Measurement of Relative Permeabilities at Low Saturation

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2014-01-30

Measurement of Relative Permeabilities at Low

Saturation using a Multi-step Drainage Process

Wang, Shengdong

Wang, S. (2014). Measurement of Relative Permeabilities at Low Saturation using a Multi-step

Drainage Process (Unpublished doctoral thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/26848

http://hdl.handle.net/11023/1336

doctoral thesis

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UNIVERSITY OF CALGARY

Measurement of Relative Permeabilities at Low Saturation using a Multi-step Drainage

Process

by

Wang Shengdong

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING

CALGARY, ALBERTA

JANUARY, 2014

© Wang Shengdong 2014

ii

ABSTRACT

The gravity drainage mechanism is important for both maximization of storage capacity of

gas and depletion of oil from oil reservoirs. A sensitivity analysis based on numerical

simulation of CO2 storage confirms that the liquid(s) relative permeabilities at low liquid

saturations and the end points are important for both reservoir simulation and volumetric

modeling of these processes.

This thesis employs the multi-step drainage process to determine the wetting phase

permeabilities close to the end points. In this process, the wetting phase production history

was modeled with fully-coupled capillary pressure, by numerical, analytical and pore-scale

modelling methods. These models leads to corresponding relative permeability calculation

methods, including numerical modeling with automatic history matching, direct estimation

using analytical modelling, and an interactive tube-bundle model correlating the pore

structure and the relative permeabilities.

The first step was to program a simulator in order to mimic the multi-step drainage process.

A group of equations were employed to model the one dimensional multi-step drainage

process according to Darcy's Law and the mass conservation equations. The equations

were solved numerically using PcSim, a program coded using the C++. This program was

used as a benchmark to all other models developed in this thesis. Using the program,

automatic history matching was introduced as a conventional method to determine the

relative permeabilities from the multi-step drainage process. Guo Tao genetic algorithm

iii

(GTGA) was successfully applied to history match the two-phase flow in a porous medium.

The results indicate the application of the GTGA is faster and more reliable than the

conventional genetic algorithm. It was also found that at low wetting phase saturation, the

permeabilities of the wetting phase dominate the production history. Following that, an

analytical method was developed to directly estimate the relative permeabilities of the

wetting phase. The newly developed analytical method simplifies the calculation of the

relative permeabilities close to the end points to a level as easy as calculation of absolute

permeability using Darcy's Equation. In addition, an interactive tube-bundle model

conceptually validated the findings and models developed in this thesis. Further

development of this model could potentially be used to history match the experimental data.

Finally, experiments carried out with both sandpacks and core samples indicate that these

methods can be applied to measure the relative permeabilities close to the end points using

a multi-step drainage process for gas/water, oil/water and gas/oil/water systems.

iv

ACKNOWLEDGMENTS

I would like to express my sincere thanks to my supervisor, Dr. Mingzhe Dong, for his

continuous guidance, encouragement and comment during this work. I have learnt a lot

from him both personally and academically throughout my studies in Canada.

I also wish to express the special thanks and appreciation to my colleagues and friends, Dr.

Jinxun Wang, Dr. Zhaowen Li and other colleagues for the helpful discussions and

suggestions concerning this work and the great helps in my life. I would like to thank Mr.

Bernie Then for the assistance in the experimental designing and manufacturing.

The financial support provided by the Department of Chemical and Petroleum Engineering

at the University of Calgary, Petroleum Technology Research Centre, and the Natural

Resource Canada are gratefully acknowledged.

Thanks for the American Chemical Society for granting free use of the copyright of the

paper “A model for direct estimation of wetting phase relative permeabilities using a multi-

step drainage process”, DOI: 10.1021/ie300638t, 2012.

v

To my dear parents, Fengyi Wang and Cuizhen Dai, for their tens of years love and support.

To my family, Lina Wu and Andrew Wang, for their love and understanding with each

ohter. To remember my grandma with a late greeting from the eastern beach of the Pacific

Ocean.

此文献给我的父母，王丰义与戴翠珍，以感谢他们几十年来的养育与支持；献给我

的家人，吴丽娜与王昊宇，以感谢对彼此的爱与理解；纪念我的奶奶，送去这份远

在大洋彼岸的迟到的祝福。

vi

TABLE OF CONTENTS

Abstract........................................................................................................................................ii

Acknowledgments ..................................................................................................................... iv

Table of Contents ....................................................................................................................... vi

List of Tables............................................................................................................................xiv

List of Figures ........................................................................................................................ xvii

Nomenclatures ........................................................................................................................ xxv

Chapter 1: Introduction .......................................................................................................... 1

1.1 Introduction .................................................................................................................. 1

1.2 Mechanisms of Gas Drainage ..................................................................................... 7

1.2.1 Gravity Drainage and Film Flow ........................................................................ 7

1.2.2 Interfacial Tension Reduction ............................................................................. 8

1.2.3 CO2/Oil Phase Behavior ...................................................................................... 8

1.2.4 Multiple-Contact-Miscible .................................................................................. 9

vii

1.2.5 Swelling Effect ..................................................................................................... 9

1.2.6 Viscosity Reduction ........................................................................................... 11

1.2.7 Blow-Down Recovery ....................................................................................... 11

1.2.8 Capillary Pressure .............................................................................................. 12

1.2.9 Irreducible Wetting-phase Saturations ............................................................. 13

1.2.10 Film Flow and Liquid Mobility at Low Saturation.......................................... 15

1.3 Sensitivity Study on CO2 Storage ............................................................................. 15

1.4 Method for Determination of Flow Functions ......................................................... 16

1.4.1 Measurement of Relative Permeabilities .......................................................... 17

1.4.2 Measurement of Capillary Pressure .................................................................. 18

1.5 Objectives and Roadmap........................................................................................... 21

Chapter 2: Numerical Modeling .......................................................................................... 23

2.1 Introduction ................................................................................................................ 23

2.2 Model Assumptions ................................................................................................... 23

2.3 Mathematical Model.................................................................................................. 25

viii

2.4 Boundary Conditions and Initial Conditions ........................................................... 26

2.5 Flow Function Representation .................................................................................. 28

2.5.1 Global Power Function Form ............................................................................ 28

2.5.2 Discrete Spline Function Form ......................................................................... 29

2.6 Description of Finite-Difference Model................................................................... 30

2.6.1 Grid System ........................................................................................................ 30

2.6.2 Inter-Block Transmissibility Calculations........................................................ 30

2.6.3 Improved IMPES Method ................................................................................. 31

2.7 Validation ................................................................................................................... 36

2.7.1 Pseudo-single-phase Flow ................................................................................. 37

2.7.2 Two-phase Flow ................................................................................................. 40

2.8 Sensitivity Analysis ................................................................................................... 43

2.8.1 Discretization ..................................................................................................... 43

2.8.2 Viscosities/Mobilities ........................................................................................ 45

2.8.3 Membrane ........................................................................................................... 46

ix

2.8.4 Saturation Profiles .............................................................................................. 49

2.9 Summary .................................................................................................................... 50

Chapter 3: History Matching using Genetic Algorithm ..................................................... 52

3.1 Introduction ................................................................................................................ 52

3.2 Methodology .............................................................................................................. 55

3.2.1 Numerical Model ............................................................................................... 55

3.2.2 Guo Tao Genetic Algorithm .............................................................................. 59

3.3 Hypothetical Test ....................................................................................................... 64

3.3.1 Comparisons of GTGA and Conventional GA ................................................ 64

3.3.2 Experimental Uncertainties ............................................................................... 68

3.3.3 Impacts of the Nonwetting Phase...................................................................... 71

3.4 Summary .................................................................................................................... 75

Chapter 4: Analytical Modelling and Direct Estimation of Relative Permeabilities ....... 77

4.1 Introduction ................................................................................................................ 77

4.2 Theoretical Development .......................................................................................... 81

x

4.2.1 Membrane Resistance Negligible ..................................................................... 83

4.2.2 Membrane Resistance Considered .................................................................... 91

4.3 Applications of Analytical Models ........................................................................... 97

4.4 Validation of Assumptions........................................................................................ 99

4.4.1 Gas phase Flow Resistance ............................................................................. 101

4.4.2 Membrane Resistance ...................................................................................... 104

4.4.3 Flow Functions ................................................................................................. 105

4.5 Computational Tests ................................................................................................ 108

4.6 Experimental Validation ......................................................................................... 112

4.7 Summary .................................................................................................................. 115

Chapter 5: Interactive Tube-Bundle Modelling................................................................ 117

5.1 Introduction .............................................................................................................. 117

5.2 Numerical Modeling Results .................................................................................. 120

5.2.1 Saturation Profiles ............................................................................................ 120

5.2.2 Production History ........................................................................................... 121

xi

5.3 Three-Tube Interactive Capillary Model ............................................................... 124

5.4 Extending to an Interacting Tube-Bundle Model .................................................. 134

5.5 Modeling of the Drainage Process.......................................................................... 139

5.5.1 Modeling of Saturation Profiles ...................................................................... 140

5.5.2 Modeling of Drainage History ........................................................................ 141

5.5.3 Modeling of Multi-step Drainage Process...................................................... 145

5.6 Conclusions .............................................................................................................. 151

Chapter 6: Experimental Validation .................................................................................. 153

6.1 Introduction .............................................................................................................. 153

6.2 Apparatus ................................................................................................................. 153

6.2.1 Multi-step Drainage Process using a Sandpack ............................................. 154

6.2.2 Multi-step Drainage Process using a Core Sample ........................................ 155

6.3 Measurement of Resistance/Permeability .............................................................. 156

6.3.1 Procedures ........................................................................................................ 158

6.3.2 Calculation of Resistance/Permeability .......................................................... 159

xii

6.4 Gas Leakage due to Gas Diffusion ......................................................................... 161

6.5 Multi-step Drainage Process using a Sandpack ..................................................... 164

6.5.1 Procedures ........................................................................................................ 164

6.5.2 Calculation of the Basic Parameters ............................................................... 165

6.5.3 Gas/Water System ............................................................................................ 167

6.5.4 Oil/Water System ............................................................................................. 171

6.5.5 Results and Discussions................................................................................... 173

6.6 Multi-step Drainage Process using Core Sample .................................................. 181

6.6.1 Procedure .......................................................................................................... 181

6.6.2 Calculation of the Basic Parameters ............................................................... 182

6.6.3 Gas/Water System ............................................................................................ 184

6.6.4 Oil/Water System ............................................................................................. 189

6.6.5 Gas/Oil/Water System ..................................................................................... 197

6.6.6 Results and Discussions................................................................................... 201

6.7 Summary .................................................................................................................. 210

xiii

6.7.1 Sandpack and Core Sample ............................................................................. 210

6.7.2 Simulation and Analytical ............................................................................... 210

6.7.3 Limitations........................................................................................................ 211

Chapter 7: Conclusions and Recommendations ............................................................... 212

7.1 Summary of Conclusions ........................................................................................ 212

7.2 Recommendations ................................................................................................... 216

Reference ................................................................................................................................. 217

Appendix ................................................................................................................................. 232

A-1 Copyright Permission ............................................................................................... 232

A-2 Interfaces of the Application for History Matching and Visualization ............ 233

A-3 Class for Numerical Simulation of Multistep Drainage Process ....................... 236

A-4 Class for Genetic Algorithm .................................................................................... 250

A-5 Class for Tube-bundle Modeling ............................................................................ 270

xiv

LIST OF TABLES

Table 1.1: Oil recoveries for the primary, secondary and tertiary recovery ........................... 4

Table 1.2: Oil recoveries for varous tertiary recovery methods .............................................. 5

Table 2.1: Core sample parameters used at simulator validation test ................................... 36

Table 3.1: Properties of the core sample and fluids in the numerical experiments .............. 65

Table 3.2: Descriptions of the four runs with an oil/water system, considering experimental

uncertainty ................................................................................................................................. 69

Table 3.3: Descriptions of the four cases the with air/water system considering

experimental uncertainty .......................................................................................................... 73

Table 4.1: Coefficients used in Corey‟s equation to calculate the relative permeability

curves ....................................................................................................................................... 100

Table 4.2: Properties of the core sample and the fluids used in numerical simulation ...... 101

Table 5.1: A comparison of the results calculated from the analytical model and the results

calculated from the interacting tube-bundle model .............................................................. 149

Table 6.1: Basic properties of the oil-wetting and the water-wetting membranes. ............ 155

Table 6.2: Parameters used for hypothetical gas diffusion calculation ............................... 163

xv

Table 6.3: Basic parameters of the sandpack used at the drainage experiment .................. 168

Table 6.4: Measurement of porosity and residual water saturation – Test 1 (Sandpack,

Gas/water System) .................................................................................................................. 169


Gas/water System) .................................................................................................................. 170

Table 6.6: Results of the resistance tests for the sandpack systems .................................... 171

Table 6.7: Measurement of porosity and the residual water saturation (Sandpack, Oil/Water

System) .................................................................................................................................... 172

Table 6.8: Basic parameters of the cores samples ................................................................ 183

Table 6.9: Basic resistance and permeability measured by resistance tests ........................ 184

Table 6.10: The permeabilities and capillary pressures calculated by the numerical

simulation and the analytical model – Gas/Water System. .................................................. 205


simulation and the analytical model – Oil/Water System (1). ............................................. 206


simulation and the analytical model – Oil/Water System (2). ............................................. 207


simulation and the analytical model – Gas/Oil/Water System. ........................................... 208

xvi

xvii

LIST OF FIGURES

Figure 1.1: Phase diagram for a binary mixture system of CO2 and Wasson oil ................. 10

Figure 1.2: Schematic diagram of porous plate method to measure capillary pressure. ...... 20

Figure 1.3: Main structure of the thesis. .................................................................................. 22

Figure 2.1: Cylindrical core sample and membrane configuration. ...................................... 24

Figure 2.2: Differential model for the numerical simulations. .............................................. 32

Figure 2.3: Capillary pressure and relative permeability curves for single-phase flow tests

.................................................................................................................................................... 38

Figure 2.4: Comparison of PcSim and CMG-IMEX 2008 for pseudo-single-phase flow. .. 39

Figure 2.5: Capillary pressure and relative permeability curves for two-phase flow tests .. 41

Figure 2.6: Comparison of PcSim and CMG-IMEX 2008 for two-phase flow. ................... 42

Figure 2.7: Comparison of the wetting phase production histories of the four cases

simulated using different time step. ......................................................................................... 44


simulated using different block numbers. ............................................................................... 45

xviii


simulated using different wetting phase viscosities................................................................ 47


simulated using different nonwetting phase viscosities. ........................................................ 48

Figure 2.11: Comparison of the wetting phase production histories of the five cases

simulated using different membrane/core conductivity ratio................................................. 49

Figure 2.12: The wetting phase (1.0cp) saturation profile along the core sample for

different non-wetting phase viscosities. .................................................................................. 51

Figure 3.1: Conceptual model for multi-step drainage experiment. ...................................... 56

Figure 3.2: Flow chart of GTGA. M individuals are selected as parents. Crossover and

mutation are combined using crossover coefficients ( 5.15.0 i ). .................................. 61

Figure 3.3: Permeability and capillary pressure curves used in the hypothetical experiments.

.................................................................................................................................................... 66

Figure 3.4: Matched permeabilities of the non-wetting phase and the wetting phase, using

conventional GA, in three different runs. ................................................................................ 67

Figure 3.5: Matched results for the hypothetical experiment using GTGA and the spline

function form............................................................................................................................. 68

xix

Figure 3.6: Relative permeabilities retrieved by GTGA with an oil/water system and

experimental measurement uncertainties. ............................................................................... 71

Figure 3.7: Relative permeabilities retrieved by GTGA with an air/water system .............. 74

Figure 4.1: Cylindrical core sample and membrane configuration. ...................................... 82

Figure 4.2: Cylindrical porous medium and the membrane configuration. The porous

medium is sealed by resin to ensure fluids flow in one direction. ......................................... 84

Figure 4.3: Capillary pressure considered as a linear function of the wetting phase

saturation in one drainage step. ................................................................................................ 87

Figure 4.4: Graphic demonstration for the solutions for the eigenvalues ......................... 95

Figure 4.5: The capillary pressure curve and the relative permeability curves calculated

using Corey‟s equation. .......................................................................................................... 100

Figure 4.6: Comparisons of the wetting phase recovery history of the hypothetical model,

the full analytical model and the one term approximation model. ...................................... 103

Figure 4.7: Comparison of the wetting phase recovery history of the hypothetical model

with the membrane resistance, the one term approximation model and the complementary

model. ...................................................................................................................................... 106

Figure 4.8: The linear relationship between the dimensionless time and logarithm of the

dimensionless average saturation for the analytical model and numerical models. ........... 107

xx

Figure 4.9: Linear relationship between the dimensionless time and logarithm of the

dimensionless average saturation for the analytical model and numerical models ............ 109

Figure 4.10: Comparison of the results estimated directly using the analytical model and

the original hypothetical values without the membrane resistance. .................................... 110

Figure 4.11: Comparison of the original hypothetical values and the results estimated

directly using the analytical model with and without membrane resistance revising. ....... 111

Figure 4.12: Capillary pressure measured in Jennings‟ experiment. ................................... 113

Figure 4.13: Capillary pressure measured in Jennings‟ experiment: (a) step one; (b) step

two; (c) step three; (d) step four. ............................................................................................ 114

Figure 4.14: Comparisons of the results obtained using Jennings‟ automatic history

matching method and the directly estimation methods presented in this study. ................. 115

Figure 5.1: Schematic saturation profiles in a drainage process with membrane selective

sealing effect. .......................................................................................................................... 122

Figure 5.2: Schematic of the wetting phase production history in a 4-step drainage process

with the membrane selective sealing effect........................................................................... 124

Figure 5.3: Schematic of the three-tube interacting capillary model: pressure and fluid

distribution. ............................................................................................................................. 125

xxi

Figure 5.4: Schematic of the three-tube interacting capillary model: fluid interaction in the

three-tube interacting capillary model. .................................................................................. 127

Figure 5.5: Pressure profiles in the three-tube interacting capillary model. ....................... 131

Figure 5.6: Results of the sample calculations with the three-tube interacting capillary

model. ...................................................................................................................................... 133

Figure 5.7: Schematic of an interacting tube-bundle model. ............................................... 135

Figure 5.8: Tube radius distribution satisfying truncated Weibull distribution. ................. 139

Figure 5.9: Wetting phase saturation profiles in the interacting tube-bundle model. ........ 142

Figure 5.10: Wetting phase production history of the interacting tube-bundle model with

different numbers of capillary tubes. ..................................................................................... 143

Figure 5.11: Relationship of R 1ln and time calculated using the interacting tube-

bundle model with different numbers of capillary tubes. ..................................................... 144

Figure 5.12: Capillary pressure curve calculated from the pore size distribution and the

drainage pressures used in the 5-step drainage experiment. ................................................ 145

Figure 5.13: Cumulative wetting phase production history in the 5-step drainage process

modeled by the interacting tube-bundle model. .................................................................... 146


modeled by the interacting tube-bundle model. .................................................................... 147

xxii

Figure 5.15: Wetting phase recovery histories at each drainage step in the 5-step drainage

process. .................................................................................................................................... 147

Figure 5.16: Calculated relative permeabilities of water from the analytical model and the

interactive tube bundle model. ............................................................................................... 150

Figure 6.1: Schematic diagram for the multi-step drainage process using a sandpack. ..... 154

Figure 6.2: Schematic diagram of the multi-step drainage system using a core sample. ... 156

Figure 6.3: A photo of the apparatus for the multi-step drainage process using core sample.

.................................................................................................................................................. 157

Figure 6.4: An example for the permeability/resistance test data processing. .................... 158

Figure 6.5: Linear regression to calculate conductivity of target flow system. .................. 161

Figure 6.6: Schematic of the gas diffusion model through the membrane. ........................ 164

Figure 6.7: The experimental and the calibrated cumulative water production histories. . 174

Figure 6.8: The experimental cumulative water production histories and history matching

results of oil/water system in sandpack. ................................................................................ 175

Figure 6.9: History matching results of gas/water system in sandpack. ............................. 176

Figure 6.10: Calculation of the relative permeabilities to water for the gas/water system in

sandpack. ................................................................................................................................. 177

xxiii

Figure 6.11: Calculation of the relative permeabilities to water for an oil/water system in

sandpack. ................................................................................................................................. 178

Figure 6.12: Capillary pressure measurement in the sandpack............................................ 179

Figure 6.13: Comparisons of the relative permeabilities of the wetting phase in the

sandpack obtained from simulation and analytical equation. .............................................. 180

Figure 6.14: Water production history at a three step gas/water system. ............................ 185

Figure 6.15: Gas diffusion rate measured in a multi-step drainage experiment with

gas/water. ................................................................................................................................. 186

Figure 6.16: Water production history of the gas/water system and the results of history

matching. ................................................................................................................................. 187

Figure 6.17: Calculations of the relative permeabilities to water using the analytical method

for the gas/water system. ........................................................................................................ 188

Figure 6.18: Cumulative water production history in a four-step drainage process using

oil/water (1) system. ............................................................................................................... 190

Figure 6.19: Cumulative water production history in a five-step drainage process using

oil/water system. ..................................................................................................................... 191

Figure 6.20: Water production history and the results of history matching....................... 192

Figure 6.21: Water production history and the results of history matching........................ 193

xxiv

Figure 6.22: Comparison of history matched results using Corey‟s equation and monotone

cublic spline functions. ........................................................................................................... 194

Figure 6.23: The calculations of the relative permeabilities to water using the analytical

method for the water/oil (1) system. ...................................................................................... 195

Figure 6.24: Calculations of the relative permeabilities to water using the analytical

method for the water/oil (2) system. ...................................................................................... 196

Figure 6.25: Cumulative kerosene production history in a four-step drainage process. ... 198

Figure 6.26: Diffusion of air through the membrane in kerosene. ..................................... 199

Figure 6.27: Kerosene production history and the results of history matching .................. 200


method for the gas/water/oil system. ..................................................................................... 201

Figure 6.29: Comparisons of the capillary pressure curves measured for different fluid

systems. ................................................................................................................................... 203

Figure 6.30: Comparisons of the relative permeabilities calculated from numerical

simulation for different systems............................................................................................. 204

Figure 6.31: Comparisons of the relative permeabilities of the wetting phase obtained from

simulation and analytical equation. ....................................................................................... 209

xxv

NOMENCLATURES

English Letters

A Cross-sectional area of the core sample/sandpack

kA Cross-sectional area of the thk tube in ITBM

c Gravity to capillary pressure factor

1e Inlet boundary of the core sample/sandpack

2e Outlet boundary of the core sample/sandpack

F Fitness of an individual in genetic algorithm

k Slope of the fitted straight line in the analytical model

K Permeability of the core sample/sandpack

mK

Permeability of the membrane

rnK

Relative permeabilities of the nonwetting phase

rwK

Relative permeabilities of the wetting phase

L Length of the core sample/sandpack

mL

Length of the membrane

klL , Length of different section at the left side of the ITBM

krL , Length of different section at the right side of the ITBM

xxvi

kl Length of different sections in the 3 tube interactive model

nN

Exponent of the nonwetting phase relative permeability in Corey's Equation

wN

Exponent of the wetting phase relative permeability in Corey's Equation

pcN

Exponent of the capillary pressure in Corey's Equation

N Total number of tubes displaced by the nonwetting phase in one drainage step

gN

Total number of tubes initially filled with the nonwetting phase

wN

Total number of tubes initially filled with the wetting phase

LeftN

Total number of tubes broken through at the left side

RightN

Total number of tubes broken through at the right side

kP

Pressure at different location of the 3 tube interactive capillary model

nP

Pressure of the non-wetting phase

wP

Pressure of the wetting phase

inP

Non-wetting phase pressure at the inlet

outP

Wetting phase pressure at the outlet

'outP

Wetting phase pressure at the interface of the membrane and the core

sample/Sandpack

cP

Capillary pressure

max

cP

Capillary pressure at wrS

xxvii

mP

Pressure drop along the membrane

kcP , Capillary pressure in different tubes of ITBM

klP , Pressure at different tubes at the left side of ITBM

krP , Pressure at different tubes at the right side of ITBM

PV Pore volume of the core sample/sandpack

wq

Wetting phase flow rate

nq

Non-wetting phase flow rate

wmq

Wetting phase flow rate in the membrane

Q Total flow rate in the 3 tube interactive capillary model

simQ

Cumulative wetting phase production from simulation model

expQ

Cumulative wetting phase production from experiments

R Oil recovery in an imbibition process

*R Normalized oil recovery

0R

Maximum oil recovery

pvR

Recovery in the units of pore volume

kR

Radius of different tubes in the three tube interactive model

NR

Radius of the smallest tube which will be broken though in this drainage step

1wR

Radius of the largest tube which will not be broken though in this drainage step

xxviii

nS

Saturation of the non-wetting phase

wS

Saturation of the wetting phase

gS

Saturation of the gas phase

wrS Wetting phase saturation at

max

cP

nrS

Non-wetting phase residual saturation

wiS

Initial wetting phase saturation at the beginning of one drainage step

wfS

Water saturation behind an imbibition front

wDS

Normalized saturation of the wetting phase

wDS Normalized average saturation of the wetting phase

t Drainage/imbibition time

Dkt

Normalized drainage time without the relative permeability to the wetting phase

Dt Normalized drainage/imbibition time

T Total number of experiment records

wT

Transmissibility of the wetting phase

nT

Transmissibility of the non-wetting phase

wV

Cumulative volume the wetting phase

tV

Total cumulative produced volume of the wetting phase in one single drainage step

x Coordinate along the core sample/sandpack

xxix

Dx Normalized coordinate along the core sample

x Block size of the 1-demension simulator

Greek Symbols

Interfacial tension between gas and water

ji

Mobility index, i is the index of the tubes; j is the phase index for gas or water

w Mobility of the wetting phase

n Mobility of the non-wetting phase

m Hydraulic conductivity of the membrane

n Viscosity of the non-wetting phase

w Viscosity of the wetting phase

g Viscosity of the gas phase

Porosity of the core sample/sandpack

Conductivity ratio of the core sample/sandpack to the membrane

1

CHAPTER 1: INTRODUCTION

1.1 Introduction

The gas drainage process, a process whereby a non-wetting phase displaces a wetting

phase, is a common process in petroleum engineering, for applications such as the gas

flood, NCG(non-condensable gas) injection, CO2 storage and thermal SAGD (steam

assisted gravity drainage) process. The relative permeabilities at low wetting phase

saturation and the end psoints are important to both the oil recovery and gas storage

capacity.

In recent decades, global warming, characterized by the continuous increase in the average

atmospheric temperature at the earth surface since the mid-twentieth century, is generally

believed to be caused by increasing concentrations of greenhouse gases, which result from

human activities such as the burning of fossil fuel and deforestation. CO2 is one of the

leading members of the greenhouse gas club. With the development of economy and

industry, the amount of CO2 emitted by the human activities will rapidly increase.

According to the International Energy Outlook reference case in 2007, the world CO2

emission is expected to rise from 26.9 billion metric tons in 2004 to 33.9 billion metric

tons in 2015 and 42.9 billion metric tons in 2030. To control the emission of CO2, the

2

Kyoto Protocol aiming to reduce the concentration of greenhouse gases that cause global

warming was signed in Kyoto, Japan in December 1997.

According to the Organization of Economic Cooperation and Development statistics,

Canada is the third-worst of 30 countries when it comes to CO2 emissions per capita and

ninth-worst on emissions on a GDP basis. The Kyoto Protocol set targets for reducing

greenhouse gas emissions starting in 2008. Canada agreed to cut its emissions by six

percent below the level in 1990 by 2012. In order to realize this target, one of the most

promising options is to capture, inject and store CO2 in underground geological formations.

Oil fields, gas fields, saline formations, unminable coal seams, and saline-filled basalt

formations have been suggested as sequestration sites. Among these candidate geological

formations for CO2 sequestration, depleted oil reservoirs are the most appealing option for

the following reasons (Wang et al., 2009). First, depleted oil reservoirs have been

extensively investigated during the oil exploitation stage. To sequester CO2 safely in

geological time requires a clear understanding about the geological structure of the

sequestration site. Second, the underground and the surface infrastructure (wells,

equipment, and pipelines) are already available and could be modified and used for CO2

injection. Third, injection of large amounts of CO2 can potentially increase oil recovery.

By increasing oil recovery, injection of CO2 in depleted oil reservoirs can effectively offset

the expenditures of the sequestration.

Canada, as a major petroleum production country, has great potential to sequester CO2 in

depleted oil reservoirs. Oil pools in Alberta have been characterized and evaluated for

CO2-EOR and CO2 sequestration by several researchers (Bachu and Steward, 2002; Bachu

3

and Shaw, 2002; Bachu, 2002; Bachu and Shaw, 2003). Based on their results, there are

9,128 pools in total in Alberta, including light, medium and heavy oil pools in 2001. Of

these, CO2 sequestration capacity for 8,159 pools was estimated and 929 pools were

considered unsuitable for CO2 sequestration due to data loss, steam injection and the

commingled of pools (Bachu and Steward, 2002). The total capacity of these oil pools is

2,857×106 ton, with 2,468×10

6 ton in light and medium oil pools and 389×10

6 ton in heavy

oil pools.

Although CO2 sequestration in depleted oil reservoirs, as a post-EOR process, provides an

appealing option to sequester green house gas in the atmosphere, there are also some

challenges (Dong et al., 2008). These challenges all require the investigation of fluid flow

abilities during gas storage at low liquid saturations.

First, the ultimate oil recovery of non-wetting gas flooding requires investigating the fluid

flow mechanisms at low liquid saturations. The ultimate oil recoveries of most recovery

methods are summarized in Table 1.1 and Table 1.2 (Schumacher, 1980; Brock and Bryan,

1989; Green and Willhite, 1998; Kon Wyatt et al., 2002). There is still more than 30% oil

left in the oil reservoirs after most tertiary oil recovery methods. The remaining oil after

CO2 sequestration is one of the obstacles to employ depleted oil reservoirs as sequestration

sites. Therefore, before we put sequestration projects into practice, we must have a clear

understanding about what residual oil saturation can be achieved in the reservoirs after

sequestration. From another aspect, if a large amount of CO2 is injected into the reservoirs

and sequestered over geological time, will the remaining oil flow to the bottom of the

reservoir in long time run and be recovered in the future? Moreover, the evacuation of oil

4

could provide more available space, thus enhancing the sequestration capacity of the

selected reservoirs and lowering monitoring and management costs. Therefore, reducing of

oil saturation requires studying the fluid flow mechanism during CO2 sequestration.

Table 1.1: Oil recoveries for the primary, secondary and tertiary recovery

Methods Recovery

Primary 5%-20%

Water Flooding 30%-45%

Tertiary recovery 30%-75%

Second, to enhance CO2 storage capacity requires a clear understanding of the fluid flow

mechanisms at low liquid saturations. Depleted oil reservoirs usually have undergone

primary, secondary (waterflooding) and enhanced (such as chemical or CO2 flooding) oil

recovery processes. In both the secondary and enhanced oil recovery processes, a

considerable amount of water is injected into the reservoirs to displace oil, maintain

reservoir pressure, or improve sweep efficiency (in water-alternating-gas, miscible and

immiscible process). As most oil reservoir formations are water wet, the retention of a

large portion of the injected water in the reservoirs is a common consequence of secondary

and enhanced oil recovery processes. The available space in most depleted oil reservoirs

for storing CO2 is quite limited because a large portion of the reservoir is occupied by the

5

injected water, edge water and bottom water. The available space for CO2 in free

supercritical gas phase is significantly reduced. CO2 storage capacity of depleted oil

reservoirs is remarkably reduced if CO2 is just sequestered as solution gas in brine.

Therefore, it is very significant to remove remaining water from depleted oil reservoirs to

achieve a maximum storage capacity.

Table 1.2: Oil recoveries for varous tertiary recovery methods

Methods Recovery

Immiscible gas drive CO2 40-60%

Miscible gas flooding

CO2 40-65%

Enriched Gas 40-65%

Chemical

Polymer 50-70%

Alkali 45 to 70%

ASP 45 to 75%

Finally, study of fluid flow mechanisms of CO2 sequestration is of significance to evaluate

the risk of CO2 sequestration project. In geologic time frame, CO2 will never keep as a

stationary fluid in the reservoir. Micro scale film flow, mass transfer and capillary

phenomena, together with macro scale gravity differentiation, macro convection, and fluid

6

migration may decrease the safety of CO2 sequestration project. tTese processes all involve

fluid flow at low liquid saturations.

In summary, to reduce the concentration of CO2 in the atmosphere, and thus reduce the

global green house effect, requires the study of CO2 sequestration technology in depleted

oil reservoirs. To decrease costs, reduce risks, and enhance the capacity of CO2

sequestration in depleted oil reservoirs require an accurate understanding of fluid flow

mechanisms at low liquid saturations during CO2 sequestration processes.

Although there are many methods available to measure the capillary pressure, none are

currently used directly under high pressure. Meanwhile, the drainage process is also a

dynamic process involving multiphase fluid flow in porous media. No method has been

mentioned till now to obtain relative permeability curves during capillary test.

The above challenges require investigating the fluid flow mechanism at low liquid

saturations during CO2 sequestration. Capillary pressure and relative permeabilities are

crucial properties to multi-phase statics and dynamics of fluids in porous media. When a

considerable amount CO2 is injected in a reservoir, the reservoir fluids will flow at

relatively low liquid phase saturations, at which, the liquid phase relative permeability is

relatively low while the capillary pressure is relatively high. The gravity-assisted-film flow

will also help to improve the mobility of liquid phases at low saturation. Although the

mechanism of CO2 enhanced oil recovery has been studied for decades, it is still very

difficult to use conventional core flood method to measure the relative permeabilities at

low liquid saturations.

7

Gas drainage processes are very popular in both petroleum engineering such as CO2

storage, gas flood, steam assisted gravity drainage, NCG injection, and environment

engineering such as gas invaded into a vague zone. In this chapter, literature review was

conducted on the mechanisms of gas drainage process. The dominant factors for a gas

drainage process were analyzed using numerical simulation and the flow functions

(capillary pressure and relative permeability) measurement methods was then listed and

compared with each other.

1.2 Mechanisms of Gas Drainage

Among all gas drainage process, CO2 injection may be considered as one of the most

complex processes. CO2 flooding, as one of the most important enhanced oil recovery

methods, has already been studied for decades. The mechanisms of CO2 enhanced oil

recovery are discussed as belows.

1.2.1 Gravity Drainage and Film Flow

Lab experiments with high-pressure visualization micromodels were used to investigate the

recovery mechanisms of CO2 flood in the past decade (Sohrabi et al., 2000; Sohrabi et al.,

2008). Sohrabi et al. (2008) conducted experiments at conditions of very low gas–oil

interfacial tension, negligible gravity forces, and a water-wet porous medium. The

visualizations of pore-scale fluid distribution and displacement mechanisms during the

recovery of residual oil by near-miscible hydrocarbon gas and simultaneous water and gas

injection (SWAG) were presented. They demonstrated that a significant amount of residual

8

oil left behind after waterflood can be recovered by both near-miscible gas and SWAG

injection. In particular, they showed that in both processes, the recovery of the contacted

residual oil continues behind the main gas front and ultimately all of the oil that can be

contacted by the gas will be recovered. Combined with gravity drainage, the mechanism

should be more effective.

1.2.2 Interfacial Tension Reduction

Reduction of interfacial tension between oil and water by the solution of CO2 was observed

by several researchers (Farouq, 1985; Camal, 1986). Camal observed that for heavy oil

with dissolved CO2 of 50-100 m3/m

3, a 30% reduction of interfacial tension can be

expected. For water-wetting reservoirs, reduction of interfacial tension enhances the

mobility of the oil, reduces the residual oil saturation and improves final oil recovery.

