Generating Relative Permeability

8/3/2019 Generating Relative Permeability

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UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

DETERMINATION OF RELATIVE PERMEABILITY FROM WELL PRODUCTION BY

CONSIDERATION OF FLUID TYPE, FORMATION HETEROGENEITY, AND SKIN

FACTOR

A THESIS

SUBMITTED TO THE GRADUATE FACULTY

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE

By

AHMED ZARZOR HUSSIEN AL-YASERI

Norman, Oklahoma

2010


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DETERMINATION OF RELATIVE PERMEABILITY FROM WELL PRODUCTION BY

CONSIDERATION OF FLUID TYPE, FORMATION HETEROGENEITY, AND SKIN

FACTOR

A THESIS APPROVED FOR THE

MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL ENGINEERING

BY

___________________________________

Dr. Faruk Civan, Chair

___________________________________

Dr. Deepak Devegowda

___________________________________

Dr. Bor-Jier Shiau


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Copyright byAHMED Z. HUSSIEN Al-YASERI 2010

All Rights Reserved.


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DEDICATION

This thesis is dedicated to my parents, brothers and sisters for their love and support and to my

wife for her encouragement and patience. All of my success is because of them.

My Mother

Mothers are the lovely and greatest persons in the life. I love my mother because she is the one

who gave the life. Always she took care of me, she stayed awake all the time to make sure that I

am alright, and she got tired all the day for my comfort. She spent her life to raise me up. When I

was a child, she was feeding me. Usually, my mother advice me and provided me the good

guidances from her experience in life.

I love you mom!


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ACKNOWLEDGMENTS

I would like to thank Dr. Faruk Civan, chairman of my committee, for his advice and

suggestions that help me to complete the present work. Also, I would like to thank Dr.

Deepak Devegowda and Dr. Bor-Jier Shiau for their contributions and time as members

of my committee.

Thanks again for Dr. Deepak Devegowda for his assistance and suggestions in

implementation of reservoir simulator.

Thanks to Dr. Tibor Bodi and Dr. Peter Szucs, University of Miskolc, Hungary for their

assistance in providing the data from their paper.

Special thanks to the faculty and staff of the Mewbourne School of Petroleum and

Geological Engineering, especially Shalli Young and Sonya Grant for their kindness and

willingness to help when needed.

I wish to acknowledge and thank many people for their cooperation and support during

my stay at the University of Oklahoma.

Above all, I would like to thank God for the love, support and blessings in my life.


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v

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... v

LIST OF FIGURES .........................................................................................................vi

ABSTRACT ....................................................................................................................xvi

1. BACKGROUND ...........................................................................................................1

2. REVIEW OF THE TOTH ET AL. METHOD FOR DETERMINATION OF

RELATIVE PERMEABILITY FROM WELL PRODUCTION DATA ..................16

3. GENERATION OF SIMULATED WELL PRODUCTION DATA BY A

COMMERCIAL RESERVOIR SIMULATOR ...........................................................37

4. EVALUATION OF THE TOTH ET AL. METHOD FOR RADIAL FLOW

USING SIMULATED PRODUCTION DATA.............................................................45

4.1 EFFECT of RESERVOIR SIZE on TOTH et al. METHOD.......................................45

4.2 THE EFFECT OF (P) VALUE ON THE TOTH Et Al. METHOD ........................60

4.3 EFFECT OF THE VISCOSITY ON THE TOTH ET AL. METHOD .......................78

4.4 EVALUATING THE TWO CRITICAL EQUATIONS FOR THE TOTH ET AL.

METHOD .......................................................................................................................102

4.5 CONCLUSIONS ......................................................................................................109

5. CONSIDERATION OF SKIN FACTOR AND HETEROGENEITY

DURING THE ESTIMATION OF RELATIVE PERMEABILITY

CURVES FOR UNSTEADY STATE RADIAL DISPLACEMENT.....110

5.1 EFFECT OF SKIN FACTOR ............................................................................ .....110

5.1.1 SKIN FACTOR AND CAPILLARY PRESSURE EFFECT ................................116


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5.1.2 NEGATIVE SKIN FACTOR ................................................................................119

5.2 EFFECTS OF HETEROGENEITY ON RELATIVE PERMEABILITY

CURVES..........................................................................................................................121

5.3 CONCLUSIONS ......................................................................................................130

6. MODIFICATION AND GENERALIZATION OF THE TOTH ET

AL. METHOD FOR COMPRESSIBLE AND INCOMPRESSIBLE

FLUIDS AND EVALUATION BY MEANS OF A RESERVOIR

SIMULATOR ............................................................................................131

6.1 FORMULATIONS OF RELATIVE PERMEABILITIES UNDER LINEAR

FLOW............................................................................................................................. 134

6.2 FORMULATION FOR RELATIVE PERMEABILITIES UNDER RADIAL

FLOW.............................................................................................................................138

6.3 FORMULATION FOR FLUID SATURATIONS....................................................139

6.4 EVALUATION OF THE NEW TECHNIQUE USING A RESERVOIR

SIMULATOR..................................................................................................................143

6.5 APPLICATION AND VERIFICATION ..................................................................144

6.6 CONCLUSIONS ......................................................................................................146

RFERENCES.................................................................................................................159

APPENDIX A NOMENCLATUR ...............................................................................166


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LIST OF TABLES

Table 2.1 Petrophysical data for example 2.1..................................................................32

Table 2.2 Petrophysical data for example 2.2 ..................................................................37

Table 4.1 Petrophysical parameters for the case studies ..................................................46

Table 4.2 Production data for example 4.1.1 ...................................................................47

Table 4.3 Np, Wp and Wi for example 4.1.1 ...................................................................48

Table 4.4 Constant parameters for examples (4.1.1), (4.12), (4.13) ................................56

Table 4.5 Petrophysical parameters for the case studies ..................................................60

Table 4.6 Petrophysical parameters for the case studies .................................................69

Table 4.7Constant parameters for (P) =34.45 bar.........................................................78

Table 4.8Constant parameters for (P) = 74.98 bar........................................................78

Table 4.9 Petrophysical parameters for the case studies .................................................79

Table 4.10 Production data for the core sample ...............................................................95

Table 4.11 Use Toth et. al method to recalculate relative permeability curves ...............95

Table 4.12 Petrophysical parameters for example 4.3.5 ..................................................98

Table 5.1 Petrophysical parameters for example 5.1 ......................................................114

Table 6.1 Petrophysical parameters for example 5.1, 5.2, 5.3 and 5.4 .........................147


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LIST OF FIGURES

Fig.(1.1) Permeability definition ......................................................................................11

Fig.(1.2) General relative permeability curves..................................................................11

Fig.(1.3) Drainage and Imbibition displacement ..............................................................12

Fig.(1.4) Steady state and unsteady state method for core sample ...................................12

Fig.(1.5) Three section core for Penn -State method ........................................................13

Fig.(1.6)General Welges plot .........................................................................................13

Fig.(1.7) Pressure volume relationship .............................................................................14

Fig.(1.8) Fluid density vs. pressure for different fluid types ............................................14

Fig.(1.9) Flow regimes .....................................................................................................15

Fig.(2.1) Oil and water Production data for example 2.1 .................................................31

Fig.(2.2) Displacement equation for example 2.1.............................................................32

Fig.(2.3)Welges plot for example 2.1 .............................................................................32

Fig.(2.4) Relative permeability ratio curve for example 2.1.............................................33

Fig.(2.5) Relative permeability curves for example 2.1 ...................................................33

Fig.(2.6) Water fractional curve (after breakthrough time) for example 2.2 ...................34

Fig.(2.7) Displacement equation for example 2.2 ............................................................35

Fig.(2.8) Cumulative water influx for example 2.2 ..........................................................35

Fig.(2.9) Relative permeability ratio curve for example 2.2 ............................................36

