ELASTO-PLASTIC DEFORMATION AND FLOW ANALYSIS IN ...

ELASTO-PLASTIC DEFORMATION AND FLOW

ANALYSIS IN OIL SAND MASSES

by

THILLAIKANAGASABAI SRITHAR

B. Sc (Engineering), University of Peradeniya, Sri Lanka, 1985

M. A. Sc. (Civil Engineering) University of British Columbia, 1989

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIESDepartment of

CIVIL ENGINEERING

We accept this thesis as conforming

to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

April, 1994

© THILLAIKANAGASABAI SRITHAR, 1994

In . presenting this thesis in partial fulfilment of the requirements for an advanced

degree at the University of British Columbia, I agree that the Library shall make it

freely available for reference and study. I further agree that permission for extensive

copying of this thesis for scholarly purposes may be granted by the head of my

department or by his or her representatives. It is understood that copying or

publication of this thesis for financial gain shall not be allowed without my written

permission.

(Signature)

_______________________

Department of Civil Engineering

The University of British ColumbiaVancouver, Canada

Date - A?R L 9 L

DE-6 (2188)

Abstract

Prediction of stresses, deformations and fluid flow in oil sand layers are important

in the design of an oil recovery process. In this study, an analytical formulation is

developed to predict these responses, and implemented in both 2-dimensional and

3-dimensional finite element programs. Modelling of the deformation behaviour of

the oil sand skeleton and modelling of the three-phase pore fluid behaviour are the

key issues in developing the analytical procedure.

The dilative nature of the dense oil sand matrix, stress paths that involve decrease

in mean normal stress under constant shear stress, and loading-unloading sequences

are some of the important aspects to be considered in modelling the stress-strain

behaviour of the sand skeleton. Linear and nonlinear elastic models have been found

incapable of handling these aspects, and an elasto-plastic model is postulated to

capture the above aspects realistically. The elasto-plastic model is a double-hardening

type and consists of cone and cap-type yield surfaces. The model has been verified

by comparison with laboratory test results on oil sand samples under various stress

paths and found to be in very good agreement.

The pore fluid in oil sand comprises three phases namely, water, bitumen and gas.

The effects of the individual phase components are considered and modelled through

an equivalent fluid that has compressibility and hydraulic conductivity characteristics

representative of the components. Compressibility of the gas phase is obtained using

gas laws and the equivalent compressibility is derived by considering the individual

contributions of the phase components. Equivalent hydraulic conductivity is derived

from the knowledge of relative permeabilities and viscosities of the phase components.

Effects of temperature changes due to steam injection are also included directly

11

in the stress-strain relation and in the flow continuity equations. The analytical

equations for the coupled stress, deformation and flow problem are solved by a finite

element procedure. The finite element programs have been verified by comparing the

program results with closed form solutions and laboratory test results.

The finite element program has been applied to predict the responses of a hor

izontal well pair in the underground test facility of Alberta Oil Sand Technology

and Research Authority (AOSTRA). The results are discussed and compared with

the measured responses wherever possible, and indicate the analysis gives insights

into the likely behaviour in terms of stresses, deformations and flow and would be

important in the successful design and operation of an oil recovery scheme.

111

Table of Contents

Abstract ii

List of Tables x

List of Figures xi

Acknowledgement xvi

Nomenclature xvii

1 Introduction 1

1.1 Characteristics of Oil Sand 4

1.2 Scope and Organization of the Thesis 8

2 Review of Literature 10

2.1 Stress-Strain Models 10

2.1.1 Stress-Strain Behaviour of Oil Sands 11

2.1.2 Stress-Strain Models for Sand 19

2.1.2.1 Elasto-Plastic Models 20

2.1.2.2 Constituents of Theory of Plasticity 22

2.1.3 Stress Dilatancy Relation 23

2.1.4 Modelling of Stress-Strain Behaviour of Oil Sand 24

2.2 Modelling of Fluid Flow in Oil Sand 25

2.3 Coupled Geomechanical-Fluid Flow Models for Oil Sands 27

2.4 Comments 30

iv

3 Stress-Strain Model Employed

3.1 Introduction

3.2 Description of the Model

3.3 Plastic Shear Strain by Cone-Type Yielding

3.3.1 Background of the Model

3.3.2 Yield and Failure Criteria

3.3.3 Flow Rule

3.3.4 Hardening Rule

3.3.5 Development of Constitutive Matrix [C8] .

3.4 Plastic Collapse Strain by Cap-Type Yielding


3.4.2 Yield Criterion

3.4,3 Flow Rule


3.4.5 Development of Constitutive Matrix [Cc]

3.5 Elastic Strains by Hooke’s Law

3.6 Development of Full Elasto-Plastic Constitutive Matrix

3.7 2-Dimensional Formulation of Constitutive Matrix

3.8 Inclusion of Temperature Effects

3.9 Modelling of Strain Softening by Load Shedding

3.9.1 Load Shedding Technique

3.10 Discussion

4 Stress-Strain Model - Parameter Evaluation and

4.1 Introduction

4.2 Evaluation of Parameters

4.2.1 Elastic Parameters

4.2.1.1 Parameters kE and n

32

32

35

37

37

42

47

48

51

55

55

57

58

58

59

61

62

• 65

• 67

68

70

72

74

74

74

75

75

Validation

v

4.2.1.2 Parameters kB and m

Evaluation of Plastic Collapse Parameters

Evaluation of Plastic Shear Parameters

4.2.3.1 Evaluation of ij and L2

4.2.3.2 Evaluation of and )

4.2.3.3 Evaluation of KG, np and R1

4.2.4 Evaluation of Strain Softening Parameters

4.3 Validation of the Stress-Strain Model

4.3.1 Validation against Test Results on Ottawa Sand

4.3.1.1 Parameters for Ottawa Sand

4.3.1.2 Validation

4.3.2 Validation against Test Results on Oil Sand

4.3.2.1 Parameters for Oil Sand

4.3.2.2 Validation

4.4 Sensitivity Analyses of the Parameters

4.5 Summary

76

4.2.2

4.2.3

79

80

82

82

83

86

87

88

91

96

96

101

107

109

114

5 Flow Continuity Equation 115

5.1 Introduction 115

5.2 Derivation of Governing Flow Equation 116

5.3 Permeability of the Porous Medium 123

5.4 Evaluation of Relative Permeabilities 124

5.5 Viscosity of the Pore Fluid Components 132

5.5.1 Viscosity of Oil 132

5.5.2 Viscosity of Water 134

5.5.3 Viscosity of Gas 136

5.6 Compressibility of the Pore Fluid Components 136

5.7 Incorporation of Temperature Effects 140

vi

5.8 Discussion. 142

6 Analytical and Finite Element Formulation

6.1 Introduction

6.2 Analytical Formulation

6.2.1 Equilibrium Equation

6.2.2 Flow Continuity Equation

6.2.3 Boundary Conditions

6.3 Drained and Undrained Analyses

6.4 Finite Element Formulation

6.5 Finite Elements and the Procedure Adopted

6.5.1 Selection of Elements

6.5.2 Nonlinear Analysis

6.5.3 Solution Scheme

6.5.4 Finite Element Procedure

6.6 Finite Element Programs

6.6.1 2-Dimensional Program CONOIL-Il .

6.7 3-Dimensional Program CONOIL-Ill

7 Verification and Application of the Analytical Procedure

7.1 Introduction

7.2 Aspects Checked by Previous Researchers . .

7.3 Validation of Other Aspects

7.4 Verification of the 3-Dimensional Version

7.5 Application to an Oil Recovery Problem

7.5.1 Analysis with Reduced Permeability .

7.6 Other Applications in Geotechnical Engineering

144

144

145

146148148149152158

158

159

162

164

166

166

167

168

168168175181

183

203208

vii

8 Summary and Conclusions 216

8.1 Recommendations for Further Research 219

Bibliography 220

Appendices 242

A Load Shedding Formulation 242

A.1 Estimation of {LO}LS 243

A.2 Estimation of {F}Ls 245

B Relative Permeabilities and Viscosities 247

B.1 Calculations of relative permeabilities 247

B.1.1 Relevant equations . 247

B.1.2 Example data . . 249

B.1.3 Sample calculations . 249

B.2 Viscosity of water 250

B.3 Viscosity of hydrocarbon gases (from Carr et al., 1954) 252

B.3.1 Example calculation 254

C Subroutines in the Finite Element Codes 258

C.1 2-Dimensional Code CONOIL-Il 258

C.1.1 Geometry Program 258

C.1.2 Main Program 259

C.2 3-dimensional code CONOIL-Ill 261

D Amounts of Flow of Different Phases 264

E User Manual for CONOIL-Il 270

E.1 Introduction 270

E.2 Geometry Program 272

viii

E.3 Main Program.275

E.4 Detail Explanations 292

E.4.1 Geometry Program 292

E.4.2 Main Program 295

F User Manual for CONOIL-Ill 304

F.1 Introduction 304

F.2 Input Data 305

F.3 Example Problem 1 319

F.3.1 Data File for Example 1 320

F.3.2 Output file for Example 1 321

ix

List of Tables

4.1 Summary of Soil Parameters 75

4.2 Soil Parameters for Ottawa Sand at Dr = 50% 94

4.3 Details of the Test Samples 101

4.4 Soil Parameters for Oil Sand 107

5.1 Parameters needed for relative permeability calculations 133

7.1 Parameters for Modelling of Triaxial Test in Oil Sand 178

7.2 Model Parameters Used for Ottawa Sand 181

7.3 Parameters Used for Thermal Consolidation 184

7.4 Parameters Used for the Oil Recovery Problem. . 192

7.5 Soil Parameters Used for the Example Problem 209

B.1 Viscosity of water between 0 and 1000 C 251

B.2 Viscosity of water below 00 C 251

B.3 Viscosity of water above 1000 C 251

D.1 Average Viscosities and Temperatures in Different Zones 266

D.2 Initial Saturations and Mobilities of Water and Oil 268

D.3 Calculation of Flow and Saturations with Time 269

D.4 Saturations and Mobilities of Water and Oil after 300 Days 269

E.1 Element Types 294

E.2 Time Increment Scheme 300

E.3 Load Increments 302

x

List of Figures

1.1 Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978) 2

1.2 In-situ Structure of Oil Sand (after Dusseault,1980) 6

1.3 Undrained Equilibrium behaviour of an Element of Soil upon Unload

ing (after Sobkowicz and Morgenstern, 1984) 7

2.1 Fabric of Granular Assemblies (after Dusseault and Morgenstetn, 1978) 12

2.2 Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseault

and Morgenstern, 1978) 13

2.3 Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987) 14

2.4 Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agar

et al., 1987) 15

2.5 Effect of Temperature on Stress-Strain Behaviour (after Agar et al.,

1987) 16

2.6 Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et al.,

1987) 18

3.1 A Possible Stress Path During Steam Injection 34

3.2 Components of Strain Increment 36

3.3 Mobilized Plane under 2-D Conditions 38

3.4 Spatial Mobilized Plane under 3-D Conditions 40

3.5 Yield and Failure Criteria on TSMp— °sMp Space . . 43

3.6 Matsuoka-Nakai and Mohr-Coulomb Failure Criteria 45

3.7 Effect of Intermediate Principal Stress (After Salgado (1990)) . . . 46

3

3.8 (TsMp /osMP) Vs — (desMp /d7sMp) for Toyoura Sand (after Matsuoka,

1983) 47

3.9 Flow Rule and The Strain Increments for Conical Yield 49

3.10 TSMp/o5Mp Vs YsMP for Toyoura Sand (after Matsuoka, 1983) . . . 50

3.11 Isotropic Compression Test on Loose Sacramento River Sand (after

Lade, 1977) 56

3.12 Conical and Cap Yield Surfaces on the o — o3 Plane 57

3.13 Possible Loading Conditions 63

3.14 Modelling of Strain Softening by Frantziskonis and Desai (1987) . . 69

3.15 Modelling of Strain Softening by Load Shedding 71

4.1 Evaluation of kE and ii 77

4.2 Evaluation of kB and m 78

4.3 Evaluation of C and p 80

4.4 Evaluation of and L 83

4.5 Evaluation of ) and it 84

4.6 Evaluation of G1, and ‘quit 85

4.7 Evaluation of K0 and np . 86

4.8 Evaluation of , and q 88

4.9 Grain Size Distribution Curve for Ottawa Sand (after Neguessy , 1985) 89

4.10 Stress Paths Investigated on Ottawa Sand 90

4.11 Variation of Young’s moduli with confining stresses 91

4.12 Plastic Collapse Parameters for Ottawa Sand 92

4.13 Failure Parameters for Ottawa Sand 93

4.14 Flow Rule Parameters for Ottawa Sand 94

4.15 Hardening Rule Parameters for Ottawa Sand 95

4.16 Results for Triaxial Compression on Ottawa Sand 97

4.17 Results for Proportional Loading on Ottawa Sand 98

xii

4.18 Results for Various Stress Paths on Ottawa Sand 99

4.19 Grain Size Distribution for Athabasca Oil Sands, (after Edmunds et

al., 1987) . . . 100

4.20 Determination of kB and m for Oil Sand 102

4.21 Plastic Collapse Parameters for Oil Sand 103

4.22 Failure Parameters for Oil Sand 104

4.23 Determination of and np for Oil Sand 105

4.24 Flow Rule Parameters for Oil Sand 106

4.25 Results for Isotropic Compression Test on Oil Sand 108

4.26 Results for Triaxial Compression Tests on Oil Sand 110

4.27 Results for Tests with Various Stress Paths on Oil Sand . . . 111

4.28 Sensitivity of Parameters C,p,A and p 112

4.29 Sensitivity of Parameters KG, np, R1 and i 113

5.1 One dimensional flow of a single phase in an element 117

5.2 Typical two-phase relative permeability variations (after Aziz and Set

tan, 1979) 125

5.3 Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979) 127

5.4 Comparison of calculated and experimental three-phase oil relative per

meability (after Kokal and Maini, 1990) 130

5.5 Comparison of calculated and experimental relative permeabilities us

ing power law functions 131

5.6 Experimental and predicted values of viscosity (after Puttagunta et al.,

1988) 135

6.1 Finite Element Types Used in 2-Dimensional Analysis 160

6.2 Finite Element Types Used in 3-Dimensional Analysis 161

6.3 Flow Chart for the Finite Element Programs 165

xiii

7.1 Stresses and Displacements Around a Circular Opening for an Elastic

Material (after Cheung, 1985) 170

7.2 Comparison of Observed and Predicted Pore Pressures (after Cheung,

1985) 171

7.3 Comparison of Observed and Predicted Strains (after Cheung, 1985) 172

7.4 Results for a Circular Footing on a Finite Layer (after Vaziri, 1986) 173

7.5 Stresses and Displacement in Circular Cylinder (after Srithar, 1989) 174

7.6 Pore Pressure Variation with Time for Thermal Consolidation (after

Srithar, 1989) 176

7.7 Undrained Volumetric Expansion (after Srithar, 1989) 177

7.8 Finite Element Modelling of Triaxial Test 179

7.9 Comparison of Measured and Predicted Results in Triaxial Compres

sion Test 180

7.10 Comparison of Measured and Predicted Results for a Load-Unload Test

in Ottawa Sand 182

7.11 Finite Element Mesh for Thermal Consolidation 184

7.12 Comparison of Pore pressures for Thermal Consolidation 185

7.13 A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991 187

7.14 Plan View of the UTF (after Scott et al., 1991) 188

7.15 Vertical Cross-Sectional View of the Well Pairs 189

7.16 Finite Element Modelling of the Well Pair 191

7.17 Temperature Contours in the Oil Sand Layer 193

7.18 Pore Pressure Variations in the Oil Sand Layer 195

7.19 Comparison of Pore Pressures in the Oil Sand Layer 196

7.20 Horizontal Stress Variations in the Oil Sand Layer 197

7.21 Vertical Stress Variations in the Oil Sand Layer 198

7.22 Stress Ratio Variations in the Oil Sand Layer 199

xiv

7.23 Comparison of Horizontal Displacements at 7 m from Wells 200

7.24 Vertical Displacements at the Injection Well Level 201

7.25 Total Amount of Flow with Time 202

7.26 Individual Flow Rates of Water and Oil 204

7.27 Total Amount of Oil Flow 205

7.28 Pore Pressure Variation for Analysis 2 206

7.29 Stress Ratio Variation for Analysis 2 207

7.30 Details of the Cases Analyzed 210

7.31 Variation of Pore Pressure Ratio for Case 1 212



A.1 Strain Softening by Load Shedding 242

B.1 Prediction of pseudocritical properties from gas gravity . . 253

B.2 Viscosity of hydrocarbon gases at one atmosphere 254

B.3 Viscosity ratio vs pseudo-reduced pressure 255

B.4 Viscosity ratio vs pseudo-reduced temperature 256

D.1 Zones involved in Fluid Flow. . 265

E.1 Nodes along element edges . . 290

E.2 Element types 293

E.3 Plane Strain Condition 296

E.4 Axisymmetric Condition . . . 296

F.1 Available Element Types . . . 306

F.2 Finite element mesh for example problem 1 319

xv

Acknowledgement

The author is greatly indebted to his supervisor Professor P. M. Byrne for his guid

ance, valuable suggestions and the encouragement throughout this research.

The author wishes to express his appreciation to the members of the supervisory

committee for reviewing the manuscript and making constructive criticisms. Appre

ciation is also extended to Mr. Jim Grieg for his valuable helps on the computer

aspects.

The author expresses his gratitude to his wife, Vasuki, for her support and toler

ance of the odd working habits of a graduate student.

The author would like to thank his colleagues in Dept. of Civil Engineering , in

particular, Uthayakumar and Hendra for sharing common interest.

Finally, the fellowship awarded by the University of British Columbia and the

research grant provided by Alberta Oil Sand Technology and Research Authority

(AO STRA) are gratefully acknowledged.

xvi

Nomenclature

B bulk modulus

B pore pressure shape function derivatives

displacement shape function derivatives

C plastic collapse modulus

CEQ equivalent compressibility

D stress-strain matrix

E Young’s modulus

e void ratio

F body force vector

f plastic collapse yield function

initial plastic shear parameter

Gt tangent plastic shear parameter

H Henry’s constant

I, 12 and 13 stress invariants

K0 plastic shear number

k Darcy’s permeability of the porous medium

kB bulk modulus number

Young’s modulus number

kEQ equivalent hydraulic conductivity

kh permeability in horizontal direction

kmi mobility of phase ‘1’

kmT total mobility

xvii

kri relative permeability of phase ‘1’

krog relative permeability of oil in oil-gas system

kr relative permeability of oil in oil-water system

relative permeability of oil at critical water saturation

permeability in vertical direction

l, l, and l direction cosines of o to the x, y and z axes

M constrained modulus

m bulk modulus exponent

mz,my and m direction cosines of o-2 to the x, y and z axes

N shape functions for pore pressures

N shape functions for displacements

n Young’s modulus exponent

n, n, and n2 direction cosines of o3 to the x, y and z axes

np plastic shear exponent

P pore pressure

Pa atmospheric pressure

capillary pressure

p plastic collapse exponent

q strain softening exponent

failure ratio

S saturation

normalized saturation

residual oil saturation

S critical water saturation

t time

U displacement vector

V volume

xviii

W plastic collapse work

Greek letters

coefficient of volumetric thermal expansion

cEQ equivalent coefficient of thermal expansion

shear strain

Kronecker delta

El, 62 and 63 principal strains

plastic collapse strains

6e elastic strains

plastic shear strains

volumetric strain

stress ratio

failure stress ratio at atmosphere

8 temperature

strain softening constant

flow rule slope

proportionality constant

p flow rule intercept

viscosity of phase ‘1’

P30,0 viscosity of oil at 30°C and at 0 gauge pressure

v Poisson’s ratio

normal stress

1, 2 and u3 principal stresses

mean normal stress

r shear stress

xix

(6m mobilized friction angle

Subscripts

f failure state

g gas phase

j partial derivative with respect to j

MP mobilized plane

o oil phase

SMP spatial mobilized plane

ult ultimate state

w water phase

Superscripts

c plastic collapse condition

e elastic condition

plastic shear condition

xx

Chapter 1

Introduction

The oil contained in oil sand deposits in northern Alberta is one of the major resources

in Canada. These deposits underlie an area of about 32,000 square kilometres with

estimated in-place reserves of 146.5 million cubic meters (Mosscop, 1980). Much of the

oil exists as high viscosity bitumen in Arenaceous Cretaceous formations, primarily in

the Athabasca oil sand deposits (see figure 1.1). Approximately 5% of these deposits

are found at depths less than 50 m and the rest are encountered at depths from 200

to 700 m.

Oil recovery schemes involve open pit mining in the shallow oil sand formations,

and in-situ extraction techniques such as tunnels and well-bores in the deep oil sand

formations. In the in-situ extraction procedures some form of heating is often required

as the very high viscosity of the bitumen makes conventional recovery by pumping

impractical. In-situ thermal methods such as steam injection through vertical well-

bore have been used and are relatively effective for the recovery of heavy oil from

deep seated formations. There have been, however, numerous well casing failures and

instability problems reported during field injection trials. During steam injection,

high pore fluid and stress gradients are created around the well-bore which can lead

to the instability and collapse of the well casing. Therefore, to understand the mech

anisms involved and to design these oil recovery schemes rationally and economically,

analyses which capture the complex engineering characteristics of the oil sand are

necessary.

Analyzing the problems related to oil sands is somewhat different from analyzing

1

Chapter 1. Introduction 2

United States of Amenca

Figure 1.1: Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978)

Northwest Territories


a general geotechnical problem because of the nature of the oil sand and the recovery

process involved. Oil sand comprises four phases; solid, water, bitumen and gas,

whereas, a general soil consists of three phases; solid, water and air. The presence of

bitumen and gas makes the analytical procedures for oil sands different and difficult.

Oil recovery by steam injection will cause changes in temperature and their effects

are also of prime concern. The changes in temperature induce changes in volume and

pore fluid pressure which in turn affect the engineering properties such as strength,

compressibility and hydraulic conductivity. When there is an increase in temperature,

if the volume change of the pore fluid components is greater than that of the voids

in the soil skeleton, there will be an increase in pore pressure and consequently a

reduction in effective stress. The effective stresses may become zero and liquefaction

may occur, if the oil sand is subjected to rapid increase in temperature and if an

undrained condition prevails.

The deformation and flow behaviour of oil sand is governed by several factors.

However, it can be categorized into two major constituents; the behaviour of pore

fluids and the behaviour of sand skeleton. An analytical model for the oil sand was

first developed by Harris and Sobkowicz (1977); It was later extended by Byrne

and Grigg (1980), Byrne and Janzen (1984) and Byrne and Vaziri (1986). However,

these analytical models consider a linear or nonlinear elastic behaviour for the sand

skeleton. Oil sand is very dense in its natural state and shows significant dilation

upon shear. The linear and nonlinear elastic models are not capable of modelling the

dilation effectively. Furthermore, steam injection and subsequent recovery will lead

to loading and unloading cycles and for realistic modeffing an elasto-plastic model

is necessary. In this study, a double hardening elasto-plastic model is postulated for

the sand skeleton based on the models by Nakai and Matsuoka (1983) and by Lade

(1977), and it is very effective in handling the dilation.

With regard to the pore fluid behaviour, Byrne and Vaziri (1986) considered the


individual contributions of the pore fluid components in the compressibility but not in

the hydraulic conductivity. In this research work, the relative permeabilities of water,

bitumen and gas are considered and an equivalent hydraulic conductivity is derived

to model the pore fluid behaviour appropriately. The equivalent compressibility term

as proposed by Byrne and Vaziri (1986) is also included.

The effects of temperature changes in stresses and volume changes have been di

rectly included in the governing equilibrium and flow continuity equations. It should

be noted that the equation of thermal energy balance is not considered in the analyt

ical model. However, the temperature-time history which is obtained from a separate

heat flow analysis or by some other means is considered as an input to the analytical

model and, the effects of these temperature changes on the stress-strain behaviour

and the fluid flow are evaluated.

An analytical procedure considering all these aspects has been developed and

incorporated in the 2-dimensional finite element code CONOIL-Il. In order to analyze

the three dimensional effects a new 3-dimensional finite element code CONOIL-Ill is

also developed.

1.1 Characteristics of Oil Sand

Since the oil sand is different form a general soil, it is appropriate to present some

brief descriptions about its unusual characteristics. Oil sand can be considered as a

four phase geological material comprising solid, water, bitumen and gas. The two

dominant physical characteristics of the oil sand are the quartz mineralogy and the

large quantity of interstitial bitumen. The quartz grains of the oil sand are 99% water

wet as the water phase forms a continuous film around it. A larger portion of the

pore space is filled with bitumen and since bitumen and water form continuous phases,

gas can only exists in the form of discrete bubbles (free gas). However, significant

quantities of gas can also exist in the dissolved state in the pore fluid. An illustration


of oil sand structure (Dusseault, 1980) is shown in figure 1.2.

In its natural state, oil sand is very dense, uncemented, fine to medium grained

and exhibits high shear strength and dilatancy. It shows low compressibility charac

teristics compared to normal dense sand of similar mineralogy. The extremely high

viscosity of bitumen makes the effective hydraulic conductivity very low and causes

the oil sand to behave in an undrained manner.

Another unusual characteristic of oil sand is its behaviour upon unloading. Be

cause of the very low effective hydraulic conductivity, oil sand behaves in an undrained

manner, however, it responds quite differently compared to the undrained behaviour

of a normal sand. Above the liquid-gas saturation pressure (U119), oil sand behaves

like a normal sand (path I of figure 1.3). A decrease in confining stress will result in a

decrease in pore pressure and the effective stress remains constant. When the level of

confining stress decreases below the liquid-gas saturation pressure, the dissolved gas

in the pore fluid comes out of solution and causes the pore fluid to become very com

pressible. At this point, the soil matrix commences to take the load and the effective

stress decreases while the pore pressure stays constant (path J). As the effective stress

decreases, the soil skeleton compressibility increases and becomes comparable to the

pore fluid compressibility. Then, the pore fluid takes the load and the pore pressure

starts to decrease again (path K). At some stage, the effective stress becomes zero

and the physical consequences of this process are significant increase in volume and

a marked reduction in shear strength. Plots of pore pressure versus total stress for

saturated (path M), unsaturated (path L) and gassy soils (path J-K) are shown in

figure 1.3. A comprehensive study of the gas exsolution phenomenon upon unloading

can be found in Sobkowicz and Morgenstern (1984).

Tj

I-.

(b I-.

U) I

I-.

0

CC

CC

--

m

rcn

oa;

-’ -C

—m

0o

m

-

a

C

-

,oo a..

C-


U

Uj/gD(I,C,,w0

LAJ

00

I ..°

atm

Figure 1.3: Undrained Equilibrium behaviour of an Element of Soil upon Unloading(after Sobkowicz and Morgenstern, 1984)

//..—. u=o,

____,_,_

o-c=o J

IN SITUSTRESS

/

TOTAL STRESS

CEGASSEDPORE FLUID

0

FINE SOIL


1.2 Scope and Organization of the Thesis

The objective of this study is to present a better analytical formulation for the stress,

deformation and flow analysis in oil sands, from a geotechnical point of view. The

analytical model is developed on the premise that the oil sand is a four phase material

comprising solid, water, bitumen and gas.

In developing the analytical formulation the key issues are; a stress-strain model

for the sand skeleton, the compressibility and permeability characteristics of the three-

phase pore fluid, the effects of temperature, and the overall analytical and finite

element procedure. Discussions on these issues highlighting the previous research

works in the literature are given in chapter 2.

The main feature in a deformation analysis is the stress-strain model employed. In

this study, a double-hardening elasto-plastic model is postulated. The fundamental

details of the stress-strain model and the development of the constitutive matrix

using plastic theories are described in chapter 3. The parameters required for the

stress-strain model, procedures to obtain them, the sensitivity of these parameters

and the verification of the stress-strain model against laboratory results are presented

in chapter 4.

One of the major concerns in the analytical formulation presented in this study

is the modelling of the multi-phase fluid. Chapter 5 describes the development of the

flow continuity equation, considering the contributions from all the fluid phase com

ponents, in detail. Inclusion of temperature effects in the flow continuity equation is

also given in this chapter. Inclusion of the temperature effects in stress-strain relation

is explained in chapter 3. Details concerning the overall analytical procedure and its

implementations in 2-dimensional and 3-dimensional finite element formulations are

given in chapter 6.

Verifications and the validations of the developed formulation are presented in

chapter 7. Some specific problems where closed form solutions are available and some


laboratory experiments are considered and the results are compared. Application to

an oil recovery process by steam injection is presented and the results are analyzed

in detail. Possible applications of the developed formulation for general geotechnical

problems are discussed and an example problem is also given.