1.2.3 CO2/Oil Phase Behavior

PVT tests show that CO2 is not as miscible with most crude oils as hydrocarbon solvents,

but it is soluble with crude oil and water at reservoir temperature and pressure. Figure 1.1

presented the phase diagram of CO2 mixed with Wasson oil containing dissolved gas at 32 ℃

(Orr et al., 1982). The original oil without CO2 is a liquid at pressures above 900 psi and

splits into liquid and gas below this pressure. A mixture containing 40 mol percentage CO2

forms a single liquid phase at pressures above 1,350 psi and a liquid phase and a gas phase

at lower pressures. At high CO2 concentrations, the phase behaviour is more complex. At

low pressures, liquid and gas phases form. With the increase of pressure, the gas phase,

9

which contains CO2 and light hydrocarbon gases, condenses into a second liquid phase.

There is a narrow pressure range over which two liquids and a vapour coexists. Above

these pressures, two liquids form, a CO2-rich liquid and an oil-rich liquid. At higher

temperatures, the two-phase region has the same general shape, but three-phase region will

not show up. Instead, the CO2-rich phase is a low-density gas at low pressures and a dense

supercritical phase at high pressures. From the phase diagram, it can be seen that when the

mole percent of CO2 is larger than 70%, there will be two phases, either a system with

liquid/vapour or a system with liquid/liquid.

1.2.4 Multiple-Contact-Miscible

The swelling factor for oil is relatively small. If swelling factor alone accounted for the

recovery of oil in CO2 flooding, the incremental recovery of oil would be relatively low.

However, under the right circumstances, CO2 can be multiple-contact-miscible with oil

(Hutchinson and Braun, 1961). In a multiple-contact-miscible, CO2 mixes with oil

continuously and the mixing process is repeated until the critical tie line is reached and

there will be two-phase region showing up. But if the oil composition or PVT conditions

are different, multiple-contact-miscible cannot be developed.

1.2.5 Swelling Effect

CO2 is a quite soluble gas in crude oil at typical reservoir pressure, but not miscible in all

proportions at any pressure. When CO2 is mixed with oil, at the very beginning it just

simply dissolves in the oil phase as solution gas. The solution of CO2 in the oil phase

10

results in expansion in the volume of oil/gas mixture. Simon (1965) introduced generalized

correlations for the CO2 solubility and swelling effect. Basically, at low CO2 mole fraction,

the dissolving effect happens, the volume of the binary mixture increases and the oil

expands as an additional CO2 solution. As CO2 mole fraction increases, a second CO2-rich

phase shows up and this phase extracts light hydrocarbons from the oil and causes the oil

phase to become denser and more viscous.

Figure 1.1: Phase diagram for a binary mixture system of CO2 and Wasson oil at 32 ℃ (Orr,

1982).

0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100

Mole Percent CO2, %

Pre

ss

ure

, p

si

Liquid Phase

(CO2 Dissoved in Oil)

Two Liquid Phases

(CO2 and Oil

in Each Phase)

Two Liquid + Vapor

Liquid (Oil + CO2)

+ Vapor (CO2 + Light Component)

11

1.2.6 Viscosity Reduction

The dissolved CO2 can decrease the viscosity of oil phase before the appearance of the CO2

rich phase. When two phases form, the CO2 rich phase extracts light component from oil,

which makes oil rich phase more vicious the it otherwise would be. Although the viscosity

of the mixture of oil and dissolved CO2 is lower than that of the original oil, unfortunately,

the viscosity of the CO2-oil mixture is still much higher that of the CO2 rich phase.

Therefore, the performance of a CO2 flood on a reservoir scale is also influenced by the

macroscopic behaviour of the displacement regions occur as CO2 pushes oil. The sharp

viscosity fingering effect is the main problem in CO2 flooding process.

1.2.7 Blow-Down Recovery

The energy stored by CO2 when it goes into solution with an increase in pressure is

released when the pressure is decreased after flooding and continues to drive the oil to the

well bore (Kamal, 1986). The blow-down recovery following an immiscible CO2 flood is

very effective in recovering additional oil during the later stage of an oil reservoir life. The

solution gas drive mechanism is involved in this process. However, this mechanism is not

relevant for a gas storage process when pressure is maintained.

CO2 sequestration is a post-EOR process. There are some significant differences between

CO2 EOR and CO2 sequestration (Wang et al., 2009): (1) the target of CO2 sequestration is

to store as much CO2 as possible safely in a long period, while EOR process requires

12

minimizing the usage of CO2 but recovering more oil. (2) CO2 sequestration is a process

involving great amount of non-wetting gas injection, which will result in the reservoir

fluids flow at high capillary pressures and low liquid phase relative permeabilities. (3) CO2

sequestration is a slow and long-term process. The roles of gravity drainage and film flow

become dominant over a long period of time.

The maximization of CO2 storage capacity and benefits in the underground formations

have attracted more and more attention in recent decade (Vidiuk and Cunha 2007; Wang et

al., 2009; Jahangiri, 2010). Gas injection into underground formations is usually a process

that non-wetting phase displaces wetting phase(s). At low wetting phase saturation, the

capillary pressure becomes essentially high and the relative permeability of wetting phase

becomes relative low. Capillary pressure and relative permeability are dominating

functions required to model the fluid flow and distribution in porous media. Accurate

capillary pressure and relative permeabilities data are thereby required for a successful

simulation to maximize the CO2 storage capacity and benefits in these underground

formations.

1.2.8 Capillary Pressure

Most of the potential sequestration sites, such as depleted gas reservoirs, depleted oil

reservoirs, and especially underground aquifer formation, have great percentage of brine.

These brines mainly come from injected water in the improved oil recovery period, the

connate water, and edge or bottom water supplement. When CO2 is injected into depleted

oil reservoirs or underground aquifer formations, a drainage process that non-wetting phase

13

displace wetting phase happens. As the decrease of liquid saturation, the capillary pressure

became relative high and the relative permeabilities of the liquid phase(s) become

essentially low. Reservoir simulation showed that the capillary pressure magnitude directly

influence the CO2 storage capacity in these underground formations (Wang et al., 2009;

Wang and Dong, 2011). Macroscopically speaking, these flow functions affect the fluid

distributions and fluids motilities. Microscopically speaking, the capillary pressure can

hold the brine and oil in the corner of the pore structure. Therefore, the magnitude of

capillary pressure has very important influence on the storage capacity.

1.2.9 Irreducible Wetting-phase Saturations

Typically drainage capillary pressure curves indicate an irreducible saturation. It is

assumed that wetting phase loses the hydraulic continuity and has no conductivity below

this situation. It is generally believed that water can only flow above this saturation.

However, experiments researching on the residual wetting phase in porous media which

have been carried out since 1970s show that there should be no specific irreducible wetting

phase saturation for most real porous media. The preferentially wetting phase in porous

media is always continuous and conductive along the pore edges and grooves, even at very

low saturation. The existence of the residual liquids was discussed and modelled by Bryant

and Johnson (2001). There is no remarkable irreducible wetting phase saturation for most

real porous media. The existence of sub-irreducible water saturation in some natural gas

reservoirs or even without water indicates the possibility that the CO2 reservoirs with sub-

irreducible liquid saturations can exist stably for geologic time. Specific additional

14

documentation on sub-irreducible water saturation reservoirs is provided by Katz et

al.(1982). The reasons for these sub-irreducible water saturations are believed to be a

combined effect of dehydration, desiccation, compaction, mixed wettability, gravity

drainage, and diagenetic effects occurring during geologic time (Gupta, 2009).

Morrow(1970; 1971) studied systematically the residual wetting phase in porous media by

experiments using packing of clean sands and microspheres, investigating the influence of

interfacial tension, viscosity, fluid density, viscoelasticity, contact angle, mix wettability,

particle shape, pore shape, permeability, porosity and particle size distribution on the

magnitude of irreducible wetting phase saturation. He concluded that all the above factors

have very minor impact on the residual wetting phase saturations. The magnitude of

irreducible wetting phase saturations is dominated by heterogeneity of pore structure. The

irreducible wetting phase saturations in the sphere packing are in the range of 6% to 10%,

much lower than the connate water saturations commonly observed in practice. The later

experimental works by Dullien et al.(1986; 1989) provided evidence that wetting phase has

the hydraulic continuity along the edges and grooves in porous media even at very low

saturation. They conducted experiments based on sandstone samples and etched glass bead.

The residual wetting phase saturation can reach as low as 1% in the etched glass bead

packs. The residual wetting saturation in Berea sandstone reached 10% by the application

of capillary pressure of about 1 atm. Raising the capillary pressure to 27 atm, the wetting

saturation was reduced to 5%. The lowest value of residual oil saturation was 0.8%.

15

1.2.10 Film Flow and Liquid Mobility at Low Saturation

The extra low liquid saturations in Dullien et al.‟s (1986; 1989) experiments indicates that

even below residual saturation measured in the lab, the liquid(s) still can have mobility.

This mobility is believed to be caused by film flow. As the saturation decreases, the fluid

flow in a porous media takes place in two steps, bulk liquids flow below the gas/liquid

interface and film flow above the interface. Chatzis (1988) did micromodel study and

found that both flow mechanisms are influenced by the wetting film left behind the

gas/liquid interface. Stead-state flow has already studied by several researchers. Ransohoff

and Radke (1988) numerically solved the steady-state corner flow problem and defined

resistance factors to the flow for various corner configurations. Patzek and Kristensen

(2001) took the contact angle into consideration and proposed a universal curve for the

corner flow in arbitrary angular tube. Unsteady-state flow was first proposed by Bird et

al.(1960) by providing a solution for thin-film 2D flow using analytical approaches. Dong

et al.(Dong et al.1994; Dong, 1995) used finite element method to solve oil film flow over

water film problem in angular cross-section. Xu (2008) combined bulk flow and film flow

together and examined effect of the capillary pressure at the bottom interface.

1.3 Sensitivity Study on CO2 Storage

Some preliminary studies regarding how to maximize the CO2 storage capacity have

already been done in 2009 (Wang et al., 2009). Core flooding tests and core-scale

16

simulations were conducted to evaluate influencing parameters on core scale storage

capacity during CO2 sequestration. It has been concluded that:

(1) The gravity drainage effect plays a significant role in any gas storage/drainage process

in thick reservoirs and reservoirs with dip angles. A vertical displacing direction pattern is

recommended for well pattern design.

(2) It is confirmed that the irreducible water saturation varies under different core flood

inclination angles and the endpoint of the relative permeability curve varies with the fluid

flow direction and the injection rate. Conventional core flood tests are not reliable to

measure the end points.

(3) Orthogonal experiment results show that capillary pressure/relative permeabilities, fluid

flow direction and average pressure are the top three factors that have impacts on gas

saturation. For any gas drainage process, the flow functions, especially the capillary

pressure and relative permeabilities directly determine the gas storage capacity and oil

displacement efficiency.

(4) More studies about the measurement of the relative permeability close to the irreducible

water saturation are necessary.

1.4 Method for Determination of Flow Functions

Since 1950, there are several different methods for determinations of the relative

permeabilities and the capillary pressure reported in the literature. Generally, capillary

17

pressure and relative permeabilities are measured independently. According to the

apparatus and corresponding mechanisms, the measurement of relative permeabilities was

categorized under two types as forward and reverse methods, while capillary pressure

measurements generally include five catalogues: porous plate method, mercury injection

method, centrifuge method, dynamic method and vapour pressure method.

1.4.1 Measurement of Relative Permeabilities

Forward methods are the methods by which relative permeabilities can be obtained directly

from the coreflood test results, either steady-state experiments or unsteady-state

experiments. The steady-state method is the most direct way to measure the relative

permeabilities from coreflood tests (Marle, 1981). In order to obtain a complete set of

relative permeabilities, the fluid mixture is injected under different flow fractions. The

relative permeabilities are calculated by pressure drops and corresponding flow rates till

the production flow rates is identical as that at the inlet. This method is simple but very

time-consuming. The unsteady-state methods, such as JBN method (Johnson, et al., 1959),

have been widely used and well developed because the unsteady-state methods is time-

effective and can be finished very quickly, although it is more rigorous compared to the

steady-state methods. However, most forward methods require applying some

experimental conditions ignore capillary pressure such as Penn-State method (MacAllister

et al., 1990), high rate methods (Maini et al., 1989), stationary-liquid method (Ning and

Holditch, 1990) or an ancillary capillary curve measure in another experimental run.

18

However, if the capillary pressure is high, these experimental conditions are not easy to be

applied.

Reverse methods by automatic history matching the coreflood production data make

simultaneous determination of capillary pressure and relative permeability possible

(Richmond and Watson, 1990; Oak, 1990). These methods require measuring the pressure

drop and production history in coreflood experiments. One dimensional simulator

combined with a non-linear regression algorithm is used to extract the observed parameters

by history matching the experimental results. Many researchers (Dane et al., 1998; Ucan,

et al., 1998; Watson et al., 1998a; Watson et al., 1998b; Esam, 2002a; Esam, 2002b) did

experiments based on relative permeabilities test and meanwhile the capillary pressure

curves are tested simultaneous. Among them, Ayub et al.(2001) provided a set of complex

system and systematic methodology to measure capillary pressure and relative

permeabilities at the same time. Relative permeabilities measurements recently have

achieved great improvement.

1.4.2 Measurement of Capillary Pressure

The porous plate method (Bruce and Welge, 1947) is one of the conventional methods to

measure the capillary pressure of porous media. As shown Figure 1.2, the wetting fluid is

forced out of the sample through the porous plate and collected in the U-tube in the

drainage curve and imbibes into the core and displaces the non-wetting fluid by decreasing

the capillary pressure. The main disadvantage of this method is very time consuming.

Similarly, Kalaydjian (1992) conducted a horizontal three-phase capillary pressure curves

19

measurement in both drainage and imbibition on an outcrop water wet sample and on

unconsolidated water wet cores. The two ends of the coreholder were equipped with semi-

permeable membranes. The gas phase was injected through the lateral surface using two

inlets. His measurements results shown that both drainage and imbibition capillary

pressures are the function of the three saturations. The spreading coefficient effects on both

drainage and imbibition capillary pressure curves are quantified.

Mercury injection method was first adapted by Purcell (1949). In the test, mercury was

injected into the dry and evacuated porous media as a drainage process, while the

withdrawal of mercury presents the imbibition process. The measurement of capillary

pressure by mercury injection is fast and can be applied to irregularly shaped rock sample.

But the disposal of mercury and damage of the sample limit the application of this method

to some extent. Centrifuge method appeared in 1951, the centrifugal force due to the

revolving of the device produce the pressure difference to conduct imbibition and drainage

processes. The capillary data get from the centrifugal method is relatively accurate and

quick, but complex analysis required can lead to calculation errors and the experimental

device is relatively expensive. The vapour pressure method was presented by Melrose,

(1990) and Melrose et al. (1991).

20

Brine

Oil

Nitrogen Pressure

Spring

Core

Seal Oil

Ruler

Figure 1.2: Schematic diagram of porous plate method to measure capillary pressure.

21

1.5 Objectives and Roadmap

The PhD. thesis focuses on the variant modelling methods of the multi-step drainage

process. Using the models developed in the dissertation, the relative permeability can be

estimated when measuring capillary pressure in a multi-step drainage process. The details

topics include, as shown in Figure 1.3:

(1) Develop a numerical program to simulate the multi-step drainage process. The purpose

of the program is (a) virtual numerical experiments (b) automatic history matching (c)

benchmark the analytical model and the pore scale model.

(2) Apply the reversed method using automatic history matching and genetic algorithm to

simultaneously obtain relative permeabilities when measuring the capillary pressure using

multi-step drainage process.

(3) Derive analytical model to directly estimate the relative permeabilities using the data

from multi-step drainage process.

(4) Build pore-scale/tube-bundle models to investigate the fluid flow mechanisms in the

multi-step drainage process.

(5) Carry out laboratory experiments to measure the capillary pressures and relative

permeabilities at low liquid phase saturations especially near and below the conventional

residual oil saturation and irreducible water saturation.

22

Figure 1.3: Main structure of the thesis.

23

CHAPTER 2: NUMERICAL MODELING

2.1 Introduction

This chapter presents the mathematical equations describing unsteady state two-phase flow

in porous media during the multi-step drainage process. The basic assumptions are

introduced to model the multi-step drainage process and build the mathematical model.

The general mass conservation equations and its boundary conditions and initial conditions

are then listed in details. The detailed numerical solution procedure for the two phase flow

drainage model considering the membrane sealing effect is presented. The grid system,

inter-block transmissibility treatments, and solving methods are introduced and a one

dimension numerical simulator, PcSim, was developed using C++. Commercial simulation

packages were used to validate the accuracy of the simulator. In order to choose

appropriate parameters for the further investigations, sensitivity analyses about grid system,

relative permeabilities and membrane hydraulic conductivity were carried out.

2.2 Model Assumptions

As shown in Figure 2.1, the geometry of the experimental model used for the multi-step

drainage process is a thin cylindrical core sample whose rim is sealed by resin. The outlet

of the core sample connects to a thin plastic membrane. Within the breakthrough pressure

24

of the membrane, only wetting phase is allowed to flow through the membrane. Gas is

injected into the sample from the inlet and the wetting phase is expelled from the outlet by

the injected gas. A multi-step drainage process consists of several single drainage steps. In

each step, the wetting phase saturation starts from equilibrium status, then the gas phase

pressure is increased to a new level and the wetting phase recovery history are recorded.

)(WaterOutlet

imper

mea

ble

imperm

eable

membrane

)(GasInlet

Figure 2.1: Cylindrical core sample and membrane configuration. Core sample is sealed by

resin to keep the fluid flow in vertical direction.

25

Since the saturation variation is small within one drainage step and the porous plate is

relative thin, the following assumptions are made to develop the model:

(1) Fluid flows satisfy Darcy‟s law and there is no fluid flow in horizontal direction;

(2) The core sample and the membrane are homogeneous;

(3) Liquid phases are saturated by gas and the solubility of the gas in the liquids phase is

unchanged in the capillary test pressure changing range;

(5) Gravity is neligible comparing to capillary pressure;

(6) There is no flow in the membrane for the non-wetting gas phase;

(7) The breakthrough pressure of the membrane is not exceeded;

(8) There is no chemical reaction, dispersion, or adsorption.

2.3 Mathematical Model

The mathematic model a multistep drainage process can be described as a two-phase

incompressible flow in one dimension governed by the following mass balance equations

(Gottfried et al., 1966):

t

SqP

KK n

nn

n

rn

...................................... (2-1)

26

t

SqP

KK w

ww

w

rw

..................................... (2-2)

Two auxiliary equations are:

wnc PPP ..................................................... (2-3)

1 nw SS ...................................................... (2-4)

where K is the permeability of the core sample; rnK and rwK are relative permeabilities

for non-wetting phase and wetting phase; n and w are viscosities for non-wetting phase

and wetting phase; is the porosity of the core sample; nS and wS are saturations of non-

wetting phase and wetting phase; nP and wP are pressures of non-wetting phase and

wetting phase.

2.4 Boundary Conditions and Initial Conditions

For one drainage step, the following equations are used as initial conditions and boundary

conditions. When 0t , the initial wetting phase saturation in the core sample is wiS and

wetting phase pressure equals to the pressure at the outlet outP . Therefore, the initial

conditions are written as follows:

witw SS 0| ...................................................... (2-5)

27

outtw PP 0| ..................................................... (2-6)

where wiS is the initial wetting phase saturation and

outP is the wetting phase pressure at

the outlet. At the inlet, the boundary conditions are written as:

01

ewq ........................................................ (2-7)

inen PP 1

....................................................... (2-8)

where, wq is the wetting phase flow rate. inP is the pressure at the inlet. At the outlet, the

boundary conditions are different from conventional coreflood test model. The boundary

conditions are written as:

02

enq ........................................................ (2-9)

'2 outew PP ................................................... (2-10)

mwewoutout qPP 2

' ........................................... (2-11)

where 1e and 2e represent inlet boundary and outlet boundary, respectively; inP is the non-

wetting phase pressure at the inlet; 'outP is the wetting phase pressure at the interface of the

membrane and the core sample; m is the hydraulic conductivity of the porous membrane

and pipeline system, which was measured before the multi-step drainage test and is defined

as:

28

ww

mm

q

P

.................................................... (2-12)

where mP is the viscous pressure drop to generate a flow rate of

wq , which both are

function of time.

2.5 Flow Function Representation

2.5.1 Global Power Function Form

There exist several global functional representations for the relative permeability and the

capillary pressure as functions of wetting phase saturation. Siddiqui et al. (1999)

summarized and compared twelve models by experimental, semi-analytical and numerical

approaches and found that every model had its own advantages and limitations. However,

the most commonly employed global functional representation is to describe the relative

permeabilities and the capillary pressure as power functions of the normalized wetting

phase saturation, which are written as follows,

nN

wrnr

wnr

wrrnrnSS

SSSKK

1

1 .................................. (2-13)

wN

wrnr

wrwnrrwrw

SS

SSSKK

1 .................................. (2-14)

29

pcN

wrnr

wrw

ccSS

SSPP

1

max ...................................... (2-15)

where wrrn SK is the relative permeability of the non-wetting phase at the initial wetting

phase saturation wrS ; nrrw SK is the relative permeability of the wetting phase at the

residual non-wetting phase saturation nrS ;

max

cP is the capillary pressure value at wrS .nN ,

wN and pcN are exponents of the non-wetting phase relative permeability, the wetting

phase relative permeability and the capillary pressure equations, respectively.

In this study, discrete capillary pressure data can be obtained directly from the multi-step

drainage experiments. The magnitude of capillary pressure at a given saturation can be

calculated by monotone cubic spline interpolation using the discrete capillary pressure data.

Only parameters wiro SK , wiS , nN , and wN were considered for history matching the

production data.

2.5.2 Discrete Spline Function Form

It was pointed out that the global functional representation for flow functions cannot reach

better matching results than the discrete spline function form (Ucan, et al., 1997). It is also

expected that it is not reasonable to express a flow function in a global function form

because these global equations are only empirical regressions. In addition, a specific

discrete spline function with favourable control point values can completely replace a

global function. Therefore, the spline functional representations such as linear

interpolation, cubic spline interpolation, B spline interpolation and monotone cubic spline

30

interpolation were extensively investigated in this study. These four numerical

interpolation methods were applied to interpolate a capillary pressure system with five

discrete points. Linear interpolation can keep the monotonicity of capillary pressure but the

curve is not smooth, while cubic spline method and B spline method have monotonicity

problem in spite of smooth interpolation curve. Only monotone cubic interpolation can not

only keep the monotonicity but also the smoothness of the original capillary pressure

function. Therefore, the monotone cubic interpolation method was used to interpolate the

discrete capillary pressure data in this study to insure a reasonable fluid flow in the core

sample.

2.6 Description of Finite-Difference Model

2.6.1 Grid System

If one dimension block-centered grid system was adopted to differentiate the differential

equations group presented in above chapter. Block-centered was chosen because it is easier

to control the volumetric equivalence. The grid system is described in Figure 2.2. For the

first and last blocks in the grid system, volumetric modification factors equal to 0.5 were

considered to decrease the error caused by the boundary conditions and the block-centered

grid system.

2.6.2 Inter-Block Transmissibility Calculations

The mobilities of the wetting phase and the non-wetting phase at block n were defined as:

31

n

rn

n

nKKn

............................................... (2-16)

w

rw

w

nKKn

............................................... (2-17)

Considering the block dimension, we defined the dynamic transmissibility as

12

1

2

1

nxnx

nnT o

o

..................................... (2-18)

12

1

2

1

nxnx

nnT w

w

..................................... (2-19)

2.6.3 Improved IMPES Method

The IMPES method was employed to solve the partial differential equations, which was

first proposed by Sheldon et al. (1959) and Stone and Garder (1961). This method is to

create a single pressure equation by a combination of ancillary equation of saturation. The

pressure equation was solved implicitly and the saturation equation is solved explicitly.

Chen et al. (2004) presented an improved IMPES method to solve the partial differential

equations. This method differentiates one large pressure time step into small saturation

steps using an adaptive control strategy. Their study showed that this method is effective

and efficient for the numerical simulation of two-phase flow and it is strong capabilities to

32

solve two-phase coning problems. In this study, as it is similar to a two-phase gas coning

problem, the simulation algorithm has the same convergent issue, although there is no non-

wetting phase production at the outlet of the system. The improved IMPES prese Chen et

al. (2004) presented was successfully used to improve the simulation stability and save

computation time.

0nP 1NPn

0wP 1NPwinQ outQ

nPn

nPw

L

Figure 2.2: Differential model for the numerical simulations. It is a one-dimension model

with N blocks; L is the length of the core sample; Q s are the flow rates from the inlet

and outlet; the last block N is the single phase resistance block considering the porous

membrane and pipe line system.

To apply improved IMPES method, we add Eq. (2-1) to Eq. (2-2):

t

S

t

SqqP

KKP

KK wn

wnw

w

rw

n

n

rn

Substituting (2-3), the following equation is obtained:

33

0

wnw

w

rw

n

n

rn qqPKK

PKK

..................... (2-20)

Similar to conventional IMPES method, the improved IMPES method is expected to have

a linear equation system after differentiation written as,

1

2

1

0

2

1

11

222

111

00

N

N

w

N

n

NN

NNN

d

d

d

d

q

P

P

q

ac

bac

bac

ba

..................... (2-21)

For the 0th

block, Equation (2-20) is differentiated as follows:

12

10

2

1

100

12

10

2

1

100

xx

PP

xx

PPq ww

w

oo

o

100100 wwwooo PPTPPTq

110010 cocoinwooino PPPPTPPTq

1000100 ccwoinowo PPTPTPTTq ............... (2-22)

The coefficients in the first row of (2-21) are as follows:

10 a

00 Tb

00 c

10000 ccwoin PPTPTd

34

For the 1st block, Equation (2-20) is differentiated as:

22

11

2

1

211

22

11

2

1

211

12

10

2

1

100

12

10

2

1

100

xx

PP

xx

PP

xx

PP

xx

PP ww

w

oo

o

ww

w

oo

o

211211100100 wwwooowwwooo PPTPPTPPTPPT

22111211

1100010

cocowooo

cocowooino

PPPPTPPT

PPPPTPPT

22111211

110010

cocowooo

cocoinwooino

PPPPTPPT

PPPPTPPT

1002110

21101

ccwccwoin

oo

PPTPPTPT

PTPTT

. (2-23)

The coefficients in the second row of the coefficient matrix in Eq. (2-21) are as follows:

011 TTa

11 Tb

01 c

10021101 ccwccwoin PPTPPTPTd

For the other blocks in the system, the flow equation can be differentiated as:

12

1

2

1

1

12

1

2

1

1

2

11

2

1

11

2

11

2

1

11

nxnx

nPnPn

nxnx

nPnPn

nxnx

nPnPn

nxnx

nPnPn

ww

w

oo

o

ww

w

oo

o

111

11111

nPnPnPnPnTnPnPnT

nPnPnPnPnTnPnPnT

cwcwwooo

cocowooo

35

111

111

nPnPnTnPnPnT

nPnTnPnTnTnPnT

ccwccw

ooo ........................... (2-24)

Correspondingly, the coefficients are:

nTnTan 1

nTbn

1 nTcn

111 nPnPnTnPnPnTd ccwccwn

For the block 2N :

12

12

2

1

122

12

12

2

1

122

22

13

2

1

233

22

13

2

1

233

NxNx

NPNPN

NxNx

NPNPN

NxNx

NPNPN

NxNx

NPNPN

ww

w

oo

o

ww

w

oo

o

22122233

22333

NPNTNPNTPNTNPNPNT

NPNTNTNPNT

cwcowoutccw

oo

............................................................ (2-25)

For block 1N :

12

12

2

1

122

12

12

2

1

122

NxNx

NPNPN

NxNx

NPNPNq ww

w

oo

o

36

122122 NPNPNTNPNPNTq wwwooo

1222

1122

NPNPNPNT

NPNPNPNTq

wcow

cwoo ......... (2-26)

For the membrane block N , according to Darcy‟s Law:

m

outw

w L

PNPKq

1

.......................................... (2-27)

2.7 Validation

In order to validate the simulation results of PcSim, numerical simulations were run based

on the model defined by the parameters listed in Table 2.1.

Table 2.1: Core sample parameters used at simulator validation test

Parameters Value

Length 10 cm

Permeability 1 Darcy

Pressure Drop 0.1 atm

Viscosity 1 cp

Cross Area 16 cm

37

Two groups of flow functions with different relative permeability curves and capillary

pressure curve were employed, which were plotted in Figure 2.3 and Figure 2.5,

respectively. Figure 2.3 shows completely misible two phases flow to mimik pseudo-

single-phase flow that can be modelled by Darcy's Equation. Capillary pressure between

two phases is zero. Figure 2.5 presents a typical two phase flow with two relative

permeability curves and a capillary pressure difined according to Corey's Equation.

2.7.1 Pseudo-single-phase Flow

The model was first examined by comparing with analytical solutions calculated using

Darcy‟s Law. in order to satisfy the assumptions of Darcy‟s Law, the following

assumptions were applied: (1) the interfacial tension between the non-wetting phase and

the wetting phase is negligible and there is no capillary pressure; (2) viscosities of the

nonwetting phase and the wetting phase are assumed to be identical; (3) as shown in Figure

2.3, the relative permeabilities of the two phases are straight line, their summation is equal

to 1 and the total mobility does not vary with saturation. Two cases were tested as follows.

(1) The hydrodynamic conductivity of the membrane was assumed to be infinite. Darcy‟s

Law was applied to the core sample directly and the flow rate was calculated as 0.16 cm3/s,

which is exactly the same as the value calculated by PcSim. This case is also simulated by

CMG-IMEX. As shown in Figure 2.4 (a) and (b), both the saturation profile and the

production history are completely identical.

38

(2) The hydrodynamic conductivity was assumed to be 1.0×10-5

Darcy/cm, which is far

smaller than that of the core sample. Consequently, the pressure drop between the inlet and

outlet is mainly applied to drive the fluid flowing through the membrane. The PcSim also

obtained exactly the same results as these calculated from Darcy‟s Law.

Figure 2.3: Capillary pressure and relative permeability curves for single-phase flow tests.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

Kro

Krw

39

(a)

(b)

Figure 2.4: Comparison of PcSim and CMG-IMEX 2008 for pseudo-single-phase flow. (a)

Saturation profile at 5 minutes along the x direction; (b) Cumulative production history.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10

Satu

rati

on

Position,cm

PcSim

CMG

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6

Cu

m. O

il, c

m3

Time,min

PcSim

CMG

40

2.7.2 Two-phase Flow

If the interfacial tension between the non-wetting phase and the wetting phase were

considered, as shown in Figure 2.5, typical water-wetting relative permeability curves and

capillary pressure curves were employed. No direct analytical solution is available to the

model this process with a remark capillary pressure impact. Theforefore, in order to

benchmark the simulator developed in the work, a commercial software package CMG-

IMEX 2008 was employed to validate PcSim. However, IMEX 2008 fails to model the

single phase flow in the membrane, because zero non-wetting phase permeability is not

allowed in the simulator. Only the saturation profiles and the cumulative production curve

before the displacement front reaches the membrane was used to compare the results. The

results are plotted in Figure 2.6. It can be seen that both the saturation profile and the

cumulative production curve before the displacement front reaches the membane from

IMEX and PcSim are exactly the same. These comparions confirm that the PcSim is

reliable to simulate a two phase flow process.

41

Figure 2.5: Capillary pressure and relative permeability curves for two-phase flow tests.

0

0.5

1

1.5

2

2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Pc

, atm

Kr

Sw

Kro

Krw

Pc

42

(a)

(b)

Figure 2.6: Comparison of PcSim and CMG-IMEX 2008 for two-phase flow. (a)

Saturation profile at 5 minutes along the X direction; (b) Cumulative production history.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10

Satu

rati

on

Position, cm

PcSim

CMG

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

Cu

m. P

rod

ucti

on

, cm

3

Time, min

PcSim

CMG

43

2.8 Sensitivity Analysis

2.8.1 Discretization

When a differential equations is solved using a numerical method, there are always some

concerns about the discretazation error. For this study, the discretization errors are mainly

caused by discretization of the length and discretization of time. In order to select

appropriate parameters for fast but relatively accurate calculation results, the sensitivity

analysis regarding time step and block number were carried out.

Four cases using different time steps were run and as depicted in Figure 2.7, the results of

the wetting phase recovery histories for all four cases are the same. It indicates that if the

algorithm is able to converge, the largest time step is recommended during the simulations.

Four cases with different block numbers were run and the wetting phase production history

for all cases are drawn in Figure 2.8. It can be seen that although the production histories

are not exactly the same, the results calculated with ten blocks is acceptable. Compared to

the cases with different viscosity in the following section, the discretization error can

caused by block number is negligible. Therefore, in all simulation cases in this study, the

block number is 10.

44


simulated using different time step. The time steps in the four cases are 0.1 s, 0.01 s, 0.001

s and 0.0001 s, respectively.

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

Cu

m. L

iqu

id, c

m3

Drainage Time, s

Δt=0.1

Δt=0.01

Δt=0.001

Δt=0.0001

45


simulated using different block numbers. The block numbers in the four cases are 5, 10, 20

and 1000, respectively.

2.8.2 Viscosities/Mobilities

The objective of this study is to measure the relative permeabilities according to the

wetting phase production history. Therefore, the production history must be sensitive to the

relative permeabilities so that the relative permeabilities can be determined with a

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

Cu

m.

Liq

uid

, c

m3

Drainage Time, s

Block Number=5

Block Number=10

Block Number=20

Block Number=1000

46

confidence. The relative permeability may vary during the drainage process and the

permeability of the core sample simultaneously changes the mobility of the wetting phase

and the non-wetting phase. Therefore, viscositiy variation also reflects the variation of the

the relative permeabilities. In the test run, the wetting phase and the non-wetting phase

permeabilities are identical and hold a consant during the simulation. As shown in Figure

2.9, the production histories are greatly affected by the viscosity of the wetting phase. The

wetting phase production rate decreases with increasing viscosity of the wetting phase. The

offsetting of the wetting phase cumulative production rate show an exponential relationship

to the variation of the wetting phase viscosity. However, Figure 2.10 indicated that the

wetting phase production history is not sensitive to the non-wetting phase viscosity. When

the non-wetting phase viscosity goes to zero, the production curve should be able to

converge to a curve completely controlled by the wetting phase mobility.

2.8.3 Membrane

The resistance of the membrane, including the pumpline system, was tested in this section.

The difference between drainage process and free imbibitions process is that one puts a

porous membrane or plate to stop non-wetting phase from breaking through. This plate has

higher capillary threshold pressure and the resistance should not be ignored. Therefore, the

conductivity ratio of the membrane to the core sample varies from 1, 0.1, 0.01 and 0.001.

The wetting phase production histories are depicted in Figure 2.11. It can be seen that the

resistance of the membrane has a very large effect on the wetting phase production history.

In every experiment, the resistance of the membrane should be measured carefully because

47

the packing of the system with different vertical forces changes the thickness and

permeability.


simulated using different wetting phase viscosities. The viscosities of the wetting phase in

the four cases are 0.5 cp, 1.0 cp, 2.0 cp and 4 cp, respectively.

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

Cu

m. L

iqu

id, c

m3

Drainage Time, s

Visw=0.5 cp

Visw=1.0 cp

Visw=2.0 cp

Visw=4.0 cp

48


simulated using different nonwetting phase viscosities. The viscosities of the nonwetting

phase in the four cases are 0.01 cp, 0.04 cp, 0.07 cp and 0.10 cp, respectively.

