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

I

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.

II

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.

III

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

IV

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

V

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

VI

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

VII

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

VIII

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

IX

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

X

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

Investigation of wellbore storage effect on analysis of well test data 1

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


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).


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


‘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:


2Dxfkt0.000264

ft xct

ϕµ=

(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. 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:

41

DxffDfD.wk

451.2 tPD =

(2.2)

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

kkk f

fD =

(2.3)

ffD

xww =

(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

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


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. 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:

21

.

= DxfD tP π (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).

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


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.

ppww

T SShhS +=

(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.


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 )

32

9.2452

Φ=

mc

qBk ts

µµ

(2.7)


3

=

r

s

r

v

kk

kk

(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]


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%).

mft

ft

mtft

ft

VcVc

VcVcVc

+

=+

=)()(

)()()(

φφ

φφφ

ω

(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.

f

mw k

kr 2αλ =

(2.10)


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]


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

21

1

)()()(

tt

t

VcVcVc

φφφω+

=

(2.11)

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

21

12

)()()(khkh

khrw += αλ

(2.12)


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:

21

1

)()()(khkh

kh+

=κ (2.13)

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]


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.

dtdp

BCq wf

wb24

−= (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]

pVC∆∆

= (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


cVC w= (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.

2894.0

wtD hrc

CCφ

= (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

wsf qqq += (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.


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:

D

D

ct

=Dp (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:

DD 0.02s) (0.041 Ct += (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:

0.01 0.1 1 10 100 10001

10

100

C =0.01C=0.1

C=1 C=10


DD 3.5s) (60 Ct += (3.8)

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;


• 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.


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:

)(0002637.0)( 2 pmk

ctpm

t

∇=∂

∂φ

µ

(3.9)

We introduce the pseudotime:

where tc

pIµ1)( =

(3.10)

or, better, the normalized version:

where t

reft

cc

pIµµ )(

)( =

(3.11)

Where the diffusion equation becomes:

)()(

0002637.0)( 2 pmck

tpm

reftps

∇=∂∂

µφ (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

∫=t

wfps dpItt0

))(()( ττ

∫=t

wfps dpItt0

))(()( ττ


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.


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:

+−+∆

∆

−==∆−∆ scrkt

tCkhqBtptp

wwfws 87.023.3log)(log 116.162 )0()( 22 φµ

µ (4.1)

Where the C2-term is defined as:

=∆+= )0(1000528.0 22 tp

Lg

r

khC wft

ρµ

(4.2)

Combining Eqs. 4.1 and 4.2 into a plotting function format, we obtain:

)(log)hr 1(11

)]0()([

2

tmtf

tC

tptpsl

wfws ∆+=∆=

∆

−

=∆−∆ (4.3)


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:

∆

−

=∆−∆=

tC

tptpy wfws

2

11

)]0()([ (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:

slmqBkh µ 6.162= (4.5)

And the skin factor can be estimated using:

+−

=∆= 23.3log)hr 1( 151.1 2

wsl crk

mtfs

φµ (4.6)


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].

)()()( tptqtp u≈∆ (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]

+−

∆=

∆−−∆

srctk

khtqqptp

wto

swfws 87.023.3)(log6.162)(

)(2

,

φµµ

(4.8)

Eq. 4.8 indicates that a plot of )log(.

)()( , tvs

tqqptp

o

swfws ∆∆−−∆

(4.9)

should be linear with slope equal to

khm µ6.162

=′ (4.10)

The skin is determined from

+∆

−∆−

−∆= 23.3)(log

)()(

515.1 2,

wto

swfws

rctk

tqqptp

sφµ

(4.11)


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:


qN

t pmb = (4.12)

The wellbore storage-based, material balance time function for the pressure buildup case

is given as:

][11

1

1 ,

,,,

wswbs

wswbs

BUwbs

BUwbspBUmb

ptd

dm

pm

t

qN

t∆

∆−

∆−∆=

−=∆ (4.13)

And the wellbore storage-based, rate-normalized pressure drop function for the pressure

buildup case is given as:

wsws

wbsBUwbs

wsBUs p

ptd

dm

qpp ∆

∆∆

−=

−∆

=∆][11

11 ,

, (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

∫ −′=Dt

DsDDDwD dtpqtp0

)()()( τττ

(4.15)

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

inverse Laplace transform it becomes


DDDDsD ttp βα=)( (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: qb

qD taq = (4.17)

pbpw tap =∆ (4.18)

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

the expression: B

s Atp =∆ (4.19)

Where

)1()1()1(+Γ+Γ

+Γ=

Bbaba

Aqq

pp

And

qp bbB −=


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:

DtDD etq 1)( β−−= (during wellbore storage distortion) (4.20)

A similar approximation can be used for pressure buildup data:

DtDD etq )( β−= (4.21)

The convolution integral is given as

∫ −′=Dt

DsDDDwD dtpqtp0

)()()( τττ

(4.22)

The β-deconvolution formula, which computes the undistorted pressure drop function

directly from the wellbore storage affected data, is given as:

D

DwDDwDDsD dt

tdptptp )(1)()(β

+=

(4.23)

And in terms of field variable

dtwpd

wpsp)(1 ∆

+∆=∆α

(4.24)


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

Initial Pressure(psi) Temperature(˚F)

Pi(psi) T(˚F) 5000 210

Table5. 2.Reservoir initial condition

Fig5. 1.input reservoir characteristic Fig5. 2.input Temperature an Pressure


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).

