General Principles of Bottomhole Pressure Tests

GENERAL PRINCIPLES OF BOTTOM-HOLE PRESSURE TESTS

BHP BALANCE ACHIEVED OBJECTIVE Test Analyst Proposal Writer

Field Operators

BHP TEST ACHIEVED OBJECTIVE Prof. Mike Obi Onyekonwu Petroleum Engineering Department University of Port Harcourt, Nigeria.

1:23:09 PM

-1-

PREFACE Bottom-hole pressure test is one of the most economical methods of obtaining information required for reservoir management. The test involves measuring the bottom-hole pressure at the sandface under specified rate conditions. The bottomhole pressure contractors (wireline operators) are responsible for running the test while the reservoir engineers analyze the tests. For the past twelve years, we have been analyzing pressure tests for companies in Nigeria. The analysis involves using sophisticated software and models to obtain reservoir parameters that will not only match obtained pressure, but that are in agreement with known information about the reservoir. From our experience, we observed that more than fifty percent of the errors in BHP analysis could be due to ignorance on the part of the field operators responsible for measurements. Therefore, there is need for everyone involved with BHP test, to understand the purpose of the test and factors that can affect the test. This is what we call the total concept approach to BHP tests and analysis. The total concept approach requires that all parties involved in BHP operations - BHP proposal writer, BHP contractor, production staff, reservoir engineers and users of BHP test data - need to understand the principles of BHP tests. This book provides required general information for all the parties involved in BHP test and analysis. Chapter 1 of the book is on general introduction and definitions while Chapter 2 is on field practices. The field practices cover gauges, running procedure and procedure for data quality check. Chapter 3 is on basis for BHP analysis. Subjects such as units, flow phases and flow regimes are discussed. Chapter 4 is on analysis models while Chapter 5 dwells on skin factor and concepts. Chapter 6 is on analysis methods. The conventional and type-curve methods are covered. Chapter 7 is on sensitivity on factors affecting BHP tests. Factors such as leaks, rate variation and gauge problems are considered. Chapter 8 is on field cases. Both good and bad tests are presented. The bad tests could be a result of man-made problems and problems resulting from malfunction of equipment. For the bad ones, solutions are proffered on how to avoid future mistakes. Chapter 9, on the theory of welltesting, is for the academic minded group that is not only interested in how, but why. The fundamental equation, diffusivity equation, is derived and solutions presented for different conditions. Concepts like superposition are also discussed. Finally, there is analogy between liquid and gas flow equations. Chapter 10 is on bottom-hole pressure analysis in horizontal wells. Several equations are presented and procedures for using the equations outlined. Finally some field cases on horizontal wells are presented. Materials for this book were taken from manuals that we have used successfully in teaching three different short courses – “Practical Aspects of Bottom-Hole Pressure Tests, General Principles of Bottom-Hole Pressure Tests and Theory of Welltesting.” These courses were organized for participants with different backgrounds. We are therefore certain that anyone who needs some knowledge on bottom-hole pressure tests will find this book useful. In this era of multidisciplinary approach to problems, this book will give reservoir and non-reservoir engineers a good working knowledge of bottom-hole pressure tests. Mike Onyekonwu 2

ACKNOWLEDGEMENT For me, writing a book is simply a way of expressing my understanding of a subject. We therefore owe our teachers, students and collegues a lot in contributing to our knowledge on the subject. We are grateful to Late Professor H. J. Ramey (Jr) of Stanford University, California – the father of Well Testing – for teaching us a difficult subject in a simple way. We are also grateful to Steve Rice and Jim Willetts of Shell Petroleum Development Company who challenged us and helped us acquire much needed practical experience on this subject. Tom McAllister who took over from Steve and Jim has continued to encourage us. We appreciate his contributions to our continued growth. We are also particularly grateful to petroleum and production staff of Shell Petroleum Development Company, Warri, for their support and use of our services.

I also thank my good friend, Dr G. A. Okpobiri, who was my co-consultant when we taught the first Well Testing course. I appreciate the immense contributions made by Laser Engineering Consultants staff and particularly Mr. Obi Ekeh and Mr. Cornel Udoh for being part of the analysis team. Mrs. Margaret Anele and Mr. Sam Jumbo of Laser Engineering did a good job at typesetting the book. I am equally indebted to my students both in the oil industry and University of Port Harcourt for allowing us use them as the guinea pigs. Their contributions are indeed invaluable. I also thank Professor G. K. Falade who taught us the first course on “Well Testing” at University of Ibadan. I am equally indebted to Professor Chi Ikoku for creating a nice working environment at the University of Port Harcourt and therefore made it possible for me to continue practicing engineering. Finally, I thank my dear wife, Tina, for always standing by me. May God reward everyone. This is actually our book.

Mike Onyekonwu

3

ABOUT THE AUTHOR Include photgraph! Dr Mike Onyekonwu has a B.Sc. (First Class Honours) in Petroleum Engineering from University of Ibadan, Nigeria.

He also has an MS and Ph.D. degrees in

Petroleum Engineering from Stanford University.

Dr Onyekonwu is a Senior Lecturer and former Head of Petroleum Engineering Department, University of Port Harcourt. He is a member of University Senate. Dr Onyekonwu is the founder and Managing Consultant of Laser Engineering Consultants, Nigeria.

Dr Onyekonwu worked for Shell Petroleum Development

Company Nigeria and Stanford University Petroleum Research Institute, California.

Dr Onyekonwu is a registered engineer and a member of different professional bodies. He consults for Shell, Mobil, Elf, NNPC, Agip and other oil operating and service companies. His area of specialization includes welltest analysis, reservoir simulation, recovery methods and computer applications.

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TABLE OF CONTENTS PAGE PREFACE ............................................................................................................................ COURSE ADVERTISEMENT ............................................................................................

ii iii

1

1

INTRODUCTION .......................................................................................... 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Objectives of BHP Survey .. .............................................................................. Uses of BHP Derived Information..................................................................... Common Types of BHP Tests ........................................................................... Ideal Conditions and Information Derived from Test........................................ Importance of Sticking to Ideal Condition for Test ........................................... Uses of Information Derived from BHP Tests................................................... Well Test Equipment ......................................................................................... Electronic Gauges and Problems ...................................................................... Flowing Gradient/Buildup/Static Gradient(Fg/Bu/Sg) Survey.......................... Flowing Gradient/Buildup/Static Gradient Survey Proposal............................. Useful Hints on Proper Testing of Wells ........................................................... Practical Hints.................................................................................................... Gauge Quality Check Procedure........................................................................ Roles of Field Staff In BHP Survey................................................................... 1.14.1 Roles of Production Staff...................................................................... 1.14.2 Roles of BHP Contractor Staff .............................................................

1 1 2 3 4 6 7 11 11 19 23 23 27 28 32 33

2.

BASIS OF ANALYZING BOTTOM HOLE PRESSURE TESTS ............................ 2.1 Flow Phases ...................................................................................................... 2.2 Features of Different Phases .............................................................................

34 34 35

3.

ANALYSIS OF BOTTOM HOLE PRESSURE TESTS ...........................................

46

4.

EFFECT OF CERTAIN FACTORS ON ANALYSIS OF SIMULATED DATA ..... 64 4.1 Analysis of Ideal BHP Data .............................................................................. 64 4.2 Effect of Gauge Accuracy and Datum Correction ............................................ 73 4.3 Effect of Noise .................................................................................................. 78 4.4 Effect of Gauge Sensitivity ............................................................................... 98 4.5 Effect of Rate Variation .................................................................................... 98 4.6 Effects of Leaks ................................................................................................ 112 4.7 Effect of Interference/Leak ............................................................................... 112

5.

FIELD CASES ........................................................................................................... 5.1 Good Test .......................................................................................................... 5.2 Effect of Gauge Movement ............................................................................... 5.3 Effect of Gauges Off-Depth .............................................................................. 5.4 Effect of Reporting Wrong Rates ..................................................................... 5.5 Effect of Ineffective Shut-in/Well not flowing before Shut-in .......................... 5.6 Effect of Leak ................................................................................................... 5.7 Effect of Gauge Oscillations/Sensitivity Problems ........................................... 5.8 Effect of Gas Phase Segregation ....................................................................... 5.9 Effect of Liquid Interface Movement ............................................................... 5.10 Effect of Gaslift ................................................................................................ 5.11 Effect of Short Buildup or Flow Period ............................................................

6.

CLASS DISCUSSION ............................................................................................... 154

7. 8.

FLOWING AND STATIC GRADIENT SURVEYS ................................................. 165 SKIN FACTOR 8.1 What is Skin Factor............................................................................................ 172

117 117 117 125 125 131 132 135 140 144 144 144

5

8.2 8.3 8.4 8.5 8.6 8.7

Causes of Skin ................................................................................................... Classification of Pseudoskins ............................................................................ Calculation of Pseudoskins ................................................................................ Relationship Between Total Skin and Pseudoskins ........................................... Pressure Change Due to Skin............................................................................. Relationship Between Skin and WIQI ............................................................... Exercises ............................................................................................................

174 174 176 180 181 181 182

APPENDIX A: TYPICAL PROPOSAL .............................................................................. 184 APPENDIX B: PROPOSAL WITH WRONG INSTRUCTION ......................................... 192 APPENDIX C: MISCELLANEOUS MATERIAL...............................................................

6

1. INTRODUCTION Oil well tests are made for numerous reasons and the type of test required depends on the objective of the test. Common well tests include: (a) Potential test (b) Gas-oil ratio test (c) Productivity test (d) Bottom-hole pressure test Potential test involves measurement of the amount of oil and gas a well produces during a given period (normally 24 hours or less) under certain conditions fixed by regulatory bodies. The information obtained from these tests is used in assigning a producing allowable of the well. The gas-oil ratio test is made to determine the volume of gas produced per barrel of oil so as to ascertain whether or not a well is producing gas in excess of permissible limit. Bottom-hole pressure test involves measurement of sandface pressure and flowrate variation with time. Such tests are quite economical to run and they yield valuable information about the reservoir characteristics and well characteristics. Hence, bottom-hole pressure tests are usually referred (Earlougher, 1977; 1982) to as welltests. Productivity tests are made on oil wells and include both the potential test and the bottom-hole pressure (BHP) test. The purpose of this test is to determine the effects of different flow rates on the pressure within the producing zone of the well and thereby establish producing characteristics of the producing formation. In this manner, the maximum potential rate of flow can be calculated without risking possible damage to the well which might occur if the well were produced at its maximum possible flow rate. In this book, the term welltest will be used for bottom-hole pressure test unless otherwise stated. In this chapter, the purpose of well testing, types of well tests and well test equipment are discussed. In addition, other practical aspects of BHP tests such as test procedure and equipment problems are discussed.

1.1 OBJECTIVES OF BHP SURVEY Bottom-hole pressure tests are conducted to obtain data that can be used for the following purposes: •

•

Determine Well Parameters Skin Productivity Index Wellbore storage constant Fluid distribution in wellbore Flowing pressures in wellbore Static gradients Determine Reservoir Parameters Average pressure in the drainage area Permeability Distance to boundaries 7

-

Vertical/Horizontal permeability Gas/oil contacts

•

Determine Dynamic Influence of other Wells/Aquifer

•

Assess Changes Since Previous Survey Changes in datum pressure Changes to damage skin Changes in drainage area (from a drawdown test) Confirm boundaries

1.2

USES OF BHP DERIVED INFORMATION Results obtained from BHP tests are used for the following purposes:

* * * * *

Reservoir Surveillance Determination of Stimulation Candidates Gaslift Optimisation Input for Reservoir Simulation Material Balance Calculation

Examples of benefits from BHP test compiled by a client are given in Table 1.1. The benefits were realised by using results from BHP tests for good well and reservoir surveillance. 1.3 COMMON TYPES OF BOTTOM-HOLE PRESSURE (BHP) TESTS Common types of bottom-hole pressure tests include the following: (a) Drawdown test (b) Injectivity test (c) Buildup test (d) Falloff test (e) Interference/pulse tests (f) Others The definitions of the tests and the rate and pressure profiles during the test are as follows: Table 1.1: Benefits from BHP Surveys ACTIVITIES Well Surveillance Stimulation (abort 5 jobs, contribute to finding 5 more) Gaslift Optimization (10% improvement of target at $1/bbl) Reservoir Surveillance Sand F4.0/F4.1X Production (3 Mbopd) Sand-X Block (new well cancelled) Dump Creek (10% of the 6 fewer wells required) Well -11 (sidetrack raise trajectory) Sand D5.0X Development (horizontal well changed to recompletion) (10% of 8 well campaign) Total

SAVINGS ($ million)

0.8 1.0

1.0 3.0 3.0 1.0 3.0 4.0 16.8 8

1. Drawdown Test: Involves measuring the variation of sandface pressure with time while the well is flowing. For a drawdown test, the well must have been shut in to attain average pressure before production commences for the test. The rate and pressure profiles during drawdown test are in Fig 1.1.

Pwf q

0

time

0

time

Fig. 1.1: Rate and Pressure Profiles During Drawdown Test

2.

Injectivity Test: This is the counterpart of a drawdown test and involves measuring the variation of sandface pressure with time while fluid is being injected into the well. The rate and pressure profiles during drawdown test are in Fig 1.2. q 0

Pwf time Injection (-ve q) time

Fig. 1.2: Rate and Pressure Profiles During Injectivity Test 3. Buildup Test: Involves measuring the variation of sandface pressure with time while well is shut-in. The well must have flowed before shut-in. Figure 1.3 shows the rate and pressure profiles during the flow and buildup periods. Buildup tests are more common and will be the main subject of our discussion. Pw q Buildup

Buildup

Drawdown Drawdown

0 0

time

0

time Shut-in Time

Fig. 1.3: Rate and Pressure Profiles During Drawdown and Buildup 4. Falloff Test: This is the counterpart of buildup test and it involves measuring the variation of sandface pressure with time while well is shut-in. In this case, some fluid must have been injected into the well before shutting. Figure 1.4 shows the rate and pressure profiles during the injection and falloff periods. 9

q

Pw

0

time Injection

Injectivity

Falloff

Shut-in

time Fig. 1.4: Rate and Pressure Profiles During Injectivity and Falloff Test 5. Interference Test: Unlike the first four tests (drawdown, injectivity, buildup, falloff) which are tests involving only one well (single well tests), the interference test involves the use of more than one well (multiple well test). During interference tests, pressure changes due to production or injection or shut-in at an active well is monitored at an observation well. The active well and the observation well are shown in Fig 1.5. Only one active well is required, but there could be more than one observation well. Interference tests are primarily used to establish sand continuity between the active and observation wells. In situation where more than one observation well is used, interference test can be used to find (Ramey and ) maximum and minimum permeability and their directions.

q>0 q=0 q> k). The fracture is considered to have an infinite permeability and therefore there is no pressure drop during flow in the fracture. The pressure profile in this case is shown in Figure 4.4.

This reservoir goes through a linear flow, followed by a pseudo-radial flow before the boundary effect. However, no pseudo-radial flow will appear if xf/xe = 1. This is shown in Figure 4.5.

4.2.2 Uniform Flux Fracture

In this case, fluid enters the fracture at uniform flowrate per unit area of fracture face so that there is a pressure drop in the fracture. Features of uniform flux fracture are similar to the infinite conductivity case shown in Figures 4.4 and 4.5. 4.2.3 Finite conductivity fracture

In this case, fluid flows within the fracture and there is a pressure drop along the length of the fracture. Features of this case are shown in Figure 4.6.

74

Fig. 4.4: Pressure and Derivative for Well on A Single Vertical Infinite Conductivity Fracture.

Fig. 4.5: Effect of xf/xe on Well on a Single Vertical Infinite Fracture

75

Fig 4.6: Log – Log Graph of Data from Well with Finite Conductivity Fracture

4.3

DOUBLE POROSITY RESERVOIR

The double porosity reservoir is simply a fissured (naturally fractured) reservoir and is shown in Figure 4.7.

Fig. 4.7: Fissured Reservoir

76

The main feature of this reservoir is that the pore space is divided into two distinct media: the matrix, with high storativity and low permeability, and the fissures with high permeability and low storativity. Flow between the fissure and matrix can occur under pseudo steady state on transient state. However, the former is more common. In addition to fissured reservoirs, double-porosity models can also represent layered reservoirs in which one layer has a permeability that is much higher than the other. This is shown in Fig 4.8. Fluid essentially reaches the wellbore through the layer with higher permeability.

k1 k2 >> k1

Fig. 4.8: Layered Reservoir

Layered reservoirs are also modeled with double permeability model (Bourdet, 1985), but the features of double permeability model are similar to the double porosity model. Warren and Root (1963), de Swaan (1976), Bourdet and Gringarten (1980) and Gringarten (1984) published double porosity solutions. Informations that may be deduced with the double porosity model are as follows: k

=

permeability

s

=

skin factor

Cs

=

wellbore storage constant

ω

= ratio of the storativity in the most permeable medium to that of the total reservoir

77

=

λ

( ∅ VCt ) fissure (f) ( ∅ VCt ) fissure (f) + ( ∅ VCt ) matrix (m)

= Inter porosity flow coefficient

α rw2 k m = k Some parameters in the equations are defined fas follows: km

=

permeability of matrix or least permeable layer

kf

=

permeability of fracture or most permeable layer

V = bulk volume

ratio of the total volume of one medium (matrix or fissure) to the

α

characteristics of the geometry of the interporosity flow.

=

Figure 4.9 shows features of a double porosity reservoir with pseudo-steady state interporosity flow. Features in Fig 4.9 are explained as follows: 1.

At early time, only the fissures are detected. A homogeneous response corresponding to the fissure storativity and permeability may be observed.

78

Fig. 4.9 Features of Double Porosity Reservoir

2.

When the interporosity flow starts, a transition period develops. This is seen as an inflection in the pressure response and a “valley” in the derivative.

3.

At the end of the transition, the reservoir acts as a homogeneous medium, with the total storativity and fissure permeability. A few things to note while analyzing test with double porosity models are as follows:

(a)

Wellbore storage may mask all indications of heterogeneity.

(b)

The depth of the transition valley is a function of ω. When ω decreases (low fissure storativity) the valley is more pronounced and the transition starts early.

(c) The time the transition ends is independent on ω. (d) The time when transition occurs is a function of λ. When λ increases (higher km/kf ), the transition occurs earlier. The time the transition ends is proportional to 1/λ only. (e) The value of ω can be less than or equal to one. The double porosity system degenerate to single porosity system when ω = 1. (f) The values of λ are usually small (≈ 10-3 to 10-10). If λ is larger than 10-3, the level of heterogeneity is insufficient for dual porosity effects to be important. The system then acts as a single porosity reservoir. (g)

If inter porosity flow is transient, the “valley” is less evident.

4.3.1 Practical Hints on Double Porosity Models The double-porosity model can either be used for a fissured reservoir on a multi layer reservoir with high permeability contrast between the layers. As a result, it is not possible, from the shape of the pressure versus time curve alone, to distinguish between the two possibilities. The following practical hints will be of help in distinguishing the systems. Common features are also highlighted. (a)

If well is damaged, an increase in C, after an acid job and the resulting high value of wellbore storage constant are characteristic of fissured formation. This is because when the well is damaged, most of the fissures intersecting the wellbore are plugged and do not contribute to wellbore volume. On the other hand, there is no significant change in the wellbore storage constant following an acid job in a multi layer reservoir.

(b)

Double-porosity reservoirs have skin value for non-damaged well that is lower than zero. In reality, double-porosity reservoirs exhibit pseudo skins, as created by hydraulic fractures. A skin of –3 is normal for non-damaged wells in formations with double-porosity behaviour. Acidized wells may have skins as low as –7, whereas a 79

zero skin usually indicates a damaged well. A very high wellbore-storage constant and a very negative skin should suggest a fissured reservoir, even if the well exhibits homogeneous behaviour. (c)

4.4

The parameters ω and λ may change with time for the same well depending on the characteristics of the reservoir fluid. The reason is that ω and λ both depend on fluid properties, not just on rock characteristics. The parameters ω and will definitely change as pressure falls below bubble point. BOUNDARY MODELS

It is impossible to cover all possible boundary models. Appendix C taken from Middle East Well Evaluation Review shows features of different analysis models including different boundary models. Although it is not absolutely correct to use models derived during drawdown for buildup analysis, but for practical purposes, it is accepted. Pressure and pressure derivative obtained during buildup show similar features seen in drawdown tests. Figure 4.10 published by Economides (1988) for different models shows this. 4.5

MODEL SELECTION

Two key steps in the process for estimating reservoir properties from pressure/production data are as follows: 1.

Selection of an appropriate reservoir model.

2.

Estimation of parameters with the chosen model.

The selection of an appropriate model requires selecting appropriate set of material and energy balances for the physical processes involved, as well as the fluid properties and reservoir geometry. The problem in choosing the most appropriate reservoir model is that several different models may apparently satisfy the available information about the reservoir. That is, the models may be consistent with available geologic and petrophysical information and seem to provide more or less equivalent matches of the measured pressure/production data.

80

Fig. 4.10: Different Models used for Buildup Analysis Watson et al (1988) suggested a method of model selection, which is summarized as follows: 1.

Select candidate models that are consistent with all available information about the reservoir. A pool of candidates may be formed as a hierarchy of models as shown in Figure 4.11. The number of independent parameters to be estimated from the models is also shown.

2.

Using a parameter estimation (automatic history matching) method, estimate the independent parameters.

3.

Using the calculated independent parameters, calculate the expected pressure and production data.

81

4.

Compare the calculated data with actually obtained data. The correct model is the one that minimizes the difference between calculated and actual data in the least square sense. Weighting factors can be included.

Although the model can be chosen at the end of Procedure 4, there is still need to find out whether a simpler model can be used. This is because pressure and production data are not known with certainty. Also with a simpler model, fewer numbers of unknowns are calculated. A model that has too many parameters for the given set of data will often result in parameter estimates that have large errors associated with them. The reason for this is that the estimation process using models with too many parameters tends to be poorly conditioned in that many different set of parameter values tend to give essentially equivalent fits to the data. Consequently small measurement errors may result in large errors in parameter estimates. In deciding whether a simpler model can be used, Watson et al (1988) suggest using an F-test to find out whether the estimated parameters are very different from known values of such parameters for the simpler model. For example, if at end of Procedure 4, a double-porosity model is chosen, calculated values of ω and λ are compared with 1 and 0 which correspond to a simpler single porosity model. A hypothesis test is done for chosen level of significance.