Fig.(2.10) Relative permeability curves for example 2.2 .................................................36

Fig.(3.1) Three dimension shape from Eclipse with six injection wells and one production

well in the center (=60o) ................................................................................................39

Fig.(3.2) Two dimension shape from Eclipse with six injection wells and one production


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well in the center (=60o) .................................................................................................39

Fig.(3.3) Two dimension shape from Eclipse result .........................................................40

Fig.(3.4) Three dimension shape from Eclipse result .......................................................41

Fig.(3.5) Three dimensional shape from Eclipse result ....................................................42

Fig.(3.6) One slice shape for Radial flow .........................................................................43

Fig.(3.7) Radial flow but only for one slice (From Eclipse result) ...................................43

Fig.(3.8) Time vs. rate for one slice and the whole reservoir ............................................43

Fig.(4.1) Assumed relative permeability curves for examples 4.1.1, 4.1.2, and 4.1.3 .....46

Fig.(4.1.1) Production data from Eclipse for example 4.1.1 ............................................46

Fig.(4.1.2) Displacement equation for example 4.1.1 ......................................................49

Fig.(4.1.3) Cumulative water volume produced for example 4.1.1 ..................................49

Fig.(4.1.4) Relative permeability ration (assumed and calculated) for example 4.1.1 .....50

Fig.(4.1.5) Relative permeability curves (assumed and calculated) for example 4.1.1 ....50



Fig.(4.1.8) Cumulative water volume produced for example. 4.1.2 .................................52

Fig.(4.1.9) Relative permeability ration (assumed and calculated) for example 4.1. 2 ....52

Fig.(4.1.10) Relative permeability curves (assumed and calculated) for ex. 4.1.2 ..........53

Fig.(4.1.11) Production data from Eclipse for example 4.1.3 ..........................................53

Fig.(4.1.12) Displacement equation for example 4.1.3 ....................................................54

Fig.(4.1.13) Cumulative water volume produced for example. 4.1.3 ...............................54

Fig.(4.1.14) Relative permeability ration (assumed and calculated) for ex. 4.1.3 ............55

Fig.(4.1.15) Relative permeability curves (assumed and calculated) example 4.1.3 .......55


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Fig.(4.1.16) Grid size for example 4.1.1 from Eclipse .....................................................57




Fig.(4.1.20) Relative permeability ration (assumed and calculated) for example 4.1.3

after reducing the grid size ................................................................................................59

Fig.(4.1.21) Relative permeability curves (assumed and calculated) for example 4.1.3

after reducing the grid size ................................................................................................59

Fig.(4.2. 1) Production data from Eclipse for example 4.2.1 ...........................................61



Fig.(4.2.4) Relative permeability ration (assumed and calculated) for ex. 4.2.1...............62

Fig.(4.2A.5) Relative permeability curves (assumed and calculated) for ex. 4.2.1...........63



Fig.(4.2.8) Cumulative water volume produced for example 4.2.2 .................................64

Fig.(4.2.9) Relative permeability ration (assumed and calculated) for ex. 4.2.2 ..............65




Fig.(4.2.13) Water fraction for example 4.2.3 after breakthrough time ...........................67

Fig.(4.2.14) Cumulative water volume produced for example 4.2.3 ................................67

Fig.(4.2.15) Relative permeability ratio (assumed and calculated) for ex.4.2.3 ..............68


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Fig.(4.2.16) Relative permeability curves (assumed and calculated) for ex. 4.2.3.. ........68




Fig.(4.2.4) Relative permeability ration (assumed and calculated) for ex. 4.2.1 ..............71

Fig.(4.2.5) Relative permeability curves (assumed and calculated) for ex. 4.2.1 ........72

Fig.(4.2.6) production data from Eclipse for example 4.2.2 .............................................72



Fig.(4.2.9) Relative permeability ration (assumed and calculated) for ex. 4.2.2 .............74




Fig.(4.2.13) Water fraction for example. 4.2B.3 after breakthrough time .......................76

Fig.(4.2.14) Cumulative water volume produced for example. 4.2.3 ..............................76

Fig.(4.2B.15) Relative permeability ration (assumed and calculated) for ex. 4.2B.3 ......77




Fig.(4.3.3) Water fraction for example 4.3.1 after breakthrough time .............................81

Fig.(4.3.4) Cumulative produced water volume for example 4.3.1 ..................................81

Fig.(4.3. 5) Relative permeability ration (assumed and calculated) for ex. 4.31 ..............82

Fig.(4.3. 6) Relative permeability curves (assumed and calculated) for ex. 4.3.1 ...........82


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Fig.(4.3.9) Water fraction for example 4.3.2 after breakthrough time .............................84


Fig.(4.3. 11) Relative permeability ration (assumed and calculated) for ex. 4.3.2 ...........85

Fig.(4.3. 12) Relative permeability curves (assumed and calculated) for ex. 4.3.2 .........85




Fig.(4.3. 16) Relative permeability ration (assumed and calculated) for ex. 4.3.3 ...........87

Fig.(4.3. 17) Relative permeability curves (assumed and calculated) for ex. 4.3.3 .........88

Fig.(4.3.18) Cumulative recovery of oil versus cumulative volume of injected fluid for

different viscosity ratio (Eclipse result data) ....................................................................89

Fig.(4.3.20) Displacement eq. for different examples with different viscosity ratios ......89

Fig.(4.3.21) Cumulative produced water volume for different examples with different

viscosity ratios ..................................................................................................................90

Fig.(4.3.22) Water saturation distribution as a function of distance between injection and

production wells for (a) ideal and (b) non-ideal displacement ........................................90

Fig.(4.3.23) Water cut vs. production rate for heavy and light oil reservoir ....................91

Fig.(4.3.24) Total production vs. water cut for the last examples from Eclipse result data

for different viscosity ratios ..............................................................................................92

Fig.(4.3.25) Effect of viscosity ratio on relative permeability curves ..............................93

Fig.(4.3.26) Viscosity vs. residual oil ...............................................................................93


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Fig.(4.3.27) Relative permeability curves by JBN method for different viscosity

ratio....................................................................................................................................94



Fig.(4.3. 30) Relative permeability ratio for example 4.3.4 .............................................97

Fig.(4.3. 31) Relative permeability curves for example 4.3.4 ..........................................97

Fig.(4.3.32) Assumed relative permeability curves as input data in Eclipse for example

4.3.5....................................................................................................................................99


Fig.(4.3.35) Cumulative water volume produced for example 4.3 5 ..............................100

Fig.(4.3.34) Displacement equation for example 4.3.5 ..................................................100

Fig.(4.3. 37) Relative permeability curves (assumed and calculated) for ex. 4.3.5........101

Fig.(4.3. 36) Relative permeability ration (assumed and calculated) for ex. 4.3.5 .........101

Fig.(4.4.1) Displacement equation for field data (well A-225) ......................................103

Fig.(4.4.2) Displacement equation for field data (well A-710) ......................................103

Fig.(4.4.3) Displacement equation for simulated example by use Eclipse .....................104

Fig.(4.4.4) Wedges plot for simulated example by Eclipse ..........................................105

Fig.(4.4.5)Toths plot for simulated example by Eclipse ..............................................105

Fig.(4.4.6)Welges plot for field data for well A-225 ...................................................106

Fig.(4.4.7)Welges plot for field data for well A-710 ...................................................106

Fig.(4.4.8) the value of (b1=1.0068) for simulated example by Eclipse ........................108

Fig.(4.4.9) the value of (b1=1.0106) for the same pervious simulated example by Eclipse

..........................................................................................................................................108


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Fig.(5.1) Skin Zone .........................................................................................................111

Fig. (5.2) Three dimensional shapes by Eclipse .............................................................114

Fig.(5.3) Production data by Eclipse for example 6.1 ....................................................114