Chapter 8 summarizes the important findings of this research work. Some com

ments on the aspects which warrant further investigation are also stated in this chap

ter.

Chapter 2

Review of Literature

The research work carried out in this study can be broadly classified under the fol

lowing topics; stress-strain model for the oil sand, modelling of flow characteristics of

the three-phase pore fluid; and the analytical and finite element formulations. There

fore, it is appropriate to present a review on the previous research works under these

subheadings. The intention of the literature review presented in this chapter is not

to critically assess each and every research work but to give an overall picture, and

to set the stage to discuss the work carried out in this study.

2.1 Stress-Strain Models

The stress-strain behaviour of the oil sand skeleton is essentially the stress-strain

behaviour of a dense sand. This conclusion was not widely accepted until the com

pletion of series of research programs at the University of Alberta in the late 1970s

and in 1980s. In particular, the perception of bitumen as a cementing material was

widely held until the last decade, as many geologists and petroleum engineers failed

to recognize the geomechanical behaviour of the sand skeleton. It is now recognized

that the oil sands must be considered as a particulate material and its behaviour can

be described by an appropriate stress-strain model. Before going into a detailed re

view of the stress-strain models, it will be useful to describe the observed stress-strain

behaviour of oil sands. The next subsection summarizes the stress-strain behaviour

of oil sands in laboratory experiments.

10

Chapter 2. Review of Literature 11

2.1.1 Stress-Strain Behaviour of Oil Sands

Dusseault (1977) showed that the Athabasca oil sands have an extremely stiff struc

ture in the undisturbed state, accompanied by a large degree of dilation when loaded

to failure and subsequent yield. This was attributed to its extreme compactness which

provides extensive grain-to-grain contact. The grain orientations of the oil sand are

compared schematically to ideal and rounded sand grains in figure 2.1. The angular

ity of the Athabasca sand grains illustrate why significant dilation can be expected

as the sand is sheared.

Dusseault and Morgenstern (1978) studied the shear strength of Athabasca oil

sands and stated that the Mohr-Coloumb failure envelope is not a straight line but

curvilinear. The residual and peak shear strengths measured in direct shear tests are

shown in figure 2.2. The curvilinear nature is said to be due to the dilatancy and the

grain surface asperity.

Agar et al. (1987) carried out extensive testing on Athabasca oil sand to study the

effects of temperature, pressure and stress paths on shear strength and stress-strain

behaviour. Figure 2.3 shows the effect of stress paths on stress-strain behaviour.

Six different triaxial stress paths were investigated which are shown in figure 2.3(a).

Typical stress-strain curves for these stress paths are plotted in figure 2.3(b). These

curves illustrate the influence of stress paths on peak deviator stress and stress-strain

behaviour. It can be seen from the figure that the dilatancy is more pronounced on

certain stress paths (see paths B and C), and at lower effective confining stress than

at higher stress levels (compare paths C and D).

Figure 2.4 shows the shear strength of Athabasca oil sand compared to dense

Ottawa sand. The shear strength of oil sand is greater than that of dense Ottawa

sand at lower effective confining stress levels. However, at higher stress levels, the

strengths of these two materials apparently converge.

Figure 2.5 shows the effect of temperature for a drained triaxial compression test.


(a) Hexagonal close-packed spheres.Point contacts.

(b) Densely packed rounded sand.Point contacts, with some straight contacts (arrows)

(c) Athabasca oil sandPoint contacts, with many straight and

interpenetrative contacts (arrows)

Figure 2.1: Fabric of Granular Assemblies (after Dusseault and Morgenstern, 1978)


a.

U,0

L0

(U0-c(0

Figure 2.2: Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseaultand Morgenstern, 1978)

Three different samples

o o Peak strength• Residual strength

•

0 200 400 600 800

o normal stress, kPa

1000 1200

(a) Various Stress Paths

b

>0

(b) Stress-Strain Behaviour

Figure 2.3: Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987)

I;


28

24

20

16

a

20

16

12

8

4

00.5

12

0 4 8 12

./7O1 (MPa)

16—0.5

0.5 1.0 1.5

e (%)


60

a)

aUCDU,

ina,40

U)CI

U

-C‘/,— 300a)U)C

20

Figure 2.4: Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agar etal., 1987)

. LEGEND

D

.Athobasca Oil Sand v

ATHABASCA OIL SAND (This Study)OTTAWA SAND (This StudyDIJSSEAUT & MORGENSTERN(1978)SOBKOWICZ (1982)DUNCAN & CHANG (1970)

.

1 2 3 4 5 6 7 8

Effective Confining Stress c (MPa)


The effect of temperature on the stress-strain behaviour does not seem to be signifi

cant. For some other stress paths, it appeared that the temperature has considerable

influence on the stress-strain behaviour. However, Agar et al (1987). concluded that

the differences in the stress-strain behaviour at various temperatures are small. They

attributed the observed differences to the disturbances in sampling and the mate

rial heterogeneities. The test results appeared to be far more sensitive to sample

disturbances than heating.

0 0.5 1.0

e (%)

Figure 2.5: Effect of Temperature on Stress-Strain Behaviour (after Agar et al., 1987)

Kosar (1989) continued Agar’s work and tested various oil sands and noted some

essential differences in the geomechanical behaviour. Kosar claimed that in addition

20

16

12

4

0

04

1.5 2.0


to temperature, pressure and stress paths, the grain mineralogy, geological environ

ment of deposition and the geological history are the major factors affecting the

geomechanical behaviour. The maximum shear strength and the stress-strain moduli

of Athabasca oil sands are much greater than those of Cold Lake oil sand reflecting

the grain mineralogy and the geological factors. Athabasca oil sands consist of a

uniformly graded, predominantly quartz sand, whereas, Cold Lake oil sands contain

several additional minerals which are weaker. Figure 2.6 shows typical drained tn-

axial compression test of these two oil sands. Athabasca oil sand exhibits dilatant

behaviour but the Cold Lake oil sand does not. In the Athabasca oil sand, the increase

in volume change during shear is also accompanied by strain softening behaviour in

the post peak region. The Cold Lake oil sand shows contractive behaviour and the

reason for this is the presence of weaker minerals. The weaker minerals are prone

to grain crushing at the applied stress levels. Because of these weaker minerals, the

geomechanical behaviour of Cold Lake oil sand changes with temperature as well.

Athabasca oil sands, on the other hand, do not show significant changes in behaviour

at different temperatures.

Wong et al. (1993) pointed out that testing of oil sand samples should include

some important stress paths which are expected to be encountered in the field. They

carried out detailed testing on Cold Lake oil sand which includes stress paths with

increasing and decreasing pore pressures under constant total stress. This results in

load-unload-reload stress paths in terms of effective stress ratio. They identified four

different modes of granular interactions namely; contact elastic deformation, shear

dilation, rolling and grain crushing for the observed geomechanical behaviour. They

also noticed grain crushing in Cold Lake oil sand when the effective confining stress

increased above 8 MPa.


6-

Mairjmshearsfl-ength = 16.9 MPa

5.I

a—4.OUPa

4 / Athabasca (Agar. 1984)

0

3• : Mi,m shaer strength • 6.9 MPa

I2. :

Athabasca £ - 2200 MPa

‘7 CoidLake

S

Axial Strain (%)

Figure 2.6: Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et aL,1987)

Chapter 2. Review of Literature i9

Therefore, the modelling of oil sand behaviour should include two significant fea

tures; non-recoverable strains and dilatancy. A realistic model must take the deforma

tion history into account, particularly if the stresses are to be cycled through loading

and unloading. The elasto-plastic formulation incorporates these features naturally.

There are a number of elasto-plastic stress-strain models available for sands in the

literature and a brief review of those are presented next.

2.1.2 Stress-Strain Models for Sand

A number of models have been proposed in the literature for the stress-strain be

haviour of sand. Most of them make use of the well developed classical theories of

elasticity and plasticity either separately or in a combined form. These theories are

based on the observations made on materials that can be described in the context of

continuum mechanics. To adopt these theories to model the stress-strain behaviour

of sand, they have to be modified. Different modifications are made to capture dis

tinguished features of sand behaviour and thus, different models are proposed by

different researchers. One of the difficult features of sand behaviour to model has

been the shear induced volume change.

Basically, constitutive models can be classified into two categories; linear or in

cremental elastic models and elasto-plastic models. In the theory of elasticity, the

state of stress is uniquely determined by the state of strain so that the stress-strain

response of an elastic models is independent of the stress path. The simplest elastic

model would be the isotropic linear elastic model which requires only two material

parameters. Incremental elastic models (Duncan and Chang (1970), Duncan et al.

(1980)) are the most commonly used because they can capture the nonlinearity and

are easy to use. Essentially, the incremental elastic models also require only two pa

rameters when analyzing a load increment. However, to update these two material

parameters with stress levels and to model the nonlinearity additional parameters are


necessary. Generally, in elastic models, the shear and normal stresses and strains are

uncoupled from each other. Byrne and Eldrige (1982) incorporated the shear volume

coupling effects in the incremental elastic models using a stress dilatancy equation.

Reviews of the existing elastic and elasto-plastic constitutive models are avail

able in the literature as state-of-the-art papers, special workshops and international

symposia. Ko and Sture (1980) provided a clear summary of the most important

models as of 1980 and described the methods needed to obtain their coefficients.

Chen (1982) described and analyzed what is meant by different levels of elasticity.

He also described some of the elasto-plastic models most commonly used for soils.

Scott (1985) presented a very lucid treatise on plasticity and stress-strain relations.

A series of workshops held at McGill University (1980), University of Grenoble (1982)

and Case Western University (1987) and the international symposia (ASCE sympo

sium, Florida, 1980; International Symposium, Deift, 1982) provide better insights

into the different stress-strain models.

Since an elasto-plastic model is proposed in this study, a brief review of elasto

plastic models and the related theories are presented next.

2.1.2.1 Elasto-Plastic Models

The theory of plasticity has been developed on the basis of observed stress-strain

behaviour of metals. Since soils exhibit plastic non-recoverable strains, the theory

of plasticity provides an attractive theoretical framework for the representation of

the stress-strain behaviour of soils. However, there are major differences such as the

presence of voids and the tendency for volume change during shear that distinguish

soils from metals (Lade, 1987).

In the elasto-plastic models, the strain increment is decomposed into an elastic

component and a plastic component. The amounts of elastic and plastic strains will

vary with the level of loading and unloading. The elastic strain increment is obtained


using the theory of elasticity and the plastic strain increment is obtained from the

theory of plasticity.

Drucker et al. (1955) were the first to treat soils as work hardening materials.

The yield surface that they postulated consists of a Mohr-Coloumb surface and a cap

which passes through the isotropic compression axis. Most of the elasto-plastic models

evolved from this study. The Cam-Clay model (Roscoe et al., 1958) introduced the

concept of critical state and presented an equation for the yield surface considering

energy dissipation. Prevost and beg (1975) used the critical state line concept in

their model, but defined two yield surfaces, one for volumetric and shear deformation

and the other for shear deformation alone. The Cam-Clay model has been used in

one form or another by many researchers, for example, Adachi and Okamo (1974),

Pender (1977), Nova and Wood (1979) and Wilde (1979).

The models of Lade and Duncan (1975) and Matsuoka (1974) contain features of

the Mohr-Coloumb criterion and incorporate the influence of intermediate principal

stress. The yield and failure surfaces are assumed to be described by similar functions

so that both surfaces have similar shapes. Lade (1977) introduced a yielding cap in

order to control the plastic volumetric strain making his model a double hardening

one. Vermeer (1978) also used a double hardening model. He divided the plastic

strain into two parts; one is described by means of a shear surface and the shear

dilatancy equation and the other is strictly volumetric.

Multiple yield surface plasticity theory has also been used to predict soil behaviour

(Iwan(1967), Prevost (1978, 1979)). In computations, this theory requires that the

positions, sizes and plastic moduli of each of the yield surfaces be stored for every

integration point, which is very tedious and therefore not very commonly used.


2.1.2.2 Constituents of Theory of Plasticity

In the theory of plasticity, existence of a yield function, a potential function and

a hardening function are necessary to relate the plastic strain increments to stress

increments mathematically. The yield function defines the stress conditions causing

plastic strains. The yield surface represented by the yield function encloses a volume

in the stress space, inside of which the strains are fully recoverable. Only stress

increments directed outward form the yield surface cause plastic strains. A stress

increment directed outward from the yield surface causes an expansion or translation

of the yield surface. During yielding, the state of stress remains on the yield surface

which is known as the consistency condition. A state of stress outside the yield surface

is not possible.

The direction of plastic strain increment is defined by the potential function which

is referred to as flow rule. If the potential function and the yield function are the

same, the flow rule is said to be associative. If these functions are different, then the

flow rule is non-associative.

The amplitude of the plastic strain increment is specified by the hardening func

tion. In plasticity, two types of hardening have been distinguished; isotropic hardening

and kinematic hardening. In a model undergoing isotropic hardening, the yield sur

face expands radially about the fixed axes. When the yield surface translates without

changing its size, the model undergoes kinematic hardening.

Once the constituents of the theory of plasticity are defined, the plastic strain

increment, can be calculated from,

=— n (2.1)

where,

Lo- - applied stress increment

n, - vector defining the unit normal to yield surface at the stress point


- vector defining the unit normal to potential surface at the stress point

H - plastic resistance

2.1.3 Stress Dilatancy Relation

The stress dilatancy theory derived from theoretical considerations has been used

extensively in stress-strain modeffing of sand. The stress dilatancy theory proposed

by Rowe (1962,1971) can be considered a remarkable effort to explain the shear de

formation behaviour. After Rowe, a number of other researchers published theories

to model the dilatancy following different approaches (Murayama (1964), Matsuoka

(1974), Oda and Konishi (1974), Nemat-Nasser (1980)). A noticeable difference be

tween Rowe’s theory and the other theories is that Rowe’s theory is independent of

the spatial distribution of interparticle contacts. Rowe’s theory considers that sliding

occurs on certain favourably oriented contact planes. The orientation of the sliding

planes will be such as to minimize the rate of dissipation of energy in sliding friction

between particles with respect to energy supplied.

Matsuoka (1974) developed the stress dilatancy relationship through a microscopic

point of view. He carried out shear tests by using cylindrical rods to model the

shearing mechanism of soil particles. From the fundamental measurements of the

angle of the interparticle contact, interparticle force and the angle of interparticle

friction, he developed a relationship between the shear resistance and the dilatancy.

Lade’s (1977) model incorporates the dilatancy through a empirical relation ob

tained by curve fitting. The equation relates a dilation parameter to the amount of

plastic work.

Nemat-Nasser (1980) presented an equation to describe the volumetric behaviour

of soil upon shearing which is based on the mechanics of the relative motion of the

grains at the micro level. The equation was obtained by considering the rate of

frictional losses and the energy balance.


2.1.4 Modelling of Stress-Strain Behaviour of Oil Sand

Modelling of the geomechanical behaviour of oil sand along with the pore fluid be

haviour, so as to describe gas exsolution and other related aspects was first presented

by Harris and Sobkowicz (1977). They considered a linear elastic model for the sand

skeleton behaviour.

A nonlinear elastic model with shear dilation was proposed by Byrne and Grigg

in 1980 to model the oil sand skeleton behaviour. Their model is based upon an

equivalent elastic analysis using a secant modulus and a single step loading. This was

subsequently extended by Byrne and Janzen (1984) who used an incremental tangent

modulus rather than a secant modulus. Vaziri (1986) basically used the same model

as Byrne and Janzen to represent the stress-strain behaviour of oil sand.

In the above cited references, the dilative behaviour of the material is incorporated

through a procedure borrowed from thermoelasticity. This method involves applying

equivalent nodal loads to predict the correct volume changes. Srithar et al. (1990)

pointed out that the thermoelastic approach encounters shortcomings specially in a

consolidation type of analysis. It predicts unrealistic oscillating results when large

time steps are considered. Furthermore, the computer algorithm necessitates two

levels of iterations; one for stress calculations, and the other for shear induced volume

change corrections. Wan et al. (1991) stated that the method of including dilation

through thermoelastic approach may lead to a decrease in effective mean normal stress

0m while in a pressuremeter test, dilation is always accompanied by an increase in

Tortike (1991) stated that cyclic steam simulation imposes cyclic loads on the oil

reservoir. He further suggested that a realistic stress-strain model should have the

capability to model the loading and unloading behaviour. He adopted Hinton and

Owen’s (1977) elasto-plastic model which includes a Mohr-Coloumb failure envelope

and an associated flow rule.


Wan et al. (1991) also recognized the cyclic loadings caused in the recovery

process by steam injection and proposed an elasto-plastic model for oil sand. Their

model is based on Vermeer’s (1982) elasto-plastic model. They used Matsuoka and

Nakai (1982) equation to represent the yield and failure surfaces, and a Ramberg

Osgood type hardening function. The model involves a non-associated flow rule and

the potential function is based upon Rowe’s stress dilatancy equation. However,

their model cannot predict the plastic volumetric behaviour for stress paths involving

compression with constant stress ratio.

2.2 Modelling of Fluid Flow in Oil Sand

In petroleum reservoir engineering, multiphase fluid flow has been analyzed by a

number of researchers without consideration of the geomechanical behaviour of the

oil sand matrix. The first clear attempt to use a finite element method for fluid

flow in porous medium that appeared in petroleum engineering was by Javandel and

Witherspoon (1968). They considered a single phase isothermal fluid flow through

an isotropic homogeneous porous medium. The numerical solutions were compared

with the analytical solutions for infinite, bounded and layered radial systems with

constant flow rate or pressure constraints and were found to be in good agreement.

Solutions for two-phase isothermal fluid flow problems using variational and finite

element methods were presented by various researchers (for example: Settari and

Price, 1976; Huyakorn and Pinder, 1977a; Spivak et al., 1977; Settari et al., 1977;

Lewis et al., 1978; White et al., 1981). Spivak et al. (1977) presented a formulation

for multi-dimensional, two-phase, immiscible flow using variational method. They

compared variational and finite difference methods and concluded that the variational

method is more efficient than the finite difference method. Galerkin’s procedure was

successfully applied to the analytical formulation of the governing equations in the

presence of favourable and unfavourable mobility ratios. Numerical dispersion at the


front was less in both cases than with the finite difference method. Also, in the

variational method, grid orientation effects were not observed.

Guibrandsen and Wile (1985) used Galerkin’s scheme directly for two-dimensional,

two-phase flow. The Newton-Raphson method was used to linearize the weighted

form, which was approximated in time by backward Euler differences. The spatial

domain was divided into rectangles and approximated by byliner functions. A sharper

front was noticed when the capillary pressure was not simply a constant function of

saturation, but oscillations in the solution still occurred downstream in the front.

However, no serious solution instability occurred.

Ewing (1989) proposed a mixed element scheme for solving pressure and velocity

in miscible and immiscible two-phase reservoir flow problems. Velocity was chosen

as the primary variable to ensure that it remains a smooth function throughout the

domain, despite step changes in reservoir properties governing the flow.

Faust and Mercer (1976), Huyakorn and Pinder (1977b), Voss(1978) and Lewis et

al. (1985) are some of the researchers who analyzed two-phase fluid flow under non-

isothermal conditions. Lewis et al. (1985) used the Galerkin method to solve the water

flow and energy equations in two dimensions. Byliner elements were used to model

hot water flooding for thermal oil recovery. Linear and higher order elements were

used to model the heat losses from the reservoir in all directions. Artificial diffusion

was introduced along streamlines to negate any grid orientations. The solutions were

found efficiently at the end of each time step using an alternating direct solution

algorithm.

The solution for multiphase fluid flow problem using finite elements was first pre

sented by McMichael and Thomas (1973). They analyzed a three-phase isothermal

flow in a two dimensional domain subdivided into linear finite elements. Reportedly,

no difficulties were encountered in finding the solution at each time step. The evalua

tion of all the reservoir properties at each quadrature point for numerical integration


appeared to obviate the need for upstream weighting for numerical stability. However,

this result is not in accordance with later studies of the multiphase flow problem by

the finite element method.

Tortike (1991) presented a detailed literature review on modelling of fluid flow

under isothermal and non-isothermal conditions. He solved the three-phase ther

mal flow problem using finite differences. He also tried to develop a fully coupled

geomechanical fluid flow model, but was not successful as the results were unstable.

It appears that in most of the research work in petroleum engineering, the flow

in oil sand is modelled by two phase system (water and bitumen) with reasonable

accuracy. However, these models solve only the fluid flow problem and do not con

sider the geomechanical behaviour. Therefore, the effects of stress distribution and

deformation in the oil sand matrix are not included in these models.

2.3 Coupled Geomechanical-Fluid Flow Models for Oil Sands

Some models in petroleum reservoir engineering include the effects of deformations

in oil sand matrix through poroelasticity. Geertsma (1957) combined the approaches

of Biot (1941) and Gassman (1951) to develop the equations of poroelasticity in a

more straightforward manner. He clearly defined and related the rock bulk and pore

compressibilities, and described the boundary conditions and procedure to determine

the correct parameters defining the compressibilities. Geertsma (1966) reviewed the

applications of poroelasticity in petroleum engineering. An analogy is presented be

tween poroelastic and thermoelastic theories, to take advantage of the many solutions

under different boundary conditions that have already been published. The concept

of the nucleus of strain for volume elements was described and it has been applied

to predict surface displacements. It should be noted however, the poroelastic theory

does not consider the effect of stress distribution through a porous medium.

Raghavan (1972) derived a one dimensional consolidation equation coupled with


fluid flow and compared his results with Terzhaghi’s solution. The general solution

was obtained from the partial differential equations describing the flow of fluid and

material displacement using a transform to convert it to an ordinary differential equa

tion. He also presented a significant review of the literature to that time.

Finol and Farouq Ali (1975) analyzed a two-phase flow model using finite differ

ences which included the effects of compaction on fluid flow and the prediction of

surface displacements. The problem was formulated by two discretized equations for

oil and water flow, and one analytical equation for poroelasticity which was numer

ically integrated. The variation of permeability and porosity was considered in the

analysis as the effect of compaction on ultimate recoveries. The authors concluded

that the ultimate recoveries of oil increased with compaction.

Harris and Sobkowicz (1977) derived an analytical model from a more geotechni

cal point of view. They presented a coupled mathematical model for the fluid flow

and the geomechanical behaviour of oil sand. The model was developed mainly to

analyze excavations, immediate foundation settlements and underground openings in

oil sands. Since these scenarios involve short term conditions, and because of the

high viscosity of the bitumen, their model was only concerned with the undrained

response. The authors claimed that the short term conditions govern the design in

the above circumstances.

Byrne and Grigg (1980), and Byrne and Janzen (1984) extended Harris and

Sobkowicz’s formulation. Byrne and Janzen also included the fully drained condition

in their analysis. Their analysis procedure involved an effective stress approach in

which the stresses in the sand skeleton were computed using a finite element scheme.

The pore fluid pressures were computed from the gas laws together with volume

compatibility between fluid and skeleton phases.

Vaziri (1986) coupled the equilibrium equation and the flow continuity equation

and analyzed the transient conditions as a consolidation problem. He included the


thermal effects on stresses, hydraulic conductivity and volume change and presented

a two dimensional finite element formulation. The fluid flow was considered as a

single phase one. The effects of different phase components on compressibility were

taken into account by means of an equivalent compressibility. Vaziri followed the

thermoelastic approach to model temperature effects. This approach appeared to

predict unrealistic oscillating results. Srithar (1989) incorporated the temperature

induced stresses and volume changes directly in the governing equilibrium and flow

continuity equations and presented a better formulation of Vaziri’s model.

Dusseault and Rothenberg (1988) reviewed the effect of thermal loading and pore

pressure changes around a wellbore on dilation and permeability. They described

the physical process of deformation in terms of particulate media. They concluded

that effective water permeability would increase one or two orders of magnitude with

dilation as the thickness of the water film coating the grains would increase by a

factor of two. The authors continue to document the changes likely from shear failure,

including the localization of shear and the growth of the shear zone from the edge of

a hydraulic fracture due to the altered stress state and the increased pore pressures.

Settari (1988), Settari et al. (1989) described a model to quantify the leak-off

rates for fracture faces in oil sand. The authors used a nonlinear elastic model and a

two-phase isothermal flow in their analysis. The nonlinear response was shown to give

a different pressure distribution than the linear elastic one. Settari (1989) extended

their earlier model to thermal flow.

Fung (1990) described a control volume finite element approach for coupled isother

mal two-phase fluid flow and solid behaviour. He adopted a hyperbolic stress-strain

law with Rowe’s stress dilatancy theory.


Schrefler and Simoni (1991) presented the equations for two-phase flow in a de

forming porous medium, which are, a linear momentum balance for the whole mul

tiphase system and continuity equations for solid-water and solid-gas systems. Aux

iliary equations included water saturation constraint (S + S9 = 1), and the ef

fective stress equation. Three combinations of solution variables were considered ({ U, F,(,, P}, {U, P,P9}, {U, P, S}). Among these the best convergence was found

when using the combination of { U, P, P9}.Tortike (1991) attempted to develop a fully coupled three dimensional formulation

for thermal three-phase fluid flow with geomechanical behaviour of oil sand. He was

not successful and concluded that the formulation is very tedious and too unstable.

As a second approach, he carried out separate analyses of soil behaviour using finite

elements and thermal fluid flow by finite difference and combined the results. He

found the second approach to be successful and useful.

Recently Settari et al. (1993) presented a model to study the geomechanical

response of oil sand to fluid injection and to analyze the formation parting in oil

sand. They used a generalized form of the hyperbolic model for material behaviour.

They also approximated the multiphase fluid flow by means of an effective hydraulic

conductivity term. The value of the effective hydraulic conductivity term was found

by matching the results of the single phase model with the rigorous multiphase flow

model. The authors further examined the behaviour of the constitutive model at low

effective stress ranges and concluded that the frictional properties at low effective

stresses control the development of the failure zone around the injection well and the

fractures.

2.4 Comments

The following are some of the important facts that can be extracted from the literature

review. In the models reviewed, except for Tortike (1991), all other models use elastic


models. Cyclic loads are more common in the oil recovery procedures such as the

cyclic steam simulation. The cyclic loading unloading behaviour cannot be modelled

by elastic models. Dilative behaviour is an important feature in oil sands. Modelling

of dilation through a thermoelastic approach is inefficient and may lead to unrealistic

oscillating results. Temperature effects and the multiphase nature of the pore fluid are

very important aspects to be considered in an analytical model. The multiphase flow

models with poroelasticity used in petroleum reservoir engineering do not consider

the effect of stress distribution through the porous medium.

Chapter 3

Stress-Strain Model Employed

3.1 Introduction

In developing a procedure to analyze the geotechnical aspects of oil sands, appropri

ate modelling of the deformation behaviour of oil sand is the most important issue.

Basically, modeffing of oil sand behaviour can be divided into two parts; modeffing of

the behaviour of pore fluid and modeffing of the behaviour of the sand skeleton. In

this chapter, modelling of sand skeleton behaviour is described in detail. Modelling

of pore fluid behaviour is explained in chapter 5.

As explained in section 2.1.1, oil sand is very dense in its natural state and exhibits

significant shear induced volume expansion or dilation. The dilation in the sand

skeleton will increase the pore space and hence increase the permeability and reduce

the pore pressure. These changes will have significant effect in the overall deformation

and flow predictions. Therefore, realistic modeffing of dilation is important.

Generally, oil recovery methods are cyclic processes which will cause the sand

skeleton to undergo loading and unloading sequences resulting in irrecoverable plastic

strains. This necessitates the use of an elasto-plastic stress-strain model. There are a

number of models available in the literature as discussed in chapter 2. Among these,

the model proposed by Matsuoka and his co-workers has been chosen as the basis for

the stress-strain model employed in this study for the following reasons.

1. The failure criterion is based on stress ratio rather than shear stress. This

would realistically model the behaviour when the soil undergoes a decrease in

32

Chapter 3. Stress-Strain Model Employed 33

mean normal stress with constant shear stress (see figure 3.1) which is a possible

scenario in oil recovery process with steam injection.

2. It is based on microscopic analysis of the behaviour of sand grains and not by

curve fitting.

3. It considers the effect of the intermediate principal stress.

4. It appeared to predict the experimental data best based on the proceedings of

the Cleveland workshop on constitutive equations for granular materials (Sal

gado, 1990). A modified version of this model has been extensively used in the

University of British Columbia (Salgado (1990), Salgado and Byrne (1991)) and

gave very good predictions.

The stress-strain model employed in this study is an improved version of the model

used by Salgado (1990). Improvements to Salgado’s model have been made in three

aspects.

1. Changes proposed by Nakai and Matsuoka (1983) regarding the strain increment

directions are implemented.

2. A cap type yield criterion is added to model the constant stress ratio type

loadings accurately.

