reservoir models, and analysis methods. This chapter presents a recommended systematic workflow to use when interpreting a pressure-transient test. The recommended workflow is intended to be applicable to most pressure-transient testing situations. As a result, it is unlikely that any test will require every step in the workflow. Further, some tests may require additional steps that are not included in this workflow. Finally, although the steps are presented in the order they would normally be performed, it is often necessary to iterate between steps to complete the interpretation. In brief, the workflow comprises the following steps: 1. Collect the data necessary for the interpretation, Section 11.2. 2. Review, quality control (QC), and prepare the data for interpretation, Section 11.3. 3. Deconvolve the test data, Section 11.4. 4. Identify the flow regimes present in the test data, Section 11.5. 5. Select the reservoir model to use for interpretation, Section 11.6. 6. Estimate the parameters that characterize the model using manual straight-line and log-log methods, Section 11.7. 7. Simulate or history-match the pressure response, Section 11.8. 8. If appropriate, calculate confidence intervals, Section 11.9. 9. Interpret the estimated model parameters, Section 11.10. 10. Validate the results, Section 11.11. Section 11.12 provides a worked-out example, offering an alternative interpretation for a published field example. 11.2 Collect Data The first step in analyzing a pressure-transient test is to collect the data necessary to interpret the test. Tables 11.1 and 11.2 list the data required for interpreting most pressure-transient tests. 11.2.1 Geology and Geophysics. Geology and geophysics are two of the most important sources of information used in well test interpretation. Since geology, geophysics, and well testing all interpret different aspects of the same underlying physical object, the models used for all three disciplines should be mutually consistent. Structure Maps. A structure map of the top of the formation being tested is one of the best places to start when analyzing a pressure-transient test. Structure maps may include information about locations and distances to faults and other boundaries, distances to fluid contacts, and location of offset wells. Isopach Maps. Isopach maps may be constructed on gross formation thickness, net sand thickness, net-pay thickness, or net hydrocarbon thickness. Isopach maps provide an excellent indication of expected changes in reservoir quality away from the well.

274 Applied Well Test Interpretation TABLE 11.1—DATA REQUIRED FOR WELL TEST INTERPRETATION Geology and Geophysics

Rate Data

**Structure maps

*Detailed rate data before test for the least of

Cross Sections

Entire rate history

**Distances to boundaries

10x duration of test

Distances to fluid contacts

Time to reach PSSF

**Expected reservoir or sand body size

User Horner tp approximation for earlier data

**Evidence of natural fracturing

Well Data

**Evidence of layering

Drilling report

Petrophysics

Mud weight

Logs

**Workover history

Gamma ray or SP

Stimulation treatments

Porosity

Recompletions

Borehole image log

Type of completion

Cement bond log

Type of stimulation

Interpreted logs

Matrix acid

*Average porosity

Hydraulic fracture

*Average fluid saturations

Acid frac

*Net-pay

Type of artificial lift

Lab data

**Wellbore schematic

**Pore volume compressibility

Casing/liners/tubing

Relative permeability data Reservoir Data

Packers/bridge plugs Fish

**Temperature

**Toolstring schematic

**Initial pressure

Placement and depth of gauges

**Datum elevation

Location of downhole shut-in tool

Reservoir fluid contacts Fluid Property Data

Previous pressure-transient tests Offset wells

Data for correlations—Oil

Bottomhole locations

API gravity

Production/injection history

Gas/oil ratio and gas gravity

Pressure-transient tests

Separator conditions Data for correlations—Gas Separator gas gravity Impurities content Yield and condensate gravity Separator conditions Data for parameter estimation *Viscosity *Formation volume factor

Test Data **Daily reports *Rate vs. time (oil, gas, water) **Flowing and static gradient surveys Gauge Data **Gauge type and serial number Date of most recent calibration *Pressure and temperature vs. time

*Compressibility

**Start date/time

**Bubblepoint pressure/dewpoint pressure

End date/time

* Required. ** Strongly recommended.

Cross Sections. Cross sections provide information similar to that provided by gross formation thickness isopach maps. Cross sections also indicate the presence of other zones that may be in communication with the tested zone. Other Geological Information. Other geological information that may provide insight useful in well test interpretation include lithology, depositional environment, expected reservoir size and connectivity, grain size and sorting, degree of cementation or other forms of diagenesis, evidence of natural fracturing, evidence of heterogenieties laterally and vertically, estimates of distances to faults and other boundaries, and estimates of distances to oil/water, gas/oil, or gas/water contacts.

Well Test Interpretation Workflow 275 TABLE 11.2—RECOMMENDED WORKFLOW FOR WELL TEST INTERPRETATION Collect Data

Select Reservoir Model

Review, QC, and Prepare Data for Interpretation

*Engineering data

Review geology, geophysics, and petrophysics Review well history and offset well data *Review rate vs. time data Compare gauge data **Synchronize gauges **Multiple downhole gauges **Wellhead tubing and annulus gauges *Review p and T data Identify erratic readings Identify unexplained noise *Select primary gauge Most accurate gauge? Gauge with lowest thermal sensitivity? Gauge closest to perfs? Gauge with highest frequency data? *Synchronize rate data with primary gauge Identify time/pressure at start of each test period Sync rate changes with pressure data *Convert pressures to midpoint of perforations Calculate plotting functions Pseudopressure/pseudotime functions Rate-normalization Superposition time functions Calculate pressure change and derivatives for each test period **Identify non-reservoir phenomena Phase segregation/variable WBS Non-Darcy skin factor Movement of liquids past gauge Tidal effects/diurnal temperature effects Gauge precision artifacts Interference from offset wells Abrupt leaks Evaluate whether or not test is interpretable

Type of completion Diagnostic plot Flow regime sequence External data *Geology *Geophysics Petrophysics Estimate Model Parameters **Straight-line methods Radial flow Linear flow Volumetric behavior Spherical flow Bilinear flow **Log-log methods Type-curve analysis Manual log-log analysis *History matching **Manual history match Automatic history match Confidence intervals *Validate assumptions with residual, lag plots If applicable, calculate confidence intervals Validate Results *Reality Check Check consistency between flow periods **Wellbore storage *Permeability **Check consistency with previous tests **Calculate radius of investigation Beginning, end of each flow regime *Simulate pressure response *Compare simulated and measured pressures

Deconvolve Data

*Cartesian graph of all flow periods to ensure correct reservoir volume

Identify Flow Regimes

*Log-log graph of each test period to ensure correct permeability and skin

*Log-log diagnostic plot

**Semilog graph of each test period

Radial flow—horizontal derivative

**Specialized graphs of each test period

Linear flow—½ slope

Compare estimated with external data

Volumetric behavior—unit slope Spherical flow— –½ slope Bilinear flow—¼ slope Specialized diagnostic plots * Required. ** Strongly recommended.

11.2.2 Petrophysics. Petrophysics provides both information for interpretation and data for parameter estimation in well testing. Although interpreted openhole wireline logs are used in virtually every well test interpretation, cased-hole logs and lab data are also important sources of information for both interpretation and parameter estimation. Wireline Logs. A standard logging suite provides information necessary to estimate porosity, water and gas saturation, and net-pay thickness. The spontaneous potential and gamma ray logs provide an indication of the

276 Applied Well Test Interpretation

shaliness of the formation, which is used in estimating net-pay thickness. The density, neutron, and sonic logs can be used to estimate porosity; the density and neutron logs can be used together to estimate gas saturation. The various resistivity logs may be used to estimate water saturation. Other logs provide supplemental information that may help in selecting an appropriate reservoir model or to detect the presence of conditions that may affect the test. Borehole image logs detect the presence of natural fractures. Cement bond logs may be useful for determining whether or not there is communication behind pipe between different zones. Interpreted Logs. The raw log data should be processed by a log interpretation specialist, unless the well test analyst is also an expert in log interpretation. At a minimum, interpreted log data must include average porosity and fluid saturations over the tested interval, as well as net-pay or net formation thickness. Although porosity is not used in estimating permeability from data in infinite-acting radial flow, it is used in estimating all properties involving reservoir dimensions, including radius of investigation, distance to boundaries, channel width, fracture half-length, drainage area, and even skin factor. The gas, oil, and water saturations are used to calculate the total compressibility. The total compressibility, like the porosity, is used in estimating all properties involving reservoir dimensions. Because of the difficulty of estimating net-pay thickness, h, some commercial software packages report the permeability thickness product, kh, as the primary result rather than the permeability, k. While there is merit to this approach, net-pay thickness must also be used to calculate estimates of many other parameters in well test interpretation, including skin factor, s, radius of investigation, ri, distances to boundaries, Le, and drainage area, A. The net-pay thickness used in the interpretation should always be included in the final well-test report. Lab Data. In addition to laboratory-measured porosity and permeability data, two types of special core analysis are particularly important for well testing, pressure-dependent porosity, and relative permeability. Core porosity data is often used for calibration of porosity logs. Core porosity measured at ambient pressure may be higher than porosity at reservoir stress conditions. However, the decrease in porosity because of stress is almost always smaller in magnitude than the uncertainty in other reservoir properties used in the pressuretransient analysis. Core permeability data can be useful, provided several factors are taken into account. Ideally, the total permeabilitythickness product, kh, from core over the tested interval should be close to kh from pressure-transient analysis. However, for this to be achieved in practice, core recovery should be close to 100%. In addition, core permeability must be corrected to in-situ conditions, with corrections for Klinkenberg effect, overburden pressure, and connate water saturation (Jones and Owens 1980). In abnormally pressured low-permeability reservoirs, in-situ permeability may be 100 times smaller than core permeability measured at ambient pressure. Further, core permeability does not include the contribution to flow capacity from natural fractures and, thus, may be orders of magnitude lower than well test-derived permeability. Finally, permeability measurements at the core-plug or even whole-core scale (centimeters to inches) may not be comparable to permeabilities measured at the reservoir scale (100s to 1000s of feet). Pressure-dependent porosity data may be used to estimate pore volume compressibility. A rough estimate of pore-volume compressibility can be obtained from the slope of a graph of the natural logarithm of the stressed porosity on the y-axis vs. net overburden pressure on the x-axis. For a more accurate estimate of pore volume compressibility, the lab data, which are taken under hydrostatic conditions, must be converted into uniaxial strain compressibility data more representative of reservoir conditions (Anderson 1988). Relative permeability data are useful for analyzing reservoirs in which a transient encounters a fluid contact or other change in saturation during a test. A change in saturation may appear as either a radial or linear composite boundary. Relative permeability data are necessary to provide a complete interpretation of the mobility and storativity ratios that describe composite reservoir behavior. 11.2.3 Reservoir Data. Reservoir data required for well test interpretation include formation temperature, initial pressure, datum elevation, and location of fluid contacts. Temperature. Temperature is required for estimating fluid properties from correlations. Temperature may be obtained from openhole logs or from downhole electronic pressure gauge data. If temperature measurements are not available for the formation of interest, it may be necessary to correct the temperature from the formation nearest to the tested zone using a known or estimated temperature gradient for the region. Typical temperature gradients are between 1 and 2°F/100 ft of depth (Levorsen 1967). Initial Pressure. For the initial test on a new well, the initial reservoir pressure will normally be one of the objectives of the test. However, a rough estimate of the pressure is useful for validation of the results of the analysis. Reservoir pressure is also necessary for estimating fluid properties from correlations for use in pressure-transient analysis.

Well Test Interpretation Workflow 277

Datum Elevation. The datum elevation for a reservoir is normally chosen after the hydrocarbons have been mapped with enough confidence to estimate the location of the hydrocarbon pore volume weighted average depth for the reservoir. Fluid properties for material balance analysis should be calculated at the pressure at the datum elevation. If average reservoir pressure is to be obtained from a test, the gauge pressure must be converted to the appropriate reservoir datum before use in material balance calculations. Reservoir Fluid Contacts. Locations of any gas/oil, oil/water, or gas/water contacts provide information about distances to expected changes in mobility that may appear as a composite reservoir boundaries during a pressuretransient test. The elevation of reservoir fluid contacts must also be known to convert pressures from mid-point of perforations to reservoir datum. 11.2.4 Fluid Property Data. Fluid property data are required to estimate virtually every parameter describing any reservoir model. The important fluid properties for parameter estimation are formation volume factor, compressibility, and viscosity, all of which may be obtained from correlations or from laboratory tests. In addition, the bubblepoint or dewpoint pressure should always be considered during well test interpretation. Data for Correlations. Accurate correlations are available for most of the fluid properties required in parameter estimation. Oil property correlations require, the API gravity of the stock-tank oil, the producing gas/oil ratio and the gas gravity, in addition to the reservoir temperature and pressure. More accurate estimates of fluid properties may be obtained if separator conditions are also available. Gas property correlations require the separator gas gravity and nonhydrocarbon impurities content, the condensate yield, and the API gravity of the condensate. As with oil, more accurate estimates of fluid properties may be obtained if separator conditions are available. Water property correlations require the brine salinity and bubblepoint pressure. For a gas reservoir, or for a saturated oil reservoir, the bubblepoint pressure for the water phase is usually taken to be the pressure in the gas phase. For an undersaturated oil reservoir, the bubblepoint pressure for the water phase is usually assumed to be the same as that of the oil phase. Fluid Properties Parameter Estimation. The three fluid properties required for well test interpretation are viscosity, formation volume factor, and compressibility. Whether obtained from laboratory data or correlations, all three properties are subject to uncertainty. Viscosity is especially problematic for heavy oils. There is some indication that oils with measured viscosities higher than 1 cp are non-Newtonian (McCain et al. 2011), and therefore, laboratory measurements of viscosity (and correlations based on laboratory data) may not accurately represent fluid behavior in the reservoir. Gas viscosities are almost always calculated from a correlation, most often the Lee et al. (1966) correlation. The formation volume factor may be estimated either from correlations, or calculated from laboratory pressure, volume, temperature (PVT) data. For black oils and single-phase gases, the bubblepoint formation volume factor estimated from the better correlations has an average absolute relative error of 2% or less (McCain et al. 2011). Compressibility. Fluid compressibility is particularly important for estimating radius of investigation, distances to boundaries, drainage areas or volumes, and hydrocarbons in place. Oil compressibility increases by approximately a factor of 10 as the pressure decreases through the bubblepoint. Gas compressibility is approximately equal to the reciprocal of pressure and is thus strongly pressure-dependent. Formation Volume Factor. The formation volume factor may be estimated either from correlations, or calculated from laboratory PVT data. For black oils and single-phase gases, the bubblepoint formation volume factor estimated from the better correlations has an average absolute relative error of 2% or less (McCain et al. 2011). Bubblepoint Pressure or Dewpoint Pressure. For an oil reservoir, the bubblepoint pressure is particularly important for well test interpretation. The bubblepoint pressure may be used to improve the accuracy of some of the fluid property correlations. Perhaps more important, the large change in oil compressibility at the bubblepoint pressure will cause estimates of distances to boundaries to be high by a factor of 3, and oil in place to be too high by a factor of 10, if a test is analyzed incorrectly assuming the oil is undersaturated. Finally, the pressures during a test should be compared to the bubblepoint pressure to determine whether the pressure response might be affected by multiphase phenomena. A free gas saturation near the well may reduce the oil mobility, causing an apparent skin factor. Or, compression of a small free gas saturation may cause decreasing total compressibility during a buildup. This decreasing total compressibility may in turn give a steeper than unit slope derivative on the diagnostic plot or an increasing derivative on the primary pressure derivative plot. For a retrograde gas reservoir, the gas compressibility is much less affected by the dewpoint pressure. However, multiphase effects may also affect a test on a retrograde gas reservoir because of the development of a nearwellbore condensate ring, and the resulting apparent skin factor, when the flowing bottomhole pressure is below the dewpoint pressure.

278 Applied Well Test Interpretation

11.2.5 Rate Data Prior to Test. Some of the most important data for well test interpretation is the rate history before a test. The question that often arises is “How much of the rate history is really necessary?” For a well that has been producing for several years, it seems obvious that daily production early in the life of the well can have no impact on the pressure response. On the other hand, ignoring rate history can lead to misinterpretation of rate-history-induced pressure changes as reservoir boundaries. We recommend that detailed rate data for time period preceding the test for a total duration of the least of the following: (1) duration of the entire rate history, (2) 10 times duration of test, and (3) time to reach pseudosteadystate flow. If linear flow is predominant during the test, it may be necessary to include rate history before the test for a period as much as 100 times longer than the duration of the test. 11.2.6 Well Data. A wide variety of well data are either necessary or useful in well test interpretation. Workover History. The analyst should be familiar with the workover history for the well. It would be rather embarrassing to discover that the well you thought was unstimulated had been hydraulically fractured or that the well has been recompleted in a different zone from the one you thought was the test interval! Stimulation treatment design and daily reports and recompletion history may all be relevant to well test interpretation. Stimulation treatment design should indicate the skin factor or fracture half-length expected from the stimulation; the stimulation treatment report may indicate whether problems occurred that would have prevented the design value from being achieved. Workover history would include the zones that have produced and when each zone was completed and/or abandoned. Type of Completion. The type of completion gives a strong indication of the range of skin factor to be expected, whether positive or negative. The type of completion also determines the wellbore model to use for model selection, whether a vertical well, a hydraulically fractured well, or a horizontal well. Completion types are openhole, cased-hole/perforated, gravel-packed, or frac-packed; partial-penetration, limited-entry, or fully penetrating; vertical-well, deviated-well, horizontal-well, or multilateral-well. Type of Stimulation. The type of stimulation, if any, will also indicate the range of skin factor to be expected. Stimulation types include matrix acid, acid fracture, and hydraulic fracture. Type of Artificial Lift. Artificial lift can have a significant effect on the pressure behavior during flow periods, and the nature of wellbore storage (WBS) after shut-in. Types of artificial lift include rod pump, electric submersible pump, gas lift, and plunger lift. Wellbore Schematic. The wellbore schematic showing the wellbore configuration during the test should always be available to the analyst doing the interpretation. The schematic should include the depths and sizes of the tubing, casing, and liners; perforated interval(s), with shot size and density; location of screens, if any; type and location of crossover subs, packers, and bridge plugs; presence and nature of any fish in the wellbore; and depth of the rathole. Tool-String Schematic. The tool-string schematic should show the placement and depth of each gauge and where the gauges are located within the gauge carrier(s). If the test were run with downhole shut-in, the tool-string schematic should also show the location of the downhole shut-in tool. Wireline Formation Test Data. Wireline formation test data provide estimates of pressure and permeability for comparison with conventional pressure-transient test results. Previous Pressure-Transient Tests. A complete interpretation of a pressure-transient test should include a comparison with the results of previous tests on the same interval, and an explanation of any differences in skin factor, permeability, reservoir model, or distances to boundaries. Offset Wells. Offset well data should include bottomhole locations, completed intervals, production and/or injection history, and pressure-transient tests. Daily Reports. Daily reports for drilling, cementing, completion, and stimulation may provide information about reservoir characteristics; problems encountered may suggest problems with wellbore integrity. Mud weight can be used to provide a rough estimate of formation pressure. Mud weight can also be used to determine if there was pressure communication with the mud column during a drillstem test. 11.2.7 Test Data. Test data should include the test design document, the daily report for test operations, choke size and flow rates or volumes vs. time for all phases, a schematic of the surface equipment in use during the test, and tubing and casing pressure data vs. time during the test, in addition to the gauge data. A flowing gradient survey is often run when the gauge is run in during the flow period before a buildup test, and a static gradient survey is often run at the end of a buildup. The gradient surveys may be used to obtain fluid levels and densities necessary to correct pressures to midpoint of perforations. 11.2.8 Gauge Data. In addition to the pressure and temperature vs. time recorded by each gauge, the gauge type, serial number, and date of most recent calibration should be available. If pressure is recorded vs. elapsed time,

Well Test Interpretation Workflow 279

the date and time the gauge started recording is often essential to synchronize the data with the test rate data as well as pressure data from other gauges. 11.3 Review and Quality Control Data One of the most important steps in well test interpretation is quality checking and validating the data before continuing with the interpretation. Mattar and Santo (1992) recommend that 50% of the analyst’s time should be spent examining, validating, and reconciling the raw data. 11.3.1 Estimate Errors in Rock and Fluid Properties. One of the primary sources of errors in parameters estimated during well test interpretation is uncertainty in the input data. Often, the effect of uncertainty in the net sand thickness on the permeability is indicated by reporting a permeability-thickness product, kh, rather than values for the individual properties. However, it is necessary to review all of the data to determine which property estimates are likely to have an adverse effect on the interpretation. Spivey and Pursell (1998) summarized the effects of errors in input properties on the ensuing parameter estimates for typical well test interpretation scenarios. The scenarios assume correct flow regime and model identification, and that straight-line, type-curve, or history-matching methods are used appropriately. Table 11.3 shows how input errors affect the results for a vertical well with constant WBS and skin factor, in a homogeneous, isotropic, reservoir with one or more boundaries. The first row in the table shows that an error in the input viscosity will affect only the calculated permeability. Although the calculated permeability is used in further calculations, it always appears in the term k µ , so the errors in viscosity and permeability cancel for the remaining parameter estimates. Error in net thickness affects not only permeability, but skin factor (albeit only slightly), distances to boundaries, and drainage area. Note that the wellbore radius affects only the skin factor; even then its effect is quite small. A factor of two error in wellbore radius (e.g., failure to convert diameter to radius) causes only a shift in the skin factor by ln(2), or approximately 0.69, almost never a difference that will result in a different reservoir management decision. Table 11.4 shows the effects of errors in input data on results of reservoir limits testing, Eqs. 2.78 through 2.80.

TABLE 11.3—EFFECT OF ERRORS IN INPUT PARAMETERS ON PARAMETER ESTIMATES, VERTICAL WELL IN HOMOGENEOUS, ISOTROPIC RESERVOIR WITH ONE OR MORE BOUNDARIES (SPIVEY AND PURSELL 1998) Results Input

kcalc

Cwb calc

scalc

Lx calc

min = a mtrue

a ktrue

—

—

—

Acalc —

f in = a f true

—

—

strue + 0.5ln(a)

Lx true/α

Atrue/α

ct in = a ct true

—

—

strue + 0.5ln(a)

Lx true/α½

Atrue/α

hin = a htrue

ktrue/α

—

strue + 0.5ln(a)

Lx true/α½

Atrue/α

qin = a qtrue

a ktrue

a Cwb true

strue – 0.5ln(a)

α½ Lx true

a Atrue

Bin = a Btrue

a ktrue

a Cwb true

strue – 0.5ln(a)

α Lx true

a Atrue

rw in = a rw true

—

—

strue + ln(a)

—

—

½

½

TABLE 11.4—EFFECT OF ERRORS IN INPUT PARAMETERS ON PARAMETER ESTIMATES, RESERVOIR LIMITS TEST Results Input

Vp calc

Vb calc

Acalc

Ncalc

f in = a f true

—

Vb true/α

Atrue/α

—

ct in = a ct true

Vp true/α

Vb true/α

Atrue/α

Ntrue/α

hin = a htrue

—

—

Atrue/α

—

qin = a qtrue

a Vp true

a Vb true

a Atrue

a Ntrue

Bin = a Btrue

a Vp true

a Vb true

a Atrue

—

(1–Sw)in = a (1–Sw)true

—

—

—

a Ntrue

280 Applied Well Test Interpretation

In any problem where one of the objectives is to estimate distances to boundaries, drainage areas, or fluid in place, the total compressibility is likely to have the largest error of any of the input data. Unfortunately, typical values for compressibility tend to be an order of magnitude lower than worst-case values, causing overoptimistic estimates of fluid in place. Pore volume compressibility is usually around 4 × 10–6 psi–1 for consolidated rock, having porosity in the range of 20 to 30%, in normally pressured reservoirs. However, either low- or highporosity rock, geopressured reservoirs, or unconsolidated formations can have compressibilities of 50 to 150 × 10–6 psi–1. Similarly, undersaturated oil compressibility is typically 4 to 12 × 10–6 psi–1, while compressibility for the same oil at pressures below its original bubblepoint pressure will be 40 to 120 × 10–6 psi–1. It is essential to obtain accurate pore-volume and fluid compressibility estimates before booking reserves based on well test interpretation alone. 11.3.2 Review Geology and Petrophysics. Review the data collected in Sections 11.2.1 and 11.2.2, with a view to understanding how the geology and petrophysics might affect the pressure-transient response. Of particular interest are grain size and sorting, evidence of layering or natural fractures, heterogeneities, expected reservoir compartment size and connectivity, and distances to boundaries. 11.3.3 Review Well History and Offset Well Data. Review the data collected in Section 11.2.6. 11.3.4 Review Rate-Time Data. The first step in quality checking the data is to review the rate-time data, most easily done graphically on a Cartesian scale. The following questions should be addressed while reviewing the rate-time data. Is the data complete? The rate-time data needed for well test interpretation include the rate history before the test in addition to the flow rates measured during the test. One interpretation mistake (that seems to occur far more often than it should) is to include in the analysis only the rates measured while the pressure gauges are in the wellbore. For multiphase flow, are the fluid ratios constant before the test? During the test? Constant fluid ratios suggest that saturations are not changing with time. This allows analytical methods, such as the Perrine-Martin and multiphase pseudopressure methods, to be used to interpret the well test. Are there any offset wells that might affect the pressure response? The worst-case scenario is an offset well producing just long enough for its radius of investigation to reach the test well during the test. As shown in Chapter 2, the rate of change of pressure at a given distance is greatest at the time the radius of investigation reaches that point in the reservoir. For a gas well, is the production rate high enough to lift liquids? A quick estimate of the critical rate to lift liquids may be obtained from the equation qcrit ≅ 7 ID 2 FWHP, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.1) where qcrit is the critical rate to lift liquids in Mscf/D, ID is the inside diameter of the casing or tubing in inches, and FWHP is the flowing wellhead pressure in psia. A more accurate estimate of the critical rate may be obtained using the methods proposed by either Turner et al. (1969) or Coleman et al. (1991). 11.3.5 Review Gauge Data. Review the gauge data on a Cartesian scale. Identify the beginning and end of each flow period. Other phenomena that may be identified include: (1) running in the hole, (2) flowing or static gradient survey stops, (3) swabbing, (4) perforating, (5) other rig activity, and (6) pulling out of the hole. It may be helpful to refer to the test report and the rate history while reviewing the gauge data. Review pressure and temperature data from each gauge to identify erratic readings or unexplained noise. Erratic temperature data may indicate problems with the pressure data as well. Does the pressure pass through a bubblepoint or dewpoint during the test? For a gas well, does the pressure fall below twice the pressure downstream of the choke? If so, pressure fluctuations downstream of the choke may affect flow rate through the choke and thus the pressure recorded by the gauge. For an exploration well, is there evidence of depletion? 11.3.6 Compare Gauge Data. When there are multiple gauges, data from all of the gauges should be compared. Mattar and Santo (1992) advise the analyst to “compare all the pressure recorders for inconsistencies; do not rely on one recorder when one or more backup recorders are used.”

Well Test Interpretation Workflow 281

Synchronize all of the gauges so the beginning of each flow and buildup period coincide. Once the gauge data have been synchronized, select one gauge as the primary gauge. Factors to consider include 1. Which gauge has the best data? 2. Which gauge is the most accurate? 3. Which gauge has the fastest response/stabilization time? 4. Which gauge is least sensitive to temperature fluctuations? 5. Which gauge is closest to the perforations? 6. Which data has the most complete data? 7. Which gauge has the highest frequency data? 8. Which gauge is right? This is not a democracy. Do not rely on “majority rules” (Mattar and Santo 1992). Calculate the difference curves between the primary gauge and the remaining gauges. Review the difference curves to identify periods when a liquid level is moving between the gauges or other phenomena are causing the difference in pressure to change over time. Identify the cause of the change. Synchronize and compare bottomhole gauge data with wellhead tubing and annulus gauges. Compare tubing and annulus gauge data to identify any leaks or unintended communication between the tubing and the annulus. 11.3.7 Synchronize Rate Data With Primary Gauge. Pressure data is recorded at much higher frequency (usually as little as 1 to 15 seconds apart) than are rate data. Thus, the beginning of each flow and shut-in period can be most accurately determined from the pressure data. Using the pressure data from the primary gauge, identify the beginning of each flow and shut-in period. Synchronize the rate history so the beginning of each flow and shut-in period coincides with those identified from the pressure data. In some cases, the pressure data may show that rate changes have occurred that have not been recorded. If the rate changes are obviously significant in magnitude or duration, it may be possible to estimate the unknown change in rate from data in WBS or in infinite-acting radial flow, assuming the WBS coefficient or permeability, respectively, can be estimated from other test periods with known flow rates. Identify the time and pressure at the beginning of each test period. Correct identification of the pressure at the beginning of each test period (or more accurately, the pressure at the end of the previous flow or shut-in period) is essential to getting an accurate estimate of skin factor. 11.3.8 Calculate Plotting Functions. Normally, the software will do this for you. But you need to know what plotting functions the software is calculating, how the software calculates the plotting functions, and more importantly, the reason for and limitations of the plotting functions. For oil well tests, the pressure plotting function options may include pressure or rate-normalized pressure change. Time-plotting function options for buildup tests may include elapsed time, Agarwal multirate equivalent time, logarithmic superposition time function, or Horner time ratio. Time plotting options for flow tests may include elapsed time and material balance time. For gas well tests, pressure plotting function options may include pressure-squared, pseudopressure, or adjusted pressure. Time-plotting function options may include pseudotime, normalized time, or adjusted time; adjusted equivalent time; adjusted superposition time function; or adjusted Horner time ratio. Time plotting functions for flow tests may include elapsed time and adjusted material balance time. 11.3.9 Calculate Pressure Change and Derivatives for Each Test Period. As with the plotting functions, the software will normally do this for you. And as with plotting functions, you need to know what the software is calculating, how the software is calculating it, and why the software is calculating it. At a minimum, the pressure change, the primary pressure derivative, and the logarithmic derivative of pressure must be calculated for each flow period. 11.3.10 Identify Non-Reservoir Phenomena. The next step is to review the pressure data from the primary gauge on a log-log scale. Often, phenomena that are not initially apparent on a Cartesian scale can be quickly recognized from the pressure derivative curve on a log-log scale. This review should be repeated for each test period. WBS Phenomena. Identify the WBS unit-slope line. Identify variable WBS phenomena that may be caused by a sudden change in WBS coefficient, phase segregation, or thermal effects. Non-Darcy Skin Factor. For a buildup or falloff test in a well with a large positive skin factor, is the derivative during the transition out of WBS much steeper than predicted for a well with constant WBS and skin factor? If so, the well may have significant non-Darcy skin factor.