Fig.(5.4) Relative permeability curves for different Skin values, example 6.1 ..............115

Fig.(5.5) Relative permeability ratios for different Skin values, example 6.1 ...............115

Fig.(5.6) Capillary pressure vs. water saturation ............................................................116

Fig.(5.7) Relative permeability ratios for different Skin values with capillary pressure

..............................................................................................................................117

Fig.(5.8) Relative permeability curves for different Skin values with capillary pressure

effect ...............................................................................................................................117

Fig.(5.9) Relative permeability with Skin equal to zero with and without capillary

pressure include ..............................................................................................................118

Fig.(5.10) Relative permeability curve with and without capillary pressure by

experimental work. .........................................................................................................119

Fig.(5.11) Cross section for the reservoir with skin zone ...............................................120

Fig.(5.12) Relative permeability curves for different negative Skin values when the

viscosity is 10cp ..............................................................................................................120

Fig.(5.13) The distribution for the permeability by Eclipse (Mean = 100, S.D = 30) ....122

Fig.(5.14) Histogram for random permeability values (Mean = 100, S.D = 30) ............123

Fig.(5.15) Relative permeability curves for homogenous (k=100 md) and heterogeneous

(Mean=100, S.D. =30) ....................................................................................................123

Fig.(5.16) Relative permeability ratios for homogenous (k=100md) and heterogeneous

(Mean=100, S.D. =30) ....................................................................................................124


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Fig.(5.17) The distribution for the permeability by Eclipse (Mean = 100, S.D = 30) with

short channel ...................................................................................................................124

Fig.(5.18) Relative permeability curves for homogenous (k=100 md), heterogeneous

(Mean=100, S.D. =30) and heterogeneous with short channel (k=400 md and 1000 md)

..............................................................................................................................125

Fig.(5.19) 2 D. distribution for the permeability by Eclipse (Mean = 100, S.D = 30) with

short channel ...................................................................................................................125

Fig.(5.20) Relative permeability ratios for homogenous (k=100 md), heterogeneous

(Mean=100, S.D. =30) and heterogeneous with short channel (k=400 md and 1000 md)

..............................................................................................................................126

Fig.(5.21) Histogram for random permeability values (Mean = 500, S.D = 100) ..........126

Fig.(5.22) Relative permeability curves for homogenous (k=500 md), heterogeneous

(Mean=500, S.D. =100) and heterogeneous with short and long channel (k=2000 md and

3000 md) .........................................................................................................................127

Fig.(5.23) The distribution for the permeability by Eclipse (Mean = 500, S.D = 100) with

long channel ....................................................................................................................127

Fig.(5.24) Relative permeability ratios for homogenous (k=500, k=100 md),

heterogeneous (Mean=100, 500, S.D.=30,100) and heterogeneous with short and long

channel (k=2000, 1000, 3000 md) ..................................................................................128

Fig.(5.29) The distribution for the permeability by Eclipse (Mean=100, S.D=30) with

different long channels ....................................................................................................128

Fig.(5.25) Relative permeability curves for homogenous (k=100 md) and different long

channels ...........................................................................................................................129


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Fig.(5.26) Relative permeability ratios for homogenous (k=100 md) and different long

channels ...........................................................................................................................129

Fig.(6.1)Pressure versus 1/B ........................................................................................147

Fig.(6.3) Two dimensional shapes by Eclipse ................................................................148

Fig.(6.2) Linear displacement .........................................................................................148

Fig.(6.4) Three dimensional shape by Eclipse ................................................................148

Fig.(6.6) Power low equation for Toth et al. example 5.1 ..............................................149

Fig.(6.5) Displacement equation for Toth et al. example 5.1 ..........................................149

Fig.(6.8) Relative permeability ratios for example 5.1by using different methods ........150

Fig.(6.7) Power low equation for the new method example 5.1 .....................................150

Fig.(6.10) Production data for example 5.2 from Eclipse ..............................................151

Fig.(6.9) Relative permeability for example 5.1by using different methods ..................151

Fig.(6.12) Cumulative injected vs. time for Toth et al. example 5.2 ..............................152

Fig.(6.11) Displacement equation for Toth et al. example 5.2 .......................................152

Fig.(6.14) Relative permeability for ex. 5.2 by using different methods with assumed

values...............................................................................................................................153

Fig.(6.13) Relative permeability ratios for example 5.2 by using different methods with

assumed values ................................................................................................................153

Fig.(6.16) Production data for example 5.3 from Eclipse ..............................................154

Fig.(6.15) Relative permeability for example 5.2 by using different methods with

assumed values ................................................................................................................154

Fig.(6.18) Relative permeability for exAMPLE 5.3 by using the new method with

assumed values.................................................................................................................155


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Fig.(6.17) Relative permeability ratios for example 5.3 by using different methods with

assumed values ................................................................................................................155

Fig.(6.20) Production data for example 5.4 from Eclipse ....................................... .......156

Fig.(6.19) Assumed Relative permeability for example 6.4 ...........................................156

Fig.(6.21) Relative permeability ratios for example 5.4 by using the new method and

Welge method ..................................................................................................................157


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ABSTRACT

Accurate estimation of relative permeability is essential for reliable reservoir history

matching and decision-making, and effective reservoir management. This information is

also critical for the design, implementation, and monitoring of enhanced oil recovery

processes.

This study presents an effective method for acquiring the relative permeability

data directly from the well production data in oil reservoirs where water or gas acts as the

fluid phase displacing oil. The methodology presented in this study provides convenient

interpretation formulae which are applicable to unsteady-state, two-phase, immiscible,

and both compressible and incompressible fluids. The total mobility and the mobility

ratio of the immiscible fluids are related to the characteristic parameters of the

displacement process and the cumulative injected fluid pore volume following Toth et al.

These parameters are then incorporated into a general correlation function which allows

for analytically estimation of the relative permeability functions. The present approach

produces unique estimation of the relative permeability functions and is more practical

than the previous approaches which rely on computationally complicated history

matching procedures, often suffer from the non uniqueness issue of the obtained relative

permeability data.

As an extension of the Toth et al. method, the analytic method developed here

determines the relative permeability functions for compressible and slightly-compressible

fluids uniquely considering the effects of formation heterogeneity and skin factor on the

well production data. This approach provides the estimates of the relative permeability

functions, representative of the macroscopic two-phase flow behavior in the formation


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around the well. The Toth et al. method and its present modification and extension are

evaluated using the simulated well production data generated by means of a commercial

reservoir simulator under various production and reservoir conditions.

It is demonstrated that the drainage area, pressure drop and fluid viscosity have

significant effects on determined relative permeability curves using Toth et al. method for

radial flow. The Toth et al. method works satisfactorily even for heterogeneous reservoirs

and there is skin effect. The skin factor has a significant effect on the relative

permeability curves which increases when the oil viscosity increase or capillary pressure

increase. The effect of heterogeneities on the relative permeability curves is negligible.

However, the effect becomes significant when there are channels. The developed method

is a very simple, general and accurate method that is applicable for both incompressible

and compressible fluids (gas or liquid).


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CHAPTER 1

BACKGROUND

Introduction

Relative permeability is essential information required for evaluation, history matching,

effective management, and characterization of multiphase flow in a petroleum reservoir.

Thus, accurate and representative estimates of the relative permeability functions are

critical. Many methods have been proposed for determination of the relative permeability

curves. The goals of this study include the evaluation of the Toth et al. method (1998,

2001, 2005, 2006) and modification for compressible fluids. The Toth et al. method is

one of the simplest direct methods used to estimate relative permeability by processing

production data. It can give accurate and average estimations for relative permeability

compared with other methods that depend on the fluid flow test data obtained with core

samples to estimate the relative permeability curves. However, the Toth et al. method

was derived for incompressible fluids. This limitation is circumvented by developing a

new method for compressible fluids.