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

Cu

m. L

iqu

id, c

m3

Drainage Time, s

Visn=0.01 cp

Visn=0.04 cp

Visn=0.07 cp

Visn=0.10 cp

49

Figure 2.11: Comparison of the wetting phase production histories of the five cases

simulated using different membrane/core conductivity ratio. The conductivity ratios of the

membrane to the core sample in the five cases are infinity, 1, 0.1, 0.01 and 0.001,

respectively.

2.8.4 Saturation Profiles

Figure 2.12 shows typical saturation profiles of the wetting phase in a single-step drainage

process for different fluid system. Figure 2.12 (a) shows the saturation profile with a

system where the nonwetting phase viscositiy is significant higher than that of the wetting

phase, such as mercury/air system. The fluid flow directions of the nonwetting phase and

the direction that the wetting phase saturation decreasing are the same, thus the process is

0

0.5

1

1.5

2

2.5

3

3.5

1 10 100 1000 10000

Cu

m. L

iqu

id, c

m3

Drainage Time, s

No Membrane

Cond. Ratio=1

Cond. Ratio=0.1

Cond. Ratio=0.01

Cond. Ratio=0.001

50

called “forward flooding”; Figure 2.12 (c) shows the saturation profile with a system where

the non-wetting phase viscosity is significant lower than that of the wetting phase. The

saturation of the wetting phase drops from the outlet, thus this process is called “reverse

flooding”; For a system that the viscosity of the non-wetting phase is comparable to the

viscosity of the wetting phase, as shown in Figure 2.12 (b), the saturation of the wetting

phase drops from both ends of the porous medium. The drainage process is then called

“bidirectional flooding”.

2.9 Summary

A program, PcSim, was developed using Visual C++ 6.0 to solve the above numerical

model. The equations were solved using IMPES (Implicit Pressure Explicit Saturation)

method and a one dimensional model with the 9-27 grid blocks. The simulation was run at

a desktop computer with 2.0 GHz CPU and 4 GB memory for couples of minutes. To

benchmark results of the C++ program, saturation and pressure profiles in the core sample

before the non-wetting phase front reaches the membrane were examined by using a

commercial simulation package (CMG-IMEX). Sensivity analysis was run regarding grid

size, time size, viscosities, membrane resistance, and saturation profiles were carried out.

51

(a)

(b)

(c)

Figure 2.12: The wetting phase (1.0cp) saturation profile along the core sample for

different non-wetting phase viscosities. (a) 10 cp; (b) 0.5 cp; (c) 0.01 cp.

0.5

0.52

0.54

0.56

0.58

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Satu

ratio

n

Position, cm

1s 3s 13s 55s 144s

0.5

0.52

0.54

0.56

0.58

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Satu

ratio

n

Position, cm

1s 3s 13s 55s 144s

0.5

0.52

0.54

0.56

0.58

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Satu

ratio

n

Position, cm

1s 3s 13s 55s 144s

52

CHAPTER 3: HISTORY MATCHING USING GENETIC

ALGORITHM

3.1 Introduction

A clear understanding of the mechanisms in a gas/liquid drainage process is important for

both gas flooding and gas storage. In recent years, CO2 storage in underground formations

has provided an attractive option to mitigate the concentration of the greenhouse gas

(Bachu,2002). Although there have been many studies focusing on how to maximize the

gas storage capacity in underground formations (Jahangiri and Zhang, 2010),

determination of the relative permeabilities at high capillary pressures and low liquid(s)

saturation is important for maximizing gas storage capacity in underground formations

(Richmond and Watson, 1990; Akin and Demiral, 1998).

One of the most conventional ways to measure the capillary pressures and the relative

permeabilities is a coreflood test. In this process, one/two phase(s) is/are injected from one

end of a core sample or a sandpack and both phases are produced from the other end. The

production data are collected and used to calculate the relative permeabilities. These

measurements can be categorized as steady-state method and unsteady-state method. The

steady-state methods are time-consuming, while the measurement using an unsteady-state

methods, such as the JBN method (Johnson et al., 1959), requires applying some

53

experimental conditions to make capillary pressure negligible, conducting an ancillary

capillary measurement or automatic history matching the coreflood production data to

simultaneously determine the capillary pressures and the relative permeabilities (Kerig and

Watson, 1986; Watson et al.,1988; Richmond and Watson, 1990; Ucan et al, 1997; Watson

et al., 1998). However, at low wetting phase saturation, high capillary pressure and fairly

low relative permeability to the wetting phase make the measurement of the relative

permeability more and more difficult using a conventional coreflood test.

The multi-step drainage process provides another way to determine the relative

permeabilities when measuring the capillary pressure. The most commonly used method is

the inverse method, which involves history matching the production data during the

drainage process. Jennings (1983; 1988) used a gradient-based optimization algorithm for

automatic history matching and to calculate the relative permeabilities in an air/kerosene

drainage process. A similar methodology has also been applied by soil science researchers

(Eching et al., 1993; Eching et al., 1994; Liu et al., 1998; Chen et al., 1999). Although

reduced by Eching (1994) through incorporating the measurement of the pressure at the

middle point of the soil column, the non-uniqueness issue is always a concern for these

inverse methods. The causes for the non-uniqueness issue of an inverse method can be

summarized as: (1) instinct non-uniqueness (2) algorithm non-uniqueness; (3) sensitivity

non-uniqueness. The instinct non-uniqueness is the non-uniqueness caused by the problem

itself, such as a linear equation with two unknowns. The algorithm non-uniqueness is

defined as the non-uniqueness caused by the optimization method, which easily converges

to a local optimum value and lead to multiple solutions. The sensitivity non-uniqueness is

54

caused by the sensitivity of the output to the input. If output is not sensitive to certain input

parameters, any experimental error, or even the some numerical truncated errors will

change the estimated value. Although the authors of the above publications believed that

the results calculated from the automatic history-matching were unique, it was also pointed

out that, for a general two-phase coreflood test, using only external production data and

pressure drop data cannot guarantee the uniqueness of the estimated flow functions (Ucan

et al., 1997). Therefore, the uniqueness of the solution and the reliability of the results by

history matching a multi-step drainage process require examination.

Due to concerns that the conventional gradient-based optimized methods cannot guarantee

that the calculation converges to the global optimum solutions and that they cause the

algorithm non-uniqueness, the conventianal Genetic Algorithm (GA) was applied as an

alternative for automatic history matching coreflood tests (Ucan et al., 1997; Sun and

Mohanty, 2003; Tokuda et al., 2004). Although conventional GA has many advantages,

such as global optimization and no Jacobian matrix or other gradient calculations, it is

either possible to converge to a local optimum solution (namely premature convergence)

due to inappropriate parameter selection (Reed and Minsker, 2000), or to have relatively

low efficiency for converging to the optimum solution. Therefore, various improved

conventional GAs have been proposed and have shown remarkable improvement to the

conventional GA. Guo (Guo et al., 1999) presented the Guo Tao genetic algorithm

(GTGA), a simple, robust and accurate algorithm, which can be used for inverse problems.

Wu (2004) indicated that the GTGA has a good tolerability when dealing with the

uncertainty of the observed experimental data.

55

This thesis applies genetic algorithm to analyze the uniqueness of estimation of the relative

permeability in a multi-step drainage experiment. A C++ program is coded to simulate the

two-phase flow in the multi-step drainage process, taking the membrane resistance into

consideration. The GTGA is used to reduce the algorithm non-uniqueness of the history

matching. Systematic examinations regarding the non-uniqueness issue when estimating

the relative permeabilities from the production history of a multi-step drainage process are

carried out based on the hypothetical numerical simulation. The real experimental data is

then used to validate the conclusions.

3.2 Methodology

3.2.1 Numerical Model

A schematic diagram of the apparatus used in a multi-step drainage process is shown in

Figure 3.1. To ensure the fluids flow in one direction, the edge of the thin core sample is

coated with epoxy before the experiment. The core sample is partially saturated with water

which is the wetting phase. A membrane is positioned at the bottom and is in capillary

contact with the core sample. The non-wetting phase (air) is injected from the top of the

core sample and only the wetting phase is produced from the bottom, due to the membrane.

A typical multi-step drainage process consists of several single drainage steps. Each single

drainage step starts from the equilibrium saturation distribution of the previous drainage

step. The non-wetting phase pressure suddenly increases to a higher level, the wetting

phase is produced from the core sample and its production history is recorded. In order to

56

develop the one-dimensional flow model to simulate this process, the following

assumptions are made: (1) fluid flows satisfy Darcy‟s law; (2) fluids flow in one direction;

(3) the core sample and membrane are homogeneous; (4) gravity is negligible due to the

minimal thickness of the core sample.

)(WaterOutlet

imper

mea

ble

imperm

eable

membrane

)(GasInlet

Figure 3.1: Conceptual model for multi-step drainage experiment.

The mathematic model is governed by the following mass balance equations:

t

SqP

KK nw

nwn

nw

rnw

................................... (3-1)

t

SqP

KK w

ww

w

rw

.................................... (3-2)

57

where K is the absolute permeability of the core sample; rnwK and

rwK are the relative

permeabilities for the non-wetting phase and the wetting phase; nw and

w are the

viscosities for the non-wetting phase and the wetting phase; is the porosity of the core

sample; nwS and

wS are the saturations of the non-wetting phase and the wetting phase;

nwP and wP are the pressures of the non-wetting phase and the wetting phase.

In order to solve the partial differential equations above, another two equations are

required as follows:

wnwc PPP ................................................... (3-3)

1 nww SS .................................................... (3-4)

where cP is the capillary pressure. To solve the equations, the following initial condition

and boundary conditions are considered:

When 0t , the wetting phase saturation in the core sample is wiS and the wetting phase

pressure is equal to the pressure at the outlet, outP . Therefore, the initial conditions are

written as follows:

witw SS 0| ..................................................... (3-5)

where wiS is the initial wetting phase saturation the beginning of one drainage step and

outP is the wetting phase pressure at the outlet. At the inlet, there is no flow for the wetting

58

phase and the non-wetting phase pressure is equal to the inlet pressure; the boundary

conditions are the written as:

01

ewq ....................................................... (3-6)

inen PP 1

...................................................... (3-7)

At the bottom of the core sample, there is no non-wetting phase flow, thus:

02

enq ....................................................... (3-8)

'2 outew PP ..................................................... (3-9)

mwewoutout qPP 2

' ........................................... (3-10)

where 1e and 2e represent the inlet and outlet boundaries, respectively; inP is the non-

wetting phase pressure at the inlet; 'outP is the wetting phase pressure at the interface

between the membrane and the core sample;m is the hydraulic conductivity, which is

defined as:

ww

m

mq

P

.................................................... (3-11)

where mP is the viscous pressure drop to yield a flow rate of

wq in the membrane. In a

real experimental process, this hydraulic conductivity includes the membrane, the tubings

and the valves in a multi-step drainage experiment. Before the core sample is mounted in

59

the capillary cell, several tests are run to measure the hydraulic conductivity, m , of the

experimental system.

In this model, both relative permeabilities and capillary pressure are considered as

functions of saturation. The representations of relative permeabilities and capillary pressure

are generally categorized into two types: the global functional form and the discrete spline

function form. Corey‟s functions (Corey and Rethjens, 1956) of the normalized wetting

phase saturation are widely accepted and extensively used to express the relative

permeabilities and the capillary pressures due to their simplicity and remarkable physical

significance of each term in the equations. Later studies showed that these equations were

not sufficient to represent the fluid flow (Kerig and Watson, 1987). It was pointed out

(Ucan et al, 1997) that the global functional representation could not reach better matching

results than a discrete spline function form, such as linear interpolation, cubic spline and B

spline. In these interpolation methods, linear interpolation is simple and can keep the

monotonicity, but it may lose the smoothness. Other functions can obtain a good

smoothness, but they cannot keep the monotonicity when the number of discrete points is

limited. A monotone cubic spline method presented by Fritsch and Carlson (1980), can

keep both smoothness and monotonicity; thus, it is used in this study to interpolate the

permeability and capillary pressure as a function of saturation.

3.2.2 Guo Tao Genetic Algorithm

The conventional GA that mimics biological evolution is claimed to be a global

optimization method. However, it easily converges to local minimum value due to

60

inappropriate parameter selection or fitness function definition. Many changes have been

proposed to improve its robustness and reliability, including either improving the evolution

strategy or the fitness calculation. GTGA changes the concept of generation-by-generation

evolution to individual-by-individual evolution. It randomly, or using certain strategies,

picks several parents from the population space to generate one new individual. If the

fitness of the new individual is bigger than the fitness of the worst individual in the

population, the worst individual is then replaced by the new individual. The flow chart of

GTGA is shown in Figure 3.2.

The detailed process of this algorithm is summarized as follows:

(1) Initialize the population space P with N individuals.

(2) Select M individuals randomly, or using certain strategies, from current population

space P .

(3) Determine a new individual. The calculation uses a linear combination of the M

parental individuals with random weighting factors, i , ranging from -0.5 to 1.5. The

application of weighting factors considers the crossover effect, as well as the mutation

effect, among the parental individual.

(4) If a new individual is better than the worst individual, update the population space by

replacing the worst individual with the new individual.

61

GTGA is a method in which all individuals “climb” to the optimum value together. The

advantage of this method is the best individual will always be kept in the population space.

To accelerate the convergence speed, it is recommended that the individuals with higher

fitness are used as the parents.

Initialize first generation space P(0)

Calculate its fitness

Select M individuals as parents for the new generation

(Randomly or according to fitness)

Crossover

M

i

ijiNj ChromChrom0

,1, 5.15.0 i

Is the new individual is

better than the worst one?

Replace the worst individual by the

new individual

The best individual

satisfies the end condition? End

Start

Yes

No

Yes

No

Figure 3.2: Flow chart of GTGA. M individuals are selected as parents. Crossover and

mutation are combined using crossover coefficients ( 5.15.0 i ).

62

Chromosomes Coding

In the multi-step drainage process, discrete capillary pressure data are directly measured in

a multi-step drainage process. The capillary pressure at any water saturation is calculated

by monotone cubic spline interpolation using the discrete capillary pressure data. Only the

relative permeabilities of each phase are unknowns and need to be calculated by automatic

history matching. Therefore, the chromosome coded with a series of genes, real numbers,

represents a group of relative permeability curves, either as Corey‟s form or the monotone

cubic spline-function form. For the global function form, the chromosomes of each

individual consisted of six pieces of genes, which represent wiS , nwrS , wn , nwn , wirnw SK ,

nwrrw SK . Divided by 10max n , the parameters wn and nwn were converted to a real

number between 0 and 1. For discrete interpolation functional forms, the gene chain of

each individual is composed of )1(2 N genes, where N is the number of drainage

experiment steps. Each gene represents a relative permeability datum. The first 1N

chromosomes are used for storing the relative permeability data of the non-wetting phase,

while the following 1N chromosomes are used to store the relative permeability data of

the wetting phase. Based on the assumption that relative permeability is a number between

zero and one, all genes are bounded from zero to one.

Fitness Definition

Automatic history matching of a coreflood experiment is a non-linear, least-square

optimization problem (Sigmund and McCallery, 1979). The relative permeabilities and

capillary pressure were calculated by minimizing the following objective function:

63

T

t

N

n

simS

T

t

simQ

T

t

simP SSQQPPF0 0

2

exp

0

2

exp

0

2

exp .... (3-12)

where subscripts sim and exp represent simulation and experimental data, respectively.

P , Q , and S are the weighting factors for pressure drop, cumulative production and

internal saturation. T and N are time step number and block number in the simulation.

At every step during the multi-step drainage experiment, the non-wetting phase pressure at

the inlet and the wetting phase pressure at the outlet are known parameters. Meanwhile, the

saturation profile in the core sample is hard to measure accurately, due to the limited core

sample thickness. Therefore, only the single phase production history was taken into

consideration as the objective function. The fitness function is then simplified as:

T

t

sim QQF0

2

exp ............................................ (3-13)

Considering the pore volume and experimental data number, the fitness is normalized to:

T

PV

QQ

F

T

t

sim

0

2

exp

.......................................... (3-14)

where PV is the total volume of drained water for the core sample in one drainage step. T

is the total number of the sampling points in one drainage step.

64

3.3 Hypothetical Test

Hypothetical numerical experiments were carried out using the known capillary pressures

and relative permeabilities, which are plotted in Figure 3.3. The properties of the core

sample and the fluids used in the simulation are listed in Table 3.3. Water and kerosene,

whose viscosities are 1.00 cp and 0.70 cp, were used as the wetting phase and the non-

wetting phase, respectively. At the beginning of the hypothetical test, a numerical

simulation was run based on the known capillary pressures and relative permeabilities to

generate the production history of water as the target for history matching. Then, automatic

history matching was carried out using conventional GA or GTGA to tune the relative

permeabilities and match the water production history. By comparing the input and output

relative permeabilities, the effectiveness of the optimization method, the uniqueness of the

estimated relative permeabilities and the effect of the non-wetting phase in the drainage

process can are evaluated.

3.3.1 Comparisons of GTGA and Conventional GA

Both conventional GA and GTGA were applied as the automatic optimization algorithm in

order to estimate the relative permeabilities. For conventional GA, the population size was

100, the crossover rate was 0.70, the mutation rate was 0.01 and the maximum generation

was 1,000. For the GTGA, the population size was 100, the mating pool size M was 20

and the maximum number of calculation was 10,000.

65

Table 3.1: Properties of the core sample and fluids in the numerical experiments

Parameter Hypothetical Value

Core sample length 1.00 cm

Core sample cross-sectional area 20.00 cm2

Core sample porosity 20 %

Core sample permeability 100 md

Wetting phase viscosity 1.00 cp

Non-wetting phase viscosity 0.70 cp

Three conventional GA runs were carried out and the calculated relative permeabilities for

these runs are plotted in Figure 3.4. The lines represent the hypothetical relative

permeabilities, while the points are the calculated values by automatic history matching. It

can be seen that all three cases failed to converge to the hypothetical values within 1,000

generations, with the total number of calculation being 100,000. The best matching values

were found at generations 132, 495, and 124, respectively. The corresponding fitness was

26.4, 42.8 and 23.1. It can be demonstrated that conventional GA efficiently locates the

domain of an optimum value, but has some difficulties converging to the global optimum

value effectively. Compared to conventional GA, as shown by circles and squares in Figure

3.5, GTGA presented a high calculation effectiveness and the solution converges to the

hypothetical values within 10,000 generations, the number of total calculations being

29,564. To achieve the same fitness as the second case using conventional GA, GTGA

66

consumed only 946 generations and 3,893 calculations. Therefore, the GTGA is not only

efficient when it comes to locating the domain of a global optimum, but it is also able to

effectively converge to the optimum value. With a properly designed optimization

algorithm, the relative permeabilities of both the wetting phase and the non-wetting phase

can be estimated by only matching the wetting phase production history. In other words,

the history matched solution is unique. However, note that the wetting phase production

data are from a hypothetical simulation output is ideal and relative uncertainties are below

1×10-8

.

Figure 3.3: Permeability and capillary pressure curves used in the hypothetical

experiments. The relative permeability data was calculated using the Corey model with the

following parameters: nwrS =0.2; wiS =0.15; nwn =2; wn =2; wirnw SK =0.9; nwrrw SK =0.4.

0

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Ca

pilla

ry p

res

su

re, a

tm

Re

lati

ve

Pe

rme

ab

ilit

y

Wetting Phase Saturation

Krn

Krw

Pc

67

Figure 3.4: Matched permeabilities of the non-wetting phase and the wetting phase, using

conventional GA, in three different runs. Lines represent the known relative permeabilities,

while squares and circles represent the estimated values.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1

Kr

Sw

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1

Kr

Sw

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1

Kr

Sw

68

Figure 3.5: Matched results for the hypothetical experiment using GTGA and the spline

function form. Solid lines represent the hypothetical relative permeabilities, while circles

and squares represent the estimated values.

3.3.2 Experimental Uncertainties

The water production data in the above section is ideal numbers with an unreasonably

small measurement uncertainty, which is not true in reality. In order to simulate the

uncertainties, the production data was multiplied by a random factor from 98% to 102%

and rounded to 0.001 cc or 0.01 cc for different uncertainty levels. Spline function form or

global function form is used as a different interpolation method for the relative

permeabilities. Combining these methods and different levels of measurement uncertainties,

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 0.9 1

Kr

Sw

Krn, Hypothetical

Krn, MPGA, 3893 calculations

Krn, MPGA Final

Krw, Hypothetical

Krw, MPGA, 3893 calculations

Krw, MPGA Final

69

four cases were carried out; the detailed descriptions of these four runs are listed in Table

3.2.

Table 3.2: Descriptions of the four runs with an oil/water system, considering experimental

uncertainty

ID Flow Function Matched

Permeability Uncertainty Results

1 Spline Function Oil and water 0.001cc/0.005 CP Figure 3.6-A

2 Spline Function Oil and water 0.01cc/0.05 CP Figure 3.6-B

3 Corey‟s Equation Oil and water 0.001cc/0.005 CP Figure 3.6-C

4 Corey‟s Equation Oil and water 0.01cc/0.05 CP Figure 3.6-D

CP: the final cumulative production of the wetting phase at each drainage step.

By applying GTGA to history match the four cases, the matched results are shown in

Figure 3.6. As can be seen from Figure 3.6–A and Figure 3.6–B, the matched results for

the relative permeabilities of the wetting phase are acceptable, but for the non-wetting

phase if the uncertainty is higher than 0.01 cc (5% total production), the results are not

acceptable. In order to indicate that the result we obtained is not a local optimum value, the

hypothetical relative permeabilities were also employed as an individual at the beginning

of GTGA calculation. Due to the characteristic of GTGA, the hypothetical value will

70

always stay in the population if it is an optimum value. However, it was found that the

optimum solutions did not stay at the hypothetical relative permeability, but shifted away

slowly to the real optimum values corresponding to the production data with uncertainty.

Therefore, for a hypothetical real case with limited measurement precision and random

experimental error, it is not reliable to obtain two-phase relative permeabilities

simultaneously by matching only the wetting phase production history. It was also noticed

that the low permeabilities, for either the non-wetting phase or the wetting phase, were

easy to match and converge earlier to the hypothetical values, which means that the phase

with lower mobility dominates the wetting phase recovery.

The global function representation was also employed to retrieve the relative permeabilities.

The results are depicted in Figure 3.6-C and Figure 3.6-D. The overall results are better

than the cases with spline function representation. For the 0.005 PV case, the Corey‟s

model is accurate enough to obtain the hypothetical value. For the 0.05 PV case, although

the wetting phase permeability is matched very well, for the non-wetting phase the results

are not acceptable. The difference between hypothetical values to the matched value

increases as the wetting phase saturation decreases. If the relative permeabilities were

governed by Corey‟s equation, 5% uncertainty is accurate enough to retrieve the relative

permeability.

71

Figure 3.6: Relative permeabilities retrieved by GTGA with an oil/water system and

experimental measurement uncertainties.

3.3.3 Impacts of the Nonwetting Phase

The above discussion indicates that the water production is more sensitive to the phase

with lower mobility. For a gas/water system, since the mobility of gas is much higher than

that of water, the contribution of gas to the production of water may be negligible. To

verify this hypothesis, the viscosity of the non-wetting phase was reduced from 0.70 cp to

0.02 cp and another four runs were carried out; the details are listed in Table 3.3. Spline

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

B

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

D

72

function form was used to represent the relative permeabilities in all cases. In the last two

cases, the relative permeabilities of gas were taken as a constant one and only the relative

permeabilities of water were tuned to match the production history.

The relative permeabilities obtained in the four cases were plotted in Figure 3.7. It can be

seen from Figure 3.7-A that with an air/water system, although the relative permeabilities

of water can be estimated successfully, the relative permeabilities of gas are not acceptable.

Moreover, as shown in Figure 3.7-B, when experiment uncertainty is considered, the

estimated relative permeabilities of gas phase is completely wrong. On the other hand, the

water production can be matched with arbitrary gas phase permeability. The relative

permeabilities of water can be estimated independently without considering permeabilities

of gas. The gas relative permeabilities were taken as a constant one in Test 7 and 8 in order

to make the effect of gas phase negligible. As depicted by Figure 3.7-C and Figure 3.7-D,

all permeabilities of the wetting phase were successfully retrieved. Hence, the water phase

permeabilities are unique if the relative permeability of gas phase was negligible and even

the experimental uncertainty was considered.

73

Table 3.3: Descriptions of the four cases the with air/water system considering

experimental uncertainty

ID Flow Function Matched

Permeabilities Accuracy Results

5 Spline Function Gas and water No uncertainty Figure 3.7-A

6 Spline Function Gas and water 0.01cc/0.05 CP Figure 3.7-B

7 Spline Function Water only, Krg=1 No uncertainty Figure 3.7-C

8 Spline Function Water only, Krg=1 0.01cc/0.05 CP Figure 3.7-D

CP: the final cumulative production of the wetting phase at each drainage step.

74

Figure 3.7: Relative permeabilities retrieved by GTGA with an air/water system: (A)

accurate recovery data and relative permeabilities of both water and air were matched; (B)

limited accurate recovery data with experimental error and relative permeabilities of both

water and air were matched; (C) accurate recovery data, only relative permeabilities of

water was matched and relative permeability of air was taken as one; (D) limited accurate

recovery data with experimental error, only relative permeabilities of water was matched

and relative permeability of air was taken as one.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

B

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

D

75

3.4 Summary

In this chapter, GTGA is successfully applied to estimate the relative permeabilities by

automatic history matching the production history in a multi-step drainage process with

ideal data. Taking the experimental uncertainty into considerations, the uniqueness of the

solution is analyzed by hypothetical numerical model and a lab experimental study.

Our investigations indicate that GTGA is much more effective than conventional GA.

GTGA could successfully find and escape from the local maximum values and finally

converge to the global maximum value. Both the global function form (Corey‟s model) and

the local spline function form (monotone cubic interpolation) can be used to represent the

flow functions. The global function form converges more easily, while the monotone cubic

interpolation is more realistic. Considering the uncertainty of real lab experiments, it is

considered to be impossible to obtain reliable results for the relative permeabilities of the

highly mobile phase. An experimental study with kerosene/water system confirmed that

this method is of high reliability for low mobility phase.

Using GTGA and history matching of the wetting phase production history during a multi-

step drainage process, the relative permeabilities of the non-wetting phase and the wetting

phase are only simultaneously retrievable when (1) the mobilities of the non-wetting phase

and the wetting phase are comparable and (2) the experimental uncertainty and precision is

controlled within certain level; the relative permeability of the non-wetting phase can be

taken as arbitrary number and the unique relative permeabilities of the wetting phase can

76

be obtained, when (1) the wetting phase saturation of the system is low or (2) the viscosity

ratio of the non-wetting to the wetting phase is small, such as an air/water system.

77

CHAPTER 4: ANALYTICAL MODELLING AND DIRECT

ESTIMATION OF RELATIVE PERMEABILITIES

4.1 Introduction

CO2 sequestration (Bachu, 2002) in geological formations provides an appealing option to

reduce CO2 concentration in the atmosphere, and thereby mitigates the impacts of the

greenhouse gas. The sequestration injection of CO2 gas into the geological formation is a

process where the non-wetting gas phase displaces the wetting liquid phase(s). Near the

end of this process, the capillary pressure of the two phases becomes comparatively high

and the relative permeability of the wetting phase becomes very low. A clear

understanding concerning the capillary pressure and the relative permeabilities at low

wetting phase saturation can assist to maximize CO2 sequestration capacity in geological

formations.

The porous plate method (Bruce, 1947), which is referred to the multi-step drainage

process in this thesis, is one of the conventional methods used to measure the drainage-type

capillary pressure versus liquid saturation relationship of a porous medium. In this process,

the wetting phase recovery history is determined by the capillary pressure, and the relative

permeabilities, simultaneously. The capillary pressure versus saturation curve is

78

determined by displacing the wetting fluid point by point, at each of which, pressure

equilibrium is established. The challenge is how to determine the relative permeabilities in

this process.

One-step drainage experiments were conducted using an air–wetting soil column system,

combined with a parameter optimization algorithm to simultaneously estimate the capillary

pressures and the relative permeabilities (Parker and Welge, 1985). A gradient-based

automatic history matching method was used by Jennings (Jennings, 1983; Jennings et. al,

1988) with a one-dimensional (1-D) numerical simulator to obtain the wetting phase

relative permeabilities in a core sample. Chen et al. (Chen et. al, 1999) considered oil-water

flow in a multi-step drainage process and obtained the relative permeabilities to both oil

and water simultaneously by numerical simulation and history matching. However, all the

above inverse methods using history matching require an accurate simulator and an

effective history matching algorithm. Moreover, the main concern associated with the

inverse function estimation method is that the result of history matching is not unique.

Although said concern was more or less mitigated by simultaneously measuring internal

capillary pressure (Eching and Hopmans, 1993) and extending the one-step drainage

process to a multi-step drainage process (Van Dam, 1994), the uniqueness of the matched

results still requires systemic examination. Eching et al. (1994) and Liu et al. (1998) used

the wetting phase production data at the time periods immediately following an increase in

non-wetting phase pressure to estimate permeability functions (Eching et al, 1994; Liu et

al., 1998). However, the flow rate changes rapidly at the beginning of a drainage process.

If simple analytical equations can be developed to match the entire recovery history, not

79

only will there be no further concern about the uniqueness of the results, but the relative

permeabilities of the wetting phase can be directly estimated.

As early as the 1950s, some models for calculating the relative permeabilities using a

capillary pressure curve were developed for imbibition-type displacement based on the

assumption that the wetting phase and the non-wetting phase flow independently in the

small pores and the large pores, respectively (Wyllie and Spangler, 1952; Wyllie and

Gardner, 1958). This assumption may not always be valid because, at low wetting phase

saturation, the wetting phase stays at the corner of the large pore structure and starts

contributing to the flow rate. For imbibition, analytical or empirical equations modeling the

volume variation of an imbibed wetting phase into a non-wetting-phase-saturated core

sample have been presented since 1950s. Handy‟s equation (Handy, 1960) is usually used

to describe the variations of the volume of imbibed wetting phase as a function of the

imbibition time:

tSKKP

AVw

wfrwc

w

222

.......................................... (4-1)

where A is the cross-section area of the core sample; wV is the volume of water imbibed

into the core sample; and w are the porosity of the core sample and the viscosity of

water, respectively. t is the imbibition time; wfS is the water saturation behind the

imbibition front; rwK and cP are the relative permeability of water and the capillary pressure

at wfS . It should be noted that the imbibition volume of the wetting phase cannot go to

infinity when the time goes to infinity (Li and Horne, 2001). Other models describing the

dict://key.0895DFE8DB67F9409DB285590D870EDD/empirical%20formula

80

process of spontaneous water imbibition into water-wet rocks have been suggested. The

first model was presented by Aronofskyn et al.(1958). The variation of oil recovery as a

function of time is governed by:

teRR 10.................................................. (4-2)

where R is the oil recovery, 0R is the limit toward which the recovery converges, is a

constant giving the rate of convergence, and t is the imbibition time. This model was

based on the following two empirical assumptions: 1) cumulative oil recovery from a small

volume of core converges to a finite limit; and 2) the convergence rate and the convergence

limit do not change during the imbibition process. Based on Aronofsky‟s model, several

modifications or derivations of the parameter were later reported (Mattax and Kyte,

1962; Lefebvre, 1978; Reis, 1992; Ma et al., 1997). Recently, Li and Horne (Li and Horne,

2006) developed the following model to describe the relationship between the water

recovery history and imbibition time in a free imbibition process where water imbibed into

a gas-saturated core sample.

DtR eeR

**1 ................................................ (4-3)

where, *R is the normalized recovery (pvcRR * );

pvR is the recovery in the units of pore

volume and c is a parameter associated with the ratio of gravity to capillary pressure.

However, all of these models are used for a one-step imbibition process. Multi-step

displacement experiments, either imbibitions process or drainage process, require a porous

plate or a thin membrane to control the pressure for each saturation point. There is no

81

analytical model for the multi-step drainage process, especially when considering the

resistance of the porous plate or the membrane.

This chapter presents a new analytical model, which describes the wetting phase recovery

history in a multi-step drainage process. From the regression of the wetting phase recovery

curve at each pressure step, the wetting phase permeabilities at that saturation point can be

determined. In order to examine the model and its assumptions, numerical simulations are

conducted for one-dimensional two-phase drainage-type displacement. Additionally,

hypothetical experiments are carried out to test and validate the analytical model. Finally,

the experimental data reported by Jennings are used to further validate this method.

Comparisons of the results of the model in this chapter and the results presented in

Jennings‟ work point to the effectiveness and validity of the model proposed in this chapter.

4.2 Theoretical Development

As shown in Figure 4.1, the model for the multi-step drainage process is a cylindrical core

sample, with its side surface sealed. The outlet of the core sample connects to a thin semi-

permeable membrane. Within the breakthrough pressure of the membrane for the gas phase

(non-wetting phase), only the wetting phase is allowed to flow through the membrane. Gas

is injected into the sample through the top surface and the wetting phase is expelled from

the lower side. An entire multi-step drainage process consists of several single drainage

steps. In each step, the wetting phase saturation starts from the previous equilibrium state,

82

the gas phase pressure at the outlet increases to a new level, and the wetting phase recovery

history is recorded.

)(WaterOutlet

imper

mea

ble

imperm

eable

membrane

)(GasInlet

Figure 4.1: Cylindrical core sample and membrane configuration.

Since the saturation variation is small within a drainage step and the porous disc is relative

thin, the following assumptions are made in developing the analytical model:

(1) The porous medium is homogeneous;

(2) The effect of gravity is negligible compared to the capillary forces, due to the thinness

of the core sample;

(3) The flow of fluids in the core sample is governed by Darcy‟s law;

(4) At the end of each drainage step, the saturation of the wetting phase reaches an

equilibrium state;

83

(5) The pressure drop of gas phase along the drainage direction within the porous sample is

negligible, due to high gas mobility;

(6) Wetting phase relative permeability within a drainage step is considered as a constant;

Within a drainage step, the capillary pressure is linearly dependent on the wetting phase

saturation.

Assumptions (1)-(4) are reasonable for any Darcy‟s flow in a porous medium, while

rationales of assumptions (5)-(6) are based on the numerical analysis in the later sections of

this chapter and will be discussed then.

4.2.1 Membrane Resistance Negligible

If it is assumed that the hydraulic resistance of the membrane is negligible, an analytical

model which describes the correlation of the wetting phase recovery and the drainage time

can be derived. As shown in Figure 4.2, the length of the core sample is L , the thickness

of the membrane is mL , inP is the non-wetting (gas) phase pressure at the inlet and outP is

the wetting phase pressure at the outlet.

84

inn PPInlet ,

outPOutlet ,x

L

0x

imper

mea

ble

im

perm

eable

membranemL

Figure 4.2: Cylindrical porous medium and the membrane configuration. The porous

medium is sealed by resin to ensure fluids flow in one direction. Pn is the inlet pressure of

the nonwetting phase; Pout is the outlet pressure of the wetting phase.

The ith step drainage process starts from the initial equilibrium water saturation

iwS . The

gas phase pressure has sudden increase from i

inP to 1i

inP at the beginning of the ith

drainage process. The wetting phase saturation correspondingly decreases, and finally

reaches 1i

wS at the end of the ith drainage process.

For one-dimension two-phase flow in a homogeneous core sample, combining the material

balance equation and the Darcy‟s equation gives the following continuity equation for the

wetting phase:

85

t

S

x

PKK

x

ww

w

rw

....................................... (4-4)

where t

is the drainage time, x is the coordinate along the core sample, K and are the

instinct permeability and porosity of the core sample, respectively, w is the viscosity of

the wetting phase, rwK , wP and wS are the relative permeability, the pressure and the

saturation of the wetting phase in the core sample, respectively, at time, t , and position, x .