282 Applied Well Test Interpretation

Movement of Liquids Past Gauge. Are the abrupt changes in the derivative that may indicate movement of liquids between the perforations and the gauge? If so, carefully review the pressure data on a Cartesian scale to confirm the phenomenon. Although this is a wellbore phenomenon, it may occur at any time during a test. It is not uncommon to observe the phenomenon twice in a single test, with liquid moving first one direction, then the other. If there are multiple gauges at different depths, compare the timing of the abrupt change in slope of the pressure curve for gauges at different depths to confirm liquid movement as the cause of the abrupt change. Pressure Oscillations. Are there oscillations in the pressure? In high-rate oil wells, momentum effects may cause short period oscillations (usually less than a minute per cycle) that decay rapidly. In long-term tests where pressure is changing very slowly, earth tides may be observed, with a 12:25 minute period and an amplitude that varies over a 14-day cycle. In climates with a large difference in wellhead temperature between daytime and nighttime, diurnal temperature fluctuations may cause irregular oscillations with a 24-hour period. Gauge Precision Artifacts. Identify artifacts in the derivative that may have been caused by low-gauge precision or the way the derivative is calculated. Because the derivative is calculated from the pressure, any feature observable in the derivative must be caused either by a change in the pressure or by the way the derivative is calculated. Carefully inspect the raw pressure data corresponding to any suspected artifacts in the derivative to determine if they represent a physical phenomenon or are simply calculation artifacts. Changing the smoothing parameter or the way the derivative is calculated may also help identify artifacts. Abrupt Leak. An abrupt leak may occur at any time during a test, either because of mechanical failure or because a valve was opened inadvertently. Any abrupt change in the pressure, in the slope of the pressure (i.e., the derivative), or in the slope of the derivative must be caused by some wellbore phenomenon rather than reservoir behavior. 11.3.11 Account for Non-Reservoir Phenomena. Once a non-reservoir phenomenon has been identified, the interpretation must somehow account for the phenomenon. Three options are to (1) ignore it, (2) model it, or (3) remove it from the data. In many cases, non-reservoir phenomena can simply be ignored, omitting any affected data from use in the interpretation. This is feasible when the phenomenon is of short duration, occurs early in the test, and causes only a temporary change in the pressure trend. It may also be feasible for situations such as earth tides or diurnal pressure fluctuations that have small-amplitude oscillations about the trend, provided the test duration is long enough that the oscillations tend to average out. In other cases, such as phase segregation, it may be possible to choose a WBS model that can match the nonreservoir phenomenon. In a few cases, it may be necessary to correct the data to remove the effects of the phenomenon. For example, a moving liquid level between the gauge and the completion can cause a permanent offset in pressure between the time at which the last flowing pressure before shut-in, pwf , was recorded, and the time during buildup at which the shut-in pressures used to estimate permeability and skin factor were recorded. Unless the data are corrected for the effects of the fluid movement, the permanent pressure offset will cause the skin factor to be incorrectly estimated. 11.3.12 Look for Evidence of Depletion and/or Interference From Offset Wells. Late-time behavior may be affected by depletion or interference from offset wells. These phenomena must be identified and accounted for during interpretation. Depletion. It is particularly important to identify depletion during initial testing of an exploration or appraisal well. Traditionally, depletion has been identified from a decrease in the extrapolated pressure p* between consecutive shut-in periods. A concave downward curve in the buildup derivative may also be an indication of depletion, as shown in the following example. Fig. 11.1 shows a synthetic drillstem test on a gas well, with an initial 30-minute flow period at 2,500 Mscf/D, followed by a 3-hour shut-in, a 6-hour flow period at 5,000 Mscf/D, and a second 12-hour buildup. The full-scale graph, Fig. 11.1, does not appear to show any signs of depletion. The log-log graphs for the first buildup and for both drawdowns showed no evidence of any boundaries. The drawdown periods were noisier than the buildup periods, which may have contributed to failure to observe boundaries. The second buildup, Fig. 11.2, shows a concave downward derivative at late time, similar to the drawdown pressure response for a well in a reservoir with a circular constant pressure boundary. As shown in Figs. 8.31b and 8.31c, the derivative for a buildup test in a closed reservoir closely resembles that for a drawdown test in a reservoir with a circular constant-pressure boundary. Unless there is compelling geological evidence to support a large increase in mobility to explain a constant pressure boundary, a concave downward derivative, as in Fig. 11.2, should be considered evidence for depletion.

Well Test Interpretation Workflow 283

10100 10000

Pressure, psi

9900 9800 9700 9600 9500 9400

0

5

10

15

20

25

Elapsed Time, hr Fig. 11.1—Depletion example.

Pressure Change and Derivative, psi

1000

100

10

1

0.1 0.001

0.01

0.1 1 Equivalent Time, hr

10

100

Fig. 11.2—Depletion example, log-log graph for second buildup. Derivative shows depletion.

A detailed inspection of the Cartesian graph with the pressure scale zoomed in to emphasize the late-time behavior of the buildups shows clear evidence of depletion, Fig. 11.3. Any initial test on an exploration or appraisal well with multiple buildup periods should be checked for depletion in this way. Comparison of the Horner plots for the two buildup periods, Fig. 11.4, again exhibits depletion during the second buildup. Interestingly, the depletion is indicated by the Horner graph flattening at late time rather than by the extrapolated pressure p*, which has traditionally been used as a diagnostic for depletion. Although it may seem counterintuitive, a buildup test in a closed circular reservoir will begin to exhibit depletion about a half log cycle earlier than a drawdown test; see Figs. 8.16 and 8.17. Interference From Offset Well. Interference caused by production or injection from offset well will be almost negligible early in a buildup test and will be most noticeable at the end of the test. If not identified, interference may be incorrectly interpreted as a reservoir boundary effect. During a buildup test, interference from a producing well will cause the pressure buildup to flatten faster, and possibly even reverse direction, toward the end of the buildup. The derivative curve will curve steeply downward, similar to the pressure response for a constant pressure boundary or a buildup test in a closed reservoir. However, the pressure derivative response from an offset producer may become negative and disappear from the log-log plot, while the constant pressure or closed reservoir pressure derivative will never become negative.

284 Applied Well Test Interpretation

10005

Pressure, psi

10000

Depletion

9995

9990

9985

9980

0

5

10 15 Elapsed Time, hr

20

25

Fig. 11.3—Depletion example, detail. 10005

BU 1 Fit, BU 1 BU 2 Fit, BU 2

Shut-In Pressure, psi

10000

9995 Depletion 9990

9985

9980

1

10 100 Horner Time Ratio

1000

Fig. 11.4—Depletion example, Horner plots for first and second buildup periods. Second buildup shows depletion.

On the other hand, interference from an offset injector will cause the pressure to rise faster than that for the same well without interference, and the derivative will deviate upward from that for the no-interference case. 11.4 Deconvolve Data See Section 8.6.3, Practical Aspects of Deconvolution. 11.5 Identify Flow Regimes After the data have been quality-checked, and optionally deconvolved, the next step is to identify any flow regimes exhibited during the test. The primary tool for flow-regime identification is the log-log diagnostic plot. The flow-regime specific diagnostic plots may also be used for flow-regime identification. The flow-regime specific straight-line plots should be used only to confirm flow-regime identification. 11.5.1 Log-Log Diagnostic Plot. The standard log-log diagnostic plot is a graph of the pressure change and the logarithmic derivative of pressure vs. time on a log-log scale. The logarithmic derivative will exhibit a characteristic slope for each flow regime, as discussed in more detail in Chapter 6, and summarized in Table 11.5. 11.5.2 Flow-Regime Specific Diagnostic Plots. The two primary advantages of the flow-regime specific diagnostic plots are that they give a horizontal line for data exhibiting the flow regime for which they are constructed, enabling rapid visual identification of the flow regime, and that they allow the slope of the corresponding flow-regime specific straight-line graph to be read directly from the horizontal portion of the flow-regime specific diagnostic plot.

Well Test Interpretation Workflow 285 TABLE 11.5—SUMMARY OF FLOW REGIME DIAGNOSTIC CHARACTERISTICS Flow Regime

Slope of Logarithmic Derivative on Log-Log Plot

Radial

0

Linear

½

Flow-Regime Specific Diagnostic Plot dp vs. Dt d ln t dp dt½

Volumetric

1

dp dt

Spherical

–½

vs. Dt

dp dt

–½

Bilinear

¼

dp dt

¼

vs. Dt

vs. Dt vs. Dt

Flow-Regime Specific Straight-Line Analysis Plot p vs. log(Δt) p vs. Δt½ p vs. Δt p vs. Δt–½ p vs. Δt¼

11.5.3 Flow-Regime Specific Straight-Line Plots. The flow regime specific diagnostic plots should be used only for confirmation of the correct flow-regime identification as determined from the standard log-log diagnostic plot and the flow-regime specific diagnostic plots. For each flow regime, the start and end of the flow period should be determined from the standard diagnostic plot or the appropriate flow-regime specific diagnostic plot, then transferred to the flow-regime specific straight-line plot. The data on the flow-regime specific straight-line plot should exhibit a straight line. If the data do not fall on a single straight line as expected, review the pressure data and the derivative calculation to determine the cause of the apparent presence of the flow regime on the diagnostic plot. Flow-regime identification should always be based on the derivative on the standard diagnostic plot or on the flow-regime specific diagnostic plots. The flow-regime specific straight-line analysis plots should be used only for confirmation of the correct flow regime. 11.6 Select Reservoir Model Selection of an appropriate reservoir model is essential to obtaining a valid well test interpretation. In our experience, major interpretation errors are far more likely to be caused by using an inappropriate reservoir model or by analyzing the wrong part of the data than by using any specific analysis method. Parameter estimates obtained from using the wrong model or analyzing the wrong part of the data are at best meaningless. At worst, they lead to poor reservoir management decisions. Major errors in interpretation are more likely to be caused by using the wrong reservoir model or by incorrectly identifying the flow regime than by using any specific analysis method. Parameter estimates from the wrong model or the incorrect flow regime are at best meaningless; at worst, they lead to poor reservoir management decisions. 11.6.1 The Ideal Reservoir Model. An ideal reservoir model is one that meets all of the following criteria: (1) it includes all important physical phenomena; (2) it fits the entire observed rate and pressure history; (3) it is consistent with available geology, geophysics, and petrophysics; and (4) it provides a useful reservoir characterization (i.e., it helps make better reservoir management decisions). Of these criteria, providing a useful reservoir characterization is the only essential criterion*. It costs money to design, conduct, and analyze a pressure-transient test. That expense is justified only by the value of the information obtained from the test. The value of information for a well test interpretation using a model that does not lead to better reservoir management decisions is zero. Kikani (2009) provides an extended discussion of the value of information obtained from well testing. Includes the Important Physical Phenomena. An ideal reservoir model includes all important physical phenomena, including the wellbore, the completion, the reservoir medium, and the reservoir boundaries. The inner boundary comprises the wellbore, the completion, and any damage to the well. The reservoir medium comprises the bulk reservoir properties, whether homogeneous, anisotropic, naturally fractured, or layered. *

The Arps (1945) decline equation is a classic example of a model that meets the final criterion without satisfying any of the other criteria.

286 Applied Well Test Interpretation

Careful consideration should be given to what physical phenomena are important, and how they should be modeled. A gas reservoir with strong aquifer pressure support may show little or no decline in reservoir pressure over several years, suggesting that the aquifer is acting as a constant pressure boundary with respect to production. However, a constant pressure boundary would be entirely inappropriate for pressure-transient analysis. Instead, a composite reservoir model with a significant reduction in mobility would be appropriate. WBS phenomena may include constant or variable WBS, phase segregation, and thermal effects. Skin factor may include constant skin, rate-dependent skin, such as non-Darcy flow, or time-dependent skin such as cleanup or increasing damage. The well may be vertical, deviated, horizontal, or multilateral. It may be completed open hole, cased and perforated, gravel packed, or frac packed. The well may have one or more hydraulic fractures. The reservoir may be homogeneous, it may have anisotropic permeability, it may have layering, or it may be naturally fractured. It may have single- or multiphase flow. It may have gas adsorbed on the matrix, as in coalbed methane and naturally fractured shales. Reservoir boundaries may be sharply delineated, such as sealing faults or unconformities, or gradual, such as pinchouts. They may involve parallel boundaries or intersecting boundaries, or they may represent changes in both mobility and storativity. Fits the Rate and Pressure History. The second criterion of an ideal reservoir model is that it must fit the complete rate and pressure history. In other words, given the complete rate history, the model should predict a pressure response that matches the complete pressure history for the well. The simulated pressure and pressure derivative should agree with the observed pressure and pressure derivative for all flow and shut-in periods, on each graphical scale, whether Cartesian, log-log, semilog, or other specialized scales, as appropriate for the flow regimes observed on the diagnostic plot. If the test data have been deconvolved, the deconvolved unit-rate pressure response satisfies this criterion by virtue of the nature of the deconvolution process itself. Is Consistent With Geology, Geophysics, and Petrophysics. The third criterion of an ideal reservoir model is that it be consistent with the available geological, geophysical, and petrophysical data. All of the various sources of data about the reservoir, whether from transient testing, seismic surveys, or wireline logs, describe a single physical reservoir system. Just as well test interpretation is an inverse problem, so are geological interpretation, seismic interpretation, and log interpretation. Because each of these disciplines focuses on only a subset of data available for the reservoir system, ensuring the model used for each discipline is consistent with those for other disciplines removes much of the ambiguity inherent in the inverse problem. Provides a Useful Reservoir Characterization. The final criterion that must be met by any reservoir model is that it provides a useful reservoir characterization (i.e., one that leads to better reservoir management decisions). Typical results of pressure-transient analysis include estimates of completion and reservoir properties, such as initial pressure, permeability, skin factor, fracture half-length, and distances to reservoir boundaries that may be used to forecast production, estimate reserves, select optimum operating strategy, and identify stimulation candidates. The right reservoir model must provide a reservoir characterization that leads to better reservoir management decisions. Tools for Model Selection. The process of model selection is based primarily on two sources of information, engineering data and external data. 11.6.2 Engineering Data. Engineering data to be considered when selecting a reservoir model include the type of completion, the diagnostic plot, and the analyses of previous pressure-transient tests. Type of Completion. The type of completion will normally determine the “inner boundary condition” for the reservoir model. Even here, the engineer should use good judgment in selecting the inner boundary model. For example, a well with a frac-pack completion will usually not exhibit the characteristic behavior for a hydraulically fractured well. On the other hand, a well intersecting or near a single natural fracture may exhibit the same flow regimes as a hydraulically fractured well. Some of the early tests on horizontal wells were analyzed using type curves for hydraulically fractured wells. If the analyst is using commercial or in-house pressure-transient testing software, he may be limited to the models in the software available to the analyst. For example, a well with a partial penetration completion is often modeled as a fully penetrating completion with a positive geometric skin factor. A deviated well is often modeled as a vertical well with a negative geometric skin factor.

Well Test Interpretation Workflow 287

Diagnostic Plot and Flow Regime Sequence. The diagnostic plot, especially the type and sequence of flow regimes present, is the most common tool for reservoir-model selection. However, the diagnostic plot must not be relied on as the only source of information for model selection. As discussed in Chapter 7, many reservoir models have similar pressure responses. Some reservoir models, such as the well in a rectangular reservoir and the well in a radial composite reservoir, can fit many different pressure responses. Previous Tests on the Same Well. Previous tests are a good source of comparative information and provide at least a starting point for model selection. If the previous test lasted much longer than the test being analyzed, flow regimes during the earlier test may be reasonably assumed to be present in the current test. On the other hand, if the current test is significantly longer than the first test, flow regimes observed in the current test may allow refinement of the model selected for the first test, and may even suggest that the first test should be reinterpreted with a different reservoir model. Changes may occur between tests that require different models for analysis. These changes may occur at any portion of the test, during the early-, middle-, or late-time regions. During the early time region, the most significant change is likely to be caused by stimulation, such as hydraulic fracturing or acidizing. Changes may also occur because of increasing damage. Increasing damage to a hydraulically fractured well may even be severe enough that the fracture “heals” so that the characteristic fracture pressure response is no longer apparent. Stimulation may even cause a change in the model used for reservoir medium. For a naturally fractured carbonate, a pre-acid test may show the reservoir behaving like a pseudosteady-state dual porosity model, while a post-acid test might change the connectivity between the fractures and the matrix to make a transient dual porosity model more appropriate. During the late-time region, changes in boundaries are most likely to be caused by fluid movement in the reservoir, such as encroachment of an aquifer into a water-drive gas reservoir, or development of a secondary gas cap. 11.6.3 External Data. Geology, geophysics, and petrophysics are the primary sources of external data used for reservoir model selection. Geology and geophysics provide information about reservoir shape, lateral extent, and nature of boundaries, evidence for natural fracturing, layering, and possible reservoir compartmentalization. The reservoir model selected for analysis should honor those aspects of the geology, geophysics, and petrophysics that have a significant effect on the reservoir pressure response. 11.7 Estimate Model Parameters Once a reservoir model has been selected, the process of estimating the parameters that describe the model is fairly straightforward. Most of the parameter estimation methods can be classified as straight-line methods, loglog methods, and simulation/history matching methods. Although any of these methods may be used to obtain estimates of reservoir properties, the recommended workflow is to use straight-line or log-log methods to get rough estimates of reservoir parameters, then refine those rough estimates using manual or automatic history matching. Use straight-line methods and/or log-log methods to get rough estimates of reservoir parameters. Use manual or automatic history matching to refine the rough estimates and to ensure that the chosen model fits the complete rate and pressure history. 11.7.1 Straight-Line Methods. In straight-line methods, pressure and/or time-plotting variables are calculated so that data in a specific flow regime fall on a straight line. The slope and intercept of the straight line are used to estimate reservoir parameters specific to the flow regime and model selected for interpretation. Straight-line methods have several advantages and disadvantages. The biggest advantage is that straight line methods are simple and easy to use. Used correctly under the right circumstances, straight-line methods give accurate estimates of reservoir properties. Unfortunately, straight-line methods also have several disadvantages as well. First, for each straight-line analysis method, a specific flow regime must be present to use the method. If the flow regime is not present, straight-line analysis is of limited value. Second, straight-line analysis methods use only the data within a single flow regime; all remaining data is ignored. There is no guarantee that the results from straight-line analysis from two different flow regimes will be consistent. If a flow regime is incorrectly identified, the results from straight-line analysis are meaningless. Finally, straight-line methods for any test other than a constant rate drawdown require the use of flow-regime specific superposition time functions. Implicit in each superposition time function is the assumption that the flow regime in question is the only flow regime present in each superposition term.

288 Applied Well Test Interpretation

In practice, the superposition time function is applicable as long as any superposition terms not exhibiting the flow regime may be considered constant for the duration of the portion of the test data used for analysis. For radial flow, this last condition is usually met provided the last constant-rate flow period before a buildup is at least 10 times the duration of the buildup. The only general way to account for a variable rate history with straight-line analysis is to deconvolve the data first, then apply straight-line analysis to the appropriate portions of the deconvolved unit-rate pressure response. Straight-line methods have the following disadvantages: 1. For each straight-line method, the corresponding flow regime must be present. 2. Straight-line methods use only the data in a single flow regime; all remaining data is ignored. 3. Superposition time functions assume that only the corresponding flow regime exists. Radial Flow—p vs. log ∆t. Data in radial flow usually provide an estimate of the geometric mean permeability in the plane of flow. For a vertical well, the geometric mean permeability is given by kh =

kx k y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.2)

Accurate estimation of permeability from data in radial flow to a vertical well requires a knowledge of the net reservoir thickness, h. Linear Flow—p vs. ∆t½. If the permeability in the direction of flow is known, data in linear flow provide an estimate of the width of the flow path (e.g., channel width, effective lateral length, fracture half-length). If the width of the flow path is known, data in linear flow give the permeability in the direction of flow. Often neither the width of the flow path nor the permeability in the direction of flow are known, giving a range of possible interpretations. As with radial flow, accurate interpretation of data in linear flow requires a knowledge of the net reservoir thickness, h. Volumetric Behavior—p vs. ∆t. Data in volumetric behavior give the volume-compressibility product of some component of the reservoir system. That component may be the wellbore, it may be the entire reservoir, or it may be a small compartment in a much larger reservoir system. Accurate estimation of pore volume from data in volumetric behavior requires a knowledge of the total compressibility, ct. Spherical Flow—p vs. ∆t-½. Data in spherical flow may be used to obtain an estimate of the geometric mean permeability in all three coordinate directions: ks =

3

kx k y kz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.3)

In contrast to radial and linear flow, data in spherical flow do not require a length be known to estimate permeability, simply whether the flow is spherical or hemispherical. Bilinear Flow—p vs. ∆t¼. For a well with a finite-conductivity fracture, either natural or hydraulic, data in bilinear flow may be used to estimate the fracture conductivity, wkf, provided the formation permeability k is known. 11.7.2 Log-Log Methods. Log-log methods rely on a graph of the pressure change and the logarithmic derivative vs. a superposition time plotting function on a log-log scale. As with straight-line methods, the superposition time function implicitly assumes that a single flow regime describes all terms in the superposition. Manual Log-Log Analysis. Manual log-log analysis has almost the same advantages and disadvantages as straight-line analysis. Perhaps the only additional advantage of manual log-log methods over straight-line methods is that the flow regime identification and manual log-log analysis are performed using a single graph, making it less likely that the wrong data will be used for analysis. Type Curve Analysis. Type curve analysis uses the log-log graph of pressure change and pressure derivative, or field data graph, along with a set of solutions to the diffusivity equation for the selected reservoir model, or type curve, in dimensionless form. In a typical application, the vertical position of the field data relative to the type curve is used to estimate the reservoir permeability, while the horizontal position is used to estimate the WBS coefficient or some reservoir length, such as fracture half-length. The shape of the type curve that best matches the field data is used to estimate a third reservoir parameter, such as skin factor. Log-log type curve analysis can be a powerful tool. However, type curve analysis using even a moderately complex reservoir model requires the use of computer-generated type curves. A variable flow-rate history may be taken into account either using a superposition time plotting function, or using superposition type curves. Superposition time functions have the same limitations for type curve analysis as for straight-line analysis. Superposition type curves must be generated specifically for a given rate history.

Well Test Interpretation Workflow 289

11.8 Simulate or History-Match Pressure Response This is one of the most important steps of the interpretation workflow, as it closes the loop on the interpretation. However, it is also one of the steps most frequently omitted. 11.8.1 Simulation. The most general way to account for both variable rate history and complex reservoir models is the use of analytical or numerical simulation along with manual or automatic history matching. Analytical simulation avoids the assumptions that have to be made to use straight-line or type-curve methods, such as the use of superposition time functions, or pressure and time transforms. Numerical simulation can be used to model a wide variety of boundary conditions and/or nonlinear phenomena for which analytical solutions are not available. Analytical Simulation. Analytical simulation uses the principle of superposition in time to calculate the pressure response for a given flow-rate history from the dimensionless solution to the diffusivity equation for constant rate production for a particular set of inner and outer boundary conditions and reservoir media. Analytical simulation is rigorously applicable only for single-phase systems with small and constant compressibility. However, under the right conditions, the use of appropriate pseudopressure and pseudotime functions allow use of analytical simulation for single-phase gas reservoirs as well, even taking into account stress-dependent permeability (Al-Hussainy et al. 1966) and porosity, as well as desorption of gas from the matrix in coalbed methane and naturally fractured shale reservoirs (Spivey and Semmelbeck 1995). Numerical Simulation. Numerical simulation can model virtually any reservoir phenomenon, linear or nonlinear, including multiphase flow, vertical and lateral heterogeneities, and boundaries of arbitrary shape. However, numerical simulation is significantly more time consuming in terms of both data preparation and history matching. 11.8.2 Manual History Matching. In manual history matching, the analyst fixes the known reservoir properties and varies the values of the unknown reservoir properties until the calculated pressure response matches the observed pressure response. Manual history matching can be performed with either analytical or numerical simulation. General Principles. History matching should incorporate the full-rate history using superposition. One of the primary reasons for using simulation/history matching is to confirm results obtained from straight-line and type-curve methods. Use of superposition time functions, such as the Horner time ratio or the Agarwal equivalent time, should be avoided. Use straight-line methods and type-curve matching to get rough estimates of reservoir properties before beginning the manual match process. Use external data to fix as many parameters as possible, then vary the remaining unknown parameters to obtain a match. Work from early time to late time, varying only one or two parameters at a time. Systematically, calculate the change in each parameter necessary to move the simulated pressure curve the direction and distance necessary to obtain a match; don’t just type in numbers at random and hope to get a match. Match the WBS period, then the middle time region, then the nearest reservoir boundaries, and finally the reservoir volume or initial pressure. Use the log-log plot while matching the early and middle time regions. Use the log-log and the Cartesian plot together, along with flow-regime specific straight-line plots if appropriate, for matching late-time data. Use the residual plot to fine tune the match. Recommended Procedure. The following steps usually allow a manual match to be obtained fairly quickly. In Steps 1 through 4, use only the log-log plot of pressure change and pressure derivative during the match. Use both the log-log plot and the Cartesian plot for Step 5. After completing the match for each step, freeze the value of the parameter modified during that step for all subsequent steps. 1. Adjust the WBS coefficient to match the WBS-dominated data. Increase the WBS coefficient to move the simulated pressure response unit slope line to the right (or down); decrease it to move the unit slope line to the left (or up). Use the tic marks on the time axis to estimate how much to change the WBS coefficient. To calculate the new WBS coefficient necessary to shift the unit slope line horizontally:  ∆t  Cnew =  w , obs  Cold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.4)  ∆t w , sim  To shift the unit slope line vertically:  ( t ∆p′)   ∆p  w , sim  Cold =  w , sim  Cold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.5) Cnew =    ∆pw , obs   ( t ∆p′) w , obs 

290 Applied Well Test Interpretation

2. Adjust the permeability to make the simulated pressure derivative match the horizontal part of the field data derivative. Increase the permeability to move the simulated pressure derivative downward, increase permeability to move the derivative upward. Use the tic marks on the pressure change axis to estimate how much to change the permeability. If the field data derivative does not exhibit a clearly defined horizontal period, it will be virtually impossible to get a unique estimate of permeability. If the field data derivative does exhibit a horizontal period corresponding to infinite-acting radial flow, the simulated derivative must match the field data derivative, or the permeability estimate will not be accurate. Use the following equation to calculate the new permeability required to shift both pressure and derivative curves vertically:  ( t ∆p′)  r , sim  kold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.6) knew =   ′ ∆ t p ( )  r , obs  3. Adjust the skin factor to make the simulated pressure change match the observed field data pressure change, during the period corresponding to infinite-acting radial flow. snew = sold +

(∆pr , obs − ∆pr , sim ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.7) 2 ( t ∆p′)r

4. Adjust distances to boundaries to make the simulated pressure derivative match the field-data derivative. Increase the distance to a boundary to move the corresponding change in the derivative to the right; decrease the distance to move the simulated boundary pressure response to the left. Use the time axis to determine how much to change the distance. To shift the pressure derivative response to the right on the time scale by a factor of 4, increase the distance to the boundary by a factor of 2. To shift the response right by 10%, increase the distance to the boundary by 5%.  ∆t obs Lnew =   ∆tsim

  Lold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.8) 

5. Using the Cartesian plot, adjust the initial pressure and/or the reservoir volume to get overall match. Increase the initial pressure to shift the simulated pressure upward by a constant amount; increase the reservoir volume to increase the simulated pressure response at the end of the test while leaving the simulated response at the beginning of production unchanged (in effect, “rotating” the pressure response counterclockwise). 6. To change the permeability and skin factor simultaneously while keeping the productivity index fixed, estimate the new skin factor from k  snew = ( sold + 8)  new  − 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.9)  kold  11.8.3 Automatic History Matching. After rough estimates of reservoir parameters have been obtained using some combination of straight-line analysis, type-curve analysis, and manual history matching, automatic history matching may be used to fine-tune the estimates. Automatic history matching, sometimes called automatic typecurve matching, uses a nonlinear regression algorithm to find the combination of model parameters that minimize an objective function defined as 2  N  f (α ; t j ) − p j  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.10)   F (α ) = ∑  σj  j = 1 

where pj is the pressure recorded at time tj, sj is the standard deviation of the pressure measurement for point j,  α is the vector of the parameters of the model function f, f (α ; t j ) is the pressure predicted by the model function  at time tj, and F (α ) is the objective function to be minimized. Nonlinear regression algorithms can only find a local minimum to the regression problem. However, in matching a field data set with a given model, the goal is to find values of the parameters that give the global minimum. One of the best ways to ensure that automatic history matching finds the desired global minimum rather than another local minimum is to use good starting estimates for the model parameters.

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The following steps are recommended for automatic history matching: 1. Carefully select the data to match; don’t attempt to match data during phenomena that are not included in the model. For example, do not attempt to match data in the transition between the WBS-dominated period and infinite-acting radial flow with a constant WBS/constant skin-factor model if the shape of the transition is not that of a well with constant WBS and skin. 2. Use straight-line methods, type-curve matching, and/or manual history matching to get an approximate match before starting the automatic history-matching process. 3. Simulate the entire rate history; do not use superposition time functions. This is particularly important when confidence intervals are to be calculated. 4. Use external data to fix as may parameters as possible, then vary only the remaining unknown parameters to get a match. 5. Vary all unknown parameters simultaneously for the final match. 11.8.4 Evaluate the Match. Whether using numerical or analytical simulation, the match must be evaluated graphically to determine whether or not it is satisfactory. Evaluate the match using several different graphs rather than relying on a single graph. Cartesian Graph. The Cartesian graph of pressure vs. time is particularly sensitive to the quality of the match of the large scale reservoir features such as reservoir volume and reservoir pressure. On the other hand, the Cartesian graph tends to hide mismatches in the pressure response at early times that might be caused by poor estimates of permeability, skin factor, and WBS coefficient. Log-Log Graph. The logarithmic derivative of pressure is especially sensitive to WBS coefficient, permeability, lateral changes in reservoir quality, and reservoir boundaries. The pressure change curve is sensitive to WBS coefficient, permeability, and skin factor. However, compression of the logarithmic scale for large pressure changes may obscure important behavior. Because the derivative is computed from the pressure, it contains no information not already present in the pressure curve. The pressure curve should be examined closely whenever there is an anomaly in the shape of the derivative. Semilog Graph. In addition to its role in straight-line analysis for infinite-acting radial flow, the semilog graph is also useful for evaluating the pressure match, since it avoids the compression of the pressure scale. Because it uses the same time scale as the log-log graph, anomalies in the derivative can easily be correlated with the corresponding parts of the pressure data. Specialized Graphs. The specialized straight-line graphs for any identified flow regimes should be reviewed to ensure the simulated pressure response matches the field data during those flow regimes. The flow-regime-specific log-log diagnostic plots are especially useful for ensuring that the slopes of the simulated and observed pressure responses are in good agreement. Residual Plots. Residual plots are graphs of the difference between the measured and simulated pressure responses. Because they emphasize small differences between the measured and simulated pressure responses, residual plots play much the same role in model/match validation as the logarithmic derivative does in flow regime identification. Residual plots may be constructed using any desired time scale, but the Cartesian and semilog scales seem to be the most useful. Graphing both the pressure and derivative residuals on a single graph is also useful, but the analyst should always bear in mind that the derivative is calculated from the pressure data and contains no additional information. Look for trends, especially late-time trends. The pressure and derivative residuals should both have slopes of zero during the most important parts of the test. 11.8.5 Manual History Matching Example. This following example illustrates the manual history matching workflow. We need to determine an appropriate permeability for the first simulation run. Using the pressure change during the production period preceding the buildup, we can estimate the permeability from the single phase productivity index, assuming the skin factor is 0, as k= =

1000µ BJ 1000µ B q = h h ∆p

( 575) (1000 ) ( 0.42) (1.48) (11.55) ( 4, 900 − 2, 226.8)

md. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.11) = 11.6 md. We also need a WBS coefficient. In general, it seems to be easier to start out with a WBS coefficient that is too low, rather than one that is too high. We will arbitrarily choose a WBS coefficient of 0.01 psi.