3. Modelling of strain softening is added.

A detailed description of the stress-strain model, development of the constitutive

matrix in a general three dimensional Cartesian coordinate system, its implementation

in three dimensional, two dimensional plane strain and axisymmetric conditions are

presented in this chapter. It should be noted that effective stress parameters are

implied throughout this chapter and the prime symbols are omitted for clarity.


Cl)Cl)

2Failure Envelope

(Increasing Steam Injection Pressure)

Normal Stress

Figure 3.1: A Possible Stress Path During Steam Injection


3.2 Description of the Model

Generally the total strain increment, de of a soil element can be expressed as a summa

tion of an elastic component, dee and a plastic component, den. In the stress-strain

model developed in this study, the plastic component is further divided into two

parts; a plastic shear component, de8 (the strain increments caused by the increase in

stress ratio) and a plastic volumetric or collapse component, dcc (the strain increment

caused by the increase in mean principal stress). Figure 3.2 schematically illustrates

these elastic, plastic shear and plastic collapse components of the total strain in a

typical triaxial compression test.

Mathematically, the total strain de can be expressed as,

de = dc9 + dcc H- dee (3.1)

These different strain components can be calculated separately; the plastic shear

strains by plastic stress-strain theory involving a conical type yield surface, the plastic

collapse strains by plastic stress-strain theory involving a cap type yield surface and

the elastic strains by Hooke’s law.

From the stress-strain theories, the strain components can be written as

{de8} = [C8] {do}

{de} [Ce] {th}

{dc6} = [Ce] {d} (3.2)

where [C8], [Cc] and [Ce] are the constitutive matrices corresponding to plastic shear,

plastic collapse and elastic strains. Combining equations 3.1 and 3.2 a stress-strain

relation for the total strain can be obtained as follows:

{de} = [[C8] H- [CC] + [CC]] {do}


I

cizwwUU

C,,C,,U

z

IC?,

ci

I-U

-J0>

Figure 3.2: Components of Strain Increment


= [C] {do} (3.3)

The theories involved in developing the [C8], [Cc] and [Ce] matrices in general

Cartesian coordinate system are explained in the next sections and at the end, the

full elasto-plastic constitutive matrix [C] is formed according to different loading

conditions.

In developing a finite element formulation, the stress-strain relation is generally

expressed as

do = [D] dE (3.4)

The above equation is an inverse of equation 3.3. Once the [C] matrix is known, the

[D] matrix can be easily obtained as the inverse of [C].

3.3 Plastic Shear Strain by Cone-Type Yielding


The stress-strain relationship for the plastic shear strain is developed based on the

cSpatial Mobilized Plane’ concept by Nakai and Matsuoka (1983). Before going into

the three dimensional conditions, a brief description of the concept of mobilized plane

in two dimensional conditions is given to provide a better insight.

The concept of mobilized plane was first developed by Murayama (1964). The

term ‘Mobilized Plane (MP)’ refers to the plane where the shear-normal stress ratio

(rMp/crMp) is the maximum. This is the plane on which slip can be considered to

occur. The 2-D representation of this plane is shown in figure 3.3 (a). This plane

makes an angle of (45° + m/2) to the major principal stress plane, where q is the

mobilized friction angle. The Mohr circle for the stress conditions and the mobilized

friction angle are shown in figure 3.3 (b).

Chapter 3. Stress-Strain Model Employed

C,,(I,bJcC’,

TMbJ

C,,

Q3

2-D MobilizedPlane

(a)

(b)

38

Q

NORMAL STRESS

Figure 3.3: Mobilized Plane under 2-D Conditions


From a large number of tests and from the analysis of the shear mechanism of

granular material in a microscopic point of view, Murayama and Matsuoka (1973)

proposed a relationship between the shear-normal stress ratio (TMp /crMP) and the

normal-shear strain increment ratio (dMp/d7Mp) on the mobilized plane as,

rp (_d6MP+ (3.5)MP \ d-yf )

where ) and i are constant soil parameters. Equation 3.5 forms the basis for the

developments of the constitutive models later by Matsuoka and his co-workers.

Under general three dimensional conditions, the stress state of a soil element can

be characterized by the three principal stresses o, 02 and o. Mohr circles for these

three stresses can be drawn as shown in figure 3.4 (a) and three mobilized friction

angles, ml2,4m23 and ç3 can be obtained. These mobilized friction angles can be

expressed by the following equation:

tan(450+) Z (i,j=1,2,3;ucT) (3.6)

Using these mobilized friction angles, a 3-D plane ABC can be constructed as

shown in figure 3.4 (b). This plane ABC is considered to be the plane where the

soil particles are most mobilized and is called the ‘Spatial Mobilized Plane (SMP)’.

Under isotropic stress condition (o = = 03) the mobilized plane will coincide with

the octahedral plane and will vary with possible changes in stresses. The direction

cosines of the SMP are given by the following equation:

a=

(i = 1,2,3) (3.7)

where 11,12 and 13 are the first, second and third effective stress invariants and ex

pressed by the following equations in terms of principal stresses or the stresses in the

general coordinate system.


r 13

m12

o•1

(a)

1

Ia;cI-f-———---———--y.— Spatial

Mobilized

V’ Plane

6 O3

—

- B7

m23450+

2

A7’ L5•+2(b)

Figure 3.4: Spatial Mobilized Plane under 3-D Conditions


‘1 = O1+02+03 =

12 = 12 + 0203 + O301 = 0x0y + 0y0z + 0z0 — — T2—

13 = °y°z +2TTyzTz — OT —— OzTy (3.8)

The general stress-strain relationship will be developed basically from the rela

tionship of the stresses on the SMP and the strain components to the SMP. The

normal stress (oSMP) and the shear stress (TSMp) on the SMP can be obtained from

the following equations:

SMP = o1a + o2a + o3a = 3 (3.9)

and

\/111213 — 9ITSMp = /(oi —o2)2a?a + (o2 —o3)2aa + (o — O1)21

= ‘2(3.10)

The shear-normal stress ratio, i can be expressed as

= TSMp = I1I2 —913(3.11)

SMP 913

By assuming that the direction of the principal stresses and the direction of the

principal strain increments are identical, which is the common assumption in plastic

ity, the normal and the parallel components of the principal strain increment vector

to the SMP (dcsMp and d7sMp) are given by

dEsMp = dea1 + dea2 + dEa3 (3.12)

and

d7sMp = i,J(dEa2 — d€a1)2+ (deas — dca2)2-- (d€ai — d€1a3)2 (3.13)


It should be noted that before Nakai and Matsuoka (1983), Matsuoka used the

normal and shear strain increments on the SMP rather than components of the prin

cipal strain increments to the SMP. After a thorough investigation of the theories

involved, Nakai and Matsuoka (1983) concluded that the average sliding direction of

the soil particles coincides with the direction of the principal strain increment vector

and not with the direction of the strain increment vector on the SMP. They denoted

their earlier model as SMP (Matsuoka and Nakai, 1974, 1977) and the new model as

SMP’. The concepts used in this study follow the SMP model.

In the theory of plasticity, the stress-strain relation is formulated from a yield

function, a plastic potential function (or a flow rule) and a strain hardening function.

The model developed by Matsuoka does not explicitly define these functions. How

ever, those can be formulated and the constitutive matrix can be derived easily as

explained in the next subsections.

3.3.2 Yield and Failure Criteria

The yield criterion defines the boundary between the elastic and plastic zones. A

family of yield surfaces in the TSMp — 0SMP space is shown in figure 3.5. These yield

surfaces are given by the following equation:

77—3\/tanmi2+ tan m23 + tanqm13 = k (3.14)

where i TsMp /0sMP, q5m are the mobilized friction angles and k is a constant.

The ‘current’ yield surface corresponding to the stress state at a point in a mass of

soil is defined by the maximum stress ratio mobilized at that point during its history

of loading. For instance, assume the current yield surface is represented by line A and

the stress state of the point is represented by P (see figure 3.5), the shaded area will be

the current elastic region corresponding to that yield surface. In a loading sequence,

if the stress state of the point moves to Pu within the elastic region, only elastic


Failure Surface

B

Yield Surfaces

A

P...

ElastIc Region

°SMP

Figure 3.5: Yield and Failure Criteria on TsMp — 05MP Space


strains will occur and it represents an unloading condition. If the stress state moves

to FL which is outside the elastic region, there will be elastic and plastic strains. The

yield surface will be dragged along to a new yield surface represented by line B and

the elastic region will expand up to line B. This corresponds to a loading condition.

The limit or the boundary of the yield surfaces will be the failure surface which

is given by the following equation:

tan f12 + tan f23 + tan f13 = kf (3.15)

where is the failure stress ratio and are the failure friction angles. Salgado

(1990) claims that the failure stress ratio is dependent on the normal stress on the

SMP at failure, and that a better agreement with the laboratory data will be obtained

if the failure stress ratio is expressed by the following equation:

(asMP)f= — log10 (3.16)

where

- failure stress ratio at (osMp ) = 1 atmosphere

- decrement in failure stress ratio for 10 fold increase in (oSMp )

The failure surface on the octahedral plane and in the 3-D space is shown in

figure 3.6. The Mohr-Coulomb failure surface is also shown in the figure and it can

be seen that the Mohr-Coulomb and Matsuoka-Nakai failure surfaces coincide for the

triaxial conditions (compression and extension) but differ for any other stress path.

The Matsuoka-Nakai failure criterion considers the effect of the intermediate principal

stress. This effect is shown as the difference between the failure friction angles for

Matsuoka-Nakai and Mohr-Coloumb criteria with b-value in figure 3.7. The triaxial

compression condition will correspond to b-value = 0 and triaxial extension condition

will correspond to b-value = 1.


01

MOHR-COULOMB\

MATSUOKA- NAKAJ

(a) Octahedral Plane

01

/1II\ #\/L\’ “\/ A

1/ \%(II ,C/

p7C

0

(b) 3-Dimensional Stress Space

Figure 3.6: Matsuoka-Nakai and Mohr-Coulomb Failure Criteria


8-

7-

6-

TX

5..7400

-a- 3Q0:

4- .

-a-20° E

2

I0o

1- .

00 0.2 0.4 0.6 0.8

b-VALUE

çb is the failure friction angle in triaxial conditionsis the failure friction angle in Matsuoka-Nakai failure criterion

Figure 3.7: Effect of Intermediate Principal Stress (After Salgado (1990))


Figure 3.8: (TSMp/oSMp) Vs —(dEsMp/d7sMp) for Toyoura Sand (after Matsuoka,1983)

At a particular stress state, the ratio of the normal strain to the shear strain to

the SMP (dEsMp /d7SMp) is given by the following equation:

3.3.3 Flow Rule

The flow rule defines the direction of the plastic strain increments at every stress

state. Matsuoka’s model does not explicitly give a plastic potential function defining

the direction of plastic strain increment. Instead, a relationship for the amount of

plastic strain increment components is given, and in fact, this relationship will give

the direction of the plastic strain increment vector. An example of this relationship

obtained from triaxial compression and extension tests for Toyoura sand is shown in

figure 3.8 which is essentially a straight line. This straight line relationship holds for

all densities.

1.0

02

be”a-2

08

0.6

0.4

0.2

-0.4 -0.2 0 0.2 0.4

- ESMp “YSMP

0.6


[—dESMP’\?7= i , 1+11 (3.17)

\a7sMpJ

where A and t are soil parameters and is the stress ratio on the SMP.

Rewriting the above equation yields,

d6sMp(3.18)

d7sMp A

Equation 3.18 implies that the plastic strain increment vector will not be perpen

dicular to the yield surface and therefore the flow rule is nonassociative. For <

(desMp /d7sMp) will be positive which means there will be an increase in volumetric

strain for an increase in shear strain which implies contractive behaviour. For i > u,

(dEsMp/d7sMp) will be negative which indicates dilative behaviour. Figure 3.9(a)

shows the flow rule and the regions of dilative and contractive behaviour and figure

3.9(b) shows the corresponding results as desMp versus d7sMp.


The hardening rule defines how the threshold of yielding changes with plastic strain, or

in other words how the yield stress state changes with plastic strain. In Matsuoka’s

model, the plastic shear strain to the SMP (7sMp) is considered as the hardener.

Therefore, a relationship between i which defines the stress state and the plastic

shear strain to the SMP, 7sMP, will form the hardening rule. Matsuoka defines the

hardening rule by an empirical equation as follows:

7SMP = 7o exp (, —

(3.19)\P’ /.‘J

where i and i’ are constant soil parameters. The parameter Yo is assumed to be a

function of mean principal stress (crm) and expressed as follows:

7o -yo + Cd log10 (--) (3.20)°mi


(a)

dEsMp

d7SM P

Dilation

71>11Contraction

“<It

(b)

Ti

Dilation

ContractionA

1

(dSMp\dy5i,jp

Figure 3.9: Flow Rule and The Strain Increments for Conical Yield


where Cd is a constant, omj is the initial mean principal stress and yoi is the value of

7o at 0m = 0mi An example of the hardening rule is shown in figure 3.10, which is

obtained from triaxial compression and extension tests on Toyoura sand (Matsuoka,

1983).

1.0392 kN/m2

o comp.• ext. •

2.0 3.0 4.0

Figure 3.10: rsMp/OsMp Vs YsMP for Toyoura Sand (after Matsuoka, 1983)

However, the equation 3.19 given by Matsuoka is not used in this study. Instead,

the relationship proposed by Salgado (1990) is used because, the parameters in his

relationship are more meaningful and it is easier to implement in an incremental finite

element procedure. Salgado (1990) defines the hardening rule using the hyperbolic

nature of the relationship and following the procedure by Konder (1963) as

7SMP(3.21)

+7SMP

G,. 1luUwhere


G,, - initial slope of the i— 7sMP curve

- stress ratio (TSMp/JSMp)

‘Tlult - asymptotic value of the stress ratio

By differentiating equation 3.21, the plastic shear strain increment /7sMP can be

obtained as,

dy5Mp = d (3.22)

where is the dimensionless tangent plastic shear parameter. This parameter is

dependent on both normal stress on SMP (crsMP) and the stress ratio. can be

evaluated by a similar procedure as given by Duncan et al. (1980) as follows:

= G(1 — Rf __)2 (3.23)1i

and

= KG(osMP)

(3.24)

where

- plastic shear number

np - plastic shear exponent

Pa - atmospheric pressure

- stress ratio

R1 - failure ratio (7f/ij,zt)

3.3.5 Development of Constitutive Matrix [CS]

The development of plastic shear constitutive matrix in terms of general Cartesian

stress and strain components from the yield criterion, hardening rule and the flow

rule is described in this section. The hardening rule (equation 3.22) and the flow rule

(equation 3.18) give the following:


dysMp = —di1 (3.25)

IL—?’desMp d-y.9 (3.26)

Substituting equation 3.25 in equation 3.26 will give,

dEsMp =—

(IL ?‘) d (3.27)

By assuming that the directions of the principal stresses and the directions of the

principal strain increments are the same, the direction cosines of desMp are given by

a=

(i = 1,2,3) (3.28)

If it assumed that the direction of d7sMp and the direction of TsMp coincide, then

the direction cosines of d7sMp are given by

— SMP 0jI2 — 3131,: = = (3.29)

TSMp /o- ‘2 (I 12 — 913)

where 11,12 and 13 are stress invariants as given by equation 3.8. The plastic principal

strain increments due to shear can be obtained from the following equation.

de = a desMp H- b d7SMP i = 1,2,3 (3.30)

By substituting equation 3.25 and equation 3.27 into equation 3.30,

dE=+.i)d?’ (3.31)

Equation 3.31 can be written in matrix notation as

{defl = {M12} d?’ (3.32)


where M1=

+

The general Cartesian strain increments can be obtained by multiplying the prin

cipal strain increment vector by the transformation matrix, as given by the following

matrix equation:

dE l m n

de8 12 m2V

d€8d8 12 m2 n2z z z z

(3.33)d7 2l7,l, 2mm 2nn

dc2l,l 2mm2 2nn2

2l1 2m2m3

where

l, l,, and l - direction cosines of o to the x, y and z axes

m, m and m - direction cosines of 02 to the x, y and z axes

, and n - direction cosines of 03 to the x, y and z axes

Equation 3.33 can be written in matrix form as

{de8} = [MT] {dc} (3.34)

Substitution of equation 3.32 into equation 3.34 yields

{de8} = [MT] {M1} di1 (3.35)

From equation 3.11 the stress ratio on the SMP, is given by

/1112 — 913=

91(3.36)


By considering the invariants in terms of Cartesian stresses (equation 3.8) and

differentiating equation 3.36 with respect to Cartesian stresses the following equation

can be obtained for di7:

,

I 77 Id=

{do}

T

‘213 + 1113 (o, + o) — 1112 (o,o — r) do

1213 + 1113 (o + o) — ‘112 (o °•r — T) doy

— 1 1213 + I113(0 + o)—IiI2(t717y — r2) do-i

— 18iiI dr

—2IlI3r — 2IlI2(rr — dr

—2IlI3T — 2IiI2(rr2— or)

= {M2}T{do} (3.37)

where superscript T denote the transpose of the matrix.

Substituting equation 3.37 in equation 3.34 gives

{d68} = [MT] {M1} {M2}T {do} (3.38)

This can be further written as

{d68} = [C8] {dcr} (3.39)

where [C8] is the plastic shear constitutive matrix and will be given by

[C8] = [MT] {M1} {M2}T (3.40)


3.4 Plastic Collapse Strain by Cap-Type Yielding


The plastic stress-strain theory with the conical yield surfaces described in the pre

vious section is not capable of predicting the behaviour of soil under proportional

loading. In that model, the yield surfaces are constant stress ratio lines and therefore,

for a stress path having constant stress ratio, only elastic strains will be predicted.

However, the laboratory experiments show that proportional loading with increasing

stresses causes some plastic deformation.

An additional yield surface which forms a cap on the earlier conical yield surface is

considered to circumvent this deficiency as explained in this section. The stress-strain

relationship for predicting the plastic collapse strains was developed by following the

concepts of the cap-type yielding given by Lade (1977).

As explained in section 3.2, it is reasonable to assume that the plastic collapse

strains are produced by the increase in mean normal stress and the plastic shear

strains will be associated with the shear stresses. However, under general loading con

ditions, it is difficult to separate the plastic shear and plastic collapse strains because

both will occur simultaneously. Therefore, the development of the cap-type yield

model is based on the isotropic compression tests where no plastic shear strains are

produced. Figure 3.11 shows the typical results for loading, unloading and reloading

conditions in an isotropic compression test. The elastic strains which are recoverable

can be calculated using Hooke’s law are also shown in figure 3.11. Then, the collapse

strains can be obtained by subtracting the elastic strains from the total strains.

In order to model the plastic collapse behaviour, a yield criterion which forms a

cap at the open end of the conical yield surface is used. The yield criterion and the

hardening functions for the cap-type yield are explained in the following subsections.

The stress-strain relation for the plastic collapse strain is formulated following the


E

C

‘1,w

IC,,

0

0C’)

56

Figure 3.11: Isotropic Compression Test on Loose Sacramento River Sand (after Lade,1977)

VOLUMETRIC STRAIN, eq,, (‘‘


general theory of plasticity.

3.4.2 Yield Criterion

The yield criterion which defines the onset of plastic collapse strain is given by

f =— 212 (3.41)

where I and 12 are the first and second stress invariants as given in equation 3.8.

The yield criterion which is defined by equation 3.41 represents a sphere with centre

at the origin of the principal stress space which forms a cap at the open end of the

conical yield surface. Figure 3.12 shows the conical and the cap yield surfaces in

01 Hydrostatic Axis

Conical Yield Surface

Plastic Collapse StrainIncrement /ector

Spherical Yield Cap

Iasti Regior


03

Figure 3.12: Conical and Cap Yield Surfaces on the o—

03 Plane


the o-1 — 03 plane. The elastic region at any particular stress state will be bounded

by these two yield surfaces. As f increases beyond its current value, the yield cap

expands, soil work hardens and collapse strains are produced. It should be noted that

there are no bounds on the cap yield surface and yielding according to equation 3.41

does not result in eventual failure. The failure is entirely controlled by the conical

yield surface.

3.4.3 Flow Rule

Under isotropic compression, an isotropic soil shows equal strains in all three principal

directions. Therefore, the direction of strain increment vector should coincide with

the hydrostatic axis pointing outwards from the origin (see figure 3.12). To satisfy

this conditioi-i the plastic potential function must be identical to the yield function.

This implies the flow rule is associative and will be given by the following equation:

de = (3.42)8o.ij

where is the proportionality constant which gives the magnitude of the plastic

collapse strain and can be determined from the hardening rule.


The hardening rule gives a relationship between the yield function and the plastic

strain, defining how the yield function changes with plastic strain. For the cap yield

model, Lade (1977) developed an empirical relationship between the plastic collapse

work (We) and the yield function. The plastic collapse work is a function of plastic

collapse strains and given by

= J {}T{dE} (3.43)


The relationship between the plastic collapse work and the yield function is given

by

= CPa()P

(3.44)

where C and p are dimensionless constants and called the collapse modulus and the

collapse exponent respectively.

The proportionality constant LSX which gives the magnitude of the plastic collapse

strain increment can be obtained as follows. The increment in plastic collapse work

can be expressed as

dW = {}T {dec} (3.45)

Substitution of equation 3.42 into equation 3.45 gives

= (3.46)

Since the yield function f is a homogeneous function of degree 2, it can be shown

that

= 2f (3.47)

From equations 3.46 and 3.47, can be given as

= dWC(3.48)

3.4.5 Development of Constitutive Matrix [CC]

The constitutive matrix relating the plastic collapse strains and the stress increments

can be developed as described below. Substitution of equation 3.48 in equation 3.42

gives


c dW af(. )

Jc O3

By differentiating equation 3.43, dW can be obtained as

= C p a

()121

d (3.50)

and it can be further written as

dW = A df (3.51)

where A = (f)P_1

df will be obtained by differentiating 3.41 as,

df =

T

2o do

2o do

2o do= (3.52)

4r dr

4r dr

4Tz dr

By combining equations 3.49, 3.51 and 3.52 the following equation can be obtained:

A 8f 8fde = —dokj (3.53)2f 8kl

In terms of Cartesian components of stress and strain the above equation can be

written as


oo- 2or 2OTzm do

d 2or 2or22, do

dE = o 2o-2r 2o-r 2u2r do-i

d79 f 4r dT

d-y Symmetry 4r2 4r2r dr

d7 4r2 dr3,

In short matrix notation the constitutive matrix for the plastic collapse strain can be

written as

{Cc]= {8fc}{afc}T

3.5 Elastic Strains by Hooke’s Law

The elastic strains which are recoverable upon unloading can be evaluated using

Hooke’s law by considering the soil as an isotropic elastic material. In matrix notation,

the elastic strains can be given by

{dee} = [Ce] {do} (3.56)

In Cartesian components the above matrix equation can be written as

de 1 —v —v 0 0 0 do

de 1—v 0 0 0 do,

d 1 1 0 0 0 do2

(3.57)d- 2(1H-v) 0 0 dr

d72 Symmetry 2(1 + v) 0 dr2

d 2(1 + v) dr2


where E is the tangential Young’s modulus obtained from the unload-reload portion

of a stress-strain curve. i-’ is the Poison ratio which can be calculated from Young’s

and bulk moduli as

v= (i_&) (3.58)

E and B are assumed to be stress dependent and given by the following equations:

E = kE Pa()fl

(3.59)

B = Pa () (3.60)

where,

kE - Young’s modulus number

- bulk modulus number

n - Young’s modulus exponent

n - bulk modulus exponent

3.6 Development of Full Elasto-Plastic Constitutive Matrix

In the previous sections, the constitutive matrix is formed individually for different

components of strain. One of the major advantages of having the strain components

separated is that it is easy to model the different loading conditions. Depending on

the loading condition, the relevant strain components can be included and the corre

sponding full elasto-plastic constitutive matrix can be formed. The loading conditions

can be classified into four cases which are shown in figure 3.13 on the i — o plane.

Case I

Case I indicates a loading condition where there is an increase in stress ratio as

well as in mean stress. In this case, all three; the plastic shear, plastic collapse and


a1 Failure Surface

Ill

lastc’Rag ion

• 7

-.7.

Hydrostatic Axis


Failure Surface

a3

Figure 3.13: Possible Loading Conditions


elastic strains will be present. Then, the full elasto-plastic constitutive matrix will be

given by

[C] [[C8] + [CC] + [Ce]] (3.61)

Case II

This case considers a loading condition where there is an increase in stress ratio

and a decrease in mean stress. Here, only plastic shear and elastic strains will occur.

The full constitutive matrix will comprise those two matrices only, i.e.,

[C] = [[C8] + [CC]] (3.62)

Case III

Case III considers the loading conditions where there is a decrease in stress ratio

and an increase in mean stress. In this case, plastic collapse and elastic strains will

occur and the corresponding full constitutive matrix will be

[C] = [[CC] + [CC]] (3.63)

Case IV

Case IV indicates a complete unloading condition where there will be decrease in

both stress ratio and mean stress. Under these conditions, only elastic strains will be

recovered. Therefore, the full elasto-plastic constitutive matrix will be the same as

the constitutive matrix for the elastic strains, i.e.,

[C] = [CC] (3.64)


3.7 2-Dimensional Formulation of Constitutive Matrix

Generally 2-dimensional plane strain and axisymmetric analyses are more often car

ried out than 3-dimensional analyses because 3-D analysis require tedious work to

generate the relevant input data and more computer time for execution. The consti

tutive matrix for 2-D plane strain and axisymmetric conditions can be obtained easily

by imposing the appropriate boundary conditions on the 3-D constitutive matrix. A

general stress-strain relation under 3-d conditions can be given as

where C3 are

Plane Strain

Assume that the horizontal and vertical axes in the 2-D conditions are defined by

x and y. Then, all the terms associated with yz and zx and r) will

have no effect in the 2-D plane strain analysis. Hence, equation 3.65 can be reduced

to

C11 C12 C13

— C21 C22 C23

C31 C32 C33

C41 C42 C43

plane strain boundary

dc C11 C12 C13 C14

C21 C22 C23 C24

= C31 C32 C33 C34

C41 C42 C43 C44

d’y2 C51 C52 C53 C54

C61 C62 C63 C64

the components of the constitutive

C15 C16

C25 C26

C35 C36

C45 C46

C55 C56

C65 C66

matrix.

dr

do

do

dr2

drza,

(3.65)

de

dc

d6

d7

Now, by imposing the

C14 do

C24 da

C34 do

C44 dr

condition that = 0, do

(3.66)

can be


written as

do2 = — + do-!, + dT) (3.67)

Substitution of equation 3.67 in equation 3.66 yields:

de C1 C’2 C’3 do

de = C;1 C;2 C;3 do-u (3.68)

d C;1 C;2 C3 d

where

ri — (V C13C31 — r

_____

. C13C34‘-‘11 — ‘-‘11

—

LI12 — L112— C33 ‘ ‘—‘13 LI4

—

_______

— — C2C31 .—

(1 C,C32 .— ,- C2C3

‘—‘21 — ‘-‘21 C33 ‘ ‘-‘22 — ‘-‘22— C33 ‘ ‘-‘23 — ‘—‘24

— c,3f_I,, — f_I C43C31 . — f_I C43C32 . f_I

—f_I C43C34

— ‘-‘41— C33 ‘ ‘—‘32 — L142

— c33 ‘ ‘-‘33— C33

In the above 2-D formulation, the 3-D characteristics will not be lost and the

effect of the intermediate principal stress is still considered. The intermediate stress

can be obtained using equation 3.67.

Axisymmetric

In case of axisymmetric conditions, the modifications are much simpler. Suppose

the x-axis is redefined as radial (r-axis), y-ax.is as circumferential (0-axis) and z-axis

(vertical) is kept the same. Under axisymmetric conditions, d’yre, 7ez, r and r will

not have any influence and hence, equation 3.65 can be reduced to

dEr C11 C12 C13 C64 do.

de8 C21 C22 C23 C64 do-8(3.69)

de2 C31 C32 C33 C64 do-i

C61 C62 C63 C64 drrz


3.8 Inclusion of Temperature Effects

The effects of temperature changes in oil sand and the works by previous researchers

to include these effects in the analytical procedures were described in chapter 2. The

approach used by Srithar and Byrne (1991) is followed here. This involves additional

terms in the stress-strain relation and in the flow-continuity equation. The changes

which have to be made in the stress-strain relation are explained in this section.

Inclusion of temperature effects in the flow-continuity equation is described in section

5.8.

The incremental stress-strain relation can be written as

{de} = [C]{do} (3.70)

where [C] is the elasto-plastic constitutive matrix. If there is an increase in the

temperature, the sand matrix will expand and there will be additional strains. Then,

equation 3.70 will become

{d} = [C]{do} — {de8} (3.71)

where {dee}T {a d, a8 d6, a8 d6, 0, 0, 0} and a8 is the linear thermal expansion

coefficient of the sand grains and d6 is the change in temperature. It should be noted

that compressive strains are assumed positive.