If w

rw

w

KK

is defined as the wetting phase mobility and w is assumed be a constant

within a single drainage step, the following equation can be obtained:

t

S

x

P ww

w

2

2

................................................ (4-5)

The initial conditions for the pressure and saturation of the wetting phase are:

outw PtxP 0, ................................................ (4-6)

and

)(0,

i

ww StxS ............................................... (4-7)

where outP is the wetting phase pressure at the outlet and i

wS the initial equilibrium water

saturation at the beginning of this drainage step. As there is no wetting phase flowing into

the core sample at the inlet, the boundary conditions for the wetting phase are written as:

86

0,0

tx

x

Pw ................................................ (4-8)

At the outlet of the core sample, if the resistance of the membrane is ignored, the boundary

condition is written as:

outw PtLxP , ................................................ (4-9)

As shown in Figure 4.3, the capillary pressure is assumed to vary linearly as saturation

changes within a single drainage step, and the wetting phase saturation decreases from

)(i

wS to )1( i

wS during one drainage step.

The capillary pressure increase and the wetting phase saturation decrease in a drainage step

are represented by, respectively:

i

c

i

cc PPP 1

............................................. (4-10)

1

i

w

i

ww SSS ............................................ (4-11)

If the resistance to gas phase flow is negligible, the pressure in the gas could be considered

as a constant, that is:

inn PP ...................................................... (4-12)

The capillary pressure at x is equal to the pressure difference between the gas phase and

the liquid phase. Therefore:

87

txPPtxP cnw ,, ........................................... (4-13)

wS

cP 11

, i

c

i

w PS

i

c

i

w PS ,

StepOne

Figure 4.3: Capillary pressure considered as a linear function of the wetting phase

saturation in one drainage step.

As shown in Figure 4.3, the capillary pressure at any time t and any location x has the

following relationship to the wetting phase saturation:

1

1

1

1,,

i

wwi

w

i

w

i

c

i

ci

cc StxSSS

PPPtxP ...................... (4-14)

Substituting Eq.s (4-10) to (4-13) into Eq. (4-14) gives:

88

11,,

i

ww

w

ci

cinw StxSS

PPPtxP .................... (4-15)

Substituting Eq. (4-15) into Eq. (4-5) gives:

t

S

x

S

S

P ww

w

cw

2

2

....................................... (4-16-1)

Substituting Eq. (4-15) into Eq. (4-8) gives:

0,0

tx

x

Sw ........................................... (4-16-2)

Substituting Eq. (4-15) into Eq. (4-9) and considering:

outin

i

c PPP 1

The first boundary condition changes to:

1,

i

ww StLxS ......................................... (4-16-3)

Similarly, substituting Eq. (4-15) into Eq. (4-10) the initial condition changes to:

)(0,

i

ww StxS .......................................... (4-16-4)

Eq. (4-16-1) to Eq. (4-16-4) can be normalized by defining the following dimensionless

variables:

89

w

i

wwwD

S

SSS

1

............................................. (4-17)

L

xxD ...................................................... (4-18)

and

tSL

Pt

w

cw

D

2

................................................ (4-19)

Substituting the following expressions for water saturation, distance and time into Eq.

(4-16),

1

i

wwwDw SSSS

Lxx D

D

cw

w tP

SLt

2

Eq. (4-16-1) then changes to:

D

cw

w

i

wwwD

D

i

wwwD

w

cw

tP

SL

SSS

Lx

SSS

S

P

2

1

2

12

)(

or

90

D

wD

D

wD

t

S

x

S

2

2

............................................. (4-20-1)

Boundary conditions of Eq. (4-16-2) and Eq. (4-16-3) are:

0

,,1

1

1

i

w

i

w

i

ww

DDwDSS

StLxStxS ....................... (4-20-2)

0

,0,0

1

1

i

w

i

w

i

wwDD

D

wD

SS

StxS

L

xtx

x

S ................ (4-20-3)

The initial condition of Eq. (4-16-4) becomes:

1

0,0,

1

1

i

w

i

w

i

ww

DDwDSS

StxStxS ....................... (4-20-4)

In order to obtain an analytical solution to Eq. (4-20-1) to Eq. (4-20-4), the separation of

variables method is applied and the solution is expressed as:

0

2

cos2

,n

t

Dn

n

DDwDDnextxS

, where ,....1,0,2

12

n

nn .... (4-21)

The average saturation in the core sample can be expressed as:

02

1

00

22 2cos

2

n

t

n

D

n

t

Dn

n

wDDnDn edxexS

.................... (4-22)

where,

91

47.22

2

2

0

21.222

32

2

1

69.612

52

2

2

The solution of Eq. (4-22) can be approximated by taking the first term,

D

D

tt

wD eeS

2

20 2

22

0

2

22

.................................. (4-23)

Taking logarithm of the both sides of the above equation gives:

8ln

4ln

22 DwD tS ....................................... (4-24)

4.2.2 Membrane Resistance Considered

If the resistance of the membrane at the outlet boundary is not negligible compared to that

of the core sample, the boundary condition at the lower end of the core sample (the

interface between the core sample and the membrane) will change. At this interface, the

wetting phase flow rate in the core sample is governed by Darcy‟s Law (Darcy, 1856) as:

92

tLxx

PKKtLxq w

w

rw

w ,,

................................ (4-25)

If the membrane is considered as a homogeneous porous medium, the wetting phase flow

rate in the membrane is expressed as:

m

outw

w

m

wmL

PtLxPKq

,

...................................... (4-26)

where, tLxPw , is the wetting phase pressure at the interface of the core sample and the

membrane. wmq is the wetting phase flow rate in the membrane. From material balance:

wmw qtLxq , ............................................... (4-27)

The boundary condition at the interface is, therefore, written as:

tLxx

PKK

L

PtLxPK w

w

rw

m

outw

w

m ,,

....................... (4-28)

Eq. (4-28) is further simplified by applying Eq. (4-15) at the boundary. When Lx , Eq.

(4-15) can be rearranged as:

w

i

ww

c

i

cinwS

StLSPPPtLP

1

1 ,,

Considering the definition of wDS :

DDwDc

i

cinw txSPPPtLP ,1,)1(

........................... (4-29)

93

DDwDcoutw txSPPtLP ,1, ................................. (4-30)

Substituting Eq. (4-30) into Eq. (4-28):

DD

D

wDc

rwDDwDc

m

m txx

SP

L

KKtxSP

L

K,1,1

DD

D

wDrw

m

m

DDwD txx

SK

LK

LK

txS ,1,1

..................... (4-31)

If a dimensionless parameter , which essentially means the conductivity ratio of the core

sample to the membrane, is defined as:

m

m

LK

LK

.................................................. (4-32)

The new boundary condition is written as:

DD

D

wD

rwDDwD txx

SKtxS ,1,1

........................... (4-33)

The partial differential equation and its boundaries conditions are summarized as:

D

wD

D

wD

t

S

x

S

2

2

.............................................. (4-34-1)

94

DD

D

wD

rwDDwD txx

SKtxS ,1,1

........................ (4-34-2)

0,0

DD

D

wD txx

S ......................................... (4-34-3)

10, DDwD txS ........................................... (4-34-4)

The solution for Eq. (4-34-1) is in the form of:

Dt

DDDDwD exBxAtxS2

sincos,

........................... (4-35)

Applying boundary condition (4-34-3) to Eq. (4-35):

00cos0sin2

DteBA ............................ (4-36)

B has to be zero to make Eq. (4-36) satisfied for every , Eq. (4-35) is thus changed to:

Dt

DDDwD exAtxS2

cos,

...................................... (4-37)

Applying boundary condition ((4-34-3) to Eq. (4-37) gives:

DD t

Drw

t

D exAKexA22

sincos

That is:

rwKctg .................................................. (4-38)

95

Eq. (4-38) is a non-linear equation, whose solution can be obtained numerically. Figure

4.4(a) depicts the solutions of Eq. (4-38) schematically, which are the cross-points of the

straight line rwKy and the curves ctgy .

(a) (b)

Figure 4.4: Graphic demonstration for the solutions for the eigenvalues in Eq. (4-38)

: (a) the first three solutions; (b) the first solution in domain 2/,4/ .

The general solution to Eq. (5-37) can be expressed as:

0

2

cos2

,n

t

Dn

n

DDwDDnextxS

................................ (4-39)

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 3.14 6.28 9.42

96

where n is the thn solution of Eq. (4-38).

The average wetting phase saturation over the entire core sample is:

02

1

00

22 2cos

2

n

t

n

D

n

t

Dn

n

wDDnDn edxexS

.................... (4-40)

Similar to Eq. (4-22), the solution of Eq. (4-40) can also be approximated by taking the

first term. However, there is no direct analytical solution for the first eigenvalue 0 . Here,

an approximation method to calculate 0 is provided. Noticing that the conductivity of the

core sample is usually smaller than the membrane and rwK is always less than one, we can

say that 1rwK . If 1rwK at the domain of

2,

4

, as shown Figure 4.4-(b), the

function 0ctg is approximated as:

002

ctg ................................................. (4-41)

Substituting Eq. (4-41)into Eq. (4-38), we get:

rwK

1

1

20

................................................ (4-42)

Eq. ((4-40) changes to:

D

rwD

tKrwt

wD eK

eS2

2

20 14

2

2

2

0

182

......................... (4-43)

97

Taking logarithm of the both sides of the above equation gives:

2

2

2

2

18ln

14ln

rw

D

rw

wD

Kt

KS

......................... (4-44)

4.3 Applications of Analytical Models

Eqs ((4-22) and ((4-40) presented in the previous sections are models with and without

considering the membrane resistance effect, respectively. By taking the first term as an

approximation, both models change to linear equations, Eq.s (4-24) and (4-44). It is also

noticed that when rwK

approaches zero, Eq. (4-44) converges to Eq. (4-24), which means

that, if the resistance of the membrane is negligible, these two equations are consistent with

each other.

For the application of this model, at each drainage step we define another dimensionless

time, Dkt , as:

tSL

PKt

ww

c

Dk

2

2

2 ......................................... (4-45)

The dimensionless wetting phase saturation wDS is defined as:

twwD VVS 1 ................................................. (4-46)

98

where wV is cumulative produced volume of the wetting phase and

tV is the total volume

expelled by the non-wetting phase at the end of this drainage step. Eq. (4-24) and Eq. (4-44)

then become, respectively:

8lnln

2DkrwwD tKS ..................................... (4-47)

and

22

12

4674.2ln

1ln

rw

Dk

rw

rwwD

Kt

K

KS

......................... (4-48)

The slopes of the linear regression are dominated by the permeability of the wetting phase

at this step. In order to estimate the relative permeability, Dkt and wDS are plotted as a

semi-log graph. The slope of the curve k is determined by linear regression of the

experiment points. To estimate the effect of the membrane, a dimensionless parameter is

calculated:

k

LK

LK

k

m

m

................................................ (4-49)

If k is small enough, k can be directly used as an estimation of the wetting phase relative

permeability. Otherwise, the following equation is required to revise the permeability to

the wetting phase:

99

21 kkK rw ................................................ (4-50)

4.4 Validation of Assumptions

A program was developed using Visual C++ 6.0 to solve the above numerical model. The

equations were solved using IMPES (Implicit Pressure Explicit Saturation) method and a

one dimensional model with the 27 grid blocks. The simulation was run at a desktop

computer with 2.0 GHz CPU and 4 GB memory for couples of minutes. To benchmark

results of the C++ program, saturation and pressure profiles in the core sample before the

non-wetting phase front reaches the membrane were examined by using a commercial

simulation package (CMG-IMEX). In order to confirm the validity of the analytical model

and the rationality of its assumptions, numerical simulations for a single drainage step were

carried out. At the beginning of this drainage step, the system was assumed to have reached

an equilibrium state, and the average saturation of the wetting phase was 0.40. To start the

drainage process, the non-wetting phase pressure at the outlet was increased from 0.20 atm

to 0.32 atm. The wetting phase saturation was, consequently, reduced from 0.40 to 0.36

during the drainage process, and the recovery history was recorded.

As depicted in Figure 4.5, the capillary pressure curve and the relative permeability curves

were calculated using Corey‟s equations (Corey, 1954). The coefficients used in Corey‟s

equations are listed in Table 4.1. Other properties of the core sample and the fluids adopted

in the numerical simulations are listed in Table 4.2.

100

Table 4.1: Coefficients used in Corey‟s equation to calculate the relative permeability

curves

Coefficients used in Corey‟s equation Value

Snr - Non-wetting phase residual saturation 0.2

Swr - Wetting phase residual saturation 0.15

Nn - Exponent of non-wetting phase 2.0

Nw - Exponent of wetting phase 2.0

Krnmax - Non-wetting phase maximum relative permeability 0.9

Krwmax - Wetting phase maximum relative permeability 0.4

Figure 4.5: The capillary pressure curve and the relative permeability curves calculated

using Corey‟s equation.

0

0.2

0.4

0.6

0.8

1

1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Cap

illa

ry p

ressu

re,

atm

Rela

tive P

erm

eab

ilit

y


Krn

Krw

Pc

101

Table 4.2: Properties of the core sample and the fluids used in numerical simulation

Parameter Hypothetical Value

Core sample length 1.00 cm

Core sample cross-sectional area 16.00 cm2

Core sample porosity 20 %

Core sample permeability 100 md

Wetting phase viscosity 1.00 cp

Non-wetting phase viscosity 0.02 cp

Membrane Conductivity 60 d/cm

4.4.1 Gas phase Flow Resistance

In order to investigate the impacts of the non-wetting phase resistance, the viscosity of air

at standard conditions was used as the non-wetting phase viscosity. The viscosity of the

non-wetting phase was then taken as 0.02 cp, and its relative permeabilities are shown in

Figure 4.5. The wetting phase permeability was taken as a constant, while the membrane

hydraulic conductivity was assigned a large number to make the resistance of the

membrane negligible. The cumulative production volume of the wetting phase obtained

from simulation is plotted as circles in Figure 4.6-(a). For comparisons, the calculated

cumulative production volume of the wetting phase using the full analytical solution, Eq.

102

(4-22) and the one term approximation solution, Eq. (4-24) were also plotted. These results

were shown as a dash line and solid line, respectively.

It can be seen that the cumulative production volume of the wetting phase obtained from

numerical simulation considering the non-wetting phase resistance, and the curve using

analytical model without considering the non-wetting phase resistance, are almost identical.

This implies that the wetting phase recovery history is not sensitive to the non-wetting

phase if its mobility is large enough, and thus confirms the assumption in developing the

analytical model that the non-wetting phase resistance is negligible. Furthermore, the

results of the one-term approximation model shows a little difference at the very beginning

of the process, but at later stages, the approximation model shows the same results as the

numerical solution and the full analytical model. The one term approximation model is a

reasonable approximation of the full analytical model. Moreover, when both the

cumulative production volume of the wetting phase and time in Figure 4.6(a) are

normalized and re-plotted as DwSln versus Dt graph in Figure 4.6(b), a perfect linear

relationship between these two variables shows up. As we discussed in the above section,

the slope of the straight line is determined by the wetting phase relative permeability.

103

(a)

(b)

Figure 4.6: Comparisons of the wetting phase recovery history of the hypothetical model,

the full analytical model and the one term approximation model.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 10 100

Cu

mu

lati

ve V

olu

me (cc)

Time (s)

Numerical Simulation

Analytical Solution

One Term Approximation

0

2

4

6

8

10

12

0 1 2 3 4

-Ln

Sw

D

tD

Numerical Simulation

Analytical Solution

One Term Approximation

104

4.4.2 Membrane Resistance

In the conventional porous plate method, the porous plate is used to prevent the non-

wetting phase from breaking through and being produced from the outlet. This porous plate

usually has high gas entry pressure and low permeability. Although Jennings (Jennings,

1983) replaced the porous plate with a plastic membrane and reduced the experimental

time to some extent, the resistance of the membrane may still not be negligible. Therefore,

so as to investigate the impacts of the membrane, the membrane hydraulic conductivity

was taken as reasonable number (60 md/cm) in the numerical simulations. The cumulative

production volume of the wetting phase obtained from numerical simulation is plotted as

circles in Figure 4.7(a). For comparisons, the cumulative production volume of the wetting

phase calculated using the analytical solution neglecting membrane, Eq. (4-22) and the

analytical solution considering membrane, Eq. (4-40), are plotted together as a dash line

and solid line, respectively. Both the cumulative production volume of the wetting phase

and drainage time are normalized and plotted as DwSln versus Dt graph in Figure 4.7(b).

It can be seen from Figure 4.7(a) that, if the resistance of the membrane is neglected, the

wetting phase is recovered faster and the slope of the straight line in Figure 4.7(b) is bigger.

Hence without considering the resistance of the membrane, using Eq. (4-22), the relative

permeability of the wetting phase is under-estimated from the wetting phase recovery

history. The estimated permeability combines the resistance of the core sample and the

membrane together. However, when the membrane resistance is considered, using Eq.

(4-40), the results of the analytical model are consistent with the numerical model. In the

105

case when the resistance of the membrane is not negligible, the model with the membrane

should be used to revise the results.

4.4.3 Flow Functions

The real capillary pressure is not linearly dependent on the wetting phase saturation and the

real relative permeabilities of the wetting phase also vary at different saturations. In the

numerical simulations, the relative permeabilities of the wetting phase and the capillary

pressure were calculated using monotone cubic spline interpolation (Fritsch and Carlson,

1980). The monotone cubic spline interpolation remains the monotonicity and smoothness

of the flow functions, simultaneously. Four simulations were run for four single drainage

steps and the corresponding saturation ranges were 0.45-0.40, 0.40-0.35, 0.35-0.30 and

0.30-0.25, respectively. For the analytical model, the arithmetic mean of the wetting phase

permeabilities was used as the constant relative permeabilities, rwK . The capillary pressure

gradient was calculated by the ratio of the capillary pressure change to the saturation

change in the corresponding drainage step.

106

(a)

(b)

Figure 4.7: Comparison of the wetting phase recovery history of the hypothetical model

with the membrane resistance, the one term approximation model and the complementary

model.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 10 100

Cu

mu

lati

ve V

olu

me (cc)

Time (s)

Numerical Solution with Membrane

Analytical Model without Membrane

Analytical Model with Membrane

0

2

4

6

8

10

12

0 1 2 3 4

-Ln

Sw

D

tD

Numerical Solution with Membrane

Analytical Model without Membrane

Analytical Model with Membrane

107

The results from simulations and from the analytical model are shown in Figure 4.8, which

depicts the relationship of the logarithm of the dimensionless average saturation versus the

dimensionless time. It can be seen that both the simulation results and the analytical results

show a straight line relationship, but the curves for the analytical model do not match with

the curves of the simulation model. This means the slopes of these straight lines are not

good approximations of the arithmetic mean of the wetting phase relative permeabilities in

the corresponding drainage step.

Figure 4.8: The linear relationship between the dimensionless time and logarithm of the

dimensionless average saturation for the analytical model and numerical models. From the

left to the right, the curves are wetting phase recovery for the drainage steps with the

wetting phase saturation ranges for 0.45-0.4, 0.4-0.35, 0.35-0.3 and 0.3-0.25.

0

2

4

6

8

10

12

14

16

0 50 100 150 200 250 300 350 400 450

-Ln

(Sw

D)

tD

Simulation : Krw is a function of Sw

Analytical Solution: Krw is the average relative permeability over a single drainage step

108

From Figure 4.8, it can be noted that the linear relationship shows up at the later stage of a

drainage process. Using the average saturation and the capillary gradient mounted on the

entire saturation range may not be reasonable. Therefore, the wetting phase permeabilities

and the capillary pressure gradient at the ending saturation were used in the analytical

model. The results were re-plotted in Figure 4.9. The simulation results do not match the

analytical results, but the slopes of the curves, directly determined by the relative

permeability, are consistent. The fitted results are shown as the linear equations in Figure

4.9. The slopes of these equations are the estimated relative permeabilities and the

corresponding relative permeabilities used to generate these results in the simulation are

0.059, 0.038, 0.021, and 0.009. All relative errors are smaller than 0.5%.

4.5 Computational Tests

To further validate the effectiveness of the analytical model, a virtual numerical multi-step

drainage experiment was carried out using the numerical model developed above. The

hypothetical multi-step drainage experiment consists of 10 drainage steps, and the initial

wetting phase saturation is 0.7. The hydraulic conductivity of the membrane is 1.0

Darcy/cm. The maximumrwK is 0.025. Since

rwK is less than 0.1, the relative

permeabilities were estimated using the no-membrane model.

109

Figure 4.9: Linear relationship between the dimensionless time and logarithm of the

dimensionless average saturation for the analytical model and numerical models From the

left to the right, the curves are for the drainage steps with wetting phase saturation of 0.45-

0.4, 0.4-0.35, 0.35-0.3 and 0.3-0.25.

The hypothetical relative permeabilities used as input in the simulations, and the estimated

relative permeabilities obtained by analytical model, are both plotted in Figure 4.10. The

hypothetical values are plotted as a solid line, while the estimated values are plotted as

circles. It can be seen that, using the direct estimation method, the relative permeabilities

obtained are consistent with the hypothetical values. However, as rwK increases, the no-

membrane model without the membrane resistance may underestimate the relative

permeabilities.

-LnSwD = 0.058tD - 0.377-LnSwD = 0.037tD + 0.282

-LnSwD = 0.021tD + 0.523

-LnSwD = 0.009tD + 0.379

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600

-Ln

(Sw

D)

tD

Simulation : Krw is a function of Sw

Analytical Solution: Krw is the relative permeability at the ending saturaion of a single drainage step

110

Figure 4.10: Comparison of the results estimated directly using the analytical model and

the original hypothetical values without the membrane resistance.(1) Solid line: the

hypothetical relative permeability curve used in the numerical simulation; (2) Circles:

estimated relative permeabilities at each drainage step.

In the second hypothetical experiment, the conductivity of the membrane was reduced to

0.1 Darcy/cm. The corresponding maximum rwK is 0.25, which is larger than 0.1.

Another 10-step hypothetical drainage experiment was conducted, and the wetting phase

relative permeabilities were calculated using the membrane model. Figure 4.11 shows that,

at the relative permeability range less than 0.35, the values estimated by both the model

neglecting the membrane and the model considering the membrane are consistent with the

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Krw

Sw

Hypothetical

Estimated

111

hypothetical values. But, as the relative permeabilities increase, the error caused by the

resistance of the membrane is more and more significant. However, with the model

considering the membrane resistance, the relative permeabilities are almost the same as the

inputted hypothetical values.

Figure 4.11: Comparison of the original hypothetical values and the results estimated

directly using the analytical model with and without membrane resistance revising. (1)

Solid line: the hypothetical relative permeability curve used in the numerical simulation;

(2) Circles: estimated relative permeabilities at each drainage step without membrane

resistance; (3) Rectangles: estimated relative permeabilities at each drainage step

considering membrane resistance.

0

0.05

0.1

0.15

0.2

0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Krw

Sw

Hypothetical

Neglecting membrane

Considering membrane

112

Several hypothetical experiments were then run, and the results indicated that an estimated

rwK value can be used to determine which model should be used to estimate the relative

permeability. If the estimated rwK is less than 0.05, the model neglecting membrane is

reliable enough to estimate the relative permeability. If the estimated rwK is larger than

0.05, the model considering membrane is required to estimate the wetting phase relative

permeabilities from the wetting phase recovery data.

4.6 Experimental Validation

In order to further validate the effectiveness of the model, Jennings‟ experimental data

(Jennings, 1983) were used to calculate the wetting phase relative permeabilities, using the

model developed in this study. The capillary pressure curves measured in Jennings‟

experiment are shown in Figure 4.12. In his study, a 4-step drainage experiment was

carried out and the relative permeabilities to water were obtained by automatic history

matching.

In order to calculate the relative permeabilities to water, a graph of DwSln versus t was

plotted for each step, as shown in Figure 4.13. It can be seen that in each step, the curve

has a linear part, except for step 2. It is shown that, the drainage time for step 2 is shorter

than the other three steps, thus the valid period of the model developed in this study has not

been reached yet. If the drainage time was longer, the linear relationship should be more

established. Using the slopes at each plot and both the model neglecting membrane and the

model considering membrane, the relative permeabilities to water were calculated.

113

Figure 4.12: Capillary pressure measured in Jennings‟ experiment.

The results are plotted in Figure 4.14. The consistent results showed with the numerical

hypothetical test: at low saturation range, the three relative curves have the same value;

while at high water saturation range, the results considering membrane resistance have a

better result. If we assume that Jennings‟ simulation results can represent the true values of

the relative permeabilities to water, it is indicated that the direct estimation method

presented in this thesis is valid and effective, because neither simulation nor automatic

history matching are required.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

Pc,

atm

Sw

114

(a) (b)

(c) (d)

Figure 4.13: Capillary pressure measured in Jennings‟ experiment: (a) step one; (b) step

two; (c) step three; (d) step four.

y = 0.00149x + 1.32729

R² = 0.99463

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500

-Ln

(Sw

D)

Time, s

Experimental Data

Regression

y = 0.00340x + 1.82990

R² = 0.95373

0

0.5

1

1.5

2

2.5

3

3.5

4

0 200 400 600

-Ln

(Sw

D)

Time, s

Experimental Data

Regression

y = 0.00132x + 0.82863

R² = 0.99325

0

1

2

3

4

5

6

0 1000 2000 3000 4000

-Ln

(Sw

D)

Time, s

Experimental Data

Regression

y = 0.00047x + 0.58818

R² = 0.99724

0

1

2

3

4

5

6

0 2000 4000 6000 8000 10000

-Ln

(Sw

D)

Time, s

Experimental Data

Regression

115

Figure 4.14: Comparisons of the results obtained using Jennings‟ automatic history

matching method and the directly estimation methods presented in this study.

4.7 Summary

An analytical model to describe the wetting phase recovery history in the multi-step

drainage process was developed in this study. This model can be used to directly estimate

the wetting permeabilities by linear regression of the wetting phase recovery history in

each single step. The results of the model are consistent with the results of the one-

dimensional numerical simulations. Comparisons of the analytical model and numerical

simulation indicate that:

0.00001

0.0001

0.001

0.01

0.1

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Krw

Sw

Jennings' Simulation Results

Neglecting Membrane

Considering Membrane

116

(1) The resistance of the non-wetting phase is negligible when estimating the relative

permeability of the wetting phase at low wetting phase saturation;

(2) The capillary pressure gradient around the ending saturation should be used, and the

estimated relative permeability of the wetting phase is the permeability at the ending

saturation;

(3) A dimensionless parameter was created to determine if the resistance of the membrane

is negligible. If it is not, a complementary model is developed to modify the wetting phase

permeabilities and get better results.

(4) Numerical hypothetical tests and laboratory experimental data verification

demonstrated that this method is effective for the direct estimation of the wetting phase

relative permeability.

117

CHAPTER 5: INTERACTIVE TUBE-BUNDLE

MODELLING

5.1 Introduction

The tube-bundle model, with various sizes of cylindrical tubes, was developed in the 1940s

and 1950s to simulate the pore size distribution of a real porous medium (Purcell, 1949;

Yuster, 1951; Scheidegger, 1953). In this model, the radius of one tube at the axial

direction of the model is kept identical, which made it particularly useful in the

interpretation of pore size distribution from the mercury injection capillary pressure test.

This model was then extensively studied, most notably by Scheidegger (1953), who

examined varying the radius of each tube along the length which was referred to as a

serial-type model, and by Payatakes et al.(1973), who looked at periodically constricted

tubes. More recent applications of this model presented by Bartley and Ruth (1999, 2001,

2002) and Dahle et al.(2005) indicate that this model is still the most successful model

used to simulate the porous media. However, as shown in later sections of this chapter, due

to the selective sealing effect of the membrane for the non-wetting phase, for a drainage-

type process with an air/water system in which the wetting phase (water) has lower

mobility, the wetting phase saturation drops from both ends of a one-dimensional core

118

sample. The conventional tube-bundle model has some difficulties in modeling this process

properly.

Based on the observation of multiphase flow in real porous media, Dong et al.(1998, 2005,

2006) pointed out that the most common and important feature of the conventional tube-

bundle model was that no interaction between the fluids in different tubes was considered.

Noting this, they introduced a new interacting tube-bundle model, which allowed for

interaction between the fluids flowing in different tubes. The advance of the oil/water

meniscus in a waterflood in an imbibition process and a Buckley–Leverett type front, at

higher injection rates, was successfully predicted using this model. The concept of the

interacting tube-bundle model was then experimentally validated by Unsal et al.(2007a,

2007b, 2009), who conducted extensive experimental studies in a model containing two

non-circular tubes, with connections between them through the gap between the rod and

the plate. Wang et al.(2008) presented detailed studies of fluid transfer between tubes in

interacting tube-bundle models, and it was concluded that the fluid transfer takes place in

close vicinity of the oil/water meniscus, under very small pressure drops. Interacting

capillary models were also constructed by Wang and Dong (2011) using triangular tubes,

in which the wetting phase was allowed to reside in the edges of the noncircular tubes.

Trapping of the nonwetting phase in imbibition processes was studied in an interacting-

serial type tube-bundle model.

The porous plate method (Bruce and Welge, 1947), which is referred to as the multi-step

drainage process in this thesis, is one of the conventional methods used to measure the

capillary pressure versus liquid saturation relationship of a porous medium. In this process,

119

a thin porous plate is used to prevent the nonwetting phase from breaking through within

its entry pressure, which is referred to as the selective sealing effect in this thesis. In order

to mitigate the effect of its resistance, the porous plate is replaced by a plastic membrane

(Jenning, 1983). In order to easily model this process either by analytical derivation or

numerical simulation, one-dimensional fluid flow geometry is required. Consequently, the

side surface of the porous medium is sealed either by resin, for porous medium (Jenning,

1983), or a wall of a coreholder for a soil column (Parker et al., 1985). In the multi-step

drainage process, the wetting phase production history is determined by the capillary

pressure and the relative permeabilities, simultaneously. While the capillary pressure can

be measured directly, many studies have been conducted to determine the relative

permeabilities in terms of wetting phase production history (Eching and Hopmans, 1993;

Liu et al., 1998; Chen et al.1999; Wang et al.2012). All of these studies considered

capillary pressure and the relative permeabilities as independent functions, but, in fact, they

impact each other to some extent (Wyllie, 1952, 1958). The tube-bundle models provide a

bridge to connect these two functions and help understand the fluid flow mechanisms in

porous media.

In order to model the multi-step drainage process, the effects of the membrane should be

handled properly. Due to the thinness of the membrane, its resistance can be negligible, but

the selective sealing effect on the nonwetting phase not only prevents the nonwetting phase

from breaking through, but, for a one-dimensional geometry, causes the saturation of the

wetting phase decreases from both ends of the porous medium; at the inlet, the wetting

phase was displaced by the nonwetting phase and, at the outlet, the wetting phase is

120

discharged from the system and leaves the nonwetting phase accumulating, thereby

reducing the wetting phase saturation. Simulation results indicate that for a gas/water

system at low water (the wetting phase) saturation, a decrease in the water at the outlet

dominates the water production history. Without considering the interaction between the

capillary tubes, conventional tube-bundle models cannot properly model the saturation

drop at the outlet. Dong‟s interactive tube-bundle model, which considers the interaction

among tubes in an imbibition process, cannot properly handle the membrane selective

sealing effect in drainage process. An extended interactive tube bundle model should be

considered to model this process in the right manner.

In this thesis, a numerical simulation model was developed to benchmark the entire

modeling process. A new interacting capillary model with three tubes was developed based

on Dong‟s interactive two-tube model (Dong et al., 1998), so as to qualitatively model the

selective sealing effect of the membrane. The three-tube model was then extended to an

interactive tube-bundle model with hundreds of tubes. With this model, the saturation

profiles along the porous medium, the wetting phase production history, and the multi-step

drainage process were successfully modeled.

5.2 Numerical Modeling Results

5.2.1 Saturation Profiles

Simulation results showed that a typical saturation profile of the wetting phase in a single-

step drainage process, with the membrane selective sealing effect, shown schematically in

121

Figure 5.1(b), consists of three regions: (1) a transition region of length 1l , due to the

invasion of the nonwetting phase; (2) a region of uniform saturation of the wetting phase of

length 2l ; (3) another transition region of length 3l , due to the discharge of the wetting

phase and the accumulation of the nonwetting phase. According to the ratios of the

viscosities, and of the relative permeabilities of the nonwetting phase to the wetting phase,

as shown in Figure 5.1(a) and Figure 5.1(c), Region 1 or Region 2 may not show up

significantly in a drainage process. If Region 1 does not show up, the nonwetting phase is

injected from the inlet, and the wetting phase saturation decreases from the outlet, the

displacement process is referred to as “reverse flooding”. If Region 2 does not show up,

and the directions of the nonwetting phase injection and the wetting phase saturation

decreasing are the same, the process is called “forward flooding”. If both regions show up,

significantly, the drainage process is called “bidirectional flooding”.

5.2.2 Production History

A multi-step drainage process was simulated, using the program developed in Chapeter 2,

at different capillary pressures and different relative permeabilities; the wetting phase

production histories are plotted in Figure 5.2, where it can be seen that all wetting phase

recovery curves converge to one and, as saturation decreases, the production rate slows.

122

wS

x

1wS

2wS

2l1l

(a)

wS

x

1wS

2wS

2l1l 3l

(b)

wS

x

1wS

2wS

2l 3l

(c)

Figure 5.1: Schematic saturation profiles in a drainage process with membrane selective

sealing effect. (a) Forward flooding; (b) Bidirectional flooding; (c) Reverse flooding.

123

Simulation results (Wang et al.2012), also shown in Chaphter 3, indicate that for an

air/water system, where water has lower mobility than that of gas, the production of water

is determined by the mobility of water. Wang et al. (2012) presented an analytical model,

based on a drainage process with an air/water system to describe the relationship between

the normalized saturation of water and drainage time, which is written as:

8ln

2ln

2

2

2

t

SL

PKKS

ww

crw

wD ............................... (5-1)

where,

twwD VVS 1 ................................................... (5-2)

In Eq. (5-1) and (5-2), wV is the cumulative wetting phase recovery and

tV is the total

volume expelled by the nonwetting phase at the end of this drainage step; K and are the

absolute permeability and the porosity of the core sample, respectively; L is the length of

the core sample; rwK is the relative permeability to water; w is the viscosity of water at

the experiment condition; wc SP is the gradient of the capillary pressure at the

corresponding saturation. The agreement of the analytical model and the above numerical

simulation model indicate that the analytical model is reliable and reasonable. The slope

term of Eq. (5-1) which can be used to calculate the relative permeability of water, rwK , at

the corresponding saturation, is defined as:

124

ww

crw

SL

PKKk

2

2

2 .............................................. (5-3)

Figure 5.2: Schematic of the wetting phase production history in a 4-step drainage process

with the membrane selective sealing effect.

5.3 Three-Tube Interactive Capillary Model

For the conventional tube-bundle model, which can only allow the nonwetting phase

discharging from the inlet and does not account for interactions between tubes, it is

125

considered hard to model “bi-directional” and “reverse flooding”, which mostly happened

in the gas drainage process. In order to model these phenomena, a conceptual interacting

capillary model with three tubes was developed. As shown in Figure 5.3, the model

consists of three capillary tubes, 1, 2 and 3, of different radii, 1R ,

2R and 3R , respectively,

where 321 RRR , and all are of equal length, L . The corresponding capillary pressures

are 1Pc ,

2Pc and 3Pc , where 321 PcPcPc . Gas is used as the nonwetting phase and

water is used as the wetting phase.

Tube1

Tube3

Tube2

1l 2l 3l

Inlet Outlet

Pc Pc

Gas

Water

outPinP

1P 2P

Figure 5.3: Schematic of the three-tube interacting capillary model: pressure and fluid

distribution.