292 Applied Well Test Interpretation

Fig. 11.5 shows the first simulation run. Obviously, the field data show the well has a significant degree of damage. To improve the match, we need to move the derivative down by approximately 1 log cycle or, in other words, increase the permeability by a factor of 10. However, the pressure change seems to be about right, so we need to increase the skin factor to keep the same productivity index. The WBS coefficient also seems to be too low, perhaps by a factor of 2. To increase the permeability by a factor of 10 without changing the productivity, so we calculate the new skin factor from k  snew = ( sold + 8)  new  − 8  kold   120  = ( 0 + 8)   −8  12  = 72. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.12)

Pressure Change, Logarithmic Derivative, psi

Fig. 11.6 shows the second simulation run, with permeability increased to 120 md, skin factor increased to 72, and WBS coefficient increased to 0.02 bbl/psi. It looks like we need to increase the permeability another 20% to shift the derivative downward into better alignment with the field data. 10000

1000

k = 12 md s=0 D = 0 D/STB C = 0.01 bbl/psi

100

Shift down 1 log cycle

10

1 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr

Pressure Change, Logarithmic Derivative, psi

Fig. 11.5—Run 1, permeability estimated from drawdown assuming zero skin factor. 10000

1000

k = 120 md s = 72 D = 0 D/STB C = 0.02 bbl/psi

100

Shift down 20%

10

1 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.6—Run 2, permeability increased by a factor of 10; skin factor adjusted to maintain same productivity index.

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Fig. 11.7 shows the log-log graph for Run 3. The derivative match at the end of the test is about as good as we can do with manual history matching. The residual graph in Fig. 11.8 shows that the simulated pressure change curve is about 125 psi below the observed pressure change. The log-log graph, Fig. 11.7, is much less sensitive to differences between the simulated and observed pressure change, because of the compression of the pressure on the log-log scale. Because the derivative match appears reasonable, we will change the skin factor to match the pressure change. The additional skin factor required to account for a pressure difference of 125 psi may be calculated from the defining equation for skin factor as 0.00708 kh ∆p 0.00708kh ∆ ∆ss = = ∆p qB µ qBµ ( 0.00708 0.00708) (144 144 ) (11.5 11.5) (125) = 125 = 1.48 0.42 ( 575 ) ( ) ( 575 1.48 0.42)

= 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.13) = 4.1

Pressure Change, Logarithmic Derivative, psi

For the next run, we increase the skin factor from 72 to 76.1. 10000 k = 144 md S = 72 D = 0D/STB C = 0.02 bbl/psi

1000

100

10

1 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.7—Run 3, permeability increased by an additional 20%.

800 600

Pressure residual Derivative residual

Residual, psi

400 200 0 −200 −400

125 psi low

−600 −800 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.8—Run 3, pressure residual shows that the calculated pressure change is too small, while the derivative is good.

294 Applied Well Test Interpretation

Figs. 11.9 and 11.10 show the log-log and residual graphs for the final match with the constant WBS and skin factor model. The residual graph, Fig. 11.10, shows the pressure and derivative curves matching very well from 20 hours through the end of the test. To obtain a better match, we must incorporate additional phenomena to describe the shape of the transition from the WBS unit slope line to the horizontal infinite-acting radial flow derivative. One option would be to use a variable WBS model, but the unit slope line seems too well defined to justify this option. In a buildup test, a high rate-dependent skin factor can cause a steep derivative during the transition out of WBS, so we replace the Darcy skin factor with the equivalent non-Darcy skin; the required non-Darcy flow coefficient D is given by D=

s 76.4 = = 0.1323 D STB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.14) q 575

Pressure Change, Logarithmic Derivative, psi

The log-log graph for Run 5 is shown in Fig. 11.11. The match of the WBS period and from 10 hours through the end of the test is quite good, but the steep derivative is not matched very well. We can shift the transition to the left by decreasing the WBS coefficient; we graphically estimate the required decrease as 20%.

10000

1000

k = 144 md S = 76.1 D = 0D/STB C = 0.02 bbl/psi

100

10

1 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.9—Run 4, final match with constant WBS and skin factor model.

800 Pressure residual Derivative residual

600

Residual, psi

400 200 0 −200 −400 −600 −800 0.001

0.01

0.1

1

10

100

Shut-In Time, hr Fig. 11.10—Run 4, pressure change and derivative residual plot.

1000

Pressure Change, Logarithmic Derivative, psi

Well Test Interpretation Workflow 295 10000

1000

k = 144 md s=0 D = 0.1323 D/STB C = 0.02 bbl/psi

100

10

1 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.11—Run 5, replace Darcy skin with equivalent non-Darcy skin.

Figs. 11.12 and 11.13 show the log-log and residual graphs, respectively, for the final match using the nonDarcy skin model. Decreasing the WBS coefficient improved the match of the transition period, at the expense of a poorer match during the WBS-dominated period. We could probably obtain a better match by introducing a variable WBS model with a 20% decrease in WBS. However, is there any likelihood that the resulting match would give us any additional information that would dictate a different remediation strategy? The match of the last 1½ log cycles, or roughly last 97% of the test on a linear scale, is excellent; any alternate match would have to match this portion of the test as well or better. Given that the effects of small decrease in WBS will decay very rapidly, the effort required to obtain a match of the entire test does not seem to be justified. 11.9 Calculate Confidence Intervals If the right conditions are met, confidence intervals for individual parameters or a joint confidence region for the solution may be calculated. A two-step process is recommended. First, validate the assumptions on which confidence limits are based. If the assumptions are valid, calculate the confidence limits.

Pressure Change, Logarithmic Derivative, psi

11.9.1 Key Assumptions. Nonlinear regression involves finding the set of parameters that minimizes the objective function given by Eq. 11.10. For nonlinear regression to be useful, the parameters of the model function must 10000

1000

k = 144 md s=0 D = 0.1323 D/STB C = 0.016 bbl/psi

100

10

1 0.001

0.01

0.1

1

10

100

Shut-In Time, hr Fig. 11.12—Run 6, WBS coefficient decreased by 20%. Final match.

1000

296 Applied Well Test Interpretation 200

100

Pressure residual Derivative residual

Residual, psi

0 −100 −200 −300 −400 0.001

0.01

0.1

1

10

100

1000

Shut-In Time, hr Fig. 11.13—Run 6, pressure change and derivative residual plot. Final match.

provide accurate estimates of reservoir properties. Used correctly, confidence intervals can quantify the level of accuracy of the parameters obtained through nonlinear regression. The following assumptions form the basis for calculating confidence intervals for parameter estimates obtained by nonlinear regression. 1. The measured data comprise the sums of two components, a deterministic component and a noise component. 2. The deterministic component is described exactly by the model function. In other words, for some combination of parameters (the true solution), the model function reproduces the deterministic component of the measured data exactly. 3. The noise component, representing the total measurement error, is the result of a random process (i.e., measurement errors are randomly distributed). 4. Measurement errors are independently distributed. 5. Measurement errors are normally distributed with zero mean and uniform variance. If the above assumptions are satisfied, it can be shown that the solution of Eq. 11.10 is a maximum likelihood estimator for the true solution. To develop confidence intervals as normally used, another assumption must be made: 6. Near the solution point for Eq. 11.10, the model function can be accurately approximated by a linear combination of the model parameters. These assumptions should be kept in mind when calculating or using confidence limits. 11.9.2 Common Misconceptions. Horne emphasizes the requirement that the reservoir model be correct (Horne 2009): “A confidence interval does not define the range of possible values for an unknown parameter; it defines the range over which one can have confidence in the estimate, assuming that both the data and the reservoir model are correct.” Unfortunately, even small, seemingly innocuous, deviations from ideal behavior are sufficient to render confidence limits meaningless, or worse, dangerously misleading (Spivey 2013). The standard log-log diagnostic plot of pressure change and pressure derivative is not sensitive enough to detect the subtle differences that indicate that an approximate model provides a poor match of the data, thus the resulting confidence limits are invalid. Assuming that the model used for nonlinear regression is correct, without adequate validation, gives rise to the following misconceptions:

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Misconception 1: Match All of the Data To Get the Best Answers. Reality—Unless the model is exact, matching all of the data actually distorts the answer. It is better to restrict the range of data used for the match to only that part of the data that can be matched accurately with an approximate model. Misconception 2: The True Answer Lies Within the Confidence Intervals. Reality—If the measurement error is known, and if confidence intervals are calculated using the known measurement error, confidence intervals obtained using an approximate model give a range about the biased answer that would be obtained by matching the (unknown) deterministic portion of the data with the approximate model. Misconception 3: Nonlinear Regression Can Extract Good Answers From Bad Data. Reality—“Garbage in, garbage out.” Theoretically, if all of the assumptions are met, including knowing the exact model, then accurate results can be obtained from data entirely within a transition period (such as the transition from WBS to infiniteacting radial flow). Practically, we simply do not have exact models. As a consequence, we cannot salvage data lying entirely within transition periods using approximate models. Misconception 4: Narrow Confidence Intervals Verify Match Is Acceptable. Reality—Confidence limits are dangerously misleading if an approximate model is used without first verifying that the residuals are independently, randomly distributed. A recent Monte Carlo simulation study found that out of 360 series of runs matching synthetic data sets with approximate models, in about one-third of the cases, a match with an approximate model failed either the chi-square goodness-of-fit test or the runs randomness test (Spivey 2013). Without exception, every case that failed either the chi-square test or the runs test passed published heuristic criteria for evaluating confidence intervals. On the other hand, most of the cases that passed both chi-square and runs tests failed the confidence interval criteria. Misconception 5: Wide Confidence Intervals for One or More Parameters Invalidates Entire Match. Reality—If the model is accurate, wide confidence intervals on a single variable indicate that that parameter alone cannot be determined uniquely from the match. Wide confidence intervals on two or more parameters may indicate that neither individual parameter can be uniquely determined, or that some combination of those parameters tend to offset each other (e.g., permeability and skin factor for data in pseudosteady-state flow). The parameters that do have narrow confidence intervals are uniquely determined, provided the model is correct. In many cases, a more rigorous analysis of confidence intervals may reveal that one of the parameters is very poorly determined, but the others are accurately estimated. 11.9.3 Examples—Approximate Models. Spivey (2013) studied the consequences of violating the standard assumptions, specifically the assumptions that (1) the model is exact and (2) the errors are normally distributed. He selected several models that were slight variations of the constant WBS and skin factor in an infinite-acting reservoir; the alternative models deviated from the standard model by only 2 to 5 psi (0.7 to 2% of the pressure change during the flow period preceding the buildup). He then performed a Monte Carlo simulation study, generating synthetic data sets with the alternate models. He used the standard WBS and skin model to match the synthetic buildups, varying initial pressure pi, permeability k, skin factor s, and WBS coefficient C to get the best fit. He also calculated confidence limits for each of the parameters. Example 1. Figs. 11.14 and 11.15 show the semilog and log-log graphs, respectively, for one of the matches. From the semilog graph, Fig. 11.14, the match appears to be almost perfect, certainly better than the vast majority 5000 Obs Fit

pws, psi

4900

4800

4700

4600 0.0001

0.001

0.01

0.1 ∆t, hr

1

10

Fig. 11.14—Model 2a, 0–1 hour, Run 1, semilog graph.

100

298 Applied Well Test Interpretation

∆p, t(dp/dt ), psi

1000 Obs Fit 100

10

1 0.0001

0.001

0.01

0.1 ∆t, hr

1

10

100

Fig. 11.15—Model 2a, 0–1 hour, Run 1, log-log graph.

of field data set matches. The test appears to have reached the semilog straight line, thus assuring us that we can get an accurate estimate of permeability. The log-log graph, Fig. 11.15, appears to confirm the quality of the match, although the horizontal derivative is not as well defined as, and does not last as long as, we would prefer. If this were a field data set, it is unlikely anyone would question the results of this match. The quality of the match is further supported by the statistics shown in Table 11.6. The confidence limits are all well within published heuristic guidelines (Horne 2009). So, is this the correct answer? Let us dig a little deeper. Fig. 11.16 shows the residual plot, a graph of the differences between the pressure calculated from the match and the observed pressure. If the model were exact, the residual plot would show a horizontal band of points scattered above and below the horizontal axis. In this case, the residual plot shows a definite trend, with the trend crossing the horizontal line four times (not coincidently, the same as the number of parameters used in the match). This is a clear indication that the model is an approximate model. The differences between the (unknown) true model and the approximate model used for the match are three or four times larger than the random noise in the data. TABLE 11.6—RESULTS OF HISTORY MATCH, MONTE CARLO SIMULATION, MODEL 2A, 0–1 HOUR, RUN 1 Parameter

Matched Value

Confidence Interval

Heuristic Guideline

pi

4988.97 psi

± 0.68 psi

± 3 psi

k

353.2 md

± 1.50%

± 10%

S

14.85

± 0.31

±1

C

0.049 bbl/psi

± 0.27%

± 10%

0.6

pfit - pobs, psi

0.4 0.2 0 −0.2 −0.4 −0.6 0.0001

0.001

0.01

0.1 ∆t, hr

1

10

100

Fig. 11.16—History Match, Model 2a, 0–1 hour, Run 1, residual plot.

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The lag plot, shown in Fig. 11.17, confirms that the residuals are correlated, violating one of the assumptions used to develop the theory for confidence limits. So, how do the results from the match compare with the values used to generate the data set? Table 11.7 shows a comparison between the values obtained from the match and the true values. All of the four parameters are in error by 7 to 20 times the calculated confidence limits. The parameter having both the smallest confidence interval and the least error, the WBS coefficient, is the one parameter we are least interested in. Table 11.8 shows summary statistics for 1,000 Monte-Carlo simulation runs of the same scenario. This example exposes Misconceptions 1 through 4, showing that (1) matching all of the data may cause significant errors; (2) the true answer may lie far outside the confidence intervals obtained with an approximate model; (3) nonlinear regression cannot extract good answers from data entirely within a transition; and (4) narrow confidence intervals alone do not guarantee that a match is acceptable. Example 2. Table 11.9 shows summary statistics for a second set of runs with the same model, over a time range of 0.5 to 3 hours. The statistics for this set of runs show almost perfect estimates for initial pressure, permeability, and skin factor, with very narrow confidence intervals, while the WBS coefficient is almost 20% too low, with a confidence interval of ±30%. The residual plot for the first run in this set, Fig. 11.18, shows that the 0.6

(pfit - pobs)i–1, psi

0.4

0.2

0

−0.2 −0.4 −0.6 −0.6

−0.4

−0.2 0 0.2 (pfit - pobs)i, psi

0.4

0.6

Fig. 11.17—History match, Model 2a, 0–1 hour, Run 1, lag plot.

TABLE 11.7—RESULTS OF HISTORY MATCH COMPARED TO TRUE VALUES, MODEL 2A, 0–1 HOUR, RUN 1 True Value

Confidence Interval

Actual Error

4988.97 psi

5000 psi

± 0.68 psi

–11.03 psi

353.2 md

273.5 md

± 1.50%

29.1%

S

14.85

10

± 0.31

4.85

C

0.049 bbl/psi

0.05 bbl/psi

± 0.27%

–1.93%

Parameter

Matched Value

pi K

TABLE 11.8—SUMMARY STATISTICS FOR MONTE CARLO SIMULATION, MODEL 2A, 0–1 HOUR Value

Median

Bias

Average Confidence Interval

Monte Carlo Confidence Interval

pi

5000 psi

4988.81 psi

–11.19 psi

± 0.65 psi

± 0.28 psi

k

273.5 md

354.4 md

29.6%

± 1.45%

± 0.63%

S

10

14.91

4.91

± 0.30

± 0.09%

C

0.05 bbl/psi

0.049 bbl/psi

–1.82%

±0.26%

±0.11%

Parameter

300 Applied Well Test Interpretation TABLE 11.9—SUMMARY STATISTICS FOR MONTE CARLO SIMULATION, MODEL 2A, 0.5–3 HOUR Value

Median

Bias

Average Confidence Interval

Monte Carlo Confidence Interval

pi

5000 psi

4999.83 psi

–0.17 psi

± 1.08 psi

± 1.09 psi

k

273.5 md

274.4 md

0.47%

± 2.50%

± 2.49%

Parameter

S

10

10.08

0.08

± 0.42

± 0.42

C

0.05 bbl/psi

0.041 bbl/psi

–17.4%

± 30.10%

± 29.15%

0.3

pfit - pobs, psi

0.2 0.1 0 −0.1 −0.2 −0.3 0.1

1 ∆t, hr

10

Fig. 11.18—History match, Model 2a, 0.5–3 hour, Run 1, residual plot. 0.3

(pfit - pobs)i–1, psi

0.2

0.1

0

−0.1 −0.2 −0.3 −0.3

−0.2

−0.1 0 0.1 (pfit - pobs)i, psi

0.2

0.3

Fig. 11.19—History match, Model 2a, 0.5–3 hour, Run 1, lag plot.

residuals are evenly distributed in a uniform band around the horizontal axis, while the lag plot, Fig. 11.19, shows that the residuals are uncorrelated. The linearized 95% confidence intervals agree very well with 95% confidence intervals obtained from Monte Carlo analysis. This example exposes Misconception 5, showing that wide confidence intervals for a single parameter do not invalidate the entire match, provided the model is accurate over the range of data used for the match. 11.9.4 Validate Assumptions. If confidence limits on the parameters are required, the assumptions on which they are based must be tested. The randomness assumption may be checked using the following suite

Well Test Interpretation Workflow 301

of plots: (1) a scatter plot of the observed and model responses vs. time, (2) a scatter plot of the residuals vs. time, and (3) a lag plot of the residuals (Spivey 2013). Scatter Plot of the Observed and Model Responses vs. Time. This should include the log-log plot of pressure change/pressure derivative, the semilog and/or other straight-line plots, and the Cartesian plot. These are the plots normally used to evaluate model validity, but they must be supplemented by the more sensitive methods discussed below. Scatter Plot of the Residuals vs. Time. This plot should be generated with both linear and log-log scales for the time axis. If the assumptions are met, the scatter plot should show an even distribution of residuals in a band of uniform width centered on the x-axis. Ideally, the width of the band would be consistent with the measurement error comprising gauge resolution and environmental noise. In the worst case scenario, the residuals will fall on a sinuous curve crossing the x-axis the same number of times as the number of parameters in the model, as in Fig. 11.16. When this occurs, the interpretation model does not adequately describe the field data. It may be necessary to restrict the range of data used for the match to a part of the data that is adequately described by the model, as in Fig. 11.18. This plot is also very useful for identifying subtle tidal and diurnal effects that would not otherwise be evident. Lag Plot of the Residuals. The lag plot is used to detect whether consecutive residuals are independent. When the residuals are independent, the lag plot will show a random cloud of points centered on the origin, with roughly equal density of data points in each of the four quadrants, as in Fig. 11.19. Concentration of data points in the first and third quadrants on the lag plot, as in Fig. 11.17, indicates that the model does not provide an adequate description of the field data. If the above plots show that the assumptions that support the use of confidence limits are satisfied, the confidence interval may now be estimated. If the assumptions are not met, it may be possible to find a model that better describes the data, or it may be possible to restrict the range of data used for the fit to that part of the pressure response during which the model does describe the data. If neither of these two alternatives give a model that meets the assumptions, confidence limits must not be calculated. 11.9.5 Calculate Confidence Intervals. If the underlying assumptions are valid, calculate the confidence intervals for each parameter. In many cases, it may be impossible to find a model that is consistent with the geology and matches the data well enough to satisfy the standard assumptions. If this occurs, confidence limits should not be calculated. Mosteller and Tukey (1977) address this issue in the context of statistics in general when they comment. “Statistics courses tend not to emphasize indication*; instead, they concentrate upon how to express the uncertainties of particular kinds of indications. In impressing the student with the importance of assessing uncertainties** when one can (and can afford the effort), it is perhaps inevitable that they may give the impression that indications whose uncertainties have not been assessed are worthless. This is, of course, just not so. We need to assess uncertainties vigorously and often, but we also need to look at indications of unassessed certainty, especially where assessment is impossible or uneconomical.” The raison d’être of well testing is, of course, to help us make better reservoir management decisions. Reservoir management inherently requires making decisions on the basis of incomplete information. As well test interpreters, we have a responsibility to provide the best answers we can, including confidence intervals when appropriate, but refraining from giving confidence limits when they are not. 11.10 Interpret Model Parameters One step that is often overlooked in pressure-transient interpretation is to interpret the parameters obtained from the parameter estimation process in terms of physical reservoir properties. This is particularly true of the composite reservoir models and the dual porosity reservoir models. 11.10.1 Composite Models. The radial and linear composite reservoir models are defined in terms of the distance to the interface between two regions, along with two parameters describing the contrast in reservoir properties in the two regions, usually mobility ratio, M, and either diffusivity ratio, D, or storativity ratio, S, defined as M = ( k µ )2 ( k µ )1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.15) *

Parameter estimates. Calculating confidence intervals.

**

302 Applied Well Test Interpretation

S = (φct )2 (φct )1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.16) and D = M S = ( k φµct )2 ( k φµct )1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.17) respectively, where the well is located in Region 1. Often, an analyst will present a match obtained using some combination of mobility and storativity ratios that fits the observed rate and pressure history, without making any attempt to explain the ratios in terms of reservoir rock and fluid properties. This practice is strongly discouraged for three reasons. First, using M and S simply as matching parameters that may be varied independently ignores the fact that both M and S are functions of reservoir rock and fluid properties and must not be varied arbitrarily. Second, the ratios M and S are not directly comparable to any external data, thus cannot be cross checked with external data without interpreting the ratios in terms of reasonable rock and fluid properties. Third, the mobility and storativity ratios M and S by themselves have only limited use in making reservoir management decisions. Recommended practice for using a composite reservoir model is to interpret the mobility and storativity ratios M and S in terms of reasonable rock and fluid properties, taking into account differences in lithology, fluid saturation, total compressibility, and relative permeability between the two regions. The distance to the interface between the regions should then be cross checked with external data (e.g., using mass balance to estimate the distance to the oil zone in a test on a water injection well). 11.10.2 Dual Porosity Models. The dual porosity models are usually described in terms of a storativity ratio, w, and an interporosity flow coefficient, l. The storativity ratio w is defined as

ω=

V f φ f ctf V f φ f ctf + Vmφm ctm

=

(V φct ) f , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.18) (V φct ) f + (V φct )m

while the interporosity flow coefficient l is defined as

λ = αrw 2

km , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.19) k

where

α=

4 j ( j + 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.20) L2

Unfortunately, although the parameters w and l are defined in terms of fracture spacing and porosity, there seems to be very little indication in the literature that w and l are useful for estimating physical reservoir properties. Indeed, in one of the early papers on application of the dual porosity models, Crawford et al. (1976) state that “little interpretative use was made of the values for the parameters l and w” and “the parameters l and w derived from the model have only limited value for calculating fracture spacing and porosity.” Thirty years later, the situation seems to be unchanged; Narr et al. (2006) state, “Although dual-porosity well test analysis presents a clear set of analytical tools, in practice its usefulness is limited.” Recommended practice for naturally fractured reservoirs is to first consider alternative models that may also fit the pressure-transient response, such as the well near a single high-conductivity natural fracture (Cinco-Ley et al. 1976) or a finite-conductivity fault (Abbaszadeh and Cinco-Ley 1995), while providing a clear physical interpretation of the model parameters. If no other model can be found that fits the pressure response, then attempt to relate the dual porosity parameters to physical reservoir properties. 11.10.3 Convert Average Pressure to Datum. If average reservoir pressure is to be used for material balance calculations, the average drainage area pressure obtained from well test interpretation must be converted to pressure at the reservoir datum elevation (Brownscombe and Conlon 1946). Converting pressure at gauge depth to datum is a two-step process. First, the gauge pressures must be converted to pressure at the top of perforations using a wellbore fluid hydrostatic gradient, taking into account the fact that there may be multiple fluid gradients (gas, oil, oil/water

Well Test Interpretation Workflow 303

emulsion, or water) between the gauge and the completion. This conversion may be done with the raw gauge data, or after the drainage area pressure (referenced to the gauge depth) has been estimated. Second, the pressure at top of perforations should be converted to datum elevation using a hydrostatic reservoir pressure gradient. This conversion may again require the use of multiple fluid gradients. 11.11 Validate Results Another important step in the workflow is to validate the results of the interpretation. The validation process includes (1) performing simple reality checks to make sure that results are reasonable, (2) checking for consistency for estimates of the same parameter from different flow periods, (3) comparing test results with those from previous tests on the same well, (4) calculating the radius of investigation at the beginning and end of each flow regime, (5) simulating the complete rate and pressure history and comparing the simulated and observed pressure histories, and (6) comparing parameter estimates with external data. If the analysis passes all of the internal validity checks, yet fails the comparison of parameter estimates with external data, it may be necessary to re-evaluate the external data to determine if an alternate interpretation of the external data is reasonable. If after re-examination of alternative interpretations of external data there are still inconsistencies, the selected model should be rejected, and interpretation should be attempted using a different model. 11.11.1 Reality Check. Reality checks serve to allow immediate rejection (or at least re-examination) of results that are obviously wrong. Skin Factor. For wells that have been stimulated, the skin factor cannot be more negative than given by the equation r  smin = − ln  e  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.21)  rw  where smin is the most negative skin factor possible, re is the drainage radius, and rw is the wellbore radius. The skin factor is not likely to be more negative than –3 to –5. Table 11.10 shows the expected skin factor for several types of completions, under favorable conditions. Local experience may suggest that higher skin factors are appropriate for a particular field or formation, or that skin factors lie within a more restricted range. However, the likelihood that significantly more negative skin factors occur is very low. There is no limit to how high damage skin factors can be. In many cases, a well will not flow before a breakdown treatment or other stimulation treatment is performed. The skin factor in these cases is effectively infinite. Large, but finite, skin factors become increasingly difficult to measure at values larger than a few hundred. One useful rule of thumb is that the reservoir pressure drop during pseudosteady-state flow for typical well spacing is roughly equal to the additional pressure drop because of a damage skin of 7. Thus, a skin factor of 700 would mean that the additional pressure drop because of skin is roughly 100 times the total pressure drop in the reservoir, or equivalently, that the pressure drop in the reservoir is only 1% of the total pressure drop during pseudosteady-state flow.

TABLE 11.10—EXPECTED SKIN FACTOR FOR VARIOUS COMPLETION TYPES UNDER FAVORABLE CONDITIONS Type of Completion

Typical Skin Factor Range*

Openhole completion, homogeneous reservoir

0

Openhole completion, naturally fractured reservoir

–1 to –3**

Acid treatment, homogeneous reservoir

–1 to –3

Acid treatment, naturally fractured reservoir

–3 to –7**

Small hydraulic fracture treatment

–3 to –5

Large hydraulic fracture treatment

–4 to –6

Cased-hole gravel pack

+8 to +20

Openhole gravel pack

+2 to +10

Frac pack

0 to +8

* These ranges are for favorable conditions. Local experience may suggest higher skin factors are more typical for a given area, but significantly lower skin factors are very unlikely. ** After Gringarten (1984).

304 Applied Well Test Interpretation

WBS Coefficient. Normally, the WBS coefficient (or the adjusted WBS coefficient for a gas well) will be no larger than 0.05 to 0.2 bbl/psi, unless some part of the reservoir, such as a natural fracture network or highpermeability streaks of very limited areal extent, contributes to the effective WBS coefficient. Distance to Boundaries and Drainage Area/Volume. Unrealistic estimates of distances to boundaries and/or drainage area may indicate a large error in the total compressibility (e.g., using undersaturated oil compressibility instead of saturated oil compressibility), or vice versa, or assuming low formation compressibility in stresssensitive formations. 11.11.2 Check Consistency Between Flow Periods. If manual or automatic history matching is used to refine parameter estimates as recommended, this step is redundant. WBS Coefficient. If the WBS coefficients from different test periods are inconsistent, are there actual differences in the WBS coefficient (e.g., downhole shut-in for one test period and surface shut-in for another)? WBS coefficient in a naturally fractured carbonate may increase by an order of magnitude or more following an acid stimulation treatment (Gringarten 1984). Consistently higher apparent WBS coefficient during flow periods than during buildup periods may be caused by non-Darcy skin factor (Spivey et al. 2004). Unresolved differences in the WBS coefficient for different flow periods may indicate an error in flow rate for one or both test periods. Permeability. Unless there are saturation changes in the reservoir or permeability is stress dependent, permeability should be the same for each buildup. Differences in permeability between test periods may be indication of errors in flow rate. If two periods should give the same WBS coefficient and permeability, and they differ by the same amount in the same direction (e.g., period B has 20% higher WBS coefficient and 20% higher permeability than period A) an error in flow rate is probably the cause. This can be graphically checked very easily by graphing the rate normalized pressure change and pressure derivative for each test period on a single log-log graph. If the data for one test period appear shifted up or down relative to the other test periods while retaining the same overall shape, the flow rate for the inconsistent test period is probably too low or too high, respectively. Skin Factor. The WBS coefficient and permeability should be checked before comparing skin factors to make sure that flow rates are consistent between test periods. If the WBS coefficients and permeabilities are consistent, then the skin factors should be compared. There are a number of possible reasons for changes in skin factor between test periods, including rate-dependent skin factor, cleanup causing decreasing skin with time, damage causing increasing skin factor with time, and transition from single-phase to two-phase flow as the pressure drops below the bubblepoint or dewpoint pressure in the near-wellbore region. Skin factors for multiple test periods may be quickly compared by a graph of the rate-normalized pressure change and pressure derivative vs. time on a log-log scale. If the rates have been reconciled, the derivatives for the all test periods should follow a single trend, while the pressure change curves will coincide only if the skin factors for the corresponding test periods are roughly the same. Boundaries. If boundaries are seen during two different test periods, the boundaries should be consistent in type and distance. 11.11.3 Check Consistency With Previous Tests. Checking for consistency with previous tests is similar to checking for consistency between test periods, keeping in mind that major changes, especially in the bulk reservoir and boundaries, are more likely to have occurred between tests months or years apart. 11.11.4 Calculate Radius of Investigation at Beginning and End of Each Flow Regime. Calculate the radius of investigation at the beginning and end of each flow regime using the radius of investigation equation ri =

kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.22) 948φµct

11.11.5 Compare Simulated Flow Rate/Pressure History With Measured Pressures. If manual or automatic history matching was not used to refine parameter estimates as recommended, the parameters estimated using straight-line methods or log-log methods must be verified by simulating the complete flow rate/pressure history and comparing the simulated pressures with the observed pressure data. The comparison must include a Cartesian graph of the entire pressure history, log-log graphs of each individual test period, and flow-regime specific straight-line graphs of any flow regimes interpreted with straight-line analysis. 11.11.6 Compare Parameter Estimates With External Data. As an additional check, the parameter estimates obtained from the interpretation should be compared with the available external data.

