Investigation of Wellbore Storage Effects on Analysis of Well Test Data

INVESTIGATION OF WELLBORE STORAGE EFFECTS ON ANALYSIS OF WELL TEST DATA By Omid Shahbazi

Supervisor: Dr.A.Hashemi

A Thesis Submitted To Department of Petroleum Engineering For Fulfilment Of Bsc Degree Of Petroleum Engineering

MAY 2011

Abstract Usually flow rate of wells is controlled from the surface and the pressure is measured at the bottom hole. At the beginning of Drawdown test or Buildup test, the production is due to expansion of fluids and we could not use the radial flow equations for this period of time. After any change of surface rate, there is a time lag between the surface production and the sand face rate. This effect is called wellbore storage. The analysis/interpretation of wellbore storage distorted pressure transient test data remains one of the most significant challenges in well test analysis .there is two models for wellbore storage 1. Constant wellbore storage 2. Changing wellbore storage For the elimination of wellbore storage effects in pressure transient test data, a variety of methods using different techniques have been proposed .The objective of this project is to investigate the wellbore storage effects in oil wells. We wish to determine the extent of this effect using a test design for different type of reservoir and well model.

I

ACKNOWLEDGEMENTS

I want to express my gratitude to my supervisor Dr.A.Hashemi for his unfailingly positive attitude,encouragement,trust and support all long. I would also like to express my gratitude to everyone who offered friendship and encouragement along the way.

II

Nomenclature

Dimentionless variables CD

dimensionless wellbore storage coefficient

kfD

dimensionless fracture permeability

pD

dimensionless pressure

tD

dimensionless time

tDxf

dimensionless time based on the fracture length

wfD

dimensionless fracture width

Field variables ap

Constant for the pressure drop model

aq

Constant for the rate model

A

Constant for the power law deconvolution model

bp

Constant for the pressure drop model

bq

Constant for the rate model

B

Constant for the power law deconvolution model

Bo

Formation volume factor, bbl/STB

C

Wellbore storage coefficient, bbl/psi

C2

Arbitrary constant, hr-1

cf

Final fluid compressibility,psi-1

ci

Initial fluid compressibility,psi-1

ct

Total compressibility, psi-1 III

h

Reservoir net pay thickness, ft

hw

Perforated thickness, ft

k

Reservoir penneability, md

kf

Fracture permeability, md

kH

Horizontal permeability, md

km

Matrix blocks permeability, md

kr

Radial permeability, md

ks

Spherical permeability, md

kV

Vertical permeability, md

m(p)

Pseudo pressure, psia2/cp

msl

Slope of semilog plot, psi/hr

mwbs

Slope of wellbore storage dominated regime, psi/hr

Np

Cumulative oil production, STB

p

Reservoir pressure, psi

pwf(Δt=0)

Wellbore pressure at the time of shut-in, psia

pu

Constant rate pressure response,psi

pwf

Flowing bottomhole pressure, psia

pws

Shut-in bottomhole pressure, psia

q

Volumetric production rate, STB/D

qwb

Wellbore unloading fluid rate,STB/D

qsf

Sandface rate, STBID

rw

Wellbore radius, ft

S

Skin coefficient, or saturation

IV

Sm

Matrix skin

Spp

Geometrical skin of partial penetration

ST

Total skin

Sw

Skin over the perforated thickness

t

Producing time, hr

tps

Pseudotime,hr

Vw

Wellbore volume, bbl

w

Fracture width, ft

xf

Fracture half-length, ft

Zw

Distance to the lower reservoir limit, ft

Δt

Shut-in time, hr

Δps

Wellbore pressure drop for deconvolved, constant rate data,psi

Greek α

Geometric coefficient in λ or "beta-deconvolution" variable(field variable), hr-1

β

beta –deconvolution variable, hr-1

Δ

Difference

ϕ

Porosity, fraction

κ

Mobility ratio

λ

Interporosity (or layer) flow coefficient

μ

Oil viscosity, cp

ω

Storativity ratio

V

ρ

Fluid density, lb/cuft

Subscript D

Dimensionless

m

Matrix

pp

Partial penetration

ps

Pseudo

sf

Sandface

w

Observed data (variable rate)

wbs

Wellbore storage

wf

Flowing well conditions

ws

Shut-in well conditions

VI

TABLE OF CONTENTS

ABSTRACT .............................................................................................................................. I ACKNOWLEDGEMENTS .................................................................................................. .II NOMENCLATURE .............................................................................................................. III TABLE OF CONTENTS .................................................................................................... VII LIST OF FIGURES .............................................................................................................. IX LIST OF TABLES .................................................................................................................. X Chapter 1 .................................................................................................................................. 1 INTRODUCTION ..................................................................................................................... 1 Chapter 2 .................................................................................................................................. 3 REVIEW OF WELL TEST ANALYSIS .................................................................................. 3 2.1.Pressure Transient Tests ................................................................................................... 3 2.2.Well Model....................................................................................................................... 3 2.2.1.Vertical Well.............................................................................................................. 4 2.2.2.Fractured Model ........................................................................................................ 4 2.2.3. Partial Penetration..................................................................................................... 7 2.3. Reservoir Model .............................................................................................................. 9 2.3.1. Homogenous ............................................................................................................. 9 2.3.2. Dual Porosity ............................................................................................................ 9 2.3.3. Dual Permeabillity .................................................................................................. 11 Chapter 3 ................................................................................................................................ 14 WELLBORE STORAGE MODELS ....................................................................................... 14 3.1. Constant Wellbore Storage............................................................................................ 14 3.2. Changing Wellbore Storage .......................................................................................... 17 3.2.1. Use of a changing wellbore storage analytical model ............................................ 18 3.2.2. Use of pseudotime .................................................................................................. 19 3.2.3. Use of a numerical model ....................................................................................... 20 Chapter 4 ................................................................................................................................ 21 METHODS FOR THE ANALYSIS OF WELLBORE STORAGE DISSTORTED WELL TEST DATA ............................................................................................................................ 21 VII

4.1. Russell Method ............................................................................................................. 21 4.2. Rate Normalization ....................................................................................................... 23 4.2.1. Gladfelter Rate Normalization................................................................................ 23 4.2.2. Fetkovich Rate Normalization ................................................................................ 24 4.3. Material Balance Deconvolution................................................................................... 24 4.4. Power Deconvolution .................................................................................................... 25 4.5. β - Deconvolution .......................................................................................................... 27 Chapter 5 ................................................................................................................................ 28 INVESTIGATION OF WELLBORE STORAGE EFFECT ON WELL TEST DATA USING TEST DESIGN ........................................................................................................................ 28 5.1.Oil well Data .................................................................................................................. 29 5.1.1. Constant wellbore storage, Homogenous reservoir, Vertical well, Infinit acting .. 30 5.1.2. Constant wellbore storage, Homogenous reservoir,H.C fracture, Infinit acting .... 31 5.1.3. Constant wellbore storage, Homogenous reservoir,L.C fracture, Infinit acting..... 32 5.1.4. Constant wellbore storage, Homogenous reservoir, Limited entry well, Infinit acting ................................................................................................................................ 33 5.1.5. Constant wellbore storage, Double porosity reservoir, Vertical well, Infinit acting ..................................................................................................................................... ….34 5.1.6. Constant wellbore storage,Double permeability reservoir,Vertical well,Infinit acting ................................................................................................................................ 35 Chapter 6 ................................................................................................................................ 36 CONCLUSION AND RECOMMENDATION ....................................................................... 36 6.1.Conclusion...................................................................................................................... 36 6.2.Recommendation............................................................................................................ 36 REFRENCES ......................................................................................................................... 37 APPENDIX A ......................................................................................................................... 39 APPENDIX B ......................................................................................................................... 40 APPENDIX C ......................................................................................................................... 42