In a drainage process, only the largest pores in a porous medium are broken through by gas,

at first. To account for this, initially the largest tube, Tube 1, is filled with gas and the other

two smaller tubes are filled with water. A capillary equilibrium was maintained by an

external pressure drop at the two ends of the model, which is equal to 2/21 PcPc . In

126

addition, it is assumed that there is a water-wet membrane at the outlet end of the model, in

order to prevent gas from breaking through, and that the membrane is thin enough that its

resistance is negligible.

At t = 0, the inlet of the model is exposed to the gas phase and the pressure of the gas

suddenly increases to 2/32 PcPc . Consequently, Tube 2 is broken through by the gas

phase. Following the assumptions of the two tube interactive capillary model (Dong et al.,

1998), the following criteria are applied: (1) for the same phase, the pressures in different

tubes, but at the same position (x), are the same; (2) if two conjunct tubes are filled with

different fluid, there is no fluid exchange between gas and water along the tube; (3) Fluid

exchange happens at any gas/water meniscus.

This model differs from Dong‟s model (Dong et al., 1995) in that Tube 2 is broken through

at any location and at any time, so long as the pressure difference between gas and water is

greater than 2Pc , instead of only at the inlet. As shown in Figure 5.3 and Figure 5.4, it is

assumed in this model that over the length, 1l , the water pressure is the same in Tube 3 at

any x, and that the gas pressure in Tube 1 and Tube 2 varies linearly. Over the length, 3l ,

the gas pressure is the same in Tube 1 and Tube 2 at any x, and the water pressure in Tube

3 varies linearly. In the middle zone of length 2l , both the gas pressure and the water

pressure, in all tubes, vary linearly. There are two menisci in Tube 2 and, at both menisci,

the gas pressure is larger than the water pressure by the value of capillary pressure 2Pc .

The gas pressure at Meniscus 1 is 1P and the water pressure at Meniscus 2 is 2P . Assuming

127

the total fluid rate is Q , the viscous pressure drops across different sections of the model

during the drainage process can be obtained by applying the Hagen-Poiseuille equation.

1l 2l 3l

Inlet Outlet

11Q

12Q

13Q

21Q 31Q

22Q

23Q

32Q

33Q

A B

Figure 5.4: Schematic of the three-tube interacting capillary model: fluid interaction in the

three-tube interacting capillary model.

Because there is only gas flow in Tube 1 and Tube 2 in Section 1l , the Hagen- Poiseuille

equation is written as:

1

1

4

2

4

1

88 l

PPRRQ in

gg

......................................... (5-4)

For Section 2l , Tube 1 contains gas, Tube 2 and Tube 3 are filled with water. Thus:

2

21

4

3

2

21

4

2

2

21

4

1

888 l

PPPR

l

PPPR

l

PPPRQ c

w

c

w

c

g

128

2

21

4

3

4

2

4

1

888 l

PPPRRRQ c

wwg

............................... (5-5)

For Section 3l , there is only water flow in Tube 3. Thus:

3

2

4

3

8 l

PPRQ out

w

.............................................. (5-6)

The linear Eq. (5-4), (5-5) and (5-6) can be solved, simultaneously, for Q , 1P , and

2P ,with

known 1l and 3l . The pressure profiles in the three tubes are plotted in Figure 5.5. At the

next time step, 1l and 3l can be calculated explicitly according to the flow rate in each tube.

As shown in Figure 5.4, 1l and 2l can be determined by the movement of the two

meniscuses, A and B. For Meniscus B, because there is no fluid exchanging between tubes

which contain different fluids (gas and water) and there is no gas flow in Section 3l , the

gas flow, 21Q , will flow into Tube 2 at Point B and move Meniscus B leftwards. This

means that the flow rate 21Q decides the movement of interface B.

21QQB ........................................................ (5-7)

For Meniscus A, because the total gas rate flowing into the whole system is

232221 QQQQ , and the flow exchange only occurs at point A and point B, the flow

rate determining the movement of interface A is 232221 QQQQQA , or at the

interface A, gas invades in a flow rate of 1211 QQ , and part of this rate flows into Tube 1

at a rate of 21Q , and the left part of the this rate is used to move the interface A. Therefore:

129

2322211211 QQQQQQA ..................................... (5-8)

The ratio of the movements of the two capillary interfaces can be used to describe the

relative movement of these two interfaces, which are defined as:

2322

21

QQ

Q

Q

Q

A

B

.................................................. (5-9)

Considering,

2

21

4

121

8 l

PPPRQ c

g

......................................... (5-10)

2

21

4

222

8 l

PPPRQ c

w

........................................ (5-11)

2

21

4

323

8 l

PPPRQ c

w

........................................ (5-12)

Eq. (5-9) can be derived as,

4

3

4

2

4

1

RR

R

Q

Q

g

w

A

B

............................................. (5-13)

130

When gw , AB QQ , Meniscus B moves faster than A. When gw ,

AB QQ ,

meniscus A moves faster than B. The other term, 4

3

4

2

4

1

RR

R

, essentially represents the

relative permeability ratio of gas to water.

By defining,

j

i

ji

R

8

4

..................................................... (5-14)

where, i is the index of the tubes; j is the phase index for gas and water. Eq. (5-4), (5-5)

and (5-6) can be derived as:

1211 PPQl ingg ..................................... (5-15-1)

2132131 PPPllLQ cwwg .................... (4-15-2)

outw PPQl 233 ......................................... (4-15-3)

To build a transient model, first we take 1l and 3l as zero, and calculate 1P and 2P explicitly.

Then, the increment of 1l and 3l at each time step, t , can be calculated using:

tl

PPP

R

RtVl c

g

B

2

21

2

2

4

13

8 ......................... (4-15-4)

131

tl

PPP

R

R

R

RtVl c

ww

A

2

21

2

2

4

3

2

2

4

21

88 ................ (4-15-5)

The calculation will stop when 02 l , which means the two meniscuses merge and the

drainage process is completed.

Figure 5.5: Pressure profiles in the three-tube interacting capillary model.

132

To examine the three-tube interacting capillary model, a sample calculation is carried out.

In the sample calculation, the radii of the three tubes are assumed to be 1.0 μm, 2.0 μm and

3.0 μm, respectively. The viscosity of water is 1.0 cp, and the viscosi ty of the nonwetting

phase varies, 50 cp, 5 cp and 0.5 cp. The pressure of the nonwetting phase before the

drainage process is 28 kPa, and it increases to 41 kPa after the drainage process starts. The

positions of the two meniscuses, A and B, for the three cases with different viscosities, are

drawn in Figure 5.6. It can be seen that with the same tube size and, thus, the same relative

permeabilities for the nonwetting phase and the wetting phase, the viscosity ratio

determines the movement of the two meniscuses. When the viscosity ratio of the

nonwetting phase to the wetting phase is large enough, e.g. a mercury/air system, the

movement of Meniscus A is much faster, “forward flooding” shows up and the cumulative

production of water is decided by the nonwetting phase and its relative permeability. When

the viscosity ratio is small enough, e.g. an air/water system, the movement of the Meniscus

B is faster, “reverse flooding” appears and the cumulative production of the wetting phase

is mainly contributed by the wetting phase and its relative permeability. When the viscosity

ratio is comparable, e.g. a kerosene/water system, “bidirectional flooding” happens, and

the production of water is determined by both the nonwetting phase and the wetting phase.

This confirms that when gas is used as the nonwetting phase and water is the wetting phase,

the cumulative wetting phase production is determined by the viscosity and the relative

permeability of the wetting phase (Wang et al.2012).

133

t=1 sec

t=10 sec

t=25 sec

(a)

t=1 sec

t=5 sec

t=15 sec

(b)

t=1 sec

t=5 sec

t=15 sec

(c)

Figure 5.6: Results of the sample calculations with the three-tube interacting capillary

model. Wetting phase viscosity is 1 cp and nonwetting phase viscosities are: (a) 50 cp;

(b) 5 cp; (c) 0.5 cp.

134

5.4 Extending to an Interacting Tube-Bundle Model

The above three-tube interacting capillary model only can be used for qualitative analysis

of the drainage process with the membrane selective sealing effect. In this section, the

above model is extended to an interacting tube-bundle model, which consists of M

cylindrical capillary tubes of different radii, iR ( MRRRR 321 ), and all of the

same length, L .

The schematic of the interacting capillary bundle model is shown in Figure 5.7. Similar to

the three-tube interacting capillary model, it is assumed that, initially, the larger tubes

( MSRRR S 1,, 21 ) are filled with gas (the nonwetting phase) while the other smaller

tubes ( NSRRR MSS 1,, 21 ) are filled with water (the wetting phase). The left hand

end of the model is connected to a container to supply gas at a constant pressure, and the

right hand end of the model is connected to water. What should be emphasized here is that

it is assumed there is a water-wet membrane at the right hand end of the model to prevent

gas from breaking through.

135

Gas

Water

ZoneTrans .

Tube1g

TubeNg

Tube1

TubeN

Tube1w

TubeNw

1,lL

nleftlL ,nrightrL ,

2,rL1,rL

1cP 1cP

2cP 2cP

3cP

Figure 5.7: Schematic of an interacting tube-bundle model.

Unlike the conventional non-interacting tube-bundle model, the positions of oil–water

menisci in the capillaries of the interacting tube-bundle model are dependent on each other

because of pressure equilibration (Dong et al., 2005). Gas cannot flow out of the system,

due to a membrane selective sealing effect.

As with the three-tube interacting capillary model, gas is assumed to break through from

any location, as long as the pressure difference reaches the breakthough pressure of a

certain tube. Therefore, as shown in Figure 5.7, this modified model consists of three zones:

the Gas Zone ( Ng tubes), which represents the largest tubes having been broken through

136

by gas in the previous drainage step; the Transaction Zone( N tubes), which represents the

tubes being broken through at current drainage step; the Water Zone ( Nw tubes), which

represents the tubes that will not be broken through at current drainage step. It should be

noted that NwNNgM , where M is the total number of tubes in the model. There

are LeftN sections at the left hand of the model and RightN sections at the right hand of the

model. For the Gas Zone, the hydraulic conductivity is defined as:

Ng

i g

i

g

RM

1

4

8

............................................... (5-16)

where iR is the i th tube in the Gas Zone. For the Water Zone, the hydraulic conductivity is

defined as:

Nw

i w

i

w

RM

1

4

8

............................................... (5-17)

where iR is the i th tube in the Water Zone. According to the Hagen-Poiseuille equation, as

shown in Figure 5.7, the first section on the left part of the tube-bundle model is written as:

1,

1,

1

4

8 l

linnLeft

i g

ig

L

PPRMQ

.................................. (5-18)

For the second section:

137

1,2,

22,11,

4

1,2,

2,1,1

1

4

8

8

ll

clcl

w

N

nLefti w

i

ll

llnLeft

i g

i

g

LL

PPPPM

R

LL

PPRMQ

............... (5-19)

Similarly, for the thj section:

1,,

,,1,1,

2

4

1,,

,1,1

1

4

8

8

jljl

jcjljcjl

w

N

jnLefti w

i

jljl

jljljnLeft

i g

ig

LL

PPPPM

R

LL

PPRMQ

......... (5-20)

For the section in the middle, 1 LeftNj :

m

nRightrnLeftl

w

N

i w

i

m

nRightrnLeftl

gL

PPM

R

L

PPMQ

,,

1

4,,

8

........... (5-21)

For the thj sections counted from the left hand, when kNNj RightLeft 2 , where k

varies from RightN to 2:

1,,

1,1,,,

2

4

1,,

1,,1

1

4

8

8

krkr

kckrkckr

w

N

knRighti w

i

krkr

krkrknRight

i g

i

g

LL

PPPPM

R

LL

PPRMQ

.......... (5-22)

For the last section at the right hand, 1 RightLeft NNj :

138

1,

1,1,

1

4

8 r

outcr

w

N

nRighti w

i

L

PPPM

RQ

.......................... (5-23)

The equation group can be simplified as:

1,1,1, lingl PPMQL ........................................... (5-24)

jcjljcjljwjljljgjljl PPPPMPPMQLL ,,1,1,,,1,,1,, ....... (5-25)

nRightrnLeftlnLeftwnRightrnLeftlnLeftgm PPMPPMQL ,,1,,,1, .............. (5-26)

1,1,,,,1,,,1,, kckrkckrjwkrkrjgkrkr PPPPMPPMQLL ...... (5-27)

outcrnRightnLeftwr PPPMQL 1,1,1,1, ................................ (5-28)

The array P is gas pressure and the meniscus positions L represent the saturation

distribution along the tube. Therefore, like the traditional IMPES method, these equations

can also be solved implicitly for pressure and explicitly for meniscus positions. From the

pressure distribution, the flow rate at each section, for both gas and water, can be

calculated. Assuming that the flow rate of gas is nRightnLeft to1 iQoi, the velocity of

the movement of the meniscus can be calculated by:

k

oioi

ikA

QQV 1

................................................. (5-29)

where, kA is the cross-sectional area of the thk tube. Positions of all meniscus at the next

time step is updated explicitly by applying the position increment to each meniscus.

139

5.5 Modeling of the Drainage Process

In order to validate the extended interacting tube-bundle model, a model with hypothetical

parameters was built. Then simulations were carried out in order to model the typical

behavior of the drainage process in a porous medium with the membrane sealing effect.

The length of the interacting tube-bundle model is 1.00 cm; the tube size distribution

satisfies truncated Weibull distribution, as shown in Figure 5.8, with mRaverage 0.6 ,

mR 30max and mR 0.1min ; the interfacial tension is 72.60 dyne/cm, the tubes are

strong water wetted and the contact angle is assumed to be 0. Water is used as the wetting

phase and its viscosity is 1.00 cp at the simulation conditions.

Figure 5.8: Tube radius distribution satisfying truncated Weibull distribution.

140

5.5.1 Modeling of Saturation Profiles

The three-tube capillary model can only qualitatively model the “reverse flood” and

“bidirectional flood”. In this section, the interactive tube-bundle model, which consists of

50 tubes, was used to model the saturation profiles in a porous medium. Radii distribution

of these 50 tubes satisfies truncated Weibull distribution. The nonwetting phase pressure

increases from the previous 25kPa to a new level of 52kPa. Correspondingly, the tubes are

broken through from the 13th

to the 41st tube.

Figure 5.9 shows the wetting phase saturation profiles along the model, at selected

drainage times, for three cases with different nonwetting phase viscosities: 500 cp, 5 cp

and 0.05 cp, respectively. Correspondingly, the viscosity ratios of the nonwetting to

wetting phase are 100, 1, and 0.01. Note, at a very high nonwetting phase viscosity, as

shown in Figure 5.9(a), the saturation of the wetting phase drops from the inlet end of the

model. This is generally consistent with the results of a conventional coreflood test.

Conversely, at a very low nonwetting phase viscosity, as shown in Figure 5.9(c), the

wetting phase saturation drops from the outlet end of the model. As the process progresses,

the decreased saturation spread to the inlet end of the model and the “reverse flood” shows

up. At an intermediate nonwetting phase viscosity, wetting phase saturation decreases from

both the inlet and outlet of the model, as shown in Figure 5.9(b). It can also be noticed that,

early on, the saturation at the inlet decreases faster and, later, the saturation at the outlet

decreases faster. The reason for this is that the decreasing of wetting phase saturation

reduces its relative permeability, and the lowered relative permeability of the wetting phase

141

gradually converts the “forward flooding” to “reverse flooding”. This is completely

consistent with the results from the numerical simulation.

5.5.2 Modeling of Drainage History

The wetting phase production in the three-tube capillary model stops once the two menisci

in the middle tube merge and the middle tube is filled with the nonwetting phase. However,

the wetting phase production does not have such an end point in a real drainage process.

This artificial end pont could be caused by the limited number of the tubes used in the

three-tube capillary model. In this section, the total number of the tubes in the model is

increased from 3 to 9, 27, 81 and 243, and correspondingly, the number of the tubes broken

through in the drainage process increases from 1 to 3, 9, 27, and 81. All tubes are assumed

to have the same wettability. The viscosities of the nonwetting phase and the wetting phase

are identical and both equal to 1.0 cp. At the beginning of the drainage process, the

pressure of the nonwetting phase at the inlet is calculated using the following equation:

1

4'

wN

inRR

P

................................................. (5-30)

where NR is the smallest tube which will be broken though in this drainage step and 1wR is

the largest tube which will not be broken though in this drainage step.

142

(a)

(b)

(c)

Figure 5.9: Wetting phase saturation profiles in the interacting tube-bundle

model.Nonwetting phase viscosities are: (a) 500 cp; (b) 5 cp; (c) 0.05 cp, respectively.

143

The wetting phase production history is plotted in Figure 5.10. M is the total number of

tubes in the model, N is the number of the tubes broken through in the drainage step.

From Figure 5.10, it can be seen that the wetting phase production, calculated by the three-

tube capillary model, stops when the two menisci in the middle tube merge. However, as

the tube number increases, time required to fill all tubes in the drainage process increases.

It is reasonable to expect that when the total number of the tubes, N , goes to infinity, the

curve will converge to the production curve in a real drainage process.

Figure 5.10: Wetting phase production history of the interacting tube-bundle model with

different numbers of capillary tubes. M is the number of all tubes, N is the number of the

tubes broken through in the drainage step.

The analytical equation (Wang et al., 2012) shows that there is a linear relationship

between wDSln and t , where wDS is the normalized saturation of the wetting phase at

144

one drainage step defined by Eq. (5-2) and t is the drainage time. The relationship of

wDSln and t was plotted in Figure 5.11. All curves have a linear section at the beginning,

and then rapidly increase towards the end. This rapid increase in each curve is because, at

that time, only one or two tubes are contributing to the drainage process. All other tubes

have already been filled with the nonwetting phase. All models are downgraded to a three-

tube model. However, it is reasonable to expect that, as the number of the tubes in the

model increases, the length of the straight line section will be extended, and the curve will

eventually converge to a straight line when the number of tubes, N , goes to infinity.

Figure 5.11: Relationship of R 1ln and time calculated using the interacting tube-

bundle model with different numbers of capillary tubes. M is the number of all tubes; N is

the number of the tubes broken through in the drainage process.

145

5.5.3 Modeling of Multi-step Drainage Process

In order to model the multi-step drainage process for a gas/water system, the extended

interacting tube-bundle model is expanded to a model that consists of 243 tubes. The

viscosity of water (the wetting phase) is 1.0 cp, while the viscosity of gas (the nonwetting

phase) is 0.1 cp. The capillary pressure curve of gas/water for this model, which is plotted

in Figure 5.12, is calculated from the pore size distribution defined in Figure 5.8.

Figure 5.12: Capillary pressure curve calculated from the pore size distribution and the

drainage pressures used in the 5-step drainage experiment.

A hypothetical 5-step drainage experiment was carried out based on this model. The

drainage pressures used in the hypothetical experiments are represented by circles in

Figure 5.12. The corresponding saturation changes of each drainage step are 0.7-0.4, 0.4-

0.2, 0.2-0.1, 0.1-0.04, and 0.04-0.005, respectively. In each drainage step, around 30 tubes

146

were broken through by gas, and the cumulative water production history was recorded and

depicted in Figure 5.13. The normalized water recovery histories at each step were

calculated and plotted in Figure 5.14. It can be seen that all production curves have the

same shape as the experimental and simulation results in previous publications (Jennings,

1983; Eching and Hopmans, 1993; Liu et al., 1998; Chen et al.1999; Wang et al., 2012).


modeled by the interacting tube-bundle model.

147


modeled by the interacting tube-bundle model.

Figure 5.15: Wetting phase recovery histories at each drainage step in the 5-step drainage

process.

148

Moreover, as shown in Figure 5.15, all curves for the relation of wDSln versus time, t ,

have a linear section, which is consistent with Eq. (5-1). The slopes of the linear sections

can be calculated by linear regression and the results are listed in the forth column of Table

5.1.

The following group of equations (Wang, 2010) is applied to calculate the absolute

permeability, the water saturation and the relative permeabilities of each phase in the

interacting tube-bundle model, consisting of N tubes, of which k tubes are filled with gas

and the others filled with water. The absolute permeability of the model can be calculated

using:

ARKN

i

i 81

4

............................................... (5-31)

where iR is the radius of the thi tube and A is the cross-sectional area of the model. The

water saturation in the model is calculated by:

N

i

i

N

ki

iw RRS1

2

1

2............................................. (5-32)

where iR is the radius of the thi tube and rgK and rwK are the relative permeabilities of

gas and water, respectively. By applying the following equations, the relative

permeabilities of gas and water can also be calculated:

N

i

i

k

i

irg RRK1

4

1

4 ............................................ (5-33)

149

N

i

i

N

ki

irw RRK1

4

1

4 ............................................ (5-34)

Table 5.1: A comparison of the results calculated from the analytical model and the results

calculated from the interacting tube-bundle model

Water

Saturation

(Sw)

Capillary

Pressure

(Pc)

Pc Gradient

(∆Pc/∆Sw)

Regressed

Slope

(k)

Analytical

Calculated

Permeability

Relative

Permeability

(Krw)

0.005 0.5558 31.15 0.2256 0.000235 0.000231

0.040 0.3316 2.593 0.3247 0.004060 0.003990

0.100 0.2595 0.7354 0.4656 0.02052 0.01950

0.200 0.2115 0.3563 0.5670 0.05160 0.05190

0.400 0.1664 0.1700 0.8500 0.1621 0.1610

Using Eq. (5-31), the absolute permeability of the model in the hypothetical test is

calculated as 12.5 Darcies; From the capillary curve in Figure 5.12, the capillary gradient

at the tested saturation, ∆Pc/∆Sw, can also be calculated and listed in the third column of

Table 5.1. By applying Eq. (5-13), the relative permeabilities of water for the model are

calculated and listed in the fifth column of Table 5.1. From the other aspect, the relative

permeabilities of water at the corresponding saturations of each drainage step are also

calculated by Eq. (5-34). These values are listed in the last column of Table 5.1. As shown

150

in Figure 5.16, the relative permeabilities calculated by the analytical model. Based on the

simulation results using the interacting tube-bundle model, are identical to the results

calculated directly from the tube-bundle model. In other words, for a porous medium like a

bundle of tubes, the modeling method is accurate and the interactive tube-bundle model

completely replicates the multi-step drainage process. This indicates that the modeling

method is reasonable, and the interacting tube-bundle model is more applicable when

modeling the multiphase fluid in porous media. However, most porous media are not as

simple as a bundle of tubes. It is believed that by further incorporating the serial-type

model and the triangle tubes to simulate water trapping mechanisms, this model should be

able to be applied for history matching of the multi-step drainage process.

Figure 5.16: Calculated relative permeabilities of water from the analytical model and the

interactive tube bundle model.

0.0001

0.001

0.01

0.1

1

0.0001 0.001 0.01 0.1 1

Krw

Cal

cula

ted

fro

m A

nal

ytic

al M

od

elin

g

Krw Calculated from Pore Size Distribution

151

5.6 Conclusions

The porous plate method is one of the most conventional ways to measure the capillary

pressure in a porous medium. The relative permeabilities can also be estimated by

analyzing the wetting phase production histories at each step. The distinct feature of this

method is the application of a porous plate, or membrane, to prevent the nonwetting phase

from being discharged from the porous medium, which is referred to as the selective

sealing effect in this chapter. Although the conventional tube-bundle model can

successfully model the capillary phenomenon, it has some difficulties when it comes to

properly modeling the wetting phase production history, because, in a drainage-type

process, the sealing effect of the membrane significantly changes the multiphase flow

pattern. In this chapter, an extended interactive tube-bundle model was developed and

applied to model this process. First, in order to qualitatively model this process, a new

three-tube interacting capillary model was developed and the “reverse flood” and

“bidirectional flood” were conceptually modeled. It also explains in concept why, in this

process, the phase with lower mobility determines the wetting phase production history.

After that, the three-tube interacting capillary model was extended to a complex interactive

tube-bundle model, consisting of hundreds of tubes. The saturation profiles along the

porous medium, the wetting phase production curves, and the multi-step drainage process

were all successfully modeled. The application of the interactive tube-bundle model

indicates that it can better represent the pore structure of a porous medium than the

conventional non-interactive tube-bundle models.

152

The multi-step drainage process used to simultaneously measure the capillary pressure and

relative permeabilities is modeled by the extended interacting tube-bundle model

developed in this chapter. All phenomena in this process, such as “reverse flooding”,

“bidirectional flooding”, the relationship between wDSln and time t , and the multi-step

drainage process can be successfully modeled. It is proved that the interacting tube-bundle

model more accurately represents the pore structure and fluid flow dynamics in a porous

medium. By incorporating the serial-type model and the triangle tubes to simulate water

trapping mechanisms, this model should be able to history match the multi-step drainage

experiments and build the relation between the capillary pressures and the relative

permeability curves.

The interacting tube-bundle model provides another way to model the one-dimensional

multiphase flow. The advantage of the interacting tube-bundle model is that it is able to

obtain an accurate displacement front and saturation profile along the model. However, the

disadvantage is that this method considers the porous medium as a tube-bundle with a

limited number of tubes. This is the reason for the 100% wetting phase recovery in a

drainage process and 100% water cut in the waterfood simulation within limited time

(Dong et al., 2005). To solve this problem, a dynamic tube-bundle model, which can

dynamically rearrange the number and size of the tubes according to capillary pressure

curve in a displacement process, is suggested.

153

CHAPTER 6: EXPERIMENTAL VALIDATION

6.1 Introduction

Three different models to describe the multi-step drainage process were developed and

extensively discussed in the above chapters, including: numerical modeling, analytical

modeling and interactive tube-bundle modeling. In addition, a method using automatic

history matching with a Guo Tao genetic algorithm and direct estimation using analytical

modelling were developed to estimate the relative permeabilities of the wetting phase by

matching the production data in a multi-step drainage process. In this chapter, the multi-

step drainage experiments in the lab were carried out using both sandpacks and core

samples to validate the effectiveness of these methods. The multi-step drainage

experiments were carried out using oil/water, gas/water and gas/oil/water systems to

investigate the practical application of the different methods on different fluid systems.

6.2 Apparatus

Multi-step drainage process can be carried out using both sandpack and core sample. The

apparatus and schematic diagram are shown in the following section.

154

6.2.1 Multi-step Drainage Process using a Sandpack

The schematic diagram of the experimental apparatus for the multi-step drainage process

using a sandpack is shown in Figure 6.1. The entire system consists of a coreholder, a

ruler, a nylon tube with a valve and a computer controlled high accurate balance.

Membrane

Distributor

Sandpack

Figure 6.1: Schematic diagram for the multi-step drainage process using a sandpack.

A fabric membrane was positioned between the metal distributor and the sandpack. The

basic properties of the membrane are listed in Table 6.1. The sandpack was positioned

horizontally to make the gravity effect negligible. The pressure difference between the two

ends of the sandpack was generated by the hydraulic pressure in the nylon tube connected

to the outlet of the sandpack.

155

Table 6.1: Basic properties of the oil-wetting and the water-wetting membranes.

Oil-wetting Water-wetting

Pore Size (µm) 0.22 0.10

Water Flow Rate(cm3/min/cm

2) 6.7 2.5

Thickness (µm) 125 125

Porosity (%) 70 70

Threshold Pressure (atm) 3.45 4.8

6.2.2 Multi-step Drainage Process using a Core Sample

As shown in Figure 6.2 and Figure 6.3, the entire system for a multi-step drainage process

using core sample consists of a capillary pressure cell, a pressure gauge, a buffering

cylinder, a tubing pump and a digital balance. Air/kerosene was used as the non-wetting

phase and water as the wetting phase. The pressure of the non-wetting phase was

controlled by pumping air into a buffering cylinder using a tubing pump and monitored by

a pressure gauge. The produced water was collected in the buffering cylinder at the

ambient condition and the production history was recorded by a computer-controlled

digital balance, which was covered by a plastic box to reduce the impacts of the indoor air

flow. The readability of the balance is 0.01g and the data was collected every 6 seconds.

156

Capillary pressure cell

Computer Controlled Balance

P

Pressure Gauge

Tubing Pump

Cylinder

V-1

V-2

V-3

E-1

Figure 6.2: Schematic diagram of the multi-step drainage system using a core sample.

6.3 Measurement of Resistance/Permeability

Before beginning the multi-step drainage experiment, it is necessary to measure the

resistance/permeability of the following three systems:

(1) The system with the plastic tubing, the valves, and the core holder;

(2a) System (1) + membrane, sandpack case;

(2b) System (1) + core sample, core sample case;

157

(3) System (1) + membrane + sandpack/core sample;

With the above measurements, the resistance of the experimental system and the

membrane can be calculated. The permeability of porous medium, being the sandpack or

the core sample, can be calculated as well. The calculated resistance/permeability will be

incorporated in both automatic history matching and analytical modeling.

Figure 6.3: A photo of the apparatus for the multi-step drainage process using core sample.

158

6.3.1 Procedures

The experiment procedures and calculations are shown as follows:

1. Assemble the system without the membrane and the porous medium;

2. Pump water at different rates and record the cumulative water flow through the system,

as shown in Figure 6.4. Record the pressure difference at the corresponding rate using the

pressure gauge connected to a computer. This process is referred to as a multi-rate flow test

in this work.

Figure 6.4: An example for the permeability/resistance test data processing.

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000

Cu

m. V

olu

me,

cm

3

Time, s

159

3. For the system (2a), assemble the system (1) with the membrane and perform a multi -

rate flow test. For the system (2b), assemble the core sample with system (1) and perform

the multi-rate flow test.

4. Assemble the entire system including the plastic tubing, the valves, the core holder and

the porous medium. Carry out the multi-rate flow test. Record the pressure and cumulative

water production volume.

6.3.2 Calculation of Resistance/Permeability

For Darcy‟s flow, flow rate Q in a porous medium is proportional to the pressure

difference ΔP, which is expressed as:

QRP h ...................................................... (6-1)

where hR is the hydrodynamic resistance, which is defined as the pressure difference

applied at two ends of a porous medium to generate a unit rate of flow. For one

dimensional flow in a system consisting of N segments:

N

iih

N

i

itotal RQPP11

......................................... (6-2)

The system in this work consists of the pipeline, the membrane, and the porous medium:

umporousmedihmembranehpipehtotal RRRQP ___ ........................ (6-3)

For Darcy's equation in a porous medium:

160

L

PA

KQ

w

................................................... (6-4)

combined with Eq. (6-3):

umporousmedih

w

AR

LK

_

............................................... (6-5)

pipehR_

, membranehR _ , and umporousmedihR _ can all be measured step by step using the procedures

described in Section 6.3. The absolute permeability can be calculated using Eq. (6-5).

As shown in Figure 6.4, each linear portion in the cumulative production data recorded by

the computer controlled balance represents a constant rate flow with a certain pressure

difference. The slope of a linear portion is the flow rate under the corresponding pressure.

All slopes are then plotted together, as shown in Figure 6.5. Linear regression gives the

resistance of the corresponding system. By considering the viscosity of water and the

geometry of the system, the permeabilities of the porous media and the equivalence

permeabilities of the pipeline and membrane, the equivalence to the same diameter as the

porous media, can be calculated.

161

Figure 6.5: Linear regression to calculate conductivity of target flow system.

6.4 Gas Leakage due to Gas Diffusion

Figure 6.7 shows a typical water production history in a multi-step drainage process in an

air/water system. The expectation was that water production would stop increasing once

the system reached capillary equivalence. However, as can be seen, the cumulative

production of water continues to increase at the last drainage step. Initially, this was

believed to be caused by gas leakage in the system. Efforts to prevent the leakage were

taken, but did not completely mitigate it.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2

ΔP

, atm

Flow Rate, cm3/s

162

Coming back to the fundamentals of model, this part of the cumulative production is

actually caused by the diffusion of air in water: the gas going through the membrane due to

diffusion occupies some space under the membrane and expels some water. Consequently,

the water production increases. At the beginning of this process, the impact of gas

diffusion will not be observed, when the pressure difference is low. Later, when the

capillary pressure is high, the pressure difference leading to gas diffusion increases.

Meanwhile, the water production rate decreases. The impact of the gas diffusion on the

water production becomes non-negligible. The following shows an example to indicate the

magnitude of the gas diffusion by assuming the parameters listed in Table 6.2.

As shown in Figure 6.6, according to Fick‟s Law, the diffusion equation through a porous

medium can be calculated by the following equation:

l

CDAqdiff

.................................................. (6-6)

where diffq is the air flow rate due to diffusion; D is air diffusion coefficient in water; A is

the cross sectional area of the porous medium; is porosity; is density of gas; is

tortuosity; C is the concentration of gas in water; l is the length of the porous medium.

From the above equation, the calculated gas diffusion rate (0.288cm3/hour) has a similar

magnitude as the water volume produced in the drainage process. However, unlike the

water volume produced in the drainage process, the gas diffusion rate is constant for a

certain pressure difference. There are two ways to handle this issue: (1) include the

constant gas diffusion rate into history matching or (2) subtract the linear part of the

163

cumulative production curve to extract the cumulative water production due to the gas

drainage. In order to apply the analytical method, the systems using gas as the non-wetting

phase were calibrated using the second approach to handle the impacts of the gas diffusion

effects.

Table 6.2: Parameters used for hypothetical gas diffusion calculation

Parameter Value

Room Temperature, T 25 ℃

Thickness, b mm1.0 ( m4101 )

Area, A 216cm (241016 m )

Solubility of air in water at 1 atm, 1C 0.023 kg/m

3

Solubility of air in water at 2 atm, 2C 0.045 kg/m3

Diffusion coefficient of gas in water, D sm /102.3 29

Porosity, 0.2

Tortuosity, w 1.5

Density, 1.1839 kg/m3

164

batm2

atm1

membrane saturatedWater

Figure 6.6: Schematic of the gas diffusion model through the membrane.

6.5 Multi-step Drainage Process using a Sandpack

6.5.1 Procedures

The procedure for a multi-step drainage experiment with a sandpack using gas/water and

oil/water systems is described in this section. Before the multi-step drainage experiment,

the basic parameters, such as permeability, porosity, geometry, and the resistivity of the

membrane and pipeline were measured. The step-by-step workflow is summarized as:

(1) Measure the weight of the empty core holder, chm .

(2) Assemble the system without packing the sand and measure the resistances of the

pipeline without membrane and with membrane, respectively.

(3) Wet-pack the core holder with water and sand.

(4) Vibrate the core holder for 2 hours to ensure the sandpack is tightly packed.

165

(5) Measure the weight of the core holder and the saturated sandpack as spm .

(6) Assemble the core holder into the system and measure hydrodynamic resistance of the

sandpack.

(7) Carry out the multi-step drainage experiment. The pressure is controlled by lowering

the outlet of the nylon tube gradually. For a gas/water system, the inlet end of the core

holder is connected to the free atmosphere. For an oil/water system, the inlet end of the

core holder is connected to a beaker full of kerosene.

(8) After the drainage experiment, the majority of the wet sand is dug out and its weight is

measured as 1wm .

(9) The wet sand is dried for 12 hours by hot air from a hair dryer until no weight variation

is observed. The weight of the dry sand is measured as 1dm .

(10) The remaining wet sand adhering to the core holder is washed, collected, dried and its

weight is measured as 2dm .

6.5.2 Calculation of the Basic Parameters

Porosity

The porosity of the sandpack is calculated in terms of the mass difference between the

water saturated sandpack and the dry sandpack. The mass of the water saturated sandpack

166

is chspw mmm . The mass of the dry sandpack is 21 ddd mmm . Porosity is therefore

calculated by:

chw

dw

V

mm

.................................................. (6-7)

where chV is the volume of the core holder, which equals the bulk volume of the sandpack.

w is the density of water at experimental conditions.