Well Test Interpretation Workflow 305

Skin Factor. The skin factor should be consistent with the stimulation history of well. A large negative skin for an unstimulated well may indicate the presence of either natural fractures (Gringarten 1984) or high permeability streaks or channels (Corbett et al. 1996). Skin factors more negative than –6 are not likely to occur because of the large rwa required to give a high negative skin factor. Even for a well with a massive stimulation treatment, a large negative skin factor may not be observed because of the large drainage area required to see infinite-acting radial flow before encountering boundaries. In typical low-permeability reservoirs where long hydraulic fractures are used, well spacing is usually small enough that pseudoradial flow never occurs. WBS Coefficient. A WBS coefficient obtained from pressure-transient analysis should be close to the value calculated from wellbore geometry. A test-derived WBS coefficient significantly higher than the value calculated from wellbore geometry may be caused by the presence of natural fractures in good communication with the wellbore or a high-permeability streak with limited areal extent. A test-derived WBS coefficient significantly smaller than that calculated from wellbore geometry may indicate that phase segregation has caused the apparent WBS coefficient to be lower than the true value. Permeability. Permeability should be compared to core permeability data, if available. Core permeability significantly higher than that obtained from pressure-transient analysis suggests that overburden and saturation corrections have not been applied, the wrong reservoir model has been used, or the infinite-acting radial flow regime has been incorrectly identified. Core permeability significantly lower than that obtained from pressure transient analysis suggests that the reservoir may be naturally fractured. Distance to Boundaries and Drainage Area/Volume. Inconsistencies between estimates of distances to boundaries and/or drainage area derived from well test data and those derived from external data may indicate an error in either permeability or in total compressibility. Large errors in total compressibility may arise from using undersaturated instead of saturated oil compressibility, or vice versa, or from assuming low-pore-volume compressibility in stress-sensitive formations. An inconsistency in distance to a fault estimated from well test data and that obtained from seismic data may arise from the boundary observed in the pressure transient data being caused by a fault not clearly resolved by the seismic data. Average Reservoir Pressure. Because the reservoir pressure typically declines over the life of a well, it may be more difficult to cross validate than other parameters. For new wells, average reservoir or drainage area pressure may be compared to estimates from wireline formation testing, from regional pressure gradients, or even from mud weight. For existing wells, the average drainage area pressure should be compared to regional trends or results of earlier tests. For any test, all necessary datum corrections must be made before comparing pressure data from different sources. When using initial pressure as a history-matching parameter, keep in mind that the reservoir model used for the match must accurately describe reservoir behavior for the entire duration of the production period as well as the test period for the resulting initial pressure estimates to be valid. The average reservoir pressure at the beginning of the test must be calculated from the initial pressure estimate, the model reservoir volume, and the observed production history using material balance. 11.11.7 Critical Evaluation of Well Test Interpretation. So far, we have discussed the work flow for well test interpretation in the context of doing a complete interpretation. We recommend elements of the work flow also be applied for reviewing a well test interpretation report prepared by another analyst. Although the full work flow is not appropriate, because we rarely have enough data to redo the interpretation from scratch, there is a long list of questions that should be asked. 1. What was the purpose of the test? 2. Does the interpretation pass a reality check? Are the reservoir rock and fluid data reasonable? Are the parameter estimates within the range of values expected for the type of well or reservoir in question? Are any out-of-range values noted and explained? 3. Is the interpretation internally consistent? If the interpretation is not internally consistent, the interpretation is likely to be invalid. 4. Does the interpretation take into account geology, geophysics, and petrophysics? 5. Are the bottomhole pressure and time at the beginning of each test period identified correctly? 6. How much of the rate history is included in the interpretation? 7. Are the flow regimes, if any, identified correctly? Does each identified flow regime last at least one-third to one-half log cycle? Are the flow regimes in a logical sequence? 8. Are wellbore and reservoir phenomena identified and handled (or ignored) correctly? Do not try to match wellbore phenomena with a reservoir model.

306 Applied Well Test Interpretation

9. Is the interpretation model appropriate? Is it consistent with the geology and petrophysics? Does it explain the pressure behavior, especially any significant late-time trends? 10. Are there other models that might be appropriate that were not considered? 11. Are the analysis methods used appropriate? 12. For straight-line methods, does the straight-line fit the correct part of the data? If a superposition time function is used, is it used appropriately? (Remember that superposition time functions assume that a single flow regime adequately describes the pressure response for all superposition terms.) 13. For type-curve methods or history matching, does the field data derivative show a distinct horizontal line? If so, does the horizontal part of the model pressure derivative response overlay the field data derivative? Does the model derivative response overlay the field data derivative during other well-defined flow regimes? Field data often show deviations from ideal behavior during transitions, so it is less important to match data during such transitions. 14. For type-curve methods or history matching, do the model pressure change and derivative responses agree with the field data during the last one-third to one-half log cycle? This is especially important if the test ends before reaching infinite-acting radial flow. Estimating permeability in such situations should be treated as extrapolation. Although unique estimates of permeability and skin factor cannot be determined, lower bounds may be obtained by finding the smallest permeability and skin factor that match the final portion of the test data, provided WBS is constant, the reservoir is homogeneous, and there are no nearby reservoir boundaries or fluid contacts. 15. If the interpretation includes estimates of distances to boundaries, reservoir pore volume, or fluid in place, are the correct fluid and pore volume compressibilities used? 16. What is the radius of investigation at the beginning and end of the middle time region? What is the radius of investigation at the end of other flow regimes present in the test? 17. Does the interpretation include a simulation of the pressure response using the full-rate history? It is especially important to simulate the complete rate history, instead of using superposition time functions, if the well is still in WBS at the end of the flow period, or if the duration of the shut-in period is comparable to or longer than the producing time. 18. Do the bottomhole pressures from the simulation match the field data at the beginning of each flow and shut-in period? Keep in mind that the log-log graph shows pressure change rather than pressure. 19. Are enough different graphs presented to evaluate the match? While a log-log graph is a concise summary of the pressure response, important phenomena may be overlooked if other graphs are not used. At a minimum, graphs comparing field data and model response should include a log-log graph of pressure change and logarithmic derivative vs. time, a Cartesian graph of pressure vs. time, and a semilog graph of pressure vs. time. A semilog graph of pressure change and derivative on a logarithmic scale vs. time on Cartesian scale is recommended when the late-time match is critical. 20. If confidence intervals are presented, is there a residual graph? If so, are the residuals randomly distributed about the x-axis? If the residuals are not randomly distributed about the x-axis, the model function does not reproduce the deterministic component of the field data, and the resulting confidence intervals are meaningless. 11.12 Field Example In this section, we take a second look at a published field example to illustrate the thought process involved in doing a critical evaluation of a well test interpretation, as discussed in Section 11.11.7. Ramey offers a novel interpretation of an unusual field example of an extended shut-in on a pumping well, where one of the primary conclusions is that the pump valves are leaking (Ramey 1992). Table 11.11 gives reservoir rock and fluid property data for this example; Table 11.12 gives the reported buildup pressure data. The subject well was a low-rate pumping well producing from an underpressured reservoir at 13,000 ft. The objective of the test was to determine the cause of the low productivity of the well relative to its offsets. 11.12.1 Published Interpretation. The log-log pressure response is shown in Fig. 11.20. The pressure rose almost 200 psi in the first 24 hours, then continued to rise linearly for 300 hours. The rapid early pressure increase was attributed to liquid in the tubing above the pump leaking back into the annulus. The additional liquid would be equivalent to a slug test. Thus, the pressure response caused by the added liquid should decay very quickly. To account for the pressure change caused by the added liquid, the flowing bottomhole pressure was adjusted upward to 1,952 psi, and the first three points of the buildup were deleted, giving the log-log pressure

Well Test Interpretation Workflow 307 TABLE 11.11—FIELD EXAMPLE, RESERVOIR ROCK, AND FLUID PROPERTY DATA Property

Value

Units

Property

Value

Units

Property

Value

Units

q

36

STB/D

h

165

ft

f

0.17

tp

3912

hour

m

0.5058

cp

pi

2688

Psi

rw

0.4

ft

Bo

1.333

bbl/STB

ct

5 × 10–6

1/psi

TABLE 11.12—FIELD EXAMPLE, REPORTED PRESSURE BUILDUP DATA Δt, hour

Δt, hour

pws, psi

pws, psi

Δt, hour

pws, psi

Δt, hour

pws, psi

0

1812.53

90

2004.16

941

2285.42

8053

2658.08

20

1831.43

100

2009.29

1157

2332.80

8169

2658.53

25

1927.24

110

2014.43

1421

2384.72

8408

2659.58

30

1954.37

120

2019.56

1747

2434.85

8653

2660.50

35

1974.85

130

2024.69

2146

2484.00

8778

2660.83

40

1978.31

147

2033.38

2637

2529.93

9035

2660.63

45

1980.90

181

2050.72

3240

2568.66

9166

2660.90

50

1983.51

223

2071.72

3982

2598.66

9433

2661.48

55

1986.09

274

2095.92

4892

2626.00

9570

2661.85

60

1988.68

336

2123.29

7387

2656.50

9708

2662.17

65

1991.26

413

2154.49

7494

2656.17

9849

2662.58

70

1993.85

507

2186.60

7602

2656.14

75

1996.42

624

2214.57

7712

2656.65

80

1999.00

766

2246.61

7938

2657.74

Pressure Change, psi

10000

1000

100

10 10

100

1000 Shut-In Time, hr

10000

100000

Fig. 11.20—Field example, buildup pressure response (adapted from Ramey 1992).

response shown in Fig. 11.21. A history match of the adjusted pressure data gave the published interpretation, shown in Table 11.13. 11.12.2 Review. Two observations suggest that there may be a problem with the published interpretation: the estimated WBS coefficient is much larger than expected, and there are internal inconsistencies in the interpretation. Reality Check. The estimated WBS coefficient, 3 bbl/psi, immediately raises a red flag. A pumping well will normally have a WBS coefficient caused by a rising liquid level, which for a typical casing size would be around 0.05 bbl/psi. In the discussion, a fluid rise of 1,000 ft is associated with a pressure change of 300 psi, implying a

308 Applied Well Test Interpretation

Pressure Change, psi

10000

1000

100

10 10

100

1000 Shut-In Time, hr

10000

100000

Fig. 11.21—Field example, buildup pressure response with adjusted pwf (adapted from Ramey 1992).

TABLE 11.13—FIELD EXAMPLE, PUBLISHED RESULTS FROM REGRESSION WITH ADJUSTED FIELD DATA Parameter

Value

Confidence Interval

Units md

k

0.111805

± 0.0193642

s

–1.73821

± 0.638584

C

3.03355

± 0.139022

bbl/psi

fluid gradient of 0.3 psi/ft. A WBS coefficient of 3 bbl/psi with a fluid gradient of 0.3 psi/ft would require a casing capacity of 0.9 bbl/ft. (For comparison, 5½ in. 17 lbm/ft casing has a capacity of 0.02324 bbl/ft.) A WBS coefficient of 3 bbl/psi may be appropriate for a naturally fractured reservoir where the fractures are in good communication with the wellbore, but it is too high to be explained by a liquid level rising in casing or casing-tubing annulus. Looking closer at the data, Table 11.12 shows the reported test elapsed time to be in hours. If the elapsed time data are in hours, the total test duration is 9,849 hours, or 410 days. It would be unusual to run a buildup test on a low-productivity well for 410 days. The first buildup data point in the table is at 20 hours, with subsequent points reported at 5 hour intervals. Again, this is unusual. Another anomaly is that the test is affected by WBS throughout the buildup, never reaching a semilog straight line or encountering a boundary. This would be unusual in a 410-day test, although consistent with the high-WBS coefficient and low permeability. Internal Inconsistencies. In arguing that the rapid rise in pressure at the beginning of the test was caused by fluid leaking from the tubing into the annulus through the pump valves, the capacity of the casing-tubing annulus was estimated as 58 bbl/1,000 ft. Combined with a fluid gradient of 0.3 psi/ft, this gives a WBS coefficient for a rising liquid level of Cwb =

58 bbl 1000 ft = 0.193 bbl/psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.23) 0.3 psi ft

While higher than the 0.05 bbl/psi more typical of a rising liquid level, this is a reasonable value for 8-in. casing. However, it is more than an order of magnitude lower than the estimated value of 3 bbl/psi. If the WBS coefficient is 3 bbl/psi, 52 barrels of fluid leaking from the tubing to the casing-tubing annulus would cause a pressure increase of only ∆pleak =

52 bbl = 17.3 psi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.24) 3 bbl psi

a much smaller pressure change than the 300 psi change in pressure observed immediately after shut-in.

Well Test Interpretation Workflow 309

Given the internal inconsistencies, we are left with two choices; either (1) the published WBS coefficient of 3 bbl/psi is wrong, or (2) the WBS coefficient is not caused by a rising liquid level. If the WBS coefficient is incorrect, we must explain how an error of an order or magnitude or more could occur. If the WBS coefficient is not caused by a rising liquid level, we must seek another explanation for the sudden rise in pressure at the beginning of the test. 11.12.3 Alternative Interpretation. In reviewing the data and original interpretation, we find several points of concern: (1) the anomalous WBS coefficient; (2) the unusual sampling intervals and duration of the buildup test; and (3) the unusually long duration of WBS. Could these be related? Is it possible the elapsed shut-in times are in minutes rather than hours, as reported in Table 11.12? The WBS coefficient of 3.0 bbl/psi is 60 times a typical value of 0.05 psi. Is this just a coincidence? A first sampling point of 20 minutes, with subsequent samples at 5-minute intervals, would not be unexpected. A total test duration of 9,849 minutes would be 164.15 hours, or slightly less than 7 days, a reasonable duration for a buildup test on a low-productivity well. It would not be unusual for WBS to last 7 days in a lowproductivity well. We will continue attempting an interpretation under the assumption at the times in Table 11.12 are minutes instead of hours. (It should be stressed that this assumption is based solely on the observation that it makes the WBS coefficient and test duration fall within typical ranges. If this were a field data set, it would be imperative for the interpreter to confirm the assumption before proceeding.) Type-Curve Match. Fig. 11.22 shows a preliminary match using equivalent time, assuming shut-in times are in minutes instead of hours. Because of the distortion of the pressure curve, the interpretation relies very heavily on the shape of the derivative. The WBS coefficient is 0.06 bbl/psi, within the range of values typical of wells with a rising liquid level. Further, the initial pressure from the match is very close to the reported initial pressure of 2,688 psi. This is about as far as straight-line and type-curve methods can take us. Is there another way to incorporate the hypothesis of a leaky pump? If we knew the rate at which fluid leaked back into the well, we could model the pressure increase by using superposition to model the fluid leakage as an injection rate. But, we don’t know the rate. Is there any way to estimate the rate? Can we estimate the rate from the buildup pressure data? Estimating Leak Rate From WBS Coefficient. If we assume the WBS coefficient of 0.06 bbl/psi is constant, we can calculate the rate of fluid influx into the wellbore from the rate of change of pressure with time. For a normal test, this would provide an estimate of the afterflow rate. However, in this case, the fluid influx also includes the rate at which liquid leaks from the tubing into the annulus. Fig. 11.23 shows the calculated influx rate for the first 100 hours of shut-in. The calculated rate spikes very briefly at over 1,000 STB/D, then drops rapidly to about 33 STB/D, then declines almost exponentially.

Pressure Change, Logarithmic Derivative, psi

10000

1000

k = 1.5 md s = 55 C = 0.06 bbl/psi pi = 2696.9 psi

100

10

1 0.1

1

10 100 Equivalent Time, hr

1000

10000

Fig. 11.22—Field example, alternative interpretation, preliminary type-curve match.

310 Applied Well Test Interpretation

Estimated Influx Rate, STB/D

10000

1000

100

10

1

0

10

20

30

40

50

60

70

80

90

100

Shut-In Time, hrs Fig. 11.23—Field example, estimated influx rate assuming C = 0.06 bbl/psi.

Table 11.14 shows the calculated influx rates for the first 2 hours of the buildup. We set the actual beginning of shut-in to 19 minutes after the reported time zero, giving the corrected elapsed shut-in times in Column 2. The estimated influx rates in Column 5 are calculated from qinflux =

24C dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.25) Bo dt

After 45 minutes, the calculated influx rates dropped to approximately 33.5 STB/D. Assuming this represented the true afterflow rate, we adjusted the calculated influx rates upward to give the corrected influx rates shown in Column 6: qinflux,corr =

36 qinflux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.26) 33.5

Finally, we subtracted the assumed afterflow rate of 36 STB/D from the corrected influx rate to give the estimated leak rate shown in Column 7. Run 1—Simulation With Preliminary Parameter and Leak Rate Estimates. We used a single-layer, single-phase, single-well finite-difference reservoir simulator to calculate the buildup pressure response, initializing the model

TABLE 11.14—FIELD EXAMPLE, ESTIMATED LEAK RATES Δt, min

Δtcorr, min

pws, psi

dpw/dt, psi/hr

qinflux, STB/D

qinflux, corr, STB/D

qleak, STB/D

0

0

1812.53

20

1

1831.43

1134.0

1225.0

1316.4

1280.4

25

6

1927.24

1149.7

1242.0

1334.7

1298.7

30

11

1954.37

325.6

351.7

377.9

341.9

35

16

1974.85

245.8

265.5

285.3

249.3

40

21

1978.31

41.5

44.9

48.2

12.2

45

26

1980.90

31.1

33.6

36.1

0

50

31

1983.51

31.3

33.8

36.4

0

55

36

1968.09

31.0

33.4

35.9

0

60

41

1988.68

31.1

33.6

36.1

0

65

46

1991.26

31.0

33.4

35.9

0

70

51

1993.85

31.1

33.6

36.1

0

75

56

1996.42

30.8

33.3

35.8

0

80

61

1999.00

31.0

33.4

35.9

0

Well Test Interpretation Workflow 311

with the estimates of reservoir parameters from manual type curve analysis. We treated the leak as a brief injection period, with the estimated leak rates from Table 11.14, and the first two flow rates combined and treated as a single rate. Fig. 11.24 shows the log-log graph of the measured and simulated pressure responses. (Interestingly, with the shifted time origin, the pressure change curve looks very similar to that of a well with an abrupt increase in WBS (see Fig. 5 of Ramey and Agarwal 1972). The match looks very good, at least on the log-log scale, except for the apparent WBS period from 0.4 to 10 hours, where the simulated pressure derivative lies below the field derivative. Run 2—Adjust WBS Coefficient to Match Unit Slope Line. To move the simulated derivative upward to match the observed derivative, we estimated the derivative at 1 hour for both measured and simulated curves, then estimated the WBS coefficient that would make the two coincide from Cnew = =

t ( dp dt )sim

t ( dp dt )obs

Cold

( 25.68) 0.06 ( ) (31.03)

= 0.0497 bbl psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.27) Fig. 11.25 shows the log-log graph for Run 2. The lower WBS coefficient gives a better match of the WBS period from 0.4 to 4 hours, but now the calculated pressure change curve is too high. Run 3—Reduce Leak Rates to Match Pressure Change in Early Buildup. Since the early pressure rise is caused by the leak, we reduced the leak rates to get a better match of the pressure change curve. For the first, 6-minute flow period: qleak,new = =

∆pobs qleak,old ∆psim

(114.71) 1295.7 ( ) (149..314 )

= 995.4 STB D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.28) The remaining flow period were also adjusted downward by the same factor. Fig. 11.26 shows the log-log graph for Run 3. Fig. 11.27 is a graph of the pressure and pressure derivative residuals for Run 3. The derivative residual is negative, indicating that the simulated derivative is too low. Runs 4 and 5—Reduce Permeability and Skin to Get Better Match of Derivative at End of Test. We decreased the permeability to get a better match of the derivative at the end of the test. We did not want to change the flowing bottomhole pressure, which was very close to the measured pressure, so we adjusted both the permeability and

Pressure Change, Logarithmic Derivative, psi

10000

1000

k = 1.5 md S = 55 C = 0.06 bbl/psi pi = 2688 psi

100

10

1 0.01

0.1

1 10 Shut-In Time, hr

100

1000

Fig. 11.24—Field example, Run 1, preliminary estimates of parameters and leak rate.

312 Applied Well Test Interpretation

Pressure Change, Logarithmic Derivative, psi

10000

1000

k = 1.5 md s = 55 C = 0.0497 bbl/psi pi = 2688 psi

100

10

1 0.01

0.1

1 10 Shut-In Time, hr

100

1000

Fig. 11.25—Field example, Run 2, WBS coefficient adjusted to match derivative unit slope line.

Pressure Change, Logarithmic Derivative, psi

10000

1000

k = 1.5 md s = 55 C = 0.0497 bbl/psi pi = 2688 psi

100

10

1 0.01

0.1

1 10 Shut-In Time, hr

100

1000

Fig. 11.26—Field example, Run 3, leak rates modified to match early pressure change data.

the skin factor. We first tried a permeability of 1.0 md and a skin factor of 34 for Run 4 (not shown), but that did not change the derivative as much as we had expected. So, we used a permeability of 0.75 md and a skin factor of 23.5 for Run 5: snew = ( sold + 8) = ( 55 + 8)

knew −8 kold

( 0.75) − 8 (1.5)

= 23.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.29) Fig. 11.28 shows the log-log graph for Run 5. Fig. 11.29 is a semilog graph of the residuals for 0.01 to 10 hours, while Fig. 11.30 is a Cartesian graph of the residuals for the entire test. Although the log-log graph shows good agreement between the observed and simulated pressure responses, the residual graphs show that there is room for improvement during both the WBS period and the end of the test.

Well Test Interpretation Workflow 313 80 Pressure residual Derivative residual

60

Residual, psi

40

Poor match

20

0

−20 Poor match −40 −60

0

50

100 Elapsed Shut-In Time, hr

150

200

Fig. 11.27—Field example, Run 3, pressure and derivative residuals.

Pressure Change, Logarithmic Derivative, psi

10000

1000

k = 0.75 md s = 23.5 C = 0.0497 bbl/psi pi = 2688 psi

100

10

1 0.01

0.1

1 10 Shut-In Time, hr

100

1000

Fig. 11.28—Field example, Run 5, permeability and skin tweaked to match late derivative.

Run 6—Tweak Leak Rates to Match Early Pressure Change, Skin to Match Late Pressure Change. The derivative residual in Fig. 11.29 during the apparent WBS unit slope line from 0.4 to 0.7 hour is almost perfect, indicating that the WBS coefficient is good, but the residual for pressure change curve rises during the leak period, 0 to 0.4 hour, to about 1.4 psi, then remains constant until about 1 hour. We can adjust the leak flow rates downward slightly to compensate: A wellbore pressure that is too high by 1.4 psi indicates that too much fluid has entered the wellbore; we can estimate the extra fluid with the WBS coefficient as ∆N p = =

C ∆p Bo

( 0.0497) (1.4 )

1.333 STB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.30) = 0.0522 STB.

314 Applied Well Test Interpretation 20

15

Pressure residual Derivative residual

Residual, psi

10

1.4 psi high

5

0

Good match

−5

−10 0.01

0.1 1 Elapsed Shut-In Time, hr

10

Fig. 11.29—Field example, Run 5, pressure residual mismatch, 0.01–10 hour.

80 Pressure residual Derivative residual

60

Residual, psi

40 Good match

20

0

-20 9.6 psi low -40

-60

0

50

100 Elapsed Shut-In Time, hr

150

200

Fig. 11.30—Field example, Run 5, pressure residual mismatch, linear scale.

So, we adjusted the leak rates downward to make the total leak volume smaller by 0.0522 STB. We increased the skin factor to make the pressure change curve fit the end of the data. To increase the pressure by 9.6 psi, we increased the skin factor by ∆s = =

0.00708kh ∆ ( ∆p) qBoµ

( 0.00708) ( 0.75) (165) 9.6 ( ) (36) (1.333) ( 0.5058)

= 0.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.31)

Well Test Interpretation Workflow 315

Run 6, with a permeability of 0.75 md and a skin factor of 23.85, gave the residuals plots shown in Figs. 11.31 and 11.32. Run 7—Adjust Initial Pressure to Match Observed Flowing Bottomhole Pressure. The final step is to adjust the initial pressure so the calculated flowing bottomhole pressure matches the measured flowing bottomhole pressure. The adjustment is given by pi , new = pi , old + ( pwf , obs − pwf , sim ) = 2, 688 + (18812.53 − 1789.91) = 2710.62 psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.32) 20 Pressure residual Derivative residual

15

Resi duall, ps i

10

5

Good match

0

Good match

-5 5

-10 0.01

0.11

10

Elapsed Shut-In Time, hr Fig. 11.31—Field example, Run 6, final match of pressure and derivative residuals, early data.

80 Pressure residual Derivative residual

60

Residual, psi

40

20

0

-20

Good match

-40

-60

0

50

100

150

200

Elapsed Shut-In Time, hr Fig. 11.32—Field example, Run 6, final match of pressure and derivative residuals.

316 Applied Well Test Interpretation

For a single-phase oil well, Eq. 11.32 should give the right answer without iterating. For a gas well, or an oil well with multiphase flow, it may be necessary to iterate to get the desired accuracy. If it is necessary to iterate, we recommend that you review, and tweak if necessary, the matches of the other portions of the test response. Discussion. It must be stressed that this alternate interpretation is based on an unsupported conjecture (i.e., that the wrong units were used for the time column in the buildup pressure data). That we were able to get a good match of the data with this conjecture is not sufficient to prove that the conjecture is valid. In fact, the same approach may be used to interpret the test using the original units, giving a match that is equally good. Matching the data using the original units is left as an exercise for the reader. For this example, we have focused on matching the apparent WBS unit slope line, from 0.4 to 4 hours, and the last part of the test, from 128 to 168 hours. The last 40 hours of the final match shows almost perfect agreement with the pressure response, as well as good agreement with the derivative. In addition, the derivative residual is not showing an obvious trend upward or downward, indicating that the second derivative of the simulated pressure response is also in reasonable agreement with the field data. The last 40 hours is the most important part of the test to match because it is the part that is least affected by wellbore phenomena and most influenced by reservoir behavior. It does not appear possible to improve the rest of the match, between 4 and 128 hours, without introducing additional phenomena to describe the shape of the pressure response. The kink in the derivative between 7 and 20 hours might be caused by an increase in WBS, perhaps associated with the liquid level rising above the top of the liner. Without having the wellbore schematic, this is, of course, just conjecture. If one were available, we could (and should) compare the calculated height of the fluid column at the time of the apparent increase in WBS coefficient with the wellbore schematic. We could (and should) also compare the WBS coefficients calculated from the liner and casing IDs to see if the increase in the WBS coefficient is consistent with a rising liquid level moving from the liner to the casing. We considered alternative matches, with permeabilities of 1.5 and 3.0 md, but the 0.75 md match presented here was the only case where we were able to match both pressure change and derivative during the final 40 hours of the test. If the alternative interpretation is correct, the low productivity is caused by significant damage skin, and the well is a candidate for stimulation. Assuming the alternative interpretation, with permeability of 0.75 md and skin factor of 23.85, is correct, we can estimate the production rate that could be achieved if a workover could remove the damage skin: s + 8 qnew =  old  qold  snew + 8  =

( 23.85 + 8) 36 ( ) ( 0 + 8)

= 143 STB D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11.33) This rate will only be achievable if the well can continue to produce at the same flowing bottomhole pressure post-stimulation. Because the well is producing by rod pump, the condition of fixed bottomhole pressure seems reasonable, so 143 STB/D may be a good estimate of post-workover productivity. 11.13 Summary The recommended workflow for well test interpretation comprises the following steps: 1. Collect the data necessary for the analysis. Data that may be required include geological and geophysical interpretations; petrophysical data, including both raw and interpreted logs and lab data; reservoir data; fluid property data, well data, including type of completion, workover history, artificial lift, wellbore schematic, and production history; and test data, include test report, gauge pressure and temperature data, and test rate data. 2. Review, quality control, and prepare data for interpretation. 3. Deconvolve the test data. 4. Identify the flow regimes present in the test data. 5. Select the reservoir model to use for interpretation. 6. Estimate the parameters that characterize the model, using manual straight-line and log-log methods. 7. Simulate or history-match the pressure response. 8. If appropriate, calculate confidence intervals. 9. Interpret the estimated parameters. 10. Validate the results. 11. Report and archive the results.

Well Test Interpretation Workflow 317

Nomenclature A = drainage area, ft2 c = compressibility, psi–1 FWHP = flowing wellhead pressure, psia h = formation thickness, ft ID = tubing or casing inside diameter, in. k = permeability, md = distance to boundary, ft Le M = mobility ratio, dimensionless p = pressure, psia = critical rate to lift liquids, Mscf/D qcrit = wellbore radius, ft rw = drainage radius, ft re = wellbore radius, ft rw = apparent wellbore radius, ft rwa s = skin factor, dimensionless = largest possible negative skin factor, dimensionless smin S = storativity ratio, dimensionless t = time, hour V = volume fraction, dimensionless a = matrix coefficient, ft–2 l = interporosity flow coefficient, dimensionless m = viscosity, cp f = porosity, fraction w = storativity ratio, dimensionless Subscripts 1 = Region 1 (well is located in Region 1) 2 = Region 2 f = fracture h = horizontal m = matrix s = spherical t = total x,y,z = coordinate directions

Chapter 12

Well Test Design Workflow “Perilous to us all are the devices of an art deeper than we possess ourselves.” —Gandalf, Lord of the Rings: The Two Towers, by J.R.R. Tolkien “Measure what is measurable, and make measurable what is not so.” —attributed to Galileo Galilei 12.1 Introduction While the purpose of well test interpretation is to develop an improved reservoir characterization to assist in making better reservoir management decisions, the purpose of well test design is to ensure that the test data are interpretable and that the objectives of the test are met. This chapter focuses on well test design from a subsurface point of view (i.e., estimating the flow rate and test duration required to achieve the desired test objectives). Planning a test for an exploration well typically involves personnel from multiple operating and service companies, with many different areas of expertise (Nardone 2009); a full discussion of the issues involved is beyond the scope of this text. Section 12.2 discusses several common test design scenarios, and the specific issues likely to be encountered in each scenario. Sections 12.3 through 12.13 discuss the workflow step by step. The workflow is illustrated with an example in Section 12.14. Because of the high cost of testing exploration and appraisal wells, the workflow is directed toward these two scenarios. The same basic workflow is applicable to most other well testing scenarios but may be adjusted as needed for specific requirements. As with test interpretation, test design often requires an iterative approach before reaching the final design. The workflow is composed of the following steps: 1. Identify the test objectives, Section 12.3. 2. Consider whether the necessary information can be obtained at lower cost, in less time, or with less environmental impact with other tools, Section 12.4. 3. Collect the data necessary for the design, Section 12.5. 4. Determine preliminary estimates of the reservoir properties to be obtained from the test, Sections 12.6 and 12.7. 5. Estimate the duration necessary to meet the objectives of the test, Sections 12.8 and 12.9. 6. Estimate the test rate and determine the flow rate sequence, Section 12.10. 7. Estimate the magnitude of the expected pressure response, Section 12.11. 8. Select appropriate number, placement, and types of gauges, Section 12.12. 9. Simulate the test, Section 12.13. For reference, the workflow is summarized in Table 12.1. 12.2 Typical Test Design Scenarios In this section, we consider several different well testing scenarios. These scenarios differ widely in their objectives, in the amount of data available, in the expense of designing and conducting a test, and the investment decisions to be made depending on the test results. In some scenarios, testing is a primary focus of the activity; in some scenarios, testing may be conducted on a routine basis; in yet other scenarios testing is conducted as needed to diagnose a specific problem.

320 Applied Well Test Interpretation TABLE 12.1—RECOMMENDED WORKFLOW FOR WELL TEST DESIGN Define Test Objectives

Estimate Flow Rate

Fluid sampling

Low enough to avoid bubblepoint or dewpoint

In-situ permeability

High enough to lift liquids

Distances to boundaries

High enough to get clean pressure response

Hydrocarbons in place

Low enough to maintain critical flow

Average drainage area pressure

Low enough to sustain rate for entire flow period

Consider Alternatives to Well Testing

Determine Flow Rate Sequence

Wireline formation testing

Estimate Magnitude of Pressure Response

Log-derived permeability estimates Rate-transient/production data analysis Collect Data Estimate Reservoir Properties Permeability

Infinite-acting radial flow Linear flow Pseudosteady-state flow Select Gauges

Skin factor

Operating temperature and pressure range

Wellbore storage coefficient

Accuracy

Distances to boundaries

Resolution

Hydrocarbons in place

Drift

Estimate Time to Reach Flow Regimes…

Stabilization time

Time to end of wellbore storage

Simulate Test

Surface shut-in

Simulate wellbore hydraulics

Bottomhole shut-in

Simulate reservoir pressure response

Time to reach specific flow regimes

Analyze simulated pressure response

Time to reach desired radius of investigation

Does test design satisfy test objectives?