VIII

LIST OF FIGURES Fig2. 1.Responses for a well with wellbore storage and skin in an infinite homogeneous reservoir ..... 4 Fig2. 2.Schematic of fractures ................................................................................................................ 4 Fig2. 4.loglog responce of finite conductivity fracture ........................................................................... 5 Fig2. 3.Flow Pattern of finite conductivity fracture................................................................................ 5 Fig2. 6.loglog responce of infinite conductivity fracture ........................................................................ 6 Fig2. 5.Flow Pattern of infinite conductivity fracture............................................................................. 6 Fig2. 7.Flow pattern of partial penetration ............................................................................................. 7 Fig2. 8.Schematic of partial penetration ................................................................................................. 7 Fig2. 9.Log-log response Sensitivity to anisotropy kV/kH....................................................................... 8 Fig2. 10.Semi-log response Sensitivity to anisotropy kV/kH ................................................................... 8 Fig2. 11.Fissured system production ...................................................................................................... 9 Fig2. 12.Pressure and derivative response for a well in double porosity reservoir............................... 10 Fig2. 13.Sensitivity to ω in double porosity reservoir .......................................................................... 10 Fig2. 14.Sensitivity to λ in double porosity reservoir ........................................................................... 11 Fig2. 15.Double permeability reservoir ................................................................................................ 11 Fig2. 16.loglog Responce of double porosity reservoir when two layers are producing ...................... 12 Fig2. 17.loglog Responce of double porosity reservoir when layer 2 is producing............................. 12 Fig2. 18.log log responce sensitivity to κ with high storativity contrast .............................................. 13 Fig2. 19.log log responce sensitivity to κ with low storativity contrast................................................ 13 Fig3. 1.Schematic diagram of well and formation during pressure build-up ........................................ 14 Fig3. 2.Wellbore storage effect. Sand face and surface rates ............................................................... 15 Fig3. 3.Wellbore storage log-log responses .......................................................................................... 16 Fig3. 4. log-log plot of pressure drop versus time ................................................................................ 16 Fig3. 5.Production; increasing storage ................................................................................................. 17 Fig3. 6.Build-up; decreasing storage …………………………………………………………………17 Fig4. 1. Schematic plot showing determination of the correct C2 value………………………..….....22 Fig5. 1.Input Temperature an Pressure…..............................................................................................28 Fig5. 2.Input reservoir characteristic……………………………………………….………………....28 Fig5. 3.schematic of model chosen in test design……………………………………………………..29 Fig5. 4.test design screen……………………………………………………………………………...29 Fig5. 5.Sensitivity to C for homogenous reservoir……………………………………………………30 Fig5. 6.Sensitivity to C for high conductivity fracture…………………………………………….….31 Fig5. 7.Sensitivity to C for low conductivity fracture ………………………….…………………….32 Fig5. 8.Sensitivity to C for limited entry well…………………………………………………….......33 Fig5. 9.Sensitivity to C for Double porosity reservoir………………………………………………...34 Fig5. 10.Sensitivity to C for Double permeability reservoir…………………………………………..35 IX

LIST OF TABLES Table5. 1.Reservoir properties ................................................................................................. 28 Table5. 2.Reservoir initial condition ....................................................................................... 28 Table5. 3.Fluid properties ........................................................................................................ 29 Table5. 4.Production data ........................................................................................................ 29 Table5. 5.Start of flow regimes for limited entry well ............................................................ 33 Table5. 6.Start of flow regimes for double porosity reservoir ................................................. 34 Table5. 7.Start of flow regimes for double permeability reservoir..………..…………….….35

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Investigation of wellbore storage effect on analysis of well test data

Chapter 1

INTRODUCTION During a well test, a transient pressure response is created by a temporary change in production rate. The well response is usually monitored during a relatively short period of time compared to the life of the reservoir, depending upon the test objectives. For well evaluation, tests are frequently achieved in less than two days. In the case of reservoir limit testing, several months of pressure data may be needed. When a well is opened, the production at surface is first due to the expansion of the fluid in the wellbore, and the reservoir contribution is negligible. After any change of surface rate, there is a time lag between the surface production and the sand face rate. This effect is called wellbore storage

[1]

. For instance, during the beginning of a

buildup test (often referred to as "afterflow"), wellbore storage affects the pressure and flowrate in such a way that these rates rapidly fall below the measurement threshold of the tools, which then record a no-flow period.

This scenario causes a loss of

information with regard to the behavior in the wellbore and in the reservoir. In the presence of such limitations, well test interpretation techniques have been developed to analyze the wellbore storage distorted pressure response — using only pressure transient data (which are recorded with higher accuracy than the well flowrates). For the elimination of wellbore storage effects in pressure transient test data, a variety of methods using different techniques have been proposed. An approximate "direct" method by Russell[2] "corrects" the pressure transient data distorted by wellbore storage into an equivalent pressure function for the constant rate case. Rate normalization techniques [Glatfelter. Al[18], Fetkovich and Vienot [19]] have also been employed to correct for wellbore storage effects and these rate normalization methods were successful in some cases. The application of rate normalization requires measured sandface rates, and generally yields a shifted results trend that has the correct slope (which should yield the correct permeability estimate), but incorrect intercept in a semilog plot (which will yield an incorrect skin factor). Material balance deconvolution (an enhancement of rate normalization) is also thought to require continuously varying sandface flowrate measurements. Power and Beta

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Investigation of wellbore storage effect on analysis of well test data

deconvolution are another methods that compute the undistorted pressure drop function directly from the wellbore storage affected data. First ,we should review the well test analysis methods for some reservoir and well models that we are going to investigate wellbore storage effect for them. Second ,two models for wellbore storage(constant and changing) will be discussed in chapter 3. Finally, we will review methods for eliminating this effect also investigate wellbore storage sensitivity to change in well model and reservoir model. Here we change model parameters to see the sensitivity of model to these parameters. Main parameter that we change is Wellbore storage coefficinet.we want to investigate the effect of wellbore storage coefficient on pressure response of models(including reservoir model and well model).

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Investigation of wellbore storage effect on analysis of well test data

Chapter 2

REVIEW OF WELL TEST ANALYSIS 2.1.Pressure Transient Tests Oil well test analysis is a branch of reservoir engineering. Information obtained from flow and pressure transient tests about in situ reservoir conditions are important to determining the productive capacity of a reservoir. Pressure transient analysis also yields estimates of the average reservoir pressure. The reservoir engineer must have sufficient information about the condition and characteristics of reservoir/well to adequately analyze reservoir performance and to forecast future production under various modes of operation. The production engineer must know the condition of production and injection wells to persuade the best possible performance from the reservoir. Pressures are the most valuable and useful data in reservoir engineering. Directly or indirectly, they enter into all phases of reservoir engineering calculations. Therefore accurate determination of reservoir parameters is very important. In general, oil well test analysis is conducted to meet the following objectives: • To evaluate well condition and reservoir characterization • To obtain reservoir parameters for reservoir description • To determine whether all the drilled length of oil well is also a producing zone • To estimate skin factor or drilling- and completion-related damage to an oil well. Based upon the magnitude of the damage, a decision regarding well stimulation can be made[17].

2.2.Well Model Well geometries are generally assessed in the first part of the well test / production response, after wellbore effects have faded. Specific flow regimes related to the well geometry may allow the engineer to assess well parameters, sometimes in complement of some reservoir parameters. 2.2.1.Vertical Well The simplest model is a vertical well fully penetrating the reservoir producing interval. This is the model used to derive the basic equations . This model is sometimes called

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Investigation of wellbore storage effect on analysis of well test data

‘wellbore storage & skin’, reference to the original type-curves of the 1970’s. The reason is that the two only parameters affecting the log log plot response will be the wellbore storage and the skin factor. On the log-log plot, the shape of the derivative response, and with a much lower sensitivity the shape of the pressure response, will be a function of the factor CD.e2s .

Fig2. 1.Responses for a well with wellbore storage and skin in an infinite homogeneous reservoir [1]

2.2.2.Fractured Model The interpretation of well tests from such wells must therefore consider the effects of the fracture; indeed, often tests of fractured wells are conducted specifically to determine fracture properties so that the effectiveness of the fracture stimulation operation can be evaluated. The case of common practical interest is of a vertical fracture of length xf, fully penetrating the formation .