Water Saturation

For the gas/water system, the remaining water in the sandpack after the drainage process is

calculated by measuring the remaining water in the first part of the wet sand. The

remaining water in the first part of the sand is 111 dwrw mmm . The remaining water in the

entire sandpack can be calculated by:

1

1

21 d

rw

dd

rw

m

m

mm

m

............................................... (6-8)

By multiplying both sides by 21 dd mm :

21

1

1dd

d

rwrw mm

m

mm ........................................... (6-9)

The remaining water saturation is calculated by:

167

dw

rwrw

mm

mS

................................................ (6-10)

then:

211

211

ddwd

ddrw

rwmmmm

mmmS

...................................... (6-11)

For a system with air as a non-wetting phase, the volume of the remaining water will be

calibrated by the total produced water for mass balance. For an oil/water system, it is

difficult to measure the remaining water in the sandpack using the mass difference method,

but both the volume of cumulative produced water and the cumulative injected oil can be

measured. The average value calculated from these two methods is used to estimate the

water saturation.

6.5.3 Gas/Water System

Initially, air and water were used as the non-wetting phase and the wetting phase,

respectively. Prior to the experiment, the dimensions and the weight of the core holder

were measured. The basic parameters of the sandpack are listed in Table 6.3. The sandpack

is 7.00 cm in length and 4.10 cm in diameter. Its cross-sectional area and its volume is

calculated as 13.20 m2 and 92.37 m

3, respectively. The dead volume in the distributor is

estimated to be 1.0 cm3.

168

Table 6.3: Basic parameters of the sandpack used at the drainage experiment

Parameter Value Unit

Length 7.00 cm

Diameter 4.10 cm

Area 13.20 cm2

Volume 92.37 cm3

As listed in Table 6.4 and Table 6.5, the porosity of the sandpack was measured as 0.370

based on masses difference; the viscosity of water is 1.0 cp at laboratory conditions and the

viscosity of kerosene was measured as 0.80 cp. From the data listed in Table 6.6, the

absolute permeability of the sandpack was calculated as 2.50 Darcies and the hydraulic

conductivity of the pipeline and the membrane was calculated as 66.1 md/cm. A four-step

drainage experiment was carried out with a gas/water system using the sandpack and the

water production history was recorded and plotted as blue circles in Figure 6.7. Gas

diffusion effect discussed in the above section was considered to correct the cumulative

production of water. After substracting the water volume produced due to gas diffusion, the

corrected cumulative production was plotted as red circles in Figure 6.7.

169


Gas/water System)

Test one Value Unit

Coreholder Weight 2095.48 g

Coreholder & Wet Sand 2319.03 g

Dead Volume 1.00 cm3

Wet Sand Weight 222.55 g

DrySand & Beaker 403.49 g

Beaker 215.09 g

DrySand Weight 188.40 g

Water in place 34.15 g

Porosity 0.370

170


Gas/water System)

Test Two Value Unit

Highest pressure 0.182 atm

Beaker weight 118.24 g

Beaker&Half-wet Sand 302.12 g

Beaker&Dry Sand 297.55 g

Liquid Left 4.57 g

Dry Sand 179.31 g

Liquid Left 2 4.80 g

Final Water Saturation 0.141 g

171

Table 6.6: Results of the resistance tests for the sandpack systems

System Unit Test G/W Test O/W

Pipeline atm/(cm3/s) 0.1308 0.1368

Membrane atm/(cm3/s) 0.7734 0.7134

Sand atm/(cm3/s) 0.1915 0.1968

Permeability Darcy 2.50 2.77

6.5.4 Oil/Water System

The oil/water system was also used in order to validate the applicability of the methods

presented in the thesis. As listed in Table 6.3, the same core holder as the gas/water system

was used with a thickness of 7.00 cm and a diameter of 4.10 cm; the porosity of the

sandpack was measured as 0.368; the viscosity of water is 1.0 cp at laboratory conditions

and the viscosity of kerosene was measured as 0.80 cp. From the data listed in Table 6.6,

the absolute permeability of the sandpack was calculated as 2.77 Darcies and the hydraulic

conductivity of the pipeline and the membrane was calculated as 80.0 mD/cm. A four-step

drainage experiment was carried out based on the oil/water system and the water

production history was recorded and plotted as squares in Figure 6.8. For oil/water system,

the diffusion effect is negligible.

172

Table 6.7: Measurement of porosity and the residual water saturation (Sandpack, Oil/Water

System)

Porosity & Residual Liquid Calculation

Porosity

Coreholder Weight 2096.20 g

CH & Wet Sand 2321.12 g

Dead Volume 1.00 cm3

Wet Sand Weight 223.92 g

DrySand & Beaker 405.20 g

Beaker 215.30 g

DrySand Weight 189.90 g

Water in place 34.02 g

Porosity 0.368

Final Saturation

Cumulative Water 28.80 cm3

Cumulative Oil 29.40 cm3

Saturation 0.145

173

6.5.5 Results and Discussions

Figure 6.7 shows the measured cumulative water production and the calibrated cumulative

water production, considering the gas diffusion and the residual water saturation in the

sandpack. As can be seen from Figure 6.7, about 2.0 cm3 volume difference is observed

between measured water production and the calibrated water production, which is

essentially caused by the gas diffusion. The last step on the measured curve in the drainage

process is a straight line with a constant slope where the cumulative production is

dominated by the gas diffusion. In this step, the multi-step drainage process using sandpack

and gas/water system has significant measurement uncertainty caused by the gas diffusion.

However, as shown in Figure 6.8, for the water/oil system neglecting the diffusion of oil in

water, the curves are much smoother and no significant impacts of diffusion or leakage are

indicated at the last step. Moreover, it is also seen that the residual water saturation in the

oil/water system is very close to the calibrated residual water saturation in the gas/water

system. Because the density of kerosene and water are very close, the residual water

saturation was only calculated from the cumulative production of water and the cumulative

injection of kerosene.

The automatic history match method is used to estimate the relative permeabilities of water

considering the resistance of the membrane. The history matched results of the wetting

phase production for gas/water and oil/water systems are plotted in Figure 6.8 and Figure

6.9, respectively. Using automatic history matching, the production histories for both

gas/water and oil/water systems were matched very well.

174

Figure 6.7: The experimental and the calibrated cumulative water production histories. The

calibrated the data considers gas diffustion volume and the residual saturation at the core

sample.

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

Cu

mu

lati

ve W

ate

r, c

m3

Time, hour

Calibrated Data (Diffusion, Residual Saturation)

Experimental

175

Figure 6.8: The experimental cumulative water production histories and history matching

results of oil/water system in sandpack.

0

5

10

15

20

25

30

35

0 5 10 15 20

Cu

mu

lati

ve W

ate

r, c

m3

Time, hour

Experimental Data

History Match Results

176

Figure 6.9: History matching results of gas/water system in sandpack.

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

Cu

mu

lati

ve W

ate

r, c

m3

Time, hour

Experimental Data

History Match Results

177

Analytical method was also used to calculate the relative permeabilities to water for

different liquid systems. As shown in Figure 6.10, given enough time, the first 3 steps in

the gas/water system all exhibit linear sections. For the last step of the gas/water system,

there are not enough valid data points to plot a straight line, thus it is not considered in the

analytical model. For oil/water system, as plotted in Figure 6.11, linear relationship is

observed for all four steps. The relative permeabilities to water were then calculated from

the slopes of these straight lines for different systems.

Figure 6.10: Calculation of the relative permeabilities to water for the gas/water system in

sandpack.

y = 1.4187E-01x + 9.1049E-01R² = 9.9696E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20

-Ln

(Sw

D)

Td

Step 1

y = 1.5531E-02x + 2.5106E-01R² = 9.9713E-01

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250

-Ln

(Sw

D)

Td

Step 2

y = 1.0236E-03x + 4.0918E-01R² = 9.9630E-01

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 500 1000 1500

-Ln

(Sw

D)

Td

Step 3

178

Figure 6.11: Calculation of the relative permeabilities to water for an oil/water system in

sandpack.

The capillary pressures measured for both oil/water and gas/water systems are plotted

together in Figure 6.12. The capillary pressure for the oil/water system is slightly lower

than that for the gas/water system; this is due to the different interfacial tension and contact

angle of the systems. However, the residual water saturations at the end of the drainage

experiments are very close. The relative permeabilities calculated from the numerical

method and the analytical methods are shown in Figure 6.13. There is a very good

consistency between these two methods; however, the closed-form analytical method is

much easier to apply to the relative permeabilities calculation.

y = 9.0404E-02x + 3.8125E-01R² = 9.9696E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35

-Ln

(Sw

D)

Td

Step 1

y = 7.3161E-03x + 4.4570E-01R² = 9.9314E-01

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300

-Ln

(Sw

D)

Td

Step 2

y = 2.0894E-03x + 5.4568E-02R² = 9.9661E-01

0

0.5

1

1.5

2

2.5

3

0 200 400 600 800 1000 1200

-Ln

(Sw

D)

Td

Step 3

y = 6.4086E-04x - 2.0105E-01R² = 9.8259E-01

0

0.5

1

1.5

2

2.5

3

0 1000 2000 3000 4000 5000

-Ln

(Sw

D)

Td

Step 4

179

Figure 6.12: Capillary pressure measurement in the sandpack.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1

Cap

illa

ry P

ressu

re,

atm

Water Saturation, Frac

G/W

O/W

180

Figure 6.13: Comparisons of the relative permeabilities of the wetting phase in the

sandpack obtained from simulation and analytical equation.

0.00001

0.0001

0.001

0.01

0.1

1

0.00001 0.0001 0.001 0.01 0.1 1

Krw

, An

alyt

ical

Krw Simulation

O/W G/W

181

6.6 Multi-step Drainage Process using Core Sample

The above sections illustrate the successful application of measurements of the relative

permeabilities of water using the multi-step drainage process in the sandpack. A similar

apparatus and procedure was designed to apply the multi-step drainage process on core

samples. The detailed experiment procedures are listed as follows.

6.6.1 Procedure

For the multi-step drainage process applied on core samples, fluids systems using

gas/water, oil/water and gas/oil/water were all tested. Generally, the detailed procedures

for these three different fluids system are very similar and summarized as follows:

(1) Measure the viscosities of kerosene and water; viscosity of air is assumed negligible;

(2) Measure the basic geometry dimensions of the core sample;

(3) Dehydrate the core sample using hot air for about 24 hours and measure its dry weight

(4) Saturate the core with water and measure its wet weight.

(5) Calculate the porosity using the mass difference between the wet weight and the dry

weight of the core sample.

(6) Run a multi-rate flow test to measure the resistivity and the permeabilities.

182

(7) In order to build the capillary equilibrium in the core sample and displace the free water

in the capillary cell, inject the non-wetting phase (air/kerosene) into the system at a

pressure slightly higher than the ambient pressure for several hours, until no water

production is observed at the outlet.

(8) The multi-step drainage starts from the above system. At the very beginning of a

drainage step, increase the injection pressure to the design value, and record the water

production at the outlet of the system with the digital balance every 6 seconds until no

more wetting production is observed within half an hour.

(9) Increase the injection pressure step by step, and record the water production history at

each step every 6 seconds.

(10) For the experiments using gas/water system, take the core sample out of the core

holder after the last-step. Measure the weight of core sample in order to check the residual

water saturation for mass balance.

6.6.2 Calculation of the Basic Parameters

Before all the experiments, the dimensions of all core samples were measured. Following

that, the porosity of each core sample was measured using the same mass difference

methods used for the sandpack. The measured length, diameters and the porosity are listed

in Table 6.8. The porosities measured are all very close to 0.2.

183

Before any drainage experiments, the multi-rate flow test was run to measure the

permeability and the resistance of the system. As listed in Table 6.9, the permeability of

the core sample varies from 72 mD to 103 mD. The resistance of the membrane is 5-6

times higher than that of the core sample itself, resulting in high uncertainty for the

measurement of the relative permeability curve at high water saturation. However, the

normalized parameters can be calculated using Eq. (4-32) and Eq. (4-49). Therefore, as

long as the measured relative permeabilities are lower than 0.1, there is no need to calibrate

the regressed results using the analytical model. However, the resistance of the membrane

and the pipeline has been considered when calculating the relative permeabilities to water

using the automatic history match method.

Table 6.8: Basic parameters of the cores samples

Gas/Water Oil/Water Oil/Water Gas/Oil/Water

Diameter, cm 4.570 4.600 4.600 4.650

Thickness, cm 0.770 0.800 0.800 0.800

Bulk Volume, cm3 12.630 13.295 13.295 13.585

Area, cm2 16.395 16.611 16.611 16.974

Pore Volume, cm3 2.470 2.632 2.632 2.630

Porosity 0.196 0.198 0.198 0.194

184

Table 6.9: Basic resistance and permeability measured by resistance tests

Resistance/Permeability Gas/Water Oil/Water Oil/Water Gas/Oil/Water

Pipeline, atm/(cm3/s) 0.074 0.079 0.083 0.079

Membrane, atm/(cm3/s) 2.751 3.681 3.800 2.600

Resistance, atm/(cm3/s) 2.825 3.761 3.883 2.679

Core Sample, atm/(cm3/s) 0.424 0.568 0.577 0.650

Permeability, Darcy 0.101 0.085 0.083 0.072

6.6.3 Gas/Water System

One set of the multi-step drainage experiments was run with a gas/water system using core

sample. As listed in Table 6.8, the porosity of the core sample was measured as 0.196; the

thickness was 0.77 cm and the diameter was 4.91 cm; the viscosity of water was 1.0 cp at

lab conditions and the viscosity of kerosene was measured as 0.80 cp. From the data listed

in Table 6.9, the absolute permeability of the core sample was calculated as 101 mD and

the hydraulic conductivity of the pipeline and the membrane was calculated as 32.7 mD/cm.

A four-step drainage experiment was carried out and the water production history was

recorded and plotted as circles in Figure 6.14.

185

Figure 6.14: Water production history at a three step gas/water system.

From the last two steps of Figure 6.14, there is continuous gas production showing up, as a

result of the air diffusion through water-saturated membrane. If assuming that under certain

pressure difference, the air diffusion rate is a constant, the slope of the linear portion could

be used to calculate the gas diffusion rate. The calculated gas diffusion rates at pressure

differences of each step were plotted in Figure 6.15. As can be seen, the gas diffusion rate

is linearly dependent on the pressure difference. The regressed linear equation was then

used to calculate the gas diffusion rate at different pressure and correct the water

production history.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5000 10000 15000 20000 25000 30000 35000 40000

Cu

mu

lati

ve V

olu

me, cm

3

Time, s

186

Figure 6.15: Gas diffusion rate measured in a multi-step drainage experiment with

gas/water.

After calibration considering the gas diffusion, the production data resulting exclusively

from the drainage process was plotted as circles in Figure 6.16. Two sets of automatic

history matching were run using the methods developed in the above section in terms of

two different relative permeabilities function representations, respectively: Corey‟s

equations and the monotone cubic spline function. The water recovery history data and the

matched value obtained from these two history matching runs are all shown in Figure 6.16.

0

0.000005

0.00001

0.000015

0.00002

0.000025

0 0.2 0.4 0.6 0.8 1 1.2

Dif

fusio

n R

ate

, cm

3/s

Pressure, atm

Diffusion Rate(cm3/s)

187

It can be seen that both Corey‟s equation and the monotone cubic spline function obtained

reasonable matching results. The analytical method was also employed to estimate the

relative permeabilities to water from the results of the drainage process as shown in Figure

6.17. Finally, the calculated permeabilities using analytical method are plotted in Figure

6.30 together with the permeabilities measured with the other fluid systems.

Figure 6.16: Water production history of the gas/water system and the results of history

matching.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5000 10000 15000 20000 25000 30000 35000 40000

Cu

mu

lati

ve P

rod

ucti

on

, cm

3

Time, s

Experiment

HM Corey's Equations

HM Monocubic

188

Figure 6.17: Calculations of the relative permeabilities to water using the analytical method

for the gas/water system.

y = 2.1415E-04x + 2.8238E-01R² = 9.9866E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 2000 4000 6000 8000 10000 12000 14000

-Ln

(Sw

D)

Td

Step 1

y = 5.1209E-05x + 2.3097E-01R² = 9.9987E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 10000 20000 30000 40000 50000 60000 70000

-Ln

(Sw

D)

Td

Step 2

y = 2.3294E-05x - 1.1075E+00R² = 9.9805E-01

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50000 100000 150000 200000 250000

-Ln

(Sw

D)

Td

Step 3

189

6.6.4 Oil/Water System

Two sets of the multi-step drainage experiments were carried out using water/oil systems

with the same core sample. As listed in Table 6.8, the porosity of the core sample was

measured as 0.198; the thickness was 0.80 cm and the diameter was 4.6 cm; the viscosity

of water is 1.0 cp at lab conditions and the viscosity of kerosene was measured as 0.80 cp.

From the data listed in Table 6.9, the permeability of the core sample was calculated as 83

mD and 85 mD from two different runs. A four-step and a five-step drainage experiment

were carried out and the water production history was recorded and plotted as circles in

Figure 6.18 and Figure 6.19, respectively.

Based on the first set of experiments, two history matching runs were carried out using

GTGA with two different relative permeability function representations, respectively:

Corey‟s equations and the monotone cubic spline function. The water production data from

the experiment, as well as the matched values calculated from these two history matching

runs, are all plotted in Figure 6.20 and Figure 6.21. It can be seen that both Corey‟s

equation and the monotone cubic spline function obtained reasonable well matching results.

However, as plotted in Figure 6.22, the calculated relative permeabilities to kerosene (the

non-wetting phase) using the monotone cubic spline function and Corey‟s Equation are

significantly different from each other, which means any curve between these two curves

would get similar matching results. The conclusion drawn in the above chapters, that

permeabilities for the non-wetting phase are non-unique, or at least of high uncertainty.

When the water saturation is less than 0.4, the relative permeabilities of water are almost

the same using these two function representations. It is confirmed that considering

190

laboratory measurement uncertainty, the relative permeabilities of the wetting phase

measured in a multi-step drainage experiment at low saturation is reliable and unique.

Figure 6.18: Cumulative water production history in a four-step drainage process using

oil/water (1) system.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5000 10000 15000 20000 25000 30000

Cu

mu

lati

ve V

olu

me, cm

3

Time, s

Cum. Vol.

191

Figure 6.19: Cumulative water production history in a five-step drainage process using

oil/water system.

0

0.2

0.4

0.6

0.8

1

1.2

0 5000 10000 15000 20000 25000 30000

Cu

mu

lati

ve V

olu

me, cm

3

Time, s

192

Figure 6.20: Water production history and the results of history matching.

0

0.2

0.4

0.6

0.8

1

1.2

0 5000 10000 15000 20000 25000 30000

Cu

mu

lati

ve P

rod

ucti

on

, cm

3

Time, s

HM Monocubic


Experiment

193

Figure 6.21: Water production history and the results of history matching.

0

0.2

0.4

0.6

0.8

1

1.2

0 5000 10000 15000 20000 25000

Cu

mu

lati

ve P

rod

ucti

on

, cm

3

Time, s

HM Monocubic


Experiment

194

Figure 6.22: Comparison of history matched results using Corey‟s equation and monotone

cublic spline functions.

The analytical method was also used to estimate the relative permeabilities for these two

set of experiments. The results are shown in Figure 6.23 and Figure 6.24, respectively. All

steps in both experiments show a perfectly straight line section, given enough time. Finally,

the calculated permeabilities for these two sets of experiments using the analytical method

are plotted in Figure 6.30 together with the permeabilities measured from other fluid

systems.

195


method for the water/oil (1) system.

y = 6.2261E-04x + 2.7266E-01R² = 9.9918E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 1000 2000 3000 4000 5000

-Ln

(Sw

D)

Td

Step 1

y = 1.0740E-04x + 1.1649E-01R² = 9.9932E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 5000 10000 15000 20000 25000 30000

-Ln

(Sw

D)

Td

Step 2

y = 3.5721E-05x + 2.6436E-01R² = 9.9879E-01

0

0.5

1

1.5

2

2.5

0 10000 20000 30000 40000 50000 60000

-Ln

(Sw

D)

Td

Step 3

y = 1.3400E-05x + 2.6737E-01R² = 9.9633E-01

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20000 40000 60000 80000 100000 120000

-Ln

(Sw

D)

Td

Step 4

196

Figure 6.24: Calculations of the relative permeabilities to water using the analytical

method for the water/oil (2) system.

y = 1.7079E-03x + 8.9756E-02R² = 9.9768E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000

-Ln

(Sw

D)

Td

Step 1

y = 4.4212E-04x + 4.6615E-01R² = 9.9841E-01

0

0.5

1

1.5

2

2.5

3

0 1000 2000 3000 4000 5000 6000

-Ln

(Sw

D)

Td

Step 2

y = 8.7228E-05x + 6.5303E-01R² = 9.9145E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 5000 10000 15000 20000 25000 30000

-Ln

(Sw

D)

Td

Step 3

y = 6.4762E-05x + 1.9691E-01R² = 9.9976E-01

0

0.5

1

1.5

2

2.5

3

0 10000 20000 30000 40000

-Ln

(Sw

D)

Td

Step 4

y = 2.0363E-05x + 1.9459E-01R² = 9.9979E-01

0

0.5

1

1.5

2

2.5

0 20000 40000 60000 80000 100000

-Ln

(Sw

D)

Td

Step 5

197

6.6.5 Gas/Oil/Water System

Multi-step drainage experiments using three-phase systems were carried out with

gas/oil/water system. Air was used as the gas phase, kerosene was used as the oil phase and

water was used as the water phase. The core sample was first saturated with water, and

then the system was equipped with an oil-wet membrane saturated by oil, which only

allows oil to flow through at the experimental conditions. The core holder was positioned

upside down, and then kerosene was injected to displace water. The produced water was

carefully collected and used to calculate the remaining water in the sample. The core

holder was then positioned right side up and a multi-step drainage process similar to that

employed for the gas/water system was carried out on an gas/oil/water system.

One set of multi-step drainage experiments was carried out for validation with an

gas/oil/water system. As listed in Table 6.8, the porosity of the core sample was measured

as 0.194; the thickness was 0.80 cm and the diameter was 4.65 cm; the viscosity of water is

1.0 cp at lab conditions and the viscosity of kerosene was measured as 0.80 cp. From the

data listed in Table 6.9, the absolute permeability of the core sample was measured as 72

mD and the hydraulic conductivity of the pipeline and the membrane was calculated as

30.6 mD/cm. A four-step drainage experiment was carried out and the kerosene production

history was recorded and plotted as circles in Figure 6.25.

198

Figure 6.25: Cumulative kerosene production history in a four-step drainage process.

From the last two steps of Figure 6.25, as has been observed in the gas/water system

drainage process, there are significant gas diffusion effects, due to the air diffusion through

the kerosene-saturated membrane. As was done for the gas/water system, the linear portion

was considered as a diffusion effect. The gas diffusion rates at different pressure

differences were plotted at Figure 6.26. The gas diffusion rate is linearly dependent on the

pressure difference. The regressed linear equation was then used to correct the liquid

production history. The corrected production data due to the drainage process was plotted

as circles in Figure 6.27.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Cu

mu

lati

ve V

olu

me, cm

3

Time, s

199

Figure 6.26: Diffusion of air through the membrane in kerosene.

Automatic history matching, using GTGA, was used to calculate the relative permeabilities.

Mono cubic spine function was used to represent the relative permeabilities in the

automatic history matching to estimate the relative permeabilities, which is shown as the

purple line in Figure 6.27. Figure 6.28 shows the regression results using the analytical

methods for each step. All four steps show a linear relationship between given enough time.

Finally, the calculated permeabilities using the analytical method are plotted in Figure 6.30

together with the permeabilities measured from other fluid systems.

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

0.00004

0.000045

0.00005

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Dif

fusio

n R

ate

, cm

3/s

Pressure, atm

Diffusion Rate(cc/s)

200

Figure 6.27: Kerosene production history and the results of history matching

201


method for the gas/water/oil system.

6.6.6 Results and Discussions

The designed multi-step drainage experiments can be run with both sandpacks and core

samples with different fluids systems, including water/oil, gas/water and gas/water/oil.

The relative permeabilities were calculated using both the automatic history matching

method and analytical model developed in this work.

y = 6.5554E-03x + 3.1678E-01R² = 9.9802E-01

0

0.5

1

1.5

2

2.5

3

3.5

0 100 200 300 400 500

-Ln

(Sw

D)

Td

Step 1

y = 8.2013E-04x + 6.5837E-01R² = 9.9720E-01

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 1000 2000 3000 4000 5000

-Ln

(Sw

D)

Td

Step 2

y = 1.9657E-04x + 4.1558E-01R² = 9.9827E-01

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5000 10000 15000 20000

-Ln

(Sw

D)

Td

Step 3

y = 3.4832E-05x + 4.4808E-01R² = 9.9669E-01

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20000 40000 60000 80000 100000

-Ln

(Sw

D)

Td

Step 4

202

The measured capillary curves are plotted together for different systems as shown in Figure

6.29. Although, the interfacial tension and contact angle might be different for different

systems, generally the two-phases-systems have very similar capillary pressure curves,

except the entrance pressure is different. The three-phase-system has slightly higher

residual liquids saturation because the residual liquid consists of both residual water and

residual oil. It is expected that with spreading oil, the residual liquid saturation should be

much lower than measured in the experiment. As shown in Figure 6.29, the slope of the

low saturation range for the core sample is not as steep as that of the sandpack, because the

core sample has a more uniform pore size distribution than the sandpack. Due to a higher

contact of the grains in the core sample, the residual water saturation is slightly lower than

that of the sandpack.

All the relative permeabilities measured in the system were plotted together as shown in

Figure 6.30. It can be seen that the relationship between the logarithm of the relative

permeabilities and the wetting phase saturation is generally linear. For all systems using

water as the wetting phase, the slope of the curve is very similar. This indicates that the

relative permeability of the wetting phase only depends on the water phase itself. The slope

of the water/oil (2) system is slightly offset from the gas/water system and the water/oil (1)

system, this could be due to the uncertainty of the measurement. For the gas/oil/water

system, the slope of the curve is steeper, which indicates a faster drop of the relative

permeabilities and a faster drainage of the middle phase, which is oil. As can be seen from

Figure 6.30, the extrapolation of the straight line indicates the residual water saturation in

the core sample can potentially drops to a level of 5% or even lower.

203

Figure 6.29: Comparisons of the capillary pressure curves measured for different fluid

systems.

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0.000 0.200 0.400 0.600 0.800 1.000

Cap

illa

ry P

ressu

re,

atm


Gas/Water

Water/Oil (1)

Water/Oil (2)

Gas/Oil/Water

204

Figure 6.30: Comparisons of the relative permeabilities calculated from numerical

simulation for different systems.

Table 6.10 to Table 6.13, and Figure 6.31 show the comparison of the relative

permeabilities of the wetting phase calculated from automatic history matching using

simulation and the analytical method developed in this work. For different experiments

using a sandpack or core samples in different fluid systems, the analytical method

developed in this work has good consistency with the results of numerical simulation. The

analytical method developed in this work can be used directly to calculate the

1.000E-06

1.000E-05

1.000E-04

1.000E-03

1.000E-02

1.000E-01

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800

Rela

tive P

erm

eab

ilit

y


Gas/Water

Water/Oil (1)

Water/Oil (2)

Gas/Oil/Water

205

permeabilities with linear regression, which allows more flexibility and reliability to

measure the permeabilities at low and ultra-low saturations.


simulation and the analytical model – Gas/Water System.

Gas/Water

Sw Krnw Krw Pc Analytical

0.150 - 2.330×10-5

1.000 2.329×10-5

0.214 - 4.355×10-5

0.500 5.175×10-5

0.295 - 2.052×10-4

0.300 2.141×10-4

0.442 - 7.435×10-4

0.200 -

1.000 - - 0.150 -

206


simulation and the analytical model – Oil/Water System (1).

Oil/Water (1)

Sw Krw Pc Analytical

0.123 1.956×10-5

1.500 1.340×10-5

0.188 5.602×10-5

0.700 3.572×10-5

0.249 1.345×10-4

0.350 1.074×10-4

0.340 4.692×10-4

0.202 6.226×10-4

0.564 1.205×10-2

0.102 -

1.000 - 0.082 -

207


simulation and the analytical model – Oil/Water System (2).

Oil/Water (2)

Sw Krw Pc Analytical

0.118 1.670×10-5

1.600 2.084×10-5

0.172 5.402×10-5

1.000 6.476×10-5

0.217 1.027×10-4

0.600 8.723×10-5

0.297 3.967×10-4

0.300 4.421×10-4

0.415 2.315×10-3

0.150 1.708×10-3

0.540 3.464×10-3

0.100 -

1.000 - 0.080 -

208


simulation and the analytical model – Gas/Oil/Water System.

Gas/Oil/Water

Sl So Krw Pc Analytical

0.245 0.112 2.900×10-5

1.200 4.517×10-5

0.293 0.168 1.750×10-4

0.500 1.894×10-4

0.360 0.247 8.200×10-4

0.250 7.844×10-4

0.440 0.341 4.580×10-3

0.150 6.944×10-3

0.680 0.624 2.571×10-2

0.090 -

1.000 1.000 - 0.060 -

209

The minimum value of the estimated permeabilities using the current design apparatus can

be at an order of 10-5

.With a specially designed more accurate apparatus, this method can

measure the permeabilities to a magnitude of 10-6

. Theoretically, as long as the capillary

pressure can be measured, the relative permeability can be calculated.

Figure 6.31: Comparisons of the relative permeabilities of the wetting phase obtained from

simulation and analytical equation.

0.000001

0.00001

0.0001

0.001

0.01

0.1

0.000001 0.00001 0.0001 0.001 0.01 0.1

Krw

, A

naly

tical

Krw, Simulation

Water/Oil (1) Water/Oil (2)

Gas/Oil/Water Gas/Water

210

6.7 Summary

6.7.1 Sandpack and Core Sample

The multi-step drainage experiment works suceeded both a sandpack and a core sample.

Generally, all multi-step drainage experiments using core samples have much smoother

production curves than those using sandpacks. The main reason for this is that the

sandpack has higher heterogeneity and longer length. The horizontally positioned sandpack

cannot generate a uniform saturation profile along the sample. Meanwhile, the gap between

the core holder and the sandpack leads to a highly permeable channel. This does not affect

the calculation of the relative permeabilities to the wetting phase at a low saturation range.

However, because the sandpack is made up of highly uniform sands, the capillary curve is

much steeper than that of the core sample over the low saturation range. The steep slope

makes it difficult to reliably measure the saturation changes in the low range.

6.7.2 Simulation and Analytical

Reverse methods using automatic history matching and the analytical method developed in

this work were successfully used for estimation of the relative permeabilities of the low

mobile phase and had consistent results. Reverse methods using automatic history

matching can be used for uncertainty analysis using the properties of the Guo Tao genetic

algorithm, whereas the closed form analytical method is more easily applied in the

estimation of the relative permeabilities of the wetting phase directly. Linear regression can

significantly reduce uncertainty caused by the measurement.

211

6.7.3 Limitations

Due to gas diffusion, as can be seen from the experiment, without further modification, the

designed experiment cannot be directly used to measure the permeabilities of a CO2/oil

system. Due to the high solubility of CO2 in oil and high diffusivity of CO2, the volume

produced due to gas diffusion will be hundreds of times higher than the volume produced

by the drainage process. In order to improve that, new methods should either be used to

eliminate the gas diffusion or accurately measure the gas volume due to diffusion.

212

CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS

7.1 Summary of Conclusions

In this work, the multi-step drainage process was extensively modeled using numerical

simulation, analytical modeling, and pore scale modeling using an interactive tube-bundle

model. Two methods were developed to estimate the relative permeability of water. A

Guo Tao genetic algorithm was used to estimate the relative permeabilities by automatic

history matching. An analytical model was also developed to directly estimate the relative

permeabilities of the wetting phase. Both numerical simulation and experimental validation

indicates that the results of these two methods agree with each other. The new development

tube bundle model was successfully applied to qualitatively model the drainage process,

the saturation profile and production history of a multi-step drainage process was

successfully modeled. The major conclusions from this work are listed below:

(1) A program was developed using Visual C++ 6.0 to model a multi-step drainage process.

The program was used for the further analysis, history matching and analytical model

benchmark. Sensitivity analysis and benchmark using commercial software indicate that:

The program developed with the scope of this work is suffiently accurate to

represent the multi-step drainage process.

213

Implicitly processing of the membrane, significantly increased calculation speed

and the accuracy.

The improved IMPES method is good enough to solve the equation numerically

within one minute and an acceptable numerical error.

(2) GTGA is successfully applied to estimate the relative permeabilities and its uncertainty

by automatic history matching the production history in a multi-step drainage process. The

study indicates that:

GTGA is much more effective than conventional GA in searching for the global

optimum value. GTGA is also can be used for the uncertainty analysis.

Considering the uncertainty of real lab experiments, it is considered to be

impossible to obtain reliable results for the relative permeabilities of the highly mobile

phase.

Using GTGA and history matching of the wetting phase production history during a

multi-step drainage process, the relative permeabilities of the non-wetting phase and the

wetting phase are only simultaneously retrievable when (1) the mobilities of the non-

wetting phase and the wetting phase are comparable and (2) the experimental uncertainty

and precision is controlled within certain level.

The relative permeability of the non-wetting phase can be taken as arbitrary number

and the unique relative permeabilities of the wetting phase can be obtained, when (1) the

wetting phase saturation of the system is low or (2) the viscosity ratio of the non-wetting to

the wetting phase is small, such as an gas/water system.

214

(3) An analytical model to describe the wetting phase recovery history in the multi-step

drainage process was developed in this study. A closed form equation was developed to

directly estimate the wetting phase relative permeabilities. A dimensionless parameter was

created to determine if the resistance of the membrane is negligible. If it is not, a

complementary model is developed to modify the wetting phase permeabilities and get

better resultsThe development and comparisons of the analytical model and numerical

simulation indicate that:

The resistance of the non-wetting phase is negligible when estimating the relative

permeability of the wetting phase at low wetting phase saturation;

The capillary pressure gradient around the ending saturation should be used, and

the estimated relative permeability of the wetting phase is the permeability at the ending

saturation;

Numerical hypothetical tests and laboratory experimental data verification

demonstrated that this method is effective for the direct estimation of the wetting phase

relative permeability.

(4) The inactive-tube-bundle model was used to model the multi-step drainage process as

well. The modeling process indicates that:

All phenomena in this process, such as “reverse flooding”, “bidirectional flooding”,

the relationship between wDSln and time t , and the multi-step drainage process can be

successfully modeled.

215

It is proved that the interacting tube-bundle model more accurately represents the

pore structure and fluid flow dynamics in a porous medium.

By incorporating the serial-type model and the triangle tubes to simulate water

trapping mechanisms, this model can potentially history match the multi-step drainage

experiments and build the relation between the capillary pressures and the relative

permeability curves.

(5) The laboratory experiments were used for validate the models and methods developed

in above sections:

The multi-step drainage experiment succeeded for both sandpacks and core samples.

It also succeeded for gas/water, oil/water and gas/oil/water system.

Both reverse method using automatic history matching and analytical method

developed in this work were successfully used for estimation of the relative permeabilities

of the low mobile phase and obtained very consistent results.

The reverse method can be used for uncertainty analysis using the properties of the

Guo Tao genetic algorithm. However, the closed form analytical method is much more

easy to use for direct estimation of the relative permeabilities of the wetting phase. Linear

regression can significantly reduce uncertainty caused by the measurement.

Due to the gas diffusivity, as can be seen from the experiment, without further

modification, the designed experiment cannot be directly used to measure the

permeabilities of CO2/Oil system.