Time to prove minimum hydrocarbons in place …Or, Estimate Time to Verify Economics Estimate minimum J Draw horizontal line for J Estimate minimum OOIP Draw unit-slope line for OOIP Economic well must have derivative below horizontal line, and cross unit-slope line to confirm economic reservoir

12.2.1 Exploration and Appraisal Wells—Conventional Drillstem Testing. One of the most important testing scenarios is the exploration well drillstem test (DST). While the term DST originated in the practice of using the drillstring for production from a temporary completion, most DSTs today are conducted through tubing. In areas where the reservoir pressure and fluid type are not known, drillpipe may be used only if the well will be shut in downhole so that no reservoir fluid is allowed to flow to surface (McAleese 2000). Objectives. The three primary objectives for exploration and appraisal well DSTs are to collect fluid samples, to estimate reservoir pressure, and to estimate in-situ permeability. Because a DST is conducted with a temporary completion, the skin factor obtained from a DST has limited relevance to predicting well performance following a permanent completion. High rig costs make it impractical to run DSTs long enough to confirm minimum economic reservoir size. As a result, exploration well testing does not usually include extended testing to estimate reserves (Barnum and Vela 1984). Data acquired during drillstem testing will be used to make decisions about whether to abandon the zone or to set casing, to conduct an extended well test, or to drill additional wells. Data Available. Data availability is limited for an exploration well. Much of the design and planning for a testing program in an exploration well is based on geology, geophysics, and economic targets. Additional data become available as the well is drilled and logged. However, by the time this data have been acquired and interpreted, time constraints prevent extensive revision to the test design. Thus, the preliminary design phase is likely to be based on the minimum economic case. During the preplanning phase, the data available are essentially the same data supporting the decision to drill (i.e., regional geology, 3D seismic interpretation, and economic potential). Seismic interpretation provides an

Well Test Design Workflow 321

estimate of gross thickness and depth of the candidate zone. Regional geology may provide estimates of expected pore pressure and temperature gradients. Burial history and temperature give a good idea of the type of fluid likely to be present. Additional data available after the well is drilled and logged may include mud weight, drill cuttings, cores, logging-while-drilling data, conventional wireline logs, and Wireline formation tests (WFTs) data. Logs may be used to identify specific zones of interest to be tested within the gross candidate interval. WFT data may be used to select target zones for and improve design of conventional DSTs (Kumar et al. 2010). Although the test objectives, design, and cost are likely to be similar for an appraisal well, there is much more data available. Interpreted logs, lab tests of fluid samples and cores, and test results from the discovery well should all be available by the time the test design for an appraisal well is to be finalized. Other Considerations. Because of the high cost of testing exploration wells, downhole shut-in is frequently used to reduce the time required to reach infinite-acting radial flow and to reduce the likelihood and severity of wellbore effects such as phase segregation. The high cost also justifies the widespread use of surface readout to allow tests to be terminated early if the objectives of the test are met (or if the zone proves to be uneconomic). Even with surface readout, tests may be terminated prematurely if appropriate criteria for early termination are not clearly understood and specified. To avoid having to repeat a test in the event of gauge failure, multiple gauges are virtually always used to provide redundancy, with some operators reportedly using as many as 18 gauges in a single test string. At least one vendor offers gauges that store identical copies of the data in four separate, nonvolatile memory banks to ensure data integrity (Metrolog 2012). 12.2.2 Exploration and Appraisal Wells—Extended Well Testing. Extended well testing is conducted to estimate fluid in place, investigate reservoir boundaries, evaluate reservoir connectivity, to gain insight into reservoir drive mechanisms, or to obtain other information that cannot be obtained through short-term testing. Long-term tests of onshore oil wells may also be conducted as a source of early revenue (McAleese 2000). In most other situations, the expense of conducting extended testing limits its usefulness. If extended well testing is anticipated, planning for drillstem testing should include preplanning for longterm testing, which should be included in an overall testing program. The results from WFTs and conventional DSTs may be used to determine whether to conduct long-term testing and, if testing is warranted, to finalize the test design. 12.2.3 Development Well. Development wells may or may not be tested on a routine basis depending on anticipated reservoir management strategies. During the early stages of field development, one objective may be to estimate skin factor to optimize completion strategies for subsequent wells. Another objective may be to collect permeability data for future use in reservoir simulation. A third objective might be to evaluate reservoir connectivity in anticipation of implementation of enhanced recovery. If testing is systematically conducted on each new well, and if the test objectives remain the same, the test design may be updated based on the success (or failure) of tests on earlier wells, taking into account variations in log-derived petrophysical data. If periodic testing is likely, installation of permanent gauges should be considered as an option during well design and construction. 12.2.4 Producing Well. Tests on producing wells are likely to fall into one of two categories, scheduled and unscheduled. Periodic scheduled testing is often conducted to estimate average drainage area pressure for material balance calculations. Scheduled testing may also be performed to monitor movement of a fluid contact. Unscheduled testing is usually conducted to diagnose specific productivity or injectivity problems. If the candidate well or its offsets have been tested before, it is not uncommon to simply re-use the same test design. A better practice would be to review the test results to see if the test objectives were achieved, and adjust the test design accordingly. As Frailey et al. (1994) point out, “It is not sufficient to identify what was done before, but to find out what worked before.” Wells producing by gas lift should be shut-in downhole to prevent test failure from inadvertent opening of a gas-lift mandrel (Frailey et al. 1994). Wells producing by rod pump are most conveniently tested by acoustic fluid-level measurement. Testing a pumping well with memory gauges would require the following time-consuming process: • Shut the well in to pull the pump. • Hang the gauge carrier below the pump. • Re-install the pump.

322 Applied Well Test Interpretation

• Resume pumping to reach stabilized flow (see Section 12.10.3). • Shut-in well for buildup. • Pull the pump and remove the gauge carrier. • Rerun the pump and resume production. 12.2.5 Well With Rate-Dependent Skin Factor. Although rate-dependent or non-Darcy skin factor is often easily identified from a single buildup (Section 10.7), the best way to quantify rate-dependent skin factor is with a multirate test. Multirate tests are routinely conducted on gas wells in conventional reservoirs, but may also be run on oil wells if non-Darcy flow is suspected. Traditionally, multirate tests have been run and analyzed as deliverability tests. High-frequency data from electronic gauges also allow multirate tests to be interpreted as pressure-transient tests. Analyzing a multirate test as a deliverability test provides an inflow performance relationship, or inflowperformance-relationship curve, describing the relationship between wellhead or bottomhole flowing pressure and production rate for a given average drainage area pressure. The same test data may be analyzed as a transient test, yielding estimates of non-Darcy flow coefficient, as well as permeability and skin factor. Typically, deliverability analysis focuses on behavior of the flow periods of a multirate test, while transient analysis focuses on the behavior of the shut-in periods. Multirate tests are normally run as a series of flow and shut-in periods. To minimize the effects of earlier flow periods, the flow rate is increased for each successive flow period. Traditionally, the final flow period in a deliverability test continues long enough to reach stabilized or pseudosteady-state flow so that the stabilized deliverability curve may be drawn. The final extended flow period is usually run at a lower flow rate. Analyzing a multirate test as a transient test allows the stabilized performance to be estimated, assuming the drainage area size and shape are known, thus avoiding the need for an extended flow period. For either deliverability analysis or transient analysis, it is important that both flow and shut-in periods are long enough to exhibit at least one-third to one-half log cycle of infinite-acting radial flow following the end of wellbore and near-wellbore effects. The minimum flow rate should be above the minimum rate to lift liquids (see Section 12.10.1). 12.2.6 Low Productivity/Injectivity. Low productivity or injectivity is always a concern. Before appropriate remedial action can be taken, the cause of the low productivity or injectivity must be determined. Is the initial productivity lower than expected, or is there a loss of productivity over time? Is low-initial productivity caused by near-wellbore damage or an ineffective completion, by low-effective permeability, by low-initial reservoir pressure, or by reservoir boundaries? Is loss of productivity caused by reservoir depletion, by damage to the completion, or by a loss of effective permeability? Is loss of injectivity caused by sandface damage, or simply by growth of an invaded zone of reduced mobility around the injection well (a favorable mobility contrast for displacement)? The objectives for a test to determine the cause of low or loss of productivity or injectivity must address all of the likely causes. Thus, appropriate objectives would include estimating skin factor, effective permeability, average drainage area pressure. In the case of low-initial productivity or injectivity, an additional objective would be to determine whether or not there are any nearby boundaries that were either unrecognized, or believed not to restrict flow. For an injection well in a waterflood project, the objectives would include estimating the near-wellbore damage, estimating the mobilities of the invaded and uninvaded zones, and estimating the size of the invaded zone. Chu (1996) presents a field case of an injection well in which the boundary between the invaded and uninvaded zones begins to affect the pressure response less than one-half hour after the beginning of the falloff test (See Fig. 7.1). 12.2.7 Stimulated Well. Wells that have been stimulated may be tested for different reasons. In a new area, multiple wells may be tested before and after stimulation to identify stimulation best practices for the formation. When a new stimulation treatment is introduced, whether to reduce cost or improve performance, testing may be conducted to determine whether the new treatment is better than existing best practices. A well that has been recently stimulated, with poorer results than expected, presents essentially the same issues as a well with lowinitial productivity discussed in the previous section. Wells with long hydraulic fractures in low-permeability reservoirs may require months or years to reach infinite-acting radial flow. Because unique estimates of effective fracture half-length require the permeability to be known, the testing program should include both pre-fracture and post-fracture tests. The pre-frac test is used to estimate formation permeability, which is used in interpreting the post-frac test to estimate effective fracture half-length.

Well Test Design Workflow 323

12.2.8 Annual Pressure Survey. Material balance, one of the primary tools of reservoir engineering, requires frequent, high-quality estimates of reservoir pressure for best results. Many operators conduct annual pressure surveys to estimate average drainage area pressure for each well, or for a representative sample of wells in the field, so that an appropriate hydrocarbon pore volume weighted average reservoir pressure may be calculated for use in material balance. Because pressures from multiple wells are to be combined, it is especially important to correct test pressures to a common datum before averaging. The datum elevation should be close to the hydrocarbon pore volume weighted average depth. The material balance method assumes that the average pressure represents the reservoir conditions at a single point in time. While it is impractical to test all wells in a field at the same time, the pressure survey should be completed over a short period of time for best results. Important well test scenarios include • Testing exploration and appraisal wells to obtain fluid samples and estimate formation permeability and reservoir pressure. • Extended well testing to determine reservoir limits. • Testing development wells to optimize completion strategies, quantify areal variations in permeability, or to evaluate reservoir connectivity. • Testing producing wells for reservoir monitoring or to diagnose specific problems. • Multirate testing to quantify rate-dependent skin factor. • Testing to determine the cause of low productivity or injectivity. • Testing stimulated wells to evaluate stimulation treatment effectiveness. • Conducting an annual pressure survey for material balance calculations. 12.3 Define Test Objectives The first step in well test design is to identify the objectives of the test. Common objectives include obtaining fluid samples for laboratory pressure-volume-temperature analysis, estimating effective permeability at in-situ conditions, quantifying the degree of damage or evaluating stimulation treatment effectiveness, estimating distance to and investigating nature of reservoir boundaries, estimating hydrocarbons in place, pore volume, or drainage area, estimating initial or average drainage area pressure, evaluating reservoir connectivity, or even satisfying regulatory obligations. 12.3.1 Fluid Sampling. Although fluid sampling is not a pressure-transient objective, it is often a major objective that must be considered as part of an overall testing program. Obtaining fluid samples is especially important during exploration and appraisal phases for volatile oils or gas condensates, if high structural relief suggests that compositional gradients are likely, or if field development plans include the possibility of compositional modeling or miscible flooding. 12.3.2 In-Situ Permeability. Estimating in-situ permeability is one of the primary objectives of pressuretransient testing. Permeability must also be estimated as an intermediate step in estimating distances to boundaries, fracture half-length, and fracture conductivity, and to a lesser extent, skin factor. The most reliable estimates of horizontal permeability require data in infinite-acting radial flow. It is difficult to obtain accurate estimates of permeability without at least one-third to one-half log cycle of infinite-acting radial flow. Although it is theoretically possible to estimate permeability from data entirely within the transition from wellbore storage to infinite-acting radial flow, deviations from ideal behavior during the transition make such estimates subject to considerable uncertainty (Spivey 2013). Rough estimates of vertical permeability may be obtained from data in spherical flow. More accurate estimates may be obtained from vertical interference testing. The permeability obtained from data in infinite-acting radial flow is based on the assumption that the reservoir is isotropic. If the reservoir is anisotropic, the value obtained is the geometric average permeability in the plane of flow. 12.3.3 Skin Factor. Another primary objective of pressure-transient testing is the quantification of near-wellbore damage or degree of stimulation, most commonly expressed as a skin factor. If infinite-acting radial flow is present, a qualitative evaluation of skin factor may be obtained from the log-log graph of pressure change and logarithmic derivative. As with permeability, the most accurate estimates of skin factor are obtained from data in infinite-acting radial flow.

324 Applied Well Test Interpretation

Theoretically, skin factor may range from perhaps –8 (for a horizontal well with a long lateral in late pseudoradial flow) to infinity (for a well that is so damaged that it will not flow), the practical range of skin factor likely to be observed is approximately –5 to perhaps 250 to 500. Typically, wells that would display skin factors lower than –5 either are in low-permeability formations that require unrealistically long test durations to reach infinite-acting radial flow, or are drilled on well spacing such that interference effects begin before infinite-acting radial flow begins. In wells with high skin factors, the pressure drop across the completion is much larger than the pressure drop in the reservoir. Thus, noise in the measured pressure response represents a larger fraction of the reservoir component of the total pressure drop for wells with high skin factor. For a well with a damage skin factor of 70 to 80, the reservoir pressure drop is only about 10% of the total pressure drop; for a well with a damage skin factor of 700 to 800, the reservoir component would be roughly 1% of the total pressure drop. This phenomenon causes the relative noise in the derivative to increase as the skin factor increases. Given the same total pressure drop and measurement noise, the relative noise in the derivative for a well with a skin factor of 70 to 80 would be 10 times higher than that for a well with zero skin factor, while that for a well having a skin factor of 700 to 800 would be 100 times higher. Thus, the higher the skin factor, the more difficult it is to get accurate estimates of permeability, and in turn, skin factor itself. Always bear in mind that skin factor is a lump number that represents the combined effects of any phenomena that cause the pressure drop near the well to be different from that predicted by the model of a fully penetrating vertical well in a homogeneous reservoir, completed open hole, with neither damage nor stimulation. 12.3.4 Distances to Boundaries. The objective of estimating distance to boundaries may refer to either of two distinct concepts. The objective might be to confirm that there are no boundaries within a certain distance of the wellbore. On the other hand, the objective might be to get a quantitative estimate of the distance to a boundary and to determine the type of boundary. These two objectives require different approaches to test design. If the objective is to confirm the absence of boundaries within a radius re, the test should be designed for a test duration to give a radius of investigation equal to re; that is, ∆t =

948φµct re 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.1) k

Flow rate and gauge resolution must be selected so that the logarithmic derivative will be accurate enough to detect any boundaries that might be present within a distance re. On the other hand, if the objective is to quantify the distance to the boundary and to investigate its nature, the test must be designed for a duration sufficient to see enough of the effect of the boundary on the pressure response that the type of boundary may be inferred. A good rule of thumb to use as a starting point is 20 times the time required to reach a radius of investigation equal to the distance to the boundary. The actual test duration should be selected through simulating the pressure response expected during the test for the most likely boundary types, then analyzing the simulated test to verify that the test objective can be met. Accurate estimates of distances to boundaries require accurate estimates of total compressibility, particularly the pore-volume compressibility and oil compressibility (see Table 11.3). As with permeability estimates, estimates of distances to boundaries are based on the assumption of isotropy. If permeability anisotropy is present, but not accounted for, estimates of distances to boundaries will not be accurate. 12.3.5 Hydrocarbons in Place, Drainage Area. Estimating the hydrocarbons in place is one of the most important, yet most challenging, test objectives. The biggest obstacle to estimating hydrocarbons in place is the time required to reach pseudosteady-state flow. In a low-permeability reservoir, the time to reach pseudosteady-state flow is likely to be a year or more. When this is the case, rate-transient analysis of production data is more appropriate than pressure-transient analysis (see Section 12.4.3). The time required to reach pseudosteady-state flow depends on the permeability, viscosity, and total compressibility, as well as the drainage area shape and size and the location of the well within the drainage area. As will be shown in Section 12.9.2, the time to reach pseudosteady-state flow can be interpreted graphically as the point at which the logarithmic derivative crosses a unit slope line that is related to the original hydrocarbons in place, independent of reservoir geometry. Once the hydrocarbons in place have been estimated, if the average thickness and porosity are also known, the drainage area may be calculated. Perhaps the most important application for drainage area estimates is evaluating infill drilling opportunities. If a well is draining a much smaller drainage area than is consistent with well spacing, infill drilling may be indicated. If the calculated drainage area is significantly larger than indicated by

Well Test Design Workflow 325

the well spacing, either the oil or pore-volume compressibility estimates, or both, are almost certainly too low. If calculated drainage area is only slightly larger than well spacing, cutoffs for estimating net-pay may be on the conservative side. 12.3.6 Initial or Average Drainage Area Pressure. While initial reservoir pressure may be obtained fairly easily with either conventional testing or wireline formation testing, estimating average drainage area pressure presents unique challenges. When designing a test that has estimating average drainage area pressure as one of its primary objectives, if one of the classic methods based on semilog analysis [the Matthews-Brons-Hazebroek (MBH), Dietz, and Ramey-Cobb methods; see Section 3.4] is to be used, the test should be designed to show at least one log cycle of infinite-acting radial flow. The more data in infinite-acting radial flow, the better; estimating average reservoir pressure is essentially extrapolation. In addition, the drainage area shape, size, and location of the well within the drainage area must be known. The Dietz and Ramey-Cobb methods require the well to have produced long enough to reach pseudosteady-state flow before shut-in, while the MBH method requires that the well have produced at constant rate, either for the entire production period, or long enough to reach pseudosteady-state flow, whichever is smaller. With well testing software, using the initial pressure as a history-matching parameter, then using material balance to calculate the average pressure at the moment of shut-in is conceptually the same as the MBH method, without the restriction of constant flow rate. As with the classic methods, the drainage area shape, size, and well location must be known if accurate estimates of drainage area pressure are to be obtained. 12.3.7 Evaluate Reservoir Connectivity. One of the most important considerations when planning for pressure maintenance or enhanced recovery projects is to evaluate reservoir continuity. Investigating reservoir continuity requires the use of interference or pulse tests, multiwell tests that have not been discussed in this text. Like single-well tests, multiwell tests require much less time when an oil reservoir is still undersaturated than after the pressure has fallen below the original bubblepoint pressure. Once the pressure drops below the original bubblepoint pressure, the associated tenfold increase in compressibility increases the required test time by a corresponding factor. If eventual implementation of a pressure maintenance or enhanced recovery project is likely, evaluation of reservoir continuity should be considered early in the life of the field before reservoir pressure has fallen below the initial bubblepoint pressure. 12.3.8 Obtain Flowing or Static Gradient Survey. Flowing and static pressure gradient surveys are often run in conjunction with pressure-transient tests. A common test sequence is to conduct a flowing gradient survey while the gauge is being run in and a static gradient survey at the end of the final buildup as the gauge is being pulled out. 12.3.9 Satisfy Regulatory Obligations. Government regulatory agencies may require testing either for newly completed wells or on a periodic basis. Often, deliverability tests are required to establish absolute open flow potential, which may then be used for determining allowable production rates. The test protocol and analysis method may also be specified, and the test data and results may be publicly available. These tests have been used to estimate permeability (Lee et al. 1984; Cook 2001). Important well test objectives include • To obtain fluid samples for laboratory analysis • To estimate in-situ permeability • To quantify damage or stimulation • To estimate distances to boundaries • To estimate hydrocarbons in place or drainage area • To estimate initial or average drainage area pressure • To evaluate reservoir connectivity • To obtain a flowing or static gradient survey • To satisfy regulatory obligations 12.4 Consider Alternatives to Conventional Well Testing Is a conventional pressure-transient test necessary to obtain the desired information? Can equivalent data be obtained at lower cost? Or, for the same cost, can better data be obtained through alternatives to conventional testing?

326 Applied Well Test Interpretation

Before designing a conventional pressure-transient test, determine whether the objectives can be met at lower cost for equivalent quality data, or higher quality data can be obtained for the same cost. 12.4.1 WFTs. Under the right conditions, WFTs can provide fluid samples, along with estimates of formation pressure, formation permeability, and with the right probe configuration, vertical permeability (Whittle et al. 2003). When run in a dual-packer configuration (sometimes called a mini-DST), WFTs may achieve a radius of investigation of up to 90 m (Frimann-Dahl et al. 1998). Elshahawi et al. (2008) report considerable success replacing conventional tests with WFTs. They noted that wireline formation testing has become best practice within their company for collecting fluid samples for laboratory testing. Based on their success rate as of 2008, they expected mini-DSTs to replace conventional testing for estimating permeability as well. One big advantage of wireline formation testing is that the formation pressure and permeability can be obtained in much more detail than is possible with conventional testing. Perhaps 10 to 50 WFTs can be run for the same overall cost as a single conventional test (Elshahawi et al. 2008). Conventional testing in multilayer reservoirs requires sophisticated tools and test procedures to obtain individual layer properties. On the other hand, wireline formation testing can provide pressure and permeability profiles on multilayer reservoirs using the same tools, test procedures, and interpretation methods as used on single-layer reservoirs. Two areas in which WFTs have limited applicability are estimating skin factor and investigating reservoir boundaries. Although a skin factor is determined as part of the WFT interpretation, that skin factor has little relevance to the skin factor after the well is completed for production (Elshahawi et al. 2008). Investigating reservoir boundaries requires larger volumes of fluid be produced and longer test times than can be provided by WFTs. WFTs can provide multiple estimates of formation permeability and reservoir pressure as a function of depth for about the same cost as a single conventional test. However, WFTs cannot provide estimates of damage or stimulation skin, or investigate large portions of the reservoir. 12.4.2 Log-Derived Permeability. Estimates of permeability from logs can be very useful but generally require calibration against core data or WFT data for a specific formation. Anand et al. (2011) present a promising new approach to estimating permeability from logs, but the method is not in widespread use. 12.4.3 Rate-Transient Analysis. Rate-transient analysis of production data has become the primary tool for estimating reservoir permeability, effective fracture length, and original fluid in place for low-permeability reservoirs, thus taking the place of both pressure-transient analysis and material balance. Low permeability coupled with long hydraulic fracture length makes conventional testing impractical. Techniques used in rate-transient analysis are very similar to those used in pressure-transient testing (Agarwal et al. 1999). A number of terms describing essentially the same set of methods are used almost interchangeably. The term rate-transient analysis (Bello and Wattenbarger 2008) emphasizes the connection to pressure-transient testing, while advanced decline-curve analysis (Fetkovich 1980) emphasizes the relationship to conventional decline-curve analysis, and production-data analysis (Hazlett et al. 1986) emphasizes the type of data that is used. Production-data analysis has become the primary tool for estimating reservoir permeability, evaluating effective fracture half-length, and estimating original fluid in place for low-permeability reservoirs, taking the place of both conventional testing and material balance analysis. 12.4.4 Permanent Downhole Gauges. Installation of permanent downhole gauges during well construction offers many advantages over conventional testing. In most fields, wells must be shut in from time to time for operational reasons unrelated to testing. Pressure data collected during these shut-in periods can be interpreted just like any other buildup data. Although the initial investment may be significant, installation of permanent gauges may ultimately be quite cost effective. Early field applications of permanent gauges were plagued by poor gauge reliability, but many of the reliability issues have been resolved (Bezerra et al. 1992). Permanent downhole gauges allow frequent evaluation of changes in effective permeability caused by saturation changes or loss of permeability from rock compaction, changes in skin factor with time, decline in reservoir

Well Test Design Workflow 327

pressure with reservoir depletion, or movement of fluid contacts from aquifer encroachment or injection. Permanent downhole gauges may also be used to continuous monitoring and optimization of artificial lift performance. For best results, a permanent gauge system must record accurate rate data in addition to pressure data. During periods of system shut-in, each well should also be shut in as far upstream as possible, preferably at the wellhead, to prevent crossflow between wells within the gathering system and to reduce the duration of wellbore storage. Because of the large amount of data generated by high-frequency sampling over years of production, data management can become an issue. Installation of permanent downhole gauges allows buildup data to be collected at no additional cost during routine system shut-ins, giving frequent estimates of permeability, skin factor, average reservoir pressure, and distances to fluid contacts. Permanent downhole gauges also allow continuous monitoring and optimization of artificial lift performance. 12.5 Collect Data Well test design requires the same basic data as well test interpretation (see Table 11.1), with the addition of estimates of each of the properties that the test is designed to measure. For exploration wells, the data available initially are usually limited to the interpreted seismic data and the resulting static Earth model. After the well has been drilled, openhole logs will almost always be run, but there will be limited time for detailed log interpretation before testing begins. 12.6 Estimate Reservoir Properties To design a test to measure specific reservoir properties, we have to have at least rough estimates of the properties we are attempting to measure. In this section, we look at ways of estimating skin factor, wellbore-storage coefficient, distances to boundaries, and initial reservoir pressure. We discuss methods of estimating permeability in the following section. Test design requires rough estimates of the properties to be estimated from the test: permeability, skin factor, wellbore-storage coefficient, distances to boundaries, and reservoir pressure. 12.6.1 Skin Factor. If no other data are available, the skin factor may be estimated from the type of completion using Table 11.7. The skin factors in Table 11.7 may need to be modified to reflect local experience with the formation and type of completion. The lower end of the range for each completion type represents what is likely under ideal conditions; significantly lower skin factors are highly unlikely if not physically impossible. 12.6.2 Wellbore-Storage Coefficient. The wellbore-storage coefficient is one of the most important variables in well test design. If the test objectives can be achieved from data during the early and middle time regions (MTRs), the wellbore-storage coefficient determines the minimum test duration that will achieve the test objective. For a well filled with a single-phase fluid, the wellbore-storage coefficient may be estimated from C = cwbVwb, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.2) while that for a well with a rising or falling liquid level may be estimated from C = 25.65

Awb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.3) ρ wb

where Awb is in ft2 and rwb is in lbm/ft3, or C=

Awb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.4) ρ wb

for Awb in bbl/ft and rwb in psi/ft. One of the options to be considered during well test design is whether to use surface shut-in or downhole shut-in. The wellbore-storage coefficient should be calculated for the wellbore configuration for both surface and downhole shut-in. Downhole shut-in offers several advantages over surface shut-in. For a gas well, downhole

328 Applied Well Test Interpretation

shut-in reduces the wellbore-storage coefficient by decreasing the wellbore volume used in Eq. 12.2. For an undersaturated oil reservoir, downhole shut-in is much more likely to maintain single-phase conditions in the wellbore, allowing the use of Eq. 12.2 (with a single-phase liquid compressibility) instead of Eq. 12.3. For both oil and gas wells, downhole shut-in reduces the likelihood of changing wellbore storage caused by phase segregation. 12.6.3 Distance to Boundaries. We may be interested in distances to boundaries for different reasons. If the presence of a boundary is suspected from seismic interpretation, geology, or log data, we may be interested in verifying that the boundary is in fact where we think it is and determining whether it will provide a complete or partial barrier to flow. The same data that cause us to suspect the presence of a boundary will usually give some idea of the distance to that boundary. Or, we may be monitoring the encroachment of water in a gas reservoir with strong water drive. In this case, we may compare pressure responses over time as the aquifer encroaches. Previous tests provide a good reference point for refining the design of subsequent tests. Often, rather than estimating distance to a known or suspected boundary, we simply want to verify that there are no boundaries within a certain distance of the well. That distance is likely to be determined by economics. 12.6.4 Reservoir Pressure. Reservoir pressure may be estimated from WFTs, from offset wells, from depth and regional pressure gradient, or even from mud weight. A good estimate of reservoir pressure is particularly important when the bubblepoint or dewpoint pressure is close to initial pressure. Maintaining single-phase flow during testing may be difficult if not impossible in this situation. 12.7 Permeability Estimates Good permeability estimates are particularly crucial to successful well test design, because of the effect of permeability on the time to reach end of wellbore storage, see reservoir boundaries, or prove a minimum desired fluid in place. 12.7.1 Permeability From Data in Pseudosteady-State Flow. We start with the equation for pseudosteady-state flow from a closed circular reservoir, given by p − pwf =

 141.2qBµ   re  3 ln   − + s′ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.5) kh    rw  4

where p is the average drainage area pressure. Eq. 12.5 may be used for gas wells by evaluating the formation volume factor, B, and the viscosity, m, at the average of the average reservoir pressure and flowing bottomhole pressure, p = 0.5 ( p + pwf ). Eq. 12.5 may be rearranged to give the permeability k as k=

 141.2qBµ   re  3 ln   − + s′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.6) h ( p − pwf )   rw  4 

For single-point flow test data where the reservoir has reached pseudosteady-state flow, Eq. 12.6 may be used to estimate the permeability, provided the drainage radius re and the total skin factor s′ may be estimated. 12.7.2 Permeability From Productivity Index. If the single-phase productivity index, J, is known or can be estimated, the permeability can be estimated from k=

 141.2 JBµ   re  3 ln   − + s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.7) h   rw  4 

If the total skin factor s′ is assumed to be zero, Eq. 12.7 may be simplified* to give k≅

1000 JBµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.8) h

* For normal wellbore radius and well spacing, the term ln (re / rw ) – 3/4 is typically in the range of 6.5 to 8.5. Assuming a value of 7.08 gives the result in Eq. 12.8.