By multiplying equation 3.71 by the inverse of [C] which is referred to as the

stress-strain matrix [D] the following equation can be obtained:

[D]{dc} = {dcr} — [D]{de8} (3.72)

Rearranging the terms will give

{do} = [D]{dE} + {do8} (3.73)


where {do-8} = [D] {de8}, which is the additional term in the stress-strain relation

due to change in temperature. This term will give the induced thermal stresses.

3.9 Modelling of Strain Softening by Load Shedding

Laboratory tests on oil sand show a decrease in strength after a peak strength is

reached which is commonly referred as strain softening. The phenomenon of strain

softening or loss of strength under progressive straining occurs because of the struc

tural changes in the material such as initiation, propagation and closure of micro

cracks. Frantziskonis and Desai (1987) stated that strain softening is not a material

property of soil when it is treated as a continuum. It is rather a performance of

the structure composed of micro-cracks and joints that result in an overall loss of

strength. When the stresses and strains deviate from homogeneity, the behaviour of

a material will no longer be represented by continuum material properties. If strain

softening is assumed as a true material property, various anomalies may arise with

respect to the solution of boundary and initial value problems. These anomalies can

lead to loss of uniqueness in the strain softening part of the stress-strain response and

to numerical instabilities as shown by Valanis (1985).

A comprehensive review of strain softening is not attempted here as it is beyond

the scope of this thesis. Reviews on this subject can be found in Read and Hegemier

(1986) and Frantziskonis (1986). In this study, the strain softening phenomenon

is modelled quantitatively using the ‘load shedding’ or ‘stress transfer’ concept. In

principle the load shedding concept is similar to the model presented by Frantziskonis

and Desai (1987). They modelled the strain softening behaviour by separating it into

two parts; a non-softening behaviour of a continuum (topical behaviour) and a damage

or stress relieved behaviour with zero stiffness. The true behaviour is estimated as

an average of these two (see figure 3.14). In finding the average behaviour, the

hydrostatic component is assumed to be the same for both parts and the deviatoric


ShearStress

Ultimate

Topical Behaviour

— Average Behaviour

Strain

Figure 3.14: Modelling of Strain Softening by Frantziskonis and Desai (1987)


stress is averaged. Since the stiffness is assumed to be zero in the damage behaviour,

the deviatoric stress will be zero for that part. Thus, only the deviatoric stress from

the continuum behaviour is reduced or some of the deviatoric stress is taken away.

This is similar to the load shedding technique with constant mean stress.

In order to model the strain softening behaviour, the variation of the stress ratio

(or the strength) with the strains in the strain softening region should be established.

Here, the variation is assumed to be represented by an equation similar to that given

by Frantziskonis and Desai (1987) for their damage evolution. Thus, in the strain

softening region the stress-strain relation can be given as

= i + (ii, — ‘qr)exp{—k(ysMp—

(3.74)

where

- Residual stress ratio

- Peak stress ratio

7SMP,p - Peak shear strain

Ic, q - Constant parameters

3.9.1 Load Shedding Technique

Load shedding (Zienkiewicz et al. (1968), Byrne and Janzen (1984)) is a technique

to correct the stress state of an element which has violated the failure criterion, by

taking out the overstress and redistributing to the adjacent unfailed elements. A brief

description of how the load shedding technique is applied to model strain softening is

presented below. Details of the estimation of overstress and the corresponding load

vector are given in appendix A.

Figure 3.15 shows a typical scenario in modelling strain softening by load shedding.

The stress state of an element depicted by point P0 in the figure can move to point


‘1]

‘rip

Figure 3.15: Modelling of Strain Softening by Load Shedding

71

P1

P2

F? T

I

7r

71 7


P1 in a load increment. But the actual stress state should be point Pia and in order

to bring to this stress state, an overstress of should be removed. The overstress

will then be redistributed to the adjacent stiffer elements. During the redistribution

process, the modulus of the failed element will be defaulted to a low value so that it

will not take any more load. However, in another load increment the stress state may

move to point P2. Then again the stress state will be brought to point F2a by load

shedding. In the process of load shedding, it is also possible that some other elements

violate failure criteria and those loads also have to be redistributed. Therefore, several

iterations may be needed to find a solution where failure criteria are satisfied by all

the elements.

3.10 Discussion

Although the stress-strain model employed in this study is somewhat sophisticated, it

will not capture the real soil behaviour under certain loading conditions. For instance,

since the model assumes the material to be isotropic, it will not correctly predict the

deformations for pure principal stress rotations.

In the stress-strain model used in this study, the elastic principal strain increment

directions are assumed to coincide with the principal stress increment directions and

the plastic principal strain increment directions are assumed to coincide with the prin

cipal stress directions. Lade (1977) also stated that the principal strain increment

directions coincide with the principal stress increment directions at low stress levels

where elastic strains are predominant and coincide with principal stress directions at

high stress levels where plastic strains are predominant. Salgado (1990) presented a

critical review regarding the assumption that the direction of principal strain incre

ments coincide with the direction of principal stresses. He reviewed the results using

the hollow cylinder device by Symes et al. (1982, 1984, 1988) and Sayao (1989) and

concluded that the assumption is reasonably valid for most of the stress paths except


those that involve significant principal stress rotations.

One of the disadvantages of this model is its limited use in the past. Unlike the

hyperbolic model, information on the model parameters is very limited. The possible

range of values for some of the parameters and their physical significance are not

well defined. However, a sensitivity study on the parameters is given in chapter

4, which may be helpful to understand the physical significance of the parameters.

Another disadvantage of the model is that because of the nonassociated flow rule, it

will result in a non-symmetric stiffness matrix which requires considerable computer

memory and time. However, the frontal solution scheme used in this study will

circumvent the requirement for large memory since it does not assemble the full

stiffness matrix and requires only a small memory. Furthermore, these factors of time

and memory requirements may not be considered as disadvantages with the rapid

growth in computer capabilities.

Chapter 4

Stress-Strain Model - Parameter Evaluation and

Validation

4.1 Introduction

This chapter describes the procedures used to evaluate the soil parameters needed for

the stress-strain model and presents results verifying the stress-strain model against

measured responses in laboratory tests. The soil parameters required for the model

can be classified into four groups; elastic, plastic shear, plastic collapse and strain

softening. A summary of the parameters and their description are given in table 4.1.

The procedures used to evaluate these parameters from basic laboratory tests such

as isotropic compression and triaxial compression tests are described in section 4.2.

For the determination of some of the parameters at least two test results are necessary

to obtain a straight line fit. In those cases, it is advisable to have three or more test

results to obtain a better fit. Validations of the stress-strain model against laboratory

results on Ottawa sand and on oil sand are given in section 4.3. Sensitivity analyses

on some of the parameters have been carried out to provide some idea about their

significance and these are described in section 4.4.

4.2 Evaluation of Parameters

In this section, only the procedures for the evaluation of the parameters are given in

detail. Applications of these procedures to actual test data on Ottawa sand and on

oil sand can be found in section 4.3.

74

Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 75

Table 4.1: Summary of Soil Parameters

L Type Parameter Description

Elastic kE Young’s modulus numbern Young’s modulus exponent

kB Bulk modulus numberm Bulk modulus exponent

Plastic Shear Failure stress ratio at one atmosphereLi Decrease in failure stress ratio

for 10 fold increase in 0SMP

.\ Flow rule slope

i Flow rule interceptKG Plastic shear numbernp Plastic shear exponentR1 Failure ratio

Plastic Collapse C Collapse modulus numberp Collapse modulus exponent

Strain Softening Strain softening constantq Strain softening exponent

4.2.1 Elastic Parameters

4.2.1.1 Parameters kE and n

The elastic parameters kE and n can be determined from the unload-reload portion of

a triaxial compression test as explained by Duncan et al. (1980). To determine these

parameters, at least two unload-reload modulus values (see figure 4.1(a)) at different

mean normal stresses are necessary. The unload-reload Young’s modulus is given by

E kE Pa ()‘ (4.1)

By rearranging and taking the logarithm, the above equation can be written as

log (-) = log kE + n log (i) (4.2)

Thus, kE and n can be determined by plotting (E/Pa) against (0m/1Zba) on a log-log


plot as shown in figure 4.1(b).

In the standard triaxial compression test, the unload-reload stress path is often not

performed. In the absence of unload-reload results, kE for the unload-reload portion

can be roughly estimated from (kE) for primary loading. The values of (k) can be

found in Duncan et al. (1980) and in Byrne et al. (1987) for various soils. Duncan et

al. claimed that the ratio of kE/(kE) varies from about 1.2 for stiff soils such as dense

sands up to about 3 for soft soils such as loose sands. The value of the exponent n for

unload-reload is found to be almost the same as the exponent for primary loading.

Hence, if the value of n is known, kE can be determined from a single unload-reload

E value.

4.2.1.2 Parameters kB and m

The best way of evaluating kE and m is from the unload-reload results of an isotropic

compression test. The procedure proposed by Byrne and Eldrige (1982) is followed

here to determine these parameters. The volumetric strain and the mean stress in

the unload-reload path can be related as

= a (°m)’ (4.3)

where a and b are constants and can be obtained by plotting versus 0m on a log-log

scale as shown in figure 4.2.

Differentiation of equation 4.3 yields

ck 1 b—i (4.4)

Then, the bulk modulus B can be expressed as

B = (Om)1 (45)


q-a3 /AE1

€

(a) Unload-Reload Modulus

(E/Pa)

1000

100

‘<E ‘a

1 10(ojP)[log scale] . a

(b) Variation of E with a3

Figure 4.1: Evaluation of kE and n


0.01 -

kB = a.b(Pa)6

a m=1-b

100[log scale]

Figure 4.2: Evaluation of kB and m


The general expression for B is given by

B = kBPa()

(4.6)

By considering the similarities of equations 4.5 and 4.6, the parameters kB and m

can be obtained from a and b as

m=1—b (4.7)

kB= ab(Pa)’

(4.8)

It should be noted that the parameters kE and kB can be related by the Poisson’s

ratio v as

kB= 3(1—2zi)

(4.9)

Hence, by knowing one parameter, the other one can also be determined from

the Poisson’s ratio. Lade (1977) stated that the Poisson’s ratio for the unload-reload

path has often been found to be close to 0.2.

4.2.2 Evaluation of Plastic Collapse Parameters

Only two parameters are needed to evaluate the plastic collapse strains. These two

parameters define the hardening law and can be determined from an isotropic com

pression test. The hardening law is given by

= CPa ()‘ (4.10)

where W is the plastic collapse work, f defines the yield surface and C and p

are constant parameters to be determined. For the isotropic compression loading

condition, f and W will be given by


f = 3o (4.11)

Wc=Jcr3de (4.12)

where de = d€,, — d and de is the elastic volumetric strain.

By plotting W/P against f/P on a log-log plot, the parameters C and p can

be obtained as shown in figure 4.3.

0.01 -

[log scale]

Figure 4.3: Evaluation of C and p

4.2.3 Evaluation of Plastic Shear Parameters

In determining the plastic shear parameters, it is easier to divide them into three

groups as follows:

1. Failure parameters i and LSi


2. Flow rule parameters i and )

3. Hardening rule parameters KG, np and R1

The plastic shear parameters can be determined from all types of tests where the

principal stresses and principal strains can be obtained. By knowing the principal

stresses and strains, the stresses and strains on the spatial mobilized plane (SMP)

can be evaluated as described in section 3.3. The plastic shear parameters can then

be obtained as explained in the following subsections.

The most common laboratory shear tests performed are triaxial compression tests

and therefore, special attention is given here to describe how to obtain the plastic

shear parameters from those test results.

Firstly, the elastic and plastic collapse strains have to be subtracted to obtain the

principal plastic shear strains:

d = de1 — de — (4.13)

d€ = de3 — de — d (4.14)

Under standard triaxial compression conditions, the elastic and plastic collapse

strains can be given by

do1(4.15)

de = —vde (4.16)

de = o do1 (4.17)

d = 2A u1o3 do (4.18)

where2C (p 1—2p

— P\ a)

—

(o + 2o)2P


It should be noted that if the test samples are preconsolidated to a higher stress

and unloaded, then the collapse strains should not be subtracted.

By following the equations in section 3.3.1 and imposing the conditions for triaxial

compression loading, the stresses and the strains related to SMP can be obtained as

follows:3o-1o3

SMP = (4.19)•1 + 03

TSMp(4.20)

SMP 3

— dc/ + 2d4/dEsMp — (4.21)

/2o + o

2(deW — de/jd7sMp = (4.22)

2o1+c3

4.2.3.1 Evaluation of q’ and z

At least two tests up to failure at different confining stresses are necessary to determine

these parameters. The failure stress ratio on SMP is given by

(o-sMP)f= — Li log10Pa

(4.23)

The values of and (OsMP)f can be obtained using equations 4.20 and 4.19. By

plotting {(OSMp)f/Pal versus on a semi-log plot, i and ii can be determined as

shown in figure 4.4.

4.2.3.2 Evaluation ofi and X

The flow rule for the plastic shear is expressed by the following equation.

f—c1EsMpi J+/L (4.24)\ u7SMP J


‘if

____

—

_________ ______

1 10 100e’ a[log scale]

Figure 4.4: Evaluation of 1h and ‘i

The values of i, dEsMp and d7sMp for a triaxial compression test can be obtained

using equations 4.20, 4.21 and 4.22. The flow rule parameters and ) can be deter

mined by simply plotting versus —(desMp/d7sMp) as shown in figure 4.5.

4.2.3.3 Evaluation of KG,rIp and Rf

As explained in section 3.3.4, the hardening function is modelled by a hyperbola and

is given by

7SMP17 = (4.25)

+G.The parameters KG, np and R1 which define C and it in the hardening rule

are evaluated following the procedure by Duncan et al. (1980). Basically, there are

two steps involved in determining these parameters. The first is to determine the


‘17

— (dEMp/d4Mp)

Figure 4.5: Evaluation of ) and t

values of G, and the second is to plot those values against °5Mp to determine KG

and np. At least two triaxial compression test results are necessary to evaluate these

parameters.

Upon rearranging the terms, equation 4.25 becomes

7SMP — 1 7SMP4 26

1 7u1t

Now, by plotting (7sMp/7/) against fsMP the values of G7, and 71,jit can be deter

mined as shown in figure 4.6(b).

The failure ratio Rf is defined as

Rf (4.27)l7ult

By knowing from figure 4.6(b) and i from section 4.2.2.1 Rf can be deter

mined using the above equation.


“7

1

7SMP(a) Hardening Rule

7SMP

‘1

G1

7SMP

(b) Hardening Rule on Transformed Plot

Figure 4.6: Evaluation of G and ij


G is expressed as a function of op as

G = KG (4.28)

The parameters KG and np can be obtained by plotting G,. against (oSMp/Pa)

on a log-log plot as shown in figure 4.7.

1000‘1)0c-I(I)

0U 100

1 10 100[log scale]

Figure 4.7: Evaluation of K0 and np

MP’a

4.2.4 Evaluation of Strain Softening Parameters

To determine the strain softening parameters, it is necessary to have experimental

results which exhibit strain softening phenomenon. As explained in section 3.9, it

should be noted that strain softening is not a fundamental property of soils, rather it

is a localized phenomenon. Therefore, it is quite possible that different tests may yield

different softening parameters. In those cases, the average value can be considered

appropriate.

1

np

KQ


The strain softening region of a stress-strain curve can be given as (see section

3.8)

= ir + (ip— lir) exp{—i(7sMp

— 7SMP,p)} (4.29)

The value of the residual stress ratio is i is assumed to be equal to t which is

the flow rule intercept. This assumption is reasonable because, when i = p, the

incremental plastic volumetric strain will be zero, which implies a state of shear at

constant volume. The value of the peak stress ratio, which is the failure stress ratio,

can be obtained from equation 4.23. The peak shear strain 7SMp can be obtained

from the strain hardening relation (equation 4.25) as

7737SMP,p = (4.30)

G1 — R11

where is the initial tangent plastic shear parameter and Rf is the failure ratio.

By rearranging the terms in equation 4.29 and taking natural logarithm, it can

be shown that

in () K(7sMp (4.31)

Taking natural logarithm of equation 4.31 will give

ln [in(ij]

ln + qln(7sMp— 7sMp,p) (4.32)

Then, the parameters , and q can be determined by plotting {ln [ln ()] }against {ln(7sMp — 7SMP,p )} as shown in figure 4.8.

4.3 Validation of the Stress-Strain Model

The stress-strain model employed in this study has been verified against laboratory

results on Ottawa sand and oil sand.The triaxial test results reported by Neguessy


in [in (TZr)]

Figure 4.8: Evaluation of i and q

(1985) on Ottawa sand and by Kosar (1989) on Athabasca McMurray formation in

terbedded oil sand have been considered. The Ottawa sand is well defined. Uniform

test samples were constituted in the laboratory and the test results were very re

peatable. Oil sand samples on the other hand, were obtained from the field and

therefore the samples might not identical. The soil parameters for both sands are

obtained as explained in the section 4.2 and then the predicted and measured results

are compared.

4.3.1 Validation against Test Results on Ottawa Sand

The Ottawa sand is a naturally occurring uniform, medium silica sand from Ottawa,

illinois. Its mineral composition is primarily quartz and the specific gravity is 2.67.

The average particle size D50 is 0.4 mm and the particles are rounded. The gradation

curve of the Ottawa sand is shown in figure 4.9.

q

in(7sMp— 7SMP,p)

Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation

zp ae 4 48 100

‘ I

I I

I I

89

MEDIUM SAND

‘I

I00

140 200

80

60

C

t4Q

LEGE ND

X FRESH

20

• RECYCLED

I

0

ASTM - C - 109- 69 BAND

* MIT CLASSIFICATION

I 0.5 0.1 0.01

Diameter (mm)

Figure 4.9: Grain Size Distribution Curve for Ottawa Sand (after Neguessy , 1985)


The following test results reported by Negussey (1985) are considered here for the

determination of the relevant parameters and for the validation:

1. Resonant column tests

2. Isotropic compression tests

3. Triaxial compression tests

4. Proportional loading tests (R = o1/o3 = 1.67 and 2)

5. Tests along four different stress paths as shown in figure 4.10

SP4

300 SP3

a.

200

SP2SP1- = 2.0

SP2- (a/u=4.0spi100 SP3 - P

= 250 kPa, Constant

SP4 - P’ = 350 kPa, Constant

100 200 300 400

UH(kPa)

Figure 4.10: Stress Paths Investigated on Ottawa Sand

The test results considered here are for Dr = 50%. The maximum and minimum

void ratios of the Ottawa sand are 0.82 and 0.50 respectively.


4.3.1.1 Parameters for Ottawa Sand

91

As explained in section 4.2.1, the Young’s modulus values for different confining

stresses are plotted in figure 4.11. The values plotted in the figure are from resonant

:-

(kE)p= 1180

- . Resonant Column -

-

A Tria)_(UnIoad Reload)

— •Triaxlal (Primary Loading)

I I I

a3 a

Figure 4.11: Variation of Young’s moduli with confining stresses

column tests which yield similar values as are obtained in unload-reload tests. Also

shown in the figure are one unload-reload modulus and the Young’ modulus values

for primary loading from standard triaxial compression tests. It can be seen that the

unload-reload value agrees well with the resonant column values. The ratio of the

Young’s modulus for the primary loading condition to the unload-reload condition is

about 2.2 and the exponent for both conditions is 0.46. This agrees with the statement

by Duncan et al. (1980) that the ratio of Young’s moduli varies from about 1.2 for

dense sands to about 3 for loose sands. From figure 4.11 the values for kE and n can

10000

5000

3000

2000

1000

5000.3 0.5 1 2 3 5 10


be obtained as 2600 and 0.46 respectively. In the absence of resonant column tests,

the same values could also have been obtained from the values of primary loading at

different confining stress and one value of unload-reload.

There are no results of unload-reload conditions available in isotropic compres

sion test to determine kB and m. Therefore, the Poisson ratio is assumed to be

0.2 as suggested by Lade (1977). Hence, kB and m are obtained as 1444 and 0.46

respectively.

The plastic collapse parameters C and p are evaluated as explained in section

4.2.2 from the isotropic compression test. Figure 4.12 shows the variation of (We/Pa)

with (fe/P) for Ottawa sand and the value of C and p are equal to 0.00021 and 0.89

respectively.

0.01We/Pa

0.005

0.002

0.00 1

0.0005

0.0002

0.0001

5E-050.2 0.5 1 2 5 10 20

2

50 100

Figure 4.12: Plastic Collapse Parameters for Ottawa Sand

In order to obtain the failure parameters, as explained in section 4.2.3.1, the failure


stress ratio i vs usMp for the triaxial compression test results are plotted in figure

4.13. The failure parameters () and ZSi7 are determined as 0.49 and 0.0.

Figure 4.13: Failure Parameters for Ottawa Sand

The four triaxial compression test results are shown as vs. (—desMp/d7sMp) in

figure 4.14 to determine the flow rule parameters A and p (refer to section 4.2.3.2).

From the figure, p and A are obtained as 0.26 and 0.85 respectively.

As explained in section 4.2.3.3, for the evaluation of hardening rule parameters,

the results from the triaxial compression tests are transformed and the relevant plots

are shown in figure 4.15. The value of R1 is determined as 0.93. From figure 4.15(c),

the values of KG and np are obtained as 780 and —0.238 respectively.

Table 4.2 summarizes all the parameters for Ottawa sand at Dr = 50%.

Ti

0.6

0.5

0.4

0.3

0.2

0.1

—— —71iO49

ö::

zS7=O.O

1000.5 1 2 3 5

cTSMP/P


Table 4.2: Soil Parameters for Ottawa Sand at D = 50%

Elastic kE 2600n 0.46kB 1444

m 0.46Plastic Shear 0.49

1117 0.0X 0.85u 0.26

780np -0.238

Rf 0.92Plastic Collapse C 0.00021

p 0.89

‘7

0.6

0.5

0.4

03

0.2

0.1

0-0.3 -0.2 -0.1 0 0.1 0.2 0.3

_(dEsMp/d7sMP)

0.4

Figure 4.14: Flow Rule Parameters for Ottawa Sand


(a)0.5

0.4

___________

o3=5OkPa0.3 ,‘ ---0---

3=150kPa02 —

• a_3=50kPa

0.1 I a3=45OkPa--.“---.-

00 0.2 0.4 0.6 0.8

7SMP

1.87SMP

1.6 o

1,4

1.2

1

0.8

0.6

0.4

0.2 ‘

0 I I

0 0.2 0.4 0.6 0.87SMP

800G1

750.W 735%% ‘UP

700

650

600Jlr=0.145

550

500 I I I

0.5 1 2 3 5 10

Figure 4.15: Hardening Rule Parameters for Ottawa Sand


4.3.1.2 Validation

As a first level of validation, the four triaxial compression tests which were used to

determine the parameters, are modelled. Figure 4.16 shows the experimental results

and the model predictions and they both agree very well. This implies that the model

successfully represents the test results.

The stress-strain model is then used to predict the responses for proportional

loadings and four other stress paths as shown in figure 4.10. Figure 4.17 shows the

results for two proportional loading tests, R = o1/o3 = 1.67 and 2, and it can be

seen that the predictions and the measured responses agree very well. Figure 4.18

shows the results for four different stress paths and again the predicted and measured

results are in good agreement.

4.3.2 Validation against Test Results on Oil Sand

The test results reported by Kosar (1989) on Athabasca McMurray formation oil sand

are considered here. Tests were carried out on samples taken form the Alberta Oil

Sands Technology and Research Authority’s (AOSTRA) Underground Test Facility

(UTF) at varying depths from 152 m to 161 m. The samples consisted of medium

grained particles and were uniformly graded. Figure 4.19 shows the gradation curve

of the UTF sand and some other oil sands. In UTF sands, pockets and seams of silty

shale were present and their thickness ranged form 1 to several millimetres. The fines

content varied form 36 to 72% and the bitumen content from 4 to 9.5 % by weight.

The samples were sealed and frozen at the site to minimize the disturbance. Kosar

(1989) estimated the sample disturbance using an index developed by Dusseault and

Van Domselaar (1982) which compares the sample porosity to the in-situ porosity.

The index of disturbance was found to vary from 6 to 12% indicating reasonably good

quality samples.

The following test results from Kosar (1989) are considered for the determination


00

0

97

a 3 = 50 kPa----*--

a_3• = 5O kPa

o 3 = 250 kPa-------

800

600 - _..

0400- — .0

200 -

I I

a_3 =50 kPa

Symbols - ExperimentalLines - Analytical

(a)

0.05

0.>

‘U

0.15

0.2

0.250 0.2 0.4 0.6 0.8

E(%)a

Figure 4.16: Results for Triaxial Compression on Ottawa Sand


500

400

3000

200

100

600

500

400

0

a-

200

100

00

98

1.67

Symbols - ExperIment

Lines - AnaIytca1

I_ —— I I

0.1 0.2 0.3 0.4 0.5 0.6

Figure 4.17: Results for Proportional Loading on Ottawa Sand


600

500

400

300

200

100

>iLl

0.4

0.6

0.8

0.2

a

0 0.2 0.4 0.6 0.8 1

Figure 4.18: Results for Various Stress Paths on Ottawa Sand


>

E=

100

80

60

40

20

010.0

I I I I I 111111 I I -__

UTF Sand4

• Other McMurray - — —

Sands: - coarse I 1- medium -—— I-fine

.:::ZE

.--——

1.0 milhimetet 0.1 0.01

I I0.1 . 0.01 0.001

inches

I F F Fl F I F F F F I8 12 18 25 35 45 60 80 120 170 230 325400

U.S. mesh

Figure 4.19: Grain Size Distribution for Athabasca Oil Sands, (after Edmunds et al.,1987)


of the relevant model parameters and the validation:

1. Isotropic compression test

2. Standard triaxial compression tests

3. i constant compression

4. U constant compression

0m constant extension

It should be noted that since the samples tested were undisturbed samples from

the field, they were not identical. Table 4.3 summarizes the details of the test samples

considered.

Table 4.3: Details of the Test Samples

Test Sample Bulk Fraction by Weight (%) Void DisturbanceID Density Water Bitumen Solids Fines Ratio Index

(kg/rn3) (< 0.074rnrn) (%)Isotropic Comp. UFTOS1 1990 8.3 7.6 84.1 41.2 0.60 12.1Triaxial Comp. 1 UFTOS1 1990 8.3 7.6 84.1 41.2 0.60 12.1Triaxial Comp. 2 UFTOS3 2070 8.5 6.6 84.9 54.0 0.52 6.4Triaxial Comp. 3 UFTOS4 2120 6.4 6.5 87.1 52.9 0.45 10.00.1 Const. Comp. UFTOS1O 2060 6.6 7.3 86.1 71.9 0.50 10.80m Const. Comp. UFTOS9 1960 7.8 8.8 83.4 57.3 0.62 10.90m Const. Ext. UFTOS12 1980 7.0 9.5 83.4 37.7 0.60 9.6

4.3.2.1 Parameters for Oil Sand

The relevant parameters for the oil sand are obtained from an isotropic compression

test and three standard triaxial compression test results. Since the procedures for

obtaining the parameters are discussed in detail in section 4.2 and again briefly in

section 4.3.1.1, they are not repeated here.

Figure 4.20 shows the data for the unload-reload portion of the isotropic com

pression test and the elastic parameters kB and m are determined as 1670 and 0.36


0.03

EV

0.01

0.003

0.001

0.0003

0.0001

3E-051 10 100 1000 10000

am (kPa)

100000

Figure 4.20: Determination of kB and m for Oil Sand


respectively. The Poisson ratio is assumed to be 0.2 and kE and n are determined as

3000 and 0.36. The plastic collapse parameters C and p are obtained from the primary

loading portion of the isotropic compression test as 0.00064 and 0.61 respectively (see

figure 4.21).

/ a

1

0.3

0.1

0.03

0.01

0.003

0.001

0.0003

The failure and hardening rule parameters are obtained from the triaxial com

pression tests as explained in section 4.2.3. Figure 4.22 shows the relevant graph to

obtain the failure parameters. The hardening rule parameters are obtained as shown

in figure 4.23.

The reduced data to obtain the flow rule parameters are shown in figure 4.24. The

results from the three triaxial tests do not seem to give a unique set of parameters as

observed in Ottawa sand. This can be attributed to the differences in field samples.