Relative permeability is defined as the ratio of the effective permeability to any

specific fluid (oil, water, or gas phases) to the absolute permeability. For example, the oil

relative permeability is given by:

1.1---------------------------------------------------------------k

kk oro


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Similarity, the water relative permeability is:

1.2-------------------------------------------------------------k

kk wrw

The gas phase relative permeability is defined in the same manner.

Permeability was defined mathematically for the first time by Darcy (1856) by his

equation known as Darcys law (Ahmed, T. 2001) as shown in Eq.(1.3) below. Darcy

assumed that permeability is a rock property, has a constant value, and does not depend

on the fluid flowing through the rock (Dake L. P., 1978).

-1.3-----------------------------------------------------------dl

dpk

Where v is the flow per unit area per unit time, is fluid viscosity and l is distance.

If there is more than one kind of fluid (oil and water, oil and gas, water and gas or

oil, water, and gas) flowing inside porous rock, each fluid has its own permeability called

effective permeability (ko, kw, kg). The effective permeability depends on the saturation

for each fluid reaching the absolute permeability (k) when there is just one fluid flowing

through the porous medium. Usually, relative permeability is plotted versus the wetting

phase saturation, for example water saturation (Sw) when the rock is water-wet Fig.(1.2).

This study is limited to flow of two phases. We consider the main following fluid

gdisplacement processes. Fluid displacement can be carried out in two ways.

Ina drainage displacement process, the non-wetting fluid (oil or gas) displaces the

wetting fluid (water, for example for a water-wet porous media). Therefore, the saturation

of non-wetting phase increases forward as seen in Fig.(1.3), (Patrick W., 2001).

In an imbibition displacement process, the wetting fluid (water, for example in

water-wet porous media) displaces the non-wetting fluid (oil or gas), causing the


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saturation of wetting phase to increase as seen in Fig.(1.3).

Many methods are available for measurement of relative permeability. We can

classify them into two types: steady state and unsteady state. We can measure relative

permeability under steady-state conditions when"fixed ratio of fluids is forced through

the test sample until saturation and pressure equilibrium are established, (Alam W. U.,

1988)." Unsteady- state conditions happen when only one phase is injected into the core

to displace the second phase present in the core during the test, causing the saturation to

change continuously, (Alam W. U., 1988) as shown in Fig.(1.4).

Steady- State Methods:

The first study about relative permeability determination was conducted by Wyckoff and

Botest (Caudle, 1951). This was an experimental work for two phase flow in a sand core

sample. In addition, they examined the relations between relative permeability for some

fluids and their saturations. Leverett and Lewis (Levertt, 1941) extended the study by

Wyckoff and Botest to handle the three phase systems (oil, water, and gas). Leverett and

Lewis found that the relative permeability of water was a function of water saturation

only and it was not affected by the presence of other fluids (oil and gas). The relative

permeability of gas was low compared with the two phase system under the same gas

saturation. The oil relative permeability was unstable and more complex because it could

be low or high when compared with the two phase system under the same oil saturation.

Morse et al. (1947) developed a new method for determining relative

permeability. Their method was modified by Osoba et al. (1951). The technique used by

Henderson and Yuster (1948), Caudle et al. (1951), Geffen et al. (1951), and Morse et


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al. (1947) is known as the Penn-State method, illustrated by Fig.(1.5). We can define the

Penn-State method as " forcing fluid mixture through cores mounted in Lucite. The test

core sample is placed between two samples of similar material that are in capillary

contact. Mixing of the two fluids occurs in the first section and the boundary effect of the

wetting phase is confined to the third (outlet) section"(Alam W., 1988). The Penn-State

method applies for liquids or gas-liquid systems at either increasing or decreasing of the

wetting phase (Mehdi H., 1988).

Another technique for estimating relative permeability under steady state

conditions is the Single-Sample Dynamic Method, which was developed by Richardson

et al. (1951), Josendal et al. (1952), and Loomis and Crowell (1962). This technique

differs from the Penn-State method in the handling of the end effects and placement of

the test samples between two core samples and the two phases are injected

simultaneously through a single core, (Mehdi H., 1988).

Unsteady-State Methods:

There are a number of other methods for measuring relative permeability under the

unsteady-state condition, but the most important methods are as follows:

1-Stationary Fluid method developed by Leas et al. (1950).

2- Hassler Method, (1944).

3-Hafford method, (1951).

4- Dispersed Feed Method, (1951).

5- Johnson et al.(JBN), (1959).

6- Jones and Roszelle (JR), (1978).


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7- Marle, (1981).

8-Toth et al., (1998).

Most of these methods apply for linear displacement.

The determination of relative permeability under an unsteady-state condition can

be applied faster than the steady-state condition, but the application is mathematically

more complex. Buckley and Leverett, (1942) developed the first displacement theory,

which was later extended by Welge, (1952). Welge was the first to show how to calculate

the relative permeability ratio in case the gravity is neglected. Leverett, (1941) gave the

mathematical basis by combining Darcys law with a definition of capillary pressure to

obtain the following expression:

Wherefw2is the fractional water in the outlet stream; qt is the superficial velocity of total

fluid leaving the core; is the angle between the flow direction x and the horizontal; and

is the density difference between displacing and displaced fluids. Welge, (1952)

showed that if we ignore capillary pressure, assume flow horizontal ( = 0) and after

some mathematical manipulation, we can calculate the relative permeability ratio with the

saturation as shown below:

-1.5-------------------------------1

1

wnw

nwwnw

wnw

nwnw

k

k

dQ

dQ

qq

qf

Where

-1.4---------------------------------------

.1

)sin(1

2

o

w

w

o

c

ot

o

w

k

k

gx

P

q

k

f


26/188

6

1.7---------------------------------------------------------1

-1.6-------------------------------------------------------QQ w

nww

nw

ff

Q

From this definition offnw. Thus,

1.8---------------------------------1

1

,

,

ww

nwnwnwnw

w

nw

wr

nwr

f

f

dQ

dQ

dQ

dQ

k

k

-1.11---------------------------1

:asyieldspartsbysideleftthegintegratinThen,

1.10-----------------------------------------1

fluidwetting

on thedependingsaturationthemeasuretobalancematerialfollowingtheusedWelge

-1.9-----------------------------------------------

:assuchexpressionfieldainor

,

,

nwc

(L )S

Snwnw

nwc

L

onw

oiloil

waterwater

oilr

waterr

Q)SAL(xdSA(L)ALS

Q)SAL(dxSA

f

f

k

k

nw

ro

1.12---------------------------------------------------

usingandSthatObserving w

w

www

nw

dS

)(SdfQAxS

dSd

results in:

1.13-------------------------------11

dQ

dQQQ)SAL(

AL(L)S nwnwcnw

or in field expression


27/188

7

-1.14-------------------------------------)()1(1 oipwio fWNSVpVp

S

Welge observed the behavior of the production data before and after the

breakthrough time as shown in Fig.(1.6). He also observed that the relationship between

the cumulative recovery of non-wetting fluid (Qnw) versus cumulative volume for injected

wetting fluid (Q) in a linear immiscible displacement experiment is a straight-line (linear)

with slope equal to one that before breakthrough time (Q = QB), but after breakthrough

time (Q > QB) the relation will be as follows, (Collins, 1976):

1.17)ln( QbaQnw

Where (a) and (b) are constants for any small segment:

18.1 bdQ

dQ

Qnw

Welge also found that the curve for Qnw versus Q functioned strongly for the viscosity

ratio as well as the relative permeability curve, (Collins, 1976).

Note that Welges method can be used for both linear and radial displacements.