Well Test Design Workflow 329

12.7.3 Permeability From Data in Infinite-Acting Radial Flow. The one-point method (Lee et al. 1984) is a simple way of estimating permeability from single-point flow test data where the reservoir is still infinite-acting. For infinite-acting radial flow, the pressure response may be written as   kt 70.6qBµ   ln   + 2s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.9) 2 kh    1, 688φµct rw 

pi − pwf =

Unlike Eq. 12.5, Eq. 12.9 cannot be solved for permeability k in closed form. A fixed-point iteration method may be developed to evaluate Eq. 12.9 as follows. First, we rearrange Eq. 12.9 as pi − pwf =

 141.2qBµ   1 3 kt ln   − + s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.10) kh   377φµct rw  4 

If we define the transient-drainage radius rd as kt , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.11) 377φµct

rd =

we can rewrite Eq. 12.10 as p − pwf =

 141.2qBµ   rd  3 ln   − + s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.12) kh   rw  4 

Eq. 12.12 may be solved for k to give k=

 141.2qBµ   rd  3 ln   − + s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.13) h ( pi − pwf )   rw  4 

Eqs. 12.11 and 12.13 together provide a simple fixed-point iteration method for estimating permeability that converges extremely fast, often within two or three iterations. For a gas well with drawdown less than 10% of initial pressure, the liquid equations may be used, with the formation volume factor, B, viscosity, m, and compressibility, ct, evaluated at the initial pressure. For a gas well with a drawdown greater than 10% of initial pressure, more accurate results may be obtained by evaluating the gas formation volume factor, Bg, and gas viscosity, mg, at the average of the average reservoir pressure and flowing bottomhole pressure, p = 0.5( p + pwf ). The procedure is as follows: 1. Calculate the Horner pseudoproducing time from t p = (24 N p /q) for an oil well, or t p = (24G p /qg ) for a gas well. 2. Assume a starting estimate for permeability. Starting guesses of 10 md for an oil well and 0.1 md for a gas well should be adequate if no better estimates are available. 3. Calculate a trial value of the transient-drainage radius rd from Eq. 12.11. On the first iteration, use the estimate from Step 2. During subsequent iterations, use the most recent estimate from Step 4. 4. Using the value of rd calculated in Step 3, calculate a new estimate of permeability k from Eq. 13. 5. Repeat Steps 3 and 4 until the value of k converges. Example 12.1—One-Point Method Example. Problem: Given the data in Table 12.2, estimate the permeability using the one-point method. Solution. First, we calculate the producing time using the Horner pseudoproducing time approximation: tp =

24G p qg

=

( 24) ( 2, 093) = 8 hr 6, 278

330 Applied Well Test Interpretation TABLE 12.2—DATA FOR ONE-POINT METHOD EXAMPLE

f

0.15

h

23 ft

rw

0.33 ft

Bgi

0.810 bbl/Mscf

μgi

0.0209 cp

Cti

1.78 × 10–4 psi–1

pi

3450 psia

pwf

2387 psia

s′

0

Gp

2093 Mscf

qg

6278 Mscf/D

Iteration 1. Starting with a permeability estimate of 0.1 md, we calculate the transient drainage as kt 377φµ gi cti

rd =

( 0.1) (8) (377) ( 0.15) ( 0.0209) (1.78 × 10−4 )

=

= 61.7 ft. Next, we calculate a new estimate of permeability as k= =

 141.2qg Bgi µ gi   rd  3 ln   − + s′ h ( pi − pwf )   rw  4 

(141.2) (6, 278) ( 0.810 ) ( 0.0209) ln  61.7  − 3 + 0.0       0.33  4  ( 23) (3450 − 2387)

= 2.75 md. Iteration 2. The new estimate of permeability is much greater than the previous estimate, so we continue iterating. The new transient-drainage radius is kt 377φµ gi cti

rd =

( 2.75) (8) (377) ( 0.15) ( 0.0209) (1.78 × 10−4 )

=

= 323 ft. Next, we calculate an updated permeability estimate as k= =

 141.2qg Bgi µ gi   rd  3 ln   − + s′ h ( pi − pwf )   rw  4 

(141.2) (6, 278) ( 0.810 ) ( 0.0209) ln  323.4  − 3 + 0.0       0.33  4  ( 23) (3450 − 2387)

= 3.77 md. Iteration 3. The new estimate of permeability is about 40% larger than the previous estimate, so we will do one more iteration. The new transient-drainage radius is rd = =

kt 377φµ gi cti

(3.77) (8) . 377 0 15 ( ) ( ) ( 0.0209) (1.78 × 10−4 )

= 378.5 ft.

Well Test Design Workflow 331

The new permeability estimate is k= =

 141.2qg Bgi µ gi   rd  3 ln   − + s′ h ( pi − pwf )   rw  4 

(141.2) (6, 278) ( 0.810 ) ( 0.0209) ln  378.5  − 3 + 0.0       0.33  4  ( 23) (3450 − 2387)

= 3.86 md. The permeability from the third iteration is only 3% higher than that from the second iteration, so we accept the estimate as correct. For comparison, the permeability was estimated from the buildup following the flow period as 4.2 md, very close to the value estimated from the one-point method. 12.7.4 Limitations of the One-Point Method. The two major limitations of the one-point method are that it (1) assumes infinite-acting radial flow and (2) requires a good independent estimate of skin factor. Because the one-point method assumes infinite-acting radial flow, it will not give accurate results for flow tests that end before the end of wellbore storage or after reservoir boundaries are encountered. In developing the theory for the one-point method, the skin factor S′ was assumed to be known. A poor estimate of skin factor S′ will cause the permeability calculated from the one-point method to be inaccurate as well. However, for even a rough estimate of S′ the one-point method will normally give an order of magnitude estimate for permeability. For hydraulically fractured wells, the test duration must be long enough to reach pseudoradial flow. If the one-point method is used on data where the test duration is not long enough to reach pseudoradial flow, calculated permeabilities may be negative, even if the correct equivalent skin factor is known. 12.7.5 Other Sources. WFTs, either on the subject well or on offset wells in the same formation, are a valuable source of permeability data (Kumar et al. 2010). Anand et al. (2011) describe an innovative method for estimating permeability from log data. Permeability may be estimated from data in pseudosteady-state flow, from the productivity index, from a single rate and pressure measurement during transient flow, from WFTs, or from logs. 12.8 Estimate Test Duration To Reach Desired Flow Regime One of the most critical design parameters is the test duration. Tests on offshore exploration wells are often conducted from a drillship or mobile drilling platform; cost of testing in such situations includes the day rates for the rig for the duration of the test. It may be helpful to redraw the log-log plot with cost on the horizontal axis, as shown in Fig. 12.1. Each additional log cycle increases test cost by a factor of 10; even doubling the cost of the test buys only another third of a log cycle of data. On the other hand, a test that is not run long enough to achieve the design objective incurs sunk costs that cannot be recovered. In the following discussion, we consider the time to reach desired flow regimes for a ideal drawdown test conducted at constant rate with little or no noise, or for a buildup test on a well that has been on production for a period much longer than the test is likely to be run. In principle, an initial test on a new well may be designed for the total test duration based on these guidelines, and the unit-rate pressure response may be recovered through deconvolution. 12.8.1 Time to Beginning of MTR. To determine the time to the beginning of the MTR, we must consider three separate factors: the time to the end of wellbore storage, for unstimulated wells; the time to the end of spherical flow or hemispherical flow, for partially penetrating wells; and the time to beginning of the semilog straight-line, for stimulated wells. Time to End of Wellbore Storage, Darcy Skin. Conventionally, the time to end of wellbore storage is estimated from (Earlougher 1977) teWBS =

( 200, 000 + 12, 000 s ) µ C , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.14) kh

332 Applied Well Test Interpretation

Pressure Change Derivative, psi

10000

1000

100

10

1 0.00001

0.0001

0.001

0.01 0.1 Cost, million USD

1

10

Fig. 12.1—Log-log graph, well with constant wellbore storage and skin. The horizontal axis is shown in terms of cost of the test instead of elapsed test time.

which may be written in dimensionless form as  tD  = 60 + 3.5s, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.15)    C D eWBS or from teWBS =

200C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.16) JB

Eqs. 12.14, 12.15, and 12.16 were intended for use only with wells that have a fully penetrating completion, with constant wellbore storage, zero or positive skin factor, and no non-Darcy skin. Eqs. 12.14 and 12.15 were proposed before the introduction of the logarithmic derivative. Use of the derivative curve to identify the beginning and end of flow regimes reveals that while Eq. 12.15 works well for wells with little or no damage, it grossly underestimates the time to end of wellbore storage when there is significant mechanical skin factor, as shown in Fig. 12.2. Another way to define the time to end of wellbore storage is the time at which the derivative falls to within some desired tolerance of the horizontal line for infinite-acting radial flow. For a 20% tolerance, the time to end of wellbore storage may be estimated from**  tD  = 6.93 ln (C D e 2 s ) + 23.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.17)    C D eWBS For a 10% tolerance, the time to end of wellbore storage may be estimated from  tD  = 11.84 ln (C D e 2 s ) + 53. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.18)    C D eWBS Eqs. 12.17 and 12.18 are valid for C D e 2 s ≥ 10 3. Fig. 12.2 compares the time to end of wellbore storage calculated using Eqs. 12.15, 12.17, and 12.18. Eqs. 12.17 and 12.18 may be written in field units as

{

(

)

} khµ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.19)

teWBS = 23, 500 ln C D e 2 s + 80, 000 **

Eqs. 12.17 and 12.18 have been tested for C D e 2 S between 103 and 10250.

Well Test Design Workflow 333 100

CDe2s = 1060 1030 10 tD (dpD /dtD)

1015 108 103

1 CDe2s = 1 tD /CD = 60 + 3.5s

tD /CD = 6.93 ln(CDe2s) + 23.7

tD /CD = 11.84 ln(CDe2s) + 53.0 0.1

1

10

100 tD /CD

1000

10000

Fig. 12.2—Time to end of wellbore storage for a well with constant wellbore storage and skin factor in an infinite-acting reservoir. 100

CD e 2s ′ = 10120 1060 1030

10

1015

tD (dpD /dtD)

108

1 tD(dpD /dtD) = 0.5 tD /CD = 1.69*(60 + 3.5s′) tD /CD = 3.18*(60 + 3.5s′) 0.1

1

10

100

1000

10000

tD /CD Fig. 12.3—Time to end of wellbore storage for a buildup test on a well with constant wellbore storage and 100% nonDarcy skin factor in an infinite-acting reservoir.

and

{

(

)

} khµ C, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.20)

teWBS = 40, 000 ln C D e 2 s + 180, 000

respectively. Time to End of Wellbore Storage, Non-Darcy Skin. For a buildup test on a well with 100% non-Darcy skin across the completion (see Fig. 12.3), simple modifications of Eq. 12.15 gives good estimates for the

334 Applied Well Test Interpretation

time to end of wellbore storage. For a 20% tolerance, the time to end of wellbore storage for 100% nonDarcy flow may be estimated from†  tD  = 1.69 (60 + 3.5s′), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.21)    C D eWBS and for a 10% tolerance,  tD  = 3.18 (60 + 3.5s′). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.22)    C D eWBS In field units, Eqs. 12.21 and 12.22 become teWBS , ND =

(344, 000 + 20, 000 s′) µ C kh

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.23)

and teWBS , ND =

(647, 000 + 37, 700 s′) µ C , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.24) kh

respectively. It should be noted that Eqs. 12.21 through 12.24 are strictly for buildup tests. Wellbore storage during drawdown tests on wells with non-Darcy skin factor lasts longer than for wells with Darcy skin factor. For a 20% tolerance, the time to end of wellbore storage for a drawdown test on a well with 100% non-Darcy skin may be estimated from  tD  = 11 ln (C D e 2 s ′ ) + 41, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.25)    C D eWBS or, for a 10% tolerance, from  tD  = 18.4 ln (C D e 2 s ′ ) + 82.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.26)   C  D eWBS In field units, Eqs. 12.25 and 12.26 become

{

(

)

} khµ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.27)

)

} khµ C , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.28)

teWBS = 37, 300 ln C D e 2 s + 139, 000 for 20% tolerance, and

{

(

teWBS = 62, 400 ln C D e 2 s + 280, 000

for 10% tolerance. Time to End of Spherical or Hemispherical Flow. If a well has a limited entry or partial penetration completion, the pressure response may exhibit spherical or hemispherical flow during the transition from the wellborestorage-dominated period to infinite-acting radial flow. The time at which spherical flow ends may be calculated in oilfield units from‡ t = 300

φµct h 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.29) kv

Eqs. 12.21 and 12.22 have been tested for C D e 2 S′ between 108 and 10300. Eq. 12.29 is obtained by finding the point where the negative half-slope line for the logarithmic derivative during spherical flow crosses the horizontal line for infinite-acting radial flow.

† ‡

Well Test Design Workflow 335

In dimensionless form, Eq. 12.29 becomes t D = 0.08hD 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.30) where hD is the dimensionless net sand thickness, defined as hD =

h rw

kh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.31) kv

The end of hemispherical flow may be estimated from t = 1, 200

φµct h 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.32) kv

or, in dimensionless form, t D = 0.32hD 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.33) Time to Beginning of Infinite-Acting Radial Flow for a Stimulated Well. For a stimulated well, if the pressure derivative approaches the infinite-acting horizontal line from below, the following equation may be used to estimate the beginning of infinite-acting radial flow§: t = 94, 800

φµct rwa 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.34) k

in field units, or in dimensionless form as 25 tD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.35) = CD CDe2s Eqs. 12.34 and 12.35 may be used for wells that have been matrix acidized, or for hydraulically fractured wells with dimensionless fracture conductivity Cr of 0.5 or higher. For either case, the term CDe2s must be less than 0.1. The time to the beginning of the MTR may be controlled by • The end of wellbore storage, for a well with either Darcy or non-Darcy skin factor • The end of spherical or hemispherical flow, for a well with a partial penetration completion • The beginning of pseudoradial flow, for a well with significant stimulation 12.8.2 Time to Reach Specific Boundaries or Flow Regimes. Infinite-Acting Reservoir. If the test objectives do not include estimates of distances to boundaries or reservoir volume, the shut-in period should be designed so that (1) the pressure transient is beyond the near-wellbore region, to ensure that the permeability estimated from the test represents the in-situ permeability of the reservoir, and (2) the infinite-acting radial flow period lasts long enough to provide a good estimate of permeability. Using Eqs. 12.19, 12.24, 12.29, 12.32, and 12.34, as appropriate, ensures the test is long enough for the transient to be beyond the near-wellbore region. To be able to identify infinite-acting radial flow from the derivative curve, the test duration should be at least 3 to 5 times (one-half to two-thirds log cycle) the duration of near-wellbore effects. If permeability is one of the design objectives, test duration should be at least 3 to 5 times the duration of wellbore and near-wellbore effects such as wellbore storage, spherical flow, and stimulation.

Eqs. 12.34 and 12.35 are obtained by finding the time at which the logarithmic approximation to the Ei function becomes valid at the apparent wellbore radius. The equations have been tested with the Gringarten type curve for constant wellbore storage and constant skin factor and with the Cinco type curve for finite conductivity hydraulic fractures.

§

336 Applied Well Test Interpretation

Single No-Flow Boundary. The standard radius of investigation equation, with the distance to the boundary L substituted for ri, gives a good estimate of the time at which infinite-acting radial flow ends for a well near a single no-flow boundary: t = 948

φµct L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.36) k

To confirm the presence of, and estimate the distance to, a single-boundary, the test should be run a minimum of four times the time given by Eq. 12.36. Well in a Channel. Eq. 12.37 gives the time at which the horizontal line for the logarithmic derivative for infinite-acting radial flow intersects the half-slope line for channel linear flow: t = 300

φµct w 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.37) k

Well in a Closed Reservoir. The traditional radius of investigation equation, with the drainage radius re substituted for ri, gives the time at which the horizontal line for infinite-acting radial flow intersects the unit slope line for pseudosteady-state flow: t = 948

φµct re 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.38) k

Discussion. If reaching a particular flow regime is necessary to achieve the test objectives, the test duration should be chosen so the characteristic derivative lasts at least one-half to two-thirds log cycle. Thus, the test duration should be selected to be a minimum of 3 to 5 times the values given by Eq. 12.36, 12.37, or 12.38, as appropriate. 12.9 Estimate Test Duration Based on Economics For an exploration well, determining whether or not the formation is a candidate for further investment is usually one of the most important objectives. Making such a determination requires estimates of minimum productivity and reserves required to proceed with field development. If the recovery factor can be estimated, the minimum economic fluid in place may be calculated from the reserves. The test duration required to confirm minimum productivity index and oil in place may be estimated without having to know rock and fluid properties other than average water saturation and total compressibility, as shown in this section. 12.9.1 Minimum Economic Productivity Index. The equation for the logarithmic derivative during infiniteacting radial flow is  dp  qBµ  dpD  = 141.2 tD  t  kh  dt D IARF  dp IARF = 70.6

qBµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.39) kh

Thus, if khmin is the minimum permeability-thickness product necessary for a well to be economic, then Eq. 12.39 may be rearranged to obtain the maximum value the derivative can have during infinite-acting radial flow:  dp  qBµ ≤ 70.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.40) t   dp IARF ( kh )min If Jmin is the minimum economic productivity index in STB/D/psi, Eq. 12.7 may be rearranged to obtain  r    rw 

( kh )min = 141.2 Jmin Bµ ln  e  −

 3 + s′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.41) 4 

Well Test Design Workflow 337

Combining Eq. 12.40 with Eq. 12.41 and simplifying, we have  dp  ≤ t   dp IARF

q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.42)  r  3  2 J min ln  e  − + s′   rw  4 

If the skin factor is zero, and assuming typical well spacing*, Eq. 12.42 may be further simplified as  dp  q ≤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.43) t   dp IARF 16 J min 12.9.2 Minimum Economic Oil in Place. The Cartesian derivative during pseudosteady-state flow is given by Eq. 2.76: dp 0.234 qB 0.234 qB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.44) = = dt Ahφct Vpct If the reservoir pore volume Vp is known, the oil in place N may be calculated from N=

Vp (1 − Sw ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.45) 5.615 B

Substituting, we have dp q (1 − Sw ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.46) = 24 Nct dt Thus, the Cartesian derivative must meet the following criterion to prove the minimum required oil in place: q (1 − Sw )  dp  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.47) ≤   dt PSSF 24 N min ct Expressed in another way, the logarithmic derivative at time t must fall below and to the right of the unit slope line given by q (1 − Sw )  dp  t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.48) ≤  t  dt PSSF 24 N min ct Note that Eqs. 12.47 and 12.48 are independent of reservoir geometry. 12.9.3 Test Duration for Minimum Economics. We can combine Eq. 12.42 with Eq. 12.48 to obtain an expression for the test duration required to prove the minimum economic fluid in place in terms of the minimum economic productivity index. Equating the logarithmic derivatives in Eqs. 12.42 and 12.48, we have q (1 − Sw ) 24 N min ct

t=

q

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.49)

  re  3  2 J min  ln   − + s′    rw  4 

* For normal wellbore radius and well spacing, the term ln (re / rw ) – 3/4 is typically in the range of 6.5 to 8.5. Assuming a value of 8 gives a conservative result for this application.

338 Applied Well Test Interpretation

or t=

24 N min ct  r  3  2 J min  ln  e  − + s′  (1 − Sw )   rw  4 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.50)

Assuming the total skin factor is zero and typical well spacing, the test time t in hours required to prove the minimum economic fluid in place Nmin, for a formation with the minimum economic productivity index Jmin, is given by t=

N min ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.51) 3 2 J min (1 − Sw )

Eq. 12.51 is quite useful for preliminary scoping to evaluate feasibility of testing, since it requires only estimates of minimum economic fluid in place and productivity index, along with total compressibility and average water saturation. Eq. 12.51 does not depend on fluid properties (other than compressibility), porosity, or formation thickness. The only assumptions are constant-rate, single-phase, and infinite-acting radial flow of a slightly compressible fluid. For gas wells at small drawdowns such that the slightly compressible fluid assumption is valid (∆pmaxct < 0.1, or ∆pmax < 0.1 pi ), Eq. 12.51 may be written as t=

Gmin ct 3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.52) 2 J g ,min (1 − Sw )

where Gmin is the minimum economic gas in place in Mscf, and Jg,min is the minimum economic productivity index in Mscf/D/psi. Total compressibility is probably the one single parameter with the highest uncertainty in Eqs. 12.51 and 12.52. Particular consideration should be given to the value of total compressibility chosen for test design, especially because values for both oil and pore-volume compressibility may be an order of magnitude higher than the typical default values. Although an error in total compressibility has no effect on estimated permeability, and little effect on skin factor, it has a major effect on estimates of distances to boundaries, reservoir size, and fluid in place. 12.9.4 Graphical Interpretation. Eqs. 12.40 and 12.48 may be interpreted graphically to give a quick check of whether or not a test meets the desired criteria. The check may be applied to either constant-rate drawdown test data or deconvolved constant-rate pressure response data. On the standard diagnostic plot, Eq. 12.40 defines a horizontal line that gives the maximum acceptable value of the logarithmic derivative during infinite-acting radial flow. Eq. 12.48 defines a unit slope line corresponding to the minimum economic fluid in place. The minimum original oil in place (OOIP) is confirmed when the logarithmic derivative crosses the unit slope line. Figs. 12.4 through 12.8 show five different scenarios based on the same minimum economic oil in place and permeability-thickness product. The radius of investigation equation, Eq. 12.38, was used to calculate the time to reach a radius of investigation corresponding to the drainage radius of a closed circular reservoir having the minimum economic OOIP, assuming the minimum economic permeability. For these cases, Eq. 12.38 gave a time of 191 hours, so the test was designed for a duration of 200 hours. Fig. 12.4 shows a case where the reservoir has the minimum kh and the minimum OOIP to meet the economic criteria. The derivative follows the horizontal line corresponding to the minimum kh for about two log cycles, showing that the minimum kh criterion is met. At the end of the test, the derivative just reaches the unit slope line corresponding to the minimum OOIP, showing that the OOIP criterion is met. Fig. 12.5 shows a case where the test fails the minimum kh criterion. In this case, the derivative remained above the horizontal line corresponding the minimum kh for the entire test. The horizontal derivative identifying infiniteacting radial flow begins around 1.5 to 2.5 hours. The test could have been terminated after 12 to 24 hours, once it is evident that the minimum kh criterion will not be met. Fig. 12.6 shows a case where the reservoir exceeds both minimum kh and minimum OOIP requirements. The test could have been terminated at 90 hours, after the minimum OOIP is confirmed.

Well Test Design Workflow 339

Pressure Change, Logarithmic Derivative, psi

1000

100 Meets minimum OOIP

10 Meets minimum kh

1 0.01

0.1

1 10 Elapsed Time, hr

100

1000

Fig. 12.4—Case 1: Test meets both minimum kh and minimum OOIP criteria.

Pressure Change, Logarithmic Derivative, psi

1000

100

Fails minimum kh

10

1 0.01

0.1

1 10 Elapsed Time, hr

100

1000

Fig. 12.5—Case 2: Test fails minimum kh criterion. Test could have been terminated after 12 to 24 hours when it is apparent that the minimum kh criterion will not be met.

Pressure Change, Logarithmic Derivative, psi

1000

100

10

Meets minimum kh 1 0.01

0.1

1 10 Elapsed Time, hr

Meets minimum OOIP 100

1000

Fig. 12.6—Case 3: Test exceeds both minimum kh and minimum OOIP criteria. Test could have been terminated early after minimum OOIP is confirmed around 90 hours.

340 Applied Well Test Interpretation

Pressure Change, Logarithmic Derivative, psi

1000

100 Fails minimum OOIP

10 Meets minimum kh 1 0.01

0.1

1 10 Elapsed Time, hr

100

1000

Fig. 12.7—Case 4: Test exceeds minimum kh criterion but fails minimum OOIP criterion.

Pressure Change, Logarithmic Derivative, psi

1000

100 Meets minimum OOIP

10 Meets minimum kh 1 0.01

0.1

1 10 Elapsed Time, hr

100

1000

Fig. 12.8—Case 5: Test exceeds minimum kh criterion. Channel boundaries encountered around 50 hours; test run to completion confirms minimum OOIP.

Fig. 12.7 shows a case where the reservoir kh exceeds the economic criterion, but the drainage volume is not large enough to justify development. For this case, the test is run to completion; early termination because boundaries were encountered around 60 hours is not recommended. Fig. 12.8 shows a case where boundaries are encountered around 50 hours, but the reservoir meets the minimum OOIP criterion. In this case, the reservoir is a 4 × 1 rectangle, so the two nearby boundaries cause the derivative to deviate upward before the end of the test. Unlike the case shown in Fig. 12.7, the derivative reaches the unit slope line at the end of the test, confirming the minimum OOIP criterion is met. The test duration required to confirm minimum productivity index and minimum oil in place may be estimated without having to know detailed rock and fluid properties other than average water saturation and total compressibility. 12.10 Estimate Test Rate and Determine Flow Rate Sequence A complete specification of the test design must include an estimate of the expected flow rates and a schedule of the flow and shut-in periods comprising the test. The flow rate or flow rates must be high enough to generate a pressure response that can be measured accurately enough to meet the test objectives. On the other hand, the maximum flow rate must be low enough that it can be

Well Test Design Workflow 341

sustained for the duration of the test. Finally, there are a number of operational considerations that constrain the flow rate. The duration of each flow period is designed to investigate as much of the reservoir as required to meet test objectives in the least total test time. 12.10.1 Flow Rate and Pressure Limitations. Some rate limitations are directly related to flow rate itself. The rate may also be restricted to control sandface or wellhead pressure, or to limit the total pressure drawdown imposed on the formation. Minimum Rate to Lift Liquids. To maintain stable flow in gas wells that produce small amounts of liquid, either water or condensate, the flow rate must be higher than the minimum rate to lift liquids. A quick estimate of the critical rate to lift liquids is given by qgcrit = 7 pwhf d ID 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.53) where pwhf is the flowing wellhead pressure in psi, dID is the inside diameter of the production string in inches, and qgcrit is the critical rate to lift liquids in Mscf/D. Alternatively, Turner et al. (1969) and Coleman et al. (1991) give nomographs that may be used to estimate the critical rate. Eq. 12.53, along with the Turner and Coleman methods, assumes the well is in the mist flow regime. Turner et al. concluded that the critical rate was independent of the amount of liquid produced for wells producing at liquid-gas ratios up to 130 bbl/MMscf. Facilities Limits. If testing is to be conducted with existing surface facilities, the flow rate must not exceed the capacity of the equipment. For exploration and appraisal wells, the surface equipment will be sized in coordination with the test design. Disposal of Produced Fluids. One of the most important issues involved in testing exploration wells is disposal of the produced fluids. Produced gas is usually flared, while produced oil may be flared or stored. Government regulations may limit the rate or volume of hydrocarbons that may be flared. In high-flow-rate wells, flaring will generate significant amounts of heat that must be anticipated and managed during testing (Nardone 2009). If more oil will be produced during the test than can be flared, the oil must be stored or transported offsite. It may be necessary to provide additional storage beyond the capacity of the rig. Bubblepoint or Dewpoint Pressure. For undersaturated reservoirs, if possible, the flow rate sequence should be controlled so that the sandface pressure never falls below the bubblepoint pressure (for oil wells) or dewpoint pressure (for gas wells) during the test. In many reservoirs, the discovery pressure is very close to saturation pressure, making it difficult or impossible to maintain single-phase conditions during the test. Separator Intake Pressure. The minimum flowing wellhead pressure should be at least twice the pressure downstream of the choke to maintain critical flow, which isolates upstream pressure from downstream pressure fluctuations (McAleese 2000); Fair (2001) suggests a flowing wellhead pressure at least 2.2 times the line pressure. Although strictly applicable only for gas wells, also this is good rule of thumb for oil producers (McAleese 2000). Note that this is a constraint on the surface pressure upstream of the choke, rather than on the sandface pressure. Geomechanical Concerns. In many areas, drawdown may be restricted to reduce the risk of fines migration, sand production, or even loss of the completion. Similarly, in hydraulically fractured wells, drawdown may be limited to present damage to the fracture caused by either proppant crushing or proppant embedment. 12.10.2 Maximum Sustainable Rate. After the test duration and maximum allowable pressure drop have been determined, estimate the maximum rate that can be sustained throughout the test from qmax =

kmin h ∆pmax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.54)    k t  162.6 Bµ  log  min 2  − 3.23 + 0.869s ′     φµct rw 

The minimum expected permeability should be used to estimate the maximum sustainable flow rate. 12.10.3 Flow Rate Sequence. The flow rate sequence should be chosen to maximize the likelihood of achieving the test objectives. Unless there is good reason to have multiple flow and shut-in periods (e.g., to quantify non-Darcy flow), the test design should have one main flow period followed by a buildup period.

342 Applied Well Test Interpretation

Exploration Well. The classic DST has a short flow period, followed by a buildup, then a second, longer flow period, followed by a final buildup. • Initial flow period: During the initial flow period, the well should be produced long enough to clean up completion fluids and to reach a stabilized flow rate and fluid ratios, or 3 or 4 times longer than the estimated time to end of wellbore storage, whichever is longer. • Initial buildup: The initial buildup should be one and one-half to two times the duration of the initial flow period. • Main flow period: The main flow period should be long enough to reach the desired flow regime, or to investigate enough of the reservoir to meet test objectives, as discussed in Sections 12.8 and 12.9. • Main buildup: The main buildup should be one and one-half to two times duration of main flow period. Fig. 12.9 shows a schematic of an initial test with two flow and shut-in periods. Well With Non-Darcy Skin. To quantify non-Darcy skin factor, ideally the test should be run using a modified isochronal flow rate sequence, comprising a series of short flow periods at increasing rates, each followed by a short buildup. If one of the objectives is to see reservoir boundaries or reach pseudosteady-state flow, or to construct a deliverability curve using the modified isochronal method, an extended flow period followed by an extended buildup are then run, as shown in Fig. 12.10. This test may be interpreted either as a pressure-transient test or as a deliverability test. To obtain the most accurate deliverability curves, the flow rates should be chosen to bracket the anticipated range of production rates. Although non-Darcy skin is usually associated with gas wells, it may also be present in high-rate oil wells. The test sequence and analysis procedure for oil wells is essentially the same as for gas wells. Producing Well. Fig. 12.11 shows a schematic of a buildup test on a producing well. The well is assumed to have been on production at constant-rate and gas-liquid ratio (GLR) for a period several times the anticipated buildup duration, Period 1. The well must be shut in, Period 2, to place the gauge in the wellbore. After the gauge is in the wellbore, the well must be opened back up to re-establish stabilized flow rate and stabilized GLR, Period 3, so the flowing bottomhole pressure can be recorded before shutting in for the buildup. The gauge may be run to bottom while the well is shut in, Period 2, or after stabilized flow has been re-established, Period 3. If the flow rate is too high for the gauge to be run in safely while the well is flowing, then it must be run in during Period 2. If a flowing gradient survey is to be conducted, stabilized flow must be re-established before running in the gauge during Period 3.

Bottomhole Pressure, psi

5100

Initial buildup

Main flow

Main buildup

5000

4900

4800

Initial flow 4700

0

10

20

30 40 50 Elapsed Time, hr

60

Fig. 12.9—Initial test with two flow and buildup periods.

70

80

Well Test Design Workflow 343 8000

Bottomhole Pressure, psi

Extended flow period

Final buildup period

7500

7000

6500

6000 0

100

200

300 400 Elapsed Time, hr

500

600

700

Fig. 12.10—Multirate test with extended flow and buildup periods.

3800

1

2

3

4

Bottomhole Pressure, psi

3600

3400

3200

1 – Well on production 2 – Well shut in to run gauge in hole 3 – Re-establish stabilized flow 4 – Buildup

3000

2800

0

5

10

15 20 25 Elapsed Time, hr

30

35

40

Fig. 12.11—Buildup on producing well.

The stabilized flow rate and GLR during Period 3 should be the same rate and GLR as the well was producing during Period 1. Stabilized flow during Period 3 should be maintained long enough that the temporary shutin has negligible effect on the pressure response during the ensuing buildup. Frailey et al. (1994) recommend a minimum of two or three times the duration of the temporary shut-in period. Fig. 8.34 shows the effect of a flow period twice the duration of the shut-in on the subsequent buildup. The durations of the shut-in and stabilized flow periods should be recorded so the temporary shut-in can be accounted for during the interpretation.