Single Porosity, Infinite Acting

Single Porosity, Infinite Acting, With Skin Dual Porosity Infinite Acting

Dual Porosity Infinite Acting With Skin

Single Porosity With Skin

Dual Porosity

Dual Porosity with Skin

Fig 4.11: Model Hierarchy

82

EXERCISES 1. 2.

State the uses of pressure derivative plots. Assuming reservoir is infinite, sketch the pressure derivative for the following models: i ) Wellbore and skin model in homogeneous reservoir ii) Fractured model with well intersecting a single vertical fracture iii) Double porosity model

3.

Using buildup pressure and pressure derivative, show the distinguishing features between the following systems:

i) ii)

Well producing near a single fault and a well producing between two parallel faults. Well producing in a reservoir with gas-cap and a well producing in a well without gas-cap. Well producing in a truncated channel and a well in a channel that is not truncated. Well producing from an infinite system and well producing in a rectangle with boundaries close to flow.

iii) iv) 4.

Categorise the following into well, reservoir and boundary models: Wellbore storage and skin, Single fracture, leaky fault, horizontal well, double porosity, composite, constant pressure.

83

5. SKIN FACTOR, sT The skin factor is useful in evaluating the well condition, but it is usually applied wrongly. In this chapter, we shall discuss the basic things about skin and how to use the skin factor properly. What is Skin Factor? Factor used in BHP analysis to account for extra-ordinary pressure changes caused by flow around the wellbore Example of Normal (Ordinary) Pressure Change Around Wellbore Open Hole Completion of Entire Interval (no restriction)

ht

Fig.5.1: Unrestricted Flow at Wellbore

pw = pe −

_

pw = pw −

qBµ ln

re

rw 7.08x10 kh −3

(steady State)

 r  qBµ  ln e r − 0.75 w −3

7.08x10 kh

5.1

(pseudo-steady state) 5.2

Example of Extra-Ordinary Pressure Change Around Wellbore Partially Completed Well

hp

ht

Fig.5.2: Restricted Flow at Wellbore Due to the “stampede” at the wellbore, the equation for bottom-hole flowing pressure becomes 84

 r  qBµ  ln e r + sc  w p w = pe − −3 7.08x10 kh _

pw = pw −

(steady state) 5.3

 r  qBµ  ln e r − 0.75 + sc  w

7.08x10 −3 kh

(pseudo-steady state) 5.4

where sc is the skin factor due to partial completion. Generally,

pw = pe −

_

pw = pw −

 r  qBµ  ln e r + sT  w −3

7.08x10 kh

 r  qBµ  ln e r − 0.75 + s T  w

7.08x10 −3 kh

(steady state) 5.5

(pseudosteady state) 5.6

where sT is the total skin due to all factors that cause extra-ordinary pressure drop around the wellbore. The total skin is calculated from BHP data. Note: 1) Given the same drawdown and positive skin, well with normal pressure around wellbore will produce more than well with skin. 2) If the average pressure in the drainage areas are the same, both wells can produce at the same rate if the flowing pressure in well with skin is lower by an amount equal to 1412 . qBµ qBµ 5.7 ∆p s = sT = s . kh 7.08x10 −3 kh T This is shown in the Fig. 8. 3.

useful drawdown Pwf if no skin

∆ps Pwf if there is skin

rw

re

Fig.5.3: Effect of Total Skin on Wellbore Pressure (van Everdingen Model)

85

5.2 Causes of Skin a) Invasion of region around wellbore by drilling fluid and fines b) Dispersion of clay c) Presence of mud cake and cement d) Stimulation e) Sand Consolidation f) Presence of high gas saturation around wellbore g) Partial well completion I) Limited Perforation j) Slanting of wells k) Gravel Pack l) Sand in wellbore if flow occurs through it m) Non-Darcy effects (caused by turbulence especially in gas wells) n) etc. 5.3 Classification of Pseudoskins Factors that cause skin are classified under different pseudoskins as follows: 1) Pseudoskin due to Damage, sd: Includes Factors a to f and other factors that result to alteration (relative to the native permeability of the formation) of permeability around the wellbore. 2) Pseudoskin due to Completion (also called Mechanical Skin) sc: Includes Factor g. 3) Pseudoskin due Perforation, sp: Includes Factor i. 4) Pseudoskin due to Slanted, ssw: Includes Factor j 5) etc.

Each pseudoskin creates its own extra-ordinary pressure around the wellbore. This is shown in the figure for cases where the pseudoskins are positive. p useful drawdown

∆psc ∆psd

∆pT

∆psp

pwf rw

re

Fig.5.4: Effect of Pseudoskins on Wellbore Pressure (van Everdingen Model)

Note: 1) The total pressure drop ∆pT caused by the total skin is the sum of pressure drop caused by individual pseudoskins 2) Stimulation can only reduce pressure loss caused by pseudoskin due to damage

86

3) Total skin may not be a good yardstick for determining stimulation candidates. It is better to used pseudoskin due to damage, but preferably (personal experience) use ratio defined as follows: ∆p sd 5.8 R= _ P− p wf _

where ∆psd is the pressure loss caused by skin due to damage and P− p wf is the drawdown. Table 5.1 . shows the pressure loss due to skin and calculated R ratios. Table 5.1: Skin and Pressure Losses from Wells WELL TEST 1 2 3 4 Total Skin 18.4 3.5 26.4 192 57.86 174.2 216.6 723 ∆pT, psi Damage Skin 3.34 9.1 16.6 45 30.72 135.2 204 677 ∆psd, psi R 0.34 0.71 0.75 0.91

NO 5 6.1 9.9 -17.6 -17.5 -

6 196 119 66 115 0.95

7 63.6 367 3.15 190 0.47

Inferences from this table are as follows: 1) Using the magnitude of total skin for ranking the priority for stimulation, the wells will be ranked as follows: 6, 4, 7, 3, 1, 5, 2. Note that Well 5 does not require stimulation because the skin due to damage is negative. 2) Using the damage skin or R ratio, the wells will be ranked as follows: 6, 4, 3, 2, 1,7, 5. The added advantage of using R ratio is that it takes values from 0 to 1 and from experience we do not recommend stimulation if R ratio is less than 0.6. Tables 2 and 3 comparing pre-stimulation and post-stimulation surveys from two wells illustrate this. Table 5.2: Results of Pre-Stimulation and Post-Stimulation Tests (Well A) PARAMETER PREPOSTSTIMULATION STIMULATION Rate, STB/day 568 570 Drawdown, psi 68.16 168.72 Total skin,s 9.82 4.85 32.45 59.7 ∆p due to total skin, psi Mechanical skin 2.2 2.2 Damage skin 7.57 1.3 25.02 32.3 ∆pdamageskin, psi R - 0.367 0.191 ∆pdamageskin/Drawdown Oil PI, STB/day/psi 2.37 3.08

87

Table 5.3 : Results from Pre- and Post-Stimulation Analysis (Well B) Parameter

Pre-Stimulation

Post-Stimulation

Rate, STB/day

66

539

Total skin, s

46.74

8.58

Mechanical skin

0.2

0.2

Damage skin

46.57

8.4

Drawdown, psi

365.63

212.18

∆pdamageskin, psi

259.53

102.2

R

- 0.71

0.48

0.15

2.14

∆pdamageskin/Drawdown Oil PI

From the R ratios calculated from pre-stimulation, Well A was not a clear stimulation candidate (R < 0.6) while Well B was a good candidate (R > 0.6). The post-stimulation oil gains also confirm this. 5.4 Calculations of Pseudoskins a) Pseudoskin due to Damage, sd

k1

rw

k

r1

re

Fig.8.5: Damaged and Undamaged Zones  k  r s d =  − 1 ln 1 r w  k1 

if k > k1, sd > 0 (damage)

and

5.9

if k < k1, sd < 0 (stimulation)

Some implications of Equation. 5.9 that will give you a sense of magnitude is given in Table 5.4.

88

Table 5.4: Effect of Permeability Alteration on Pseudoskin Due to Damage k1 (% of k) Depth of Damage, r1, in (rw = 6 in) 8 24 60 120 90 0.03 0.15 0.25 0.33 50 0.29 1.39 2.30 3.00 10 2.59 12.47 20.7 26.96 5 5.46 26.34 43.7 56.9

Note that in high permeability formation, permeability reduction can be large if fines block the pore throat of the porous media. The problem with the equation for calculating pseudoskin due to damage is that depth of invasion and permeability of the damage zone are not known. Hence, we now estimate the damage skin from the total skin. This will be discussed. An equation for determining the pseudoskin due to consolidation is given as:   r − r  k − k c  rc s dc = 1 − 0.2 c w  h  ln  h p  k c  rw 

5.10

Comparing Equations 8.9 and 8.10, rc = r1, kh = k and kc = k. The term hp is the perforated interval b) Pseudoskin due to Partial Completion (Mechanical Skin), sc: This depends on

i) ratio of completed interval hp to formation thickness, hT (most important parameter) ii) location of completion relative to total thickness iii) ratio of vertical to horizontal permeability

hp

hT

Fig8 6: Effect of Partial Completion on Flow

Table 5.5 gives a sense of magnitude of how sc varies with hp/hT.

Table 5.5: Effect of Penetration Ratio on Mechanical Skin hp/hT 0.1 0.2 sc 36.9 16.4

0.3 9.6

89

There are many sources for obtaining sc, but Odeh’s equation is preferred. Photocopy containing his article and equation is enclosed. Saidikowski’s (SPE 8204 1979 SPE Conference) equation is simpler and gives reasonable answer for practical purposes. The equation is h   h kH   s c =  T − 1ln T  − 2  5.11  hp   rw k v   c) Pseudoskin due to Slanting of Well, ssw: This is always negative as slanting of wells will generally enhance productivity because more wellbore length will be exposed to flow. This is shown in Figure . The limiting case is a horizontal well.

Length available for fluid entry

Fig. 5.7: Length Available to Flow - Vertical and Slanted Well

The pseudoskin due to slanting of well depends on the angle of slant, α , and ratio of total thickness to wellbore radius, hT/rw.. It can be approximated using the equation published by Cinco, Miller and Ramey. The equation is α  = −   41 

α  −   56 

 h  log T  5.12  100rw  o and it is valid for 0 < α < 75 and h T/rw > 40 (i.e. about hT > 12 ft). Typical values of ssw for different hT/rw and angles of slant are given in Table 5.6. s sw

2 .06

1.865

Table 5.6: Pseudo Skin Due to Slanting of Well hT/ hT (ft) ANGLE (Degrees) rw rw = 0.4 ft 20 30 40 50 50 20 -0.18 -0.43 -0.79 -1.26 100 40 -0.23 -0.53 -0.95 -1.51 200 80 -0.27 -0.62 -1.11 -1.75 300 120 -0.30 -0.67 -1.21 -1.89 400 160 -0.32 -0.71 -1.27 -1.99

60 -1.85 -2.19 -2.53 -2.73 -2.88

75 -2.95 -3.47 -3.99 -4.29 -4.51

90

hT hT k H ≡ rw rw k v A graph for estimating pseudoskin due to slanting of well is enclosed. Note: If k H ≠ k v , define

d) Pseudoskin due to Perforation, sp: This depends on many parameters, but more importantly on perforation penetration and perforation density. Figure illustrates how this contributes to skin.

Fig.5.8: Flow Convergence to Perforation

The skin due to perforation can be positive (for shallow penetration) or negative (for deep penetration). The enclosed chart by Harris will be used to have a sense of magnitude. The best value is about -3 and the worst value is about 10. In calculating the pseudoskin due to perforation, most models (including Harris) neglect the pressure drop due to flow in the perforation tunnels. This pressure drop could be substantial and contribute to skin if tunnel is filled with sand or gravel. This is because there could be turbulent flow within the tunnels. Deductions from studies that consider pressure losses within the perforation tunnels are as follows: i) There is optimum perforation length for minimum skin. This is shown in Fig 5.7. The productivity ratio used in the figure is defined as r ln 1 r w PR = 5.13 r1 ln r + s p w Based on this, the skin caused by perforation tunnel being filled with gravel can be between 13 to 30. ii) Minimum skin due to perforation is smallest for highest shot density. iii) Minimum skin depends on gravel type, viscosity, rate openhole/casing diameter, etc. 8.5 Relationship between Total Skin, sT and Pseudoskins

1) Full Penetration of Entire Interval: s T = sc + sd + sp + ssw + .. 2) Partial Penetration a) Shallow Damage Skin (Damage restricted to perforated interval)

5.14

91

sT = sc +

[

hT s + s p + s sw + s i hp d

]

5.15

b) Deep Damage Skin (Damage covers entire total skin)

[

hT s + s + si h p p sw

sT = sc + sd +

]

5.16

If total skin is known (from BHP test result) and other pseudoskins calculated, the pseudoskin due to damage can be deduced from these equations.

8.6 Pressure Change Due to Skin The pressure change caused by skin can be calculated as follows:

1) Total Skin, sT: ∆p s =

1412 . qBµ sT kh T

5.17

2) Pseudoskin Due to Partial Penetration (Mechanical Skin), sc: ∆p sc =

1412 . qBµ sc kh T

3) Pseudoskin Due to Damage,sd: a) Shallow Damage Skin:

5.18

∆p sd =

1412 . qBµ sd kh p

5.19

∆p sd =

1412 . qBµ sd kh T

5.20

∆p si =

1412 . qBµ si kh p

5.21

b) Deep Damage Skin:

4) Other Pseudoskins, si:

Note that 7.08 x 10-3 is the inverse of 141.2.

92

8.7 Relationship Between Skin and Well Inflow Quality Indicator (WIQI) WIQI =

PI actual PI ideal

5.22

where: PI actual =

7.08x10 −3 kh  r  µB ln e − 0.75 + sT   rw 

5.23

and PI ideal

7.08x10 −3 kh =  r  µB ln e − 0.75 + sc   rw 

5.24

The mechanical skin is included in defining the ideal productivity index (PI) because it is “permanent” and can only changed if the completion is changed. From Equations 5.22 to 5.24, we have re ln − 0.75 + s c rw ∆p wf − ∆p sd WIQI = ≈ 5.25 re ∆p wf ln − 0.75 + s T re Exercises 1) A well was completed with a 7 in diameter casing diameter as shown in the Fig. 5.9

50 ft 150 ft

Fig. 5.9: Partially Completed Well The casing has a perforation density of 4 shots/ft and 120o phasing. It is estimated that the formation penetration is 15 inches and total skin determined from BHP analysis is 25. Considering only the pseudoskin due to partial completion and skin due to perforation,

Calculate the following: a) Skin due to damage assuming damage is shallow. b) Skin due to damage assuming damage is deep. c) Pressure change due to total skin. d) Pressure change due to damage skin due to damage assuming damage is shallow e) Pressure change due to damage skin due to damage assuming damage is deep f) Is there any difference between answers from d and e.

93

(Use k = 1000 md, q = 1000 STB/D, B = 1.10 rb/STB, µ = 0.6cp) 2) Repeat Exercise 1 for case where well is slanted at angle 60o. 3. Five pages of a report on a test considered unreliable are enclosed. The calculated parameters were not entered in the provided slots, but just written down. Even though the results are considered unreliable, but someone still wants to know the skin due to damage, pressure loss due to it and WIQI. Your job is to calculate required parameters. All data required can be found in the enclosed report. Skin Correction: In some analysis, it is difficult to match the correct flowing pressure before shut-in. This is usually due to rate or time problem. The case where the real flowing pressure was not matched is shown in the following simulation plot. P real data simulated profile (Analysis results)

∆p = analysis pwf - actual pwf flowrate Time

In a situation like this, the analysis skin was calculated with the following equation:  P1hr − p wfa ( ∆t = 0)   k   s = 1151 − log .  2  + 3.23 m  φµc t rw   

where: pwfa(∆t=0) = flowing pressure just before shut-in (used for analysis) 162.6qBµ m = semilog straight line slope = psi / cycle . kh As the analysis flowing pressure is greater than the actual flowing the actual skin will be s + ∆s where  p wfa ( ∆t = 0) − p wfr ( ∆t = 0)   ∆s = 1151 .    m

and pwfr(∆t=0) = real flowing pressure just before shut-in

??????????????????

94

6. ANAYSIS OF BOTTOM-HOLE PRESSURE TESTS There are three methods of analyzing bottom-hole pressure tests: conventional, typecurve and regression (automated type-curve) methods. To the “layman” the methods can be likened to three ways of getting an overall work dress for a worker. The conventional method is like going to a tailor who takes measurements at three points of the body and with just three measurements, the overall is made. Note that with the exception of the three measurements, other dimensions used in making the dress are extrapolations or interpolations. Also, if the tailor uses the dimension of the waist to be that of the shoulder, the dress will not fit. This underscores the importance of where the measurements were taken. The type-curve method can be likened to just walking into a shop and choosing from already made dresses. The problem here is that the exact size may not be available and one has to manage with the next lower of higher size. In high permeability-high skin Niger Delta formations, we have observed that available type-curves do not match the pressure data. The regression method is simply an automated type-curve method where the exact size that takes into account all dimensions is made. The problem here is that if you are not standing properly while the measurements are taken, the wrong overall will invariably be made. This implies that wrong pressure data will create problems during analysis with regression method if such data are not removed. The advantage of the regression method is that the correct-sized dress will be made and therefore there is no problem with the high permeability-high skin Niger Delta formations. In this section we shall discuss methods of analyzing drawdown, buildup and interference tests. Emphasis will be on the conventional and type-curve methods. However, we shall show that in an ideal situation, the three analysis methods give comparable results.

6.1 Conventional Method Bottom-hole pressure tests can be analyzed using the conventional method. The conventional analysis is based on locating the straight lines that are characteristic of the different flow phases. The desired parameters are then calculated from the slopes of the straight lines. This implies that if you put the wrong straight lines, you will get the wrong answers.

The procedure for analyzing drawdown and buildup test using conventional method is as follows: 6.1.1 Drawdown Test The transient state equation during a drawdown test is given as follows:

95

Pwf = Pi −

162.6qBµ kh

  k log t + log 2   φµct rw

   − 3.23 + 0.87 s   

6.1

Equation 6.1 forms the basis of calculating permeability and skin from drawdown test using data not influenced by wellbore storage effects. The procedure for the analysis is oultined.

Procedure for Analyzing Drawdown Test 1. Locate pressure influenced by wellbore storage effect and calculate wellbore storage constant. How? (a) Make a graph of (Pi -pwf) versus t on a log-log paper (b) Locate data with strong wellbore storage effect on the unit slope line. (c) Calculate the wellbore storage constant, Cs, with ∆p= (Pi -Pwf) and t taken from a point on the unit slope line. Use the following equation. Cs

=

qBt bbl (rb/psi) 24∆ psi

6.2

(d) Locate the data not strongly influenced by wellbore storage effect. Use the gentle slope rule or the 10t* to 50t* rule. 2.

Locate data obtained during the transient state phase and calculate permeability and skin. How? a) Plot pwf versus log t. Use a semilog paper with time graphed on the log scale axis. b) With the knowledge of the time when wellbore storage effect has died down completely, put the correct straight line that represents the good transient state behaviour.

c) Determine the slope of the semilog straight line, m, and calculate the permeability using the equation k (mD) =

−

162.6qBu mh

6.2

Note that the slope, m, is negative and therefore permeability, k, is positive. d) Calculate the total skin using the following equation: s

=

 (P − P )  k 1.1513  1hr i −log +3.232 2 φuct rw  m 

6.3

96

where: P1hr is the pressure taken from the straight line portion when flowing time is equal to one. Exercise 6.1:. The log-log and semilog plot of the pressure-time obtained during a drawdown test are shown in Fig 6.1 and Fig 6.2 (leave space for these figures ???? ) Data from the test are shown in Table 6.1 ??? (ask me). Draw the necessary straight lines and calculate Cs, k and s. Other information required for the analysis are as follows:

rw ct q

= 0.5 ft = 20 x 10-6 psi-1 = 1000 STB/D

Table 6.1: Drawdown Data Time, hr Flowing Pressure, psi .00000000E+00 3183.763 .10000000E-03 3183.245 .80000000E-03 3179.758 .20000000E-02 3174.302 .48000000E-02 3163.727 .96000000E-02 3150.896 .12000000E-01 3146.317 .18240000E-01 3138.186 .21600000E-01 3135.356 .27840000E-01 3131.839 .32400000E-01 3130.196 .39600000E-01 3128.530 .44400000E-01 3127.814 .55680000E-01 3126.796 .60000000E-01 3126.553 .88800010E-01 3125.731 .11040000E+00 3125.446 .13440000E+00 3125.224 .17760000E+00 3124.928 .24960000E+00 3124.572 .37440000E+00 3124.154 .53760000E+00 3123.788 .77760000E+00 3123.421

h = 50ft µ = 0.6 cp

Time, hr .10176000E+01 .12576000E+01 .14976000E+01 .17376000E+01 .19776000E+01 .22176000E+01 .24576000E+01 .28176000E+01 .31776000E+01 .35376000E+01 .38976000E+01 .42576000E+01 .46176000E+01 .49776000E+01 .53376000E+01 .56976000E+01 .60576000E+01 .64176000E+01 .67776000E+01 .71376000E+01 .74976000E+01 .79536000E+01

Bo = 1.125 φ = 0.25.

Flowing Pressure, psi 3123.157 3122.951 3122.781 3122.638 3122.513 3122.402 3122.303 3122.172 3122.056 3121.953 3121.860 3121.776 3121.698 3121.626 3121.559 3121.496 3121.438 3121.383 3121.330 3121.281 3121.234 3121.177

Although you have gone through the exercise manually, there are computer programmes that do the graphing and calculations in a twinkle of an eye. I used the programme in

97

analyzing this test and the results obtained are as follows: Cs = 0.93 x 10-2, k = 991.2, and s = 24.72. In a situation where the drawdown test is run long enough, the volume drained by the test well can be determined. The equation that forms the basis of this analysis is as follows:

Pwf = Pi −

0.234qB 141.2qBµ t+ φhAct kh

 A  2.2458 + 2s  ln 2 + ln CA  rw 

6.4

The implications of this equation are as follows: (a) A graph of Pwf versus t gives a straight with a slope, m=− (b)

0.234qB (psi/hr) φhAct

6.5

From the slope, the drainage volume, φhA, is calculated.