The work of Welge was later extended by Johnson et al. (JBN), (1959). They

showed how to calculate the individual relative permeabilites even in the case that the

gravity is not neglected. The equations for JBN method can be summarized (in field

expression) as:

1.16-----------------------------------------------)(

-1.15---------------------------------------------------------1

p

oi

p

pwiw

ow

V

fW

V

NSS

SS


28/188

8

and

19.11

/1

wrw

oro

Qd

IQd

fk

20.1 roow

owrw kf

fk

Where Ir is the relative injectivity, defined as:

Toth et al., (2001) developed a technique for calculating relative permeability

ratio for linear displacement. Later, Toth et al., (2001) extended their method to be

applicable for individual relative permeabilites. The methods of Welge (1952), JBN and

Toth et al. can determine the relative permeability ratio and can be applied with both

laboratory data and field data. In addition, these methods have the following common

assumptions:

(a) Sufficiently high displacement rate so that the effect of capillary end-effect can be

ignored. (b) Incompressible and immiscible fluids. (c) Unsteady-state flow. (d)

Homogenous medium. (e) Constant reservoir properties.

Subsequently, Toth et al. (2005) extended their method for radial displacement

while keeping the same assumptions of their previous work. At the same time they

assumed that there is one production well in the center of the reservoir and there is a

natural water influx or injection wells around the reservoir to cause radial displacement.

First, they started with a small core as a disk with a drill hole in the center for injected

fluid to produce from the surrounding area of that core as shown in Fig.(3.1). Their focus

-1.22---------------------------------------------------

-1.21-----------------------------------------int

AL

QSS

pk

Lq

yinjectivitial

yinjectivitI

owiw

or


29/188

9

was to check the validity of their equations for their method. Finally, Toth et al. (2005,

2006) tested their method on an actual size reservoir.

Civan and Donaldson, (1989) developed a technique to determine relative

permeability for unsteady-state displacement, which depends on Darcys law. They made

the same assumptions of the Welge and JBN methods, but also included the capillary

pressure effect.

.25.........A............................................................/1/k1

A.24.................................01

11

A.23..............................................................................................................

rnw

'

'

'

'

11

0

0

wrnwnwrww

nw

nwnwww

w

cw

nww

w

c

nwnw

nww

w

cw

S

Snw

w

w

c

nwnw

nw

Lnw

fkk

dQ

pdQp

kkA

Lq

Q

SQL

x

S

dS

dpf

kx

S

S

p

q

kAfdSdS

dpfkx

S

S

p

q

kAf

dxx

pp

w

xw

Subsequently, Civan and Evans, (1991) developed a method for estimating

relative permeability for steady-state and unsteady-state displacements based on Non-

Darcy law. This method was for compressible and immiscible fluids, and included the

capillary pressure effect. Also, they assumed that the viscosity is constant and the density

is variable with pressure. Later on, Civan and Evans, (1993) came up with a technique

for determining the relative permeability for compressible fluids. In addition, they

assumed that viscosity and density are variable with pressure by using a non-Darcy law

with capillary pressure included. However, these methods did not consider the skin factor

effect and heterogeneity effect although we expect a strong relationship between them


30/188

10

because we use the well production data to estimate the relative permeabilities.

PRESENT STUDY

In this study we will focus and extend upon the Toth et al. s (2006) study because it is a

unique, practical and direct method for estimating relative permeabilities in radial

systems and therefore it is applicable for determination of relative permeability from well

production data.

Most of the previous methods available for determining relative permeabilities

relied upon the other methods to check and validate their results. Hence, we cannot be

certain about the accuracy of these estimations because there is no real field data

available for relative permeabilies to compare with.

Our objectives in this study are as the following:

(1) Evaluation and determination of the accuracy and applicability of the Toth et al.

(2006)method. We used simulated data generated by reservoir simulation software for

this purpose.

(2) Determining under what conditions the Toth et al. (2006)method works the best.

(3) Determining the effect of reservoir parameters, essentially controlling the

performance of this method.

(4) Demonstrating these issues by several representative case studies.

(5) Extending the Toth et al. method for application involving the compressible fluids

systems.

(6) Studying the effect of skin factor and reservoir heterogeneity on the relative

permeability curves obtained by using the Toth et al. method.


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11

Fig.(1.2) General relative permeability curvesSource: introduced from (Dake L. P., 1978)

Fig.(1.1) Permeability definitionSource: introduced from (Slatt , 2006)


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12

Water in ection

* Unsteady-State method

* Steady-state Method

Oil saturatedWater

Oil

Water

Oil

Water

Oil

Fig.(1.4) Steady state and unsteady state methodFor core sample, introduced from (Patrick W. 2001)

Oil injection

* Drainage displacement

* Imbibition displacement

Water wet coreAt Sw 100%

Water injection

Water+

Oil

Water+

Oil

Fig.(1.3) Drainage and Imbibition displacementSource: the graph was introduced from (Patrick W. 2001)

Water wet coreAt Sor


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13

Fig.(1.5) Three section core for Penn -State methodSource: Introduced from ( Mehdi H., 1988)

Fig.(1.6)General Welges plotSource: introduced from (Levertt, M.C., 1941)


34/188

14

Fig.(1.8) Fluid density vs. pressure for different fluid types.Source: introduced from (Ahmed, T., 2005)

Fig.(1.7) Pressure volume relationshipSource: introduced from (Ahmed, T., 2005)


35/188

15

Fig.(1.9) Flow regimesSource: introduced from (Ahmed, T., 2005)


36/188

16

CHAPTER 2

REVIEW OF THE TOTH ET AL. METHOD FOR

DETERMINATION OF RELATIVE PERMEABILITY FROM WELL

PRODUCTION DATA

_____________________________________________________________

The radial displacement interpretation formulas introduced by Toth et al. (1998, 2001 ,

2005, 2006) for determination of relative permeability from the well production data are

presented in this chapter. This chapter explains and summarizes the various formulations

of Toth et al. presented in different studies here in a consistent manner.

The Toth et al. considered a disk shape porous sample where the displacing fluid

(water) is injected from a small hole in the center of the core to displace the displaced

fluid (oil) towards the surrounding area. Toth et al. assumed one-dimensional radial,

isothermal and unsteady-state flow of two immiscible and incompressible fluids in

homogeneous and isotropic porous media with uniform thickness. Its porosity is and

permeability is k. The thickness of the rock sample is h; the radius of the axial well is r w,

and the external radius is re. The rock sample is saturated with a fluid denoted by a

subscript k. Then, this fluid is displaced by another fluid denoted by a subscript d. The

volumetric rate of the injected fluid is qi. The effect of the capillary force is neglected

(Pc=0) during the displacement processes. The pressure at the inlet face is P e; the

pressure inside the well (fluid outlet face) is Pw. Thus, the pressure difference between

the outer and inner faces of the disk is P = Pe- Pw. Also, this method assumes that all

reservoir parameters will remain constant during the displacement. In addition, The Toth


37/188

17

et al. method is applied at and after the breakthrough time.

2. 1 Flow Equations

2. 1.1 Mantle of Radial Core as an Inlet Face

The total injected rate is assumed equal to the total production rates for phase 1 and phase

2

-2.1------------------------------------------------------------dki qqq

The radial Darcy's flow equations are given by:

-2.4---------------------------------)(22

and

2.3-------------------------------------------------------2

2.2-------------------------------------------------------2

dr

dpShkY

dr

dpkkhkq

dr

dprhkkq

dr

dprhkkq

dk

rk

d

rdi

k

rkk

d

rdd

Where the Y(Sd) is the total mobility as shown:

-2.5---------------------------------------kdk

rk

d

rdd

kk)Y(S

Next, the Levertt functions are introduced for the fractional flow equation as:

-2.8---------------------------------------------------,-1

-2.7---------------------------------------------)(

-2.6---------------------------------------------)(

kd

dk

rk

i

kk

dd

rd

i

dd

ff

SY

k

q

qf

SY

k

q

qf


38/188

18

-2.9---------------------------------------------------kk

dd

rk

rd

f

f

k

k

After rearranging Eq.(2.4), a differential equation is obtained as:

-2.10-------------------------------------------2

-r

dr

)hkY(S

qdp

d

i

The following boundary conditions are applied for the integration:

2.11---------------------------------atandat wwee rrpprrpp

Where Pe > Pw then Eq.(2.10) become

2.12-----------------------------------)(2ppp we

e

w

r

r d

i

SrY

dr

hk

q

Because Y(Sd) is a function of Sd and r is variable in Eq.(2.12); therefore, it should be

transformed as a function ofSd. For this purpose, we consider the Buckley-Leverett,

(1942) solution.