344 Applied Well Test Interpretation

If a producing well must be shut in to run gauges in the hole, the well must be brought back on stabilized production at the same rate as before the shut-in, for a minimum of two to three times as long as the shut-in period, before shutting in for the buildup period. 12.11 Estimate Magnitude of Pressure Response While the design flow rate must be less than the maximum sustainable flow rate, it should be high enough to provide a pressure response that is accurate enough to achieve the test design objectives. The pressure response may be estimated from the slopes of the straight-line graphs for the expected flow regimes, or it may be simulated using either an analytical or a numerical simulator. 12.11.1 Slope of Semilog Graph During Radial Flow. The slope of the semilog straight-line during infiniteacting radial flow may be estimated from m =

162.6qBµ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.55) kh

while the logarithmic derivative is given by  dp  70.6qBµ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.56) t   dt IARF kh If the standard deviation in the random noise in the pressure measurement is sp, it can be shown that the standard deviation of the noise in the logarithmic derivative, as calculated from the Bourdet 3-point method with smoothing parameter L, is given by

σ

dp  t   dt 

σp

=

2L

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.57)

For the derivative to appear reasonably smooth on a log-log plot, the noise in the derivative should be less than 10% of the magnitude of the derivative. The noise in the pressure response must be less than

σ p ≤ 0.1

(

 dp  2 L  t  ≅ 0.06 L m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.58)  dt 

)

For the recommended value of the smoothing parameter value L of 0.1, the noise in the pressure response must be less than

σ p ≤ 0.006m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.59) To get smooth derivatives with the Bourdet method, the gauges selected for the test must be capable of measuring changes in pressure smaller than given by Eq. 12.59. Accurate permeability estimates may be obtained from semilog analysis with noise levels perhaps an order of magnitude higher than given by Eq. 12.59; Barnum and Vela (1984) recommend gauge resolution be 10% or less of the expected slope of the semilog straight-line. However, for noise levels much higher than given by Eq. 12.59, the logarithmic derivative loses much of its diagnostic capability, making identification of the correct semilog straight-line problematic. To obtain smooth derivatives with the Bourdet method, gauges selected for a test must be capable of measuring changes in pressure smaller than given by Eq. 12.59. 12.11.2 Slope of Cartesian Graph During Pseudosteady-State Flow. The slope of a Cartesian graph of pressure vs. time during pseudosteady-state flow may be estimated either from drainage area, net-pay, and porosity using mPSS =

0.234 qB , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.60) Ahφct

Well Test Design Workflow 345

or from the oil in place using mPSS =

q (1 − Sw ) 24 Nct

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.61)

12.11.3 Slope of Square-Root-of-Time Graph During Channel Linear Flow. For a channel open at both ends, the slope of a graph of pressure vs. the square root of time may be estimated from mCLF = 8.128

qB hw

µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.62) kφct

12.12 Select Gauges The type or types, number, and placement of gauges must be carefully considered to ensure that the test objectives will be met. 12.12.1 Gauge Terminology. There are a number of different terms that are used to specify the characteristics of pressure gauges used for well testing (Veneruso et al. 1991). Calibration Range. Calibration range is the range of temperatures and pressures for which the gauge is calibrated. Pressure gauges must be calibrated before placed into service. Depending on the gauge type and duration of exposure to extreme temperature and pressure, periodic recalibration may be required. Accuracy and Resolution. Accuracy refers to the total error, from all sources combined, in the pressure reported by the gauge, while resolution refers to the smallest detectable change in pressure. Most gauges have much higher resolution than accuracy. Gauge resolution is dependent on sampling frequency. Specifications usually give the gauge resolution for a specified sampling interval, often 1-second sampling. For quartz gauges, achievable resolution is inversely proportional to the gate time.** Repeatability and Hysteresis. Both repeatability and hysteresis refer to the maximum difference between gauge readings at the same applied pressure. Repeatability is defined as the maximum difference in the pressure reading as the applied pressure is varied over two full-scale cycles, with the pressure readings taken as applied pressure changes in the same direction during consecutive cycles, as shown in Fig. 12.12. Hysteresis is also defined as the maximum difference in gauge reading at a single applied pressure but measured as the applied pressure changes in opposite directions during a single cycle, as shown in Fig. 12.13. Stabilization Time and Drift. Stabilization time and drift both refer to changes in gauge response over time that are not caused by changes in applied pressure. Stabilization time is the time required for the gauge response to return within a specified tolerance of the applied pressure following a step change in pressure or temperature.

Repeatability = max ∆r(same direction, different cycles)

Applied Pressure

p -max

∆r

Gauge Response

r-max

Fig. 12.12—Repeatability is the maximum difference in the gauge response at the same applied pressure, taken in the same direction during consecutive pressure cycles (adapted from Veneruso et al. 1991). **

Personal communication with Jason Blackburn (2013), Quartzdyne.

346 Applied Well Test Interpretation

Hysteresis = max ∆r (different directions, same cycle)

Applied Pressure

p -max

∆r

Gauge Response

r-max

Fig. 12.13—Hysteresis is the maximum difference in the gauge response at the same applied pressure, taken in opposite directions during a single pressure cycle (adapted from Veneruso et al. 1991).

6200 6195

Pressure, psi

6190 6185 6180 6175 6170 CQG HP PAINE

6165 6160

200

250

300 Time, seconds

350

400

Fig. 12.14—Pressure gauge overshoot following a step increase in pressure (Veneruso et al. 1991).

Fig. 12.14 shows the responses for three different gauges following a 5,000 psi step increase in pressure. Fig 12.15 shows the responses for the same three gauges following a 5,000 psi step decrease in pressure†. Drift is the maximum long-term rate of change in the pressure response at constant applied pressure. Drift is usually associated with inelastic deformation of the transducer caused by prolonged exposure to reservoir temperature and pressure. Transducers comprising crystalline materials, such as quartz or sapphire, are much less prone to drift than are other gauges. 12.12.2 Types of Gauges. In 1992, pressure gauges in use included mechanical gauges, conventional and sapphire strain gauges, capacitance gauges, and standard and compensated quartz gauges (Vella et al. 1992). Although most gauges in use today rely on the same basic physical principles, advances in design and construction have led to dramatic improvements in operating range, stabilization time, power consumption, and reliability. Mechanical Gauge. A mechanical gauge uses a helical Bourdon tube as the pressure sensing element. The Bourdon tube is coupled to a stylus that marks the gauge response on a strip of coated metal foil. A spring-driven Figs. 12.14 and 12.15 reflect the behavior of gauges as of 1991. Hewlett-Packard halted production of the HP 2813 transducer in 1993 (Ward and Wiggins 1997).

†

Well Test Design Workflow 347 40

35

Pressure, psi

30

25

20

15

10

CQG HP PAINE 50

100

150 Time, seconds

200

250

Fig. 12.15—Pressure gauge overshoot following a step decrease in pressure (Veneruso et al. 1991).

clock moves the metal foil to provide a reference time scale. Different clocks may be used depending on the anticipated test duration, with clocks having time ranges from three to 360 hours available (Kuster Company 2011; Sercel-GRC Corp. 2013b). Mechanical gauges have been supplanted by electronic gauges for the vast majority of oilfield applications. However, mechanical gauges are available that can function at temperatures up to 500°F for 360 hours, or up to 700°F for 24 hours (Kuster Company 2011; Sercel-GRC Corp. 2013b). Today’s electronic gauges are limited to a maximum temperature of 300°F to 400°F. Unlike electronic gauges, in which time is measured much more accurately than pressure, mechanical gauges are subject to comparable errors in both time and pressure. Accuracy of the 300°F and 500°F gauges is given as 0.2% of full scale, with a repeatability of 0.05% of full scale. Accuracy and repeatability of the 700°F gauges are given as 1% and 0.4%, respectively, of full scale (Sercel-GRC Corp. 2013b). Capacitance Gauge. In a capacitance transducer, a change in pressure causes a small change in the distance between the two conductive surfaces of a capacitor, thus causing a change in its capacitance. The change in capacitance is measured and converted to a digital signal. Although apparently no longer widely used for memory gauges, capacitance gauges are still used to monitor performance of electric submersible pumps and progressive cavity pumps for artificial lift (Sercel-GRC Corp. 2013a, 2013d). Conventional Strain Gauge. A strain gauge transducer comprises a resistive element bonded to a substrate that deforms under pressure. The deformation of the substrate causes the length and cross-sectional area of the resistive element to change, thereby changing its resistance. The change in resistance is measured with a Wheatstone bridge, then converted to a digital pressure reading. One manufacturer recommends annual recalibration for conventional strain gauges (Paine Electronics 2013). Sapphire Strain Gauge. The sapphire strain gauge is similar in concept to the conventional strain gauge, with resistive elements sputtered onto one face of a hollow box constructed from a single sapphire crystal. The mechanical properties of the sapphire crystal provides better repeatability, reduced hysteresis, and lower drift than conventional strain gauges (Schlumberger 2006). Piezoresistive Silicon Gauge. The piezoresistive silicon strain gauge is similar to a conventional strain gauge in that a change in pressure causes a change in resistance. However, the doped silicon resistive element experiences a change in resistivity with pressure, rather than a change in resistance, as in the conventional strain gauge. This gives a higher gauge factor than a conventional strain gauge, resulting in improved resolution. The silicon transducer may quite small. This gives very low thermal inertia, allowing rapid stabilization following a change in pressure or temperature. Gauges are available with sampling rates as high as 32,000 times per second for special applications (DataCan 2012). Piezoresistive memory gauges rated to 300°F/15,000 psi, with accuracy and resolution of 0.03% and 0.0003% of full scale, respectively, are available from multiple suppliers. Long-term drift is less than 3 psi/year. Silicon-on-Sapphire Gauge. Silicon-on-sapphire gauges combine a silicon piezoresistive element with a sapphire crystal substrate. A sapphire crystal is cut along a plane that gives a lattice spacing compatible with that of silicon. The silicon is epitaxially grown on sapphire face thus exposed (Sensonetics 2013).

348 Applied Well Test Interpretation

At least two vendors offer gauges described as silicon-sapphire or silicon-on-sapphire, rated to 350°F/25,000 psi, with accuracy and resolution comparable to that of piezoresistive silicon strain gauges (Nan Gall 2013; PioneerPetrotech 2013). Quartz Gauge. Standard quartz gauges measure pressure by detecting the change in frequency of a quartz crystal resonator caused by a change in applied pressure. In one configuration, the gauge incorporates three separate quartz resonators (EerNisse 2001) . The quartz resonators are cut to take advantage of the anisotropic pressure and temperature response of quartz. One of the resonators, used as a frequency reference, is cut to minimize its sensitivity to temperature. A second resonator, used as a temperature transducer, is cut to maximize its sensitivity to temperature, while the third resonator is cut to maximize its sensitivity to pressure. The pressure-sensitive resonator is exposed to wellbore pressure. The differences in frequency between the reference and pressure resonators, and between the reference and temperature resonators, are determined by a digital counter. The achievable resolution is inversely proportional to gate time. Early quartz gauges exhibited poor stabilization characteristics, in large part because the crystals have to return to thermal equilibrium following a step temperature or pressure change. This effect has been reduced by shrinking the physical size of the transducer assembly and moving the reference and temperature resonators closer to the pressure resonator (Ward and Wiggins 1997). One manufacturer recommends biannual calibration checks to determine when recalibration is necessary for quartz gauges (Quartzdyne 2004). Another supplier states that quartz transducers are stable enough that recalibration should not be necessary under normal use (Sercel-GRC Corp. 2013c). Quartz memory gauges rated to 392°F/25,000 psi, with accuracy and resolution of 0.02% and 0.00006% of full scale, respectively, are available from multiple suppliers. Long-term drift is less than 3 psi/year. Compensated Quartz Gauge. The compensated quartz gauge uses a single, dual-mode quartz resonator. The quartz is cut so one of the two vibrational modes is more sensitive to pressure, while the other is more sensitive to temperature. Use of a single crystal gives a significant improvement in stabilization time, particularly as compared to early standard quartz gauges. Disadvantages of the compensated gauge are the high cost and fragility of the dual-mode resonator (Schlumberger 2006). Compensated quartz gauges are available rated to 350°F/16,000 psi, with accuracy of 2.5 psi and resolution of 0.01 psi (Schlumberger 2013). Other Pressure Measurement Options. Other pressure measurement options that are particularly suited to certain applications include wellhead pressure measurement, acoustic fluid-level measurement, and inert-gas purged capillary tube. Wellhead Pressure Measurement. Wellhead pressure measurement with calculation of bottomhole pressure may be used for either gas or oil wells under certain conditions. Wellhead pressure measurement is widely applicable for both drawdown and buildup testing for gas wells that produce at rates high enough to lift liquids (Fair et al. 2002). Oil wells in which one phase is continuous from sandface to wellhead (single-phase, bubble, mist, or churn flow regimes) may be tested using a “constant-choke” flow test (Fair et al. 2002). Essentially the same technology can be used to measure pressure at the subsea tree in offshore wells; the bottomhole pressure is calculated as with wellhead pressure measurement (Waldman et al. 2002). Acoustic Fluid-Level Measurement. Acoustic fluid-level measurement is an economically attractive option for wells producing by rod pump (Podio et al. 1992; Rowlan et al. 2008). The wellhead pressure is recorded as a function of time. The height of the fluid level is measured as a function of time by repeated acoustic shots. The bottomhole pressure may be calculated at the wellsite from surface pressure, acoustic velocity, liquid level data, and well data (Podio et al. 1987). One major advantage of acoustic fluid-level measurement is that the afterflow rate for both oil and gas may also be calculated from the wellhead pressure and fluid-level data (Brownscombe 1982). Including the calculated afterflow rate in the analysis can significantly reduce the influence of wellbore storage (Fetkovich and Vienot 1984). Our experience with incorporating afterflow rate in analysis of pressure data from acoustic fluid-level measurement suggests that the apparent reduction in wellbore storage allows the reservoir pressure response to be observed about a log cycle earlier than if afterflow is not incorporated in the analysis. Helium-Purged Capillary Tube. Another pressure measurement option that may be appropriate in some situations is the use of a capillary tube filled with an inert gas, usually helium (Ramey 1992; Almanza et al. 2008). Pressure is measured at the surface. The bottomhole pressure is calculated from the known surface pressure, wellbore temperature profile, and density of the inert gas as a function of pressure and temperature. One of the primary advantages of the system is that it does not require any downhole electronics, thus allowing use in harsh environments where electronics have poor reliability (Almanza et al. 2008).

Well Test Design Workflow 349

12.12.3 General Considerations. The goal of gauge selection is to ensure that adequate data are acquired reliably and accurately so the test objectives can be met. Gauge selection includes specifying the number, type, and location of the gauges, along with the sampling interval to be programmed for each gauge. Operating Range. As of 2013, electronic gauges are available with temperature ratings of 150°C (302°F), 177°C (351°F), and 200°C (392°F). Equipment selection becomes increasingly important as temperatures approach the 200°C upper limit. Gauges should be selected so the maximum expected pressure is no more than 60 to 80% of the gauge’s pressure rating. Since gauge accuracy and resolution are often stated as a percentage of the full-scale reading, use of a gauge that is rated for pressures much higher than anticipated results in poorer performance than a gauge that has the correct range. Accuracy and Resolution. Accuracy is particularly important in situations in which data from multiple gauges will be compared, as in estimating fluid in place using material balance. On the other hand, resolution is more important than accuracy in many applications (e.g., estimating permeability from data in infinite-acting radial flow and obtaining smooth derivatives. Eq. 12.59 may be used to estimate the resolution necessary for smooth derivatives using the Bourdet method. Stabilization Time and Drift. Stabilization time is especially important when pressure and/or temperature is changing rapidly, or when the time scale of interest is in the range of seconds to minutes (e.g., buildup tests in high-permeability formations and wireline formation testing). Drift is particularly important when pressure changes are small and the time scale of interest is in the range of days to months (e.g., extended well testing to estimate reserves) and multiwell tests such as interference or pulse tests. Reliability. Given the expense of retesting in the event of failure to acquire data, reliability is a major concern in gauge selection. Multiple gauges may be used to ensure that at least one gauge has usable data. For routine testing, two gauges is typical. However, in high-cost tests, it is not uncommon to run four or more gauges, with some operators using as many as 18 gauges for a single test. Botto and De Ghetto (1994) discuss the probability of simultaneous failure of multiple gauges. If more than two gauges will be run, a mixture of different types of gauges should be used to reduce the likelihood of failure of all gauges by means of a single failure mode. Surface readout or nonvolatile memory preserve data taken before gauge failure. Battery failure on a gauge with volatile memory will result in loss of the entire data set. Location. The primary gauge or gauge carrier should be placed as close as possible to the completed interval to reduce the likelihood of movement of a fluid interface between the formation and the gauge during the test. In addition to bottomhole pressure data, depending on the wellbore configuration, tubing and/or annulus pressure may help diagnose wellbore-related anomalies. Tandem gauges at different depths help confirm the presence of a rising or falling liquid level distorting the pressure response during buildup. Sampling. Most gauges have the option of sampling intervals from 1 second to 18 hours. Many gauges have programmable sampling to allow high-frequency sampling at the beginning of each flow or shut-in period when pressure is changing very rapidly, with reduced sampling frequency during the latter part of the period when pressures are changing much more slowly. Because of the high memory capacity of today’s gauges, it is usually feasible to run an entire test with a fixed sampling interval. For example, for a typical memory capacity of 1,000,000 data points, a test with fixed 5-second sampling could last more than 8 weeks before running out of memory. Some gauges may be programmed to record data only when the pressure has changed by a specified amount since the previous data point. The use of a pressure change criterion is not recommended. If the threshold pressure change is too large, data loss is particularly likely in the worst possible situation, a thick, high-permeability reservoir during the MTR. 12.13 Simulate Test One of the primary goals of well test design is to ensure that the objectives of the test are met. Simulating the expected pressure response for a range of likely reservoir conditions allows the test design to be evaluated to determine whether or not it will meet the test objectives. If it is impossible to design a test that meets the stated objectives and honors other time, budget, or technical constraints, no test should be run (Frailey et al. 1994). Likewise, simulating the dynamic wellbore and flowline hydraulics help optimize equipment selection and minimize the potential for erosion, hydrate formation, and liquid loading. Early identification of these risks helps prevent problems that might otherwise delay completion of a test, at significant cost of rig time (Teng et al. 2006). Simulating the test may incorporate as many as three distinct facets: simulation of the reservoir pressure response, simulation of the wellbore/flowline system, simulation of the gauge behavior.

350 Applied Well Test Interpretation

12.13.1 Generate Synthetic Pressure Response Using Analytical or Numerical Simulator. In some situations, test simulation may be as simple as using an analytical model to generate the pressure response to the test design flow schedule using superposition. In other situations, the high cost of testing an offshore exploration well may justify use of a 3D, 3-phase numerical model. Regardless of the type of simulator used, the simulation model should honor the available data to the extent possible. Construction and calibration of a 3D, 3-phase numerical model requires more detailed reservoir data than is usually available in a exploration testing scenario. The data that are available, interpreted logs and WFT data, are often obtained very shortly before the test is to begin. 12.13.2 Simulate Wellbore and Flowline Hydraulics. For a buildup test design for a well that has been producing for some time, there is probably little need to model the wellbore and flowline hydraulics. On the other hand, the cost of testing an offshore exploration well is likely to require detailed modeling of wellbore hydraulics. Wellbore modeling may be limited to steady-state nodal analysis, or a dynamic wellbore simulator may be used to model transient or dynamic wellbore behavior. Issues of particular interest during wellbore modeling include wellbore unloading, completion cleanup, erosional velocity limits, hydrate formation, and liquid loading. Kumar et al. (2010) used WFT data to construct and calibrate a 3D, three-phase layer-cake model for detailed reservoir modeling. They used nodal analysis to evaluate whether or not the well would be able to clean up naturally using different combinations of the available tubing sizes and perforation intervals, or whether coiled tubing or gas lift would be necessary to unload the well for cleanup. Teng et al. (2006) presented a field case history of a test design for a big-bore gas well. They used a three-phase, dynamic wellbore simulator to model the transient behavior of the wellbore hydraulics to (1) study the effect of flow rate on wellbore unloading; (2) determine the minimum flow rate necessary to clean up the well, and (3) study the impact of an emergency shutdown occurring during cleanup. The authors used a steady-state Forchheimer inflow performance model to model reservoir behavior. 12.13.3 Simulate Noise and Analyze Synthetic Data Set. In thick high-permeability reservoirs, the predicted pressure response will be small. In this situation, it may be difficult to obtain an interpretable test (an enviable problem to have!). Adding a realistic amount of random noise to the predicted pressure response, then attempting to analyze the synthetic data set, is a good way of evaluating whether or not a given test design will achieve the desired objectives. However, if a single model is used to both generate and match the synthetic data set, and the test duration is the minimum that gives the “right” answers within the desired confidence interval, the actual test data are likely to be uninterpretable (Spivey 2013). Instead, the test should be run at least two to three times longer to ensure that the actual test data are interpretable using an approximate model. 12.14 Design Example In this section, we illustrate some of the steps of the well test design workflow. 12.14.1 Problem Statement. Your company has just drilled and logged a new exploration well and encountered a gas zone that is very promising. From the logs, the target zone is between 8,195 to 8,445 ft true vertical depth. The well will be tested through a temporary completion that has only 50 ft open to flow in the center of the pay interval. Rock and fluid property data are given in Table 12.3. The datum depth (true vertical depth) is 8,320 ft. Assume a normal pressure gradient of 0.465 psi/ft, a normal temperature gradient of 2°F/100 ft, and a mean annual surface temperature of 32°F. Based on the log data, the permeability is estimated to be in the range of 100 md to 1,000 md. Assume a mechanical damage skin factor of 10 (this will be magnified by the limited entry completion).

TABLE 12.3—ROCK AND FLUID PROPERTY DATA FOR DESIGN EXAMPLE Reservoir and Fluid Properties

f h

27%

Bg

250 ft

μ

0.0225 cp

rw

0.5 ft

ct

171 × 10–6 psi–1

cg

216 × 10–6 psi–1

0.810 bbl/Mscf

Well Test Design Workflow 351

The pressure drawdown must be limited to 10% of initial pressure. You are not permitted to flare more than a total of 25 MMscf of gas. The service company has two gauge options, a piezo gauge and a quart gauge. The piezo gauge has a rated resolution of 0.05 psi, while the quartz gauge has a resolution of 0.01 psi. Both gauges are available with calibration ranges of 0–5,000 psi, 0–10,000 psi, and 0–15,000 psi. All gauges are rated to 300°F. The well construction budget allows a total test duration of 72 hours. 12.14.2 Collect Data. Because this is an example problem, most of the data are given. We need to estimate the initial pressure and formation temperature, along with the vertical to horizontal permeability ratio. Pressure and Temperature. We estimate the initial pressure and temperature from dp = (8, 320 ) ( 0.465) = 3, 868.8 psi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.63) dz

pi = d

T f = Tsurf + d

 2  dT = 32 + (8, 320 )   = 198.4 °F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.64)  100  dz

The maximum drawdown and minimum flowing bottomhole pressure are given by ∆pmax = 0.1 pi = 386.9 psi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.65) and pwf min = pi − ∆pmax = 3, 868.8 − 386.9 = 3, 481.9 psi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.66) respectively. Vertical/Horizontal Permeability Ratio. We need the vertical to horizontal permeability ratio kv/kh to calculate time to end of spherical flow. In the absence of other information, we assume a typical value of 0.1. 12.14.3 Estimate Time Required To Meet Test Objectives. To obtain an accurate estimate of permeability, we need at least one-half log cycle of data in the MTR. To estimate the time at which the MTR begins, we need to estimate the time to end of wellbore storage and the time to end of spherical flow. Wellbore-Storage Coefficient. The cross-sectional area of the wellbore is Awb =

π dcsg 2 (3.14 )  6 2 2 =   = 0.1963 ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.67) 4 4  12 

We will assume a total wellbore depth of 8,600 ft, including 155 ft for the rathole, to calculate the wellborestorage coefficient. For surface shut-in, the wellbore volume is Vwb = dAwb = (8, 600 ) ( 0.1963) = 1, 688 ft 3 = 300.8 bbbl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.68) The wellbore-storage coefficient, assuming a gas-filled wellbore, is given by Eq. 12.2: Cwb = cwbVwb = ( 216 × 10 −6 ) (300.8) = 0.065 bbl psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.69) Partial Penetration Skin Factor. The geometric skin factor for a limited entry completion is given by Eqs. 10.7 through 10.12. The dimensionless parameters rD, hpD, and h1D are given by r rD = w h h pD =

 kv   kh

hp h

=

  

1

2

=

1 0.5 0.1) 2 = 0.00063325, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.70) ( 250

50 = 0.2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.71) 250

352 Applied Well Test Interpretation

and h1D =

h1 100 = = 0.4, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.72) h 250

respectively. Intermediate results A and B are given by A=

1 1 = = 2.222, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.73) h1D + h pD 4 0.4 + 0.2 4

and B=

1 1 = = 1.818. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.74) h1D + 3h pD 4 0.4 + (3) ( 0.2) 4

The geometric skin factor sp is given by  h  1  π 1 pD s p =  − 1  ln + ln    h pD  2rD h pD  2 + h pD

 A −1    B −1

1

2

   

1     1   π 1 0.2  2.222 − 1  2   + ln = − 1  ln     0.2   ( 2 ) ( 0.0006325)  0.2  2 + 0.2  1.818 − 1   = 20.28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.75)

The total skin factor is given by Eq. 10.5: st =

h 250 s + sp = (10 ) + 20.28 = 70.28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.76) hp 50

Time to End of Wellbore Storage. We will calculate the time to end of wellbore storage assuming Darcy skin factor. This will give a conservative estimate for the time to end of wellbore storage for the buildup, since any non-Darcy skin will cause wellbore storage to end earlier. The dimensionless wellbore-storage coefficient is given by Eq. 4.14: CD =

0.894 0.894 0.065) = 20.14 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.77) C= 2 ( 2 φct hrw ( 0.27) (171 × 10−6 ) ( 250 ) ( 0.5)

while the parameter CDe2s is C D e 2 s = 20.14e(2) (70.28) = 2.25 × 10 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.78) The time to end of wellbore storage is given by Eq. 12.20:

µ teWBS = 40, 000 ln (C D e 2 S ) + 180, 000 C kh = ( 40, 000 ) ln ( 2.25 × 10 62 ) + 180, 000

( 0.0225) 0.065 ( ) (100 ) ( 250 )

= 0.346 hour hour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.79) Time to End of Spherical Flow. The time to end of spherical flow is given by Eq. 12.29: teSF

( 0.27) ( 0.0225) (171 × 10 φµct h 2 = 300 = (300 ) kv ( 0.1) (100 )

−6

) (250)

2

= 1.95 hour. . . . . . . . . . . . . . . . . . . . . . . . (12.80)

Well Test Design Workflow 353

Time to Beginning of MTR. Because spherical flow lasts much longer than wellbore storage, there does not appear to be any advantage to using downhole shut-in, which will not reduce the time to reach the end of spherical flow. The spherical flow regime should last about log10 (1.95/ 0.346) = 0.75, or three-fourths of a log cycle, which should be long enough to provide a reasonable estimate of spherical permeability. Radius of Investigation. To determine the test duration, we use the minimum permeability estimate of 100 md to calculate the time to reach a radius of investigation of 5,000 ft: t = 948

( 0.27) ( 0.0225) (171 × 10 φµct ri 2 = ( 948) k (100 )

−6

) (5, 000)

2

= 246 hour. . . . . . . . . . . . . . . . . . . . . . . . . (12.81)

This is significantly longer than the 72 hour maximum for this test. With a total test time limited to 72 hours, we select a flow period of 24 hours and a shut-in period of 48 hours. For a permeability of 100 md, the radius of investigation at the end of the 24-hour flow period is ri =

kt = 948φµct

(100 ) ( 24) = 1, 560 ft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.82) 948 0 . 27 ( ) ( ) ( 0.0225) (171 × 10−6 )

Similarly, we find the radius of investigation at 48 and 72 hours to be 2,210 ft and 2,700 ft, respectively. For the upside case permeability of 1,000 md, the radii of investigation at 24, 48, and 72 hours are 4,940 ft, 6,980 ft, and 8,550 ft, respectively. 12.14.4 Estimate Test Rate and Determine Flow Rate Sequence. The design rate and duration for the test has to take two constraints into account, the maximum allowed drawdown of 386.88 psi, and the maximum gas volume to be flared, 25 MMscf. Maximum Sustainable Flow Rate. The maximum flow rate that can be sustained without exceeding the allowed drawdown is given by Eq. 12.54: kmin h ∆pmax min h    kkmin t ∆pmax 162.6 Bµ log k t 2  − 3.23 + 0.869s′   minc r  162.6 Bµ log  φµ − 3.23 + 0.869s′  t 2w     φµct rw   (100 )( 250 ) = (386.88) 100 )( 250 ) ( = (386  .88)         (100) ()24( 24) ) (100 3.23 + 0.869 ( 70.228 )  .0225)log 162..66 ) (00.8.810  3−.23 + 0.869 ( 70.28)  )  log ((162 ) ( 10 ))((00.0225 2 2 − −6 −6   × 10 .27 .0225 171 × 10 ) ( 0(.50).5)  ) ()0(.00225 ) ()171   ( 0(.027   48,,100 100Mscf Mscf D. D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.83) == 48

qmax = qmax =

(

)

Test Rate and Flow Rate Sequence. Because flaring is limited to a total of 25 MMscf, the maximum rate that we can produce for 24 hours is 25,000 Mscf/D. So, we design for a flow period of 24 hours at a rate of 25,000 Mscf/D, followed by a 48-hour buildup. 12.14.5 Estimate the Magnitude of the Expected Pressure Response. Maximum Drawdown and Minimum Flowing Bottomhole Pressure. The drawdown at the end of the flow period is given by  qBµ   k t  ∆p = 162.6 qBµ  log kminmint 2  − 3.23 + 0.869s′ ∆p = 162.6 kmin h log  φµct r2w  − 3.23 + 0.869s′  kmin h   φµct rw      25, 000 ) ( 0.810 ) ( 0.0225)   100 ) ( 24 )   ( (   100 24 25 , 000 0 . 810 0 . 0225 ( ) ( ) ( ) ( ) ( ) 3.23 0.869 log 162..66 2−3−.23 ==162 + 0+.869 28.)28 )  log ( 70(.70  .27 0.0225 171 × 10 −6 2 250) ) (100))((250 (100 ) ()0(.0225) ()171 × 10−6 ) ( 0(.50).5)      (0(.027 201..00psi. psi., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.84) ==201

(

)

giving a flowing bottomhole pressure of pwf = pi − ∆p = 3, 868.8 − 201.0 = 3, 667.8 psi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.85)

354 Applied Well Test Interpretation

For the 1,000 md upside case, the pressure drop at the end of the flow period is 20.4 psi, and the flowing bottomhole pressure is 3,848.4 psi. Slope of Semilog Straight-Line. The slope of the semilog straight-line is given by Eq. 12.55. For the minimum permeability case of 100 md, m =

162.6qBµ (162.6) ( 25, 000 ) ( 0.810 ) ( 0.0225) = 2.96 psi cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . (12.86) = kh (100 ) ( 250 )

Similarly, for a permeability of 1,000 md, the slope would be 0.296 psi/log cycle. Recommended Resolution. The recommended resolution to give a smooth derivative using a smoothing parameter 0.1 is given by Eq. 12.59. For the minimum permeability case,

σ p ≤ 0.006m = ( 0.006) ( 2.96) = 0.018 psi, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.87) while the 1,000 md case gives σ p ≤ 0.0018 psi. 12.14.6 Select Pressure Gauges. The rated resolution of the quartz gauge, 0.01 psi, meets the recommended resolution of 0.018 psi (for the 100 md case), while the resolution of the piezoresistive gauge, 0.05 psi, does not. Neither gauge has resolution high enough to meet the recommended criterion for the 1,000 md case. We can estimate the value of the smoothing parameter required to obtain smooth derivatives for the 1,000 md case by rearranging Eq. 12.58: L≥

σp 0.01 = = 0.56. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12.88) 0.06m ( 0.06) ( 0.296)

While this is higher than we would like to use, it is usable. If we accept a 20% noise in the derivative instead of 10%, we can use a smoothing parameter of 0.28, which is quite reasonable. The expected pressure of 3,869 psi would be 78% of full scale with the 5,000 psi gauge, but only 39% of full scale with the 10,000 psi gauge. If there is a chance that the actual pressures might be higher than 5,000 psi, the 10,000 psi gauge should be selected. 12.14.7 Simulate the Test. We will simulate the test for two cases. For both cases, we will use a gauge resolution of 0.01 psi. For Case 1, we will use a permeability of 100 md, for Case 2, we will use 1,000 md. Case 1—100 md. Fig. 12.16 shows the log-log graph for Case 1, with a smoothing parameter of 0.1. As predicted, spherical flow ends, and the MTR begins, around two hours. Spherical flow lasts a little more than one

Adjusted Pressure Change, Derivative, psi

1000

100

10

1

0.1 0.0001

Pressure change Derivative Derivative end effects 0.001

0.01 0.1 1 Adjusted Equivalent Time, hr

10

100

Fig. 12.16—Simulated test, permeability 100 md, gauge resolution 0.01 psi, smoothing parameter 0.1.