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) 200 500 300 0

Fig5. 4.test design screen(Saphir software) Fig5. 3.schematic of model chosen in test design(Saphir software)


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).

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

C=0.01C=0.1 C=1

C=1E-4

C=0.001


For C=0.001 bbl/psi Radial flow Starts At t=1hr



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.

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

C=0.001 C=0.01

C=0.1

C=1

1E-3 bbl/ psi (current)0.01 bbl/ psi0.1 bbl/ psi1 bbl/ psi




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.

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 :




C=0.001C=0.01

C=0.1

C=1

1E-3 bbl/ psi (current)0.01 bbl/ psi0.1 bbl/ psi1 bbl/ psi


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

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)

1st Stabilization Spherical Flow 2nd Stabilization

C=0.0001bbl/psi 0.04 2 100 C=0.001bbl/psi 0.7 4.5 100 C=0.01bbl/psi 2.8 6 100 C=0.1bbl/psi Not Be Seen Not Be Seen 100 C=1 bbl/psi Not Be Seen Not Be Seen Not Be Seen

C=0.0001bbl/psi

C=0.001bbl/psi C=0.01bbl/psi C=0.1bbl/psi C=1bbl/psi

1E-4 bbl/ psi1E-3 bbl/ psi0.01 bbl/ psi (current)0.1 bbl/ psi1 bbl/ psi


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.

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)

1st Stabilization Transition Valley 2nd Stabilization

C=0.0001bbl/psi 0.01 0.25 100 C=0.001bbl/psi 0.1 0.45 100 C=0.01bbl/psi 0.5 1 100 C=0.1bbl/psi Not be seen 4 100 C=1 bbl/psi Not be seen Not be seen 100

C=0.0001 C=0.001 C=0.01 C=0.1

1E-4 bbl/ psi1E-3 bbl/ psi0.01 bbl/ psi (current)0.1 bbl/ psi


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.

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)

1st Stabilization Transition Valley

2nd Stabilization

C=0.0001bbl/psi 0.01 0.3 100 C=0.001bbl/psi 0.1 0.7 100 C=0.01bbl/psi Not be seen 1 100 C=0.1bbl/psi Not be seen 4.2 100 C=1 bbl/psi Not be seen Not be seen 100

C=0.0001

C=0.001C=0.01 C=0.1

C=1

1E-4 bbl/ psi1E-3 bbl/ psi0.01 bbl/ psi (current)0.1 bbl/ psi1 bbl/ psi


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.


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.

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Regional/Low Permeability Reservoirs Symposium in Denver, Colorado, April 15-17

1991.

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APPENDIX A Summary of usual log-log responses

Geometry LOG - LOG TIME RANGE Shape Slope Early Intermediate Late

Radial Double

porosity restricted

Homogeneous behavior

Semi infinite reservoir

Linear

Infinite conductivity

fracture

Horizontal well

Two sealing boundaries

Bi-linear

Finite conductivity

fracture

Finite conductivity

fault

Double porosity

unrestricted with linear

flow

Spherical

Well in Partial

penetration

Pseudo Steady State

Wellbore storage

Layered no cross flow

with boundaries

Closed Reservoir

(drawdown)

Steady State

Conductive fault

Constant pressure boundary


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:

DtDD etq β−−=1)( (B.1)

Eq. (B-1) is based on the empirical observations made by Van Everdingen and Hurst.

Recalling the convolution theorem, we have:

τττ dtpqt

tp DsD'D

DDwD )()(

0)( −= ∫ (B.2)

Taking the Laplace transform of Eq. B.2 yields:

)()()( upuquup sDDwD = (B.3)

Rearranging Eq. B.3 for the equivalent constant rate pressure drop function, )(upsD , we

obtain:

)(1)()(

uquupup

DwDsD = (B.4)

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

β+−=

uuuqD

11)( (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:

DDwD

DwDDsD dttdptptp )(1)()(

β+= (B.6)


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.


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

τττ dtpqt

tp DsD'D

DDwD )()(

0)( −= ∫ (C.1)

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

)()()( upuquusp sDDwD = (C.2)

Rearranging Eq. C.2 for the equivalent constant rate pressure drop function, )(upsD , we

obtain:

)(1)()(

uquupup

DwDsD = (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

battf =)( (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

1)1()(

++Γ

= bsbasf (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


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

11

)1()( ++ =

+Γ=

qDqD bqD

bqDqD

D uC

uba

uq (C.6)

And

11

)1()( ++ =

+Γ=

pDpD bpD

bpDpD

wD uC

uba

up (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

11)1(1)( ++−

+Γ==

DqDpD uuCC

up DDbb

qD

pDsD β

βα (C.8)

qDpDD bb −=β (C.9)

)1(1+Γ

=DDqD

pDD C

Cβ

α (C.10)

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

DDDsD ttp βα=)( (C.11)

investigation of wellbore storage effects on analysis of well test data

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