The displacement equation in a radial flow system given as.

-2.15-------------------------------------------------

-2.14---------------------------------------------------,

where

-2.13-----------------------------------------2

0

2

dddd

t

ii

ddid

dS

df)(Sf

dtq(t)V

)(Sfh(t)V)(Sr

A differentiation of Eq.(2.13) gives the following:

2.16-------------------------------------------------2

2 di fd

h

Vrdr


39/188

19

The ratio of Eq.(2.16) and Eq.(2.13) gives:

2.17-------------------------------------------------------2

1

d

d

f

fd

r

dr

So, the new integral boundary conditions become:

-2.18-----at,,andat0,0,1 222 wddddddeddd rrfdfdffffrrfdff

As a result of Eq.(2.18) and substituting Eq.(2.17) into Eq.(2.12) gives:

2.19---------------------------------------------4

-0

fd2

)Y(Sf

fd

hk

qp

dd

di

After introducing a special G function and reformulating Eq.(2.19) gives:

-2.20-------------------------------------44

2

0

Ghk

q

)Y(Sf

fd

hk

qp i

f

dd

did

The time derivative of Eq.(2.20) yields:

2.21---------------------------------------

4

1

dt

dqG

dt

dGq

hkdt

d(( ii

Differentiating Eq.(2.14) twice with respect to time yields:

-2.22-------------------------------------and2

2

dt

(t)Vd

dt

dq

dt

(t)dVq iiii

After rearranging Eq.(2.20), the next two equations can be derived:

And

2.23---------------------------------------------------4

iq

hkpG


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20

-2.24-----------------------------------------dt

)(d42

2 tV

q

phk

dt

dqG i

i

i

The (qi dG/dt) term is interpreted as the following for the outlet face (denoted by

subscript 2):

-2.25-------------------------------------------2

22 dt

fd

f)Y(S

q

dt

dGq d

dd

ii

Then, applying Eq.(2.13) at the outlet face (r = rw), the time derivative is obtained as:

2.27-----------------------------------)(

2)(

)(

2

and

2.26---------------------------------------------------)(

2

2

222

2

2

i

i

wi

i

wd

i

w

d

qtV

rh

dt

tdV

tV

rh

dt

fd

tV

rh

f

Thus, using Eqs.(2.26), (2.27), and Eq.(2.25) can be reformulated to give:

2.28---------------------------------------------)()( 2

2

tVSY

q

dt

dGq

id

ii

Then, using Eqs.(2.24), (2.28), and Eq.(2.21) can be given as:

-2.29-----------------------------------)()(4

)(

)()(

)(4

4

1

2

2

2

2

2

2

2

2

tVShkY

q

dt

tVd

q

p

tVSY

q

dt

tVd

q

phk

hkdt

d((

id

ii

i

id

ii

i

We can apply Eq.(2.29) for two types of boundary conditions, (a) Pis constant and


41/188

21

thus Vi(t) is changing, and (b) qi is constant andP(t) is changing.

Case (a): Pis constant

Eq.(2.29) becomes:

-2.30-------------------------------------)(

)(42

2

3

2

dt

tVdtpVhk

q)Y(S

ii

id

Case (b): qi is constant

This means d2Vi(t)/dt2=0 in Eq.(2.29) and the sum of the fluid mobility is given by

2.31-------------------------------------------)(42

dt

pdhkt

q

)Y(S

i

d

The relative permeability functions can be determined using Eq.(2.29) with Eq.(2.30) or

Eq.(2.31), the last equation is always positive, Y(Sd2) > 0 if Vi(t) is increasing

continuously because:

2.32-----------------------------------0and02

2

dt

(t)Vdq

dt

(t)dV ii

i

Y(Sd2) > 0 should be positive. At the same time the other parameters are positive in

Eq.(2.31) except d(P)/dt < 0 that because the relative permeability and the phase

saturation of the displacing fluid are increasing forward.

2.1.2 Surface of The Radial Well as an Inlet Face

The boundary conditions are given by:

2.33-------------------------------atandat wwee rrpprrpp

If the displacing fluid is taking place inside the radial well, then the solution of the partial


42/188

22

differential Eq.(2.12) gives.

)(2e

w

r

r d

iwe

SrY

dr

hk

qppp

The difference between this equation and Eq.(2.12) is the minus sign. By applying the

boundary conditions (see Eq.(2.20)); the solution will be as follows:

2

0

2

4

df

dd

di

)Y(Sf

fd

hk

qp

The minus signis the difference between this equation and Eq.(2.19). At the same time

this equation is similar to Eq.(2.20). As a result, Eq.(2.21) - (2.31) given in section (2.1.1)

are also applicable for the displacement conditions considered here.

2. 2 Displacement Equations

Vi is referring to the volume of the displacing fluid during (t) time, Vkis the volume of the

displaced fluid during the time and Vd is the amount of displacing fluid. Thus, the

following volumetric expressions can be written.

-2.37-----------------------------------------------------V

2.36-------------------------------------------------------

2.35-------------------------------------------------------

2.34-----------------------------------------------------,-

i

0

0

0

dk

t

dd

t

kk

t

dii

VV

dtqV

dtqV

dtqV

Similarly, the following equation can be written for the flow rates:

2.38-----------------------------------------------------dkdi qqq


43/188

23

The effluent production rate and cumulative volume of the injected fluid are zero until the

breakthrough. Therefore, Eqs.(2.37) and Eq.(2.38) simplify before and at the

breakthrough time.

-2.40-------------------------------------------------------

-2.39-------------------------------------------------------

kadi

kaia

qq

VV

The fractional flow of the production fluids after the breakthrough is given by:

2.41---------------------------------------------------------di

kk

q

qf

2.42-------------------------------------------------------di

dd a

q

qf

If capillary effects are negligible, then we can consider the Welge's 6 equations, given by:

-2.43-----------------11

00 pidddd

k

dd

dd

d

d

/VV)S(S)SS(

f

SS

ff

dS

df

From Eq.(2.43) we can obtain

2.44---------------------------------------0

k

dddd

p

i

f

)S(S)SS(

V

V

Where pd VS and express the average saturation of the injected fluid and pore volume of

the core, respectively.

Substituting Eq.(2.34) and Eq.(2.35) into Eq.(2.41) gives:

2.45-------------------------------------------------------ikk

dV

dVf

The volume balance between the injected and displaced fluids over the core give the

following equations:


44/188

24

-2.46-------------------------------------------------)S( 0dp

kd

V

VS

Hence, Substituting Eqs.(2.45) and (2.46) into Eq.(2.44) gives the following equation:

-2.47---------------------------------------)/(

)()/( 0

ik

ddpk

p

i

dVdV

SSVV

V

V

2. 3 Distribution of Fluid Saturation along the Core Plug

The distribution of fluid saturation along the core plug with acceptable accuracy depends

on the information of the saturations at the inlet, outlet faces and the average saturation

over the core length, (Toth, 2006).