It is evident from figure 4.24(a) that different flow rule parameters can be obtained

1 10 100 1000 10000 100000

Figure 4.21: Plastic Collapse Parameters for Oil Sand

104Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation

0.8

1f

0.75

0.7

0.65

06

0.55

0.5I 2 3 5 10 20 30 50

Mp’a

100

Figure 4.22: Failure Parameters for Oil Sand


2000

1300

1000 -

500-

= -0.66

2000

0

100 -

0

50 I

1 2 3 5 10 20 30 50 100

MP’a

Figure 4.23: Determination of K0 and np for Oil Sand


0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4

0-0.4

7,EJ

A

0 --./- .------

0.2 0.4

o.9

0.6 0.8

(a)11

0.8

0.7

0,6

0.5

0—-0.4 -0.2

o a_3=1.OMPa

o _3 = 2.5 MPa

c,_3 = 4.0 MPa

0

—(dEsMp/d7sMP)

0.3

0.2

0.1

-0.2 0 0.2 0.4 0.6

—(dEsMp/d7sMP)0.8

Figure 4.24: Flow Rule Parameters for Oil Sand


if the individual test results are considered. However, an average set of parameters

can be obtained as shown in figure 4.24(b). The flow rule parameters are very much

governed by the volumetric strain behaviour and this will be discussed more in section

4.3.2.2. The summary of the parameters obtained for oil sand is given in table 4.4.

Table 4.4: Soil Parameters for Oil Sand

Elastic kE 3000n 0.36

kB 1670m 0.36

Plastic Shear 0.75iii 0.13\ 0.53ii 0.31

KG 1300rip -0.66R1 0.73

Plastic Collapse C 0.00064

p 0.61

4.3.2.2 Validation

Figure 4.25 shows the experimental and predicted results for loading and unloading of

the isotropic compression test. It can be seen that the results are in good agreement.

Figure 4.26 shows the experimental and predicted results for the triaxial compres

sion tests. It can be seen that the predicted and measured deviator stress versus axial

strain agree very well. The volumetric strain versus axial strain agree reasonably well

for 03 = 1.OMPa and O = 2.5MPa but not for o = 4.OMPa. This is because

the selected flow rule parameters are the average parameters and they tend to agree

closely with those two tests. It can be seen from figure 4.24 that for 03 = 4.OMPa, the

straight line relation is much different and steeper, which would have given a higher


14000

12000 - 0

10000 -

8000 - Loading

b 6000 -

Unloading

4000- 0

2000 o Line - PredictedSymbols - Measured

0I I I I

0 0.5 1 1.5 2 2.5 3

LV (%)

Figure 4.25: Results for Isotropic Compression Test on Oil Sand


value of ). As the line becomes steeper, there will be less volumetric expansion and

the overall behaviour will be more contractive. If a higher value of ) is selected, the

predictions and observations would agree well for o 4.OMPa. It is also interesting

to note that the value of the flow rule parameter i is not much different for the three

tests. The value of .t is, in fact, an indication of ultimate stress ratio or a state of

shearing with constant volume.

Results for three different stress paths; constant o, compression, constant °m

compression and constant om extension are shown in figure 4.27. The stress paths

are shown in the insert of the figure. It can be seen that the experimental and

predicted results are in good agreement.

4.4 Sensitivity Analyses of the Parameters

In order to provide a better understanding about the significance of the parameters,

sensitivity analyses on the parameters have been carried out. The parameters ob

tained for Ottawa sand were chosen as the base parameters and the significance of a

particular parameter was studied by changing only that parameter. A triaxial com

pression loading condition with the initial confining stress of 500 kPa was considered

and the results in terms of deviator stress and volumetric strain are analyzed. The

results are shown in figures 4.28 and 4.29

The plastic collapse parameters C and p are essentially an indication of isotropic

compressibility. The higher the values, the higher the predicted volumetric strains.

The parameter ). is the slope of the flow rule and it defines the change in volumetric

expansion for a change in stress ratio. A steeper slope (or higher A) will give less

volumetric expansion. The parameter 1u is the amount of stress ratio which separate

contraction and dilation (similar to ç5 in general soil mechanics). A smaller value of

will result in dilation at lower stress ratio.

The parameters KG and np define the initial slope of the hardening modulus


0

>w

10000 -

8000 -

6000 -

4000

2000

0

0.2

0.4

0.6

0.8

--: :

MPa

a_3 =.5 MPa

a_3=,MPa

Symbols - Experimental

Lines - Analytical

-0.8

-0.6

-0.4

-0.2

0 0.5 1 1.5 2 2.5 3

e(%)a

Figure 4.26: Results for Triaxial Compression Tests on Oil Sand


5,000 —

4,000 -

3,000 -

2,000

1,000

0

6

-4

I

/ SP1-I.1 Const.Comp2 / SP 2- a_v Const Comp./ SP3-i1 Conet Ext.

246a_r (MPa)

D.C-..

C’-,

I1

‘a

a

&

>WI

000

or0/

0/

______

o/ spiC

SP3

Symbols- Experimental

Lines - Analytical

-1.4

-1.2

—1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

zgD Q9OO5/O

. -ci - o o

-6 -4 -2 0 2 4 6- E_r (%)

Figure 4.27: Results for Tests with Various Stress Paths on Oil Sand


Ea (%)

(C) Effect of Parameter A

0

0

&

112

c_a (%) c_a (%)

(a) Effect of Parameter C (b) Effect of Parameter p

I

>WI

0.4 0.6 0.8 1

c_a (%)

(d) Effect of Parameter L

Figure 4.28: Sensitivity of Parameters C,p, ) and i


(c) Effect of Parameter Rf

1.2

€_a (%) E_a (%)

(a) Effect of Parameter KGp (b) Effect of Parameter rip

0

&

Ea (%) c_a (%)

(d) Effect of Parameter flf

Figure 4.29: Sensitivity of Parameters KG, np, R1 and i


G,. A higher value of will result in a stiffer deviator stress response and a lower

volumetric strain response. The parameter R1 and i define the shape and the failure

stress ratio in the hardening rule. Lower R1 and higher i will give stiffer deviator

stress response.

The elastic parameters are not considered here because they have been widely

used and their significance is well understood.

4.5 Summary

A double hardening elasto-plastic model has been postulated to model the stress-

strain behaviour of oil sands. Procedures for the evaluation of the parameters and

the validation of the proposed model have been presented in this chapter.

The model parameters are relatively easy to obtain and can be determined from

conventional isotropic and triaxial compression test results. The parameters have

physical meanings and a sensitivity study has been carried out on the parameters to

better explain their physical significance. Laboratory test results for various stress

paths have been compared with the model predictions. Measured results and predic

tions agree very well and the model predicts the shear induced dilation effectively.

From the validations presented in this chapter, it can be concluded that the proposed

model captures the stress-strain behaviour of oil sands very well.

Chapter 5

Flow Continuity Equation

5.1 Introduction

The pore fluid in the oil sand matrix comprises three phases namely gas, oil and water

and therefore, the fluid flow phenomenon is of multi-phase nature. In petroleum

reservoir engineering, the flow in oil sand is often analyzed as multi-phase flow, but

solely as a flow problem without paying much attention to the porous medium. The

most widely used model to analyze the flow in oil sand is called ‘3-model’ or ‘the

black-oil model’ (Aziz and Settari, 1979) and it makes the following assumptions.

1. There are three distinct phases; oil, water and gas.

2. Water and oil are immiscible and they do not exchange mass or phases.

3. Gas is assumed to be soluble in oil but not in water.

4. Gas obeys the universal gas law.

5. Gas exsolution occurs instantaneously.

With these assumptions, and considering the effects of stresses and temperature

changes in the sand skeleton, a flow continuity equation is derived in this chapter

from the general equation of mass conservation. However, the flow equations are not

considered separately for individual phases as in petroleum reservoir engineering. All

three flow equations are combined and a single effective equation is formulated. In

essence, the derived flow continuity equation is similar to a single phase flow equation

115

Chapter 5. Flow Continuity Equation 116

in geomechanics but the permeability and compressibility terms have been changed

to include the effects of different phase components. The flow continuity equation will

be combined with the force equilibrium equation and will be solved as a consolidation

problem as explained later in chapter 6.

5.2 Derivation of Governing Flow Equation

In this section, the flow continuity equation for a single phase in one dimension is de

rived first. Later, it is expanded to three phase flow in three dimensions. The amount

of flow of one phase component depends on the saturation and the mobility of that

particular phase. When the fluid is Newtonian and the flow is slow, as it usually is in

petroleum reservoir situations, the volumetric flux of a phase is proportional to the

potential gradient acting on it and inversely proportional to its viscosity. The coef

ficient of proportionality is the Darcy’s permeability. This is customarily expressed

as the product of the relative permeability of phase 1 (krj), and the absolute Darcy

permeability (k), of the medium to flow when a single fluid entirely fills the pore

space. Mathematically this is expressed as

VP1 (5.1)

where,

v - velocity vector (in m/s)

k - permeability matrix of the porous medium (in m2)

krt - relative permeability of phase 1 (non dimensional)

I’i - viscosity of phase 1 (in kPa.s)

P1 - pressure in phase 1 (in kPa)

Now, consider a single phase (denoted by 1) flow in one dimension (in z direction)

as shown in figure 5.1.

Chapter 5. Flow Continuity Equation

n - porosityS1 - saturation of phase 1

- velocity of phase 1 in z direction- unit weight of phase I

117

0(vj 71) dzôz

S1ndzn dz

dz

Phase ‘I’ in pore fluid

Pore fluid

— Solids

I

Figure 5.1: One dimensional flow of a single phase in an element


Weight of phase 1;

wi=nSi7jdz (5.2)

Incoming mass flux:

v yi (5.3)

Outgoing mass flux:

= + dz (5.4)

Difference between flux coming in and flux going out:

QI_Qo_O(vZz_y1)d 55dt 8z (.)

Rate of storage:

&wlO(nSl7l)d 568t ôt

For conservation of mass, the difference between the incoming and outgoing flux

should be equal to the rate of storage. Thus,

— O(vi 71) — 8(n Si 7i) 5 78z — ôt (.)

Expansion of the partial differentials in equation 5.7 gives

avzl 871 8n &Y1 8S1—7z—+v2i—=7iSi+nSi--+n7i-- (5.8)

Dividing by 7i yields

8v21 v21 871 8m S1 871 8S1——+----—-—=Si——+n—-—+n------ (5.9)

8z 71 8z 7’ at at

Now, consider all five terms in equation 5.9 separately, starting from the left hand

side.


0v218z

By Darcy’s law (equation 5.1) v can be written as

kkr 8P1vz1 = —

____

—

ILl c9z

= (5.10)

and therefore,

92P1= (5.11)

where

kmi - mobility of phase 1

k - intrinsic permeability of the porous medium

[function of void ratio; k = f(e)]

k,.1 - relative permeability of phase 1

[function of saturation; k,.1 = f(S1)]

- viscosity of phase 1

function of temperature and pressure; = f(8, F1)]

vz - velocity of phase 1 in z direction

P1 - pressure in phase 1

271 8z

The change in unit weight due to the increase in pressure can be expressed as,

871 = 71 (5.12)

where


B1 - bulk modulus of phase 1

- unit weight of phase 1

Therefore,

87j — v

yi 8z — B1 öz— kmi 6P1

5 13— B1 8z 8z

This term involves the square of the pressure derivatives and can be neglected

as small compared to the other terms (ERCB, 1975).

3. S1--

By adopting the usual soil mechanics sign convention as compressive strain and

stress positive, it is obvious that

dm = —dEn (5.14)

Thus the above term becomes

(5.15)

where

n - porosity

t -time

- volumetric strain


S1 b-y.4. n—-—

7i ôt

By using equation 5.12 this term can be written as,

51 87j 51 8P1(5.16)

718t B18t

as1.

Summation of saturations of all phase components should always be equal to

unity. Hence, when combining the equations for all the phases, the summation of

this term over all the phases will be zero. Mathematically this can be expressed

as

(5.17)

Since the final equation is to be derived by combining all the phases this term

need not to be considered in detail.

By making the changes to the terms as explained so far, equation 5.9 can be

written as

82P s1 8P1 8Sikmi -ä—-H-S1 --—n- --—n--O (5.18)

Extension of equation 5.18 to three dimensions yields

kmiV2Pi + 51L — — = 0 (5.19)


where

2 b’P ElF t92p(5.20)

Hence, the equations of flow for the three phases in oil sand, in three dimensions,

will be as follow:

for water phase;

(5.21)

for oil phase;

kmo V2P0H-50a

—ns0 ap0

—n = 0 (5.22)

for gas phase;

kmg V2P9 + S9 — — n = 0 (5.23)

where,

km - mobility

S - saturation

B - bulk modulus

and subscripts o, w and g denote oil, water and gas respectively. It should be noted

that in the formulation the capillary pressure between two phases is assumed to be

constant for the increment and therefore, it will not appear in derivatives.

Combining equations 5.21, 5.22 and 5.23 gives

(kmo+kmw+kmg) V2p+_n (++) =0 (5.24)

This can be written as


2 OPkEQ V P + — CEQ -- = 0 (5.25)

where,

kEQ - equivalent hydraulic conductivity

= kmo + kmw + kmg

CEQ - equivalent compressibility— (S0 Sw Sg

Equation 5.25 is similar to the one used by Vaziri (1986) and Srithar (1989), except

for the equivalent conductivity term. They considered the contributions from different

fluid phase components in the compressibility but not in the hydraulic conductivity.

Recently, Settari et al. (1993) have also used an effective hydraulic conductivity term

to model the three-phase fluid which is similar to the equivalent hydraulic conductivity

term derived above. The equivalent hydraulic conductivity is a function of mobilities

of the phases which in turn depend on their relative permeabilities and viscosities.

Evaluations of relative permeabilities and viscosities are described in detail in the next

sections. The equivalent compressibility is a function of saturation and bulk modulus

of individual phase components and the details of its evaluation are described in

section 5.5.

5.3 Permeability of the Porous Medium

The permeability of the porous medium (k) mainly depends on the amount of void

space. Lambe and Whitman (1969) collected considerable experimental data to study

the variation of k with void ratio. Although there was a considerable scatter in the

data, they found that there is a linear relationship between k and a void ratio function

e3/(1 + e) for a wide range of granular materials. It can be argued that various other

relationships could be established for the varition of k with e. However, without the


need for much specific details about the soil, the relationship given by Lambe and

Whitman (1969) is quite reasonable for most engineering purposes. Using Lambe and

Whitman’s relationship, at a particular void ratio of e, k can be expressed as

k =kOe/(l±e) (5.26)e/(1 + eo)

where e0 and k0 are the initial void ratio and the initial permeability of the porous

medium respectively.

5.4 Evaluation of Relative Permeabilities

Measurement of three-phase relative permeability in the laboratory is a difficult and

time consuming task. Due to the complications associated with the three-phase flow

experiments, empirical models have been used extensively in the reservoir simulation

studies. These models use two sets of two-phase experimental data to predict the

three-phase relative permeabilities. Figure 5.2 shows typical results that might be

obtained for such two-phase systems. Figure 5.2(a) shows the relative permeability

variations for an oil-water system and figure 5.2(b) shows the relative permeability

variations for a gas-oil system.

Numerous experimental studies on relative permeabilities have been reported

in the petroleum reservoir engineering literature starting from Leverett and Lewis

(1941). Many review articles have also appeared in the literature (Saraf and McCaf

fery (1981), Parameswar and Maerefat (1986), Baker (1988)) and an assessment of

these studies is beyond the scope of this thesis. However, the general conclusion from

these studies suggests that the functional dependence of relative permeabilities can

be given by

= f(S)

krg = f(S9)


0

krow

k,,,

(b) Gas-oil system

Figure 5.2: Typical two-phase relative permeability variations (after Aziz and Settari,1979)

I

kr

0

I

kr

0Swmaz

SW—,’..

(a) Oil-water system

rg_

ISgc Sgmoz

Sg0


k,.0 = f(S0) (5.27)

The function for the relative permeability of oil, k,.0, is not readily known and it

is estimated from the two-phase data for k,., and k,.09, where, k,.OW is the relative

permeability of oil in an oil-water system and k,.09 is the relative permeability of oil

in an oil-gas system. Their functional dependence are given by

k,.0 = f(SW)

k,.09 = f(S9) (5.28)

The simplest way of estimating k,.0 would be,

k,.0 = k,.0 k,.09 (5.29)

Two more accurate models have been proposed by Stone (1970), only the first of

which is considered here. In this model, Stone (1970) defines normalized saturations

as

S= —

S S (5.30)wc om

= SO SomSo Sam (5.31)

wc om

5;= 15S95

(5.32)wc am

Where, S, is called the critical or connate water saturation at which water starts

to flow. When S,, is less than S, the relative permeability of water k,., will be zero.

Sam is called the residual oil saturation at which oil ceases to flow when it displaced

simultaneously by water and gas. If S is less than 5om, k,.0 will be zero.

According to Stone (1970), the relative permeability of oil in a three-phase system

is given by

k,.0 = S (5.33)


The factors i3 and ,i3 are determined from the end conditions that equation 5.33

should match the two-phase data at the extreme points. The two extreme cases of

= 0 and SL, = give

— 1 — S

k,.09— 1 — S’9

(5.34)

(5.35)

The region of mobile oil phase (i.e. k,.0 > 0) predicted by Stone’s model I is shown

in figure 5.3 on the ternary diagram assuming increasing 5W and S9. For conditions

depicted by point outside the hatched area, the relative permeability of oil will be

zero.

100%WATER

Figure 5.3: Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979)

100%GASE1S Som

I,. 100%OIL

Aziz and Settari (1979) modified Stone’s model because Stone’s model will reduce


exactly to two-phase data only if the relative permeabilities at the end points are

equal to one, i.e., krow(Swc) = krog(Sg = 0) = 1. They suggest that the oil-gas data

has to be measured in the presence of connate water saturation. In that case, an oil-

water system at S, and an oil-gas system at S = 0 are physically identical. Both

systems will have, S = S and S0 = 1 — S at 59 = 0. At these conditions, the

relative permeabilities will be

krow(Swc) = krog(Sg = 0) = krocw (5.36)

Then, the modified form of Stone’s equations will be

k,.0 = S krocw w /39 (5.37)

k— (5.38)

—

k,.0911 c’ (. )

,.0cwI — LI9

Kokal and Maini (1990) claim that Aziz and Settari’s method has problems be

cause:

1. Measurements of two-phase oil-gas data are not necessarily obtained at connate

water saturation

2. The relative permeability at connate water saturation in an oil-water system

generally will not be equal to that in an oil-gas system

Kokal and Maini (1990) further modified Stone’s model by incorporating another

normalizing factor. After these modifications, the relevant equations needed to predict

the relative permeability of oil are


k,.0 = s (k,?09S;+k,?,&S)(5.40)

rOW4wko(1SI

a rog“9k0 (1Srog\ g

where,

k,?0 - relative permeability of oil at connate water saturation

in a water-oil system

k,?09 - relative permeability of oil at zero gas saturation

in an oil-gas system

When k,?0 = k,?og, the above model reduces to the one given by Aziz and Settari

(1979). Kokal and Maini (1990) compared model predictions against measured data

and found very good agreement. The best comparison given in their paper is shown

in figure 5.4.

From the discussion so far in this section, it can be concluded that the relative

permeabilities in three-phase system can be written as

= f(S) (5.43)

k,.9 = f(S9) (5.44)

k,.0 = f(k,.0,k,.09, Sw, so, S) (5.45)

k,.0 = f(5) (5.46)


OILExpenmental

— Calculated0.75

0.70

0.60

0.50

0.40

0.30020

0.100.01

. . ..

WATER “ “ “ “ ‘.‘ “ ‘I’ ‘ GAS

Figure 5.4: Comparison of calculated and experimental three-phase oil relative permeability (after Kokal and Maini, 1990)


a)Ea,a)>

a)

— f(S9) (5.47)

However, to implement the relative permeability variations in a numerical simu

lation the variations should be expressed as mathematical functions. Polikar et al.

(1989) suggest that these variations can be well represented by power law functions.

Thus, mathematically the variations can be given as

= C1(S1 — C2)c3 (5.48)

where C1,C2 and C3 are constants. Figure 5.5 shows a comparison of experimental

data with calculated values using the power law functions.

1.2k = 2.769 (0.80 - Sw) 1.996

k = 1.820 (Sw - 0.20) 2.735

row

1

0.8

0.6

0.4

0.2

0

‘ k rw

Symbols- ExperimentalLines - Correlation

0 0.2 0.4 0.6 0.8

w

1 1.2

Figure 5.5: Comparison of calculated and experimental relative permeabilities usingpower law functions


In summary, the relevant parameters needed to calculate the relative permeabil

ities of water, oil and gas phases are given in table 5.1. An example showing the

details of the calculations of the relative permeabilities and the resulting equivalent

permeability is given in appendix B, to provide a better understanding of the steps

involved.

5.5 Viscosity of the Pore Fluid Components

5.5.1 Viscosity of Oil

The mobility of an individual phase in a three-phase system depends on the viscosity

of the phase component. Viscosities of the fluid components are generally strong

functions of temperature and to some extent depend on the pressure as well.

Viscosity of oil plays a very important role in reservoir engineering. Crude oil

cannot flow at the ambient temperatures because of its high viscosity. The oil recovery

methods require some form of heating to reduce the viscosity and thereby increase

mobility. For example, the viscosity of Cold Lake bitumen is 20, 000 mPa.s at 30°C

and 100 mPa.s at 100°C, i.e., a 200-fold reduction at high temperature. There are

some correlations for the viscosity of oil available in the literature. Among those

correlations, the one proposed by Puttagunta et al. (1988) has been selected in this

study for the following reasons:

1. It requires only a single viscosity value at 30°C and 1 atmosphere as input data.

2. Generally, oil viscosity varies widely from deposit to deposit and this correlation

fits the viscosity variation of most bitumens reasonably well.

The correlation proposed by Puttaguntta et al. (1988) is expressed by the follow

ing equation:


Table 5.1: Parameters needed for relative permeability calculations

Parameter DescriptionConnate or critical water saturation

Som Residual oil saturation

A1,A2,A3 Parameters for variation of k with Si,,in water-oil system = A1(S — A2)A3]

B1,B2,B3 Parameters for variation of krow with 5win water-oil system [kro B1(B2 — Sw)B3]

C1,C2,C3 Parameters for variation of krg with Sgin oil-gas system [krg = C1(S9 — C2)c3]

D1,D2,D3 Parameters for variation of krog with S9in oil-gas system [k.09 = D1(D2— 59)D3]

Relative permeability of oil at connate water saturationin water-oil system

k,?09 Relative permeability of oil at zero gas saturationin oil-gas system


lfl(9,p) 2.3026( 8-30

— 3.0020] + B0 F exp(d 6) (5.49)+ 30315)

where,

b log .Lt(3o,o) + 3.0020

a = 0.0066940.b + 3.5364

B0 = 0.0047424.b + 0.0081709

d = —0.0015646.b + 0.0061814

8 - temperature in degrees Celsius

F - pressure in MFa gauge

- viscosity of oil in Fa.s at 30°C and 1 atmosphere (0 gauge)

Figure 5.6 shows the comparison of this correlation with experimental results for

Cold Lake and Wabasca bitumens. The above correlation is implemented in the

finite element program CONOIL. However there is an option in CONOIL to read and

interpolate user specified viscosity-temperature data, in case this correlation does not

hold for a particular bitumen.

5.5.2 Viscosity of Water

The viscosity of water does not change as drastically as that of oil. For instance, at

30°C the viscosity of water is 0.8 mPa.s and at 100°C, it is 0.28 mFa.s. A change

of 70°C in temperature causes a reduction in viscosity by a factor of 3 as compared

to 200 for oil. The viscosity-temperature data for water are well established and

can be obtained from the international critical tables. The viscosity of water is well

represented by the following equation:

= (b+8)(5.50)

where


50000

— empirical equation- * experimental

10000 -

‘‘500o *

SC0SI 1000TY 500

mPa

100S

50. *

10

0 20 40 50 80 100 120

TEEATUR.E, C

a) Wabasca bitumen

50000 -

\ — empirical equation* experimental

10000

VI 5000 -

SCo

4iooo.

500

P

• 100 -

so

10 I I

23 40 50 80 100 120 140

TE’ERATURE, C

b) Cold Lake bitumen

Figure 5.6: Experimental and predicted values of viscosity (after Puttagunta et al.,1988)


- viscosity of water

- temperature

a, b, n - constants

It is reasonable to assume the water phase in the oil sand will have the same prop

erties. These data from the International Critical Tables are reproduced in appendix

B and built into the computer program CONOIL. There is also an option to read and

interpolate from any other user specified data.

5.5.3 Viscosity of Gas

There is not much information available about the viscosity of gas in the recent

literature in petroleum engineering. Carr et al. (1954) carried out some work on

the viscosity of hydrocarbon gases as a function of pressure and temperature. The

viscosity of gas appears to be equally dependent on pressure and temperature, but the

variations are not very significant. for example, at atmospheric pressure and at 30°C,

the viscosity of paraffin hydrocarbon gases (molecular weight of 70) is 0.007 mPa.s

and at 200°C it is 0.0105 mPa.s, i.e., increases by only a factor of 1.5. The charts

given in Carr et al. (1954) are given in appendix B with an example calculation.

There is no correlation readily available for the data. The viscosity of the gas is

very low and hence its mobility will be very high compared to that of water and oil.

Therefore, it may not be unreasonable to assume a constant viscosity for gas (for

instance, 0.01 mPa.s). However, there is an option available in CONOIL as for water

and oil, to input any other data at the user’s choice.

5.6 Compressibility of the Pore Fluid Components

In the final flow equation derived (equation 5.25), the equivalent compressibility of

the pore fluid is defined as


CEQ =(+±) (5.51)

The bulk moduli of the water and oil can be assumed constant, though they

depend slightly on pressure. The important parameter that affects the equivalent

compressibility is the comprssibility of gas. If there is more gas present in the pore

fluid, it will be more compressible. The compressibility of gas can be determined using

the gas laws. The basic gas laws governing the volume and pressure relationships are

Boyle’s law and Henry’s law. According to Boyle’s law, under constant temperature

conditions,

P9V=w9RT (5.52)

where,

P9 - absolute pressure of gas

T - absolute temperature

R - universal gas constant

V, - volume of gas

Wg - weight of gas

Under undrained conditions, the weight of gas does not change and therefore,

equation 5.52 can be written as

(5.53)

where K is a constant.

Gas can be present in both the dissolved and free states. According to Henry’s

law (Sisler et al., 1953); the weight of gas dissolved in a fixed quantity of a liquid,

at constant temperature, is directly proportional to the absolute pressure of the gas

above the solution. Mathematically, this can be written as


0 0

(5.54)

where

- weight of dissolved gas

and the superscripts 0 and 1 refer to the initial and final conditions, respectively.

In other words, Henry’s law implies that the volume of dissolved gas in a fixed

quantity of liquid is constant at a constant temperature and at a confining pressure

F, when the volume is measured at F. Thus

Vd9 = H V0 (5.55)

where

H - Henry’s constant, which is temperature dependent and,

over a wide range of pressure, is also pressure dependent

V0 - volume of oil

Since the volume of dissolved gas is constant, free and dissolved gas components

can be combined. Then application of Boyle’s law to the entire volume yields (Fred

lund, 1976)

(5.56)

where V19 is the volume of free gas.

Rearranging the terms yields,

ID1IT? in— rgkVdg + Vf

V19—

0 — Vdg

9

By differentiating equation 5.57,


8V P91(Vdg+Vj9)558)8P9° — (Pg0)2

By adopting the sign convention that compression is positive,

1 - (av1/v1)- P(V9+v)559)Bg 8P9° — V°9(P90)2

Now,

Vd9 = ThiS0

TT0V =

=nS

Pg = Pa+F+Pc (5.60)

where,

- saturation of oil

S9 - saturation of gas

n - porosity

- atmospheric pressure

P - pressure in oil

- capillary pressure

By substituting these last expressions into equation 5.59

iP91(HS0+S)561)B9 S(P9°)2 (

Generally, in an incremental procedure the values used are estimated at the be

ginning of the increment. Therefore, from equation 5.61 the value of (S9/B9)at the

beginning of an increment can be given as

59 HS0+59562

BgPa±P+Pc


If the capillary effects are neglected (i.e. P = 0), equation 5.62 will be similar

to the one derived by Bishop and Henkel (1957). Equation 5.62 is slightly different

from the equation derived by Vaziri (1986). In Vaziri’s expression capillary pressure

was assumed to be a function of capillary radius and the capillary radius in turn

was assumed to be a function of saturation. He also included a derivative term

of capillary pressure with respect to saturation which is not significant since the

changes in saturation will be very small. In addition, having this derivative term

is inconsistent because, in his formulation to derive the flow equation, the capillary

pressure was assumed constant over an incremental step. The expression given by

equation 5.62 has a practical advantage because, in reservoir engineering, the variation

of capillary pressure with saturation is readily available, whereas the capillary radius,

critical capillary radius and surface tension values which are needed data for Vaziri’s

expression are not readily available. The capillary pressure P can be well represented

by a power function similar to the ones used for relative permeabilities.