Fig2. 2.Schematic of fractures[3]

For the purposes of fractured well analysis, we often make use of a dimensionless time tDxf based on the fracture length xf:

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Investigation of wellbore storage effect on analysis of well test data

tDxf =

0.000264 kt ϕµct xf 2

(2.1)

In well test analysis , two main fracture types are commonly considered 2.2.2.1.Finite Conductivity Fracture The most general case of a finite conductivity fracture was considered by Cinco, Samaniego and Dominguez (1978) [18] and Cinco and Samaniego (1981) [19].

Fig2. 3.Flow Pattern of finite conductivity fracture [3]

Fig2. 4.log log response of finite conductivity fracture [3]

At early time, there is linear flow within the fracture and linear flow into the fracture from the formation. The combination of these two linear flows gives rise to the bilinear flow period. This part of the response is characterized by a straight line response with slope ¼ at early time on a log-log plot of pressure drop against time since the pressure drop during this period is given by: 1 2.451 tDxf 4 k fD .w fD

PD =

(2.2)

where the dimensionless fracture permeability and width are given respectively by:

kfD =

kf k

(2.3)

wfD =

w xf

(2.4)

Following the bilinear flow period, there is a tendency towards linear flow, recognizable by the upward bending in Fig. 2.4 towards a ½ slope on the log-log plot. In practice, the ½ slope is rarely seen except in fractures where the conductivity is infinite. Finite

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Investigation of wellbore storage effect on analysis of well test data

conductivity fracture responses generally enter a transition after bilinear flow (¼ slope), but reach radial flow before ever achieving a ½ slope (linear flow).

2.2.2.2. Infinite Conductivity Fracture If the product kfD.wfD is larger than 300, then the fracture conductivity can be considered to be infinite. In the high conductivity we assume that the pressure drop along the inside of the fracture is negligible .

Fig2. 5.Flow Pattern of infinite conductivity fracture [3] Fig2. 6.log log response of infinite conductivity fracture[3]

Such highly conductive fractures are quite possible in practice, especially in formations with lower permeability. The pressure response of a well intersecting an infinite conductivity fracture is very similar to that of the more general finite conductivity fracture case, except that the bilinear flow period is not present. A high conductivity fracture response is characterized by a truly linear flow response , during which the pressure drop is given by: 1

PD =  π .tDxf  2

(2. 5)

Such a response shows as a ½ slope straight line on a log-log plot of pressure drop against time as shown by Gringarten, Ramey and Raghavan [17].

Beyond the linear flow period, the response will pass through a transition to infinite acting radial flow (semilog straight line behavior).

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Investigation of wellbore storage effect on analysis of well test data

2.2.3. Partial Penetration This model assumes that the well produces from a perforated interval smaller than the drained interval. In theory, after wellbore storage, the initial response can be radial in the perforated interval hw (Fig.2.7), shown as ‘Radial Flow’ (Fig.2.8). This will give a pressure match resulting in the mobility krhw (the subscript r stands for radial) and it can be imagined that if there were no vertical permeability this would be the only flow regime. In practice this flow regime is often masked by storage. In flow regime ‘Spherical flow’ (Fig.2.8) there is a vertical contribution to flow, and if the perforated interval is small enough a straight line of slope –½ (negative half slope) may develop in the derivative, corresponding to spherical or hemi-spherical flow. Finally, when the upper and lower boundaries have been seen, the flow regime becomes radial again, and the mobility now corresponds to the normal krh.

Fig2. 7.Flow pattern of partial penetration[1]

Fig2. 8.Schematic of partial penetration [5]

The total skin combines the wellbore skin Sw and an additional geometrical skin Spp due to distortion of the flow lines •

Spp is large when the penetration ratio hw/h or the vertical permeability kV is low (high anisotropy kH/kV).

•

For damaged wells, the product (h/hw)Sw can be larger than 100.

ST =

h S w + S pp hw

(2.6)

When the vertical permeability kV is low (low kV/kH), the start of the spherical flow regime is delayed (-1/2) derivative slope moved to the right.

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Investigation of wellbore storage effect on analysis of well test data

Fig2. 9.Log-log response Sensitivity to anisotropy kV/kH [1]

Fig2. 10.Semi-log response Sensitivity to anisotropy kV/kH [1]

During spherical and hemispherical flow linearity will develop in a plot of Δp versus 1/√∆t slope m related to the spherical permeability ∛(kx kykv )  Φµct k s =  2452.9qBµ  m 

   

2

3

(2.7)

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Investigation of wellbore storage effect on analysis of well test data

kv  k s  =  k r  k r 

3

(2.8)

2.3. Reservoir Model In Pressure Transient Analysis , reservoir features are generally detected after wellbore effects and well behavior have ceased and before boundary effects are detected. This is what we might call a Middle time response. 2.3.1. Homogenous The homogeneous reservoir is the simplest possible model assuming everywhere the same porosity, permeability and thickness. The permeability is assumed isotropic. That is, the same in all directions. The governing parameters are: kh

Permeability-thickness product, given by the pressure match.

φct

Reservoir storativity, input at the initialization of a standard test or as a result in interference tests.

S

Skin

2.3.2. Dual Porosity The double-porosity (2Φ) models assume that the reservoir is not homogeneous, but made up of rock matrix blocks with high storativity and low permeability. The well is connected by natural fissures of low storativity and high permeability. The matrix blocks can not flow to the well directly, so even though most of the hydrocarbon is stored in the matrix blocks it has to enter the fissure system in order to be produced.

Fig2. 11.Fissured system production[4]

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Investigation of wellbore storage effect on analysis of well test data

Fig2. 12.Pressure and derivative response for a well in double porosity reservoir[1]

The double-porosity model is described by two other variables in addition to the parameters defining the homogeneous model: 1. ω is the storativity ratio, and is essentially the fraction of fluids stored in the fissure system (e.g. ω=0.05 means 5%).

ω=

(φVct ) f (φVct ) f + (φVct ) m

=

(φVct ) f (φVct ) f +m

(2.9)

Fig2. 13.Sensitivity to ω in double porosity reservoir [1]

2. λ is the interporosity flow coefficient that characterizes the ability of the matrix blocks to flow into the fissure system. It is dominated by the matrix/fissures permeability contrast, km/kf.

λ = αrw 2

km kf

(2.10)

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Investigation of wellbore storage effect on analysis of well test data

Fig2. 14.Sensitivity to λ in double porosity reservoir [1]

2.3.3. Dual Permeability In the double-permeability (2k) model the reservoir consists of two layers of different permeabilities, each of which may be perforated or not. Crossflow between the layers is proportional to the pressure difference between them.

Fig2. 15.Double permeability reservoir [4]

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Investigation of wellbore storage effect on analysis of well test data

Fig2. 16.log log Response of double porosity reservoir when two layers are producing [1]

Fig2. 17.log log Response of double porosity reservoir when layer 2 is producing [1]

There is one more parameter than seen in the previous double-porosity PSS model. ω and λ have equivalent meanings. ω, layer storativity ratio, is the fraction of interconnected pore volume occupied by layer 1

ω=

(φVct )1 (φVct )1 + (φVct ) 2

(2.11)

λ , inter-layer flow parameter, describes the ability of flow between the layers

λ = αrw 2

(kh)1 (kh)1 + (kh) 2

(2.12)

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Investigation of wellbore storage effect on analysis of well test data

In addition another coefficient is introduced: κ is the ratio of the permeability-thickness product(Mobility ratio) of the first layer to the total for both layers:

κ=

(kh)1 (kh)1 + (kh) 2

Fig2. 18.log log response sensitivity to κ with high storativity contrast[1]

Fig2. 19.log log response sensitivity to κ with low storativity contrast[1]

(2.13)

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Investigation of wellbore storage effect on analysis of well test data

Chapter 3

WELLBORE STORAGE MODELS 3.1. Constant Wellbore Storage

Fig3.1.Schematic diagram of well and formation during pressure build-up[2]

Whenever a well is shut in, fluid from the formation will flow into the wellbore until equilibrium conditions are reached. Similarly, a part of the fluid produced when a well is put on production is the fluid that was present is the wellbore prior to the opening of the well. This "ability of the well to store and unload fluids"[18] is the definition of wellbore storage. qwb = −

24C dpwf B dt

(3.1)

Where qwb represents the rate at which the wellbore "unloads" fluids, and C represents the storage constant of the well. In the specific case where the wellbore storage is entirely due to fluid expansion, then the wellbore storage constant is defined by:[19] C=

∆V ∆p

(3.2)

Where ΔV is the change in volume of fluid in the wellbore — at wellbore conditions — and Δp is the change in bottom hole pressure. When the wellbore is filled with a single fluid phase, Eq. 3.2 becomes

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Investigation of wellbore storage effect on analysis of well test data

C = Vwc

(3.3)

where Vw is the total wellbore volume and c is the compressibility of the fluid in the wellbore at wellbore conditions. The use of dimensionless pressure functions in most of the derivations of this work leads to the use of a dimensionless wellbore storage coefficient, CD.