216

7.2 Recommendations

(1) Core sample heterogeneity should be considered in the future study.

(2) On the experimental design side, a special apparatus design should be carried out to

properly handle the diffused gas volume. Multistep drainage experiments at high

pressure/temperature should be carried out with a more safely designed system with more

accurate collection of the produced liquid.

(3) With more detailed considerations of a real core sample, it is recommended to test the

interactive tube-bundle model for the estimatation of the relative permeabilities by history

matching of a multi-step drainage process.

217

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232

APPENDIX

A-1 Copyright Permission

233

A-2 Interfaces of the Application for History Matching and Visualization

Main Interface:

Menus:

Results

Viewer

Genetic

Output

Itineration

output

234

Input Dialogs:

235

236

A-3 Class for Numerical Simulation of Multistep Drainage Process

// RpExperiment.h: interface for the CRpExperiment class.

//

#if !defined(AFX_RPEXPERIMENT_H__C806CD38_D6FB_4C4D_B486_8DEBED1F00AD__INCLUDE

D_)

#define AFX_RPEXPERIMENT_H__C806CD38_D6FB_4C4D_B486_8DEBED1F00AD__INCLUDED_

#if _MSC_VER > 1000

#pragma once

#endif // _MSC_VER > 1000

#include "RpDataBody.h"

class CRpExperiment : public CRpDataBody

{

public:

CRpExperiment();

virtual ~CRpExperiment();

public:

void ObtainExpData(CRpDataBody * pRpData); //get data from rp databody

void ObtainMatchingData(double * chromArray);//get data from ga chain

double GetFitness(); //read fitness

void RunCpTest(); //Run Test

void SetRunningMode(int nRunningMode);

private:

double m_dFitness;

double *sw;

double *po; double *pw;

double *su,*sy,*Tw,*To,*Tt,*pc,*a,*b,*c,*d,*x, *dx;

SRpData m_tbRpData[NUM_KR_EXP_TABLE]; //table rp data

double krsw[NUM_KR_CAL_TABLE], krkro[NUM_KR_CAL_TABLE];

double krkrw[NUM_KR_CAL_TABLE], krpc[NUM_KR_CAL_TABLE]; //relative

permeability used for calculation

int m_nRunningMode;

private:

BOOL VerifyData();

void InitKrTable();

void InitCPTest();

void CPTestWithMembrane();

void CalRpDataAtSingleSw(double sw, double &kro,double &krw,double &pc);

public:

int IndexLookup(double arr[],int n,double val);

static void CubicSpline(int nInArrNum, int nOutArrNum,double x[],double y[],double

xx[],double yy[]); static void MonoCubicInterpolator(int nInArrNum, int nOutArrNum, double x[],double y[],

double xx[], double yy[]);

void TriSolver(double *sa,double *sb,double *sc,double *sd,double *sx,int sn);

};

#endif

// !defined(AFX_RPEXPERIMENT_H__C806CD38_D6FB_4C4D_B486_8DEBED1F00AD__INCLUDED

_)

237

// RpExperiment.cpp: implementation of the CRpExperiment class.

//

//////////////////////////////////////////////////////////////////////

#include "stdafx.h"

#include "RpExperiment.h"

#include "ResLM.h" #include "math.h"

#ifdef _DEBUG

#undef THIS_FILE

static char THIS_FILE[]=__FILE__;

#define new DEBUG_NEW

#endif

//////////////////////////////////////////////////////////////////////

// Construction/Destruction

//////////////////////////////////////////////////////////////////////

CRpExperiment::CRpExperiment()

{

po=sw=pw=a=b=c=d=x=Tt=Tw=To=pc=su=sy=dx=NULL;

m_nRunningMode=RM_GA_RUN;

}

CRpExperiment::~CRpExperiment()

{ if (po!=NULL) delete[] po;

if (sw!=NULL) delete[] sw;

if (pw!=NULL) delete[] pw;

if (a!=NULL) delete[] a;

if (b!=NULL) delete[] b;

if (c!=NULL) delete[] c;

if (d!=NULL) delete[] d;

if (x!=NULL) delete[] x;

if (Tt!=NULL) delete[] Tt;

if (Tw!=NULL) delete[] Tw;

if (To!=NULL) delete[] To;

if (pc!=NULL) delete[] pc;

if (su!=NULL) delete[] su;

if (sy!=NULL) delete[] sy;

if (dx!=NULL) delete[] dx;

}

void CRpExperiment::ObtainExpData(CRpDataBody * pRpData)

{

int i;

// core and liquid data

m_core=pRpData->m_core;

m_liqWater=pRpData->m_liqWater;

m_liqOil=pRpData->m_liqOil;

//time control

m_tMax=pRpData->m_tMax;

m_tStep=pRpData->m_tStep;

m_nBlock=pRpData->m_nBlock;

m_SwChange=pRpData->m_SwChange;

238

m_TimeRate=pRpData->m_TimeRate;

m_nRpInteType=pRpData->m_nRpInteType;

//relative perm data

m_nRpCount=pRpData->m_nRpCount;

for(i=0;i<NUM_EXPCURVE_MAX+1;i++)

m_RpData[i]=pRpData->m_RpData[i]; //history data;

m_nHisNum=pRpData->m_nHisNum;

for(i=0;i<NUM_HISDATA_MAX;i++)

m_sHisData[i]=pRpData->m_sHisData[i];

//corey coef.

for(i=0;i<NUM_COREY_VAR;i++)

m_fRpCoreyCtrl[i]=pRpData->m_fRpCoreyCtrl[i];

}

void CRpExperiment::RunCpTest()

{

if (VerifyData())

{

InitKrTable();

InitCPTest(); CPTestWithMembrane();

}

}

void CRpExperiment::InitCPTest()

{

po=new double[m_nBlock];

pw=new double[m_nBlock];

sw=new double[m_nBlock];

dx=new double[m_nBlock];

Tw=new double[m_nBlock];

To=new double[m_nBlock];

Tt=new double[m_nBlock];

pc=new double[m_nBlock];

a=new double[m_nBlock+1];

b=new double[m_nBlock+1];

c=new double[m_nBlock+1]; d=new double[m_nBlock+1];

x=new double[m_nBlock+1];

sy=new double[m_nBlock+1];

su=new double[m_nBlock+1];

int i;

double swi=m_RpData[m_nRpCount-1].sw;

double inPres=m_RpData[m_nRpCount-2].pc;

for (i=0; i<m_nBlock; i++)

{

sw[i]=swi;

po[i]=inPres;

pw[i]=0;

239

dx[i]=m_core.m_fLength/(m_nBlock-1);

}

}

void CRpExperiment::InitKrTable()

{

int i,j; double dsw=(m_RpData[m_nRpCount-1].sw-m_RpData[0].sw)/(NUM_KR_CAL_TABLE-1);

double isw=m_RpData[0].sw;

double ssw;

//

switch (m_nRpInteType)

{

case KR_TYPE_LINEAR: //linear;

{

double dsw=(m_RpData[m_nRpCount-1].sw-

m_RpData[0].sw)/(NUM_KR_CAL_TABLE-1);


double ssw;

for (i=0;i<NUM_KR_CAL_TABLE;i++)

{

ssw=dsw*i+isw;

for (j=0;j<m_nRpCount-1;j++) {

if (ssw>=m_RpData[j].sw && ssw<m_RpData[j+1].sw)

{

krkro[i]= m_RpData[j].kro+

(m_RpData[j+1].kro-

m_RpData[j].kro)/

(m_RpData[j+1].sw-

m_RpData[j].sw)*(ssw-m_RpData[j].sw);

krkrw[i]= m_RpData[j].krw+

(m_RpData[j+1].krw-

m_RpData[j].krw)/

(m_RpData[j+1].sw-

m_RpData[j].sw)*(ssw-m_RpData[j].sw);

krpc[i]= m_RpData[j].pc+

(m_RpData[j+1].pc-

m_RpData[j].pc)/

(m_RpData[j+1].sw-m_RpData[j].sw)*(ssw-m_RpData[j].sw);

krsw[i]=ssw;

break;

}

}

}

krsw [NUM_KR_CAL_TABLE-1]=m_RpData[m_nRpCount-1].sw;

krkro[NUM_KR_CAL_TABLE-1]=m_RpData[m_nRpCount-1].kro;

krkrw[NUM_KR_CAL_TABLE-1]=m_RpData[m_nRpCount-1].krw;

krpc [NUM_KR_CAL_TABLE-1]=m_RpData[m_nRpCount-1].pc;

break;

}

case KR_TYPE_CUBIC: //cubic

240

{




double ssw;

for (i=0;i<NUM_KR_CAL_TABLE;i++) {

ssw=dsw*i+isw;

krsw[i]=ssw;

}

double *csw,*ckro,*ckrw,*cpc;

csw=new double[m_nRpCount];

ckro=new double[m_nRpCount];

ckrw=new double[m_nRpCount];

cpc=new double[m_nRpCount];

for (i=0;i<m_nRpCount;i++)

{

csw[i]=m_RpData[i].sw;

ckrw[i]=m_RpData[i].krw;

ckro[i]=m_RpData[i].kro; cpc[i]=m_RpData[i].pc;

}

CubicSpline(m_nRpCount,NUM_KR_CAL_TABLE,csw,ckrw,krsw,krkrw);

CubicSpline(m_nRpCount,NUM_KR_CAL_TABLE,csw,ckro,krsw,krkro);

CubicSpline(m_nRpCount,NUM_KR_CAL_TABLE,csw,cpc,krsw,krpc);

delete [] csw;

delete [] ckro;

delete [] ckrw;

delete [] cpc;

}

break;

case KR_TYPE_MONOCUBIC: //cubic

{




double ssw;


{

ssw=dsw*i+isw;

krsw[i]=ssw;

}

double *csw,*ckro,*ckrw,*cpc;


241

ckro=new double[m_nRpCount];

ckrw=new double[m_nRpCount];



{

csw[i]=m_RpData[i].sw; ckrw[i]=m_RpData[i].krw;

ckro[i]=m_RpData[i].kro;

cpc[i]=m_RpData[i].pc;

}

MonoCubicInterpolator(m_nRpCount,NUM_KR_CAL_TABLE,csw,ckrw,krsw,krkrw);

MonoCubicInterpolator(m_nRpCount,NUM_KR_CAL_TABLE,csw,ckro,krsw,krkro);

MonoCubicInterpolator(m_nRpCount,NUM_KR_CAL_TABLE,csw,cpc,krsw,krpc);

/* FILE *pf=fopen("c:\\out.txt","w");

for (int kk=0;kk<NUM_KR_CAL_TABLE;kk++)

fprintf(pf,"%g\t%g\t%g\t%g\n",krsw[kk],krkrw[kk],krkro[kk],krpc[kk]);

fclose(pf);*/

delete [] csw;

delete [] ckro;

delete [] ckrw;

delete [] cpc;

}

break;

case KR_TYPE_COREY: //power law;

{

double fCoreyCtrl[NUM_COREY_VAR];

for (i=0;i<NUM_COREY_VAR;i++)

{

fCoreyCtrl[i]=m_fRpCoreyCtrl[i];

}

fCoreyCtrl[0]=m_fRpCoreyCtrl[0]*(1-m_RpData[m_nRpCount-1].sw);

fCoreyCtrl[1]=m_fRpCoreyCtrl[1]*(m_RpData[0].sw);

fCoreyCtrl[2]=(POW_MAX)*m_fRpCoreyCtrl[2]; fCoreyCtrl[3]=(POW_MAX)*m_fRpCoreyCtrl[3];




double rw,ro;


{

ssw=dsw*i+isw;

rw=(ssw-fCoreyCtrl[1])/(1-fCoreyCtrl[1]-fCoreyCtrl[0]);

ro=(1-ssw-fCoreyCtrl[0])/(1-fCoreyCtrl[1]-fCoreyCtrl[0]);

krsw[i]=ssw;

if(ro>=0)

krkro[i]=fCoreyCtrl[4]*powf(ro,fCoreyCtrl[2]);

242

else

krkro[i]=0;

//krkro[i]=1;

if(rw>=0)

krkrw[i]=fCoreyCtrl[5]*powf(rw,fCoreyCtrl[3]);

else

krkrw[i]=0; }

double *csw,*cpc;




{

csw[i]=m_RpData[i].sw;

cpc[i]=m_RpData[i].pc;

}

MonoCubicInterpolator(m_nRpCount,NUM_KR_CAL_TABLE,csw,cpc,krsw,krpc);

delete [] csw;

delete [] cpc;

}

break;

default:

break; }

}

void CRpExperiment::CPTestWithMembrane()

{

int t, i;

int nCell=m_nBlock;

double Po=m_core.m_fPore;

double cA=m_core.m_fArea;

double qc=0;

int nStep=0;

double sumt=0;

double sumq=0;

int kk=0;//

int tsIdx=1;

double ts=0; double sumr=0;

int nt=1;

double skro,skrw,spc;//temperary relative permeablity data buffer

//membrane calculation

if (m_core.m_memCond>=1e20)

qc=0;

else

qc=m_liqWater.m_visc/(m_core.m_memCond);

for (t=0; tsIdx < m_nHisNum/*t<1*/; t++)

243

{

//obtain time step and cumulative time;

double dt=m_tStep;

bool bRcStep=FALSE;

if(sumt+dt>=m_sHisData[tsIdx].m_time)

{ dt=m_sHisData[tsIdx].m_time-sumt;

bRcStep=TRUE;

tsIdx++;

}

sumt+=dt;

//Capillary pressure && calculate transferability

for (i=0; i<nCell; i++)

{

CalRpDataAtSingleSw(sw[i], skro,skrw, spc);

pc[i] = spc;

Tw[i] = m_core.m_fPerm * skrw / m_liqWater.m_visc / (0.5*dx[i]+0.5*dx[i+1]);

To[i] = m_core.m_fPerm * skro / m_liqOil.m_visc /(0.5*dx[i]+0.5*dx[i+1]);

Tt[i] = Tw[i] + To[i];

}

//

pc[0]=pc[0]*4/3-pc[1]/3; pc[nCell-1]=pc[nCell-1]*4/3-pc[nCell-2]/3;

//pressure boundary

double inPres=m_sHisData[tsIdx-1].m_pres;

double outPres=0;

//Build Linear Equations

//first block

b[0] = Tt[0];

a[0] = 1;

c[0] = 0;

d[0] = Tt[0]*inPres - Tw[0]*(pc[0]-pc[1]);

//second block

b[1] = -Tt[1];

c[1] = 0;

a[1] = Tt[1] + Tt[0]; d[1] = Tt[0]*inPres+Tw[1]*(pc[1]-pc[2]) -Tw[0]*(pc[0]-pc[1]);

//other point

for (i=2; i<m_nBlock-1; i++)

{

b[i] = Tt[i];

c[i] = Tt[i-1];

a[i] = -b[i] - c[i];

d[i] = Tw[i-1]*(pc[i-1]-pc[i])-Tw[i]*(pc[i]-pc[i+1]);

}

//last point

244

b[m_nBlock-1] = 1;

a[m_nBlock-1] = Tt[m_nBlock-2];

c[m_nBlock-1] = - Tt[m_nBlock-2];

d[m_nBlock-1]= -Tw[m_nBlock-2]*pc[m_nBlock-2]+Tw[m_nBlock-2]*pc[m_nBlock-1];

//membrane

b[m_nBlock] = 0; a[m_nBlock] = -qc;

c[m_nBlock] = 1;

d[m_nBlock] =pc[m_nBlock-1] + outPres;

//solver equation

TriSolver(a, b, c, d, x, m_nBlock+1);

for (i=1; i<m_nBlock; i++)

{

po[i] = x[i];

pw[i] = po[i] - pc[i];

}

po[0]=inPres;

pw[0]=inPres-pc[0];

double qo=x[0]*cA; double qw=-x[m_nBlock]*cA;

//sw

sw[0] = sw[0] - (Tw[0] * cA * (pw[0] - pw[1]) * dt) / (Po * dx [0]*0.5 * cA);

for (i=1; i<m_nBlock-1; i++)

sw[i] = sw[i]

+ (Tw[i] * cA * (pw[i + 1]- pw[i]) * dt ) / (Po * dx[i] * cA)

- (Tw[i - 1] * cA * (pw[i]- pw[i - 1])* dt)/ (Po * dx[i] * cA);

sw[m_nBlock-1] = sw[m_nBlock-1]

+ qw * dt / ( Po * cA * dx[m_nBlock-1] *0.5)

+ (Tw[m_nBlock-2] * cA * (pw[m_nBlock-2] - pw[m_nBlock-1]) * dt) / (Po *

dx[m_nBlock-1] *0.5 * cA);

sumq+=-dt*qw;

if (bRcStep) {

if(m_nRunningMode==RM_SINGLE_RUN) LMOUTPUT(LM_TOPVIEW,

"%g\t%g\t%g\t%g\r\n",sumt,inPres,sumq,qo);

sumr+=(sumq-m_sHisData[tsIdx-1].m_liquid)*(sumq-m_sHisData[tsIdx-

1].m_liquid);

if(m_nRunningMode==RM_SINGLE_RUN)

{

LMOUTPUT(LM_BOTTOMVIEW, "Time Step : %g\r\n", sumt);

}

}

}

m_dFitness=1/sqrt(sumr);

245

}

void CRpExperiment::TriSolver(double *sa,double *sb,double *sc,double *sd,double *sx,int sn)

{

double li = sa[0];

int i; sy[0] = sd[0] / li;

for (i = 1; i<sn; i++)

{

su[i-1] = sb[i - 1] / li;

li = sa[i] - sc[i] * su[i - 1];

sy[i] = (sd[i] - sc[i] * sy[i - 1]) / li;

}

sx[sn-1] = sy[sn-1];

for ( i = 1; i<sn; i++)

{

int j = sn - i - 1;

sx[j] = sy[j] - su[j] * sx[j + 1];

}

}

void CRpExperiment::CalRpDataAtSingleSw(double sw, double &kro,double &krw, double &pc)

{ //linear intepolation method to calculate the rp values for single point

int i;

if (sw<krsw[0])

{

kro=krkro[0];krw=krkrw[0];pc=krpc[0];

}

else if (sw>krsw[NUM_KR_CAL_TABLE-1])

{

kro=krkro[NUM_KR_CAL_TABLE-1];krw=krkrw[NUM_KR_CAL_TABLE-

1];pc=krpc[NUM_KR_CAL_TABLE-1];

}

else

{

i=IndexLookup(krsw,NUM_KR_CAL_TABLE,sw);

kro= krkro[i]+(krkro[i+1]-krkro[i])/(krsw[i+1]-krsw[i])*(sw-krsw[i]);

krw= krkrw[i]+(krkrw[i+1]-krkrw[i])/(krsw[i+1]-krsw[i])*(sw-krsw[i]);

pc= krpc[i]+(krpc[i+1]-krpc[i])/(krsw[i+1]-krsw[i])*(sw-krsw[i]); }

}

double CRpExperiment::GetFitness()

{

return m_dFitness;

}

void CRpExperiment::ObtainMatchingData(double *rpArray)

{

int i;

if(m_nRpInteType==KR_TYPE_COREY)

{

246

for (i=0;i<NUM_COREY_VAR;i++)

{

m_fRpCoreyCtrl[i]=rpArray[i];

}

}

else {


{

m_RpData[i].kro=rpArray[i];

//m_RpData[i].kro=m_RpData[i].kro;

//m_RpData[i].kro=1;

m_RpData[i].krw=rpArray[i+m_nRpCount];

//m_RpData[i].krw=m_RpData[i].krw;

}

}

}

BOOL CRpExperiment::VerifyData()

{

BOOL bResult=TRUE;

if (m_nRpCount<=0)

{ LMOUTPUT(LM_TOPVIEW, "No Relative Permeability Data\r\n");

return FALSE;

}

return bResult;

}

void CRpExperiment::MonoCubicInterpolator(int nInArrNum, int nOutArrNum, double * x,double *y,

double *xx, double *yy)

{

int i,j;

double t,h;

double * m = new double[nInArrNum];

//calculate simple m;

m[0]= (y[1]-y[0])/(x[1]-x[0]);

m[nInArrNum-1]=(y[nInArrNum-1]-y[nInArrNum-2])/(x[nInArrNum-1]-x[nInArrNum-2]);

for (i=1;i<nInArrNum-1;i++)

m[i]=((y[i]-y[i-1])/(x[i]-x[i-1])+(y[i+1]-y[i])/(x[i+1]-x[i]))/2;

//modify m for monotone


{

double delta = (y[i+1]-y[i])/(x[i+1]-x[i]);

if (delta==0)

{

m[i]=0;

m[i+1]=0;

247

}

else

{

double alpha = m[i] / delta;

double beta = m[i+1] / delta;

if ((alpha*alpha + beta*beta) > 9)

{

double tau = 3/sqrt(alpha*alpha + beta*beta);

m[i] = tau*alpha*delta;

m[i+1] = tau*beta*delta;

}

}

}

//find xx[i] and calculate yy[i]

for (j=0;j<nOutArrNum;j++)

{

//overshoot

int ii=0;

if (xx[j]<=x[0]) ii=0;

else if (xx[j]>=x[nInArrNum-1]) ii=nInArrNum-2;

else

{


{

double multx=(xx[j]-x[i])*(xx[j]-x[i+1]);

if(multx<=0)

ii=i;

else

continue;

}

}

h=x[ii+1]-x[ii];

t=(xx[j]-x[ii])/h; yy[j] = y[ii] * (2*t*t*t - 3*t*t + 1)

+ m[ii] * h * (t*t*t - 2*t*t + t)

+ y[ii+1] * (-2*t*t*t + 3*t*t)

+ m[ii+1]*h * (t*t*t - t*t);

}

delete [] m;

}

void CRpExperiment::CubicSpline(int nInArrNum,int nOutArrNum,double x[],double y[],double

xx[],double yy[])

{

248

int i,k;

double p1,p2,p3,p4;

double P0=0,Pn=0;

int N=nInArrNum-1;

int R=nOutArrNum-1;

double *h=new double[N];

double *a=new double[N+1];

double *c=new double[N];

double *g=new double[N+1];

double *af=new double[N+1];

double *ba=new double[N];

double *m=new double[N+1];

/*First Step calculate the coefs of Linear equation system*/

for(k=0;k<N;k++)

h[k]=x[k+1]-x[k];

for(k=1;k<N;k++)

a[k]=h[k]/(h[k]+h[k-1]);

for(k=1;k<N;k++)

c[k]=1-a[k];

for(k=1;k<N;k++) g[k]=3*(c[k]*(y[k+1]-y[k])/h[k]+a[k]*(y[k]-y[k-1])/h[k-1]);

c[0]=a[N]=1;

g[0]=3*(y[1]-y[0])/h[0]-P0*h[0]/2;

g[N]=3*(y[N]-y[N-1])/h[N-1]+Pn*h[N-1]/2;

/*Solve the equation*/

ba[0]=c[0]/2;

g[0]=g[0]/2;

for(i=1;i<N;i++)

{

af[i]=2-a[i]*ba[i-1];

g[i]=(g[i]-a[i]*g[i-1])/af[i];

ba[i]=c[i]/af[i];

}

af[N]=2-a[N]*ba[N-1];

g[N]=(g[N]-a[N]*g[N-1])/af[N];

m[N]=g[N];

for(i=N-1;i>=0;i--)

m[i]=g[i]-ba[i]*m[i+1];

/*Calculate values*/

for(i=0;i<=R;i++)

{

if(xx[i]<x[0])

yy[i]=x[0];

else if (xx[i]>x[N])

yy[i]=x[N];

else

{

249

k=0;

while(xx[i]>x[k+1])

k++;

p1=(h[k]+2*(xx[i]-x[k]))*pow((xx[i]-x[k+1]),2)*y[k]/pow(h[k],3);

p2=(h[k]-2*(xx[i]-x[k+1]))*pow((xx[i]-x[k]),2)*y[k+1]/pow(h[k],3);

p3=(xx[i]-x[k])*pow((xx[i]-x[k+1]),2)*m[k]/pow(h[k],2);

p4=(xx[i]-x[k+1])*pow((xx[i]-x[k]),2)*m[k+1]/pow(h[k],2); yy[i]=p1+p2+p3+p4;

}

}

/*Release buffer*/

delete [] h;

delete [] a;

delete [] c;

delete [] g;

delete [] af;

delete [] ba;

delete [] m;

}

void CRpExperiment::SetRunningMode(int nRunningMode)

{ m_nRunningMode=nRunningMode;

}

int CRpExperiment::IndexLookup(double arr[],int n,double val)

{

int i=0;

int j=n-1;

int k=n/2;

while(i<=j)

{

k=(i+j)/2;

if(arr[k]==val)

return k;

else if(arr[k]>val)

j=k-1; else if(arr[k]<val)

i=k+1;

}

return i-1;

}

250

A-4 Class for Genetic Algorithm

// GACalculator.h: interface for the CGACalculator class.

//

//////////////////////////////////////////////////////////////////////

#if !defined(AFX_GACALCULATOR_H__2EEA45E1_DF8C_4768_8FC2_39617EA96556__INCLUDED

_)

#define AFX_GACALCULATOR_H__2EEA45E1_DF8C_4768_8FC2_39617EA96556__INCLUDED_

#if _MSC_VER > 1000

#pragma once


#include "ResLM.h" #include "individual.h"

#include "RpDataBody.h"

#include "RpExperiment.h"

class CGACalculator;

struct sGAData

{

int idx;

double fitness;

CGACalculator * pGA;

};

class CGACalculator

{

public:

CGACalculator(); virtual ~CGACalculator();

//static varables and functions

public:

static HANDLE hMutex;

//Mute value for paralell calculation

static double RandVal(double dBegin=0.0, double dEnd=1.0); //Random value generator

static DWORD WINAPI CalSingleIndividual(LPVOID lpParam); //Child thread for

SingleIndividual calculation

//SGA Calculations///////////////////////////////////////////

public:

void SGACalculation();

private:

void SGAInitPopulation();

int SGACrossOver(CHTYPE *parent1,CHTYPE *parent2,CHTYPE *child1,CHTYPE *child2);

void SGAMutation(CHTYPE *child); void SGAGeneration();

void SGASumfit();

int SGASelection();

void SGACreateTmpDataFiles();

int m_nLenChrom;

int m_nSGAPopSize;

int m_nSGAMaxGen;

251

double m_sumfitness;

bool bSpeedUp;

FILE *fpout;

std::vector <CIndividual> newpop;

//MPGA Calculations///////////////////////////////////////////

public:

void MPGACalculation();

//Multiple-parent Genetic Algorithm function

void GetImportedData(FILE *fp); //Import

restart file

void Serialize(CArchive & ar);

//Serialize file for storage

void SetCurWorkPath(CString strPath);

private:

void MPGAInitPopulation();

//Initialize population space;

void RearrangeChrom(double *val); //For interpolation

method to rearrange Kr in monotonous order

void MPGACrossOver();

//MPG crossover, mutation included

void CreateTmpDataFiles(); //Create Temporary data file for GA

//GA Control/////////////////////////////////////////////////////

public:

int m_nMaxGen; //Maximum generations number (100-10000)

int m_nPopSize; //Individual number in one generation (60-200)

int m_nChromSize; //The number of chromosome; every chromosome is a

0 to 1 random number;

int m_nPoolNum; //cross over pool

int m_nEliteNum; //elite individual number

double m_fOffset;

double m_pmutation;

double m_pcross;

CIndividual m_indBestFit; //Best individual

CIndividual m_indAimed; //Matched target

private:

int m_nCurGen; //Current control

int m_nCurCalNum; //Current Calculation

Number

std::vector <CIndividual> m_popSpace; //Population Space

double m_fAverFit; //Average fitness

int m_nGACalType;

//other properities/////////////////////////////////////////////

public:

void SpeedUp();

void GAInitialization();

252

void SGARearrange(CIndividual *critter);

void RunGACalculation();

void SetGACalType(int nGACalType) { m_nGACalType=nGACalType;}

CString m_strCurPath;

CRpDataBody * m_RpDataBody;

void DrawGraph(CDC *pDC, CRect rc);

};

#endif

// !defined(AFX_GACALCULATOR_H__2EEA45E1_DF8C_4768_8FC2_39617EA96556__INCLUDED_)

// GACalculator.cpp: implementation of the CGACalculator class.

//

//////////////////////////////////////////////////////////////////////

#include "stdafx.h"

#include "reslm.h"

#include "GACalculator.h"

#ifdef _DEBUG

#undef THIS_FILE

static char THIS_FILE[]=__FILE__;

#define new DEBUG_NEW

#endif

//////////////////////////////////////////////////////////////////////


//////////////////////////////////////////////////////////////////////

/********************************************************************

created: 2009/03/09

author: Shengdong Wang

purpose: Multiple-parent genetic algorithm

description: This is the main model of class multiple-parent genetic algorithm

*********************************************************************/

CGACalculator::CGACalculator()

{

m_nPopSize=100;

m_nChromSize = 0;

m_nMaxGen=2000;

m_nPoolNum=3; m_nEliteNum=1;

m_fOffset=0.5f;

m_nGACalType=1;

m_nSGAPopSize=100;

m_nSGAMaxGen =10000;

m_pcross=0.7;

m_pmutation=0.03;

bSpeedUp=FALSE;

}

CGACalculator::~CGACalculator()

{

253

}

//////////////////////////////////////////////////////////////////////////

/********************************************************************

Created: 2009/12/03

Author: Shengdong Wang

Purpose:

Description: Mult-parent GA Calculation main function

*********************************************************************/

void CGACalculator::MPGACalculation()

{

int i,gen;

sGAData *pGAData;

HANDLE *hThread;

CTime tTime;

CString strTime;

hMutex= CreateMutex(NULL, FALSE, NULL);

pGAData= new sGAData [m_nPopSize];

hThread= new HANDLE [m_nPopSize];

// output basic information

tTime=CTime::GetCurrentTime();

strTime=tTime.Format("%Y-%m-%d %H:%M:%S");

LMOUTPUT(LM_TOPVIEW,"--- Multi-parent Crossover Genetic Algorithm ---\r\n");

LMOUTPUT(LM_TOPVIEW,"Time: %s\r\n",strTime);

LMOUTPUT(LM_TOPVIEW,"Total Population: %d\r\n",m_nPopSize);

LMOUTPUT(LM_TOPVIEW,"Total Generation: %d\r\n\r\n",m_nMaxGen);

LMOUTPUT(LM_TOPVIEW,"Initializing First Generation\r\n");

// output file;

fprintf(fpout,"--- Multi-parent Crossover Genetic Algorithm ---\n");

fprintf(fpout,"Time: %s\n",strTime);

fprintf(fpout,"Total Population: %d\n",m_nPopSize);

fprintf(fpout,"Total Generation: %d\n",m_nMaxGen);

fprintf(fpout,"------------------------------------------------\n");

//using mult-thread to calculate the first generation

for (i=0;i<m_nPopSize;i++)

{

pGAData[i].idx =i;

pGAData[i].pGA=this;

hThread[i] = CreateThread(NULL, 0, CalSingleIndividual, &pGAData[i],

CREATE_SUSPENDED, NULL);

SetThreadPriority(hThread[i], THREAD_PRIORITY_BELOW_NORMAL);

ResumeThread(hThread[i]);

}

//wait for multi-thread over;

254

int tempNumThreads = m_nPopSize;

int tempMax = 0;

while(tempNumThreads >= MAXIMUM_WAIT_OBJECTS )

{

tempNumThreads -= MAXIMUM_WAIT_OBJECTS;

WaitForMultipleObjects( MAXIMUM_WAIT_OBJECTS, &hThread[tempMax], TRUE,

INFINITE); tempMax += MAXIMUM_WAIT_OBJECTS;

}

WaitForMultipleObjects(tempNumThreads, &hThread[tempMax], TRUE, INFINITE);

//close objects


{

CloseHandle(hThread[i]);

m_popSpace[i].fitness=pGAData[i].fitness;

}

delete [] pGAData;

delete [] hThread;

LMOUTPUT(LM_TOPVIEW,"First Generation Initialization Done\r\n");

//Mult-parent GA Calculation

for (gen=0;gen<m_nMaxGen;gen++) {

MPGACrossOver();

double sumfit=0;


{

sumfit+=m_popSpace[i].fitness;

if (m_indBestFit.fitness<m_popSpace[i].fitness)

{

int k;

m_indBestFit=m_popSpace[i];

m_indBestFit.gen=gen+1;

LMOUTPUT(LM_TOPVIEW,"\r\nFitness : %g\r\n",m_indBestFit.fitness);

LMOUTPUT(LM_TOPVIEW,"Chromosome Data:\r\n");

fprintf(fpout,"\nFitness : %g\n",m_indBestFit.fitness); fprintf(fpout,"Chromosome Data:\n");

for(k =0;k<m_nChromSize;k++)

{

LMOUTPUT(LM_TOPVIEW,"%20g\t%20g\r\n",m_indBestFit.dblChrom[k],m_indAimed.dblChro

m[k]);

fprintf(fpout,"%20g\t%20g\n",m_indBestFit.dblChrom[k],m_indAimed.dblChrom[k]);

}

fprintf(fpout,"CRDATA %d\n",m_nChromSize);

for(k=0;k<m_nPopSize;k++)

255

{

int j;

for(j=0;j<m_nChromSize;j++)

{

fprintf (fpout,"%lf\t",m_popSpace[k].dblChrom[j]);

}

fprintf (fpout,"\n"); }

}

}

m_nCurGen=gen+1;

m_fAverFit= sumfit/m_nPopSize;

LMOUTPUT(LM_BOTTOMVIEW,"Gn: %d\tCn: %d\tFn: %g\r\n",m_nCurGen,m_nCurCalNum,m

_fAverFit);

fprintf(fpout,"****** Gn: %d\tCn: %d\tFn: %g

***********\n",m_nCurGen,m_nCurCalNum,m_fAverFit);

}

//Calculation over, print results;

tTime=CTime::GetCurrentTime();

strTime=tTime.Format("%Y-%m-%d %H:%M:%S");

LMOUTPUT(LM_TOPVIEW,"\r\n--- Genetic Algorithm Done! ---\r\n"); LMOUTPUT(LM_TOPVIEW,"Time: %s\r\n",strTime);

}

void CGACalculator::SGAInitPopulation() //population initialization

{

int i,j,k = 0;

m_nLenChrom=8*sizeof(CHTYPE)*m_nChromSize;

CHTYPE mask = 1;

m_popSpace/**/.resize(m_nSGAPopSize);

newpop.resize(m_nSGAPopSize);

// chromosome initialization

for (i=0; i<m_nSGAPopSize; i++)

{

CIndividual newIndividual;

for (j=0; j<m_nChromSize; j++)

{ CHTYPE newChrom=0;

int stop=8*sizeof(CHTYPE);

for (k=0; k<stop; k++)

{

newChrom=newChrom<<1;

if (RandVal()<=0.5)

{

newChrom=newChrom|mask;

}

}

newIndividual.byteChrom[j]=newChrom;

}

256

//rearrange chromsome

newIndividual.parent[0]=0; //parents index

newIndividual.parent[1]=0;

newIndividual.fitness=0;

m_popSpace/**/[i]=newIndividual;

//m_popSpace/**/[i]=newIndividual;

}

}

void CGACalculator::MPGAInitPopulation() //population space initialization

{

int i,j;

// population space initialization

m_nCurGen=0;

m_nCurCalNum=0;

m_indBestFit.fitness = 0.0f;

m_fAverFit=0.0f;

m_popSpace.clear();

m_popSpace.resize(m_nPopSize);

for (i=0; i<m_nPopSize; i++) {

CIndividual newIndividual;

//Create random chromosome

for (j=0; j<m_nChromSize; j++)

newIndividual.dblChrom[j]=RandVal(0,1);