2.3.1 The Water Saturation Distribution During Water Injection

At the beginning of the water injection process, the water saturation in the core is at least

equal to the irreducible water saturation Swi or the somewhat higher Sw0. After the

breakthrough time (t ta), the water saturation distribution along the core can be

represented as:

48.2/

)(

2

0

BLx

ASxS ww

The parameters A and B are determined by applying the boundary conditions at a given

time. wfww Sl, SxS, Sx atand,0 max At the breakthrough time att . Thus,

-2.49-------------------------------------

00max

0max

)S(S)S(S

)S)(SS(SA

wwfww,

wowfww,


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25

2.51-----------------------------------

-2.50-------------------------------------

0max0

00max

)S)(SS(SSS

)S(S)S(S

)S(SB

wowfww,ww

wwfww,

wowf

After the breakthrough time 2wwa S, Stt ; therefore,

-2.54-----------------------------

2.53---------------------------------

2.52---------------------------------

20max0

020max

2

020max

max02

)S)(SS(SSS

)S(S)S(S

)S(SB

)S(S)S(S

)S)(SS(SA

wowww,ww

wwww,

wow

wwww,

wow,ww

Eqs.(2.51) and (2.54) express the average water saturation in porous media as the

geometric mean of the water saturation increments at the inlet and outlet faces. Generally,

after the breakthrough time, the saturation distribution of the injected fluid along the core

can be represented by:

-2.55-----------------------------------------/

)(

2

0

BLx

ASxS dd

Then, the average saturation can be expressed in the following manner:

-2.56-------------------------------------1

0

2

0 dxBx/L

AS

LS

l

dd

Note that the linear flow equations can be transformed to the radial flow equations by

applying the following coordinate transformation inferred by Civan, (2000).

-2.57----------------------------------------------------22

22

we

w

rr

rr

L

x


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26

Such that

2.59-----------------------------------------------------------

2.58-----------------------------------------------------------0

ee

ww

r L , rx

r, rx

From Eq.(2.56), it can be obtained that

-2.60---------------------------------

22

002

A

B,where b)SSb(SS dddd

The relationship given by Eqs.(2.56) and (2.60) can be combined to yield:

-2.61-----------------------------------------2002 V

Vb)SSb(SS

p

kdddd

Where (b) is integration constant defined as:

-2.62-----------------------------------------------------11

max

)S(S

bdid,

Where Sd,max refers to the maximum saturation that will be reached following an

infinite the displacing fluid throughput, and diS represents the initial displacing fluid

saturation.

By substituting Eq.(2.61) into Eq.(2.47), and then considering that the pore volume V p

remains constant and separating the variables yields:

-2.63-----------------------------------------

1 )/Vb(V/VV

)/Vd(V

/VV

)/Vd(V

pkpi

pk

pi

pi

The general solution of Eq.(2.63) yields a linear expression as:

2.64---------------------------------------------------p

i

k

i

V

Vba

V

V


47/188

27

Where a is a integration constant, denotes the fraction of the displaced fluid at the

saturation front with a value less than one ( Toth, 1998,2006). The pore volume is equal

to:

2.65-----------------------------------------------22

)hr(rV wep

So, the average saturation of the displacing fluid in the redial core sample after

breakthrough time is expressed as:

2.A.66-----------------------------------------------------p

kdid

V

VSS

The saturation of the displacing fluid at the outlet face denoted by a subscript 2 and it can

be estimated by:

2.67-------------------------------------------

2

2

V

Vba

V

V

bSS

p

i

p

i

did

The Leverett-type, (1941) fractional fluid volumes can be determined as following based

on Eqs.(2.37), (2.6), and (2.7):

-2.68-------------------------------------------------2

p

i

k

V

Vba

af

2.69-------------------------------------------------------1

And

kd ff


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28

Application To Oil and Water System

a) The cumulative oil and water productions and cumulative volume of water influx are

determined respectively by using the following equations:

2.72-------------------------------------------------------

2.71-------------------------------------------------------

2.70-------------------------------------------------------

dtqW

dtqW

dtqN

t

owii

t

owp

t

oop

The total production is equal to the water influx, thus:

2.73-----------------------------------------------------wowi qqq

b) The theoretical displacement equation used to determine the first two constants is

given by:

-2.74---------------------------------------------------V

Wba

N

W

p

i

p

i

That will be a straight line with slope b > 1 and intercept a < 1, where the constant a is

the oil fraction at the breakthrough time.

Thus, the pore volume for radial system can be estimated by:

2.75-------------------------------------------------22 )hr(rV wep

c) The water and oil fraction at the wellbore is determined by:

2.76---------------------------1and,12 wo

p

iow

ww ff

V

Wba

a

qq

qf


49/188

29

d) In the cases that the reservoirs produce under constant pressure, the total mobility can

be determined by:

2.77-----------------------------------14 1

1211

1

)hkp(b

tba

k

k)Y(S

)(b

o

ro

w

rw

W

Where a1 and b1 are some empirical constant that can be determined by fitting the

empirical power-law function as:

2.78---------------------------------------------------------11b

i taW

Note: The value of b1 must be greater than one (b1>1).

e) In the case of the reservoir producing under a constant rate. The total mobility can be

determined by different expression instead of Eq.(2.77):

-2.79---------------------------------------------4 222

bwi

wtbhka

q)Y(S

Where a2 and b2 are some empirical constants that can be determined by fitting the

empirical power law function as:

-2.80-----------------------------------------------22b

we tappp

Note that the value ofb2 must be negative (b2 < 0).

Eq.(2.80) can be applied only if the production well works perfectly efficiently so that the

skin factor s is zero. Otherwise, thePvalue in Eq.(2.77) and (2.80) should be corrected

as (Toth, 2005):

2.81-------------------------------------------------smeasured ppp

Where sp is the additional pressure drop due to skin effect, and it is given by:


50/188

30

2.82-------------------------------------------------------2kh

SqBp oos

f) The relative permeability ratio is determined by:

2.83---------------------------------------------------1 ow

ww

ro

rw

)f(

f

k

k

g) The individual relative permeability values are determined by:

-2.85-----------------------------------------------1

2.84---------------------------------------------------

)Y(S)f(k

)Y(Sfk

wowro

wwwrw

h) The water saturation is determined by:

2.86---------------------------------------------

2

p

i

p

i

wiw

V

Wba

V

W

bSS

We apply the Toth et al. method on two examples; one deals with a reservoir under

constant water injection as shown in example (2.1) below, and the other involves a

reservoir under constant pressure as shown in example (2.2).

Example 2.1

This example was introduced from Stiles, (1971) and Toth et al. (2005). This case is

under constant water injection (500 m3/d). The production data is shown in Fig.(2.1). The

reservoir properties are summarized in Table (2.1).


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31

0

100

200

300

400

500

600

0 1000 2000 3000

q,m3/d

t, d

qo

qw

Parameter Ex. 2.1

Well radius ,rw , m 0.1

Radius of well influence, re , m 155

Well head area (A),m2 7477

Pay zone thickness, h.m 29

Pore volume,Vp,m3 478500

Porosity, 0.219

Permeability,K, m

2

0.175

P,Pa Variable

Skin factor, S 0

Oil formation volume factor, Bo 1.23

Water formation volume

factor, Bw1

Oil viscosity, , pa.s 0.00132

Water viscosity, , pa.s 0.001

Irreducible water saturation,(Swi)

0.23

Table 2.1 Petrophysical data for example2.1, (Introduced from Toth et al., 2005)

Fig.(2.1) Oil and water Producction data forexample 2.1


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y = 1.9851x + 0.6718R = 0.9991

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3

Vi/Np

Vi/Vp

0.E+0

5.E+4

1.E+5

2.E+5

2.E+5

3.E+5

0.E+0 1.E+6 2.E+6 3.E+6

Np

Wi

Fig.(2.2) Displacement equation for example 2.1

Fig.(2.3)Welges plot for example 2.1


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0.1

1

10

100

0.2 0.3 0.4 0.5 0.6 0.7

krw

/kro

Sw

TBSC-method

Welge-method

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

kr

Sw

kro

krw

Fig.(2.4) Relative permeability ratio curve for example 2.1

Fig.(2.5) Relative permeability curves for example 2.1


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

fw

Sw

Craig Jr. F.