CD =

0.894C φct hrw2

(3.4)

Fig3.2.Wellbore storage effect. Sand face and surface rates [3]

The overall effect of wellbore storage can be seen in Fig 3.3

q = qsf + qw

(3.5)

At early time qsf is close to zero, as all the fluid produced at the wellhead originates in the wellbore. As time goes on, the wellbore storage is depleted, and eventually the reservoir produces all the fluid .The corresponding pressure transients due to the wellbore storage effects are seen in Fig. 3.3. It is important to recognize that, as a consequence of the wellbore storage effect, the early transient response during a well test is not characteristic of the reservoir, only of the wellbore. This means that a well test must be long enough that the wellbore storage effect is over and fluid is flowing into the wellbore from the reservoir. As we will see later, we can also overcome the problem of wellbore storage by specifically measuring the sandface flow rate qsf down hole. The form or the width of the hump is governed by the parameter group Ce2s , the position of the curves in time is governed by the wellbore storage coefficient C.

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Investigation of wellbore storage effect on analysis of well test data

100

C =0.01

C=0.1 C=1

C=10

10

1 0.01

0.1

1

10

100

1000

Fig3. 3.Wellbore storage log-log responses

From material balance, the pressure in the wellbore is directly proportional to time during the wellbore storage dominated period of the test:

pD =

tD cD

(3.6)

On a log-log plot of pressure drop versus time, this gives a characteristic straight line of unit slope (Fig. 3.4).

Fig3. 4. log-log plot of pressure drop versus time[3]

The unit slope straight line response continues up to a time given approximately by: t D = C D (0.041+ 0.02s)

(3.7)

provided that the skin factor s is positive. However, the storage effect is not over at this time, as there is a period (roughly one and a half log cycles long) during which the response undergoes a transition between wellbore response and reservoir response. Thus the reservoir response does not begin until a time:

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Investigation of wellbore storage effect on analysis of well test data

(3.8)

t D = C D (60+ 3.5s)

During the design of a test, care should be taken to ensure that the test is at least this long and usually very much longer, even if nonlinear regression techniques are to be used for the interpretation.

3.2. Changing Wellbore Storage Wellbore storage may vary. This is the case when fluid compressibility varies in the wellbore during the test operation. A typical case is tight gas reservoirs, where the pressure drop in the well will be considerable and the compressibility will vary during both production and shut-in periods. In such case the wellbore storage may vary considerably during the flow period being analyzed. As another example, an oil well flowing above bubble point condition in the reservoir may see gas coming out of solution in a wellbore below bubble point pressure. Initially oil compressibility would be dominating and this would gradually change to gas as more and more gas is produced in the wellbore. We would have the phenomenon of ‘increasing’ wellbore storage. When, the well is shut in the reverse will happen. First, the gas first dominates, and then later the oil compressibility takes over. The response will exhibit decreasing storage.

Fig3. 5.Production; increasing storage [4]

Fig3. 6.Build-up; decreasing storage [4]

Other conditions may produce changing wellbore storage: • Falling liquid level in an injector during fall-off (increasing); • Pressure dependant gas PVT during a build-up (decreasing) or production (increasing); • Diameter change in the completion in a well with a rising or falling liquid level;

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Investigation of wellbore storage effect on analysis of well test data

• Redistribution of phases in a multi phase producer (also called ‘humping’). Some analytical formulations of changing wellbore storage may be integrated in analytical and even numerical models. The two most popular formulations are from Hegeman et al[17] and Fair. The assumptions are that the wellbore storage starts at one value, and then there is a change to a second value where it remains constant. The input is final storage Cf, the ratio of initial to final storage (Ci/Cf) and the dimensionless time (α) of the change of the storage. The response of the Hegeman model was already presented in Fig. 3.4 and Fig. 3.5

3.2.1. Use of a changing wellbore storage analytical model The easiest way to match such data is the use of an analytical changing wellbore storage model. In the case above it is a decreasing wellbore storage option. Using any software, the principle will be to position the early storage and the time at which the transition takes place (in this case, the position of the hump in the derivative response). The initial model generation may be approximate, but non linear regression will generally obtain a good fit. There are two main models used in the industry. These are the Fair and the Hageman et al [17]models. The latter is more recent and more likely to match this sort of response.

However these models should be use with care, for three reasons: • These models are just transfer functions (in Laplace space) that just happen to be good at matching the real data. There is no physics behind them. They may end up with initial, final storage and transition time that make no physical sense. • These models are time related. There will be a wellbore storage at early time and a wellbore storage at late time. This is not correct when one wants to model pressure related storage. In the case of a production, the real wellbore storage at early time will correspond to the storage at late time of the build-up, and the reverse. So the superposition of a time related solution will be incorrect on all flow periods except the one on which the model was matched. This aspect is very often overlooked by inexperienced interpretation engineers.

18

Investigation of wellbore storage effect on analysis of well test data

3.2.2. Use of pseudotime This variable was, historically, the first solution used to modify the data to take into account changing wellbore storage, and transform the data response that could then be matched with a constant wellbore storage model or type-curve. The principle was that the model does not match the data, but rather the opposite, the data matches the model. Using pseudotime and considering the diffusion equation the idea was to enter in the time function the part that is pressure related, i.e. the viscosity compressibility product. The gas diffusion equation can then be re-written:

Modified diffusion equation:

∂m( p ) k = 0.0002637 ∇ 2 m( p ) ∂t φ µct

(3.9)

We introduce the pseudotime: t

t ps (t ) = ∫ I ( pwf (τ ))dτ

where

I ( p) =

0

1 µct

(3.10)

or, better, the normalized version: t

t ps (t ) = ∫ I ( pwf (τ ))dτ

where

0

I ( p) =

( µct ) ref

(3.11)

µct

Where the diffusion equation becomes:

∂m( p ) k = 0.0002637 ∇ 2 m( p ) φ ( µct ) ref ∂t ps

(3.12)

Although this is not a perfect solution the equation becomes closer to a real linear diffusion equation. In addition, the time function is essentially dependent on the gas compressibility and the pseudotime will therefore mainly compensate the change of wellbore storage in time. The replacement of the time by the pseudotime in the log log plot will therefore compress the time scale, which, on a logarithmic scale, will mean an expansion of the X axis to the right at early time. As a result, the compressed wellbore

19

Investigation of wellbore storage effect on analysis of well test data

storage shape of the derivative response will become closer to a constant wellbore storage solution.

There are two drawbacks to this approach:

•

This method modifies, once and for all, the data to match the model, and not the opposite. This excludes, for example, the possibility of comparing several PVT models on the same data. The method was the only one available at the time of type-curve matching, where models were limited to a set of fixed drawdown type-curves.

•

In order to calculate the pseudotime function one needs the complete pressure history. When there are holes in the data, or if the pressure is only acquired during the shut-in, it will not be possible to calculate the pseudotime from the acquired pressure. There is a workaround to this: use the pressures simulated by the model, and not the real pressures.