Exercise 6.2: Table (6.2 ??? ask) shows pressure data from a drawdown test in a well in the center of a square which was tested long enough to reach pseudo-steady state. Using the conventional method and data in Table 6.3, analyze the test and calculate the following: a) Formation permeability b) Skin factor c) Wellbore storage constant d) Drainage area

table 6.2 here ?? Table 6.3:

Parameters used for Analysis Parameter Wellbore radius, rw, ft Total compressibility, ct, psi-1

Porosity, φ Oil formation volume factor, Bo, rb/STB Oil viscosity, µ, cp Production rate, q, STB/D Formation thickness, h, ft

Value 0.5 20 x 10-6

0.25 1.125 0.6 1000 50

Solution: Plots from the analysis and results are shown in Fig 6.3. ????? ask

6.1.2 Buildup Test Two forms of transient state equation can be used in the analysis of buildup test. The first equation was developed by Horner (1951) by assuming that the test well produced

98

for a while (late-time state was not reached) before the well is shut in. The second form of the equation was developed by Miller, Dyes and Hutchinson (1950) by assuming that the test well produced for a long time (late-time state was reached) before the well was shut in. Based on the assumptions used in developing the equations, their applications were limited. For example, the Horner equation was used for new wells while Miller, Dyes and Hutchinson (MDH) equation was used for old wells. However, Ramey and Cobb (1971) have shown that the Horner equation is superior even when it is used for old wells. On the other hand Agarwal (1980) has shown that MDH equation can be used for new wells by using an equivalent shut-in time defined as follows: ∆t eq =

∆t

6.6

 ∆t  1 +   t  p  

Parameters in Eq. 6.6 are defined as follows: ∆teq ∆t tp

= equivalent shut-in time to be used for analysis = actual shut-in time = production time before shut in

Discussions on the Horner and MDH equations and procedure for using them follow:

Horner Equation The Horner equations for infinite and developed systems are given as follows: Pws = Pi −

 t p + ∆t  162.6qBµ  log kh  ∆t 

for infinite reservoir

6.7

 t p + ∆t  162.6qBµ  log kh  ∆t 

for finite or developed reservoir

6.8

and Pws = P * −

Note that as shut-in time, ∆t, increases, the Horner time {(tp + ∆t)/ ∆t} decreases. At infinite shut-in time, Horner time is unity (1).

MDH Equation The MDH equation is given as follows:

99

−

Pws = P +

162.6qBµ kh

  0.000264k   C A rw2    t log ∆ + log + log   φµc r 2   t w    A

  

6.9

Unknown parameters in Eqs. 6.7 to 6.9 are defined as follows: −

P P* Pi CA A

= Average pressure = false pressure = initial pressure = shape factor or geometric factor = drainage area

The shape factor depends on the location of the well with respect to the boundary. Typical shape factors are in Table 6.4. Table 6.4: Reservoir Geometries and Shape Factors Geometry and Well location Shape Factor

31.62 Well in center of circle 31.6

Well in center of square 1

21.8369

2 Well in the center of 2 to 1 rectangle 2.0769 1 2 Well in a quadrant of 2 to 1 rectangle Shape factors for other reservoir geometries and well location are given by Earlougher (1977). Irrespective of the equation used for analysis, the procedure for calculating wellbore storage constant and skin is the same. We shall therefore present a general procedure for analyzing buildup test and highlight the variations resulting from the use of any of the equations (MDH or Horner)

100

Procedure for Analyzing Buildup Test

Calculation of Wellbore Storage Constant 1.

Locate pressure influenced by wellbore storage effect. How? (a) Make a graph of ∆p = p ws − p wf ( t p ) versus ∆t on a log-log paper.

[

]

(b) Locate data with strong wellbore storage effect on the unit slope line. (c) Calculate the wellore storage constant, Cs, using ∆p and ∆t from a point on the unit slope line and the following equation: Cs

=

qB∆t rb/psi 24∆p

6.10

(d) Locate the data not strongly influenced by wellbore storage effect. Use the gentle slope rule or the 10∆t* (1 cycle) to 50∆t* 1.5 cycle) rule.

Calculation of Permeabilty MDH Plot 2. Plot pws versus log ∆t. Use a semilog paper with time graphed on the log scale axis. 3. With the knowledge of the time when wellbore storage effect has died down completely, put the correct straight line that represents the good transient state behaviour. 4. Determine the slope of the semilog straight line, m, and calculate the permeability using the equation. k (mD) =

162.6qBu mh

6.11

At the end of this step, move to Step 8 to calculate skin factor.

Horner Plot  t p + ∆t  5. Plot pws versus log   . Use a semilog paper with Horner time graphed on the  ∆t  log scale axis. The graph can be made with Horner time increasing from left to right to left. The later is usually preferred as it gives a graph with shut-in time increasing from left to right. 6. With the knowledge of the time when wellbore storage effect has died down completely, put the correct straight line that represents the good transient state behaviour. 7. Determine the slope of the semilog straight line, m, and calculate the permeability using the Eq. 6.11. 8. Extrapolate the straight line to Horner time equal to 1 (infinte shut-in time) and read off P* or Pi. They are needed for estimating the average pressure in a well that was not shut in long enough to reach average pressure.

101

Calculation of Skin 9. The correct equation is given as  P1hr − Pwf (t p )  k s = 1.1513 − log 2 m   φµct rw

  t +1   + 3.23  + log p  t     p 

6.12

However, if production time before shut-in, tp >> 1, Eq. 6.12 degenerates to  P1hr − Pwf (t p )  k s = 1.1513 − log 2 m   φµct rw

   + 3.23  

6.13

In Eqs. 6.12 and 6.13, P1hr is the pressure taken from the straight-line portion of the semilog plot at shut-in time, ∆t = 1 hour. Note that shut-in tome of 1 hr corresponds to Horner time of (tp + 1).

Note on Production Time before Shut-in, tp For a well that produced at constant rate (unusual) before shut in, tp, is the actual production time before shut in. This is illustrated in Fig 6.4.

q Rate

Time

t = tp

Fig.: 6.4: Actual Production Time Before Shut in Equal to tp

If a well did not produce at constant rate (usually the case) before shut in, tp can be calculated from cumulative production provided that a stable flow rate was attained during the flowtest prior to shutting in. The equation for calculating tp is given as follows: tp = where ∆Np qs

∆N p qs

6.14

= Cumulative production since last shut in = stabilized rate prior to shutting in

This case is illustrated in Fig 6.5. 102

qs Rate

t ≠ tp

Time

Fig.: 6.5: Actual Production Time Before Shut inis not Equal to tp

Experience has shown that for wells that produced for a long time (reached pseudosteady state or steady state) before shut in, tp, can be replaced by a time that is at least the time to reach pseudo-steady state, tpss. The Horner time will simply be defined as as [(tpss + ∆t)/ ∆t]. For high permeability formation obtained in the Niger Delta, we normally use 1000 hours. Exercise 6.3: Figures 6.6 to 6.8 log-log and semilog plots of pressure-time data obtained during a buildup test. Draw the necessary straight lines and calculate Cs, k and s. Other information required for the analysis are as follows:

rw Ct Q

= 0.5ft = 20 x 10-6 psi-1 = 1000 STB/D

h = 50ft µ = 0.6 cp and tp = 1000 hr.

Bo = 1.125 φ = 0.25.

Answers: k = 1000 mD, s = 25 and Cs = 0.009 rb/psi

Calculation of Average Pressure For wells that were not shut-in long enough to average pressure, the average pressure can be obtained if the transient state was attained. An example of such a well is shown in Figure 6.9. P*

Pws

Good Transient

Log (t +∆t/∆t)

1

103

Fig. 6.9: Horner Plot The average pressure is obtained based on the fact that P* is related to the average pressure by the equation −

P = P*−

m PDMBH (t PDA ) 2.303

6.15

where −

P = Average pressure P* = False pressure PDMBH = Dimensionless pressure defined by Mathews, Brons and Hazebroek (1954) m = Slope of the semilog straight line tpDA = Dimensionless (based on area) production time before shut-in =

0.000264kt p

φµct A

Oilfield units.

The PDMBH values depend on geometry of reservoir, well location and dimensionless production time based on area. The values for some cases can be obtained from Fig 6.10. Leave a page for this figure ???? Fig 6.10: Mathews, Brons and Hazebroek Dimensionless Pressure, PDMBH Note that Horner plot and Mathews, Brons and Hazebroek dimensionless pressure, PDMBH Were used to obtain the average pressure. The Dietz (1965) method which involves the used of MDH plot in obtaining average pressure will not be discussed. The values of PDMBH can also be calculated at certain periods using the following equations: 1.

Early production time before shut-in (tp < tpss) PDMBH = 4πtDA

2.

6.16

Long Production time before shut-in (tp ≥ t pass ) PDMBH = In CAtpDA

6.17

104

Confirm the correctness of these equations with Fig 6.10. Some facts about average pressure are as follows: For infinite system, p = p* = pi

(a)

If cumulative production is not sufficient to reduce the average pressure significantly, p* is still a good estimate of average pressure.

(b)

Average pressure decreases with increase in production time before shut in. This is shown in Fig 6.11

(c)

leave space here???? ask Fig 6.11: Relationship between Average Pressure and False Pressure

The average pressure calculated from a test well is the average pressure in the drainage area of the test well. The reservoir average pressure is then calculated as follows:

(d)

N

∑ P reservoir = i=1

pi q i

6.18

qt

where p i = average pressure from Well i. qi = production rate from Well i. qt = total production from all the wells in the reservoir. In situation where all the wells were not tested, the average pressure in the untested well can be estimate using the p - q relationship from the tested well. You have to interpolate or extrapolate.

6.1.3 Interference Test The equation that forms the basis of analyzing interference is as follows:

P (r , t ) = Pi −

162.6qBµ kh

  k log t + log 2   φµct r

   − 3.23   

6.19

105

Equation 6.19 predicts the transient state pressure at an observation point that is r distance from the active well. This equation does not include wellbore storage and skin in either the active or the observation well. The procedure for using Eq 6.19 for interference test analysis is as follows: (a) Graph P(r,t) (pressure recorded in the observation well) versus time, t, on a semilog graph. (b) The transient state data will fall on a straight line. Permeability can be calculated with straight line slope, m, with the equation: k (mD) = (c)

162.6qBu mh

Porosity or storativity can then be calculated from the equation  k  Pi − P1hr  = log + 3.23 φµ c m t  

6.20

6.21

In Eq 6.21, P1hr is the pressure at time, t = 1 hr, taken from the straight-line portion of the curve.

6.2 Type-Curve Method Gringarten (1987) defined a type curve as a graphic representation of the theoretical

solution of the fluid flow equation used in representing the test well and the reservoir being tested. For a constant-pressure test, the theoretical solution is presented in the form the change in production rate with time. This is usually used in decline curve type curves. For a constant-rate test, the theoretical solution is presented as change in pressure at the bottom of the well as a function of time. This form is normally used in bottom-hole pressure test analysis. Other types of response are also used, such as the time derivative of the bottom-hole pressure.

Type curves are derived from solutions to the flow equations under specific initial and boundary conditions. For the sake of generality, type curves are usually presented in dimensionless terms, such as a dimensionless pressure vs. a dimensionless time. A given interpretation model may yield a single type curve or one or more families of type curves, depending on the complexity of the model.

106

Type-curve analysis involves finding a type curve that “matches” the actual response of the well and the reservoir during the test. The reservoir and well parameters, such as permeability and skin, can then be calculated from the dimensionless parameters defining that type curve.

The match can be found graphically, by physically superposing a graph of the actual test data with a similar graph of the type curve(s) and searching for the type curve that provides the best fit. Alternatively, an automatic fitting technique involving a linear or nonlinear regression can be used. Figure 6.12 taken from Gringarten (1987) gives an example of a graphic type-curve match. The graph of the data is positioned over the graph of the type curves, with the axes kept parallel, so that the test data match one of the type curves. Reservoir parameters are calculated from the value of the dimensionless parameter defining the type curve being matched and from the x and y axis shifts.

Fig 6.12 here ask ???? Fig 6.12: Graphical Type-Curve Matching Process

There are many kinds of type-curves, but we shall only discuss the Theis (1935) typecurve used for analyzing interference test and Bourdet et al (1983) type-curve used for analyzing buildup and drawdown tests. The type curves are shown in Fig. 6.13 and Fig 6.14. The Theis type-curve is also known as line source solution.

Leave a page for this fig 6.13

Fig 6.13: Theis Type-Curve

Leave a page for this fig 6.14

Fig 6.14: Bourdet Type-Curve

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6.2.1 Basis of Type-Curve Analysis The basis of type-curve analysis will be illustrated using the interference type case and

application of type-curve will be extended to buildup tests.

Interference Test: The basis of type-curve analysis is the relationship between

dimensionless parameters and the non-dimensionless parameters used in interference test type-curve. The relationships are shown for dimensionless pressure and time as follows:

PD =

kh[ ∆p] 141.29Bµ

6.22

tD 0.000264kt = 2 rD φµ c t r 2

6.23

Taking log of Eqs 6.22 and 6.23 gives

log PD = log ∆p + log

log

kh 141.2qBµ

tD 0.000264k = log t + log 2 rD φµ c t r 2

6.24

6.25

Note that log(kh/141.2qBµ) and log(0.000264k/φµctr2) are constants and therefore regarded as shifts.

The implications of Eqs 6.24 and 6.25 are as follows: (1)

On a log-log paper, PD is directly related to ∆p and

tD is also directly related to t. rD2

The parameters of interest in the analysis constitute the “shifts” in the relationships.

108

(2)

A graph of ∆p versus t on a log-log paper will look exactly like that of PD versus tD on a similar log paper. rD2

(3)

∆p - t plot can then be overlain on the PD -

tD type curve. This is known as typerD2

curve matching process. Normally, the field data are plotted on transparent paper so that the match can easily be made. Details on the process of type-curve matching will be demonstrated in the class On matching the curves, any point on the type-curve ( PD and tD/rD2) and the corresponding point on the field data (∆p and t ) is known as a match point. With values of the match point, the desired parameters can be calculated with the following equations.

K=

1412 . qBµ  PD  .   h  ∆ p  m

φC t =

0.000264k  t   2  µ r2  t D / rD  m

6.26

6.27

Noe that Equations 6.26 and 6.27 are simply rearranged forms of dimensionless parameters defined by Eqs. 6.22 and 6.23.

Buildup Test (Pressure Match): Using Bourdet’s type-curve, a match could be obtained

using the pressure type-curve or the pressure derivative type-curve. Both type-curves are in Fig. 6.14. For pressure match, dimensionless parameters defined by Bourdet et al (1983) for analysis of buildup test are as follows:

PD =

Kh ∆p 141.2qBµ

tD Kh ∆t = 0.000295 CD µ C

6.28

6.29

109

CD =

0.8936 C φ C t hrw2

6.30

The procedure for using the type-cure is as follows: 1.

Graph ∆p = Pws – Pwf (tp) versus ∆t on a transparent paper with same log cycle dimensions as the type-curve.

2.

Match the real graphed data with a type-curve

3.

Obtain the match point values (∆p, ∆t, PD tD/CD) and the CDe2s of the representative type-curve

4.

With the match point values, calculate a

k from Eq. 6.28

b

C from Eq. 6.29

c

CD from Eq. 6.30

d

s using CD and CDe2s of the representative type-curve.

Buildup Test (Derivative Match): For this case, we give you the opportunity to outline the procedure.

6.3 Comparison of Analysis Methods Under ideal conditions, all analysis method will yield close results. The ideal conditions includes good data acquisition, test run long enough to reach required phases and also accurate identification of the phases. Table 6.5 show results obtained by analyzing drawdown test using conventional and type-curve methods. Data for this test are in Table 6.1. In this case, both analysis methods gave close results. Table: 6.5 DrawDown Analysis Results Parameter Conventional Method Permeability, k, md 991.2 Skin, s 24.72 Wellbore Storage, Cs, rb/psi 0.93 x 10-2

Type-Curve Method 1000 25 0.9 x 10-2

Some comments on the analysis are as follows: 1. Both analysis methods gave close results, but the type-curve method is more accurate. We know about this because the analyzed pressure data were generated with k = 1000md, s = 25 and Cs = 0.9 x 10-2 rb/psi. This implies that the type-curve method gave exact answers.

110

2. The test was run long enough as the transient state is clearly shown on the derivative plot. Table 6.6 shows results obtained by analyzing drawdown test using conventional and type-curve methods. The derivative for this analysis is shown in Fig 6.15. Both analysis methods did not give close results because the transient state is not apparent. We cannot say which resulta are correct because data are actual field data. Table 6.6: Buildup Analysis Results Parameter Conventional Method Type-Curve Method Permeability, k, md 65 293 Skin, s 12 75 The test well whose data are shown in Fig 6.15 was producing 129 (STB/day) and the gas-oil ratio (GOR) was 882 SCF/STB. With such a low production and high GOR, what do you expect?. Wellbore storage effect will last for unusually long period and the transient state phase will be masked.

Leave space for fig 6.15 ask Fig 6.15: Derivative from Test Data

Note that with the type-curve method, we do not look for straight lines corresponding to the different phase. Rather, we do a curve-fit to match the pressure responses with the ideal responses for the entire phases. When an acceptable curve-fit is obtained, the desired reservoir parameters can be calculated.

Problem Set 6 1 Pressure-time data from a buildup test are given in Table 6.7. The well and reservoir data are as follows: rw = 0.5 ft, h = 50 ft, q = 1000 STB/day, ct = 20 x 10-6 psi-1, φ = 0.25, µo = 0.6 cp, Bo = 1.125 rb/STB, tp = 1000 hr A. Using the Horner plot determine the following: a) effective permeability to oil b) total skin factor c) wellbore storage constant d) average pressure assuming that well is in the centre of a square with a drainage area of 80 acres

111

Table 6.7: Buildup Data pws, psi ∆t, hr 0 3183.76 0.0001 3184.28 0.0008 3187.77 0.0020 3193.22 0.0048 3203.80 0.0120 3221.21 0.0278 3235.69 0.0557 3240.73

∆t, hr 0.0888 0.1776 0.3774 0.5376 0.7776 1.0176 1.2576 1.9776

p, psi 3241.80 3242.60 3243.37 3243.74 3244.10 3244.37 3244.57 3245.01

2) Confirm results obtained in Question 1(a), (b) and (c) using the MDH method. 3) The pressure drop, ∆p, and the derivative of the pressure drop, ∆p’, from a buildup test are given in Table 6.8. The well and reservoir data are as follows: rw = 0.5 ft, h = 50 ft, q = 1000 STB/day, ct = 20 x 10-6 psi-1, φ = 0.25, µo = 0.6 cp, Bo = 1.125 rb/STB, tp = 1000 hr. A. Using the Horner plot determine the following: a) effective permeability to oil b) total skin factor c) wellbore storage constant B. Repeat the analysis using type-curve analysis and pressure derivative data Table 6.8: Buildup Data ∆t, hr ∆p, psi 0 0 0.0001 0.5180 0.0008 4.005 0.0020 9.4610 0.0048 20.036 0.0120 37.446 0.0278 51.923 0.0557 56.967

∆p’ 4.646 9.087 15.461 18.078 11.773 3.984

∆t, hr 0.0888 0.1776 0.3774 0.5376 0.7776 1.0176 1.2576

∆p, psi 58.032 58.835 59.609 59.975 60.34 60.605 60.811

∆p’ 1.668 1.200 1.195 1.169 0.987 0.979 -

4). An interference test was run in shallow water sand. The active well produced 466 STB/D and pressure was measured in the observation well, which was 99 ft from the active well. Estimated rock and fluid properties are as follows: µw = 1.0 cp, Bw = 1.0 bbl/STD, h = 9 ft, rw = 3 in, ct = 27.4 x 10-6 psi-1 Match points obtained from type curve matching with the pressure response are as follows: t t = 128 minutes, D2 = 10 , ∆p = 5.1 psi, and PD = 1.0 rD Using supplied information calculate permeability and porosity. 112

5) An interference test is run between two oil wells. One is produced at 477 STB/D and the pressure response is measured at a nearby (263ft) well with a high precision guage. Pressure and reservoir data are given below. Using the semilog analysis method and type curve method, find the effective permeability to oil, md, and the effective in place porosity, fraction of bulk volume. h µo Bo

= 15ft = 0.89cp = 1.10 RB/STB

rw ct

= =

0.275ft 13.5 x 10-6 psi-1

Table 6.9: Pressure Data at Observation Well t, hrs ∆p, psi 0 0 25 57.5 37.5 66.3 50 73.8 62.5 78.8 75 82.5 87.5 86.6 125 95.0

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7. SENSITIVITY ON FACTORS AFFECTING BHP ANALYSIS In this chapter, we shall discuss the effect of certain factors on analysis of bottom-hole pressure (BHP) tests. Factors considered include leaks, rate variation, sampling frequency, interference effect, wrong datum correction, and gauge sensitivity and accuracy. The effects of these factors were simulated in ideal BHP data and results from analysis of such data compared with the ideal case. We present the ideal data case and results from analysis of the data. We shall use the buildup test, which is the commom test, in illustrating these factors affect results obtained from BHP analysis. The ideal BHP test data are used as the standard because fluid and reservoir parameters used in generating such tests are known. 7.1. Analysis of Ideal BHP Data The ideal BHP data were obtained by simulating the buildup test using parameters in Table 7.1. The simulation is also called test design. Generated pressure-time data are in Table 7.2. Table 7.1:

Parameters for Test Design Parameter Wellbore radius, rw, ft Total compressibility, ct, psi-1

Design Value 0.5 20 x 10-6

Formation thickness, h, ft Porosity, φ Oil formation volume factor, Bo, rb/STB Oil viscosity, µ, cp Production rate, q, STB/D Production time before Shut-in, tp, hr Permeability, k, md Total skin, s, Wellbore storage constant, Cs

50 0.25 1.125 0.6 1000 1000 1000 25 0.009

The ideal buildup data were analyzed using the conventional and type-curve methods. The analysis procedure is as follows: A Conventional Analysis Method (i) Graph on log-log paper, ∆p versus ∆t and calculate Cs (see Figure 7.1)

(ii) Graph on semilog paper, Pws versus

t p + ∆t ∆t

(Horner plot). Calculate k and s (see

Fig. 7.2) (iii) Use values calculated in Steps (i) to (ii) to generate pressure profiles. Plot the profiles with solid lines and compare with ideal data. This comparison is shown in Figs. 7.3. What can you conclude from the comparison?