RearrangeChrom(newIndividual.dblChrom);

newIndividual.fitness=0;

newIndividual.gen=0;

m_popSpace[i]=newIndividual;

}

}

void CGACalculator::MPGACrossOver()

{

int i,j;

int * pMatePoolIdx;

double * pMateParam;

double sumA,tmpMax,tmpMin,MaxA,MinA;

bool bValidChild;

int mini=0;

int EliteNum=0;

if(m_nPoolNum>m_nPopSize) m_nPoolNum=m_nPopSize;

257

if(m_nEliteNum>m_nPoolNum) m_nEliteNum=m_nPoolNum;

if (bSpeedUp)

{

EliteNum=m_nPoolNum;

}

else EliteNum=m_nEliteNum;

pMateParam= new double[m_nPoolNum];

pMatePoolIdx = new int[m_nPoolNum];

bValidChild=false;

while(!bValidChild)

{

double minfit=1e300;

double maxfit=-1e300;

//look up smallest one;


{

if (minfit>m_popSpace[i].fitness)

{

minfit=m_popSpace[i].fitness;

mini=i; }

}

//look up elites

for (j=0;j<EliteNum;j++)

{

maxfit=-1e300;


{

int k;

bool bOut=false;

for (k=0;k<j;k++)

{

if (i==pMatePoolIdx[k]){bOut=true;break;}

}

if (bOut) continue;

if (maxfit<m_popSpace[i].fitness)

{ maxfit=m_popSpace[i].fitness;

pMatePoolIdx[j]=i;

}

}

}

// Select other crossover pool individuals

for(i=EliteNum;i<m_nPoolNum;i++)

pMatePoolIdx[i]=int(RandVal()*m_nPopSize-0.5);

// calculate pMateparam

sumA=0.0;

tmpMin=MinA=0-m_fOffset;

258

tmpMax=MaxA=1+m_fOffset;

for(i=0;i<m_nPoolNum-1;i++)

{

pMateParam[i]=(tmpMax-tmpMin)*RandVal()+MinA;

sumA+=pMateParam[i];

tmpMax=min(MaxA,MaxA-sumA); tmpMin=max(MinA,MinA-sumA);

}

pMateParam[m_nPoolNum-1]=1-sumA;

//Create new individual

CIndividual newind;

for(i=0;i<m_nChromSize;i++)

{

double val=0;

for(j=0;j<m_nPoolNum;j++)

{

val+=m_popSpace[pMatePoolIdx[j]].dblChrom[i]*pMateParam[j];

}

if(RandVal()<m_pmutation)

val=val+0.01*(0.5-RandVal());

val=max(0,val); val=min(1,val);

newind.dblChrom[i]=val;

}

RearrangeChrom(newind.dblChrom);

//calculate fitness of the new individual

CRpExperiment exp;

exp.ObtainExpData(m_RpDataBody);

exp.ObtainMatchingData(newind.dblChrom);

exp.RunCpTest();

newind.fitness=exp.GetFitness();

if (newind.fitness>minfit)

{

m_popSpace[mini]=newind;

CreateTmpDataFiles();

bValidChild=true;

}

m_nCurCalNum++; }


{

// m_popSpace[i].dblChrom[m_nChromSize-1]=1;

// m_popSpace[i].dblChrom[m_nChromSize/2-1]=0;

//

}

delete [] pMatePoolIdx;

delete [] pMateParam;

}

/////////////////////////////////////////////////////////////

259

/////Random value generator////////////////////////

double CGACalculator::RandVal(double dBegin, double dEnd)

{

double d01, rslt;

d01 = ((double)rand())/RAND_MAX; rslt=dBegin+d01*(dEnd-dBegin);

return rslt;

}

void CGACalculator::Serialize(CArchive & ar)

{

m_RpDataBody->Serialize(ar);

if(ar.IsStoring())

{

ar<<m_nMaxGen<<0.2<<0.2<<m_nPopSize<<m_nPoolNum<<m_nEliteNum<<m_fOffset;

}

else

{

double tmp;

ar>>m_nMaxGen>>tmp>>tmp>>m_nPopSize>>m_nPoolNum>>m_nEliteNum>>m_fOffset; }

}

DWORD WINAPI CGACalculator::CalSingleIndividual(LPVOID lpParam)

{

WaitForSingleObject(hMutex,INFINITE);

sGAData *pGAData=(sGAData *)lpParam;

int idx= pGAData->idx;

LMOUTPUT(LM_BOTTOMVIEW,"\t$ Create Individual %d\r\n",idx+1);

CRpExperiment exp;

exp.ObtainExpData(pGAData->pGA->m_RpDataBody);

exp.ObtainMatchingData(pGAData->pGA->m_popSpace[idx].dblChrom);

ReleaseMutex(hMutex);

exp.RunCpTest();

WaitForSingleObject(hMutex,INFINITE);

pGAData->fitness=exp.GetFitness();

LMOUTPUT(LM_BOTTOMVIEW,"\t$ Individual %d fitness : %g\r\n",idx+1,pGAData-

>fitness);

ReleaseMutex(hMutex);

return 0;

}

void CGACalculator::CreateTmpDataFiles()

{

int i,j;

FILE *fp;

260

//create temporary file

CString curKrdDir=m_strCurPath+".lmrst"; //create restart file

//rel perm data

fp=fopen(curKrdDir,"w");

fprintf(fp,"CRLENGTH %d\n",m_nChromSize);

for(i=0;i<m_nPopSize;i++)

{ for(j=0;j<m_nChromSize;j++)

{

fprintf (fp,"%lf\t",m_popSpace[i].dblChrom[j]);

}

fprintf (fp,"\n");

}

fclose(fp);

}

void CGACalculator::GetImportedData(FILE *fp)

{

}

void CGACalculator::RearrangeChrom(double *var)

{

if(m_RpDataBody->m_nRpInteType==KR_TYPE_COREY) return; int i,j;

double tmp;

for ( i=0; i<m_nChromSize/2; i++)

{

for ( j=i; j<m_nChromSize/2; j++)

{

if (var[j]>var[i])

{

tmp=var[i];

var[i]=var[j];

var[j]=tmp;

}

}

//var[i]=1;

}

// var[m_nChromSize/2-1]=0; for (i=m_nChromSize/2; i<m_nChromSize; i++)

{

for ( j=i; j<m_nChromSize; j++)

{

if (var[j]<var[i])

{

tmp=var[i];

var[i]=var[j];

var[j]=tmp;

}

}

}

261

}

void CGACalculator::SetCurWorkPath(CString strPath) {

int pos=strPath.ReverseFind('.');

if(pos>0)

m_strCurPath=strPath.Left(pos);

else

m_strCurPath="";

}

void CGACalculator::DrawGraph(CDC *pDC, CRect rc)

{

CRect rcPlot=rc;

rcPlot.DeflateRect(10,10);

pDC->Rectangle(rcPlot);

//draw best;

int i=0,j=0;

double wid,hig;

wid=rcPlot.Width(); hig=rcPlot.Height();

int ptsize=3;

int ptsize2=5;

CPen pen_red,pen_blue,pen_green,pen_back,*pOld;

pen_red.CreatePen(PS_SOLID, 2, RGB(255, 0, 0));

pen_blue.CreatePen(PS_SOLID, 1, RGB(0, 0, 255));

pen_green.CreatePen(PS_SOLID, 1, RGB(0, 255, 0));

pen_back.CreatePen(PS_SOLID, 1, RGB(0, 0, 0));

// all ones;

CIndividual *tmp=&m_indBestFit;

pOld=pDC->SelectObject(&pen_green);

double dx=wid/m_nChromSize;

for (j=0;j<m_nPopSize;j++)

{

tmp=&m_popSpace[j]; for (i=0;i<m_nChromSize;i++)

{

CRect rcPt;

rcPt.top=rcPlot.bottom-tmp->dblChrom[i]*hig-ptsize-1;

rcPt.bottom=rcPlot.bottom-tmp->dblChrom[i]*hig+ptsize;

rcPt.left=rcPlot.left+dx*(i+0.5)-ptsize;

rcPt.right=rcPlot.left+dx*(i+0.5)+ptsize;

pDC->Ellipse(rcPt);

}

}

pDC->SelectObject(&pen_red);

//original one

262

for (i=0;i<m_nChromSize;i++)

{

double xl,xr,y;

y=rcPlot.bottom-m_indAimed.dblChrom[i]*hig;

xl=rcPlot.left+dx*(i+0.5)-ptsize2;

xr=rcPlot.left+dx*(i+0.5)+ptsize2; pDC->MoveTo(xl,y);

pDC->LineTo(xr,y);

}

// best one;

pDC->SelectObject(&pen_blue);


{

CRect rcPt;

rcPt.top=rcPlot.bottom-m_indBestFit.dblChrom[i]*hig-ptsize2;

rcPt.bottom=rcPlot.bottom-m_indBestFit.dblChrom[i]*hig+ptsize2;

rcPt.left=rcPlot.left+dx*(i+0.5)-ptsize2;

rcPt.right=rcPlot.left+dx*(i+0.5)+ptsize2;

pDC->MoveTo(rcPt.TopLeft());

pDC->LineTo(rcPt.BottomRight()); CPoint pt1=rcPt.TopLeft();

pt1.Offset(rcPt.Width(),0);

CPoint pt2=rcPt.BottomRight();

pt2.Offset(-rcPt.Width(),0);

pDC->MoveTo(pt1);

pDC->LineTo(pt2);

}

pDC->SelectObject(pOld);

}

HANDLE CGACalculator::hMutex=NULL;

void CGACalculator::RunGACalculation()

{ GAInitialization();

switch (m_nGACalType)

{

case 0:

SGAInitPopulation();

SGACreateTmpDataFiles();

SGACalculation();

break;

case 1:

default:

MPGAInitPopulation(); // initial population;

CreateTmpDataFiles(); // create temporary file;

MPGACalculation();

263

}

}

void CGACalculator::SGACalculation()

{

int i,j;

sGAData *pGAData;

pGAData= new sGAData[m_nSGAPopSize];

hMutex=CreateMutex(NULL, FALSE, NULL);

CTime time;

CString strTime;

time=CTime::GetCurrentTime();

strTime=time.Format("%Y-%m-%d %H:%M:%S");

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"--- Simple Genetic Algorithm

Begins ---\r\n at %s\r\n",strTime);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Total

Population: %d\r\n",m_nSGAPopSize);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Total

Generation: %d\r\n\r\n",m_nSGAMaxGen);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Crossover

rate: %g\r\n\r\n",m_pcross);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Mutation rate: %g\r\n\r\n",m_pmutation);

fprintf(fpout,"--- Simple Genetic Algorithm Begins ---\r\n at %s\r\n",strTime);

fprintf(fpout,"Total Population: %d\r\n",m_nSGAPopSize);

fprintf(fpout,"Total Generation: %d\r\n\r\n",m_nSGAMaxGen);

fprintf(fpout,"Crossover rate: %g\r\n\r\n",m_pcross);

fprintf(fpout,"Mutation rate: %g\r\n\r\n",m_pmutation);

HANDLE *hThread;

hThread=new HANDLE[m_nSGAPopSize];

for (j=0;j<m_nSGAMaxGen;j++)

{

// LMOUTPUT(LM_TOPVIEW,"Generation %d\r\n",j+1);

for (i=0;i<m_nSGAPopSize;i++)

{ pGAData[i].idx =i;

pGAData[i].pGA=this;

SGARearrange(&m_popSpace[i]);

hThread[i] = CreateThread(NULL, 0, CalSingleIndividual, &pGAData[i],

CREATE_SUSPENDED , NULL);

SetThreadPriority(hThread[i], THREAD_PRIORITY_BELOW_NORMAL);

ResumeThread(hThread[i]);

}

//wait for multithread over;

int tempNumThreads = m_nSGAPopSize;

int tempMax = 0;

while( tempNumThreads >= MAXIMUM_WAIT_OBJECTS )

264

{

tempNumThreads -= MAXIMUM_WAIT_OBJECTS;

WaitForMultipleObjects(MAXIMUM_WAIT_OBJECTS, &hThread[tempMax],

TRUE, INFINITE);

tempMax += MAXIMUM_WAIT_OBJECTS;

}

WaitForMultipleObjects(tempNumThreads, &hThread[tempMax], TRUE, INFINITE); double fit=0;

for (i=0;i<m_nSGAPopSize;i++)

{

CloseHandle(hThread[i]);

m_popSpace/**/[i].fitness=pGAData[i].fitness;

fit+=m_popSpace/**/[i].fitness;

if (m_indBestFit.fitness<m_popSpace/**/[i].fitness)

{

m_indBestFit=m_popSpace/**/[i];

m_indBestFit.gen=j+1;

}

}

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Best Result: %lf\r\n",

m_indBestFit.fitness);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"The best individual found : %d

generation %lf\r\n",m_indBestFit.gen, m_indBestFit.fitness);

((CResLMApp*)AfxGetApp())->AddStringFmt(LM_TOPVIEW,"Average : %lf \r\n", fit/m_nSGAPopSize);

fprintf(fpout,"Best Result: %lf\r\n", m_indBestFit.fitness);

fprintf(fpout,"The best individual found : %d generation %lf\r\n",m_indBestFit.gen,

m_indBestFit.fitness);

fprintf(fpout,"Average : %lf \r\n", fit/m_nSGAPopSize);

for(int m =0;m<m_nChromSize/2;m++)

{

SRpData rpd;

double maxx=powl(2.0,sizeof(CHTYPE)*8);

rpd.sw=m_RpDataBody->m_RpData[m].sw;

rpd.pc=m_RpDataBody->m_RpData[m].pc;

/*rpd.kro=1;rpd.krw=1;

for (int k=0;k<m+1;k++)

{ rpd.kro*=bestfit.chrom[k]/maxx;

}

for (k=m;k<m_nChromSize/2;k++)

{

rpd.krw*=bestfit.chrom[k+m_nChromSize/2]/maxx;

}*/

rpd.kro=m_indBestFit.byteChrom[m]/maxx;

rpd.krw=m_indBestFit.byteChrom[m+m_nChromSize/2]/maxx;

((CResLMApp*)AfxGetApp())-

>AddStringFmt(LM_TOPVIEW,"%lf\t%lf\t%lf\t%lf\r\n",rpd.sw,rpd.kro,

rpd.krw,rpd.pc);

fprintf(fpout,"%lf\t%lf\t%lf\t%lf\r\n",rpd.sw,rpd.kro,

265

rpd.krw,rpd.pc);

}

SGAGeneration();

for (i=0; i<m_nSGAPopSize; i++) {

m_popSpace/**/[i]=newpop[i];

}

SGACreateTmpDataFiles();

}

fprintf(fpout,"Genetic Algorithm Done!\r\n");

time=CTime::GetCurrentTime();

strTime=time.Format("%Y-%m-%d %H:%M:%S");

fprintf(fpout,"\r\n--- Genetic Algorithm Done! ---\r\n at %s\r\n",strTime);

delete [] pGAData;

delete [] hThread;

}

void CGACalculator::SGAGeneration() {

int mate1,mate2,jcross,j=0;

SGASumfit();

do

{

mate1=SGASelection();

mate2=SGASelection();

jcross=SGACrossOver(m_popSpace/**/[mate1].byteChrom,m_popSpace/**/[mate2].byteChrom,newpop[j].

byteChrom,newpop[j+1].byteChrom);

SGAMutation(newpop[j].byteChrom);

SGAMutation(newpop[j+1].byteChrom);

newpop[j].parent[0]=mate1+1;

newpop[j].parent[1]=mate2+1;

newpop[j+1].parent[0]=mate1+1;

newpop[j+1].parent[1]=mate2+1;

newpop[j+1].gen=newpop[j].gen=m_nCurGen; j=j+2;

}

while (j<(m_nSGAPopSize-1));

}

int CGACalculator::SGASelection() //ÐÂ¸öÌåÑ¡Ôñ³ÌÐò

{

double sum,pick;

int i;

pick=RandVal();

sum=0;

if (m_sumfitness!=0)

{

266

for (i=0; (sum<pick)&&(i<m_nSGAPopSize); i++)

{

sum+=m_popSpace/**/[i].fitness/m_sumfitness;

}

}

else

{ i=int(RandVal()*double(m_nSGAPopSize));

}

return(i-1);

}

//ÓÉÁ½¸ö¸¸¸öÌå²úÉúÁ½¸ö×Ó¸öÌå

int CGACalculator::SGACrossOver(CHTYPE *parent1, CHTYPE *parent2,CHTYPE *child1,CHTYPE

*child2)

{

int j,k;

CHTYPE jcross;

CHTYPE mask,temp;

if (RandVal()<=m_pcross) //

{

jcross=RandVal(1,m_nLenChrom-1); //

{ if (jcross>=(k*(8*sizeof(CHTYPE))))

{

child1[k-1]=parent1[k-1];


}

else

{

if ((jcross<(k*8*sizeof(CHTYPE)))&&(jcross>((k-1)*8*sizeof(CHTYPE))))

{

mask=1;

for (j=1; j<=(int)(jcross-1-(k-1)*(8*sizeof(CHTYPE))); j++)

{

temp=1;

mask=mask<<1;

mask=mask|temp;

}

child1[k-1]=((parent1[k-1])&mask)|((parent2[k-1])&(~mask)); child2[k-1]=((parent1[k-1])&(~mask))|((parent2[k-1])&mask);

}

else

{



}

}

}

}

else

{

for (k=0; k<m_nChromSize; k++)

267

{

child1[k]=parent1[k];

child2[k]=parent2[k];

}

jcross=0;

}

return(jcross); }

void CGACalculator::SGAMutation(CHTYPE* child) //

{

int j,k,stop;

CHTYPE mask,temp=1;

for (k=0; k<m_nChromSize; k++)

{

mask=0;

if (k==m_nChromSize-1)

{

stop=m_nLenChrom-(k*(8*sizeof(CHTYPE)));

}

else

{

stop=8*sizeof(CHTYPE);

} for (j=0; j<stop; j++)

{

if (RandVal()<=m_pmutation)

{

mask=mask|(temp<<j);

child[k]=(child[k])^(mask);

}

}

}

}

void CGACalculator::SGASumfit()

{

int j=0;

m_sumfitness=0;

for (j=0; j<m_nSGAPopSize; j++)

{

m_sumfitness=m_sumfitness+m_popSpace/**/[j].fitness; }

}

void CGACalculator::SGACreateTmpDataFiles()

{

FILE * fp;

int i,j;

double maxx=powl(2.0,sizeof(CHTYPE)*8)-1;

//create temporary file

char curDir[MAX_PATH];

strcpy(curDir,m_strCurPath);

strcat(curDir,".lmrst");

fp=fopen(curDir,"w");

268

// m_RpData.OutData(fp);

//rel perm data

fprintf(fp,"KRD %d\n",m_nChromSize/2);

for(i=0;i<m_nSGAPopSize;i++)

{

for(j=0;j<m_nChromSize/2;j++) {

SRpData rpd;

rpd.sw=m_RpDataBody->m_RpData[j].sw;

rpd.pc=m_RpDataBody->m_RpData[j].pc;

rpd.kro=m_popSpace/**/[i].byteChrom[j]/maxx;

rpd.krw=m_popSpace/**/[i].byteChrom[j+m_nChromSize/2]/maxx;

fprintf (fp,"%lf\t%lf\t%lf\t%lf\n",rpd.sw,rpd.kro, rpd.krw,rpd.pc);

}

}

fclose(fp);

}

void CGACalculator::SGARearrange(CIndividual *critter)

{

CHTYPE *var=critter->byteChrom;

int i; for (i=0; i<m_nChromSize/2; i++)

{

CHTYPE tmp;

for (int j=i; j<m_nChromSize/2; j++)

{

if (var[j]>var[i])

{

tmp=var[i];

var[i]=var[j];

var[j]=tmp;

}

}

}

for (i=m_nChromSize/2; i<m_nChromSize; i++)

{

CHTYPE tmp; for (int j=i; j<m_nChromSize; j++)

{

if (var[j]<var[i])

{

tmp=var[i];

var[i]=var[j];

var[j]=tmp;

}

}

}

double maxx=powl(2.0,sizeof(CHTYPE)*8)-1;


269

{

critter->dblChrom[i]=critter->byteChrom[i]/maxx;

}

}

void CGACalculator::GAInitialization()

{ int i;

LMOUTPUTCLEAR(LM_TOPVIEW);

LMOUTPUTCLEAR(LM_BOTTOMVIEW);

srand((long)time(NULL)); // change randomize seed;

// Calculation of Chromosome Size

if(m_RpDataBody->m_nRpInteType==KR_TYPE_COREY)

{

m_nChromSize=NUM_COREY_VAR;


m_indAimed.dblChrom[i]=m_RpDataBody->m_fRpCoreyCtrl[i];

}

else

{

m_nChromSize = m_RpDataBody->m_nRpCount*2;

for (i=0;i<m_nChromSize/2;i++) {

m_indAimed.dblChrom[i]=m_RpDataBody->m_RpData[i].kro;

m_indAimed.dblChrom[i+m_nChromSize/2]=m_RpDataBody-

>m_RpData[i].krw;

}

}

// outputfile

CString curOutDir=m_strCurPath+".lmout"; //create restart file

//rel perm data

fpout=fopen(curOutDir,"w");

}

void CGACalculator::SpeedUp()

{

bSpeedUp=!bSpeedUp;

}

270

A-5 Class for Tube-bundle Modeling

// DITBM.h: interface for the CDITBM class.

//

//////////////////////////////////////////////////////////////////////

#if !defined(AFX_DITBM_H__679659F8_770C_4CB0_A192_26E1AFBD76DA__INCLUDED_)

#define AFX_DITBM_H__679659F8_770C_4CB0_A192_26E1AFBD76DA__INCLUDED_

#if _MSC_VER > 1000

#pragma once


#include "math.h"

#define PI 3.1415926

class CDITBM

{

public:

CDITBM();

virtual ~CDITBM();

public:

int nTotal; //Total Tube Number

double dLength; //Tube Length

int iStart, iEnd; //Starting and Ending Tube Index;

double dt; //Time Step

double dInitStep; //Initial positioning step

int tMax;

long nTimeStep; int nTimeTotal;

double fQTotal;

FILE * fout_His;

FILE * fout_Tubes;

double coef_x;

double sigma; //Interface Tension

double visw; //Wetting phase viscosity

double viso; //Nonwetting phase viscosity

double pin,pinl, pout; //Inlet and outlet pressure

double *dRadius; //Pointer to tube radius

double *pc; //Pointer to capillary pressure

double *lenLeft;

double *lenRight;//Left and right interface positions

int pw; //Debugging parameter

public:

void CreateTubeRadius(int iType, double * r, double parameter[]);

void start(); //Start Simulation

void initialization(); //initialization

void iteration(int iBegin, int nLeft, int nRight, double * lenLeft, double * lenRight);//calculation

void GaussJ(double *a, int n, double *b);

271

};

#endif // !defined(AFX_DITBM_H__679659F8_770C_4CB0_A192_26E1AFBD76DA__INCLUDED_)//

DITBM.cpp: implementation of the CDITBM class.

// Shengdong Wang

// Implementation of Tube bundle model // Limitation: Solve Saturation Explicitly Cause Lots of Convergence Problem

// The model should be change to full implicite model.

// All units are international standard

////////////////////////////////////////////////////////////////////

#include "stdafx.h"

#include "ditbm.h"

#include "stdarg.h"

#include "limits.H"

#include "string.h"

//////////////////////////////////////////////////////////////////////


//////////////////////////////////////////////////////////////////////

char fn[30]="S3_30";

CDITBM::CDITBM() {

// Numerical parameters

dt=0.0001;

tMax=LONG_MAX;

nTimeTotal=0;

nTimeStep=long(1.0/dt+1);

pin=13781;

pinl=6898;

pout=0;

fQTotal=0;

// Tube Configuration

nTotal=100;

pw=4;

sigma=0.004184285; // kerosene visw=1.77e-3;

viso=1e-3;

dLength=0.01;

dInitStep=0;

//member variance initialization

dRadius=new double [nTotal];

pc= new double [nTotal];

lenLeft = new double [nTotal];

lenRight = new double [nTotal];

}

272

CDITBM::~CDITBM()

{

delete [] dRadius;

delete [] pc;

delete [] lenRight;

delete [] lenLeft;

}

void CDITBM::start()

{

// initialization

initialization();

// open files

if((fout_Tubes=fopen(strcat(fn,"_Tubes.txt"),"w"))==NULL)

printf("Output File open failed!\n");

if((fout_His=fopen(strcat(fn,"_His.txt"),"w"))==NULL)

printf("Output File open failed!\n");

/*

// print to screen

printf("Modeling of multistep drainage using interactive tube bundle model:\n\n");

printf("Total tubes: %d\n", nTotal);

printf("IFT: %g\n",sigma); printf("Inlet Pressure: from %g to %g\n",pinl,pin);

printf("Outlet Pressure: %g\n",pout);

printf("Tube invaded: from %d to %d\n", iStart+1,iEnd+1);

printf("\n");

*/

// print to file out_tubes.txt

fprintf(fout_Tubes,"%d\t:Tubes\n", nTotal);

fprintf(fout_Tubes,"%g\t:IFT\n",sigma);

fprintf(fout_Tubes,"%g\t:Original Inlet Pressure\n",pinl);

fprintf(fout_Tubes,"%g\t:New Inlet Pressure\n",pin);

fprintf(fout_Tubes,"%g\t:Outlet Pressure\n",pout);

fprintf(fout_Tubes,"%d\t:Tube invaded from\n", iStart+1);

fprintf(fout_Tubes,"%d\t:Tube invaded to\n", iEnd+1);

fprintf(fout_Tubes,"\n");

fprintf(fout_Tubes,"%-10s\t%-10s\t%-10s\n","ID", "Radius","Pc");

for (int i=0;i<nTotal;i++) {

fprintf(fout_Tubes,"%-10d\t%-10g\t%-10g\n",i+1,dRadius[i],pc[i]);

}


fflush(fout_Tubes);

// iteration, calculation, this is the main function

iteration(iStart, 1, 1, lenLeft, lenRight);

// output and close files

fprintf(fout_Tubes,"Calculation done\n");

fclose(fout_His);

fclose(fout_Tubes);

273

}

void CDITBM::initialization()

{

iEnd =0;

iStart =nTotal;

nTimeTotal=0; //nCount, dmax,dmin,daver

double p[]={8e-6,1e-6,30e-6};

//CreateTubeRadius(2,dRadius,p);

//CreateTubeRadius(1,dRadius,p);

CreateTubeRadius(4,dRadius,p);

for (int i=0;i<nTotal;i++)

{

pc[i]=sigma*2/dRadius[i];

if(pinl-pout>pc[i]) iStart=i+1;

if(pin-pout>pc[i]) iEnd=i;

}

lenLeft [0] = dInitStep;

lenRight [0] = dInitStep;

}

void CDITBM::iteration(int iBtTube, int nLeft, int nRight, double * lenLeft, double * lenRight)

{ int nEq;

int i,j;

double * R1,* R2;

double * pcTube;

double * qo;

double * aa;

double * ax;

double * xx;

nEq=nLeft+nRight+1;

R1= new double[nEq];

R2= new double[nEq];

pcTube = new double [nEq-1];

xx = new double [nEq];

qo= new double [nEq]; aa= new double [nEq*nEq+nEq];

ax= new double [nEq*nEq+nEq];

//coeffiect

for (i=0;i<nLeft+1;i++)

{

double sum=0;

for (j=0;j<iBtTube+nLeft-i;j++) {

sum+= pow(dRadius[j],pw)/viso*PI/8;

}

R1[i]=sum;

274

sum=0;

for (j=nTotal-1;j>=iBtTube+nLeft-i;j--)

{

sum+=pow(dRadius[j],pw)/visw*PI/8;

}

R2[i]=sum;

}

for (i=0;i<nRight;i++)

{

double sum=0;

for (j=0;j<iBtTube+nRight-i;j++)

{

sum+=pow(dRadius[j],pw)/viso*PI/8;

}

R1[nEq- i -1]=sum;

sum=0;

for (j=nTotal-1;j>=iBtTube+nRight-i;j--)

{

sum+=pow(dRadius[j],pw)/visw*PI/8;

}

R2[nEq- i -1]=sum; }

// PC

for (i=0;i<nLeft;i++)

pcTube[i]=pc[iBtTube+nLeft-i-1];


pcTube[i+nLeft]=pc[iBtTube+i];

memset(aa,0,(nEq+1)*nEq*sizeof(double));

*(aa+nEq)=pin*R1[0];

for (i=1;i<nLeft+nRight;i++)

*(aa+(nEq+1)*(i+1)-1) = -(pcTube[i-1]-pcTube[i])*R2[i];

*(aa+nEq*nEq+nEq-1)=-(pout+pcTube[nEq-2])*R2[nEq-1];

//////////////////////////////////////////////////////////////////////////

*aa=R1[0];

for (i=1;i<nEq-1;i++)

{

*(aa+i*(nEq+1)+i-1)=-(R1[i]+R2[i]);

*(aa+i*(nEq+1)+i)=(R1[i]+R2[i]);

}

*(aa+nEq*nEq+nEq-3)=- R2[nEq-1];

for (int t=0;t<tMax;t++)

275

{

//////////////////////////////////////////////////////////////////////////

// coefficients

//////////////////////////////////////////////////////////////////////////

*(aa+nEq-1) = lenLeft[0];


*(aa+(nEq+1)*(i+1)-2)=lenLeft[i]-lenLeft[i-1];

*(aa+(nLeft+1)*(nEq+1)-2) = dLength - lenLeft[nLeft-1]-lenRight[nRight-1];


*(aa+(nEq+1)*(i+nLeft+1)-2)=lenRight[nRight-i]-lenRight[nRight-i-1];

*(aa+nEq*nEq+nEq-2) = lenRight[0];

// ax

memcpy(ax,aa,nEq*(nEq+1)*sizeof(double));

//xx

GaussJ(ax,nEq,xx);

//////////////////////////////////////////////////////////////////////////

qo[0]=xx[nEq-1];

for (i=1;i<nEq-1;i++)

{

qo[i]=(xx[i-1]-xx[i])*R1[i]/(*(aa+(nEq+1)*(i+1)-2));

}

qo[nEq-1]=0;


{

lenLeft[i]=lenLeft[i]+(qo[i]-qo[i+1])/pow(dRadius[iBtTube+nLeft-i-1],2)/PI*dt;

}


{

lenRight[i]=lenRight[i]+(qo[nEq-i-2]-qo[nEq-i-

1])/pow(dRadius[iBtTube+nRight-i-1],2)/PI*dt;

}

if(t%nTimeStep==0)

{

bool bOut=false;

printf("TIME:\t%g\t%g\t%g\n",(nTimeTotal)*dt,qo[0],fQTotal);

fprintf(fout_His,"TIME:\t%g\t%g\t%g\n",(nTimeTotal)*dt,qo[0],fQTotal);

fprintf(fout_Tubes,"%d\t%d\t",iBtTube,nLeft);


{

fprintf(fout_Tubes,"%g\t",lenLeft[nLeft-i-1]);

if (lenLeft[nLeft-i-1]<0) bOut=true;

}


276

fprintf(fout_Tubes,"%d\t%d\t",iBtTube,nRight);


{

fprintf(fout_Tubes,"%g\t",lenRight[nRight-i-1]); if (lenRight[nRight-i-1]<0) bOut=true;

}


if (bOut)

{

fprintf(fout_Tubes,"Error!\n");

printf("Error!\n");

}

}

nTimeTotal++;

fQTotal+=qo[0]*dt;

int iUnThrough = iBtTube+nLeft;

int iThrough = iUnThrough-1;

if (pc[iUnThrough]-pc[iThrough]<pin-xx[0])

{

char pos[]="left";

fprintf(fout_Tubes,"Oil break through at %g\t",nTimeTotal*dt);

fprintf(fout_Tubes,"Section: %s\t",pos);

fprintf(fout_Tubes,"Tube Index: %d\n", iUnThrough);

for (i=nLeft;i>0;i--)

lenLeft[i]=lenLeft[i-1];

lenLeft[0]=dInitStep;

nLeft++;

iteration( iBtTube, nLeft, nRight, lenLeft, lenRight);

break;

}

//right

iUnThrough=iBtTube+nRight;

iThrough=iUnThrough-1;

if (pc[iUnThrough]<xx[nEq-2]-pout)

{

char pos[]="right";

fprintf(fout_Tubes,"Oil break through at %g s\t",nTimeTotal*dt);

fprintf(fout_Tubes,"Section: %s\t",pos);

fprintf(fout_Tubes,"Tube Index: %d\n", iUnThrough);

for (i=nRight;i>0;i--)

277

lenRight[i]=lenRight[i-1];

lenRight[0]=dInitStep;

nRight++;


break;

}

// merge

if (dLength-lenLeft[nLeft-1]-lenRight[nRight-1]<0)

{

fprintf(fout_Tubes,"tube %d merges at %g\n",iBtTube+1,nTimeTotal*dt);

if (iEnd-iBtTube==0) // last tube merges, calculation done!

break;

else{

iBtTube++; nLeft--;nRight--;

if (nLeft==0)

{

nLeft=1;

lenLeft[0]=dInitStep;

}

if (nRight==0)

{

nRight=1; lenRight[0]=dInitStep;

}


break;

}

}

}

delete []R1;

delete []R2;

delete []pcTube;

delete []qo;

delete []aa;

delete []ax;

delete []xx;

};

void CDITBM::GaussJ(double *c, int n, double *x)

{

int i,j,t,k;

double p;

for( i=0;i<=n-2;i++)

{

k=i;

for(j=i+1;j<=n-1;j++)

if(fabs(*(c+j*(n+1)+i))>(fabs(*(c+k*(n+1)+i))))

k=j;

278

if(k!=i)

for( j=i;j<=n;j++ )

{

p=*(c+i*(n+1)+j);

*(c+i*(n+1)+j)=*(c+k*(n+1)+j);

*(c+k*(n+1)+j)=p;

} for( j=i+1;j<=n-1;j++ )

{

p=(*(c+j*(n+1)+i))/(*(c+i*(n+1)+i));

for( t=i;t<=n;t++ )

*(c+j*(n+1)+t)-=p*(*(c+i*(n+1)+t));

}

}

for( i=n-1;i>=0;i--)

{

for(j=n-1;j>=i+1;j--)

(*(c+i*(n+1)+n))-=x[j]*(*(c+i*(n+1)+j));

x[i]=*(c+i*(n+1)+n)/(*(c+i*(n+1)+i));

}

}

void CDITBM::CreateTubeRadius(int iType, double * r, double para[]) {

int i;

double daver = para[0];

double dmin= para[1];

double dmax= para[2];

switch(iType)

{

case 1: //dmin dmax 0-1

{

for (i=0;i<nTotal;i++)

{ r[i]=dmax-(dmax-dmin)/(nTotal-1)*(i);

}

break;

}

case 2: //truncated weibull

{

double Fxmax =1-exp(-pow((dmax-dmin)/(daver-dmin),2));

double dx=Fxmax/nTotal;


{

double Fx=dx/2+i*dx;

r[nTotal-i-1]=dmin+(daver-dmin)*sqrt(-log(1-Fx));

}

279

break;

}

case 3: //weibull

{

double dx=1.0/nTotal; for (i=0;i<nTotal;i++)

{

double Fx=dx/2+i*dx;

r[nTotal-i-1]=daver*sqrt(-log(1-Fx))+1;

}

break;

}

case 4: //self-defined

{

FILE *fp = fopen(strcat(fn,".txt"),"r");

fscanf(fp,"%d",&nTotal);


{

fscanf(fp,"%lf",&r[i]);

printf("%g\n",r[i]); }

fclose(fp);

}

}

}

Documents

Measurement of Relative Permeabilities at Low Saturation