This amounts to the same thing once the model has matched the data, and there is no hole. However it is a bit more complicated for the calculation, as the pressure at a considered time requires the pseudotime function, and vice versa.

3.2.3. Use of a numerical model The principle is to use a wellbore model which, at any time, uses the flowing pressure to define the wellbore storage parameter. In order for the model to be stable, the wellbore storage has to be calculated implicitly at each time step. As the problem is not linear, this can only be done using Saphir Nonlinear . This is by far the best way to simulate pressure related wellbore storage. However there are a couple of drawbacks:

•

The model is slower than an analytical model or a change of time variable

•

It is inflexible: once you have entered the PVT and the wellbore volume there is no parameter to control the match. The model works, or it does not.

20

Investigation of wellbore storage effect on analysis of well test data

Chapter 4

METHODS FOR THE ANALYSIS OF WELLBORE STORAGE DISSTORTED WELL TEST DATA

4.1. Russell Method (1966): this method does not provide results which can be considered useful in the context of modern well test analysis and interpretation methods. Russell made the following assumptions in the derivation of his wellbore storage "correction" solution:  Completely penetrating well in an infinite reservoir  Slightly compressible liquid (constant compressibility)  Constant fluid viscosity  Single-phase liquid flow in the reservoir  Gravity and capillary pressure neglected  Constant permeability  Horizontal radial flow (no vertical flow)  Ideal gas (for the gas cushion in the well) Russell's [2] wellbore storage correction is given as: pws (∆t ) − pwf (∆t = 0) = 162.6

qBµ kh

  k 1  − 3.23 + 0.87 s  1 −  log(∆t ) + log φµcrw2  C2 ∆t   

(4.1)

Where the C2-term is defined as: C 2 = 0.000528

1 kh   ρg + p wf (∆t = 0) 2  L  rt µ

(4.2)

Combining Eqs. 4.1 and 4.2 into a plotting function format, we obtain: [ p ws (∆t ) − p wf (∆t = 0)]  1  1 −   C 2 ∆t 

= f (∆t = 1 hr ) + msl log(∆t )

(4.3)

21

Investigation of wellbore storage effect on analysis of well test data

Russell treated the C2-term as an arbitrary constant to be optimized for analysis — in other words, the C2-term is the "correction" factor for the Russell method. As prescribed by Russell, the C2-term is obtained using a trial-and-error sequence which yields a straight line when the left-hand-side term of Eq. 4.3 is plotted versus log(Δt). Where the general form of the y-axis correction term prescribed by Eq. 4.3 is: y=

[ p ws (∆t ) − p wf (∆t = 0)]  1  1 −   C 2 ∆t 

(4.4)

A schematic of the Russell method is shown in Fig. 4.1, where we note Russell's interpretation of the effect of the C2-term (i.e., where C2 is too large and C2 is too small).

Fig.4. 1. Schematic plot showing determination of the correct C2 value[2]

Once the C2-term is established, the kh-product is estimated using:

kh = 162.6

qBµ m sl

(4.5)

And the skin factor can be estimated using:  f (∆t = 1 hr )  k s = 1.151  − log + 3.23 2 m sl φµcrw  

(4.6)

22

Investigation of wellbore storage effect on analysis of well test data

In short, the Russell method has an elegant mathematical formulation, but ultimately, we believe that this formulation does not represent the wellbore storage condition, and hence, we do not recommend the Russell method under any circumstances. 4.2. Rate Normalization 4.2.1. Glatfelter Rate Normalization Glatfelter, Tracy and Wilsey

[18]

introduced the "rate normalization" deconvolution

approach which, in their words "permits direct measurement of the cause of low well productivity." The objective of rate normalization is to remove/correct the effects of the variable rate from the observed pressure data.

Rate normalization can also be defined as an

approximation to convolution integral [18].

∆p(t ) ≈ q(t ) pu (t )

(4.7)

Where pu is the constant rate pressure response. The afterflow rate-normalized pressure equation proposed by Glatfelter et al to analyze pressure buildup data dominated by afterflow was given as [19] pws ∆(t ) − pwf , s qo − q (∆t )

=

 k (∆t ) 162.6 µ  − 3.23 + 0.87 s  log 2 kh  φµct rw 

(4.8)

Eq. 4.8 indicates that a plot of pws ∆(t ) − pwf , s qo − q (∆t )

vs. log(∆t )

(4.9)

should be linear with slope equal to m′ =

162.6 µ kh

(4.10)

The skin is determined from

  p ws ∆(t ) − p wf ,s k (∆t ) s = 1.515 − log + 3 . 23  φµct rw 2   qo − q (∆t )

(4.11)

23

Investigation of wellbore storage effect on analysis of well test data

Rate normalization has been employed for a number of applications in well test analysis. For the specific application of "rate normalization" deconvolution, we must recognize that the approach is approximate — and while this method does provide some "correction" capabilities, it is basically a technique that can be used for pressure data influenced by continuously varying flowrates. 4.2.2. Fetkovich Rate Normalization Rate normalization techniques and procedures are best illustrated by first examining their application to drawdown data. Although the nature of the rate variation of drawdown data with time is different than that of afterflow rate variation, the end result is the same. Also, drawdown rate variations generally last much longer than afterflow rate variations.[19]

Most notably, Fetkovich and Vienot

[19]

, and Doublet et al.[18] ,have demonstrated the

effectiveness of "rate normalization" deconvolution (albeit for specialized cases). In particular, for the wellbore storage domination and distortion regimes, rate normalization can provide a reasonable approximation of the no wellbore storage solution. For this infinite-acting radial flow case, rate normalization yields an erroneous estimate of the skin factor by introducing a shift on the semi log straight line (obviously, the sandface rate profile must be known). This last point, however, makes the application of rate normalization techniques very limited in our particular problem — we do not have measurements of sandface flowrate. Therefore, this method must be applied using an estimate of the downhole rate— which will definitely introduce errors in the deconvolution process.

Such issues make rate normalization a "zero-order"

approximation that is, rate normalization results should be considered as a guide, but not relied upon as the best methodology. 4.3. Material Balance Deconvolution Material balance deconvolution is an extension of the rate normalization method. Johnston[19] defines a new x-axis plotting function (material balance time) which provides an approximate deconvolution of the variable-rate pressure transient problem. The general form of material balance deconvolution is provided for the pressure drawdown case in terms of the material balance time function and the rate-normalized pressure drop function. The material balance time function is given as:

24

Investigation of wellbore storage effect on analysis of well test data

t mb =

Np

(4.12)

q

The wellbore storage-based, material balance time function for the pressure buildup case is given as: 1 ∆p ws mwbs ∆t mb, BU = = d 1 − q wbs, BU 1 − 1 [∆p ws ] mwbs d∆t Np, wbs, BU

∆t −

(4.13)

And the wellbore storage-based, rate-normalized pressure drop function for the pressure buildup case is given as:

∆p s, BU =

∆p ws = 1 − q wbs, BU 1 −

1 ∆p ws 1 d [∆p ws ] mwbs d∆t

(4.14)

Plotting the rate-normalized pressure function versus the material balance time function (on log(tmb) scales) shows that the material balance time function does correct the erroneous shift in the semilog straight-line obtained by rate normalization. We believe that the material balance deconvolution technique is a practical approach (perhaps the most practical approach) for the explicit deconvolution of pressure transient test data which are distorted by wellbore storage and skin effects. 4.4. Power Deconvolution This development assumes that variable (or constant) rate flow conditions exist in the reservoir. We will use the familiar convolution integral to develop our new deconvolution technique, The convolution integral is given as tD

′ pwD (t D ) = ∫ qD (τ ) psD (t D − τ )dτ

(4.15)

0

The development of this method requires the Laplace transform and if we find that the inverse Laplace transform it becomes

25

Investigation of wellbore storage effect on analysis of well test data

psD (t D ) = α Dt D

βD

(4.16)