TBSC method

Example 2.2

This example was introduced from Craig, (1971) and Toth et al. (2005). This example is

under constant water pressure (6800 kpa). The production data shown in Fig. (2.6) and

Fig. (2.7). The reservoir properties are summarized in Table (2.4).

Parameter Ex. 2.1

Well radius, rw , m 0.1

Pay zone thickness,

h.m

15.5

Pore volume,Vp,m3 16776

Porosity, -

Permeability,K, m2 0.0315

P, kpa 6800

Skin factor, S 0

Oil formation volume

factor, Bo

1.2

water formation

volume factor, Bw

1

Oil viscosity, , pa.s 0.001

water viscosity, ,

pa.s

0.0005

Irreducible water

saturation, (Swi)

0.25

Table 2.2 Petrophysical data forexample 2.2

(Introduced from Toth et al., 2005

Fig.(2.6) Water fractional curve(after breakthrough time)

for example 2.2


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0

2

4

6

8

10

12

0 5 10 15

Wi/Np

Wi/Np

y = 7E-05x1.1072R = 0.9993

0.E+0

5.E+4

1.E+5

2.E+5

2.E+5

3.E+5

0.E+0 1.E+8 2.E+8 3.E+8 4.E+8

Wi

Time, day

Fig.(2.7) Displacement equation for example 2.2

Fig.(2.8) Cumulative water influx for example 2.2


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1

10

100

1000

0.4 0.5 0.6 0.7 0.8

krw

/kro

Sw

TBSC

Craig Jr. F.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

kr

Sw

krwkro

Craig Jr. F.

Toth.

Fig.(2.9) Relative permeability ratio curve for example 2.2

Fig.(2.10) Relative permeability curves for example 2.2


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CHAPTER 3

GENERATION OF SIMULATED WELL PRODUCTION DATA BY

A COMMERCIAL RESERVOIR SIMULATOR

_____________________________________________________________

Eclipse TM software developed by Schlumbertger is a reservoir simulator well-known by

the oil and gas industry by over the last 25 years, and it is considered to be the leading

finite difference based reservoir simulator. Eclipse TM is a three phase and three

dimensional simulator. It can be used to simulate 1, 2 or 3 phase systems to predict and

manage fluid flow more efficiently. The reservoir simulator has been found to be the

most practical, less expensive, faster, more accurate and adequate when compared with

other methods.

The simulation software is used to generate simulated production data that

substitutes for actual field data. The reason for this is that the actual data is unsuitable for

testing of the method because of noise in the data. However, once the method has been

tested and verified by using simulated production data, this method should be available

for testing with real production data. For this purpose, we assume that the simulation

software represents the real reservoir closely, even though there may be some numerical

solution inaccuracies in the software. Most literature assumes that 10% error is expected

but we will try to avoid any errors when we use the software because the errors in the

numerical solution depend on various factors including the time step size and grid size.

The other main reason for using simulation software is because most of the previous


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methods available for determining of relative permeability curves relied upon the other

methods to check and compare their results. Consequently, we cannot be sure about their

accuracy. In addition, there is no real field data available for relative permeability to

compare with.

We simulate the radial flow system using the Eclipse TM software with a real

reservoir size. The Production well is in the center and the injection wells in the

surrounding areas.

In a radial flow system, there are three main parameters that need to be specified;

(1) re is the blocks outer radius which will divide into several grid blocks in the

simulation software, (2) is the segment angle of the grid block in radians, (3) the

number of layers (we assumed there is one layer in all our examples for simplicity).

Therefore, we started with a simple case and then developed the idea as shown in the

steps described below.

a) We assumed there are six injection wells (the angle is ( =60o , 360/6)) around the

reservoir. We used injection wells instead of aquifer because it is easy to control the

injection wells by constant rate or constant pressure and we do not need to know the

properties for the aquifer.

We can operate the system under unsteady-state by keeping the Pconstant and

letting the flow rate change or keeping the injection rate constant and letting the pressure

vary. We used the first option Pconstant for achieving more accuracy with the used the

software. We used water as the injection fluid to displace oil from one production well in

the center. We assumed that the reservoir is saturated by oil before the injection, and we

divided (re) it into five grid blocks as shown in Fig.(3.1) and Fig.(3. 2).


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Fig.(3.1) Three dimension shape from Eclipse with six injectionwells and one production well in the center (=60

o)

Fig.(3.2) Two dimension shape from Eclipse with six injectionwells and one production well in the center (=60

o)


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We can indicate from the two Fig.(3.1) and Fig.(3.2) that the injection wells are not at the

end of the last grid blocks. We even asked the software to do this but we think that

happened because we divided the reservoir into five sections only and we used a large

angle value. That can cause several problems; (1) The software result will not be accurate

because large grid blocks, (2) The injection wells are not at the end of the reservoir so can

displace the entire hydrocarbon, (3) The distance between one injection well to others

sufficiently large. Therefore, we tried another approach as shown in the next step.

b)To avoid the problems in part (a), we increased the number of the injection wells to

50. Thus, we reduced the angle to =7.2

o

because the angle is equal to 360/ (No. of

injection wells). After we modified the program in the software, we got the result as

shown in Fig.(3.3) and Fig.(3. 4).

Fig.(3.3) Two dimension shape from Eclipseresult


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We can see from Fig.(3.3) that the 50 injection wells are in the center of the last grid

blocks, and the production well is in the center.

Also, we can see from Fig.(3.4) that the 50 injection wells are in the center of the last

grid blocks and the production well is in the center.

This means that when we divide the external radius (re) to small grid blocks and

decrease the angle, it will give a better result. Therefore, we increased the number of

injection wells again to 100 and the angle became (=3.6o). We expected to get accurate

radial displacement for the reservoir fluid by making the distance between one injection

well to others sufficiently small as shown in Fig.(3.5) below.

Fig.(3.4) Three dimension shape from Eclipse result


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We can see from Fig.(3.5) that the (100) injection wells are in the center of the last grids

and the production well is in the center.

c) As we know from parts (a and b) that when we increased the injection wells and

decrease the angle can get accurate results. But at the same time, that makes our program

in the software more complex. For convenience and accuracy, we used a single slice

model as shown in Fig.(3.6) and Fig.(3.7) for most examples in this study because we got

exactly the same result when we used the whole reservoir as a model or when we used

just one slice as shown in Fig.(3.8). We still need to specify the angle value in this case;

therefore, we used (=1o), which means that there are 360 injection wells around the

reservoir.

Fig.(3.5) Three dimensional shape from Eclipse result


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Fig.(3.7) Radial flow but only for one slice(From Eclipse result)

Fig.(3.6) One slice shape for Radialflow

Injection well

h

re

Productionwell

0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000

qm3/D

Time,day

oil rate from 3D shape

water rate from 3D shapeoil rate from one silce

water rate from one silce

Fig.(3.8) Time vs. rate for one slice and the whole reservoir(The production data is from Eclipse result)


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Evaluation Technique:

The following approach is used for evaluating the Toth et al., (2006) method. The data

are generated by using the reservoir simulation software in the following manner.

1- Assume the relative permeability curves as input data.

2-Simulate the flow in the radial system by using Eclipse TM to generate the production

data as a result.

3- Recalculate the relative permeability curves by using Toth et al.method using

production data obtained from the software.

4-Compare the calculated relative permeability values with the assumed values to check

the accuracy.


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CHAPTER 4

EVALUATION OF THE TOTH ET AL. METHOD FOR RADIAL

FLOW USING SIMULATED PRODUCTION DATA

______________________

Generating Relative Permeability

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