= E1(S9 E2)E3 (5.63)

where E1,E2 and E3 are constants.

Therefore, by substituting equation 5.62 in equation 5.51, the equivalent com

pressibility can be written as

S, SL (S9+HS0)CEQ=fl (5.64)

5.7 Incorporation of Temperature Effects

The fluid flow model described so far is for isothermal conditions and does not in

clude temperature effects. The final equation obtained for multiphase flow (equation

5.25) can be considered as an equation of volume compatibility which is derived from


the basic equation of conservation of mass. If the temperature effects are included,

equation 5.25 will become (Srithar (1989), Booker and Savvidou (1985))

kEQ V2F + — CEQ + aEQ = 0 (5.65)

where

cEQ - equivalent coefficient of thermal expansion

- temperature

The equivalent coefficient of thermal expansion can be obtained by considering

the coefficients of thermal expansion of the individual soil constituents and their

proportions of the volume, i.e.,

0EQ = a8(1 — n) + nSa + flSoto + flSgQg (5.66)

where subscripts s, w, o and g denote solid, water, oil and gas respectively.

The coefficient of thermal expansion of solids, water and oil can be measured in

the laboratory. The coefficient of thermal expansion of gas can be obtained from the

universal gas law. According to gas law,

Povo Ply1= (5.67)

To evaluate the coefficient of thermal expansion, only the volume change due to

temperature change has to be considered. Thus, by assuming constant pressure

l 80

V0Vl—Vo

_____

= 80(5.68)

By adopting the usual notation

= (5.69)


Hence,

a9 = (5.70)

It should be noted that the temperature in the above equation should be absolute

temperature (i.e. in K).

5.8 Discussion

In this chapter, flow continuity equations for individual phases have been derived.

Those have been later combined and an equivalent single phase flow continuity equa

tion has been obtained. The effects of individual phases on compressibility and hy

draulic conductivity have been modelled by equivalent compressibility and hydraulic

conductivity terms. The flow continuity equation will be solved together with the

force equilibrium equation as a consolidation problem. The quantities of flow of in

dividual phases can be estimated from the total amount of flow predicted and from

the knowledge of the relative permeabilities.

In reservoir engineering, only the flow equations for the individual phases (equa

tions 5.21, 5.22 and 5.23) are generally solved and not in combined form as formulated

in this study. The saturations and fluid pressures are not assumed to be constants,

rather they are considered as the dependent variables. To analyze the flow there

will be six degrees of freedom per node and the corresponding nodal variables are

S, S0,S9, Pt,,, P0 and P9. The solution of the problem therefore requires the follow

ing three additional equations:

(5.71)

P0— P,L? = f(S, S0) (5.72)

Pg — Po f(S0,S9) (5.73)


Compared to the flow analysis in reservoir engineering, the major disadvantage of

the analytical model proposed here is that the treatment of multi-phase fluid as an

equivalent single phase fluid. This kind of analytical model is adequate for coupled

stress, deformati&n and flow analyses, but may not be effective if a detailed fluid flow

analysis is required. If detailed results about the flow are required, a separate rigorous

flow analysis may be necessary. However, the results from the stress-deformation

analysis and the rigorous flow analysis should be looked at together to obtain a

complete picture.

There are several advantages in the analytical model suggested in this study. In

reservoir engineering, the stress distribution and the deformation through the porous

medium are generally not considered. But the real problem at hand is a coupled

stress, deformation and flow problem and the proposed analytical model in this study

addresses all these concerns. The combined form of the flow continuity equation

makes the formulation simpler and significantly reduces the number of degrees of

freedom, computation time and other such factors.

Chapter 6

Analytical and Finite Element Formulation

6.1 Introduction

Oil recovery by steam injection from heavy oil reservoirs is a coupled stress, defor

mation and fluid flow problem. Therefore, a realistic analytical model should include

the fluid flow behaviour and the mechanical behaviour of the sand matrix. Modelling

of the stress-strain behaviour of the sand skeleton and the fluid flow behaviour with

multi-phase fluid has been described in chapters 3 and 5 respectively. This chapter

describes the development of an analytical model which couples the stress-strain and

fluid flow behaviour, and a solution scheme using finite element procedure.

Basically, the problem in hand is considered as a consolidation phenomenon. The

analytical models used in the consolidation analysis are mainly based on theories

developed by Terzaghi (1923) and Biot (1941). Terzaghi’s theory is restricted to a

one dimensional problem under a constant load. Biot extended Terzaghi’s theory to

three dimensions and for any arbitrary load variable with time. Both Terzaghi’s and

Biot’s theories assume a linear elastic stress-strain behaviour and an incompressible

pore fluid.

Closed form solutions for the consolidation equations have been derived by a

number of researchers, but only for very simplified geometry conditions and for linear

elastic material behaviour. For instance, De Josselin de Jong (1957) obtained a solu

tion for consolidation under a uniformly loaded circular area on a semi-infinite soil.

MacNamee and Gibson (1960) obtained solutions to plane strain and axisymmetric

problems of strip and circular footings on a consolidating half space. Booker (1974)

144

Chapter 6. Analytical and Finite Element Formulation 145

derived solutions for square, circular and strip footings. A solution for consolida

tion around a point heat source in a saturated soil mass was derived by Booker and

Savvidov (1985).

The computer aided techniques such as finite element methods have made the

consolidation analysis possible for more complicated boundary conditions and for

more realistic material behaviour. Sandhu (1968) developed the first finite element

formulation for two dimensional consolidation using variational principles. Sandhu

and Wilson (1969), Christian and Boehmer (1970) and Hawang et al. (1972) used the

finite element method to solve the general consolidation problem. Ghaboussi and Wil

son (1973) took the compressibility of the pore fluid also into account. Ghaboussi and

Kim (1982) analyzed consolidation in saturated and unsaturated soils with nonlinear

skeleton behaviour and nonlinear fluid compressibility. Chang and Duncan (1983)

took account of the variation of permeability due to the changes in void ratio and

saturation. Byrne and Vaziri (1986) and Srithar et al. (1990) included the nonlin

ear skeleton behaviour, nonlinear compressibility, variations in permeability and the

effects of temperature changes in the overall consolidation phenomenon. The analyt

ical model developed in this study, is based on Biot’s consolidation theory. However,

the analytical equations are extended to include elasto-plastic behaviour of the sand

skeleton, the effects of multi-phase fluid in compressibility and permeability and the

effects of temperature changes. The derived equations are solved by finite element

procedure using Galerkin’s weighted residual scheme. The details of the formulation

of the analytical equations and the finite element procedure are described herein.

6.2 Analytical Formulation

The basic equations governing the consolidation problem with changes in temperature

are as follows:


1. Equilibrium equation.

2. Flow continuity equation.

3. Thermal energy balance.

4. Boundary Conditions.

The thermal energy balance will give the temperature profile and its variation

with time over the domain considered. In the analytical formulation presented in

this study, the thermal energy balance is not included. It has been solved separately

with the heat flow boundary conditions by a separate program. The temperature

profile and its variation with time is evaluated and considered to be an input to the

analytical model presented in this study. However, the effects of these temperature

changes on the stress-strain behaviour and the fluid flow are included in the analytical

formulation.

6.2.1 Equilibrium Equation

Using the conventional Cartesian tensor notation, the equilibrium of a given body is

given by

— F2 = 0 (6.1)

where

- total stress tensor

- body force vector

subscript j = -

By assuming the geostatic body forces as initial stresses and considering only the

changes in body forces and stresses, the incremental form of the above equation can

be expressed as


— = 0 (6.2)

The total stresses are the sum of the effective stresses and the pore pressures.

Mathematically, this can be written as

= oj H- P Sj (6.3)

where

- effective stress tensor

P - pore pressure

- Kronecker delta

From chapter 3, the incremental stress-strain relation including the effects of tem

perature changes can be written as (see equation 3.73)

= LE,d H- Dkz (6.4)

where

- tensor relating incremental effective stress and strain

- strain tensor

- strain due to the change in temperature

The strains can be expressed in terms of displacements as

= (U + (6.5)

where

- displacement vector

Combining equations 6.3, 6.4 and 6.5 and substituting into equation 6.2 yields,

[Dk1 (Uk,I + LU1,k)] + a + ie1]— = 0 (6.6)


6.2.2 Flow Continuity Equation

The flow continuity equation for a multi-phase fluid including temperature induced

volume changes was derived in chapter 5. The final equation (see equation 5.78) can

be written in tensor notation as

[(kEQ)F] —1 +czEQ = 0 (6.7)

where

kEQ - equivalent hydraulic conductivity tensor

P - pore pressure

U - displacement vector

CEQ - equivalent compressibility

aEQ - equivalent coefficient of thermal expansion

6 - temperature

and superscript dot denotes the partial differentiation with respect to time (8 /8t).

Equations 6.6 and 6.7 are the resulting equations that have to be solved in the

consolidation analysis. In these equations, the fundamental unknowns to be solved

are the displacements, U, and the pore pressure, P. The unknowns are solved by

finite element procedure using Galerkin’s weighted residual scheme.

6.2.3 Boundary Conditions

To define the problem, both the displacement and the flow boundary conditions must

be specified. For the class of problems considered in this study, the following boundary

conditions can be specified.

For the displacement boundary conditions, a part of the surface, can be sub

jected to known applied traction, while the reminder of the surface, SD, can be

subjected to specified displacements, U, which may be zero.


For the flow boundary conditions, it is assumed that part of the boundary surface,

Sp, is subjected to specified pore pressures, F, which can be set to zero to simulate

a free draining surface. The reminder of the surface, 5q is considered impermeable,

i.e. there is no flow across the boundary.

Mathematically, these boundary conditions can be expressed as

a1n3=t for t0 (6.8)

U = (J for t 0 (6.9)

P=P for t>0 (6.10)

for t0 (6.11)

where n is the normal vector to the boundary surface and the bar symbol indicates

a prescribed quantity.

To complete the description of the problem, the initial conditions must also be

defined. At t = 0, since there is no time for the fluid to be expelled, the volume

change in the pore fluid and in the soil skeleton must be equal. Thus,

tSv = CEQ P at t = 0 (612)

6.3 Drained and Undrained Analyses

The drained and undrained analyses can be easily performed by considering only

the equilibrium equation (equation 6.6). The flow continuity equation need not be

considered under drained and undrained conditions. The drained analysis is quite

straight forward as it just involves solving the equilibrium equation. However, to

perform an undrained analysis some modifications have to be made.

Generally, the undrained response is analyzed with total stress parameters and

the analytical formulation has to be in terms of total stresses. If the pore pressures

are desired, they are commonly computed from the Skempton equation relating total


stress changes to pore pressure parameters. To use the effective stress formulation for

undrained analysis, Byrne and Vaziri (1986) adopted an approach similar to the one

proposed by Naylor (1973). In this approach, the stiffness matrix for a total stress

analysis is obtained from the effective stress parameters and from the compressibility

of the fluid components as described in this section. The solution procedure is then

carried out in the usual manner for a total stress analysis to obtain deformations. The

pore pressures can be evaluated from the computed deformations using the relative

contributions of the pore fluid and the skeleton, without the use of the Skempton

equation.

The incremental effective stresses are related to the incremental strains by the

following relationship:

{o’} = [D’] {L} (6.13)

where

{ e} - strain vector

{ o’} - effective stress vector

[D’] - matrix relating effective stress and strain

The volumetric strain can be expressed as

= {m}T{e} (6.14)

where {m}T = {1 1 1 0 0 0} , is a vector selected such that only direct strains will

be involved in the volumetric strain.

For undrained conditions, the volume compatibility requires that the volume

change in the skeleton equals the volume change in the fluid, i.e.,

= (6.15)

where


(Lc)j - volume change in the pore fluid

n - porosity

In chapter 5, an equivalent compressibility has been obtained by considering all

the fluid components. Based on this approach, the changes in pore pressure can be

expressed as

(6.16)

where

- change in pore pressure

CEQ - equivalent compressibility

Substitution of equations 6.14 and 6.15 into equation 6.16 gives

= 1{m}T{E} (6.17)

CEQ

From the definition of effective stress

{o-} = {&r’} + {m}i.P (6.18)

Substituting equations 6.17 and 6.13 in equation 6.18 yields

= [[D’l +1

{m}{m}T] {e} = [D]{e} (6.19)CEQ ‘

Equation 6.19 adds the contributions of both the skeleton and the pore fluid to

express the stress-strain relation in terms of total stress. Thus, the matrix [D] for the

total stress analysis is given by

[D] = [D’] +1

{m}{m}T (6.20)CEQ


Equation 6.19 is used in the finite element formulation for undrained conditions.

The pore pressure is not an unknown in the resulting system of equations, but is

obtained from equation 6.17, once the deformations are computed.

Byrne and Vaziri (1986) claimed that this method has definite advantages such

as adaptability for saturated or unsaturated soils and for any stress-strain relation.

In particular, this method gives stable solutions when the effective stresses go to zero

and all of the load is carried by the pore fluid. For an incompressible fluid, CEQ

becomes zero and the above formulation becomes ill-conditioned. However, this can

be overcome by setting the value of CEQ to a suitably low but finite value (Naylor,

1973).

6.4 Finite Element Formulation

The equations governing the consolidation with multi-phase fluid and temperature

effects have been derived and are given by equations 6.6 and 6.7. The best method of

obtaining a solution for these equations is to use a numerical technique such as finite

element method. The finite element procedure can be formulated in a number of ways.

For instance, Sandhu and Wilson (1969) used a Gurtin type variational principle.

Booker and Small (1975) employed a variational theorem involving Laplace trans

formations. Christian and Boehmer (1970), Carter (1977) and Small et al. (1976)

obtained the solutions through the principle of virtual work. Hwang et al. (1972)

and Chang and Duncan (1983) used the weighted residual technique to develop the

finite element formulation.

The choice of the different approaches depends on the type of the problem and the

boundary conditions on one hand, and the knowledge of the mathematics involved

on the other hand. In this study, Galerkin’s weighted residual scheme is used to

develop the finite element formulations. The weighted residual scheme is quite straight

forward, has relatively less mathematics involved, and is less error prone. In the


Galerkin scheme only a single application of Green’s theorem is needed to obtain a

set of integral equations. These equations can be easily turned into matrix form and

solved. However, it should be noted that regardless of the approach used, whether

weighted residual or variational principle, the end results will be the same.

From the previous section, the governing differential equations to be solved are,

(Uk,j + UZk)j + + — 0 (6.21)

[(kEQ) + — CEQ F + aEQ = 0 (6.22)

To develop the finite element formulation for these equations, the domain being

analyzed is subdivided into a finite number of elements. The quantities of the four

independent variables within each of the elements, U and F, are approximately repre

sented by means of shape functions and their values at the nodes. The equations 6.21

and 6.22 have at most second order derivatives of displacements and pore pressures.

However, by applying Green’s theorem, it can be reduced to first order. Therefore, to

solve the resulting integral equations the shape functions for displacements and the

pore pressure should be continuous. Hence, the displacement and the pore pressure

fields within the element can be written as

U = N e (6.23)

(6.24)

where

5eT= {S, S2, . . , S} (nodal displacements)

q = {qi, q,.. . , q} (nodal pore pressures)

U’ - displacement field

- pore pressure field


N - shape functions for displacements

- shape functions for pore pressures

U” and F” are approximate solutions and substituting these values into equations

6.21 and 6.22 will not exactly satisfy the equations, but will give some residual errors

as follow:

[Dkl ± + z1 + [D — (6.25)

[(kEQ F], + U”,1 — CEQ F” + aEQ S (6.26)

In the weighted residual scheme, these residual errors are minimized in some

fashion to give the best approximate solution. Thus, for the best solution

jwrdv=O (6.27)

where

r - residual error

w - weighting function

In Galerkin’s scheme the weighting functions are chosen to be the same as the

assumed shape functions. Then the following equations can be obtained to minimize

the residual errors r1 and r2:

j Nr1dv = 0 (6.28)

jNpr2dvzO (6.29)

The strains and the derivative of the pore pressure within an element can be

written as


(6.30)

= m’ (6.31)

B Iq6 (6.32)

where

mT ={1 1 100 0}

B & B - shape function derivatives

Green’s theorem for integration involving two functions, and over the domain

can be expressed as

J ç V dIZ= j c (V) dF

— j V V d1 (6.33)

where, I’ is the boundary around and i is the normal to the boundary.

By substituting equations 6.23 to 6.26 and 6.30 to 6.32 into equations 6.28 and

6.29, and by applying Green’s theorem, one obtains

j BDB,4Sdv + / B’mNqdv = / NTds + j NFdv - / B’Dedv

(6.34)

— / BkEq Bqdv H- / NpTmTBuSdv — / CEQN1’Ndv = — / NpTaeq6dv (6.35)

For a time increment t the above set of equations can be written in matrix form

as

[K] {i6} + [L] {q} = {A} (6.36)


[L]T {S} - t [E] {q} - [G] {q} = -{C} (6.37)

where

[K] =fBDBdv

[U =fBmNdv

rr1l p DTI r,L-’J — Jv -0p nEQ L’p

[G] =fCEQNNpdv

{LA} = f8 NTds + f NFdv— f B,D6ed

{zXC} fvNp0eqMdt

Equation 6.37 is considered over a time increment t, and therefore, the term q

in that equation has to be expressed as,

q = (1—

a)qt + aqt (6.38)

where a is a parameter corresponding to some integration rule. For example, a = 1/2

implies trapezoidal rule, a = 0 implies a fully explicit method and a = 1 gives a fully

implicit method. Booker (1974) showed that for an unconditionally stable numerical

integration a 1. In the formulation here, the value of a is assumed to be 1,

i.e. a fully implicit method. Thus, the term q in equation 6.37 can be given as,

q = qt+t = qt + q (6.39)

Substitution of equation 6.39 into equation 6.37 yields,

[U]T {8} — t [E] {qt + Lq} — [C] {q} = —{zC} (6.40)

By rearranging the terms,

[U]T {zS} — zSt [[E] — [G]] {q} = —{zC} + [E]qtt (6.41)


By combining equations 6.36 and 6.41 and writing them in a full matrix form

gives,

[K] [L] z8 1A ‘1(6.42)

[L]T [[E] Lt — [G]] Lq J [E] /.t{qt} — {tC} JBy changing the notation, equation 6.42 can be written in the usual matrix form

as,

[K] [L] ILS1 1A1(6.43)

[L]T [E’] q J LC’ Jwhere,

{LW’} = tt[E}{qt} - {C}

[E’] =t[E]-[G]

Equation 6.43 gives the matrix equation to be solved for an element. From the

element matrix equations a global matrix equation is formed and solved for displace

ment and pore pressure unknowns. Stresses and strains are then evaluated from the

displacements.

It should be noted that it may not be possible to use the above consolidation

routine to get the initial condition results, i.e. at t = 0. This is because for Lt = 0

{zC’} = 0 and [E’] = [G].

If the fluid is incompressible [G] will become zero and equation 6.43 will become ill

conditioned. For this situation an appropriate solution can be obtained by assuming

a small value for t. This will circumvent the ill-conditioning. However, a better

way to get around this problem is to use the undrained routine to obtain the initial

condition and then use the consolidation routine.


6.5 Finite Elements and the Procedure Adopted

The principal steps and the details such as obtaining shape functions, its derivatives,

formulation of stiffness matrix, numerical integration, etc., can be found in any stan

dard finite element text book. Therefore, only a summary with some discussions on

certain issues which are important for the class of problems considered in this study,

are given in this section. The developed analytical model has been incorporated in

an existing 2-dimensional finite element code, CONOIL-Il (Srithar (1989)) and also

in a new 3-dimensional finite element code, CONOIL-Ill. The following subsections

address the key aspects in the development of these finite element codes.

6.5.1 Selection of Elements

The choice of the finite elements has been an important issue when analyzing con

solidation problems. Different researchers used different element types. Sandhu and

Wilson (1969) introduced a composite element, consisting of a six-noded triangle for

the displacements expansion, and only three nodes being used for the pore pressure ex

pansion. The displacements varied quadratically over the element, while the stresses

and strains obtained by differentiating the displacements varied linearly. Since the

pore pressures are expressed in terms of three nodal values, they vary linearly too.

Therefore, the element has the same order of expansion for both stress components

and pore pressures. Yooko et al. (1971a) used several different elements, all of which

used the same expansion for the displacements and for the pore pressures. This makes

N = N for any choice of element and the relevant matrices can be derived easily.

The examples they presented include a two noded bar element, a three noded ax

isymmetric triangular ring element an a four noded rectangle. However, Yooko et

al. (1971b) had difficulties in obtaining reasonable results for the initial undrained

conditions. Sandhu et al. (1977) also compared several finite elements and concluded


that the elements which had the same expansion for displacement and pore pressures

do not give satisfactory answers at the initial stages of consolidation. However, they

claimed that at later stages of consolidation, the differences in the results for different

element types are insignificant.

Ghaboussi and Wilson (1973) used an isoparametric element of four nodes with

the standard expansion for pore pressures and two additional nonconforming degrees

of freedom for the displacement expansion. The two additional degrees of freedom are

eliminated by static condensation after the element stiffness is completed. However,

this procedure does not give the same expansion for pore pressures and stresses, but

uses a lower order expansion for pore pressures than for displacement.

In the 2-dimensional finite element code employed in this study, element types sim

ilar to those proposed by Sandhu and Wilson (1969) are used. Figure 6.1 shows the

two different triangular elements used for consolidation analysis in the 2-dimensional

code. Figure 6.2 shows the element types available in the 3-dimensional code. The

eight-noded brick element uses the same expansions for pore pressures and displace

ments, whereas the 20-noded brick element uses different shape functions for displace

ments and pore pressures.

6.5.2 Nonlinear Analysis

The solution of the nonlinear problems by the finite element method is usually

achieved by one of the following techniques:

1. Incremental or stepwise procedures

2. Iterative or Newton method

3. Step-iterative or mixed procedures

The method employed herein is a form of the mixed procedure which follows

the midpoint Runge-Kutta or modified Euler method. In this scheme, two cycles


A Displacement nodes (2 d.o.f)

Q Pore pressure nodes (1 d.o.f)

Linear strain triangle

6 displacement nodes

3 pore pressure nodes

6 nodes and 15 d.o.f.

Cubic strain triangle

15 displacement nodes

10 pore pressure nodes

22 nodes and 40 d.o.f.

Figure 6,1: Finite Element Types Used in 2-Dimensional Analysis

Chapter 6. Analytical and Finite Element Formulation

5I

161

• Corner nodes = 8

D.o.f. per node = 4

Internal nodes = 0

D.o.fper node = 0

8-Nodded Brick Element


D.o.f per node = 4

A Internal nodes = 12

D.o.f. per node = 3

20-Nodded Brick Element

8

A

157

,.

14

A

6

188

20

44 1

19Li16

s34 A-4-10

/11A

12

Figure 6.2: Finite Element Types Used in 3-Dimensional Analysis


of analysis are performed for each load increment. In the first cycle of analysis,

parameters based on the initial .conditions of the increment are used.. At the end of

first cycle, parameters at the midpoint of the load increment are computed. In the

second cycle, the midpoint parameters are used to analyze the load increment and the

final results are evaluated. To obtain more accurate results, this process would have

to be continued until the difference between successive results satisfies the specified

tolerance. Such an iterative procedure can increase the computer time drastically

and therefore, was not employed. However, an improvement in the results is made

by estimating the imbalance load at the end of second cycle and adding that to the

next load increment.

6.5.3 Solution Scheme

Selection of the method for solving the simultaneous algebraic equations is a major

factor influencing the efficiency of any finite element program, and there are variety of

solution techniques to choose from. Essentially, there are two classes of methods; one

is the direct solution methods and the other is the iterative solution methods. The

direct methods use a number of exactly predetermined steps and operations, whereas

the iterative methods make an approximation to solve the equations.

The most effective direct solution methods are basically variations of the Gaussian

elimination method. Most of the methods take advantage of specific properties of the

stiffness matrix, its symmetry, its positive definiteness or its banded nature to reduce

the number of operations and the storage requirements to accomplish a solution.

Bathe and Wilson (1970) and Meyer (1973) discussed the relative merits of the current

popular methods, and both of these references contain extensive bibliography.

The frontal solution scheme for symmetric matrices (Irons, 1970) and for unsym

metric matrices (Hood, 1976) have been employed in the finite element codes. In the

frontal solution scheme, the element stiffness matrices are assembled and solved by


Gaussian elimination and back substitution process, but the overall global stiffness

matrix is never formed. The variables are introduced at a later stage and eliminated

earlier than in most of the other direct solution methods. Since the variables are

eliminated as soon as conceivably possible, the operations with zero coefficients are

minimized and the total arithmetic operations are fewer. As a result, it is faster and

requires less core memory than band routines. In addition, it is not necessary to

apply a stringent node numbering scheme. Its efficiency is essentially a function of

element numbering.

Theoretically, the frontal solution scheme will always perform better or at least

as well as the bandwidth solving routines in terms of accuracy and efficiency (Irons,

1970; Irons and Ahmad, 1980). Some comparisons have already given in the literature

to substantiate this claim (eg: Sloan 1981; Light and Luxmore, 1977; Hood, 1976).

The frontal solution scheme is specially attractive for unsymmetric matrices because

less computer storage is required. The stress-strain model considered in this study

deals with a nonassociated flow rule which results in an unsymmetric stiffness matrix

and therefore, using the frontal solution scheme has a definite advantage.

The main disadvantage of this method is the complexity of the internal book

keeping. However, the bookkeeping is a programming problem and does not concern

the user. Another limitation of this technique may be its dependence on the element

numbering sequence. Although it is rather easier to number the elements in a logical

manner relative to numbering the nodal sequence, it does place some effort on the

user. However, the difficulty can be easily dealt with, if some form of front width

minimizer is incorporated in the program. There are different front width minimizing

schemes available such as by Sloan and Randolph (1981), Akin and Pardue (1975)

and Pina (1981). The procedure by Sloan and Randolph (1981) is built into the

2-dimensional finite element code.


6.5.4 Finite Element Procedure

A broad overview of the procedures followed in both, the 2-dimensional and 3-

dimensional programs is given in the flow chart shown in figure 6.3. The steps involved

in the finite element procedure can be summarized as follows:

1. Basic data such as the number of nodes, elements and material types are read

and the required storage is allocated for the variables.

2. All other data such as nodal coordinates, temperatures, element-nodal informa

tion and model parameters are read.

3. The initial conditions are read and the initial stresses, strains, pore pressures

and force vectors are set.

4. Relevant data for the load increment is read.

5. Force vector and the element stiffness matrices are evaluated using the moduli

based on the initial stresses.

6. The equations are solved using the frontal solution scheme. For linear and

nonlinear elastic stress-strain models, the solution scheme for symmetric matri

ces is used. For the elasto-plastic stress-strain model, the solution scheme for

unsymmetric matrices is used.

7. Increments in the stresses and strains for the load increment are calculated and

if it is the first cycle of analysis, new moduli are evaluated based on the stresses

at the mid point of the increment.

8. If it is the first cycle of analysis, steps 5 to 7 are repeated once more using new

moduli for step 5.

9. The stresses, strains and pore pressures and other relevant results are calculated

and the desired results are printed.


C Start DRead basic data and allocate storage

for principal arrays

Read and set the initial conditions

Read relevant data for theload increment

Evaluate stiffness matrix and load vector

Solve for displacements and pore pressures

Evaluate the changes in stresses,strains and pore pressures

Update relevantIs this the last cycle No

variables toof analysis’ average values

Yes

riJpdate all the results and print

No

NoLast increment?

Yes

C Stop D

Figure 6.3: Flow Chart for the Finite Element Programs


10. If the current stress state exceeds the strength envelope, or if there is strain

softening, load shedding vector is computed.

11. Steps 5 to 9 are repeated until all the elements satisfy the failure criterion or in

other words, until the load shedding is converged.

12. The imbalance loads at the end of the increment are calculated and added to

the next load increment, if any.

13. Steps 4 to 12 are repeated until all the load increment data have been analyzed.

The final states of the previous load increment are used as the initial conditions

for the next load increment.

6.6 Finite Element Programs

The finite element programs have been written in FORTRAN-77 and are portable

to any operating platforms. There are two separate programs; CONOIL-Il which

is a 2-dimensional program to perform axisymmetric and plane strain analyses and,

CONOIL-Ill which is a 3-dimensional program to perform 3-dimensional analysis. Al

though these finite element programs have been developed with special attention paid

to the problems in oil sand, they are capable of doing general drained, undrained and

consolidation analyses effectively. Both programs are capable of analyzing excavations

as well. Brief descriptions of these programs are given in this section. Applications

of the programs are discussed later in chapter 7.