114

Table 7.2: Designed Buildup Data Shut-in Time, hr Pressure, psi Shut-in Time, hr Pressure, psi ************************************************************************ 0.00000 3183.763 1.01760 3244.368 0.00010 3184.281 1.25760 3244.574 0.00080 3187.768 1.49760 3244.743 0.00200 3193.224 1.73760 3244.887 0.00480 3203.799 1.97760 3245.011 0.00960 3216.630 2.21760 3245.122 0.01200 3221.209 2.45760 3245.220 0.01820 3229.340 2.81760 3245.352 0.02160 3232.170 3.17760 3245.467 0.02780 3235.686 3.53760 3245.569 0.03240 3237.330 3.89760 3245.662 0.03960 3238.996 4.25760 3245.746 0.04440 3239.712 4.61760 3245.824 0.05570 3240.730 4.97760 3245.895 0.06000 3240.973 5.33760 3245.962 0.08880 3241.795 5.69760 3246.024 0.11040 3242.080 6.05760 3246.082 0.13440 3242.302 6.41760 3246.137 0.17760 3242.598 6.77760 3246.189 0.24960 3242.953 7.13760 3246.238 0.37440 3243.372 7.49760 3246.285 0.53760 3243.738 7.95360 3246.341 0.77760 3244.104

B. Type curve Analysis Method We used the automatic type curve analysis method. Results obtained were used in generating simulated profiles. The simulated profiles are compared with the ideal data and shown in Fig. 7.4. Table 7.3 is a summary of calculated results and the expected results. Table 7.3 Buildup Analysis Results Parameter Calculated Results Conventional Type Curve K, md 1000 1000.29 S 25.01 25.01 Cs rb/psi 1.03 x 10-2 0.8996 x 10-2 P* 3251 3251

Corrected Results

1000 25 0.9 x 10-2 3251

From Table 7.3, we conclude that for the ideal case, both analysis methods will give the expected results. We now investigate the effects of some factors on the results.

115

7.2 Effect of Gauge Accuracy and Datum Correction. The effect of gauge accuracy was investigated by introducing consistent errors in the ideal BHP data. Two cases were considered. In the first case we assumed that the gauge readings were consistently lower by 15 psi. This case may also represent a case where the pressures were measured at depth that is 40 ft shallower than the datum.

Gauge

Measurement depth h Top of perforation Z/2

Datum

Figure 7.1: Gauge position and Datum. As expected, the pressure profiles obtained from the analysis of this first case are similar to what we obtained with the idea data. The only difference is the shift in pressure level. This implies that P* and average pressure obtained in this analysis will be lower by 15 psi. Table 7.4 shows a summary of the results. Table 7.4: Effect Gauge Accuracy and Datum Correction-Case 1 Parameter Calculated Results Corrected Results Conventional Type Curve K, md 1000 1005 1000 S 25.01 25.01 25 Cs rb/psi 0.962 x 10-2 0.901 x 10-2 0.9 x 10-2 P* 3236 3236 3251

In the second part of this investigation, we introduced 5psi, 7psi and 10psi errors over certain ranges. In real situation, this may be likened to wrong calibration over certain pressure ranges. Figures 7.5 and 7.6 show the plots of pressure data for this case. Table 4.5 shows the results obtained from the analysis of this case.

Table 7.5: Effect of Gauge Accuracy -Case 2 Parameter Calculated Results Conventional Type curve K, md 1002 995.8 S 24.02 24.02 -2 Cs rb/psi 0.94 x 10 0.93 x 10-2 P* 3229 3229

Corrected Results

1000 25 0.9 x 10-2

116

The “staggering” of the pressure profile in this case is embarrassing, but we are pleased that the calculated results are close to the correct results. This does not mean that good gauge calibration is not important. Who knows the magnitude of the error you will introduce by not calibrating your gauge properly. 7.3 Effect of Noise By noise, we mean erratic pressure perturbations caused by gauge malfunction or erratic flow in the wellbore. The effect of noise was simulated by randomly introducing errors in the pressure data. Figures 7.7 to 7.8 show the pressure data after introducing the errors. Table 4.6 gives a summary of the results for this case. Table 7.6: Effect Gauge of Noise Parameter Calculated Results Conventional Type curve K, md 1003 995 S 35.42 35.42 Cs rb/psi 0.32 x 10-2 0.32 x 10-2

Corrected Results

1000 25 0.9 x 10-2

Some comments on the results are as follows: 1 Noise may cause overestimation or under-estimation of parameters. 2 Due to noise, the derivative plot may exhibit features, which may be wrongly misinterpreted as boundary effect 7.4 Effect of Gauge Sensitivity. We simulated the effect of gauge sensitivity by setting the pressure readings in some regions to be the same. This implies that the gauge could not “discern” the difference in pressure in that region. Figure 7.9 shows data for this case while Table 7.7 shows the summary of the results from the analysis.

Table 7.7: Effect Gauge of Sensitivity Parameter Calculated Results Conventional Type curve K, md 1003 987 S 24.9 24.9 Cs rb/psi 1.42 x 10-2 1.4 x 10-2

Corrected Results

1000 25 0.9 x 10-2

7.5 Effect of Rate Variation We simulated the bottom-hole pressure using design rate shown in Table 7.8. We also analyzed the test using the rate shown in Table 7.8. Both the design and analysis rates

117

are also shown in Fig. 7.10. Table 7.8 and Figure 7.10 represent a case where a well flowed at 1000 STB/D for 999 hours and then just an hour prior to shut in the rate was reduced to 500 STB/D. During the analysis, a rate of 500 STB/D was used and this implies that the complete rate history was not used. The rate variation simulated here could be caused by reducing the choke size to drop the tool. A tool whose size could obstruct flow reasonably could also cause it.

Table 7.8: Rate Schedule for Design and Analysis Time, hrs Design Rate, STB/D Time, hr 0 1000 0 999 500 1000 1000 0

q

Analysis Rate, STB/D 500 0

q

1000

500 500 Analysis Rate

Design Rate 0

0 999

1000

time

time

1000

Figure 7.10: Design Rate and Analysis Rate. The pressure profiles in this case are shown in Fig. 7.11 and Fig 7.12. The figures and analysis look reasonably, but the results shown in Table 7.9 are not correct. Table 7.9 also shows results obtained by analyzing the test using the correct rate history. Figures 7.13 and 7.14 show the match obtained using the wrong rate history and correct rate history respectively.

Table 7.9: Effect of Rate Variation Calculated Results (Wrong Rate)

Calculated Results (Correct Rate)

Parameter

k, md s Cs, rb/psi

Correct Results Conventional

Type Curve

Conventional

743.9 17.24 1.01 x10-2

736.9 17.24 1.01 x10-2

958.6 23.68 0.92 x 10-2

Type Curve 1000 25 0.91 x 10-2

1000 25 0.9 x 10-2

Figure 7.11 to 7.14 here ??? From these results, we make the following inferences:

118

1

Analyzing a bottom-hole pressure test with incorrect rate schedule will give wrong answers. Even the Horner’s approximation for handling multirate test as a single rate test did not give satisfactory results.

2

Avoid things that will cause unnecessary rate changes during test. If rate changes occur, record it properly and let the one who will analyze the test be aware of it.

3

Rate variation could lead to the beginning of another wellbore storage phase and this could be misinterpreted as a boundary. This is shown in Fig 7.15 for actual data.

Fig 7.15 ????? here 7.6 Effect of Leaks Leaks may result in the gauge measuring pressure that may be lower than the actual pressures especially when the leaking system is at higher pressure. For a buildup test, leaks may result to continuous flow and hence the sandface rate not being zero (qsf ≠ 0). In this section, we shall consider leaks caused by the sandface rate not equal to zero. The following may cause such leaks: (a) Leaking lubricator (b) Leaking packer, which may result in communication between two sand intervals.

We simulated the bottom-hole pressure test with a leak of 1% of the original rate during the buildup stage and analyzed the test with the assumption that there was no leak (qsf = 0) during the buildup period. The results of the analysis are shown in Table 7.10.

Table 7.10: Effect of Leak Parameter Calculated Results Conventional Type Curve k, md 944 964 s 22.94 23.84 -2 Cs, rb/psi 1.13 x 10 0.96 x 10-2

Correct Results

1000 25 0.9 x 10-2

Concluding, leaks of any form are not welcome. Both permeability and skin may be underestimated or overestimated. 7.7 Effect of Interference /Leak Producing wells that are around the test well can cause interference. These wells cause some pressure drop at the test well because the wells are draining the same reservoir and each well behaves as if it does not know that the other well is present. We simulated this effect by deducting the pressures due to interference from the ideal pressure data. The resulting data could also represent a case with leaks.

119

Figures 7.16 shows the log-log plot for this case and results obtained are in Table 7.11. Table 7.11: Effect of Interference Parameter Calculated Results Conventional Type curve K, md 1043 1316.7 S 26.18 34.6 -2 Cs rb/psi 1.016 x 10 0.917 x 10-2

Corrected Results

1000 25 0.9 x 10-2

Deductions from the analysis are as follows: (1) From the derivative plot, the effect of interference could be misinterpreted as constant pressure boundary effect. (2) Both analysis methods gave wrong answers, but the type-curve results are worse. This is because during the type curve analysis, every data point is given equal weighting while the conventional analysis considers only the segment of interest.

Concluding, we should not allow leaks and if there are serious interference effects, the effects must be accounted for before analyzing your data. For the real tests, all these factors may be present during one test and results obtained from analyzing the test will be wrong. We therefore advice that you avoid as much as possible any of the factors that may adversely affect BHP tests.

120

8. FIELD CASES In this chapter, we shall present some of the tests we have analyzed and problems encountered. We shall first present analysis of a test that was properly run to show that tests can actually be run properly. 8.1 Good Test A good test requires the following: (a) The gauges must be calibrated properly so that gauge readings will be reliable. (b) Accessories must be tagged for effective depth control. (c) Test programme must be correct, understood and followed.

Figures 8.1 and 8.2 show the readings of the upper and lower gauges for a good test. Table 8.1 is a worksheet that shows the temperature and pressure readings at different times during the test. From the worksheet we infer that the readings are consistent as the expected pressure differences between the gauges agree with the calculated pressure differences based on the fluid gradient between the gauges. The same deduction could be made from the temperature differences. ???

Fig. 8.1: Good Measurements with Upper and Lower Gauges During Buildup Period ??????

Fig. 8.2: Good Measurements with Upper and Lower Gauges During Entire Test Period Table 8.1: Worksheet For Gauge Quality Check General Information Field: Niger Top Gauge:WX1 Test Date: 01/01/95

Well: 1 Reservoir: E 1.0 Depth 7713 ft/CHH Bottom Gauge: WX2 Depth 7717 ft/CHH Gross (STB/D): 340 BSW (%): 0 GOR (SCF/STB): 1236

Quality Check Analysis

Differential Pressure Analysis Event

Time (hr)

Plower (psi)

Pupper (psi)

∆p (psi)

Fluid

Flowing Period Early Buildup Mid Buildup Late Buildup Static Stop

14.5

3082.69

3081.54

1.15

14.91

3240.73

3239.14

20.46

3246.29

26.43 27.82

Calculated ∆p (psi)

Difference in ∆p (psi)

oil

Pressure Gradient (psi/ft) 0.30

1.2

-0.12

1.59

oil

0.30

1.2

-0.92

3244.15

1.59

oil

0.30

1.2

-0.86

3245.73

3244.15

1.58

oil

0.30

1.2

-0.88

3005.19

3005.24

0.95

oil

0.30

1.2

-1.17

Average Difference in calculated ∆p (psi): - 0.722

121

Differential Temperature Analysis Event

Time (hr)

Tlower (°C)

Tupper (°C)

∆T (°C)

Fluid

Flowing Period Early Buildup Mid Buildup Late Buildup Static Stop

14.5

159.28

158.45

0.83

14.91

159.85

159.05

20.46

159.77

26.43 27.82

Calculated ∆T (°C)

Difference in ∆T (°C)

oil

Temp. Gradient (°C/m) 0.01

0.04

0.136

0.80

oil

0.01

0.04

0.045

158.74

0.83

oil

0.01

0.04

0.066

159.82

158.99

0.83

oil

0.01

0.04

0.034

156.15

155.39

0.76

oil

0.01

0.04

0.103

Average Difference in calculated ∆T ( C) : 0.083 Comments: The pressure gauges gave consistent readings. The average pressure offsets of -0.722 psi can be accepted as the pressure rise during the buildup was about 1565 psi. The pressure difference plot shows no evidence of phase segregation. This is clearly shown on the pressure difference plot in Fig 8.2.

Figures 8.3 to 8.5 show the plots used for the analysis. The derivative plot shows the wellbore storage phase, infinite-acting radial flow phase and late-time data influenced by the constant pressure boundary. Note that in this analysis, the pressure profiles simulated with the calculated results (solid lines) match the real data very well. In this test there were no leaks and fluid segregation effects.

???????

Fig. 8.3: Log-Log Plot of Data from Good Test ???????

Fig. 8.4: Semi-Log Plot of Data from Good Test

??????? Fig. 8.5: Simulation Plot of Data from Good Test 8.2: Effect of Gauge Movement From experience, moving gauges during flowtest prior to shutting in or during buildup renders tests useless. This is also true even if the test analyst knows that the gauges were moved. This implies that once you start the flowtest, do not move the gauges even if you have just realized that gauges are not at specified depth. Just state in your report the exact location of the gauges. On the other hand, if you must move the gauges (say to the perforations to minimize fluid interface movement) you must start the flowtest all over again on reaching the new depth.

122

Figure 8.6 shows an example in which gauges were moved prior to shutting in the well. The gauge was moved further down after the flowtest. In this case, the flow period at the new depth was not sufficient. ????

Fig 8.6: Gauge Movement Prior to Shutting in The problem with analyzing this test is that two different bottom hole flowing pressure before shut-in could be used in the analysis. The first bottom-hole flowing pressure is the one obtained prior to the gauge movement. Analysis plots obtained by using this bottomhole pressure are shown in Figures 8.7 and 8.8. Obtained results are shown in Fig 8.8. A second bottom-hole pressure before shut in that could be used in the analysis is the one prior to shutting in. That is, after the gauge was moved. The analysis plot for this case is shown in Fig 8.9. The second option looks more reasonable, but may be difficult to recognize. It may be regarded as an artefact as pressure has started building up. We recognize it because the BHP contractor was instructed to move the gauge after flowtest – a wrong instruction. An important issue in moving gauges prior to shutting in is that the correct shut-in pressure may not be used for the analysis. This is an important parameter in the calculation of skin and also in type-curve matching. Also, gauge movements can yield a “dome” that may be misinterpreted as wellbore storage phase as shown in Figure 8.8. This is because if you are running in the gauges at constant speed, pressure will vary linearly with time. During the wellbore storage phase, pressure also varies linearly with time. Therefore, do not complicate the analysis with unnecessary movement of the gauges. 8.3 Effect of Gauges off Depth We are not just interested in measured pressure, but we also want to know the depth where the pressure was measured. If depth is off by 50ft from datum in a reservoir that produces dry oil of gravity 0.35psi/ft, the error in datum pressure will be 17.5psi. This may seem small, but many reservoirs may not have that much depletion in one year. This implies that a 1-year error has been introduced in reserve calculations!

Depth control is important and that is why we insist on tagging accessories, marking wire, etc., but many operators still do not understand the importance. Poor depth control is a source of error in calculating wellbore fluid gradient. The equation for calculating the fluid gradient is as follows: Fluid Gradient =

P2 − P1 x 2 − x1

8.1

where P1 = Pressure measurement at Poin1 x1 = Vertical depth at Point 1 (along hole depth corrected for deviation) P2 = Pressure measurement at Poin2

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x2 = Vertical depth at Point 2 (along hole depth corrected for deviation) Figure 8.10 shows an example where we detected poor depth control. Both the upper and lower gauge readings are shown. After correcting for deviation, the calculated static gradients in Column 4 gave unrealistic results. For example, in some cases, a lighter fluid is found below a denser fluid. This is not physically possible and was likely caused by depth error. ?????? Leave a page

Fig. 8.10: Incorrect Static Gradient Due to Poor Depth Control The magnitude of the depth error can be estimated by introducing shifts in the depths until a satisfactory fluid gradient trend is obtained. Column 5 shows the shifts and the final fluid gradients are in Column 8. The shifts are considered as the suspected depth errors. This should be brought to the notice of the BHP contractor so that he will be careful with his depth measurements.

Also, leaks could cause faulty static gradients. This is shown in Fig 8.11. In this case, there is no remedy as both the numerator and denominator in Eq 8.1 may be affected. ?????? Leave a page

Fig. 8.11: Incorrect Static Gradient Due to Leak

We have observed some contradiction between the claimed gauge position and resulting effect on pressure. For example, with the gauge at mid perforation we do not expect any effect on pressure due to liquid interface movement.

8.4 Effect of Reporting Wrong Rates Flowrate data is as important as the pressure data. Without the flowrate data, permeability, the most important parameter, cannot be calculated from BHP test. The relationship between flowrate and permeability is given in Oilfield units by the following equation: K=

162.6qBµ mh

8.2

where: K = permeability, md q = flowrate, STB/day m = slope obtained from semilog plot of BHP data, psi/cycle B = formation volume factor rb/STB

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µ = Viscosity, cp h = drainage thickness, ft A consequence of using wrong rate is that permeabilities calculated from different tests on the same well will be inconsistent. This is not expected unless there is a process altering the permeability in the drainage area. Figure 8.12, shows the expected feature. The slope of the semilog plot is constant and therefore permeability is constant, but average pressure in the drainage area changes time. ????? Here

Fig 8.12: Consistency in Permeability and Analysis Rates Flowrate is also used in computer programs used in BHP analysis for calculating wellbore storage constant as shown for a buildup by the following equation: Cs =

qB∆t 24 ∆p

8.3

where: Cs = wellbore storage constant. ∆p = Pws-Pwf(tp) from unit slope of log-log plot of wellbore storage phase. ∆t = shut-in time corresponding to ∆p. The wellbore storage constant can also be estimated with equations that do not involve rate but depends on storage mechanism (fluid expansion, falling liquid, etc.), fluid compressibility in the wellbore, wellbore and connected volume. The wellbore storage constant calculated with these equations should agree within reasonable limits with wellbore storage constant calculated with Equation 8.3. In situations where they do not agree, it is likely that the wrong rate was used in the analysis. For example, in one of our analysis, the wellbore storage constant calculated using Equation 8.2 was 0.00176 STB/psi while estimated wellbore storage constant depending on whether tubing communicates with casing or not lies between 0.00556 to 0.043 STB/psi. This implies that the rate supplied for the analysis is off by a factor of at least three. In another case, we calculated a wellbore storage constant of 0.0071 STB/psi using Equation 8.3 while the estimated wellbore storage constant is 0.00724 STB/psi. In this case, we believe that the rate used for the analysis was correct. This could be used as a quality check for rate measurement if measured pressures are reliable. To overcome the problem of reporting the wrong rate, we recommend the following: 1. 2.

The BHP contractor should be interested in obtaining correct rate data as much as he is interested in obtaining good pressure data. The flowstation supervisors should cooperate with the BHP contractor and both should ensure that the test well is correctly hooked on to the test separator so that correct flowrate data will be obtained.

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3.

We are aware that the flowstation supervisor has scheduled dates for testing each well, but he should utilize the opportunity created by the BHP to perform a complete test on the test well. That is, obtain rate, BSW, GLR, etc.

8.5 Effect of Ineffective shut-in / Well not flowing before Shut-in If there are no speed limits, the state of many roads can be determined by how fast vehicles move on the roads. In the same manner, important reservoir parameters can be determined from BHP test from the rate at which pressure drops (as in drawdown) or rises (as in buildup) with time. This implies that during a good buildup test, pressure must rise. This will occur if the well was flowing and then effectively shut in.

The correct pressure rise during a buildup test can be achieved by flowing the well until stabilization and then completely shutting the well to flow. The implications of the last sentence are as follows: 1. During a buildup, the well must be flowed until stabilization. The stabilization condition is removed if well is surging. However, in that case, well must be flowed for at least 6 hours. 2.

If well cannot be completely shut-in due to faulty valve or wax or any other reason, the buildup part of the test should be aborted.

Table 8.2 shows the production data recorded by a BHP contractor before the well was shut for buildup. It is apparent from the flowrate data that the well was not flowing prior to shutting in. The little production he observed when he opened the well could have come entirely from the wellbore and not the reservoir. Do not forget that fluid in the wellbore is compressed and this could cause the wellbore fluid to be produced on opening the well initially. Table 8.2: Production Data Prior to Shut-in OIL TIME Meter Reading(bbl) Obser. Vol. (bbl) 12:45:00 205.70 67 13:00:00 210.00 4.30 13:15:00 248.40 38.40 14:00:00 336.70 2.90 14:15:00 355.00 18.30 14:30:00 372.50 17.50 15:30:00 391.70 0.00 15:45:00 391.80 0.10 16:30:00 391.90 0.00 16:45:00 391.90 0.00 17:00:00 391.90 0.00 17:15:00 391.90 0.00 18:15:00 391.90 0.00 18:30:00 391.90 0.00 The well was shut-in at 18:30:00

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Figure 8.13 is a pressure plot of a well that could not be shut in or the well was not flowing prior to shutting in. In this case, there was no pressure rise and the test is useless if the objective of the test is to calculate permeability and skin. ???

Fig 8.13: Ineffective Shutting in of Well It is easy to know whether a well flowed or could not be shut in during buildup test. In this case, the flowing gradients (misnomer for a dead well) and the static gradients are same. This is shown in Fig 8.14. Figure 8.15 is the static and flowing gradients for a well that was flowing and properly shut in during the buildup. Note that for this case, the static pressure at a given depth is higher than the flowing gradient at the same depth. ????? Leave a page Fig 8.14: Static and Flowing Gradients for a Well that is not Flowing

Fig 8.15: Static and Flowing Gradients for a Live Well In addition to our interest in ensuring that wells are effectively shut during buildup, we are equally interested in your shutting the well as fast as possible. Pressure profiles in Fig. 8.16 show a well that was slowly shut in. Slow shutting in of wells makes tests difficult or impossible to analyze. ??????

Fig 8.16: Slow Shutting in of Well 8.6 Effect of Leak Using the road analogy, a leak could be likened to a detour on the road. Such a detour will make it difficult to ascertain the condition of the road from the speed of the vehicles. Leaks make it impossible to analyze BHP tests because leaks cannot be quantified. Also, when there are leaks, calculated fluid gradients are wrong as shown in Fig 8.11.

During BHP tests, leaks could occur through (a) leaking lubricator. (b) communication between long string and short string or between a string and casing. This could occur through gas lift ports or non-sealing packers. (c) not completely shutting the well, etc. At early shut-in time (wellbore storage phase), the effect of leak may not be detected from pressure data because during that period, the rate at which pressure rises is usually greater than the rate at which it leaks. Therefore, a net pressure rise is observed. But, during the infinite-acting radial flow phase in formation with high permeability as in the Niger Delta, the rate at which the pressure rises will in most cases be lower than the rate at which pressure leaks. A net pressure drop is recorded and such a test cannot be

127

analyzed. Figure 8.17 shows pressure plots from a test with leak. Note that pressure difference between lower and upper gauge may be constant if there is leak. ?????

