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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.(4.1.6) Production data from Eclipse for example 4.1.2 ............................................51
Fig.(4.1.7) Displacement equation for example 4.1.2 ......................................................51
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.(4.1.17) Grid size for example 4.1.2 from Eclipse .....................................................57
Fig.(4.1.18) Grid size for example 4.1.3 from Eclipse .....................................................58
Fig.(4.1.19) Grid size for example 4.1.3 from Eclipse .....................................................58
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.(4.2.2) Displacement equation for example 4.2.1 ......................................................61
Fig.(4.2.3) Cumulative water volume produced for example 4.2.1 ..................................62
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.(4.2.6) Production data from Eclipse for example 4.2.2 ............................................63
Fig.(4.2.7) Displacement equation for example 4.2.2 ......................................................64
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.(4.2.10) Relative permeability curves (assumed and calculated) for ex. 4.2.2 ..........65
Fig.(4.2.11) Production data from Eclipse for example 4.2.3 ..........................................66
Fig.(4.2.12) Displacement equation for example 4.2.3 ....................................................66
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.(4.2.1) Production data from Eclipse for example 4.2.1 ............................................70
Fig.(4.2.2) Displacement equation for example 4.2.1 ......................................................70
Fig.(4.2.3) Cumulative water volume produced for example 4.2.1 ..................................71
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.(4.2.7) Displacement equation for example 4.2.2 ......................................................73
Fig.(4.2.8) Cumulative water volume produced for example 4.2.2 ..................................73
Fig.(4.2.9) Relative permeability ration (assumed and calculated) for ex. 4.2.2 .............74
Fig.(4.2.10) Relative permeability curves (assumed and calculated) for ex. 4.2.2 ..........74
Fig.(4.2.11) Production data from Eclipse for example 4.2.3 ..........................................75
Fig.(4.2.12) Displacement equation for example 4.2.3 ....................................................75
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.(4.2.16) Relative permeability curves (assumed and calculated) for ex. 4.2.3 ..........77
Fig.(4.3.1) Production data from Eclipse for example 4.3.1 ............................................80
Fig.(4.3.2) Displacement equation for example 4.3.1 ......................................................80
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.(4.3.7) Production data from Eclipse for example 4.3.2 ............................................83
Fig.(4.3.8) Displacement equation for example 4.3.2 ......................................................83
Fig.(4.3.9) Water fraction for example 4.3.2 after breakthrough time .............................84
Fig.(4.3.10) Cumulative water volume produced for example 4.3.2 ................................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.(4.3.13) Production data from Eclipse for example 4.3.3 ..........................................86
Fig.(4.3.14) Displacement equation for example 4.3.3 ....................................................86
Fig.(4.3.15) Cumulative water volume produced for example 4.3.3 ................................87
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.(4.3.29) Cumulative water volume produced for example 4.3.4 ................................96
Fig.(4.3.28) Displacement equation for example 4.3.4 ....................................................96
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.(4.3.33) Production data from 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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1
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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2
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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3
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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4
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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5
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
qf
Where
-1.4---------------------------------------
.1
)sin(1
2
o
w
w
o
c
ot
o
w
k
k
gx
P
q
k
f
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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
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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
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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
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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
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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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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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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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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)
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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)
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Fig.(1.9) Flow regimesSource: introduced from (Ahmed, T., 2005)
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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
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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
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-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
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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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-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
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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
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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
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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
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:
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-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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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
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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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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
qf
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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:
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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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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
______________________