In order to apply Eq. 18 to field data, i.e., time, pressure, and flowrate, we merely compute qD= q(t)/q and substitute Δpw and Δps for pwD(tD) and pws(tD). now we will describe how to use this deconvolution method, and apply it to field well test data .we should outline a general procedure to convert wellbore storage dominated data to the data that would have been obtained if storage effects were non-existent, and to recommend methods of calculating the sandface rate during afterflow.[17] 1. Obtain mwbs and Po: a. Drawdown I. mwbs is the slope of the early time data from a plot of Pwf vs. t . ii. Po = Pi , the intercept of the plot of Pwf vs. t . b. Buildup I. mwbs is the slope of the early time data from a plot of Pws vs. Δt . ii. Po = Pwf (Δt = 0), the intercept of the plot of Pws vs. Δt. 2. Compute the pressure drop function, Δpw, and the time function, t: a. Drawdown I. Δpw = Po - Pwf ii. t=t. b. Buildup I. Δpw=Pws-Po ii. t = Δte = Δt / (1 + Δt / tp) . Here, Δte is the equivalent time introduced by Agarwal. 3. Compute aq, bq, ap and bp from curve fitting , where: q D = aq t

bq

∆pw = a pt

(4.17) bp

(4.18)

4. Compute Δps where is the pressure drop for the deconvolved, constant rate data, by the expression:

∆ps = At B Where A=

a p Γ(b p + 1) aq Γ(bq + 1)Γ( B + 1)

And

B = b p − bq

(4.19)

26

Investigation of wellbore storage effect on analysis of well test data

4.5. β - Deconvolution Van Everdingen[18] and Hurst[19] demonstrated empirically that the sandface rate profile can be modeled approximately using an exponential relation for the duration of wellbore storage distortion during a pressure transient test. The van Everdingen/Hurst exponential rate model is given in dimensionless form as: q D (t D ) = 1 − e − β t D

(during wellbore storage distortion)

(4.20)

A similar approximation can be used for pressure buildup data:

qD (t D ) = e − β t D

(4.21)

The convolution integral is given as tD

′ pwD (t D ) = ∫ qD (τ ) psD (t D − τ )dτ

(4.22)

0

The β-deconvolution formula, which computes the undistorted pressure drop function directly from the wellbore storage affected data, is given as:

psD (t D ) = pwD (t D ) +

1 dpwD (t D ) β dt D

(4.23)

And in terms of field variable

1 d (∆pw ) ∆p = ∆p + s w α dt

(4.24)

27

Investigation of wellbore storage effect on analysis of well test data

Chapter 5

INVESTIGATION OF WELLBORE STORAGE EFFECTS ON WELL TEST DATA USING TEST DESIGN TECHNIQUE All Saphir analytical and numerical models may be used to generate a virtual gauge on which a complete analysis may be simulated. Simulation options taking into account gauge resolution, accuracy and potential drift can be the basis for selecting the appropriate tools or to check if the test objectives can be achieved in practice but here we change model parameters to see the sensitivity of model to these parameters. Main parameter that we change is Wellbore storage coefficinet.we want to investigate the effect of wellbore storage coefficient on pressure response of models(including reservoir model and well model).in other words we determine the effect of wellbore storage coefficient on the wellbore storage disappearing of different models and investigate the extent of this effect on model recognition. For test Design we have some basic parameter of these reservoir that is common in all cases listed below Table5. 1.Reservoir properties

Wellbore radius

Net pay thickness

Porosity

Permeability

Formation compressibility

rw(ft)

h(ft)

φ

K(md)

Cf(psi-1)

0.25

50

0.15

20

4E-6

Table5. 2.Reservoir initial condition

Initial Pressure(psi)

Temperature(˚F)

Pi(psi)

T(˚F)

5000

210

Fig5. 1.input reservoir characteristic

Fig5. 2.input Temperature an Pressure

28

Investigation of wellbore storage effect on analysis of well test data

After we introduce the basic parameters, we should calculate the fluid properties using default correlations of software(Saphir) so we need the fluid properties at reservoir initial condition. After defining of reservoir and well parameters next we can choose different model for test design and derive pressure response according to this model(including reservoir model and well model).

Fig5. 4.test design screen(Saphir software)

Fig5. 3.schematic of model chosen in test design(Saphir software)

5.1.Oil well Data As mentioned , we need the fluid properties at initial condition of reservoir . these parameters are listed below At Reservoir Condition :(T=210˚F , P=5000psi) Table5. 3.Fluid properties

Formation volume factor

Oil compressibility

Bo(bbl/STB)

Co(psi-1)

1.25

5E-5

Also well production data is required : Table5. 4.Production data

Time

Flow Rate

t (hr)

qo(STBD)

200 300

500 0

29

Investigation of wellbore storage effect on analysis of well test data

5.1.1. Constant wellbore storage, Homogenous reservoir, Vertical well, Infinite acting Fig. 5.4 with various constant wellbore storage constants is illustrated below. Pure wellbore storage is characterized by the merge of both Pressure and Bourdet Derivative curves on the same unit slope. At a point in time, and in the absence of any other interfering behaviors, the Derivative will leave the unit slope and transit into a hump which will stabilize into the horizontal line corresponding to Infinite Acting Radial Flow. The horizontal position of the curve is only controlled by the wellbore storage coefficient C. Taking a larger C will move the unit slope to the right, hence increase the time at which wellbore storage will fade. More exactly, multiplying C by 10 will translate the curve to one log cycle to the right. The figure below presents the response with wellbore storage values, C of 0.0001, 0.001, 0.01,0.1 and 1 (bbl/psi).

C=1E-4

C=0.001

C=0.01

C=0.1

C=1

Fig5. 5.Sensitivity to C for homogenous reservoir

The value of C has a major effect, which is actually exaggerated by the logarithmic time scale. When the influence of wellbore storage is over both the pressure change and the derivative merge together. Wellbore storage tends to masks infinite acting radial flow on a time that is proportional to the value of C. According to derivative curve of above figure the radial flow for C = 0.0001bbl/psi starts at t = 0.01hr and wellbore storage almost not be seen and for other value of C we have

30

Investigation of wellbore storage effect on analysis of well test data

For C=0.001 bbl/psi Radial flow Starts At

t=1hr

For C=0.01 bbl/psi

Radial flow Starts At

t=10hr

For C=0.1 bbl/psi

Radial flow Starts At

t=100hr

For C=1 bbl/psi

Radial flow not be seen

And we see that characterization of reservoir behavior for C=1 bbl/psi is impossible and wellbore storage effect distorted pressure response of reservoir. 5.1.2. Constant wellbore storage, Homogenous reservoir, H.C fracture, Infinite acting For high conductivity the linear flow should be seen in early time region .the characteristics of this flow regime is +1/2 slope and the distance of log2 between the pressure and pressure derivative curve.

1E-3 bbl/ psi (current) 0.01 bbl/ psi 0.1 bbl/ psi 1 bbl/ psi C=0.001

C=1 C=0.1 C=0.01

Fig5. 6.Sensitivity to C for high conductivity fracture

we see in the Fig 5.6 for C=0.001 according to pressure derivative curve radial flow starts at t = 0.1hr and the two curve have slope of +1/2 and the distance of log2 but if we go to larger value of C the slope tend to 1 and the distance tend to zero between two curves. For example the slope of Curve for C=0.01 bbl/psi is about +3/4 and the distance is lower than log2. For wellbore storage coefficient C ,0.1 bbl/psi, 1 bbl/psi we can not see the linear flow. Start of radial flow for different value of C is listed below : For C=0.001 bbl/psi Radial flow Starts At

t=0.1hr

31

Investigation of wellbore storage effect on analysis of well test data

For C=0.01 bbl/psi

Radial flow Starts At

t=1hr

For C=0.1 bbl/psi

Radial flow Starts At

t=10hr

For C=1 bbl/psi

Radial flow Starts At

t=100hr

And we see that characterization of reservoir behavior for C= 1bbl/psi is almost impossible. 5.1.3. Constant wellbore storage, Homogenous reservoir, L.C fracture, Infinite acting For Low conductivity the bilinear flow should be seen in early time region .the characteristics of this flow regime is +1/4 slope (both curves) and the distance of log4 between the pressure and pressure derivative curves. 1E-3 bbl/ psi (current) 0.01 bbl/ psi 0.1 bbl/ psi 1 bbl/ psi