Fig 8.17: Pressure Drop Caused by Leak Figures 8.18 and 8.19 show the semilog and log-log for a case where there was an intial leak through the lubricator. The leak was stopped by pumping the lubricator before pressure started to rise again. It is always better to avoid such problems by using tested lubricators. ?????? page 107

Fig 8.18: Semilog Plot of Data with Initial Leak from Lubricator

??????

Fig 8.19: Log-log Plot of Data with Initial Leak from Lubricator In conclusion, make sure your lubricator is not leaking and that wells are completely shut in. Leaks due to communication between strings and casing can be detected by recording the surface pressures of all the strings and casing using a surface chart pressure recorder. ??? Figures 8.20 and 8.21 show the pressure chart using in recordind surface pressure for two cases. Figure 8.20 shows that there was communication between the test string (Well 15S) and casing during the test. Pressure from this string merged with pressure in the casing after a while. Fig 8.21 is an example of a case with no communication between test string and casing. The pressures from both strings and casing followed their own trend. We recommend the use of these charts. ??? Leave a page Fig. 8.20: Surface Pressures of Short String, Long String and Casing (Case with Communication)

??? Leave a page Fig. 8.21: Surface Pressures of Short String, Long String and Casing (Case with no Communication)

??????????? 8.7 Effect of Gauge Oscillations/Sensitivity Problems We have observed strange oscillations in pressure data measured with electronic gauges. In most cases, the oscillations occur in both gauges as shown in Fig. 8.22. Definitely, such oscillations are neither due to well nor reservoir responses. We believe that the oscillations may be due to the procedure used while running the test. Example of procedures that could produce such oscillations include “jarring” the gauges and allowing the gauges hit objects in the well. The oscillations could also be caused by electronic problems that we do not understand.

?????? 128

Fig 8.22: Gauge Oscillations

Another factor that could cause “milder” oscillations is poor gauge sensitivity or temperature effects. This is shown in Fig. 8.23. The gauge gives readings that are about the correct value. An example of the effect of such oscillation on pressure data graphed on Horner plot is shown on Fig. 8.24. These oscillations cause discontinuities on the pressure derivative as shown in Fig. 8.25. The implication of this is that the different flow phases may not be clearly discerned.

????? Fig 8.23: Gauge Oscillations Due to Sensitivity Problem

???? Fig 8.24: Effect of Gauge Oscillations on Horner Plot

???? Fig 8.25: Effect of Gauge Oscillations on Pressure and Derivative on Log-log Plot

Depending on when and how the oscillations occur, oscillations can make test analysis impossible. Also, in cases where BHP tests with oscillations in data are analyzed, the oscillations cause a large band of uncertainty for the calculated parameters. 8.8 Effect of Gas Phase Segregation Liquid phase is usually produced with dissolved gas. Hence when a well is shut-in, the dissolved gas comes out of solution and moves to the top of the wellbore creating an unusual pressure rise (or “hump”) in the wellbore. This pressure “hump” is illustrated in Fig. 8.26 and Fig 8.27. ???? Fig 8.26: Gas Phase Segregation “Hump” on a Semilog Plot ????

Fig 8.27: Gas Phase Segregation “Hump” on a Simulation Plot Gas phase segregation is more common in wells with high gas-oil ratio(GOR) and can be identified using the following: (a) Pressure difference (i.e. lower gauge pressure minus upper gauge pressure) between the gauges increases with time because as gas comes out of solution, liquid between the gauges become denser.

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(b) The pressure changes due to phase segregation cause a “dip” in the pressure derivative. The dip in the pressure derivative may be erroneously interpreted as double porosity feature. Figure 8.28 illustrates this.

Dip due to Phase Segregation Dip due to Double Porosity

Log ∆p’

Log ∆t

Log ∆t

Fig: 8.28: Pressure Derivatives Showing Dips Due to Phase Segregation and Double Porosity Figure 8.29 shows an example of a real case of the effect of gas phase segregation on pressure derivative.

?????? Fig 8.29: Effect of Gas Phase Segregation on Pressure Derivative (c) On semilog plot of the pressure data, phase segregation effects cause a “hump”. This is shown on Fig 8.26. Generally, gas phase segregation is unlikely to prevent correct interpretation of test because the effect dies down quickly.

8.9 Effect of Liquid Interface Movement Quite like gas, oil and water will separate in a well that is shut in. Depending on the position of the gauges and amount of water produced, recorded pressure data can be affected by oil/water interface movement. The oil/water interface can fall below the gauges or rise above the gauges. A schematic from Lingen (1995) showing the effect on pressure for case of rising and falling contacts are in Figs. 8.30 and 8.31. Figure 8.30 shows a case where the oil/water contact was initially below the gauges and later rose above the gauges. In this case, pressure builds up, drops and then rises again. This is likely to occur in wells producing with BS&W of about 45% or more if the gauges are not close to the perforation. Figure 8.31 shows a case where the oil/water contact was initially above the gauges and letter dropped below the gauges. The pressure rises normally and then more rapidly before the normal pressure rise continues. This is more likely to occur in wells with gas with the gas pushing down the oil/water contact.

?????

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Fig 8.30: Effect of Rising Oil/Water Contact ????? Fig 8.31: Effect of Falling Oil/Water Contact

These effects can be explained by the fact that the density of the fluid in the well and between the gauges affect the rate of pressure change during buildup. Unlike the case of gas phase segregation, pressure distortion caused by liquid interface movement can last for a much longer time and therefore can make the test uninterpretable. Figures 8.32 and 8.8.33 show examples of a real case of rising liquid interface. Figure 8.32 is on a Cartesian plot while Fig. 8.33 is the same phenomenum on a semilog log plot. Liquid interface movement is an unwanted phenomenon. To eliminate or minimize it, the gauges must be at the perforations. This is especially important if the BS&W is high (BS&W > 30% - not a magic number).

???? Fig 8.32: Liquid Interface Movement on a Cartesian Plot ???? Fig 8.33: Liquid Interface Movement on a Semilog Plot

8.10 Effect of Gaslift Distortions on pressure data caused by gaslift effects are unpredictable. Intermittent opening and closing of gaslift valves during the buildup cause the distortions. The distortions could be worse if the gas supply to the well was not turned off.

Distortions caused by gaslift malfunction make flow phase indetification difficult. Hence analysis will yield wrong results. Figures 8.34 and 8.35 show two interpretations of a test with gaslift distortion. The log-log and semilog plots used in the interpretation are shown. Figures 8.34a and 8.34b show one form of interpretation and Fig 8.35a and 8.35b show another form of interpretation of the same test. Results from both interpretations are also shown. Note that the gaslift effects makes it difficult to simulate the correct pressure profiles and result obtained from analyzing such test may depend on the assumed profile. ????????

Fig 8.34 a and b Fig 8.35 a and b

8.11 Effect of Short Buildup or Flow Period We have discussed the different flow phases that a well that is shut-in goes through. We also discussed the information that may be derived from each phase. For example,

131

accurate values of skin and permeability are derived from the infinite-acting radial flow phase. In this section, we present results showing the effect of test duration. In one case, the test was run long enough to reach the infinite acting radial flow (IARF) phase while in another case, the some test data were removed to simulate a case where the test was not run enough to reach (IARF) phase. The two cases were analyzed and the results compared. In the first case where the (IARF) phase was reached, analysis was easy. Both conventional and automated type-curve methods could be used. Figure 8.36a and b show the log-log and semilog plots used in the analysis. Figures 8.37a and b show plots used in the analysis in the case where the (IARF) phase was not reached and the the skin constrained. That is, the skin was not varied during the autmatic type-curve matching. Figures 8.38a and b show plots used in the analysis in the case where the (IARF) phase was not reached and the skin not constrained. That is, the skin was varied during the autmatic type-curve matching. Results obtained from analyzing the different cases are shown in Table 8.3. Table 8.3: Effect of Buildup Period on Calculated Results. Parameter Calculated Results Long Shut-in Short Shut-in Case 1 Case 2 (contrained skin) (uncontrained skin) Permeability, md 1840 708 2090 Skin 32.4 9.09 37.8 Cs, STB/psi 0.0248 0.0237 0.025

Figure 8.36 to 8.38 here ?????? Deductions from the analysis are as follows: 1. Tests run long enough to reach a clearly defined infinite-acting radial flow phase will yield correct values of permeability and skin. 2. For a test that is not run long enough (e.g. infinite-acting radial flow phase not reached) the calculated permeability and skin will be wrong irrespective of the analysis method. 3. In these cases where the wellbore storage phase was observed, the calculated wellbore storage constants are close. 4. The semi-log plots show that if the infinite-acting radial flow phase is not reached test cannot be analyzed using the conventional method. In addition to our being interested in the duration of the buildup period, we are also interested in the duration of the flow period prior to shutting in of well. A flow period of about 2 to 6 hours (good for Niger Delta high permeability formations) with the gauges at the survey depth where the well will be shut is necessary for the following reasons:

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1. To ensure that there is a flow period that can be used for matching simulated profile with real data. 2. To ensure that the correct flowing pressure at shut-in is used. This affects calculated skin and productivity index. 8.12 Class Discussion It is not possible to present and discuss every type of problem encountered in BHP test and analysis. In some cases, we are not able to figure out what caused the problem. In such cases, we give probable cause of the problem.

In this section, we shall present some cases for discussion. Every one will participate in trying to find out what happened during the tests or the characteristics shown by the test data. Case 1: Figures 8.39a and b are the log-log and semilog plots of a test we regard as a mystery test. The mystery if the sudden rise in pressure of about 300 psi that occurred at about 3.00 am while the well was shut for overnight. The gauge depth was less than 150 ft from the maximum well depth. Now explain what caused the problem.

???

Fig 8.39: Log-log and Semilog Plots of the Mystery Test

Case 2: Figures 8.40a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is.

?????

Fig 8.40: Log-log and Semilog Plots of a Test

Case 3: Figures 8.41a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is.

?????

Fig 8.41: Log-log and Semilog Plots of a Test Case 4: Figures 8.42a and b are the log-log and semilog plots of a test with a problem. Explain what the problem is. Varying kh ?, Composite system? Etc.

?????

Fig 8.42: Log-log and Semilog Plots of a Test

Case 5 : Figures 8.43 shows the log-log plot of a test with ran with an Amerada gauge. Explain the trend.

133

?????

Fig 8. 43: Log-log of a Test ran with Amerada Gauge

Case 6 : Figures 8.44 shows the Cartesian plot of a test with a final shut-in pressure that is lower than the flowing pressure. Suggest the cause of the problem.

?????

Fig 8.44: Cartesian Plot of a Test with a Problem

Case 7 : Figures 8.45a and b show the semilog plots from tests in two adjacent wells draining the same reservoir. The permeability calculated from the semilog plots differ by a factor of about 10. Explain the big unrealistic difference.

Fig 8.45: Semilog Plots of Tests in Adjacent Wells Draining the same Reservoir One deduction from this case is the need for system approach to welltest analysis. Tests from wells in the same reservoir must be compared to determine unrealistic results. This comparison will help show the local (well) and global (reservoir) effects.

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9. THEORETICAL CONCEPTS We have given a lot of equations without showing how they were obtained. Readers who are mathematically inclined will like to know the origin of the equations. Therefore, for completeness, we decided to include a chapter on theoretical concepts of bottom-hole pressure test analysis. In this chapter, we shall derive some of the equations in Darcy units, outline method of prediction pressure for test design and throw a little light into the concept of superposition which forms the basis for generating multi-rate flow equations. 9.1 DERIVATION OF FLUID FLOW EQUATIONS To understand and correctly analyze well test data require:

(a) Understanding of physical processes involved. For example, the fluid flow processes the effect of reservoir geometries and heterogeneties. (b) Selection or production of proper mathematical equations or models for analysis of results. To satisfy conditions (a) and (b) will require understanding the mathematics of fluid flow in porous medium. Discussion on the derivation of some of the fluid flow equations follow. The basic concepts involved in the derivation of fluid flow equations are: (a) Conservation of mass equation (b) Transport rate equation (e.g. Darcy’s law) (c) Equation of state The use of these basic concepts in derivation of liquid and gas flow equations in porous medium is discussed. Flow in cylindrical coordinates is considered with flow in angular and z-directions neglected. The equations are given as follows: 9.1.1 Liquid Flow Equations A. Conservation of Mass Equation:

Mass rate in - Mass rate out = Mass rate storage B.

9.1

Transport Rate Equation q = -

k dp A µ dr

9.2

By conserving mass in an elemental control volume shown in Fig. 9.1 and substituting the transport rate equation, the following equations can be obtained:

135

 2πrhk ∂p   2πrhk ∂p  ρ  − ρ  r = − ∂r  ∂r   µ  µ

r + ∆r

+ 2 πr ∆r h

∂ (ρ∅) ∂t

9.3

Expanding the first term on the right hand side of Eq. 9.3 about “r” using Taylor’s series expansion gives: 1 ∂ rkp ∂p ∂ { } = (ρ∅) r ∂r µ ∂r ∂t

9.4

r r+∆

r r+∆ Figure 9.1: Schematics of the Reservoir and Control Volume. Equations 9.1 to 9.4 apply to both liquid and gas. For each fluid, the density or pressure term in Eq. 9.4 may be replaced by the correct expression in terms of pressure or density. For slight but constant compressibility liquid, ρ = ρsc e

c[P - psc ]

9.5

Substituting for pressure in Eq. 9.4, the final equation in terms of density is: ∅µ (c + c r ) ∂p ∂2 p 1 ∂p + = 2 ∂r r ∂r k ∂t

9.6

In terms of pressure, Eq. 9.4 is: (c + c r ) ∂p ∂2 p 1 ∂p ∂p + + c( ) 2 = ∅µ 2 ∂r r ∂r ∂r k ∂t

9.7

If pressure gradient is small everywhere in the system, Eq. 9.7 becomes: ∅µC t ∂p ∂2p 1 ∂p + = 2 r ∂r k ∂t ∂r

9.8

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Equation 9.8 is the diffusivity equation. Some deductions about the form of the derived equation are: 1.

The partial differential equation in terms of density is linear (Eq. 9.6).

2.

The partial differential equation in terms of pressure (Eq. 9.7) is non-linear but if the pressure gradient is small, the resulting equation (Eq. 9.8) becomes linear.

The assumptions inherent in the derivation of the diffusivity equation are as follows: (a) The formation is homogeneous, horizontal and of uniform thickness. (b) Flow is radial, with gravity and capillary effects negligible. (c) The fluid is considered to be of slight and constant compressibility. (d) The pressure gradient in the reservoir is considered to be small. (e) No reaction between fluid and formation matrix occurs. (f) Reservoir is at isothermal condition. (g) Flow is laminar. 9.1.2 Gas Flow Equations Equations 9.1 to 9.4 also hold for gases. In addition, the equation of state for gases is:

PM ZRT

ρ =

9.9

For real gases, isothermal compressibility, c, is defined as: 1 1 dz  c=  z dp  T P

9.10

For ideal gases, c=

1 P

9.11

Combining Eq. 9.4, Eqs. 9.9 and 9.10, the different forms of gas flow equations will be: A.

Pressure-Squared Form (p2 Form)

∇. [

kpM ∅cpM ∂p ]∆p = µZRT RT ∂t

9.13

Differentiating Eq. 9.13 and assuming that permeability is constant gives: 1 1 ∇2 p2 µZ µZ

2

d (µZ) ∅c ∂p 2 2 2 ( ∇ p ) = dp 2 kZ ∂t

9.14

Hence, 137

∇2 p2 -

d ∅µc ∂p 2 2 2 [ ln ( µ Z)] ( ∇ p ) = dp 2 k dt

9.15

Under certain conditions, Eq. 9.15 simplifies to : ∇2p2 =

φµc ∂p 2 k

9.16

dt

Equation 9.16, which is the commonly used p2 - form of gas flow equation follows from Eq. 9.15 and assuming that the pressure gradient is small everywhere in the system. For ideal gases, Eq. 9.16 holds but the assumption of small pressure gradient is not necessary. This is because for ideal gases, the gas compressibility factor z is unity and ideal gas viscosity is not dependent on pressure. The viscosity of ideal gas only depends on temperature. Hence, the second term on the left-hand side of Eq. 9.15 will always be zero. B. Pseudo -Gas Pressure Form (m (p) Form) Equation 9.13 still holds. That is

∇. [

1 ∅c ∂p 2 ∇p 2 ] = µz kZ ∂t

9.13

The pseudo-gas pressure is defined as: p

m(p) = ∫ Pp

2pdp µz

9.17

Thus, ∆m (p) =

1 ∆p 2 µz

9.18

Substituting Eq. 9.18 into Eq. 9.13 gives: ∆2 m(p) =

9.19

Equation 9.19 holds for both real gas and ideal gas with no assumptions on the magnitude of pressure gradient term. Equation 9.19 appears linear but it is not as µ and z are dependent on pressure. The resemblance between the gas flow equation (Eq. 9.19) and liquid flow equation (Eq. 9.8) form the basis of adapting liquid flow solution to gas flow problems. Summary of the equations of state and resulting flow equations are give in Table 9.1.

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Table 9.1: Summary of Flow Equations

Continuity Equation:

∂ 1 ∂ rk ∂p { ρ } = ( ρ∅ ) r ∂r µ ∂r ∂t

Equation of State

Resulting Flow Equation

Liquid

ρ = ρ sc e c[P-P

sc ]

∇2 p =

∅µct ∂p k ∂t

Gas

ρ =

∇2 p2 -

PM ZRT

d d2 p

∇ 2 m(p) =

[lnµZ] (∇ 2 p) =

∅µct ∂p 2 k ∂t

∅µc ∂ m(p) k ∂t

9.1.3 Relationship between m(p), p2 and p

The real gas pseudo-pressure is defined as: p

ψ = m(p) = ∫

Pb

2pdp µz

9.17

At low pressure (p < 2000 psi) µz = µi zi = constant Hence,

9.20a

P2 m (p) = µi zi

9.20b

for Pb = 0 Substituting Eq. 9.20 into Eq. 9.19, it can be shown that the m(p) form of the gas equation is exactly the same as the p2 form at low pressure. At high pressure (p > 3000 psi)

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Pi p = µz µi zi

9.21

Hence m(p) =

2 pi P µi zi

9.22

for Pb = O Substituting Eq. 9.22 for m(p) into Eq. 9.19, it can be shown that the m(p) form of the equation degenerates to the p-form. That is, the gas flow equation will be same as the liquid flow equation (diffusivity equation). The relationships in Eqs. 9.20a and 9.21 are shown in Figure 9.2. Note that in Figure 9.2, µz the assumptions that µz is constant at low pressure and is constant at high pressure P may be acceptable for practical purposes.

140

Figure 9.2: Variation of Gas Deviation Factor-Viscosity Product for a Real Gas (after Wattenberger and Ramey, 1968) Calculation of Real Gas Potential, m(p) The m(p) can be calculated graphically or read from Tables. The graphical method requires that p, µ, z be given and m(p) calculated from area under a curve. This is illustrated below for a case where the trapezoid method is used in calculating the area under the curve. Table 9.2: Graphical Method of Calculating m(p) P µ z Area m(p) 2P/(µz) P0 µ0 z0 2P/(µz)0 = Y0 P1 µ1 z1 A1 = 0.5(Y0 +Y1)( P1-P0) A1 2P/(µz)1 = Y1

141

P2 P3

µ2 µ3

z2 z3

2P/(µz)2 = Y2 2P/(µz)3 = Y3

A2 = 0.5(Y1+Y2)( P2-P1) A3 = 0.5(Y2 +Y3)( P3-P2)

A1 + A2 A 1 + A 2+ A 3

Pn

µn

zn

2P/(µz)n = Yn

An = 0.5(Yn-1 +Yn)( Pn-Pn-1)

A1 + A 2+ A3+…+ An

The graphical method is illustrated in Figure 9.3.

2p/(µz) m(p )

Pb

p

Figure 9.3: Graphical Method of Obtaining m(p)

Tables 9.3 and 9.4 can be used in obtaining m(p). Note that the data in the tables were generated using some assumed typical gas properties. Both tables give reduced pseudopressure defined as: Ψr =

Ψµ1 2 Pc2

9.22a

where ψ µ1 Pc

= m(p) = initial viscosity = critical pressure

From Eq 9.22a, m(p) can be calculated. The differences between values in both tables are simply due to the use of different bases. Table 9.3 uses reduced pressure of 0.2 as base while Table 9.4 uses reduced pressure of 0 as base. However, the values can be reconciled with the knowledge that

142

p

p

0.2

0.2

0

0

∫y= ∫y− ∫y

9.22b

Note that at low pressure, m(p) can simply be calculated with Eq 9.20b.