6.6.1 2-Dimensional Program CONOIL-Il

The 2-dimensional program CONOIL-Il was originally developed by Vaziri (1986)

based on the program CRISP (University of Cambridge). It was later modified by


Srithar in 1989 with an improved formulation for temperature analysis. CONOIL

II has been divided into two separate programs; the ‘Geometry Program’ and the

‘Main Program’. The main purpose of this split is to reduce the effort for the user.

The geometry program automatically generates and numbers the midside and interior

nodes. It also renumbers the elements and nodes to minimize the front width and

creates a input file for the main program, containing the relevant information about

the finite element mesh. The program also has some special features. The triple

matrix product as suggested by Taylor (1977) is adopted in the formation of stiffness

matrix, and this will eliminate all the unnecessary arithmetic operations which will

result in zero coefficients. The geometry program consists of 11 subroutines and the

main program consists of 58 subroutines. The names of the subroutines and their

functions are presented in appendix C.

Grieg et al. (1991) developed a pre/post processor package, COPP, for CONOIL

II to facilitate viewing and plotting the CONOIL-Il input and output data. COPP

is menu driven, very user friendly and provides many options for the user.

6.7 3-Dimensional Program CONOIL-Ill

The 3-dimensional program CONOIL-Ill has been developed from scratch following

the same sequence of procedures as the 2-dimensional one. However, compared to

the 2-dimensional program, the 3-dimensional program has less special features, and

it does not have a post processor yet. The 3-dimensional program comprises 43

subroutines. The names of the subroutines and their functions are given in appendix

C. A User manual and some example problems are presented in appendix F.

Chapter 7

Verification and Application of the Analytical

Procedure

7.1 Introduction

The analytical procedure described in chapter 6 has been incorporated in the finite

element program, CONOIL. The main intention of this chapter is to verify and val

idate the finite element program, and to demonstrate its applicability. The program

deals with a number of aspects such as, dilative nature of sand, three-phase pore fluid,

gas exsolution, effects of temperature changes, etc.. and the best way of verifying the

program would be to consider each aspect separately. The program is verified here

by considering some particular problems for which theoretical solutions are available.

Once verified, the program is validated by comparing some experimental results with

predictions from the program. Then, the program has been used to predict the re

sponses in a oil recovery problem. A problem concerning pore pressure redistribution

after liquefaction has also been analyzed to show the applicability of the program to

other geotechnical problems.

7.2 Aspects Checked by Previous Researchers

The two dimensional version of the finite element program CONOIL has been used at

the University of British Columbia since 1985, with improvements being made from

time to time. Cheung (1985), Vaziri (1986) and Srithar (1989) have demonstrated

the capability of the program on a number of aspects. Since those aspects are kept

168

Chapter 7. Verification and Application of the Analytical Procedure 169

intact with the improvements made in this study, those verifications and validations

are still valid. These are briefly described herein.

The general performance of the program in predicting stresses and strains has

been verified by Cheung (1985), by considering a thick wall cylinder under plane

strain conditions. Closed form solutions for this problem have been obtained from

Timoshenko (1941). The results from the program and the closed form solutions are

in excellent agreement and are shown in figure 7.1.

Cheung (1985) also validated the gas exsolution phenomenon in the program.

Laboratory test results by Sobkowicz (1982) on gassy soil samples have been consid

ered. Sobkowicz (1982) carried out triaxial tests to predict the short term undrained

response, i.e, no gas exsolution and the long term undrained response, i.e., with com

plete gas exsolution, The comparisons of the test results with the program results are

shown in figures 7.2 and 7.3. The measured and predicted results agree very well.

The overall structure of operations for a consolidation analysis has been verified

by Vaziri (1986). The closed form solution developed by Gibson et al. (1976) for

a circular footing resting on a layer of fully saturated, elastic material with finite

thickness has been considered for the verification. A comparison of the computed

results and the closed form solutions, shown in figure 7.4, demonstrate that they are

in very good agreement.

Srithar (1989) modified the procedure for thermal analysis in the original CONOIL

formulation. He verified the new formulation under drained and transient conditions.

The closed form solution presented by Timoshenko and Goodier (1951) for a long

elastic cylinder subjected to temperature changes has been considered to verify the

formulation under drained condition. The closed from solution and the finite element

results are shown in figure 7.5 and are in remarkably good agreement.

To verify the formulation for thermal analysis under transient conditions, a closed

form solution was derived by Srithar (1989) for one dimensional thermal consolidation


0e0

0

0a)

closed form

o o 0programQca)

I I I I I I I I V

0

Ic’J

246810

Radii (r/r0)

E = 3000 MPaI’ — 1/3initial stress : or = o. = 6000 kPafinal stress : o = 2500 kPainside radius : r = 1 in

Figure 7.1: Stresses and Displacements Around a Circular Opening for an ElasticMaterial (after Cheung, 1985)


I.’

.-

C

b0

0

0C

Figure 7,2: Comparison1985)

of Observed and Predicted Pore Pressures (after Cheung,

00

40 60 80 100 120 140Total Stress (kPa) (X1O’ )


0

Cl2

.4.)

s-I

a)

>0

___________________

.—

s-I-I-)

0

s-I

0xc

____________________

0

lab data

0

l.a.4-’

Cl)

-4-’

0

0

___________________________

C0 20 40 60 80

Effective SigmaP (kPa) (X101 )

Figure 7.3: Comparison of Observed and Predicted Strains (after Cheung, 1985)


I I

0.25Analytical Solution

. Finite Element0.30 - Analysis

0.35-

________

r0.40 -

045 xI30y/30DIE—i .5

— 0.00.50

I I I I

ia-4 io io_2 1.0 10Ct

V

Tv-

a)Amount of settlement

0.0 I I I

Analytical Solution0.2-

‘%.‘b%,

0 Finite Element%, %\ Analysis

0.4

Uy/B—0 vO.3 v—0.0

0.6 — D/E — 1

0.8 -

1.0 I I

ia—4 i— 10—2 ia-’ 1.0 10Ct

V7 -—

U2

b) Degree of settlement

Figure 7.4: Results for a Circular Footing on a Finite Layer (after Vaziri, 1986)


3000 —

LaSymbols — CONOIL—IlSolid lines — Closed Form

2000 —

C

Vertical Stress1000—

Radial Stressci)

(1)

0—

Hoop Stress

—1000—

Radial Distance(m)

Figure 7.5: Stresses and Displacement in Circular Cylinder (after Srithar, 1989)


with a uniform temperature rise. The closed form solution was obtained by making

analogy to the closed form solution by Aboshi et al. (1970) for a constant rate of

loading. Figure 7.6 shows the closed form solutions and the program results and they

agree very well. [n the figure, z denotes the depth at which the results are considered

and H denotes the total depth.

The performance of the program for undrained thermal analysis has been validated

by comparing the experimental results on oil sand samples in a high temperature

consolidometer obtained from Kosar (1989). Computed and measured results show

good agreement as illustrated in figure 7.7.

7.3 Validation of Other Aspects

In this research work, a new elasto-plastic stress-strain model has been developed and

incorporated in the finite element code. This will realistically model the dilation and

the loading-unloading sequences encountered in oil sands. To validate the program’s

capability to model the dilation phenomenon, the triaxial test results on oil sand

given by Kosar (1989) have been considered. The triaxial test has been modelled by

four triangular elements as shown in figure 7.8. An axisymmetric analysis has been

carried out with the relevant boundary conditions as shown in figure 7.8. The model

parameters used are listed in table 7.1. The predicted and the measured results are

compared in figure 7.9. Also shown in that figure are the results using a hyperbolic

model. It can be seen from the figure that the shear stress versus axial strain response

can be very well captured by both the elasto-plastic and hyperbolic models. But

the hyperbolic model does not predict the volumetric strain behaviour as measured,

whereas, the elasto-plastic model predicts results that match the measured values.

The triaxial test results for a load-unload-reload type loading on Ottawa sand

obtained from Negussey (1985) have been considered to validate the loading-unloading

operation of the program. The triaxial test specimen has been modelled by four


L’z/H = 0.875

30 —

0 0 0 0 ci p 0 0

0

- 2uCl)Cl,Q)

0

ci)

00000 CONOIL—Ilclosed formsolutions

0 —,

LHz/H = 0.5

20 —

C0

ci)

C

_

0 0

(1)

ci)

u-i0

00

0— i I I I I

0 1000 2000 3000 4000

Time(s)

Figure 7.6: Pore Pressure Variation with Time for Thermal Consolidation (afterSrithar, 1989)


8-

Test results /00000 CONOIL—Il /

/0

6—a)

-

Cc-

- 0C-)

a) -

E2 4—0>

> -

D -

E0

0

0

0

I I I I I I I I I I

20 70 120 170 220Temperature(° C)

Figure 7.7: Undrained Volumetric Expansion (after Srithar, 1989)


Table 7.1: Parameters for Modelling of Triaxial Test in Oil Sand

(a) Elasto-Plastic Model

Elastic kE 3000n 0.36

kB 1670m 0.36

Plastic Shear 0.72? 0.54

! 0.33KG 1300np -0.66Rf 0.80

(b) Hyperbolic Model

kE kB m R

1100 0.49 700 0.47 0.6 49 13


0

1.5 cm

Figure 7.8: Finite Element Modeffing of Triaxial Test

Chapter 7. Verification and Application of the Analytical Procedure

3500

3000

2500

‘a 20000S

‘0

1500

1000

500

‘I

WI

180

Figure 7.9: Comparison of Measured and Predicted Results in Triaxial CompressionTest

.

Elasto-Plastic

Hyperbolic

Experimental

Ea (%)

-0.2

-0.1

1

€_a (%)


elements as shown earlier in figure 7.8. The model parameters used are listed in table

7.2. The measured and predicted results agree very well as shown in figure 7.10.

Table 7.2: Model Parameters Used for Ottawa Sand

Elastic kE 3400m 0.0

kB 1888m 0.0

Plastic ‘i 0.49) 0.85IL 0.26

KG 780np -0.238R1 0.70

Modelling of the three-phase pore fluid is the other important aspect where major

improvements have been made in the analytical formulation in this study. There is

no theoretical or experimental solutions available to verify or to validate the overall

formulation for the modeffing of the three-phase pore fluid. However, validations for

the analytical representation of the relative permeabilities have been made and were

presented in chapter 5.

74 Verification of the 3-Dimensional Version

The 3-dimensional version of CONOIL is newly written following the same operational

framework as the 2-dimensional version. Since the 3-dimensional program is new, it

is necessary to check that the performance of the program in all aspects agrees with

the intended theories, as was proven for the 2-dimensional version. The problems

considered to verify the 3-dimensional code were similar to those used to verify the

2-dimensional code and all gave satisfactory results. Since the verifications are similar

to those presented in the previous sections, they are not repeated here. However, the


350

300

250

a. 200

150

100

50

0.00 0.20

Figure 7.10: Comparison of Measured and Predicted Results for a Load-Unload Testin Ottawa Sand

0.05 0.10 0.15

El (%)


verification for the thermal consolidation is described here as an example.

Figure 7.11 shows the finite element mesh of a soil column subjected to a uniform

temperature increase at a rate of 100°/hr. The boundary and the drainage conditions

are also shown in figure 7.11. The closed form solution for the pore pressure at a depth

z under one dimensional thermal consolidation is given by the following equation

(Srithar, 1989):

16 M n — 1 mrz I (m2ir2’\ ‘1p r3 T m1,3,..

—i sin2H

— exp — T (7.1)

where

p - pore pressure at distance z at time t

T - time factor

n - porosity

- change in temperature at time t

M - constrained modulus

a1 - coefficient of volumetric thermal expansion of liquid

a8 - coefficient of volumetric thermal expansion of solids

The soil properties used for this analysis are given in table 7.3. The soil is assumed

to be linear elastic. The predicted pore pressures have been compared with the

analytical solutions at two different depths, at z/H = 0.75 and at z/H 0.5. The

results agree very well as shown in figure 7.12.

7.5 Application to an Oil Recovery Problem

Having verified the performance of many aspects of the finite element program, it has

been applied to predict the response of an oil recovery process by steam injection.

The Phase A pilot in the Underground Test Facility (UTF) of the Alberta Oil Sands


im

H G

__

21®22____3

13 ® 14

11 H=lm

_1O

7

5 6

z 4.. 3

.1

A B

AB, BC, CD, DA - Totally Fixed

AE, BF, CG, DH - Vertically Free

EF, FG, GH, HE - Drain Boundaries

Figure 7.11: Finite Element Mesh for Thermal Consolidation

Table 7.3: Parameters Used for Thermal Consolidation

fl V a1 a5 k M Hm3/m3/°C m3/m3/°C rn/s MPa rn

0.5 0.25 1x103 1x105 2x106 18.3 1


35z/H = 0.75

___________________________________________

(a)-

30

-25

20

01.

0I-.

00

10

Symbols - Program5 Line - Closed form

0 I I

0 500 1000 1500 2000 2500 3000 3500

Time (s)

30z/H = 0.5 (b)4Z

Symbols - ProgramLine - Closed form

0 I I I I I I

0 500 1000 1500 2000 2500 3000 3500

Time (s)

Figure 7.12: Comparison of Pore pressures for Thermal Consolidation


Technology and Research Authority (AOSTRA) is considered herein for analysis.

The UTF uses a steam assisted gravity drainage process with horizontal injection

and production wells. A brief description of the UTF and the problem to be analyzed

are presented here. Further details about the UTF can be found in Scott et al. (1992),

Laing et al. (1992) and in AOSTRA reports on UTF.

The UTF of AOSTRA is located near Fort McMurray, Alberta, and is currently

being used to test the shaft and tunnel access concept for bitumen recovery in deep

oil sand formations. The geological stratification at the UTF comprises a number of

soil layers. However, it can be simplified as consisting of three different soil types, in

a broad sense. Devonian Waterway formation limestone exists below a depth of 165

m. Overlying the limestone is the McMurray formation oil sand which is about 40 m

thick. The top 125 m overburden can be classified as Clearwater formation shale.

A schematic 3-dimensional view and a plan view of the UTF are shown in figures

7.13 and 7.14 respectively. There are two shafts accessing the tunnels in limestone

beneath the oil sand layer. The tunnels were constructed in the limestone at a depth

of about 178 m with the roof being about 15 m below the limestone-oil sand inter

face. Three pairs of horizontal injection and production wells were drilled from the

tunnels up into the oil sands at about 24 m spacing. A vertical section of the well

pairs was instrumented with thermocouples for measuring temperatures, pneumatic

and vibrating wire piezometers for measuring pore pressures and extensometers and

incinometers for measuring horizontal and vertical displacements.

Figure 7.15 shows a vertical cross-sectional view (section A-A’ in figure 7.14) of

the three well pairs. Modelling of all three well pairs with their steaming histories

and with the detailed geological stratification would be complex as the steaming

of different well pairs started at different times. To illustrate the problem and to

demonstrate the applicability of the program in a simple manner, only one well pair

is considered here for analysis.


Figure 7.13: A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991


Shaft#1

I

TObservation Tunnel

i

Injector/Producer Wellpairs...... —

Section A-A’ —* Geotechnical A A’Cross Section

____ ____

4

Figure 7.14: Plan View of the UTF (after Scott et al., 1991)

—.

....

..

......

......

..

..

V..:•:••

..

.•

.•.

..•.

.•.•

•.•

.•.•

.•.•.•.

.......

CD

CD -c

0C.

11

gg

0<

30

CD

CD=

U)

CD

CD0

=CD

CD0.

oC;

’C;

’3

33

00


To analyze the oil recovery with one well pair, the shaded region in figure 7.15 is

modelled by finite elements. The finite element mesh consisted of 240 linear strain

triangular elements as shown in figure 7.16. Plane strain boundary conditions are

assumed. The injection and production wells are modelled by nodes with known pore

pressures. The steam injection pressure is assumed to be maintained at 2800 kPa

(1300 kPa above the in-situ pore pressure) and the production pressure is assumed to

be at 2000 kPa (500 kPa above the in-situ pressure). The parameters used have been

obtained from laboratory test results reported by Kosar (1989) and from AOSTRA

and are listed in table 7.4. The gas saturation is assumed to be zero. i.e., the pore

fluid is assumed to comprise only water and bitumen. The bitumen saturation is

assumed to be 70 %.

The temperature-time histories of the nodes have also to be specified as an input

to the program. These were obtained from the field measurements made at the

UTF. The temperature contours in the oil sand layer at different times are shown in

figure 7.17. The steam chamber which is the region in the oil sand layer where the

temperature is the same the steam temperature, grows with time as shown in the

figure. At time t = 30 days, the steam chamber extends to a distance of about 10 m

horizontally and vertically from the injection well.

Even though a larger domain is analyzed as shown in the finite element mesh,

the results are plotted only for the oil sand layer which is of primary interest. The

predicted excess pore pressures in the oil sand layer are shown in figure 7.18. The

injection and production wells are also indicated in the figure. Figure 7.18 (a) shows

the excess pore pressure contours at 10 hours after the steam injection started. Only

a small region adjacent to the injection well experiences significant changes in the

pore pressures. This correlates very well with the temperature contours at that time,

as shown in figure 7.17 (a). With time, the region of higher pore pressure expands as

shown in figures 7.18 (b) and (c) indicating the growth of the steam chamber which

250.

0—

200.

0—

-150.0

Lii

100.

0—

50.0

—

0.0

—

22/7//

/////

22/?/

;c

0.0

1I

—I

I10

0.0

II

——

200.

0

II

II

300.

0

C-. 0

Dis

tanc

e(m

)F

igur

e7J

6:F

init

eE

lem

ent

Mod

ellin

gof

the

Wel

lP

air

I-.


Table 7.4: Parameters Used for the Oil Recovery Problem

(a) Soil Parameters

Elastic 3000n 0.36

kB 1670m 0.36

Plastic Shear 171 0.75J.:l7 0.13). 0.53t 0.31

KG 1300rip -0.66R1 0.73

Plastic Collapse C 0.00064

p 0.61Other e 0.6

k(m2) 1.0 x 10—12

a8(m3/m3/°C) 3.0 x 1O

(b) Pore Fluid Parameters

B 5.0 x iOB0 2.5 x iO

cx(m3/m3/°C) 3.0 x i0cz0(m3/m3/°C) 3.0 x iO1u,o(Pa.s) 20

S 0.20.2

kro = 2.769(0.8 — S)’996

krw = 1.820(S — 0.2)2.735

EIevL

on

(m)

EIev

Lo

n(m

)E

Iev

Lo

n(m

)

cyq CD CD p CD C)

0 0 cn CD 0 (j)

p CD

p I-’.

0

NJ

NJ

-01

QD

QN

JN

J-

010

00

NJ

NJ

-01

00

00

NJ

0

(I) C,

D 0 (D 3

NJ

0

0 0 NJ

0 NJ

0

(0 C-,

f DC

J00

(0

0

(0 C-,

:3 0 3

01 0(1

1 0

0c,z


is also implied by the temperature contours in figure 7.17. The 1000 kPa excess pore

pressure contour from the field measurements is compared with the predicted contours

in figure 7.19. It can be seen from the figure that the measured zone of 1000 kPa is

larger than the predicted zone. However, the shapes of the pore pressure contours

are similar to the measured ones.

The predicted horizontal and vertical stresses are shown in figures 7.20 and 7.21.

As the steam chamber grows the soil matrix expands and since the soil is more

constrained in the horizontal direction, the horizontal stresses increase. The vertical

stresses also increase, but not as much as horizontal stresses. The pattern of the stress

contours also indicates the movement of soil and the shape of the steam chamber.

The stress ratio which is an index giving the current stress state relative to the

failure stress state is shown in figure 7.22. It appears that the shape of the steam

chamber and the corresponding temperature increases create higher shear stresses in

the region above the steam chamber. This is implied by higher stress ratios and a

maximum stress ratio is about 0.45 is predicted in the region about 15 m above the

injection well. Since the predicted stress ratios are well below unity, there would not

be any failure.

Figure 7.23 shows the horizontal displacement along a vertical line at 7 m from

the wells, at time t = 30 days. Also shown in the figure are the field measurements

made in a instrumented bore hole at about the same distance away from the wells.

It can be seen that the field measurements are slightly larger than the predictions at

some locations, but in the overall picture, the predictions are in reasonable agreement

with the measurements.

The variation of vertical displacements with the distance from the wells at the

injection well level is shown in figure 7.24. Maximum displacement of 21 mm is

predicted at a distance 15 m from the well. There is no field measurements available

that could give the results due to the steaming in a single well pair. The vertical

CD I.

0 CD CD Cl)

U)

I- CD 0 U) CD 0 C,,

CD

EIevL

on

(m)

-01

EIev

Lo

nC

m)

NJ

NJ

-N01

NJ

0U Cl

)

C-,

D C) CD 3

NJ

0

U Co C—,

D 0 CD 3

NJ

0

EIevL

on(rn)

I.U Cl) C

-, 0 CD 3

01 001 0

01 &

c3 0cJ

c30

0CC

)01


50

S

40C

0

1::60

Figure 7.19: Comparison of Pore Pressures in the Oil Sand Layer

displacement measurements were made in bore holes located in between the well

pairs and therefore, those measurements cannot be considered as the result due to

the steaming from a single well pair. Moreover, those measurements were very erratic

and a definite pattern of vertical displacements could not be inferred.

The total quantity flow with time at the production well is shown in figure 7.25.

The flow rate increases with time and it can be said that a steady state condition is

achieved after 20 days. The predicted steady state flow rate is 5.18m3/m/day. In the

initial stages of production, more water will be produced than oil because much of

the bitumen will be immobile. With time, the temperature will increase, the viscosity

of bitumen will reduce, it will become mobile and more bitumen will be produced.

As the oil recovery process continues at the steady state conditions, eventually, the

amount of bitumen produced will become less as it is replaced by water.

0 10 20 30 40 50

DsLncG (m)

oq CD 0 N 0 C,)

I- CD

EevoL

on

(m)

NJNJ

-01

EIe

v3

Lo

n(m

)N

)N

J-

01

0

0

C,,

E3 C) CD 3

NJ

0

C,)

C-, a, :3 0 CD 3

NJ

0

ci) C)

0) :3 0 CD

EIevL

on

(m)

N)

NJ

-01

00

00

NJ

0

01 0

3

(ii

0(i

i0

c3 00


- () t tlOhrs

40C

0

/6ØO30

LU

20 I

Dstance (m)0 10 20 30 40 50 60

C

0

30

LU

40

200 10 20 30

D9Lance (m)40

40

E

C

0

>30

20•10 20

DLnce (m)40 60

Figure 7.21: Vertical Stress Variations in the Oil Sand Layer


60

60

60

40

0

10 20 30 40

DsLance (m)

40C

0

30

Li

200 10 20 30

DLence (m)

40

40C

0

30

Li

200 10 20 30

DLncG (m)

40

Figure 7.22: Stress Ratio Variations in the Oil Sand Layer


60Symbols - Field Measurements

• Line - Prediction

50 •

•0

••

20 I I I I0 5 10 15 20 25

Displacement (mm)

Figure 7.23: Comparison of Horizontal Displacements at 7 m from Wells


25

20

E15

10

5

0

600 10 20 30 40 50

Distance (m)

Figure 7.24: Vertical Displacements at the Injection Well Level


160

140

120

C,,

100

0U.

o 80C

0E<60

40

20

0

Time (days)

202

0 5 10 15 20 25 30 35

Figure 7.25: Total Amount of Flow with Time


The quantity of flow given in figure 7.25 is the total flow of water and oil. Unfor

tunately, the procedure adopted in the analytical formulation will not give individual

amounts of flow directly. However, approximate estimations of the individual amounts

of flow of water and oil can be calculated by knowing the area of different temperature

zones and the relative permeabilities. Details of the individual flow calculations are

described in appendix D. The individual flow rates of water and oil with time under

steady state conditions are given in figure 7.26. The total amount of oil produced with

time in the production well is shown in figure 7.27. It should be noted that the flow

predictions presented here are approximate because of the assumptions made about

the fluid flow in the analytical model. If accurate results about the flow are required,

a separate rigorous flow analysis using a suitable reservoir model is necessary.

7.5.1 Analysis with Reduced Permeability

To show the importance of this type of analytical study, the same oil recovery problem

is analyzed with reduced permeability. The absolute Darcy’s permeability of the

oil sand matrix is reduced from 10’2m2 to 1013m2. The predicted pore pressure

contours and the stress ratio contours are shown in figures 7.28 and 7.29 respectively.

These figures can be compared with figures 7.18 and 7.22 for the previous analysis.

The pore pressure in the oil sand layer is much more than the injection pressure. This

is because the pore fluid expands more than the solids and since the permeability is

low, there is not enough time for the expanded pore fluid to escape, thus, the pore

pressure increases. The worst condition occurs after 5 days and a maximum excess

pore pressure of 2200 kPa is predicted. This increase in pore pressure will greatly

• reduce the effective stresses and may lead to liquefaction.

The stress ratios shown in figure 7.29 are also much higher compared to those

in the earlier analysis. Again, the worst condition is predicted after 5 days and a

region with stress ratio of 0.7 is shown in the figure. The same kind of results would


•.8

-

Ia)

0U

Ea)

0U-

5.5

5

4.5

4

3.5

2

1.5

1

0.5

01

2 5 10 20 50 100 200 500

Time (days)

(a) Flow Rate of Water

20

Time (days)

(b) Flow Rate of Oil

500

Figure 7.26: Individual Flow Rates of Water and Oil


E

CE

0Li.

0

0E

,2

205

350

50

40

30

20

0 50 100 150 200 250 300

Time (days)

Figure 7.27: Total Amount of Oil Flow

Co C-,

D D C) CD

U Co C-,

D C) CD 3

EIev

ton

(m)

RDRD

-01

RDRD

-R01

EIev

Lo

n(m

)F—

(L

HevL

on

cmRD

RD-

(ii

‘SC‘SC

‘SC

0CD —

3

co

0 CD fri CD Cl)

fri CD 0 ‘-I

U) I-. Cl)

r\D 0

RD 0

3

RD 0

U C/) C-,

Z3 0 (0 3

RD 0

I01 0

01 001 0

0

Chapter 7.

C

0

30

LU

S

C

0

30a)

LU

S

C

0

30

LU

Verification and Application of the Analytical Procedure 207

60

60

60

Figure 7.29: Stress Ratio Variation for Analysis 2

40

200 10 20 30

DtLnce (m)

40

40

200 10 20

DsLncG (m)

30 40

40

200 10 20

DLance (m)40


have been predicted if the permeability was kept the same and the rate of heating

increased. The detailed results show that the stress ratio of one of the elements in

the highest stress ratio region reached unity indicating shear failure. Since the region

of shear failure is small and away from the wells, it will not cause any problems.

However, if the region of shear failure is large, there will be significant deformations

and if the region extends to the wells, it may cause significant damage to the wells.

To avoid this kind of situation, the rate of heating should be reduced.

The above example illustrates the usefulness of this type of analytical treatment

for oil recovery projects. This type of analysis provides important information about

the rate of heating, possible failure zones, deformations, stability of the wells etc.,

beforehand. Without an analytical treatment, these concerns have to be tested in the

field on a trial and error basis, which would be very costly.

7.6 Other Applications in Geotechnical Engineering

Even though the finite element program CONOIL was developed for analyzing prob

lems related to oil sands, it can also be applied to other potential geotechnical prob

lems. An example problem which involves pore pressure migration after liquefaction

is described herein.

Generally, loose sands are susceptible to liquefaction in the event of an earthquake

and to prevent such liquefaction, loose sand deposits are commonly densified. The

densified zone in a loose sand deposit will only be stable provided high excess pore

pressures from the surrounding liquefied sands do not penetrate it during and after

the earthquake. This concern is examined herein with different densification schemes

used in practice.

A typical soil profile for Richmond, British Columbia, was considered in the anal

ysis. The soil profile comprised 3 m of clay crust, underlain by 15 m of loose sand and

followed by 5 m of dense sand as shown in figure 7.30. The earthquake is assumed to


generate 100% pore pressure increase in loose sand and 30% pore pressure increase in

dense sand zones. A hyperbolic stress-strain model was considered and the material

parameters used in the analysis are given in table 7.5. Three cases which represent

three different densification schemes were studied as illustrated in figure 7.30

Table 7.5: Soil Parameters Used for the Example Problem

Soil Type n kB m R1 k,, kh(m/s) (m/s)

Clay 150 0.45 140 0.2 0.7 2.5 x 10 5 x iO

Liquefied Sand 300 1.0 180 1.0 0.8 5 X 1O 1 X 10

Dense Sand 2000 0.5 1200 0.25 0.6 2.5 x 10 5 x iO

Dense Sand with Drain 2000 0.5 1200 0.25 0.6 1 x 10 1 x iO

Clay with Drain 150 0.45 140 0.2 0.7 1 x iO 1 x 1O

In case 1, densification is assumed to the full depth of the loose sand without

any drainage system. This case may represent a field condition where densification

is achieved using timber piles without any drainage provisions. In case 2, the den

sification is assumed with a perimeter drainage system. This may represent a field

situation where densification is achieved using timber piles with a perimeter drainage

system of vibro-replacement columns. In case 3, the drainage was assumed in the

densified zone. This may represent densification by vibro-replacement. In the anal

ysis, the drains were not considered on an individual basis, instead, the densified

zone with drains was modelled as a soil with an equivalent permeability. The equiv

alent permeability can be estimated from the size and spacing of the drains and the

permeabilities of the materials.