C=0.001

C=1 C=0.1 C=0.01

Fig5. 7.Sensitivity to C for low conductivity fracture

we see in Fig 5.6 for C=0.001bbl/psi according to pressure derivative curve radial flow starts at t = 1hr and the two curve have slope of +1/4 and the distance of log2 but if we go to larger value of C the slope tend to 1 and the distance tend to zero between two curves. for example the slope of Curve for C=0.01 bbl/psi is about +1/2,maybe confused with high conductivity fracture, and the distance is lower than log4.for C ,0.1 bbl/psi, 1bbl/psi the linear flow disappear and wellbore storage overcome this flow regime. Start of radial flow for different value of C is listed below : For C=0.001 bbl/psi Radial flow Starts At

t=1hr

For C=0.01 bbl/psi

Radial flow Starts At

t=10hr

For C=0.1 bbl/psi

Radial flow Starts At

t=100hr

32

Investigation of wellbore storage effect on analysis of well test data

For C=1 bbl/psi

Radial flow is not seen

And we see that characterization of reservoir behavior(Middle time Region) for C= 1bbl/psi is impossible. 5.1.4. Constant wellbore storage, Homogenous reservoir, Limited entry well, Infinite acting

C=0.0001bbl/psi C=0.001bbl/psi

C=0.1bbl/psi

C=0.01bbl/psi

C=1bbl/psi

1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi 1 bbl/ psi

Fig5. 8.Sensitivity to C for limited entry well

For the wellbore storage coefficient C= 0.0001bbl/psi wellbore storage is not seen . For the Partial penetration ,Wellbore storage will quickly mask the spherical flow regime. If we look at the curve of C=0.0001bbl/psi ,we can see the 1st stabilization occurs at t=0.04hr(In practice this flow regime is more often than not masked by wellbore storage.) then spherical flow is seen (slope = -1/2). next to spherical flow ,second stabilization is seen at t=100hr.as wellbore storage coefficient increases ,the spherical flow seen at later time until at C=1 bbl/psi spherical flow disappears . the similarity of all curves is 2nd stabilization at which all of them reach together about t= 100hr .start of 1st Stabilization, Spherical Flow and 2nd

Stabilization for different value of wellbore

storage coefficient are listed below : Table5. 5.Start of flow regimes for limited entry well

Time(hr) C(bbl/psi) C=0.0001bbl/psi C=0.001bbl/psi C=0.01bbl/psi C=0.1bbl/psi C=1 bbl/psi

1st Stabilization 0.04 0.7 2.8 Not Be Seen Not Be Seen

Spherical Flow 2 4.5 6 Not Be Seen Not Be Seen

2nd Stabilization 100 100 100 100 Not Be Seen

33

Investigation of wellbore storage effect on analysis of well test data

5.1.5. Constant wellbore storage, Double porosity reservoir, Vertical well, Infinite acting Wellbore storage will invariably mask the fissure response in the double porosity reservoir. The transition can thereby easily be misdiagnosed and the whole interpretation effort can be jeopardized. we can see for C=0.0001bbl/psi there is almost no wellbore storage and two stabilization on 0.5 and valley shaped transition are seen .1st stabilization starts at t=0.01hr if we change wellbore storage coefficient to C=0.001bbl/psi ,start of 1st stabilization shift to t= 0.1 hr.for C=0.01bbl/psi the 1st stabilization disappear immediately and we will face to transition valley .finally for C=0.1bbl/psi we can not see 1st stabilization.

C=0.0001

C=0.001

C=0.01

C=0.1

1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi

Fig5. 9.Sensitivity to C for Double porosity reservoir

At higher wellbore storage coefficients even the whole transition period may be lost such C=0.1bbl/psi 1st that stabilization and transition valley disappear.the similarity of curves with different value of wellbore storage coefficient is 2nd stabilizatiion at which all of them meet eachother at 0.5 . Table5. 6.Start of flow regimes for double porosity reservoir

Time(hr) C(bbl/psi) C=0.0001bbl/psi C=0.001bbl/psi C=0.01bbl/psi C=0.1bbl/psi C=1 bbl/psi

1st Stabilization 0.01 0.1 0.5 Not be seen Not be seen

Transition Valley 0.25 0.45 1 4 Not be seen

2nd Stabilization 100 100 100 100 100

34

Investigation of wellbore storage effect on analysis of well test data

5.1.6. Constant wellbore storage, Double permeability reservoir, Vertical well, Infinit acting For the double permeability, First, the behavior corresponds to two layers without cross flow .At intermediate times, when the fluid transfer between the layers starts, the response follows a transition regime. Later, the pressure equalizes in the two layers and the behavior describes the equivalent homogeneous total system. The derivative stabilizes at 0.5.if we look at Fig 5.10,for C =0.0001bbl/psi we have no wellbore storage and the reservoir behavior is obviously seen.

1E-4 bbl/ psi 1E-3 bbl/ psi 0.01 bbl/ psi (current) 0.1 bbl/ psi 1 bbl/ psi

C=0.0001 C=0.001

C=0.01

C=0.1

C=1

Fig5. 10.Sensitivity to C for Double permeability reservoir

for C=0.001bbl/psi we cannot see pure wellbore storage but wellbore storage is seen and the 1st stabilization starts at t=0.1hr.if we change the wellbore storage coefficient to C=0.01bbl/psi ,start of 1st stabilization shift to t=0.5hr .with increasing of value of wellbore storage coefficient to C= 0.1bbl/psi 1st stabilization disappear. For C=1bbl/psi transition disappear and we can see only 2nd stabilization. Table5. 7.Start of flow regimes for double permeability reservoir

Time(hr) C(bbl/psi) C=0.0001bbl/psi C=0.001bbl/psi C=0.01bbl/psi C=0.1bbl/psi C=1 bbl/psi

1st Stabilization 0.01 0.1 Not be seen Not be seen Not be seen

Transition Valley 0.3 0.7 1 4.2 Not be seen

2nd Stabilization 100 100 100 100 100

35

Investigation of wellbore storage effect on analysis of well test data

Chapter 6

CONCLUSION AND RECOMMENDATION 6.1.Conclusions As we have seen, the wellbore storage effect distort pressure data and make it difficult to interpret well test data. In Chapter 5 ,we have analyzed wellbore storage for different type of reservoir models and well models to see difference of wellbore storage effects between them. For homogenous model, curve would shift to right. For high conductivity and low conductivity in early time, wellbore storage have not be seen .With increasing the value of C linear and bilinear flow disappear which make it difficult to detect hydraulic fracturing . In some cases wellbore storage will prevent detection the type of fracture. In limited entry well for small value of C there is no wellbore storage but with increasing value of C the time of pure wellbore storage increases and more data is distorted. For double porosity and permeability reservoirs with increasing the value of C, there is common trend in wellbore storage effect .

6.2.Recommendation 1. For performing well test in any type of reservoir, first we should estimate the time for production or build up to see all reservoir behavior .for this we can obtain rock and fluid properties from laboratory and running a test design for estimating this time.

2. For preventing distortion of pressure data, we can set a flow rate estimator that records flow rate and pressure simultaneously .

3. In a well where there is no designed partial penetration the interpreter can easily miss the effect and as the limited entry can result in a high geometrical and thus a high total skin this can often be misdiagnosed as damage alone when coupled with the wellbore storage effect and lead to ineffective acidizing.