143

Table 9.3: Reduced Pseudo-Pressure Integral (Ψr) With 0.2 as Base PseudoReduce Values of integral for Pseudo-Reduced Temperature Tpr of d Pressur e Ppr 1.05 1.15 1.30 1.50 1.75 2.00 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 14.00

0.0257 0.0623 0.1102 0.1698 0.2418 0.3264 0.4236 0.5326 0.6546 0.7903 0.9484 1.1444 1.3671 1.5828 1.7924 1.9959 2.1926 2.3821 2.5649 2.7424 2.9147 3.0825 3.2464 3.4066 3.5633 3.7169 3.8679 4.0165 4.3788 4.7278 5.0653 5.3938 5.7144 6.0276 6.3347 6.6368 -

0.0229 0.0553 0.0971 0.1485 0.2105 0.2835 0.3678 0.4631 0.5691 0.6855 0.8126 0.9503 1.0980 1.2546 1.4191 1.5883 1.7595 1.9321 2.1071 2.2841 2.4619 2.6399 2.8172 2.9937 3.4683 3.3403 3.5094 3.6766 4.0876 4.4874 4.8766 5.2579 5.6367 6.0088 6.3897 6.7235 7.0706 7.4124 7.7495 8.0821 8.4099 8.7330 9.0520 9.3670 9.6786 9.9876 10.2936 10.5963 11.1935 -

0.0198 0.0477 0.0839 0.1283 0.1810 0.2419 0.3111 0.3889 0.4755 0.5707 0.6734 0.7838 0.9020 1.0277 1.1606 1.3001 1.4457 1.5966 1.7526 1.9138 2.0791 2.2473 2.4186 2.5935 2.7710 2.9504 3.1320 3.3153 3.7771 4.2400 4.7052 5.1693 5.6277 6.0822 6.5308 6.9714 7.4044 7.8304 8.2497 8.6632 9.0711 9.4731 9.8703 10.2635 10.6531 11.0398 11.4223 11.7998 12.1731 12.5433 12.9102 13.2735 13.6340 13.9925 14.3483 14.7011 15.3996 16.0892 16.7703 17.4427 18.1069 18.7642 19.4147 20.0588 20.9676 21.3318

0.0170 0.0409 0.0716 0.1091 0.1532 0.2037 0.2608 0.3246 0.3954 0.4734 0.5579 0.6484 0.7449 0.8473 0.9558 1.0703 1.1906 1.3464 1.4474 1.5838 1.7253 1.8712 2.0214 2.1758 2.3341 2.4957 2.6612 2.8308 3.2685 3.7223 4.1897 4.6678 5.1539 5.6459 6.1412 6.6377 7.1355 7.6343 8.1338 8.6336 9.1326 9.6297 10.1249 10.6185 11.1091 11.5957 12.0794 12.5615 13.0416 13.5194 13.9939 14.4644 14.9322 15.3980 15.8609 16.3205 17.2313 18.1318 19.0212 19.8976 20.7640 21.6238 22.4762 23.3216 24.1596 24.9921

0.0145 0.0348 0.0609 0.0927 0.1303 0.1734 0.2221 0.2763 0.3358 0.4004 0.4702 0.5452 0.6255 0.7114 0.8025 0.8983 0.9988 1.1042 1.2146 1.3293 1.4498 1.5744 1.7034 1.8370 1.9751 2.1169 2.2626 2.4123 2.8038 3.2178 3.6504 4.0997 4.5638 5.0406 5.5280 6.0234 6.5252 7.0320 7.5449 8.0622 8.5836 9.1085 9.6364 10.1665 10.6973 11.2279 11.7587 12.2897 12.8211 13.3532 13.8858 14.4187 14.9513 15.4834 16.0146 16.5447 17.6030 18.6590 19.7090 20.7507 21.7858 22.8166 23.8434 24.8616 25.8642 26.8596

0.0126 0.0303 0.0530 0.0807 0.1132 0.1505 0.1927 0.2397 0.2915 0.3483 0.4098 0.4758 0.5461 0.6209 0.7001 0.7840 0.8724 0.9653 1.0624 1.1636 1.2687 1.3777 1.4904 1.6068 1.7268 1.8504 1.9778 2.1094 2.4534 2.8178 3.2016 3.6049 4.0268 4.4663 4.9203 5.3860 5.8621 6.3472 6.8412 7.3442 7.8551 8.3739 8.8993 9.4298 9.9647 10.5034 11.0452 11.5897 12.1377 12.6897 13.2440 13.7993 14.3558 14.9128 15.4700 16.0274 17.1463 18.2662 19.3931 20.5120 21.6135 22.7156 23.8144 24.9057 25.9948 27.0862

2.50 0.0100 0.0241 0.0421 0.0640 0.0898 0.1194 0.1529 0.1902 0.2312 0.2761 0.3248 0.3773 0.4335 0.4932 0.5566 0.6235 0.6940 0.7679 0.8454 0.9264 1.0111 1.0994 1.1912 1.2862 1.3846 1.4864 1.5915 1.6998 1.9849 2.2896 2.6119 2.9516 3.3077 3.6788 4.0649 4.4664 4.8825 5.3130 5.7575 6.2150 6.6844 7.1643 7.6544 8.1543 8.6633 9.1808 9.7064 10.2398 10.7812 11.3308 11.8872 12.4497 13.0182 13.5926 14.1700 14.7499 15.9178 17.0928 18.2738 19.4614 20.6575 21.8627 23.0724 24.2820 25.4964 26.7197

3.00 0.0083 0.0200 0.0349 0.0532 0.0747 0.0993 0.1271 0.1580 0.1920 0.2292 0.2695 0.3129 0.3594 0.4090 0.4616 0.5173 0.5760 0.6378 0.7025 0.7700 0.8401 0.9144 0.9907 1.0700 1.1522 1.2373 1.3232 1.4159 1.6550 1.9144 2.1841 2.4731 2.7782 3.0994 3.4357 3.7865 4.1511 4.5286 4.9194 5.3241 5.7413 6.1699 6.6104 7.0633 7.5283 8.0049 8.4921 8.9884 9.4932 10.0062 10.5281 11.0583 11.5962 12.1421 12.6952 13.2545 14.3923 15.5560 16.7372 17.9315 19.1388 20.3556 21.5858 22.8246 24.0719 25.3268

144

Table 9.4: Reduced Pseudo-Pressure Integral (Ψr) as a Function of Tr and Pr.

145

9.2 SOLUTION OF LIQUID FLOW EQUATIONS Solutions of the liquid flow equation (diffusivity equation) are discussed in this section. Three cases - steady state, pseudo-steady state and the unsteady state (transient state) are considered. Conditions under which they are attained are also given. 9.2.1 Steady State Solution Steady state condition is attained when the rate of change of pressure is zero in the entire reservoir. That is,

∂p =0 ∂t

9.23

The condition expressed in Eq. 9.23 can be attained in a reservoir undergoing depletion if there is influx of some material (e.g. water) at the external boundary to recharge the reservoir and hence prevent the pressure from dropping as a result of the production at the well. This implies that for a reservoir to attain steady state condition, its boundary must be open to flow. However, the entire boundary does not necessarily need to be open to flow. As long as the water influx from any part of the boundary is strong enough to keep the pressure in the reservoir from changing, the system will attain steady state condition. The diffusivity equation, inner and outer boundary conditions for the steady state case are given as: Diffusivity Equation

∂ 2 p 1 ∂p 1 ∂ ∂p + = (r )=0 2 ∂r r ∂r r ∂r ∂r

9.24

Inner Boundary Condition qµ  r dp   dr  r = 2 πkh = constant w

9.25

Outer Boundary Condition

p = pe at r = re

9.26

The solution to Eq. 9.24 subject to the inner boundary condition is: p - pw =

qµ r ln 2 πkh rw

9.27

Applying the condition at the outer boundary (r = re, p = pe), Eq. 9.27 becomes: r qµ Pe - Pw = ln e 2 πkh rw

9.28

146

Rearranging, Eq. 9.28 gives 2πkh [Pe - Pw ] q = r µ e ln r w

9.29

The volumetric average pressure for any system is given as follows: VT

∫

p =

pdV

o

9.30

VT

where V = π(r 2 - rw2 )h∅

9.31

VT = π(r2e - rw2 )h∅

9.32

And

From Eq. 9.13, dV = 2πrh∅dr

9.33

We can use Eqs. 9.30 to 9.33 to determine the expression for average pressure for a system at steady state. Substituting Eqs. 9.32 and 9.33 into Eq. 9.30 and integrating gives: p = pw +

rw2 qµ  re2 1 1n r 1n r 1n rw -  w w  2 2 2 2 2 πkh  re - rw re - rw 2

9.34

For re >> rw, Eq. 9.34 becomes p = pw +

r 1 qµ [1n e - ] 2πkh rw 2

9.35

Rearranging Eq. 9.35, q =

2 πkh [p - p w ] r 1 µ[1n e ] rw 2

9.36

147

=

2π kh [p - p w ] r µ [ 1n 0.606 e ] rw

9.37

Combining Eqs. 9.28 and 9.36 gives: p e - p = (p e - p w ) - (p - p w ) =

qµ 4 πkh

9.38

Equation 9.38 shows the relationship between the pressure at the external boundary and the average pressure for a system that attained steady state. Equation 9.37 is not a very useful equation because of the difficulty in obtaining the average pressure. When a well that attains steady state is shut in, the pressure does not build up to the average pressure. The pressure builds up to the initial pressure because of influx from contiguous aquifers. And for a system that attains steady state, pe = pi. 9.2.2 Pseudo-Steady State Solution A reservoir attains pseudo-steady state if the rate of pressure decline with time is a constant. At that state, the mass rate of production is equal to mass rate of depletion. Further discussion on pseudo-steady state follows. Pseudo-steady state is also referred to as semi-steady state, quasi-steady state and in some cases, wrongly referred to as steady state.

The conditions necessary for this state to be attained are as follows: (a) Reservoir outer boundary must be closed to flow. (b) Well producing from finite drainage volume due to long production time as shown in Figure 9.4. The finite drainage volume will be in proportion to the production rate of that particular well in the drainage area. That is, a well producing at a rate of 2q will drain twice the drainage volume of a well producing at rate q.

q

2q

q Figure 9.4: Reservoir Depletion under Pseudo-Steady State Mathematically, at pseudo-steady state condition, ∂p dp = = constant ∂t dt

9.39

148

Ideally, only reservoirs with closed outer boundaries should attain pseudo-steady state but partially closed reservoir with very limited influx may attain a state, which for practical purpose may be considered to be pseudo-steady state. 9.2.2.1 Pseudo-Steady State Equation Equations for pseudo-steady state are given in this section. First, the value of the constant ∂p ( = constant) is determined. Secondly, the solution to the pseudo-steady state ∂t equation is given and lastly, the expressions for the volumetric average pressure and other related equations are derived.

A.

Derivation of Value of Constant

When pseudo-steady state condition is attained, Mass production rate = rate of mass depletion That is, qρw = − VT

d (ρ∅) dt

= − VT ρ ∅ c t

9.40

dp dt

Substituting for the total volume gives: q ρ w = - π[re2 - rw2 ] h ∅ ρ c t

dp dt

9.41

Therefore, dp -q = 2 dt ∅π (re - rw2 ) h c t

ρw ρ

From the equation of state for a constant compressibility fluid ρw = e -c [p - p w ] ρ

9.42

Substituting Eq. 9.42 into Eq. 9.41 gives dp dt

=

- q -c [ p - pw ] e 2 ∅π [r - rw ] h c t

9.43

2 e

149

For liquid of small compressibility, e

− c [ p - pw ]

≅ 1

Hence dp dt

=

=

- q ∅π [r - rw2 ] h c t 2 e

=

∂p ∂t

9.44

−q = constant Vc t

9.45

Equation 9.45 is the basis of reservoir limit test. This equation implies that if the pressure-time data obtained during pseudo-state is graphed on a Cartesian paper, a straight line whose slope is related to the well’s drainage volume is obtained. Equation 9.45 can be derived from a volume balance (a degenerate form of mass balance, see Craft and Hawkins, page 286), but such derivation does not explicitly show that an assumption was made about the fluid compressibility.

B.

Pseudo-Steady State Solution

The governing diffusivity equation is given by Eq. 9.8. At pseudo-steady state, the rate of pressure change is given by Eq. 9.44. Substituting Eq. 9.44 into Eq. 9.8, the diffusivity equation at pseudo-steady state becomes: 1 ∂ r∂p - qµ ( ) = r ∂r ∂r πkh (re2 - rw2 )

9.46

Using the inner boundary condition given by Eq. 9.25 and the fact that the pressure at any point, r, in the reservoir is p, the solution to Eq. 9.46 is ( P - Pw ) =

qµ  1 ( r 2 - r 2 + r 2 ln r / r  ) e w 2 2 2 πkh (re - rw )  2 w 

9.47

For re >> rw, Eq. 9.47 becomes P - Pw =

qµ 1 2 2 [ln r / rw r / re ] 2 πkh 2

9.48

When r = re, p = pe, then Eq. 9.48 becomes Pe - Pw =

r qµ 1 [ln e - ] 2 πkh rw 2

9.49

150

Hence q =

2 πkh [p e - p w ] r 1 µ [ln e - ] rw 2

C.

9.51

Volumetric Average Pressure

Using the same principles as in steady-state case, the volumetric average pressure when pseudo-steady state condition is attained is: p = pw +

+

rw2 qµ [ - re2 ln rw ] 2 πkh [re2 - rw2 ] 2

re2 qµ 2 1 2 {r ( r ln r e e e πkh [re2 - rw2 ]2 2 4

−

r2 1 2 1 rw ln + w ) − (re4 − rw4 )} 2 4 8

9.52

For re >> rw, Eq. 9.52 becomes p = pw +

r 3 qµ [ln e - ] 2 πkh rw 4

9.53

Hence q =

2 πkh [p - p w ] r 3 µ [ln e − ] rw 4

2 πkh [p - p w ] r µln 0.472 e rw Some deductions from the pseudo-steady state equation are: =

9.54

1.

The wellbore pressure pw, average pressure p , and the pressure at the external boundary Pe in Eq. 9.54 are dependent on time. This is not explicit in Eq. 9.54.

2.

When a system with closed outer boundaries is shut-in, the pressure builds up to the average pressure. Hence Eq. 9.54 is a useful equation.

3.

Note the similarities between Eq. 9.37 derived for steady state case and Eq. 9.51 derived for the pseudo-steady state case. The equations are summarized in Table 9.5.

151

4.

The term 0.472re in Eq. 9.54 is usually replaced with rd called the drainage radius. This is a misnomer because it gives the impression that just a part of the reservoir is being drained. Actually, the whole reservoir is drained.

Table 9:5: Summary of Stabilized Flow Equations in Darcy Units Pseudo-Steady State (re >>r)

Steady State

General relationship between p and r p − pw =

p − pw =

qµ r ln 2 πkh rw

qµ r r2 [ln ] 2 πkh rw 2 re2

Flow equations when p = pe and r = re pe − pw =

r qµ ln e 2 πkh rw

pe − pw =

qµ  re 1 -  ln 2 πkh  rw 2

p - pw =

qµ 2 πkh

Flow equations in terms of reservoir average pressure p - pw =

D.

qµ 2 πkh

 re 1 -  ln 2  rw

 re 3 -  ln 4  rw

Other Useful Pseudo-Steady State Equations

By combining the pseudo-steady state equations, the relationship between the reservoir average pressure, the initial pressure Pi and the pressure at the external boundary Pe during pseudo-steady state are obtained as follows: qt Pi - p = h ct 9.55a ∅π[re2 - rw2 ] pi – p

= (pi - p ) + ( p - pw) - (p - pw) r  qµ = [2  w  2 πkh re 

2

r 3 kt 1 r 2 + ln e - + ( )e ] 2 ∅µc t rw r 4 2 r

(pe - p ) = (pe - pw) - ( p - pw) =

9.55b

qµ 8πkh

9.56

152

pi - pe = (pi - p ) - (pe - p )

=

qµ 2kt [ 2 πkh ∅µc t re2

-

1 ] 4

9.57

9.2.3 Unsteady State (Transient State) Solution During the transient state, the rate of pressure change in the reservoir has a value which is neither zero nor a constant. Transient state behavior occurs in every system when the boundary effects are not yet felt. For example, a reservoir that is infinite (let us for now assume that there is something like that) will always be in transient state. Also, reservoirs that are finite (the boundary may or may not be closed to flow) will undergo transient behavior at early time when the boundary effects or interference due to production from and/or injection into other wells are not yet felt. The early time when the transient state behavior persists is generally known as the infinite- acting stage.

In this section, a simple analytical solution of the transient state equation is given. The solution may be obtained by Boltzman transformation discussed in Matthew and Russel (1967). The governing equation and associated boundary and initial conditions are:

Diffusivity Equation ∅µc t ∂p ∂2 p 1 ∂p + = 2 ∂r r ∂r k ∂t

9.58

Inner Boundary Condition (Constant Production) [r

qµ ∂p ] = ∂r r → 0 2πkh

9.59

w

Outer Boundary Condition

Lim p = pi as r → ∞

9.60

Initial Condition

P = Pi, when t = O

9.61

Equation 9.59 specifies that the well has a vanishing wellbore radius. That is the wellbore is considered to be a line and hence the solution to the diffusivity equation obtained with such inner boundary condition is called the line source solution. If a finite wellbore inner boundary condition is considered, a solution can be obtained by Laplace transformation.

153

Although the outer boundary condition (Eq. 9.60) indicates that the system is infinite but this is just a mathematical representation of systems where the boundary effects or effects due to interference have not been felt. Solution to diffusivity equation subject to the given boundary and initial condition is: p(r, t) = pi +

∅µc t r 2 qµ Ei - ( ) 4 πkh 4 kt

9.62

For x < 0.0025, Ei (- x) ≈ ln (γx) = ln x + 0.5772

9.63

The factor, γ, is the Euler’s constant and it is equal to 1.781. Ei(- x) is called the exponential integral function and is defined as ∞

Ei(- x) = -

∫x

e -u du u

9.64

Applying the approximation (generally called log approximation) given as Eq. 9.62, for

∅µc t r 2 < 0.0025. 4 kt

That is (

kt > 100) , ∅µc t r 2

the transient state solution becomes ( γ∅µc t r 2 ) qµ p(r, t) = pi + ln 4 πkh 4kt

9.65

At the well p(rw, t) = Pi +

( γ∅µc t r 2 ) qµ ln 4kt 4 πkh

9.66

The above solution is the line source solution also called the exponential integral solution or Theis solution. Equation 9.65 makes it possible to predict the pressure at any point r, in the reservoir due to constant rate production at the well. This equation forms the basis of interference test. Equation 9.66 makes it possible to predict the pressure at the well due to constant rate production at the well. This equation forms the basis of drawdown tests The interest in the transient state solution is due to the fact that every well, at early time, goes through the transient state. That is, the well behaves like a single well in an infinite reservoir. At later times, the effects of other wells, reservoir boundaries, aquifer influence cause it to deviate from the infinite acting behaviour. Figure 9.5 is a semilog graph of

154

bottom-hole flowing pressure versus time showing that every system irrespective of the boundary condition initially goes through the transient state phase.

Figure 9.5: Plot of Pwf versus log t for Reservoirs with Different Outer Boundary Conditions Determination of Ei Function For practical purposes, the log approximation to the Ei function holds for kt > 5 with about 2 percent error. ∅µc t r 2

When the log approximation does not hold, the function can be read from Table 9.6.

Table 9.6: Values of the Ei Functions

155

9.3 SUMMARY OF TRANSIENT, PSEUDO-STEADY AND STEADY STATE FLOW EQUATIONS The equations describing flow during the transient, pseudo-steady (semi-steady) and steady state periods have been given. A brief review of the different types of flow is given in the paper published by Matthew’s (1986).

Further discussion on transient, pseudo-steady, and steady state flows is given by Ramey (1975), Matthew and Russel (1967) and Earlougher (1977). A summary of the governing equations during the flow periods is given in Table 9.7. The equations are given in both Oilfield and Darcy’s Units. In the equations in Table 9.7, the skin effect is neglected. Table 9.7: Summary of General Flow Equations Flow Type: Transient  - ∅µct r 2  qµ P = Pi + Ei  Darcy Units:  4 πkh  4 kt   - ∅µct r 2  qBµ P = P + 70.6 E Oil Field Units: i i   kh  4 x 0.00264 kt  Practical SI Units  - ∅µct r 2  2 qBµ P = Pi + 9.33 x 10 Ei   kh  4 x 0.00264 kt 

156

Flow Type:

Darcy Units: Oil Field Units: Practical SI Units

Pseudo-Steady State qµ  re 3 P = Pw + -  ln 2 πkh  rw 4 qBµ  re 3 P = Pw + 141.2 -  ln kh  rw 4 qBµ  re 3  P = Pw + 1.866 x 10 3 ln -  kh  rw 4 

Flow Type:

Steady State

Darcy Units:

P = Pw +

Oil Field Units:

P = Pw + 141.2

Practical SI Units

qµ r ln 2 πkh rw qBµ  r  ln  kh  rw 

P = Pw + 1.866 x 10 3

qBµ  r  ln  kh  rw 

9.4 LIQUID FLOW SOLUTIONS IN DIMENSIONLESS FORMS In this section, the dimensionless forms of the diffusivity equation, steady state, pseudosteady state and transient state solutions are given Diffusivity Equation:

∂ 2 PD ∂p D 1 ∂p D + = 2 ∂rD rD ∂rD ∂t D

9.66

 ∂p  Steady State Solution  D = O  ∂t D 

q =

2 πkh [Pe - Pw ] r µ ln e rw

9.67

and Pe = Pi Then in dimensionless form

157

PDw =

2 πkh qµ

[P - Pw ] i

= ln

re rw

9.68

Generally, PD =

2πkh[ Pi − p] r = ln e qµ r

Pseudo-Steady State Solution, qµ Pi - P = 2 πkh

9.69 (∂p D ) = constant ∂t D

2  r 2 re 3 kt 1  r  w + ln - + 2      2 r 4 2  re     re  ∅ µc t rw

9.70

In dimensionless form, 2t 3 1 2 2 πkh (Pi - Pw ) = 2D + ln rDe - + rDe qµ rDe 4 2 Unsteady State (Transient State) Solution (Line Source Solution)

9.71

PDw =

P = Pi +

qµ (-∅µcr 2 ) Ei 4 πkh 4kt

9.72

In dimensionless form PD

For

(-rD2 ) 1 = Ei 2 4t D

9.73

rD2 < 0.0025, the log approximation holds. That is, 4t D PD = -

( γ rD2 ) 1 ln 2 4t D

9.74

 1 t t 0.80907  = [1n D2 + 0.80907 ] = 1.15log D2 +  2 rD rD 2.303  

9.75

Figure 9.6 is a schematic of dimensionless time for an infinite system, closed outer boundary system and constant pressure outer boundary system.

158

Figure 9.6: Dimensionless Pressure and Time for Different Reservoir Outer Boundary Conditions 9.5 COMPARISON BETWEEN LINE SOURCE SOLUTION AND FINITE WELLBORE SOLUTION The line source solution also called the exponential integral solution is very easy to obtain using the Boltzman transformation technique. The finite wellbore solution can be obtained in the Laplace space using Laplace transform technique. Solution in real space may be obtained numerically using Stefhest algorithm. The comparison between the exponential integral solution and finite wellbore solution is shown in Fig. 9.7.

Deductions that may be made from Fig. 9.7 are: 1.

2.

tD

≥ 25 (for all rD) the exponential integral solution and the finite wellbore rD2 solution are same t Also at locations for which rD ≥ 20 both solutions are also the same when D2 ≥ 0.5. rD

When

159

Figure 9.7: Line Source Solution and Finite Wellbore Solution 9.5 GENERALIZED FLUID FLOW EQUATIONS So far, we have considered a single well draining a circular reservoir. The dimensionless pressures for such case have been given for different conditions. The drainage area in many reservoirs may not be circular. Hence, the reservoir geometry must be accounted for. This is done by the introduction of Dietz shape factors, which allows a general form of inflow equation to be developed for a wide range of geometries of drainage area and positions of the well within the boundary.