Well Test Design Workflow 355

log cycle, and the MTR lasts almost a log cycle, so we should be able to estimate both horizontal and spherical permeability for this case. Semilog analysis gives a horizontal permeability of 98.95 md, a skin factor of 68.16, and an initial pressure of 3,869.9 psi, in good agreement with the values of 100 md, 70.28, and 3,868.8 psi used for the simulation. Straightline analysis of the spherical flow data give a spherical permeability of 46.8 md, and a vertical permeability of 10.5 md; again, very close to the values of 46.4 md and 10.0 md used to generate the synthetic data set. Case 2—1,000 md. Fig. 12.17 shows the log-log graph for Case 2, with a smoothing parameter of 0.1. The derivative shows the typical banding when the Bourdet 3-point method is used to calculate the derivative for data with low gauge resolution. In Fig. 12.18, the smoothing parameter has been increased to 0.28, giving a much smoother derivative, although not as smooth as that for Case 1. The spherical and radial flow regimes are readily identified, with spherical flow lasting almost a log cycle, and radial flow lasting almost two log cycles. Even with the noisy pressure data, semilog analysis of the data in radial flow gives permeability as 998.1 md, skin factor as 69.3, and initial pressure as 3,868.9 psi, in excellent agreement with the input values of 1,000 md, 70.28, and 3,868.8 psi. Straight-line analysis of the data in spherical flow gives spherical permeability of 468 md and a vertical permeability of 102.9 md, very close to the 464.2 md and 100 md used to generate the synthetic data set.

Adjusted Pressure Change, Derivative, psi

1000

100

10

1

0.1 0.0001

Pressure change Derivative Derivative end effects 0.001

0.01 0.1 1 Adjusted Equivalent Time, hr

10

100

Fig. 12.17—Simulated test, permeability 1000 md, gauge resolution 0.01 psi, smoothing parameter 0.1.

Adjusted Pressure Change, Derivative, psi

100

10

1

0.1

0.01 0.0001

Pressure change Derivative Derivative end effects 0.001

0.01 0.1 1 Adjusted Equivalent Time, hr

10

100

Fig. 12.18—Simulated test, permeability 1000 md, gauge resolution 0.01 psi, smoothing parameter 0.28.

356 Applied Well Test Interpretation

Discussion. When simulating a test with noise, be careful to generate, filter, display, and analyze the synthetic data set as closely as possible to the way field data would be treated. The following points should be kept in mind: • If the synthetic data points are spaced too far apart, the effective smoothing parameter is determined by the spacing of the data points, not by the value input to the software. Ideally, the synthetic data should be generated with the same sampling interval as will be used for the test. • The synthetic data should be rounded to the same number of decimal places as will be reported in the field data set. • As shown in Figs. 12.16 through 12.18, derivative end effects may occur when using the Bourdet 3-point method with noisy data. If the analyst is not wary, these end effects may be erroneously interpreted as reservoir boundaries. • The noise in the pressure response predicted by Eq. 12.59 includes all sources of noise, both gauge related and environmental. For this example, the noise for the buildup period was assumed to be the same as the gauge resolution. Noise during flow periods is usually several times that during shut-in periods. If this were a design for a real test, we would have looked at more than the two cases simulated here, with particular focus on sensitivity to data that have high uncertainty, and to the effect of boundaries. Here are some additional questions that we would address for this scenario: • What happens if kv/kh is 0.01 instead of 0.1? • What happens if there is a no-flow boundary 1,500 ft away from the well? • How far away can the boundary be and still be detected? • If the reservoir is closed on all sides, how small can it be and still appear to be infinite-acting during the test as designed? 12.15 Summary The recommended workflow for well test design comprises the following steps: 1. Identify the test objectives. 2. Consider alternatives to conventional testing. 3. Collect data necessary for design. 4. Determine preliminary estimates of the reservoir properties to be obtained from the test. 5. Estimate the test time required to meet the objectives of the test. 6. Estimate the test rate and determine the flow rate sequence. 7. Estimate the magnitude of the expected pressure response. 8. Select pressure gauges. 9. Simulate the test. Nomenclature A = drainage area, ft2 Awb = cross-sectional area of wellbore, ft2 B = formation volume factor, bbl/STB Bg = gas formation volume factor, bbl/Mscf C = wellbore-storage coefficient, bbl/psi ct = total compressibility, psi–1 cwb = compressibility of wellbore fluid, psi–1 dID = production string diameter, inches G = initial gas in place, Mscf Gp = cumulative gas produced, Mscf h = net-pay thickness, ft J = productivity index, STB/D/psi Jg = gas well productivity index, Mscf/D/psi k = permeability, md kh = horizontal permeability, md kv = vertical permeability, md L = distance to boundary, ft L = smoothing parameter, dimensionless

Well Test Design Workflow 357

m N Np p p pi pwf pwhf q qg qgcrit rd re rw rwa s s′ Sw t teWBS tp Vp Vwb w rwb μ f sp σ  dp  t   dt 

= slope of semilog straight-line during infinite-acting radial flow, psi/log cycle = initial oil in place, STB = cumulative oil produced, STB = pressure, psi = average reservoir pressure, psi = initial pressure, psi = flowing bottomhole pressure, psi = flowing wellhead pressure, psi = flow rate, STB/D = gas flow rate, Mscf/D = minimum gas flow rate to lift liquids, Mscf/D = transient-drainage radius, ft = drainage radius, ft = wellbore radius, ft = apparent wellbore radius, ft = skin factor, dimensionless = total skin factor, dimensionless = water saturation, fraction = time, hour = time to end of wellbore storage, hour = Horner pseudoproducing time, hour = reservoir pore volume, ft3 = wellbore volume, bbl = width of channel, ft = density of fluid in wellbore, lbm/ft3 = viscosity, cp = porosity, fraction = standard deviation of measurement error, psi = standard deviation of logarithmic derivative, psi

Subscripts CLF = channel linear flow D = dimensionless i = initial conditions IARF = infinite-acting radial flow ND = non-Darcy PSSF = pseudosteady-state flow wb = wellbore conditions WBS = wellbore storage

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Author Index A Abbaszadeh, M., 167 Acuña, J.A., 156 Agarwal, R.G., 57, 60, 88, 101, 192, 222, 311, 326 Al-Hussainy, R., 100, 103, 289 Almanza, E.A., 348 Aminian, K., 266 Anand, V., 326, 331 Anderson, M.A., 276 Arditty, P., 248 Arps, J.J., 285 Aziz, K., 103 B Barnum, R.S., 320, 344 Bello, R.O., 326 Bezerra, M.F.C., 326 Bixel, H.C., 182 Blackburn, J., 345 Blasingame, T.A., 221, 222 Bøe, A., 103 Botto, G., 349 Bourdet, D.L., 80, 81, 88, 94, 95, 190, 192, 197, 199, 260 Bourgeois, M.J., 182 Brown, K.E., 38 Brown, L.P., 14, 40 Brownscombe, E.R., 302, 348 Burgess, K., 3 C Camacho-V., R.G., 136 Carter, R.D., 181 Chang, J., 156 Chatas, A.T., 145 Chen, C.C., 143, 163, 168, 180, 181 Chin, W.C., 146 Chu, W.-C., 163, 260, 322 Cinco, H., 262–264, 266 Cinco-Ley, H., 136, 154, 167, 247, 248, 302 Coats, K.H., 256 Coleman, S.B., 280, 341 Colpitts, G.P., 54, 221 Cook, D.C., 325 Corbett, P.W.M., 305 Crawford, G.E., 181, 182, 302 D Daungkaew, S., 216 De Ghetto, G., 349 Dietz, D.N., 66 Du, K., 153 E Earlougher, R.C. Jr., 39, 170, 232, 233, 238, 249, 331 Eernisse, E.P., 348 Ehlig-Economides, C., 135, 172 Elshahawi, H., 3, 326

F Fair, C., 341, 348 Fair, W.B. Jr., 233, 236, 237, 248, 268 Fetkovich, M.J., 38, 39, 221, 222, 265, 326, 348 Frailey, S.M., 321, 343, 349 Fraim, M.L., 103 Frantz, J.H. Jr., 244, 247 Frimann-Dahl, C., 326 G Gladfelter, R.E., 188, 221, 222 Gringarten, A.C., 79, 80, 217, 303–305 Gringarten, R.E., 303 H Hawkes, R.V., 14 Hawkins, M.F. Jr., 34, 35, 259 Hazebroek, P., 181 Hazlett, W.G., 326 Hegeman, P.S., 235–237, 268 Hong, K.C., 260 Horne, R.N., 296, 298 Horner, D.R., 57, 65, 72 J Jones, F.O., 4, 276 Jones, L.G., 265 Jones, S.C., 266, 267 K Kazemi, H., 216 Kikani, J., 285 Kuchuk, F.J., 182 Kumar, V., 321, 331, 350 Kviljo, K., 231, 239 L Lane, H.S., 96 Larsen, L., 231, 239 Lee, A.L., 277 Lee, J.W., 39, 40 Lee, W.J., 111, 221, 237, 325, 329 Levitan, M.M., 181, 182, 224–226 Levorsen, A.I., 172, 276 Locke, S., 260 Loucks, T.L., 180 M Mattar, L., 43, 126, 140, 143, 172, 240, 242, 249–252, 256, 279–281 Matthews, C.S., 60, 67, 216, 233, 244, 245 McAleese, S., 320, 321, 341 McCain, W.D. Jr., 277 McLeod, H.O. Jr., 260 Meunier, D.F., 100, 101 Miller, C.C., 57, 58, 264 Mosteller, F., 301

368 Author Index N Nardone, P.J., 319, 341 Narr, W., 302 Newman, G.H., 9 Nutakki, R., 172 O Onur, M., 96 P Palacio, J.C., 222 Papatzacos, P., 261, 262 Pinson, A.E. Jr., 216 Podio, A.L., 348 Prasad, R.K., 173 Press, W.H., 96 Proett, M.A., 146 Pursell, D.A., 279 R Raghavan, R., 103 Ramey, H.J. Jr., 73, 267, 306–308, 348 Rowlan, O.L., 348 S Saleh, A.M., 267, 268 Santo, M., 279, 280 Slider, H.C., 188 Spivey, J.P., 7, 8, 10, 13, 103, 104, 237, 268–270, 279, 289, 296, 297, 301, 304, 323, 350

Stegemeier, G.L., 233, 244, 245 Stewart, G., 134, 267 T Teng, D.T., 349, 350 Tukey, J.W., 301 Turner, R.G., 280, 341 V Van Everdingen, A.F., 34, 259 Van Poollen, H.K., 173 Vela, S., 344 Vella, M., 346 Veneruso, A.F., 249, 251, 345–347 Vienot, M.E., 38, 39, 221 W Waldman, N., 348 Ward, R.W., 346, 348 Whittle, T.M., 326 Winestock, A.G., 54, 188, 221, 265 Y Yortsos, Y.C., 156 Z Zaoral, K., 256 Zoral, K., 43

subject Index A Agarwal equivalent-time method, 60–61, 64–65, 90, 91 Agarwal multirate equivalent-time method, 192–193 B bounded reservoir model circular reservoir boundaries circular constant-pressure boundary, 170–172 closed circular reservoir, 169–170 composite reservoir model infinite radial-composite reservoir, 180–182 linear-composite reservoir, 182–184 linear boundaries constant-pressure boundary, 167–168 no-flow boundary, 165–167 multiple linear boundaries infinite channel reservoir, 172–173 intersecting no-flow boundaries, 173–175 rectangular reservoir model, 175–179 types of closed boundary, 164–165 flow capacity, decrease in, 163–164 flow capacity, increase in, 163 Bourdet superposition time function, 189–190, 193–195 Bourdet three-point method, 94–95 C capacitance gauge, 347 carbonate reservoir, 163 circular constant-pressure boundary model, 170–172, 181, 211 closed circular reservoir model, 164, 169–170 compensated quartz gauge, 348 composite reservoir model infinite radial-composite reservoir, 180–182 linear-composite reservoir, 182–184 constant-rate drawdown tests, 50–51, 84–88, 103–104, 164, 190, 210, 338 constant-rate production buildup test, 57–58 Agarwal equivalent-time method, 60–61, 64–65 bottomhole pressure, 65 Horner method, 59–60, 63–64 MDH method, 58–59, 61–63 drawdown semilog analysis decrease in rate, 53, 54 graph of pressure vs. time, 50, 51 permeability, 50, 51, 53, 57 radius of investigation, 51, 53 rate normalization, 54–57 rock and fluid properties, 51, 55 skin factor, 51, 53, 57 test data, 51, 52, 56 variable-rate drawdown test, 54, 55 convolution, 219–220, 228 core analysis, 4–5

D Darcy’s law, 8, 22, 23 deconvolution applications, 222–223 generic reservoir model, 227 nonlinear phenomena, 223–224 practical aspects of, 224–226 pressure-transient test design, 226–227 development wells, 2, 321 deviated wellbore geometric skin factor, 263–264 infinite-acting homogeneous reservoir, 262 transient flow, 264–265 Dietz method, 66, 68–70 diffusivity equation continuity equation, 24, 25 Ei-function solution, 26, 27 linear flow, 26 slightly compressible liquid, 25–26 drillstem test (DST), 320–321 E earth tides, 248–249 Ei-function solution, 78 exploration-well tests, 2, 161, 217, 320, 321, 341, 342 F Fair variable-WBS model, 233–236, 240, 241 finite-thickness skin factor model, 259–260 flow rate history changing flow rate bounded reservoirs, 210–211 buildup data, 217 diagnostic plot, 211–215 entire rate history, 217 field example, 217–219 flow regime effect, 215, 216 pseudosteady-state flow, 216–217 single rate change effect, 208–210 convolution, 219–220 deconvolution applications, 222–223 generic reservoir model, 227 nonlinear phenomena, 223–224 practical aspects of, 224–226 pressure-transient test design, 226–227 rate normalization buildup tests, 221 drawdown tests, 221 production data analysis, 221–222 reservoir boundaries effect Bourdet method, 204 buildup pressure response vs. shut-in time, 203 closed circular reservoir, 202–203 diagnostic and model identification, 204 infinite-acting reservoir, 197–199

370 Subject Index infinite channel reservoir, 201–202 single no-flow boundary, 199–201 slug tests and impulse tests, 203 spatial interpretation constant rate flow/buildup sequence, 204–205 two-rate flow period, 205–208 superposition and history matching, 220 variable rate/variable pressure problems advantages and disadvantages, 188 equivalent-time function, 190–197 field example, 197 MDH and related methods, 189 superposition time function, 189–190 flow regimes and diagnostic plot bilinear flow, 152 finite-conductivity fault, 155 finite-conductivity hydraulic fracture, 154, 155 logarithmic derivative, 153 characteristics of, 159 flow regime identification, aspects of, 156, 158 fractal reservoir behavior, 156 linear flow (see linear flow) linear stabilization, 155–157 power-law function pressure response, 124–125 flow-regime-specific diagnostic plot, 125–126 flow-regime-specific straight-line analysis plots, 126 logarithmic derivative of, 125 radial flow (see radial flow) radial stabilization, 155, 156 spherical flow (see spherical flow) spherical stabilization, 156, 157 volumetric behavior closed reservoir, PSSF, 141–143 data analysis, 144–145 logarithmic derivative, 140 radial composite reservoir, 143–144 wellbore storage, 141–143 formation compressibility, 9 fractal reservoir behavior, 156 fractional radial flow (FRF), 174 G gas formation volume factor, 10, 329 gas wells buildup tests log-log type-curve analysis, 111–113, 117–118 manual log-log analysis, 113, 118–119 rock and fluid properties, 114 semilog analysis, 110–111, 116–117 test data, 114–115 drawdown tests log-log type-curve analysis, 104–105, 108, 109 manual log-log analysis, 105, 109–110 rock and fluid properties, 105 semilog analysis, 103–104, 106–108 test data, 106 gas flow equation, 99 extensions of pseudopressure, 103 pseudopressure and adjusted pressure, 100 pseudotime and adjusted time, 101–103 log-log type curve analysis buildup tests, 111–113, 117–118 drawdown tests, 104–105, 108, 109 geology and geophysics, 273–274 Gladfelter deconvolution, 221 Gringarten-Bourdet type curves buildup test analysis, 88–91 constant-rate drawdown tests, 84–88

damaged well, 79 damage/stimulation qualitative evaluation high positive skin factor, 82, 83 negative skin factor, 83–84 zero skin factor, 82 dimensionless WBS coefficient, 79 logarithmic derivatives, 80 PD vs. tD (dpD/dtD) plot, 80, 81 pwD vs. tD/CD plot, 79, 80 WBS-dominated period, 80 H Hawkins damage model, 260 Hegeman, Hallford, Joseph variable-WBS model, 235, 236 hemiradial flow (HRF) horizontal well, 130, 131 partial-penetration completion, 148 spherical permeability, 151 vertical well, 127–128 Horner graph, 59, 63, 64, 283 Horner method average drainage-area pressure, 65–66 buildup test, 59–60, 63–64 Horner pseudoproducing time, 72–73, 216–218, 329 Horner time ratio (HTR), 59–60, 63–66, 69, 70 HRF. See hemiradial flow (HRF) I infinite-acting radial flow (IARF) drawdown semilog analysis, 50–52 Ei-function solution, 26, 27 graph of pressure vs. time forms, 37 radius of investigation, 43 reservoir pressure profiles, 36 vertical well, diagnostic plot for, 127 with WBS and skin, 78–79 infinite channel reservoir model, 172–173, 201–202 infinite radial-composite reservoir model, 144, 180–183 intersecting no-flow boundaries model, 173–175 L least-squares fitting method, 95–96 linear-composite reservoir model, 182–184 linear flow, 134, 140 channel reservoir in one direction, 132, 135 in two directions, 132, 135 high-conductivity fault, 133, 136 high-conductivity hydraulic fracture, 133, 135–136 horizontal well, early linear flow, 134, 136–137 logarithmic derivative, 131 log-log diagnostic plot, 138, 139 rock and fluid property, 137 square-root of time plot, 138 test data, 137 linear stabilization, 155–157 log-log type curve analysis Bourdet three-point method, 94–95 dimensionless variables Ei-function solution, 78 IARF, 78–79 gas wells buildup tests, 111–113, 117–118 drawdown tests, 104–105, 108, 109 Gringarten-Bourdet type curves (see Gringarten-Bourdet type curves) least-squares fitting method, 95–96 manual parameter estimation, 90–93

Subject Index 371 quick two-point method, 93–94 smoothing, 96 spline curve fitting, 96 M manual log-log analysis, 91–93, 105–110, 113–119, 288 Matthews-Brons-Hazebroek (MBH) method, 49, 67–68, 70–72, 189, 216 mechanical gauge, 346–347 Miller-Dyes-Hutchinson (MDH) method buildup test, 58–59, 61–63 permeability, 68 variable rate/variable pressure problems, 189 multiple linear boundaries infinite channel reservoir, 172–173 intersecting no-flow boundaries, 173–175 rectangular reservoir model, 175–179 multiwell test, 2 N near-wellbore phenomena deviated wellbore geometric skin factor, 263–264 infinite-acting homogeneous reservoir, 262 transient flow, 264–265 finite-thickness skin factor, 259–260 partial penetration/limited entry completion geometric skin, 261–262 transient pressure response, 262 perforated completion, 260–261 rate-dependent skin factor steady-state radial flow, 265–267 steady-state spherical flow, 267–268 and wellbore storage (see wellbore storage) transient tests, 270 no-flow boundary model, 165–167 non-Darcy skin, 265–270, 281, 294, 295, 304, 322, 333–334, 342, 352 O oil formation volume factor, 10 oil reservoir bubblepoint pressure, 277 earth-tides, 248 linear constant-pressure boundary, 167 radial-composite reservoir behavior, 180, 181 radius of investigation, 32 total compressibility, 14 P PBU. See pressure buildup (PBU) permanent gauges monitoring, 3 petrophysics, 275–276 piezoresistive silicon strain gauge, 347 PLF. See pseudolinear flow (PLF) porous medium diffusivity equation continuity equation, 24, 25 linear flow, 26 slightly compressible liquid, 25–26 infinite-acting radial flow, 26, 27 principle of superposition, 26, 28 general rate history, pressure response for, 30 in space, 28 in time, 29 pseudosteady-state flow drainage area shape factor and shape factor skin, 38–39

graph of pressure vs. time forms, 37 productivity index and completion efficiency, 39–40 reservoir limits testing, 41–43 reservoir pressure profiles, 36 radius of investigation buildup test, 31 derivation of, 32–34 drawdown test, 30–31 flow rate, 31 formation permeability, 32 oil reservoir, 32 steady-state flow Darcy’s law, 22 linear flow, 22–23 radial flow, 23–25 wellbore damage and stimulation skin factor, 34–36 wellbore pressure equation, 36 wellbore storage buildup test, 45 drawdown test, 44 fluid-filled wellbore, 44–46 rising/falling liquid level, 46–47 pressure buildup (PBU), 189 pressure oscillations diurnal temperature changes, 249 earth tides, 248–249 inertial effects, 248 pressure-transient test design, 226–227 PRF. See pseudoradial flow (PRF) producing wells, 321–322 production engineering, 2–3 productivity index, 39–40 pseudolinear flow (PLF), 175 pseudoradial flow (PRF) horizontal well, 131 hydraulically fractured well, 129, 130 pseudoskin, 260 pseudosteady-state flow (PSSF), 164 Cartesian scale, 37 drainage area shape factor and shape factor skin, 38–39 graph of pressure vs. time forms, 37 productivity index and completion efficiency, 39–40 radius of investigation, 43–44 reservoir limits testing, 41–43 reservoir pressure profiles, 36 volumetric behavior closed reservoir, 141–143 log-log diagnostic plot, 144 primary pressure derivative plot, 145 Q quartz gauge, 345, 346, 348, 354 quick two-point method, 93–94 R radial flow, 140 horizontal well, diagnostic plot for early HRF, 130, 131 early radial flow, 130 late PRF, 131 logarithmic derivative, 126 log-log diagnostic plot, 138, 139 radial composite reservoir, 129 rock and fluid property, 137 square-root of time plot, 138 test data, 137

372 Subject Index vertical well, diagnostic plot for fractional radial flow, 128 HRF, single no-flow boundary, 127–128 IARF, 127 sealing faults, 128 radial flow semilog analysis average reservoir pressure Dietz method, 66, 68–70 Horner method, 65–66 MBH method, 67–68, 70–72 Ramey-Cobb method, 66, 69, 70 constant-rate production (see constant-rate production) flow rate variations, shut-in times effect of, 72 Horner pseudoproducing time approximation, 72, 73 wellbore pressure, 73–74 radial stabilization, 155, 156 radius of investigation buildup test, 31 derivation of, 32–34 drawdown test, 30–31 flow rate, 31 formation permeability, 32 oil reservoir, 32 Ramey-Cobb method, 66, 69–70 rate-dependent skin factor steady-state radial flow, 265–267 steady-state spherical flow, 267–268 and wellbore storage (see wellbore storage) rate normalization buildup tests, 221 drawdown analysis, smoothly varying rate, 54–57 drawdown tests, 221 production data analysis, 221–222 rectangular reservoir model constant-pressure boundary, 179 drawdown pressure response, 176 fault blocks, wells, 175 IARF, 178 pseudosteady-state flow, 178 reservoir geometry and well placement, 178 U-shaped reservoir, 177 reservoir boundaries effect Bourdet method, 204 buildup pressure response vs. shut-in time, 203 closed circular reservoir, 202–203 diagnostic and model identification, 204 infinite-acting reservoir, 197–199 infinite channel reservoir, 201–202 single no-flow boundary, 199–201 slug tests and impulse tests, 203 reservoir engineering, 2 reservoir limits test analysis, 41–43, 143 S sapphire strain gauge, 347 semilog analysis gas wells buildup tests, 110–111, 116–117 drawdown tests, 103–104, 106–108 radial flow semilog analysis (see radial flow semilog analysis) silicon-on-sapphire gauge, 347–348 simulate/history-match pressure response automatic history matching, 290–291 manual history matching, 289–295 match evaluation, 291 simulation, 289 single-well test, 2

spherical flow, 145 logarithmic derivative, 146 and radial flow inverse square-root of time plot, 150 log-log diagnostic plot, 149, 151, 152 radius of investigation, 151 rock and fluid property, 149 test data, 149 vertical well limited-entry completion, 147 partial penetration, 147–148 spherical stabilization, 156, 157 Spivey variable-WBS models, 237 spline curve fitting, 96 steady-state flow Darcy’s law, 22 linear flow, 22–23 radial flow, 23 Cartesian graph, 24 semilog graph, 24 stimulated well, 322 straight-line methods, 6 superposition principles, 26, 28 general rate history, pressure response for, 30 in space, 28 in time, 29 T three-zone radial-composite model, 181 “tight gas” reservoir, 9 V van Everdingen model, 259 W water formation volume factor, 10, 11 WBS. See wellbore storage (WBS) wedge reservoir model, 173 wellbore phenomena cleanup process Larsen and Kviljo cleanup model, 239 thermal effects, 239–241 time-dependent skin factor, 239 data processing errors and artifacts final flowing bottomhole pressure, 250, 251, 253 gauge precision artifacts, 253, 256 incorrect producing time, 253 shut-in time, 251, 254 time and flowing pressure, buildup test, 252–255 gas-liquid interface, movement of, 240, 241 downward movement, 242 prevention and remediation, 244 probable movement, 243 smoothing parameter, 243, 244 upward movement, 242 gauge problems gauge plugging, 249, 250 gauge sampling artifacts, 249–252 thermal sensitivity, 249, 251 unidentified pressure phenomena, 250, 252 pressure oscillations diurnal temperature changes, 249 earth tides, 248–249 inertial effects, 248 from reservoir phenomena duration of transition, 257 pressure derivative, change in, 253, 254 pressure disturbance, timing and duration of, 256

Subject Index 373 primary pressure derivative, 256–257 slope of pressure derivative, change in, 253, 254 wellbore pressure, change in, 253, 254 test operations, activities bridge plugs, 244, 246, 247 choke change and opened valve, 244–246 leak, 244, 245 pressure gauge, 246, 247 variable wellbore storage [see wellbore storage (WBS)] wellbore storage (WBS), 221 buildup test, 45, 268–269 drawdown test, 44, 269–270 fluid-filled wellbore, 44–46 rising/falling liquid level, 46–47 variable wellbore storage Fair variable-WBS/phase segregation model, 233–236 Hegeman, Hallford, Joseph variable-WBS model, 235, 236 phase segregation, 233 rising gas-liquid interface, 237–239 Spivey variable-WBS models, 237 thermal effects, 236, 237, 239–241 WBS coefficiency, changes in, 232, 233 volumetric behavior, 141–143 well test design annual pressure survey, 323 data collection, 327 development wells, 321 DST, 320–321 extended well testing, 321 flow rate and pressure limitations, 341 flow rate sequence determination, 341–343 gauges accuracy and resolution, 345, 349 acoustic fluid-level measurement, 348 calibration range, 345 capacitance gauge, 347 compensated quartz gauge, 348 conventional strain gauge, 347 helium-purged capillary tube, 348 location, 349 mechanical gauge, 346–347 operating range, 347 piezoresistive silicon strain gauge, 347 pressure measurement options, 348 quartz gauge, 348 reliability, 349 repeatability and hysteresis, 345, 346 sampling, 349 sapphire strain gauge, 347 silicon-on-sapphire gauge, 347–348 stabilization time and drift, 345–347, 349 wellhead pressure measurement, 348 illustration, 350–356 log-derived permeability, 326 low productivity/injectivity, 322 maximum sustainable rate determination, 341 permanent downhole gauges, 326–327 permeability estimates infinite-acting radial flow, 329 one-point method, 329–331 productivity index, 328 pseudosteady-state flow, 328 pressure response estimation, 344–345 producing wells, 321–322 rate-dependent skin factor, 322 rate-transient analysis, 326 recommended workflows, 319, 320

reservoir properties distances to boundaries, 328 reservoir pressure, 328 skin factor, 327 wellbore-storage coefficient, 327–328 stimulated well, 322 test duration estimation graphical interpretation, 338–340 middle time regions, beginning, 331–335 minimum economic fluid in place, 337–338 minimum economic oil in place, 337 minimum economic productivity index, 336–337 specific boundaries/flow regimes, 335–336 test objectives distances to boundaries, 324 flowing and static pressure gradient surveys, 325 fluid sampling, 323 government regulatory agencies, 325 hydrocarbons in place, 324–325 initial/average drainage area pressure, 325 in-situ permeability, 323 reservoir connectivity evaluation, 325 skin factor, 323–324 test simulation, 349–350 WFT, 326 well test interpretation applications and objectives exploration and appraisal, 2 production engineering, 2–3 reservoir engineering, 2 buildup tests vs. drawdown tests, 2 confidence intervals approximate models, 297–300 assumptions, 295–296 calculation, 301 common misconceptions, 296–297 validate assumptions, 300–301 conventional well testing core analysis, 4–5 log-derived permeability estimates, 4 permanent monitoring, 3 rate-transient analysis, 3–4 wireline formation testing, 3 deconvolve data (see deconvolution) deliverability tests vs. transient tests, 2 estimate model parameters log-log methods, 288 straight-line methods, 287–288 exploration-well tests vs. development-well tests, 2 field example alternative interpretation, 309–316 published interpretation, 306–307 review, 307–309 flow-regime identification log-log diagnostic plot, 284 specific diagnostic plot, 284 specific straight-line plot, 285 fluid properties data, 277 fluid compressibility, 11–13 formation volume factor, 10, 11 viscosity, 13–14 forward and inverse problems, 5–6 gauge data, 278–279 graph scales Cartesian pressure scale, 18 field data set, 17 logarithmic derivative curve, 18

374 Subject Index log-log scale, 16 properties of semilog and log-log scales, 15–16 interpret model parameters average pressure to datum conversion, 302–303 composite models, 301–302 dual porosity models, 302 multiwell tests vs. single-well tests, 2 porous media, fluid flow in (see porous medium) rate data, 278 reservoir data, 276–277 reservoir model engineering data, 286–287 external data, 286–287 ideal reservoir model, 285–286 review and quality control, data depletion/interference, offset wells, 282–284 gauge data, 280–281 geology and petrophysics, 280 non-reservoir phenomena, 281–282 plotting functions calculation, 281 pressure change and derivatives calculation, 281 pressure data, 281 rate-time data, 280 rock and fluid properties, 279–280 well history and offset well data, 280 rock properties net sand thickness, 10

permeability, 8, 9 pore-volume compressibility, 9–10 porosity, 7 saturation, 7–8 simulate/history-match pressure response automatic history matching, 290–291 manual history matching, 289–295 match evaluation, 291 simulation, 289 simulation/history-matching approach, 6 straight-line methods, 6 test data, 278 total compressibility, 14 type-curve methods, 6 validation process evaluation, 305–306 flow periods, consistency, 304 flow rate/pressure history vs. measured pressures, 304 parameter estimates vs. external data, 304–305 previous tests, consistency, 304 radius of investigation equation, 304 reality check, 303–304 wellbore radius, 14 well data, 278 Winestock and Colpitts method, 221 wireline formation testing, 3

Applied Well Test Interpretation - John Lee

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