36

Investigation of wellbore storage effect on analysis of well test data

REFRENCE 1. Bourdet, D. : "Well Test Analysis: The Use of Advanced Interpretation Models" ,ELSEVIER 2002 2. Russell, D.G.: "Extensions of Pressure Build-Up Analysis Methods," paper SPE 1513 presented at the 1966 SPE Annual Meeting, Dallas, Texas, 2–5 October. 3. Horne,R.N.: "Modern Well Test Analysis ", Petroway, 1995 4. Houze,O.: " Dynamic Flow Analysis " , Kappa , 2008 5. Bourdarot, G.,” Well Testing: Interpretation Methods” Translated from the French by Barbara Brown Balvet 1998 6. Glatfelter, R.E., Tracy, G.W., and Wilsey, L.E.: "Selecting Wells Which Will Respond to Production-Stimulation Treatment," Drill. And Prod. Pract., API, Dallas (1955) 11729. 7. Fetkovich, M.J., and Vienot, M.E.: "Rate Normalization of Buildup Pressure By Using Afterflow Data," JPT (December 1984) 2211–24. 8. Chaudhry, Amanat U. Oil Well Testing Handbook. Oxford (GB): Elsevier/GPP, 2004. Print. 9.

Cinco, H., Samaniego, F., and Dominguez, N.: "Transient Pressure Behavior for a

Well with a Finite Conductivity Vertical Fracture", Soc. Petr. Eng. J., (August 1978), 253-264. 10. Cinco, H., Samaniego, F.: "Transient Pressure Analysis for Fractured Wells", J. Pet. Tech., (September 1981). 11. Gringarten, A.C., Ramey, H.J., Jr., and Raghavan, R.: "Unsteady State Pressure Distribution Created by a Well with a Single Infinite Conductivity Vertical Fracture", Soc. Petr. Eng. J., (August 1974),347-360. 12. Raghavan, R.: Well Test Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1993. 13. Ramey, H.J. Jr.:"Non-Darcy Flow and Wellbore Storage Effects in Pressure Buildup and Drawdown of Gas Wells," Trans., AIME, (1965) 210, 223-233. 14. Hegeman, P.S., Hallford, D.L., and Joseph, J.A.: "Well-Test Analysis With Changing Wellbore Storage", paper SPE 21829 presented at the SPE Rocky Mountain

37

Investigation of wellbore storage effect on analysis of well test data

Regional/Low Permeability Reservoirs Symposium in Denver, Colorado, April 15-17 1991. 15. Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves—Analysis of Oil Well Production Data Using Material Balance Time: Application to Field Cases," paper SPE 28688 presented at the 1994 Petroleum Conference and Exhibition of Mexico held in Veracruz, Mexico, October 1013. 16. Johnston, J.L.: “Variable Rate Analysis of Transient Well Test Data Using SemiAnalytical Methods,” M.S. thesis, Texas A&M University, College Station, TX (1992). 17. Blasingame, T.A., Johnston, J.L., Lee, W.J., and Raghavan, R.: "The Analysis of Gas Well Test Data Distorted by Wellbore Storage Using an Explicit Deconvolution Method," paper SPE 19099 presented at the 1989 SPE Gas Technology Symposium, Dallas, TX, 07-09 June 1989. 18. van Everdingen, A.F.: "The Skin Effect and Its Influence on the Productive Capacity of a Well," Trans., AIME (1953) 198, 171-76. 19.

Hurst, W.: "Establishment of the Skin Effect and Its Impediment to Fluid Flow

into a Well Bore," Pet. Eng. (Oct. 1953) B6–B16.

38

Investigation of wellbore storage effect on analysis of well test data

APPENDIX A

Summary of usual log-log responses Geometry Radial

Linear

Bi-linear

LOG - LOG Shape Slope

Early Double porosity restricted Infinite conductivity fracture Finite conductivity fracture

Spherical Pseudo Steady State Steady State

Wellbore storage

TIME RANGE Intermediate

Late

Homogeneous behavior

Semi infinite reservoir

Horizontal well

Two sealing boundaries

Finite conductivity fault

Double porosity unrestricted with linear flow

Well in Partial penetration Layered no cross flow with boundaries Conductive fault

Closed Reservoir (drawdown) Constant pressure boundary

39

Investigation of wellbore storage effect on analysis of well test data

APPENDIX B

DERIVATION OF THE β-DECONVOLUTION FORMULATION We note that the lack of accuracy in flowrate measurements (when these exist) narrows the range of application of Glatfelter deconvolution method (i.e., rate normalization). Van Everdingen and Hurst (separately) introduced an exponential model for the sandface rate during the wellbore storage distortion period of a pressure transient test. The exponential formulation of the flowrate function is given as: q D (t D ) = 1 − e − βt D

(B.1)

Eq. (B-1) is based on the empirical observations made by Van Everdingen and Hurst. Recalling the convolution theorem, we have: pwD (tD ) =

tD

∫0

q'D (τ ) psD (t D − τ ) dτ

(B.2)

Taking the Laplace transform of Eq. B.2 yields: p wD (u) = u qD (u) p sD (u)

(B.3)

Rearranging Eq. B.3 for the equivalent constant rate pressure drop function, p sD (u) , we obtain: p sD (u) = p wD (u)

1 u qD (u)

(B.4)

The Laplace transform of the rate profile (Eq. B.1) is: qD (u) =

1 1 − u u+β

(B.5)

Substituting Eq. B.5 into Eq. B.4, and then taking the inverse Laplace transformation of this result yields the "beta" deconvolution formula: p sD (t D ) = p wD (t D ) +

1 dp wD (t D ) dt D

β

(B.6)

40

Investigation of wellbore storage effect on analysis of well test data

Where we note that Eq. (B-6) is specifically valid only for the exponential sandface flowrate profile given by Eq. B-1.

This may present a serious limitation in terms of practical

application of the β-deconvolution method.

41

Investigation of wellbore storage effect on analysis of well test data

APPENDIX C

DERIVATION OF THE POWER DECONVOLUTION FORMULATION This development assumes that variable (or constant) rate flow conditions exist in the reservoir. We will use the familiar convolution integral to develop our new deconvolution technique, The convolution integral is given as pwD (tD ) =

tD

∫0

q'D (τ ) psD (t D − τ ) dτ

(C.1)

Taking the Laplace transform of Eq. C.1 yields:

pswD (u) = u qD (u) psD (u)

(C.2)

Rearranging Eq. C.2 for the equivalent constant rate pressure drop function, p sD (u) , we obtain:

psD (u) = pwD (u)

1 u qD (u)

(C.3)

At this point we will state that it is our objective to obtain functional forms for PwD(s) and qD(s) that yield a closed form solution for psD(tD) when the inverse Laplace transform of Eq C.3 is taken. The functional form that was chosen for this solution is the power law equation where

f (t) = at b

(C.4)

In this work, Eq C.4 is applied as a piecewise approximation to the data function. We now need the Laplace transform of Eq C.4 to develop our new deconvolution solution. The Laplace transform of Eq C.4 is

f ( s) =

aΓ(b + 1) sb + 1

(C.5)

Here, Г(x) is the gamma function. We have found empirically that Eq C.5 is valid although Eq C.4 is only a piecewise continuous function. We also expect Eq C.5 to be

42

Investigation of wellbore storage effect on analysis of well test data

valid for monotonic data functions and indeed this method fails for well test data distorted by wellbore phase redistribution. Using

Eq C.4 we obtain the Laplace

transforms of the PwD(tD) and qD(tD) profiles. These relations are given as

qD (u) =

aqD Γ(bqD + 1) u

bqD +1

=

CqD u

bqD +1

(C.6)

And

a pD Γ(bpD + 1)

pwD (u) =

u

b pD +1

=

C pD u

b pD +1

(C.7)

Here the subscripts p and q denote Laplace transforms pertaining to pwD(tD) and qD(tD) respectively. Combining Eq C.3,C.6,C.7 we obtain

psD (u) =

C pD CqD u

1 b pD − bqD +1

= αD

Γ( β D + 1) u β D +1

β D = bpD − bqD αD =

C pD

(C.8) (C.9)

1

CqD D Γ( β D + 1)

(C.10)

By Eq C.4 , C.5 We find Laplace transform of Eq C.8

psD (t D ) = α Dt D

βD

(C.11)

43