The pressure drop at any point in a single-well reservoir being produced at constant rate q, is then described with the general solution: qµ PD ( t D , rD , C D , geometry ) Pi - p = 2 πkh 9.76

{

}

The dimensionless pressure in Eq. 9.76 is dependent on dimensionless time, dimensionless radius, wellbore storage dimensionless constant, and reservoir geometry. The skin factor is added to the dimensionless pressure only when calculating the pressure at the well. That is

Pi - p w =

qµ { P ( t , r , C , geometry 2πkh D D D D

)

+ S}

9.77 Practically speaking, dimensionless pressure is just a number that may be given by an equation, read from a table or a graph for different systems. The equations for calculating dimensionless pressure for some systems are given.

160

A.

Infinite (Infinite Acting) System without Wellbore Storage and Skin

 − r2  PD = - 21 Ei  4t D   D 9.78 For

tD >5 rD2

PD =

1   tD  ln   + 0.80907 2   rD2 

-

-

9.79 Equations 9.78 and 9.79 should be familiar.

B. Closed System (General Case) For all closed systems undergoing transient flow, the dimensionless pressure is given within 1 percent as:

PwD =

1 A [ln t DA + ln 2 + 0.80907 ] 2 rw

9.80 where 0.000025 < tDA < t`DA and t`DA is the time to use infinite system solution with less than 1 percent error. Values of t`DA for different reservoir geometries and well locations are given in the last column of Table 9.8 taken from Earlougher (1975). Does Eq. 9.80 imply that the transient state equation depends on drainage area? No. The area term in Eq. 9.80 cancels out if tDA is expressed as a function of tD and A. Equation 9.80 is exactly the same as Eq. 9.79 for rD= 1 (wellbore).

161

Table 9.8: Shape Factors for Various Closed Single - Well Drainage Areas

162

163

At long times, every closed system attain pseudo-steady state and the dimensionless pressure is given as: PwD = 2 π t DA +

1 A 1 2.2458 ln 2 + ln 2 2 CA rw

9.81 Equation 9.81 applies for different systems when tDA is greater than the time under the column with heading, “exact for tDA >” given in Table 9.8. With a maximum of 1% error, the time when pseudo-state will start can be read from the column with the heading “less than 1 percent error for tDA >.” This makes it possible to use the pseudo-state equation to calculate pressure in a period that ideally is considered a transition period (transition from transient to pseudo-steady state). The CA in Eq. 9.81 is the geometric or shape factor, which is also given in Table 9.8 for different drainage systems and well locations. From Eq. 9.81, a Cartesian graph of dimensionless pressure versus dimensionless time based on area gives a straight line with slope of 2π when pseudo-steady state condition is reached. C. Infinite System with Single Vertical Fracture (No Storage and Skin) (i) Uniform Flux Fracture: For uniform flux fracture, fluid enters the fracture at a uniform flowrate per unit area of fracture face so that there is a pressure drop in the fracture. Uniform flux fractures are closer to natural fractures. The dimensionless pressure for uniform fracture is:

π t Dxf

PwD =

erf

1 1 t Dxf 2

-

 1 1  Ei  2  4t Dxf 

9.82 where dimensionless time is based on half-fracture length defined as t Dxf = t D

 rw    xf 

2

The function erf is the error function defined as: 2 erf (x) = π

x

2

-u ∫ e du o

9.83 For tDxf > 10, Eq. 9.82 becomes

164

1 [ln t Dxf + 2.80907] 2

PwD = 9.84

With less than 1 percent error and for tDxf < 0.1, π t Dxf

PD =

9.85 Taking the log of Eq. 9.85 gives log PD =

1 log [t Dxf + π] 2

9.86 Equation 9.86 implies that a log-log plot of PD versus tDxf gives a straight line with a slope of 0.5. This implies that at short times, flow into fracture is linear. Linear flow is characterized by a slope of 0.5 on log-log graph of PD versus tDxf or (∆p versus t). (ii) Infinite Conductivity Fracture: In this case, the fracture is considered to have infinite permeability and therefore uniform pressure in the fracture. The dimensionless pressure for infinite conductivity fracture systems is given by Earlougher (1977) as:

PD =

1 2

π t Dxf [erf(

-0.067 Ei (

0.134 0.866 ) + erf ( )] t Dxf t Dxf

- 0.018 (-0.750) ) - 0.433 E i t Dxf t Dxf

9.87 But when tDxf > 10, PD =

1 [ln t Dxf + 2.20] 2

9.88 Also, with less than 1 percent error and for tDxf < 0.01 PD =

π t Dxf

9.89 Equation 9.89 indicates that at short times, flow into infinite conductivity fracture is also linear.

165

D. Closed System with Single Vertical Fracture (No Storage and Skin) Equations given for the infinite system with single vertical fracture apply but when the system attains pseudo-steady state. 2

PD = 2 πt DA

1 x  1 2.2458 + ln  e  + ln 2 xf  2 CA

9.90 where xe is half-length of the side of the system parallel to the fracture.

9.6

WELLTEST DESIGN

In new fields or special circumstances, it is important that tests be designed before they are run. The objectives of such test design are as follows:

1. 2. 3.

To determine the duration of the flow phases To determine the expected pressure changes. To ensure that correct equipment (especially gauges) are used.

During test design, values of well and reservoir parameters are assumed while equations are used to calculate the required parameters. This will be illustrated with an example. Example: Table 9.9 shows assumed and estimated well and reservoir parameters. Using the values in Table 9.9 and assuming well in the center of a 2:1 rectangle and 40 acre drainage area calculate the following:

(a) (b) (c) (d) (e) (f) (g)

The duration of the wellbore storage phase. Duration of the transient state period The time when pseudo-steady state will start Draw the box diagram showing the duration The maximum pressure change at the end of strong wellbore storage effect The pressure change at the end of transient The pressure change at the beginning of pseudo-steady state

Table 9.9:

Parameters for Test Design Parameter Wellbore radius, rw, ft Total compressibility, ct, psi-1 Porosity, φ Oil formation volume factor, Bo, rb/STB Oil viscosity, µ, cp Production rate, q, STB/D Formation thickness, h, ft Permeability, md Skin Factor Wellbore Storage Constant, rb/psi

Value 0.5 20 x 10-6

0.25 1.125 0.6 1000 50 1000 25 0.009

166

Solution A.

For drawdown, wellbore storage ends at time (200000 + 12000S)C s = 0.054 hrs kh / µ For well in the center of 2:1 rectangle, transient ends at a time read from Table 9.8 as tDATs = 0.025. Converting tDATs to ordinary time gives t ws =

B.

∅µc t A . t DA Ts = 0.495hrs 0.000264 k

t Ts = C.

For this system pseudo-steady state starts at time read from Table 9.8 as tDAPSS = 0.3. In absolute time, this is t =

D.

φµc t A t DA pss = 5.94 hrs 0.000264 K

Box Diagram Showing Duration for Drawdown Test Wellbore storage storage phase phase Wellbore Pseudo-steady state

Transient phase 0.054 0.495 5.94 O

E.

t(hr)

Assuming wellbore storage effect died down completely at 50t*, therefore t* =

t *D =

CD =

t ws 50 0.000264 kt * = 380.16 φµc t rw2 0.8936 C s = 128.68 φc t hrw2

167

Dimensionless pressure at end of strong wellbore storage is t *D PD = CD The pressure change now becomes ∆p =

F.

141.2qBµ kh

[P

D

+ S] = 1.9062 [2.95 + 25] = 53.28 psi

Dimensionless time at end of transient is 0.000264kt Ts = 1.74 x 105 2 φµc t rw

tD =

Dimensionless pressure at end of transient is PD =

1 2

 1  Εi   4t D 

PD =

1 2

[In t D + 0.80907]

or

Log approximation holds, for

if log approximation holds.

tD > 5 , therefore, PD = 6.44 rD2

Pressure drop at the end of transient is ∆p =

G.

141.2qBµ kh

[P

D

+ S] = 1.9062 [25 + 6.44] = 59.93 psi

Dimensionless pressure at the beginning of pseudo-steady state is PD = 2π t DA pss +

1 2

In

A + rw2

1 2

In

2.2458 = 8.626 CA

Pressure drop at the beginning of pseudo-steady state is then ∆p =

141.2qBµ [PD + S] = 1.9062 [8.626 + 25] = 64.10 psi kh

9.7 SUPERPOSITION So far, we have considered the well to be produced at constant rate and the effect due to only one well in a given reservoir system. In this section, we shall use superposition concept to account for production at different rates and also the effect of having more than one well in a given reservoir system.

168

Simply put, superposition generally means that any sum of solutions to a linear differential equation is also a solution. In Petroleum Engineering, this implies that the total pressure drop at any point in a reservoir that contains more than one well is the sum of the pressure drop due to each well acting as if it is in the reservoir alone. Hence superposition concept can be used for generation of changing flowrate or pressure boundary condition in pressure buildup analysis and water-drive material balance methods. Also superposition can be used for determining the effect of many wells in a reservoir. In this section, the application of superposition will be illustrated with examples. Superposition in both space and time will be considered. Discussion on these follows.

9.7.1 Superposition in Space Considering a reservoir containing two wells as shown in Figure 9.8, the pressure drop due to both wells at point (x,y) in the reservoir is given using superposition in space as:

Pressure drop at x, y = Pressure drop at x, y due to Well 1 +Pressure drop at x, y due to Well 2 9.91

P(x, y)

a

b

Well 2

Well 2 q1

d

q2

Figure 9.8: Superposition in Space

Using the notations in Figure 9.8, Eq. 9.91 implies: Pi - Px, y, t = (Pi - Pa, t ) + (Pi - Pb, t ) 9.92

169

Generally, (Pi - Pr, t ) =

qµ PD (rD , t D ) 2 πkh

9.93 Therefore, Pi - P (x, y, t) =

q1µ

2πkh

PD (a D , t D ) +

q2µ

2πkh

PD (b D , t D )

9.94 At Well 1, Eq 9.94 becomes Pi - Pwf1 =

q1µ 2πkh

[PD (t D ) + s1 ] +

q2µ 2πkh

PD (d D , t D )

9.95

The second term on the right hand side of equation 9.95 is the interference effect due to Well 2 at Well 1.

At Well 2, Eq 9.94 becomes Pi - Pwf2 =

q1µ 2πkh

PD (d D, t D ) +

q2µ 2πkh

[PD (t D ) + s 2 ]

9.96

The first term on the right hand side of equation 9.95 is the interference effect due to Well 1 at Well 2.

For q1 = q2 = q, Eq. 9.94 becomes 2πkh (Pi - P (x, y, t)) = PD (a D , t D ) + PD (b D , t D ) qu

9.97

At Well No. 1, Eq. 9.97 is: 2πkh qu

(Pi

- Pwf1) = PD (1, t D ) + S1 + PD (d D , t D )

9.98

and at Well No. 2, Eq. 9.97 is: 2πkh qu

(Pi

- Pwf2 ) = PD (d D , t D ) + PD (1, t D ) + S 2

9.99

Note that at the wells, the skin factor terms are added. Also, for superposition in space, the wells may have different production rates but for all wells, production must commence at the same time. Signs are changed (q is negative) for injectors.

170

Implications of Superposition in Space 1. The pressure distribution for one well producing at constant rate q located at a distance d/2 from a linear flow barrier, or one well in a semi-infinite reservoir can be simulated by super-imposing solutions of two wells. This is shown in Figure 9.9. The two wells must either be injectors or producers.

d/2

d/2

d/2

Figure 9.9: Simulation of No-Flow Boundary

2. Similarly, a single well producing near a constant pressure barrier can be simulated with two wells. In this case, one of the wells must be an injector while the other must be a producer. This is shown in Figure 9.10.

==

Figure 9.10: Simulation of Constant Pressure Boundary

3. Single well producing in closed square can be generated by considering an infinite array of wells as shown in Figure 9.11.

171

==

Figure 9.11: Simulation of a Well in a Closed Square

For Figure 9.11, if we want to compute the pressure at the well in the center of a closed square with dimension A / rw = 2000 and at time tDA = 0.1, the procedure is as follows: (i) Draw the closed square with the well at center. Then, draw a few of the infinite arrays of wells around the well of interest. A few of the infinite array of wells is required because the further the wells are from region of interest, the lesser their contribution to the total pressure effect. The array of wells is shown in Figure 9.12. (ii) Of the array of wells around the well of interest, locate wells that are of equal distance from the well of interest. This was done by assigning the same code number to wells at equal distance from well of interest as shown in Figure 9.12.

5

4

3

4

5

4

2

1

2

4

3

1

1

3

4

2

4

4

4

5

4

3

4

5

Figure 9.12: Schematic of Array of Wells.

172

(iii) Calculate the distance from the well of interest to array of wells surrounding it. The square has an area of “A” and each side of the square has a length of A . Also the following relationship holds tD 2

rD

=

kt ∅µcr 2

9.100

The above relation is important because the dimensionless pressure due to each well is a function of tD/rD2. Hence the contribution of each well is easily found from the exponential integral solution. Note that each well will act as if it is in the reservoir alone. Table 9.10 shows the calculated distances and dimensionless pressures. Table 9.10: Calculated Distances, Time and Dimensionless Pressures

t D / rD2

P (t D / rD2 )

Assigned No.

No. of Wells

Distance From Well to Well of Interest

1

4

A

tDA

.012458

2

4

2A

t DA 2

.000574

3

4

A

t DA 4

.0000025

4

8

5A

t DA 5

0

2

=

kt 2 ∅µcr

For (t DA = 0.1)

(iv) Applying the superposition principle, t  2 kh D  [Pi - Pwf ] = PD (t D ) + S + 4 PD  2 = t DA  qµ  rD 

9.101

t t t t  t  t  + 4 PD  D2 = DA  + 4 PD  D2 = DA  + 8 PD  D2 = DA  2  4  5   rD  rD  rD

The first term on the right hand side of Eq. 9.101 is the contribution of the well of interest. This contribution is calculated as follows: kt kt A A . 2 = t DA . 2 tD = = 9.102 2 ∅µcA rD ∅µcrw rD For the case of interest, it is given that

173

A / rw2 = 4 x 10 6 and t DA = 0.1

Therefore PD (t D ) = PD (0.1 x 4 x 10 6 ) = 6.8542 Neglecting the skin factor S, and substituting values in Eq. 9.101 gives PD = 6.8542 + 4 x 0.012458 + 4 x 0.000574 + 4 x 0.0000025 + 0 = 6.90634 This compares favourably with a value of 6.9063 calculated by Earlougher et al (1968). The same procedure may be used to obtain dimensionless pressures at any point in the well. 9.7.2 Superposition in Time Superposition in time is used for handling flow rate that varies with time at a given well. A schematic of how this is done is for a two-rate case is shown in Figure 9.13.

q

q

q

q2 +

=

q2-q1

q1

t

∆t

t +∆t

∆t

Figure 9.13: Superposition in Time (2-Rate Case)

In words, the pressure drop due to a well producing in an infinite reservoir at rate q1 for time “t”, and then for rate q2 for a period of “∆t” is equivalent to the sum of pressure drop due to the well producing at rate q1 for time “t + ∆t” and the same well producing at rate (q2 - q1 ) for a period of “∆t”. That is: Pi - P (r, t ) =

q1 µ (q - q ) µ PD r D , (t + ∆t ) D + 2 1 PD (rD , ∆t D ) 2πkh 2πkh

[

]

9.103

174

At the well, rD =1, P(r,t) = Pwf and the skin term will be added to give (Pi - Pwf ) =

q1 µ (q - q ) µ q µ PD [1, (t + ∆t) D ] + 2 1 PD (1, ∆t D ) + 2 S 2 πkh 2 πkh 2 πkh

9.104

If q2 is zero, Eq. 9.104 becomes

(P - P ) = q i

ws

1

µ

2πkh

{P [1, (t + ∆t) D

D

] - P [1, ∆t ]} D

9.105

D

Note that the skin term has disappeared and we have changed notation from pwf (pressure well flowing) to pws (pressure well shut-in). If we assume transient and log approximation, PD [1, (t + ∆t ) D ] = 0.5[ln(t + ∆t ) D + 0.80907]

9.106

PD [1, ∆t D ] = 0.5[ln ∆t D + 0.80907]

9.107

and

Substituting Eq 9.106 and 9.107 into Eq 9.105 gives

(P - P ) = q i

ws

1

µ  t + ∆t 

ln  2πkh  ∆t 

9.108

Equation 9.108 is the basis of the buildup test. We have derived the equations for a two-rate case. Figure 9.14 shows how superposition is done for a three-rate case. You should derive the equations for this case as an exercise.

q1

q3

q =

q2

t1

∆t2

∆t

t1 + ∆t2 + ∆t2

175

q2 – q1 +

q3 – q2

+

Figure 9.14: Superposition in Time (3-Rate Case)

EXERCISES

1. Estimate the pressures at the well located in the center of a 2:1 rectangle after the well has produced 800 STB/day of dry oil for 15 minutes and 15 days. Other data are: Pi = 3265 psi, ko = 900 md, µo = 6.2 cp, Bo = 1.02 RB/STB, h = 47 ft ∅ = 0.17, ct = 2 x 10-5 psi-1 rw = 0.50 ft A = 40 acres, s = 7 2. The dimensionless pressure at pseudo-steady state well in the center of a circular reservoir is:

Pi - Pw

2 qµ   rw  2   = 2 πkh   re  

r kt 3 1  rw  + ln e +   2 rw 4 2  re  ∅µc t rw

2

 

And the generalized dimensionless pressure for any closed system with any geometry is

where CA is the geometric factor. Assuming re >>rw, find the geometric factor for a well in the center of a circular reservoir by relating both equations. 3. Ramey and Cobb (1971) have shown that for a well situated at the center of a regular shaped drainage area, (for instance, a circle square or hexagon) the transient period is of extremely short duration. Under these circumstances, it is possible to equate the transient flow equation and pseudo-steady state equation to determine the time at

176

which pseudo-steady state condition starts. Find an expression for tDA at which this occurs for a well in the center of a circle? 4.

Neglecting skin, the pseudo-steady state equation for flow in circular reservoir is: P - Pwf =

qµ  re 3   ln -  2πkh  rw 4 

The generalized flow equation at pseudo-steady state is P - Pwf =

qµ 2πkh

1 4A   ln 2   2 γ C A rw 

where CA is the shape factor. By comparing the two equations, estimate the shape factor for a well in the centre of a circular reservoir.

177

10. BOTTOM-HOLE HORIZONTAL WELLS

PRESSURE

TESTS

IN

The objectives of bottom-hole pressure tests in vertical and horizontal wells are similar. Actually, a horizontal well can be viewed as a vertical well with infinite conductivity vertical fracture or a highly stimulated vertical well. However, BHP tests in horizontal wells are more difficult to analyze for the following reasons. 1. The horizontal wells are not perfectly horizontal as assumed by the analysis models. The wells may be snake-like as shown in Fig. 10.1.

Vertical Depth

Horizontal Distance Fig. 10.1 Snake-Like Nature of some Horizontal Wells

The consequence of the snake-like nature of horizontal wells is that waves emanating from different sections of the horizontal well at the same time may hit different boundaries. This complicates pressure response. 2. The total drilled length may not all contribute to the producing length. Some drilled length may intersect non-productive interval. The unknown non-producing length is erroneously accounted for as skin factor. 3. There are many possible flow regimes which depending on the reservoir and well conditions may or may not be properly discerned from pressure tests. In this chapter, we shall present basic information about horizontal wells and how to identify the flow regimes obtained during horizontal welltests. We shall also present flow equations and how to analyze horizontal welltests.

178

10.1 Introduction A horizontal well is a well drilled parallel to the reservoir bedding plane while a vertical well is drilled perpendicular to the reservoir plane. This is shown in Fig 10.2. With horizontal well, we can enhance reservoir contact by well and therefore hence enhance well productivity. This implies that the drainage area of the horizontal well is bigger than that of a vertical well. Joshi (1988) showed two methods of calculating the drainage area of horizontal well based on knowledge of the drainage area of a vertical well. Joshi’s procedures are presented.

Horizontal Well

Vertical Well

Fig. 10.2: Vertical and Horizontal Wells.

Drainage Area of Horizontal Well If the drainage area of a vertical well is: A

=

2 π rev

10.1

Where rev is shown in Fig 10.3.

rev

Fig. 10.3: Drainage Area of Vertical Well

The two method for the calculating the drainage area of horizontal well are as follows: 179

Method 1: This assumes that each end of the horizontal well drains a semi-circle drainage area while the length drains a rectangular drainage area as shown in Fig 10.4.

rev L

Fig. 10.4: Rectangular/Semi-Circle Drainage Area

Assuming that h ≥ rev, HWDA =

{ πr

2 ev

}

+ L(2rev )

10.2

Where HwDA L

= Drainage area of the horizontal well = Productive length of the horizontal well

Method 2: We assume an elliptical drainage area in the horizontal plane, with each end of a well as a foci of drainage ellipse as shown in Fig 10.5.

L

Fig. 10.5: Elliptical Drainage Area

In this case, the drainage area is, HWDA = πab

10.3

where

180

a

= half major axis of ellipse =

b

= half minor axis =

L + rev 2

rev

The calculated drainage areas (using different methods) are not the same. Therefore, the average of the two can be used as the effective drainage area.

Applications of Horizontal Wells (a) (b) (c) (d) (e) (f)

Some naturally fractured reservoirs Reservoir with gas/water coning problems Thin reservoirs Reservoirs with high vertical permeability In EOR projects with injectivity problems In fields (e.g. offshore) requiring limited wells due to cost or environmental problems.

Advantages of Horizontal Wells (a)

Horizontal well productivities are 2 to 5 times greater than that of unstimulated vertical well. Actually, the performance of horizontal wells depends on the effective length of the horizontal section in the formation. Higher productivity may result to early payout.

(b)

Horizontal well may intersect several fractures or compactments and help drain them effectively Reduce coning tendencies As injectors can improve sweep efficiency in EOR projects

(c) (d)

Disadvantages of Horizontal Wells (a) (b) (c) (d)

Ineffective in thick (500ft to 600ft) low permeability reservoirs. Cannot easily drain different layers Technological limitations Cost more (1.4 to 2 times) than cost of drilling a vertical well.

Dimensionless Parameters used in Horizontal Wells Dimensionless parameters are also used in horizontal wells. dimensions and coordinates in horizontal wells.

Figure 10.6 shows

181

L/2 h

z

zw x

y

Fig. 10.6: Dimensions in a Horizontal Wells

Table 10.1 shows the different dimensionless parameters in Darcy unit. Table 10.1: Dimensionless Parameters used in Horizontal Wells Dimensionless Parameters

Dimensionless Pressure in terms of h and kr Dimensionless Pressure in terms of L and k r k z Dimensionless Time in terms of L/2 Dimensionless Time in terms of h and kz Dimensionless Time in terms of ye and ky Dimensionless Time in terms of h and ye Dimensionless Time in terms of and rw

PDh =

Equations

k r h∆p 141.2q sBo µo

k r h z L∆p 141.2q sBo µo 0.0002637k r t tD = φ µc t (L / 2)2 0.0002637k z t tDz = φµoc t h2 0.0002637k y t tDy = φ µoc t y2e 0.0002637k y t tDhy = φ µoc t hy e PDL =

tDrw =

0.0002637 k r k zt φ µoc t rw2

182

'

Derivative in terms of pDL

p DL =

Derivative in terms of pDh

p Dh =

Dimensionless x, y, z Coordinate

Dimensionless Wellbore Location

Dimensionless x, y-direction boundary width Dimensionless Wellbore Length Dimensionless Wellbore Radius

'

dp DL

d ln( t D ) dp Dh

d ln( t D ) xD = 2(x-xw)/L yD = 2(y-yw)/L zD = z/h xwD = 2xw/xe ywD = 2yw/ye zwD = zw/h xeD = 2xe/L yeDL = 2ye/L kz LD = L 2h k r rwD = 2rw/L

10.2. Flow Regimes during BHP Tests In Horizontal Wells

During bottom-hole pressure tests in a horizontal well, the followoing flow regimes could be discerned: linear, radial, hemiradial, and pseudo-radial. Figure 10.7 is a box diagram showing some of the flow regimes and the order in which they occurred. Some parameters in Table 5.1 are defined as follows: Zw = vertical distance measured from bottom of payzone to the well Xw, Yw, Zw = Well location co-ordinates Xe, Ye = reservoir boundaries in x and y directions L = Length of horizontal well kr = kh = permeability in horizontal place = kxky kz = kv

=

vertical permeability

183

Wellbore Storage

Radial

Linear

Pseudoradial

Boundary Effect

Transient State

Fig. 10.7: Typical Flow Regimes in a Horizontal Well

The pressure and pressure derivative for the case shown in Fig 10.7 is shown in Fig 10.8 taken from Lichtenberger (1994).

Fig. 10.8: Pressure and Derivative for a Typical Horizontal Well BHP Test

Details on the flow regimes are as follows: Early Radial Flow: Figure 10.9 shows the early time radial flow period in a vertical plane, which develops, when the well is put initially on production. The well acts as though it is a vertical well turned sideways in a laterally infinite reservoir with thickness, L. This flow period ends when the effect of the top or bottom boundary is felt or when flow across the well tip affects pressure response. This flow regime may not develop (Kuchuk , 1995) if the anisotropic ratio, kH/kV is large.

184

Fig. 10.9: Early Radial Flow in a Horizontal Well

Many authors (Ozkan et al, 1989; Goode and Thambynayagam, 1987; Odeh and Babu, 1990; Du and Stewart, 1992; Lichtenberger, 1994 and Kuchuk, 1995) published equations for identifying the different flow regimes. Inferences from their publication show that the early radial flow can be identified using the pressure derivative or semilog plot if it is not marred by wellbore storage effects. The pressure derivative gives a zero slope as shown in Fig 10.8 while a graph of pressure versus log of time yields a straight line. The basis for the straight line is the equation:

Pi - pwf =

 162.6qBµ   k y kv t  − 3 . 23 + 0 . 87 s − A log  2 kv k y L   φµct rw   

10.4

where s = skin due to damage/stimulation. If it is positive, it is denoted as Sm, mechanical damage due to drilling and completion. Also, A is a constant given by Lichtenberger (1994) as  ky

A = 2.303 log ½  4

 kv

+

4

k v  k y 

10.5

The equation implies that a graph of Pwf versus log t gives a straight line with slope m1 =

162.6qBµ k vk y L

From this, the equivalent permeability in the vertical plane,

10.6

k v k y , can be calculated.

185

kvky =

162.6 qBµ m1L

10.7

The skin equation for the first radial flow period is   kykv Pi - P( 1 hr ) S = 1.151  - log 2 m1   φµc t r w

   + A + 3.23  

10.8

Note: 1. In arealy isotropic reservoir, kx = ky = kh 2. If effective reservoir permeability k v k y is known, the given equation can be used

in determining the effective producing length. The early-time flow regime can be short and may be completely marred by wellbore storage effect. Use of downhole shut-in tool is useful here. Time to end of wellbore storage is given by Lichtenberger (1994) as:

3.

t Eus =

(4000 + 240sm )C k h kv L / µ

10.9

where tEus sm C L µ kh

= = = = = =

kv,kx,ky =

time for end of wellbore storage effects, hours skin factor wellbore storage constant, rb/psi effective producing length of well, ft viscosity of oil, cp horizontal permeability = k x k y permeability in the vertical, x and y directions respectively.

Early radial flow ends when either of the following will occur: a. Effect of bottom or top boundary is felt. b. Flow across well tips affects pressure response Mathematically, different authors gave time to the end of early radial flow, te1, as follows: Goode and Thambynayam (1987)

190d z2.095 rw−0.095 φµc t te1 = kv

10.10

where dz is the distance of the well to the closest boundary (top or bottom) Odeh and Babu (1990) and Licthenberger (1994)

186

te1

= min

1800 d 2z φµC t  kv    125L2 φµC t  ky 

10.11

The first equation represents the time when the effect of the boundary (top or bottom) will distort early radial flow while the second equation represents time when flow across the tip of the well will distort early radial. Lichtenberger assumed areal isotropy in his equations. DU and Stewart (1992)

te1 =

min

 947 φµC t d 2z  kv    947 φµC d 2 t x   kx

10.12

where dx is nearest distance from well point to the boundary normal to the well length axis. Other parameters are as defined earlier. Although the equations by different authors are not exactly the same, but they are similar and therefore can be used as a guide. Hemiradial Flow : When the wellbore is closer to any of the no-flow boundaries, hemiradial (or hemi-elliptical) flow may develop. This produces slope doubling on the semilog plot and p1 (pressure derivatives) will plateau at twice the radial flow value.

The flow equation during the hemi-radial flow period is given by Kuchuk (1995) and Lichetenberger (1994) as Pwf = Pi - 2 x

 162.6qBµ   k H k v t  0.87S  − 3.23 + − A log 2 k H k v L   φµc t rw  2 

10.13

where A

=

 k H  d z  log 1 +  k v  rw  

dz

=

distance to the nearest boundary (top or bottom)

Lichtenberger (1994) gave the time for the end of the hemiradial flow as

187

tEhrf =

1800 d 2z φµc t kv

10.14

Implications of the Eq 10.13 are as follows: a. A plot of pwf versus log t is a straight line with slope m b. c.

k H k v = 2  

162.6 qBµ   mL 

10.15

The skin equation is  P - P(1 hr )  k k S = 2.30  i - log H 2v  φµc t rw m  

     + 3.23 + log 1 + k H  d z     k V  rw     

10.16

Intermediate Linear Flow: This flow regime may develop after the effects of upper and lower boundaries are felt at the well. Figure 10.10 shows the streamlines during the intermediate-time linear flow. This flow regime develops if the well length is sufficiently long compared with reservoir thickness and there is no constant pressure boundary.

Fig. 10.10: Intermediate Linear Flow

The flow equation during the linear flow is given as follows: Pi - Pwf =

8128 . qBµ t 141.2 qBµ + (Sz + S) Lh φµc t k y L kykv

10.17

where SZ is the pseudo-skin factor caused by partial penetration in the vertical direction and is given by different authors as:

188

Odeh and Babu (1990)  ky    180° Z w   .   − 1838  − ln sin     h  kv 

 h  rw 

SZ = ln   + 0.25 ln

10.18

Lichtenberger (1994)

 πr  k  π dz   SZ ≈  w 1 + v  sin k y  h   h  

10.19

Kuchuk (1995)

 πr SZ = ln  w  h

   1 + k v  sin πd w   k H  h  

10.20

Implications of the flow equations are as follows: a. A graph of (Pi -Pwf) versus t is a straight line b. Slope of line m2 =

8128 . qB µ Lh φc t k y

10.21

Therefore 2

2

L ky

c.

∆p

t =o

 8128 . qB µ =    hm 2  φc t =

10.22

141.2 qBµ (Sz + S) L kykv

10.23

where ∆pt=0 is the pressure drop at time equals zero. The skin due to damage, S, can therefore be calculated as SZ is known (calculated from Equation 10.18, 10.19 or 10.20) Time to end of early linear flow is given by different authors as follows: Goode and Thambynayagam (1987)

te2 =

20.8 φµc t L2 kx

10.24

Du and Stewart (1992)

189

16 φµc t L2 kx

te2 =

10.25

Odeh and Babu (1990) and Lichtenberger (1994) 160 φµc t L2 kx

te2 =

10.26

Odeh and Babu (1990) also gave the time to the start of early linear flow period as 180 D2z φµc t kv

tS2 =

10.27

where Dz (= h-dz) is the maximum distance between the well and the z-boundaries (top or bottom boundary) Pseudoradial Flow: In sufficiently large reservoir, pseudo radial flow will develop eventually as the dimensions of the drainage areain the horizontal plane becomes much larger that the effective well length. Figure 10.11 is a schematic showing the streamlines during the pseudoradial flow. This is similar to what happens in a horizontal with a vertical fracture.

Fig 10.11: Pseudoradial Flow The flow equation during the pseudoradial flow period is given as follows: Pi – Pwf =

 141.2qBµ 162.6qBµ   k xt   − A + (S Z + S ) log 2   L k y kv k x k y h   φµct L 

10.28

where “A” is a contant given by many authors as follows: A A A A

= = = =

2.023 2.5267 1.76 1.83

(Goode and Thambynayagam, 1987) (Kuchuk, 1995) (Odeh and Babu, 1990) (Lichtenberger, 1994

190

Implications of Eq 10.28 are as follows: a. Graph of Pwf versus log t is a straight line b. Slope, m3 =

162.6 qBµ kxky h

Therefore kxky =

10.29

162.6qBµ m 3h

The skin equation is S=

 1151 . L k v  Pi − P(1hr ) kx − log + A  − SZ  2 h kx  m3 φµc t L 

Note: a. Pseudo radial develops if L>>h (hD ≤ 2.5) b. If top or bottom boundary is maintained at constant pressure, no pseudo-radial flow period will occur. Instead, there is steady state flow at late time.

Time for beginning of pseudoradial flow, ts3, is given by different authors as follows: tS3

=

1230 L2 φµc t kx

tS3

=

1480 L2 φµc t kx

tS3

=

2841 φµc t L2 kx

(Goode and Thambynayagam

(Odeh and Babu, Lichtenberger used 1500)

(Du and Stewart)

Lichtenberger (1994) gave the time for the end of the pseudoradial flow as follows:  2000 φµC t ( L w / 4 + D y ) 2  k tEprf = min  1650φµC D 2H t x  kH 

10.30

where Dx and Dy are the lateral distances of the reservoir in the x and y directions respectively. Late Linear Flow Period: After, the pseudoradial flow, it is possible that a late-time linear flow period develops. The flow equation for this phase is

191

8128 . qB 2 x eh

Pi - Pwf =

 µt  141.2qBµ   + (Sx + Sz + S) L kykv  k yφc t 

10.31

where 2xe = width of reservoir Sx = pseudo skin due to partial penetration in the x direction. Implications of Eq 10.31 are obvious. Skin in Horizontal Well

The skin factor will serve the same purpose in horizontal well as it does in vertical wells. The dominant pseudoskins in horizontal wells are the pseudoskin due to damage and pseudoskin due to convergence in the z-direction. The pseudoskin due to damage is dominant because of more fluid losses resulting from larger area contacted by the well. The pressure loss due to skin is defined with respect to the formation thickness in vertical wells and well length in horizontal wells. This is shown in the following equations: Pressure loss due to skin in vertical wells: ∆ps

vertical

=

141.2qBµ S kh

10.32

Pressure loss due to skin in horizontal wells: ∆ps

Horizontal

=

141.2qBµ S kL

10.33

From Eqs 10.32 and 10.33, we infer that the pressure loss due to skin in horizontal well is much smaller than the pressure loss due to skin in vertical wells because the horizontal well length is usually longer than the formation thickness. The small pressure loss due to skin in horizontal wells does not imply that skin has small effect on horizontal well productivity because the drawdown in horizontal wells is also small. The remedial factor, R, used in vertical wells should also be used in horizontal wells to quantify the effect of skin. Problem Set A 2100ft long well is completed in a 100 ft thick formation with closed top and bottom boundaries. The estimated average horizontal permeability from several vertical well test is 1500md while the vertical permeability is 300md. The horizontal well has a diameter of 8½ in and is located 30ft from top of the sand. Other parameters are as follows:

φ = 20%, µ = 0.65cp, ct = 20 x 10-6psi-1 and s = 5. Also, assume dx = L/2

192

(a)

(b) (c) (d)

Assuming that the early radial flow will not be distorted by wellbore storage effects, determine the time when the early radial flow will end. Also, determine when wellbore storage effect will end if wellbore storage constant, c = 0.025 rb/psi. Calculate the time to start and end of the early linear flow period for the horizontal well whose parameters have been given For given reservoir and well data, calculate time required to start a pseudo-radial flow. Using the following additional data calculate the pressure change in a horizontal well. S = 25, q = 4000 STB/D and Bo = 1.05 rb/STB

Tabulated Solution to Problem Set End of early End of well- Start of early End of Start of radial flow bore storage linear flow early Pseudo Authors tel x10-3 effect tews ts2 x10-3 linear flow Radial Flow (hrs) x10-5 (hrs) (hrs) te2(hrs) ts3 (hrs) Goode et al 2.560 N/A N/A 0.159 9.402 Odeh and Babu 14.040 N/A 7.644 1.223 11.313 Du and Stewart 7.387 N/A N/A 0.122 21.717 Lichtenberger N/A 5.998 N/A N/A 11.466 N/A implies author did not give required equation

Skin due to ∆P @ the start partial pent. of Pseudo In vert. dir. Radial flow Sz ∆P (psi) N/A N/A 3.996 12.11 N/A N/A 2.15 x 10-4 10.833

10.3 Detecting Flow Regimes Using Pressure Derivative The pressure derivative is the best diagnostic tool for detecting flow regimes. This follows from the characteristic slopes of the derivatives obtained for different flow regimes on a log-log plot. Figure 10.12 shows the characteristic slope for the first radial, hemiradial and pseudoradial flow regimes in a horizontal well. Note that the derivatives have zero slopes at different levels. For linear flow, the derivative has a slope of 0.5.

pseudoradial Pressure Log Derivative Hemiradial

Early Radial

Log (time) Fig 10.12: Prssure Derivatives for the Different Radial Flow Regimes

193

Figure 10.13 taken from Kucuk (1995) shows the pressure derivative for cases with well and reservoir parameters shown in Table 10.2.

Fig 10.13 : Derivatives for Cases shown in Table 10.2 Table 10.2: Reservoir Parameters for Examples Shown in Fig 10.13 Example 1 2 3 4 5

h, ft 100 100 100 40 200

kH, md 100 100 100 100 200

KV, md 10 1 5 5 1

Lw, ft 500 500 500 500 500

zw, ft 20 20 5 20 20

rwD 0.00146 0.00389 0.00194 0.00197 0.00530

 r  k  rwD =  w  1 + H  kV   2 Lw   Deductions from the derivatives are as follows: (a) The first radial period can be seen in all cases. (b) In Example 3, the well is close to the boundary (5ft) and therefore, hemiradial flow occurred after a short duration early radial flow. (c) In Example 4, a linear flow regime manifested because the well length is much greater than formation thickness (d) In all cases, the pseudoradial flow developed Du and Stewart (1992) quantified the effect of parameters on the flow regimes. In their work, they defined dimensionless parameters as follows: PDL =

2π k z k r L . ∆p qµ

10.34

194

1 PDL =

dpDL (Pressure Derivative) d ln t DZ

10.35

tDZ

kz t φµct h 2

10.36

=

1 Values of PDL for different flow regimes are as follows: 1 PDL = 0.5 vertical radial flow 1 PDL = 1 vertical hemi-radial flow 1 PDL = LD pseudo radial flow

Du and Stewart (1992) concluded that parameters zWD (dimensionless well location in the z-direction) and LD (dimensionless wellbore length) have the dominant effect on flow regimes obtained in horizontal wells. These parameters are defined as follows: zwD = zw/h and

LD =

L kv 2h k H

Figure 10.14 shows effect of LD on flow regimes for infinite reservoir with no flow top and bottom. Horizontal well is in the center of the formation (ZWD = 0.5).

Fig. 10.14: Effect of LD in Homogeneous Laterally Infinite Reservoir With Sealed Top and Bottom

Inferences from Fig. 10.14 are as follows: 1 (i) For LD ≥ 3 the flow regimes are: Vertical radial flow (VRF: PDL = 0.5) + transition + early linear flow opposite the completed section (ELF: Gradient = ½ ) + transient 1 reservoir pseudo radial flow (PRF: PDL = LD).

195

(ii)

For LD < 3, the flow regimes are: VRF + vertical spherical flow (VSF) Gradient = - ½ + transition + PRF. The smaller LD, the longer the duration of VSF.

(iii) The smaller the LD, the shorter the duration of VRF and longer the length of PRF. (iv) For LD ≤ 0.1, no VRF at all. For a 100ft thick formation and Kz / Kr = 0.2, this implies a minimum well length of 45 ft. Figure 10.15 shows effect of ZWD. From Fig 10.15, we infer that when ZWD ≤ 0.1 (well close to one of the boundaries), there is a hemi-radial flow (HRF), between VRF and ELF with a transition in between. Figure 10.16 shows the effect of ZWD in a situation with gas cap. The dimensionless well length, LD, is large.

Fig. 10.15: Effect of ZwD in Homogeneous Laterally Infinite Reservoir With Sealed Top and Bottom

196

Fig. 10.16: Effect of ZwD in Homogeneous Laterally Infinite Reservoir with Bottom Constant Pressure Boundary and Top No Flow Boundary Inferences from the graph are as follows: (i) When ZWD < 0.5 (well closer to the bottom no-flow boundary) the flow regimes are as follow: VRF + transition + HRF + transition + constant pressure effect (rollover). (ii)

When ZWD ≥ 0.5 (well nearer the constant pressure boundary) the flow regimes are VRF + transition + rollover (constant pressure effect). (iii)The nearer the well to the constant pressure boundary, the stronger the constant pressure boundary effect.

The effect of the boundaries is similar to what obtains in a vertical well. A no - flow boundary causes stabilization at higher level with respect to the infinite reservoir case. Figure 10.17 shows the effect of lateral boundaries, which are parallel to the direction of well length.

Fig. 10.17: Effect of Lateral Boundaries in Reservoir with Sealed Top and Bottom 10.4 Field Cases In this section, we shall discuss some field examples of bottom-hole pressure tests. The objective is to see what may actually obtain in real life. Table 10.3 shows basic parameters or the different tests. Table 10.3: Parameters for Different Field Cases Parameters Values for Different Cases 1

2

3

4

197

Formation thickness, ft Well length, ft Well location, ft Oil rate STB/D Porosity, % Oil Viscosity, cp Formation volume factor, rb/STB Wellbore radus, ft Calculated Permeability, md Calculated Skin Drawdown, psi

73 1984 7 3948 29 1.97 1.128 0.4 15363 35 15

95 1387 7.4 4144 29 1.97 1.120 0.4 16320 23 10

19 232 8 685 24 0.34 1.539 0.4 1070 0 9

123 1330 16.1 4951 29 2.23 1.298 0.3 3820 -1.4 34

Discussions on the field cases follow:

Case 1: Figure 10.18 shows the pressure and pressure derivative for this case.

Fig. 10.18: Pressure and Pressure Derivative for Case 1 The pressure derivative shows wellbore storage, early vertical radial flow, early linear flow, hemiradial flow and “rollover” due to a constant pressure boundary. The hemiradial flow was inferred because of the nearness of the well to the bottom boundary and the fact that ∆p′ (VRF) ≈ 2∆p′ (HRF). The constant pressure effect was caused by gas-cap. Figure 10.18 shows that mathematical models give us good insight unto pressure and flow regime obtained during actual BHP tests.

198

Case 2: Figure 10.19 shows pressure data for Case 2. Case 2 and Case 1 are from the same reservoir in western Niger Delta. The pressure and pressure derivative for the two cases exhibit similar characteristics. The exception is the scatter in Case 2 data during the wellbore storage phase. The scatter was due to gauge shift. Case 3: Data for this case were obtained from the eastern Niger Delta reservoir. Figure 10.20 shows the pressure and derivative for this case. The identified flow regimes are as follows: wellbore storage phase + early vertical radial flow + vertical spherical flow + pseudo radial flow + “rollover” due to lateral constant pressure boundary.

Fig. 10.19: Pressure and Pressure Derivative for Case 2

199

Fig. 10.20: Pressure and Pressure Derivative for Case 3 The vertical spherical flow regime resulted because the well length is small and therefore LD < 3 (actually LD ≈ 2). This is in agreement with the finding of Du and Stewart (1992). Case 4: Data for this case were obtained from the first horizontal well in eastern Niger Delta reservoir. Figure 10.21 shows the pressure and derivative for this case.

Fig. 10.21: Pressure and Pressure Derivative for Case 4 We believe that the distortions in pressure and derivative were caused by the “snakelike” nature of the horizontal part of the well. The distortions made it difficult to clearly discern the flow regimes.

200

In field situations, there could be problem resulting because the gauge may not get to the horizontal part. 10.4 Analysis Procedure Analysis of bottom-hole pressure test in horizontal wells, requires the following a Identifying boundaries and main features such as faults, fractures, etc. from flow regimes analysis. b Estimating well/reservoir parameters and refining the model that is obtained from flow regime analysis.

The graphical type curve procedure is practically impossible for the analysis of horizontal welltest data because of the many unknowns (kH, kV, s, C, Lw, h, dz, ) even in the case of a single-layer reservoir. Thus, along with the flow regime analysis, non-linear leastsquare techniques are usually used to estimate reservoir parameters. In applying these methods, one seeks not merely a model that fits a given set of output data (pressure, flowrate, and/or their derivatives) but also knowledge of what features in that model are satisfied by the data. A flowchart showing recommended procedure for test analysis in horizontal well is shown in Fig 10.22.

Diagnose Flow Regimes

Regimes Clear

No

Any Early radial?

Test cannot be Analyzed

Yes Analyze Tests using the 3 methods

Time constraints ?

Analyze test with Regression

No

Results not accepted

Yes Simulate profile and compare Accept results with best confidence

Fig. 10.22: Flow chart of Procedure for Test Analysis in Horizontal Well.

201