The excess pore pressures for the three cases considered at various times after the

earthquake are shown in figures 7.31, 7.32 and 7.33. The excess pore pressures are

r0r0

NC

)-O

0CD

0

0 CD ()0.

. .o. CD CD 0

0 3


shown in terms of pore pressure ratio u/o0, in which u is the current excess pore

pressure, and o is the initial vertical effective stress. u/a0 = 0 represents zero pore

pressure rise and u/o0 = 1 represents 100% pore pressure rise or liquefaction. The

variations of the excess pore pressure ratios with time and distance from the centre

of the densified zone are shown in graphs (a) and (b) in the figures. Graph (a) shows

the variation at a depth of 5 m and graph (b) at a depth of 10 m. Graph (c) shows

the excess pore pressure ratio with depth along the centre line.

The results for case 1. (figure 7.31) show that the excess pore pressure in the

surrounding undensified area migrates into the densified zone. The pore pressure

ratio in the upper part of the densified zone rises to 1 which means liquefaction

will be triggered. However, below a depth of 6 m, liquefaction is not triggered and

piles penetrating below this depth could support vertical load, although significant

horizontal displacements are likely to occur. The results for case 2 (figure 7.32)

indicate that a perimeter drainage system is quite effective in preventing the migration

of high pore pressure from the loose zone into the densified zone. A maximum pore

pressure ratio of 0.5 is predicted 1 mm after the earthquake. The results for case 3

where the drainage is assumed throughout the densified zone are shown in figure 7.33.

It can be seen from the figure that the drains in the densified zone are much more

effective in preventing the migration of pore pressure in the densified zone. The pore

pressure ratio in the densified zone increases from an initial value of 0.3 at time t =

0, to 0.4 after 10 seconds and then reduces.

The conclusions from the analyses are as follows. Densification alone such as could

be achieved by driving timber piles will not prevent the high excess pore pressures

from the surrounding liquefied zone penetrating the densified zone. Such penetration

will cause liquefaction to a depth of 6 m for the conditions analyzed. Below this depth

effective stress increases and timber piles would be capable of carrying vertical load

although they could be damaged by horizontal movements. Perimeter drains could


1.2d=5m (a)

30

d=lOm (b)

0.8

Distance (m)

Pore Pressure Ratio0 0.5 1.0 1.5

2

•(c)

4- Si, /di’

6- / /z’

!e

I,,

10 - QI

I/i12 - ‘:1 t=lmin

t=3Ornin

14 -t5hrs

t =lday

16

Figure 7.31: Variation of Pore Pressure Ratio for Case 1


1.2d=5m (a)

d=lOm (b)

c05 10 15 20 25 30

Distance (m)

Pore Pressure Ratio0 0.1 0.2 0.3 0.4 0.

L(C)

4- //

A 0

: :: :b-f

12- 1 1 t=o4 1

t=lmin

14- t5rflifl

I t=5hr

______________

I

______

1€

Figure 7.32: Variation of Pore Pressure Ratio for Case 2


d=5m (a)

t=rnin

15 20 30

1.2d=lOm (b)

__

30

Distance (m)

Pore Pressure Ratio0 0.1 0.2 0.3 04

(C)7

4-

4I

G)0 /

10- 4

12- 1t 9

14-

0

Figure 7.33: Variation Of Pore Pressure Ratio for Case 3


greatly reduce the migration of excess pore pressures into the densified zone. The

provision of drainage within the densifled zone can be very effective in preventing

high excess pore pressures in the densifled zone.

A more detailed study of this problem including the effect of densification depth

is presented in Byrne and Srithar (1992). Some other applications of the program

can be found in Byrne et al. (1991a), Byrne et al. (1991b), Jitno and Byrne (1991)

and Crawford et al. (1993).

Chapter 8

Summary and Conclusions

An analytical procedure is presented to analyze the geotechnical aspects in an oil

recovery process from oil sand reserves. The key issues in developing an analytical

model are: the stress-strain behaviour of the sand skeleton; the behaviour of the

three-phase pore fluid; and the effects of temperature changes associated with steam

injection. A coupled stress-deformation-flow model incorporating these key issues is

presented in this thesis.

In modelling the stress-strain behaviour of the oil sand skeleton, shear induced

dilation is an important aspect. Such dilation can increase the hydraulic conductivity

and hence increase oil recovery. Dilation will also lead to reduced pore fluid pressure

and increased stability. The other pertinent aspect is the stress-strain response un

der stress paths involving a decrease in mean stress under constant shear stress and

loading-unloading cycles. The stress-strain models used in the current-state-of-the-

practice are linear or nonlinear elastic models which are incapable of modelling the

above mentioned aspects realistically. The major contribution of this thesis is the

development of a suitable elasto-plastic stress-strain model to capture the important

aspects. The stress-strain model postulated in this thesis is a double hardening type

consisting two yield surfaces. The model has a cone-type yield surface to predict

shear induced plastic strains and a cap-type yield surface to predict volumetric plas

tic strains. The predictions from the stress-strain model have been compared with

laboratory test results under various types of loading and are in good agreement. The

dilation, plastic strains due to cyclic loading, and the response under different stress

216

Chapter 8. Summary and Conclusions 217

paths have been well predicted by the stress-strain model.

The pore fluid in oil sand comprises water, bitumen and gas and the three-phase

nature of the pore fluid has to be recognized in modelling the behaviour of pore fluid.

In petroleum reservoir engineering, multiphase fluid flow is modelled by elaborate

multiphase thermal simulators. In this study, the effects of multiphase pore fluid

are modelled through an equivalent single phase fluid. An effective flow continuity

equation is derived from the general equation of mass conservation which is one

of the other contributions of this thesis. An equivalent compressibility term has

been derived by considering the individual contributions of the phase components.

Compressibility of gas has been obtained from gas laws. An equivalent hydraulic

conductivity term has been derived by considering the relative permeabilities and

viscosities of the individual phases in the pore fluid. The relative permeabilities have

been assumed to vary with saturation and the viscosities have been assumed to vary

with temperature and pressure. Gas exsolution which would occur when the pore fluid

pressure decreases below the gas/liquid saturation pressure has also been modelled.

Oil recovery schemes commonly involve some form of heating and therefore, tem

perature effects on the sand skeleton and pore fluid behaviour are important. Changes

in temperature will cause changes in viscosity, stresses and pore pressures and con

sequently in some of the engineering properties such as strength, compressibility and

hydraulic conductivity. In this study, the stress-strain relation and the flow conti

nuity equation have been modified to include the temperature induced effects. This

approach of including the temperature effects directly in the governing equations gave

very stable results, compared to the general thermal elastic approach.

The final outcome of this research work is a finite element program which incorpo

rates all the above mentioned aspects. The new stress-strain model, flow continuity

equation, and other related aspects have been incorporated in the existing two di

mensional finite element program CONOIL-Il. This required significant undertakings


including a new solution routine as the new stress-strain model results in an unsym

metric stiffness matrix. A frontal solution technique which requires less computer

memory has been employed to solve the resulting equations. A new three dimen

sional finite element program has also been developed following the same concepts.

The validity of the finite element codes has been checked for various aspects by com

paring the program predictions with closed form solutions and laboratory results.

The predicted results agreed very well with the closed form solutions and laboratory

results.

The two dimensional finite element code has been applied to model a horizontal

well pair in the underground test facility of AOSTRA. Results have been presented in

terms of displacements, stresses, stress ratios and amounts of flow and discussed. The

measured and predicted results have been compared wherever possible and they agree

well. A method to obtain individual amounts of flow of the pore fluid components

has also been devised.

The type of analytical study presented in this thesis, is very important in oil

recovery projects, since it could give insights into the likely behaviour in terms of

stresses, deformations and flow. For instance, the permeability of the oil sand and

the rate of heating due to steam injection have been examined in some detail. It

has been revealed that in oil sands with low permeability, higher rates of heating

would cause shear failure. If the local shear failure zone extends to the wellbore it

could cause significant damage. Information of this kind would be beneficial to the

successful operation of an oil recovery scheme.

Although the finite element program has been developed to analyze the problems

related to oil sand specifically, it can be applied to other geotechnical problems. To

demonstrate its applicability, a problem involving pore pressure redistribution after

liquefaction has been analyzed and the results are discussed.


8.1 Recommendations for Further Research

Following the work presented in this study, some aspects can be identified in this

area which require further study. Application of the finite element codes to more

oil recovery problems should be carried out to increase the credibility of the models.

The three dimensional code is newly written and even though various aspects of the

code have been verified, it has not been applied to analyze a oil recovery problem of

a three dimensional nature. The three dimensional code needs to be applied to either

a physical model test or a field problem where the responses are measured, in order

to check its capability to model three dimensional effects.

Even though the analytical formulation presented in this study includes the effects

of multi-phase fluid through equivalent compressibility and hydraulic conductivity

terms, it does not take the flow of thermal energy into account. Incorporation of

an elaborate multi-phase thermal and fluid flow model would be the most desired

enhancement though it may be a very difficult task. Previous researchers concluded

that analyzing the geomechanical behaviour and the thermal and fluid flow behaviour

separately, and combining the results by partial coupling is useful and successful.

However, a fully integrated analytical formulation may be more efficient.

Perhaps another aspect which require further study would be the stress-strain

model for the sand skeleton. The elasto-plastic stress-strain model described in this

study does not consider anisotropy effects. Modelling strain softening by load shed

ding may also be inefficient since it requires a large number of iterations. A stress

strain model which includes anisotropy and strain softening effects in a realistic man

ner is worth considering.

Fractures in the oil sand layer are sometimes encountered in the oil recovery

process by steam injection. Inclusion of modelling of fracture initiation and its prop

agation will also be beneficial.

Bibliography

[1] Aboshi, H., Yoshikuni, H. and Maruyama, S. (1970), “Constant Load

ing Rate Consolidation Test”, Soils and Foundation, vol. X, No. 1, pp 43-56.

[2] Aboustit, B.L., Advani, S.H., Lee, J.K. and Sandu, R.S. (1982),

Finite Element Evaluation of Thermo-Elastic Consolidation”, Issues on Rock

Mech., 23rd Sym. on Rock Mech., Univ. of California, Berkeley, pp 587-595.

[3] Aboustit, B.L., Advani, S.H. and Lee, J.K. (1985), “Variational Prin

ciples and Finite Element Simulations for Thermo-Elastic Consolidation”, mt.J. for Num. and Anal. Mtds. in Geomech., Vol. 9, pp 49-69.

[4] Adachi, T. and Okamo, M. (1974), “A Constitutive Equation for Normally

Consolidated Clay “, Soils and Foundations, Vol. 14, No.4, pp 55-73.

[5] Agar, J.G., Morgenstern, N.R. and Scott, J.D. (1987), “Shear Strength

and Stress Strain Behaviour of Athabasca Oil Sand at Elevated Temperatures

and Pressures”, Can. Geotech. J., No. 24, pp 1-10.

[6] Agar, J.G., Morgenstern, N.R. and Scott, J.D. (1986), “Thermal Ex

pansion and Pore Pressure Generation in Oil Sands”, Can. Geotech. J., Vol.

23, pp 327-333.

[7] Akin, J.E. and Pardue, R.M. (1975), “ Element Resequencing for Frontal

Solutions”, The Mathematics of Finite Elements and Applications, Ed: White

man, J.R., Academic Press, London, pp 535-54 1.

220

Bibliography 221

[8] Aziz, K. and Settari, T. (1979), “ Petroleum Reservoir Simulation “, Ap

plied Science Publ., London.

[9] Baker, L.E. (1988), “Three-Phase Relative Permeability Correlations “, SPE

Paper No. 17369, SPE/DOE EOR Symp., Tulsa, Okiakoma.

[10] Bathe, K. and Wilson, E.L. (1976), “Numerical Methods in Finite Element

Analysis”, Pub. by Prentice Hall.

[11] Blot, M.A. (1941), “General Theory of Three Dimensional Consolidation”,

J. of App. Physics, Vol. 12, pp 155-164.

[12] Blot, M.A. (1941(b)), “Consolidation Under a Rectangular Load Distribu

tion”, J. of App. Physics, Vol. 12, pp 426-430.

[13] Bishop, A.W. (1973), “ The Influence of an Undrained Change in Stress

on The Pore Pressure in Porous Media of Low Compressibility”, Geotechnique,

Vol. 23, No. 3, pp 4 35-442.

[14] Bishop,A.W. and Henkel, D.J. (1957), “The Measurements of Soil Prop

erties in The Triaxial Test”, 2nd Ed., William Arnold Pub 1., U.K.

[15] Booker, J.R. (1974), “The Consolidation of a Finite Layer Subject to Surface

Loading”, mt. J. of Solids Structures, Vol. 10, pp 1053-65.

[16] Booker, J.R. and Small, J.C. (1975), “ An Investigation of the Stability

of Numerical Solutions of Biot’s Equations of Consolidation”, mt. J. of Solids

and Structures, Vol. 11, pp 907-917.

[17] Booker, J.R. and Savvidov, C. (1985), “ Consolidation Around a Point

Heat Source”, mt. J. for Num. and Anal. Mtds. in Geomech., Vol. 9, pp 173-

184.

Bibliography 222

[18] Byrne, P.M. (1983), “ Static Finite Element Analysis of Soil Structure

Systems”, Soil Mech. Series, No. 71, Dept. of Civil Eng., Univ. of British

Columbia, Vancouver, Canada.

[19] Byrne, P.M. and Cheung, H. (1984), “ Soil Parameters for Deformation

Analysis of Sand Masses”, Soil Mech. Series, No. 81, Dept. of Civil Eng., Univ.

of British Columbia, Vancouver, Canada.

[20] Byrne, P.M. and Eldridge, T.L. (1982), “ A Three Parameter Dilatant

Elastic Stress-Strain Model for Sand”, mt. Sym. on Num. Mtds. in Geomech.,

Switzerland, pp 78-79.

[21] Byrne, P.M. and Grigg, R.G. (1980), “ OILSTRESS - A Computer Pro

gram for Analysis of Stresses and Deformations in Oil Sand “, Soil Mech. Series,

No. 42, Dept. of Civil Eng., Univ. of British Columbia, Vancouver, Canada.

[22] Byrne, P.M. and Janzen, W. (1984), “INCOIL- A Computer Program for

Nonlinear Analysis of Stress and Deformation of Oil Sand Masses”, Soil Mech.

Series, No. 80, Dept. of Civil Eng., Univ. of British Columbia, Vancouver,

Canada.

[23] Byrne, P.M. and Srithar, T. (1992), “Assessment of Foundation Treatment

for Liquefaction “ First Can. Symp. on Geotechnique and Natural Hazards,

Vancouver, B. C.

[24] Byrne, P.M., Skermer, N. and Srithar T. (1991a), “ Uplift Pressures

Due to Liquefaction of Sediments Stored Behind Concrete Dams “ 3rd Annual

Conf of the Can. Dam Safety Assoc., Whistler, B.C., pp 149-168.

[25] Byrne, P.M., Srithar, T. and Yan L. (1991b), “ Effects of Earthquake

Induced Liquefaction of Soils Stored Behind Concrete Dams “ 2nd mt. Conf

on Geotech. Earthquake Engr. and Soil Dynamics, Missouri Rolla.

Bibliography 223

[26] Byrne, P.M., Cheung, H. and Yan L. (1987), “Soil Parameters for Defor

mation Analysis of Sand Masses “, Can. Geotech. J, Vol. 24, No. 3, pp 366-376.

[27] Byrne, P.M., Vaid, Y.P. and Samerasekera, L. (1982), “ Undrained

Deformation Analysis Using Path Dependent Material Properties”, mt. Sym.

on Num. Mtds. in Geomech., Switzerland pp 294-302.

[28] Byrne, P.M. and Vaziri, H.H. (1986), “CONOIL: A Computer Program

for Nonlinear Analysis of Stress Deformation and Flow in Oil Sand Masses”,

Soil Mech. Series, No. 103, Dept. of Civil Eng., Univ. of British Columbia,

Vancouver, Canada.

[29] Campanella, R.G. and Mitchell, K.J. (1968), “Influence of Temperature

Variations on Soil Behaviour”, J. of SMFD, ASCE, Vol. 94, SM 3, pp 709-734.

[30] Carr, N.L., Kobayashi, R. and Burrows, D.B. (1954), “ Viscosity of

Hydrocarbon Gases under Pressure “, Trans. AIMME, Vol. 201, pp 264-272.

[31] Carter, J.P. (1977), “ Finite Deformation Theory and its Application to

Elasto-Plastic Soils”, Ph.D. Thesis, University of Sydney, Australia.

[32] Chang, C.S. and Duncan, J.M. (1983), “Consolidation Analysis for Partly

Saturated Clay Using An Elasto-Plastic Effective Stress-Strain Model”, mt. J.

for Num. and Anal. Mtds. in Geomech., Vol. 17 pp 39-55.

[33] Chen, W.F. (1982), “Plasticity in Reinforced Concrete “, McGraw Hill New

York, NY.

[34] Cheung, H.K.F. (1985), “ A Stress-Strain Model for The Undrained Re

sponse of Oil Sand”, M.A.Sc Thesis, Dept. of Civil Eng., Univ. of British

Columbia, Vancouver, Canada.

Bibliography 224

[35] Christian, J.T. and Boehmer,J.W. (1970), “Plane Strain Consolidation

by Finite Elements”, J. of SMFD, ASCE, Vol. 96, SM4, pp 1435-57.

[36] Crawford, C.B., Jitno, H. and Byrne P.M. (1993), “The Influence of Lat

eral Spreading on Settlements Beneath a Fill “ to be published in Can. Geotech.

J.

[37] Drucker, D.C., Gibson, R.E. and Henkel, D.J. (1955), “Soil mechanics

and Work-Hardening Theories of Plasticity “, Proceedings, ASCE, Vol.81, pp

1-14.

[38] Duncan, J.M. and Chang, C.Y. (1970), “Nonlinear Analysis of Stress and

Strain in Soils”, J. of SMFD, ASCE, Vol. 96, SM 5, pp 1629-1651.

[39] Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980)

Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Anal

yses of Stresses and Movements in Soil Masses”, Report No. UCB/GR/78-02,

Dept. of Civil Engr., Univ. of California, Berkeley.

[40] Dusseault, M.B. (1980), “ Sample Disturbance in Athabasca Oil Sand”, J.

of Can. Petroleum Tech., Vol. 19, pp 85-92.

[41] Dusseault, M.B. (1979), “Undrained Volume and Stress Change Behaviour

of Unsaturated Very Dense Sands”, Can. Geotech. J., Vol. 16, pp 627-640.

[42] Dusseault, M.B. (1977), “ The Geotechnical Characteristics of Athabasca

Oil Sands “, Ph.D Thesis, Dept. of Civil Engr., Univ. of Alberta, Canada.

[43] Dusseault, M.B. and Morgenstern, R.N. (1978), “ Characteristics of

Natural Slopes in the Athabasca Oil Sands “, Can. Geotech. J., Vol. 15, pp

202-215.

Bibliography 225

[44] Dusseault, M.B. and Rothenburg, L. (1988), “Shear Dilatancy and Per

meability Enhancement in Oil Sands “, Paper No. 82, 4th UNITAR/UNDP mt.Conf. on Heavy Crude and Tar Sands.

[45] Dusseault, M.B. and Van Domselaar, H.R. (1982), “ Canadian and

Venezuelan Oil Sand: Problems and Analysis of Uncemented Gaseous Sand

Sampling”, Proc. of the Engr. Found. Conf on Updating Subsurface Sampling

of Soils and Rocks and Their In-Situ Testing, Santa Barbara, California.

[46] Edmunds, N.R., Wong, A. McCormack, M.E. and Sugget, J.C.

(1987), “Design of Horizontal Well Completions - AOSTRA Underground Test

Facility “, 4th Annual Heavy Oil and Oil Sand Symposium, U. of Calgary, Paper

No. 2.

[47] Ewing, R.E (1989), “Finite Elements in Reservoir Simulation”, Proc. of the

7th mt. Conf on FEM in Flow Problems, U. of Alabama, pp 1251-1257.

[48] ERCB (1975), “The Theory and Practice of the Testing of Gas Wells “, 3rd

Edn., Energy Resources Conservation Board.

[49] Faust, C.R. and Mercer, J.W. (1976), “ An Analysis of Finite-Difference

and Finite Element Techniques for Geothermal Reservoir Simulation”, SPE

5742.

[50] Finol, A. and Farouq Au, S.M. (1975), “ Numerical Simulation of Oil

Production with Simultaneous Ground Subsidence “, SPEJ 411.

[51] Frantziskonis, G. and Desai, C.S. (1987a), “ Constitutive Model with

Strain Softening “, mt. J. Solids Structures, Vol. 23, No. 6, pp 733-750.

[52] Frantziskonis, G. (1986), “Progressive Damage and Constitutive Behaviour

of Geomaterials Including Analysis and Implementation “ Ph.D Thesis, Dept.

Bibliography 226

of Civil Engr., U. of Arizona, Tucson) Arizona.

[53] Frantziskonis, G. and Desai, C.S. (1987b), “Analysis of a Strain Softening

Constitutive Model “, mt. J. Solids Structures, Vol. 23 No. 6, pp 751-767.

[54] Fredhind, D.G. (1973), “ Discussion: Flow and Shear Strength”, Proc. of

3rd mt. Conf on Expansive Soils, Haifa, Vol. 2, pp 71-76.

[55] Fredlund, D.G. (1976), “Density and Compressibility Characteristics of Air

Water Mixtures”, Can. Geotech. J., Vol. 13, pp 386-396.

[56] Fredlund, D.G. (1979), “ Appropriate Concepts and Technology for Unsat

urated Soils”, Can. Geotech. J., Vol. 16, pp 121-139.

[57] Fredlund, D.G. and Morgenstern, N.R. (1977), “ Stress State Variables

for Unsaturated Soils”, J. of Ceotech. Eng. Div., ASCE, Vol.103, pp 447-466.

[58] Fung, L.S.K. (1990), “ A Coupled Geomechanical Multiphase Flow Model

for Analysis of In-Situ Recovery in Cohesionless Oil Sands “, CIM/SPE 90-29,

CIM/SPE 1990 mt. Tech. Metting, Calgary, Alberta.

[59] Gassman, F. (1951), “ Uber die Elastizitat poroser Medien “, Vierteljahrss

chrift der Naturforschenden Geselishaft in Zurich, Mitteilungen aus dem Insti

tut fur Ceophysik, 171.

[60] Ghaboussi, J. and Kim, K.J. (1982), “ Analysis of Saturated and Partly

Saturated Soils”, mt. Sym. on Num. Mtds. in Geomech., Zurich, pp 377-390.

[61] Ghaboussi, J. and Wilson, E.L. (1973), “Flow of Compressible Fluid in

Porous Elastic Media”, mt. J. for Num. Mtds. in Engr., Vol. 5, pp 4 19-442.

[62] Gibson, R.E., Schiffmann, R.L. and Pu, S.L. (1970), “Plane Strain and

Axisymmetric Consolidation of a Clay Layer on a Smooth Impervious Base”,

Quart. J. of Mech. and Applied Math., Vol. XXIIL Part 4, pp 505-519.

Bibliography 227

[63] Geertsma, J. (1957), “ The Effect of Fluid Pressure Decline on Volumetric

Changes of Porous Rocks “, Pet. Trans., AIME, No. 210, pp 331-340.

[64] Geertsma, J. (1966), “Problems of Rock Mechanics in Petroleum Production

Engineering “, Proc. of the 1st Cong. of li-it. Soci. of Rock Mechanics, Lisbon,

pp 585-594.

[65] Grieg, J., Byrne, P.M. and Srithar, T. (1990), “ COPP - çQnoil Rost

Erocesser, A Software Package for Pre/Post Processing CONOIL Data “, Dept.

of Civil Engr., U. of British Columbia, Vancouver, Canada.

[66] Gudehus, G. and Darve, F., eds (1982), “Proceedings of the International

Workshop on the Constitutive Behaviour of Soils”, Grenoble, France.

[67] Gulbrandsen, S. and Wille, S.O. (1985), “A Finite Element Formulation

of the Two-Phase Flow Equations for Oil Reservoirs “, SPE 13516, 8th SPE

Symp. on Reservoir Simulation, Dallas, Texas.

[68] Harris, M.C. and Sobkowicz, J.C. (1977), “Engineering Behaviour of Oil

Sand”, The Oil Sands of Canada Venezula, The Canadian Inst. of Mining and

Metallurgy, Special Vol. 1’4 pp 270-281.

[69] Hinton, E. and Owen, D.R.J. (1977), “ Finite Element Programming “,

Academic Press, London.

[70] Hood, P. (1976), “ Frontal Solution Program for Unsymmetric Matrices”,

mt. J. for Num. Mtds. in Engineering, Vol. 10, pp 379-399.

[71] Hwang, C.T., Morgenstern, N.R. and Murray, D.W. (1972), “ Ap

plication of Finite Element Method to Consolidation Problems”, Application

of FEM in Geotech. Engr., Proc. of Symp., Vicksburg, Mississippi, Vol. I1 pp

739-765.

Bibliography 228

[72] Hughes, J.M.O., Worth, C.P. and Windle, D. (1977), “ Pressuremeter

Tests in Sands”, Geotechnique, Vol. 27, No. 4, pp 455-477.

[73] Huyakorn, P.S. and Pinder, G.F. (1977a), “Solution of Two-Phase Flow

Using a New Finite Element Technique”, Applied Numerical Modelling, Ed. by

C. A. Brebbia, U. of Southampton, July, pp 375-590.

[74] Huyakorn, P.S. and Pinder, G.F. (1977b), “ A Pressure-Enthalpy fi

nite Element Model for Simulating Hydrothermal Reservoirs “, Advances in

Computer Methods for Partial Differential Equations Il, Ed. by Vichnevetsky,

IMACS (AlGA).

[75] Irons, B.M. (1970), “A Frontal Solution Program for Finite Element Anal

ysis”, mt. J. for Num. Mtds. in Engr., Vol. 2, pp 5-32.

[76] Irons, B.M. and Ahmad, S. (1980), “Techniques of Finite Elements”, Ellis

Horwood, Chichester, U.K.

[77] Iwan, W.D. (1967), “ On a Class of Models for the Yielding Behaviour of

Continuous and Composite Systems “, J. Appl. Mech., Vol.34, pp 612-617.

[78] Javandel, I. and Wtherspoon, P.A. (1968), “ Application of the Finite

Element Method to Transient Flow in Poros Media”, Soc. of Petroleum Engr.

J., Sep., pp 24 1-252.

[79] Jitno, H. and Byrne, P.M. (1991), “ Prediction of Settlement at the Cole-

brook Overpass Using 2-D Finite Element Analysis “, Dept. of Civil Engr.,

Univ. of B. C., Unpublished.

[80] De Josselin de Jong (1957), “Application of Stress Functions to Consolida

tion Problems”, Proc. of the 4th mt. Conf. on Soil Mech. and Found. Engrg.,

London, Vol. 1, pp 320-323

Bibliography 229

[81] Ko, H.Y. and Sture, S. (1981), “State of the art: Data Reduction and Ap

plication for Analytical Modelling “, Laboratory Shear Strength of Soil, ASTM

STP 740, Yong, R.N and Townsend F.C., eds, American Society for Testing

and Materials, pp 329-386.

[82] Kokal, S.L. and Maini, B.B. (1990), “An Improved Model for Estimating

Three-Phase Oil-Water-Gas Relative Permeabilities from Two-Phase Oil-Water

and Oil-Gas Data “, J. Can. Petro. Tech., Vol. 29, No. 2, pp 105-114.

[83] Konder, R.L. (1963), “ Hyperbolic Stress-Strain Response: Cohesive Soils”,

J. of SMFD, ASCE, Vol. 89, SM 1, pp 115-143.

[84] Kosar, K.M. (1989), “ Geotechnical Properties of Oil Sands and Related

Strata”, Ph.D Thesis, Dept. of Civil Eng., Univ. of Alberta, Edmonton, Canada.

[85] Kosar, K.M., Scott, J.D. and Morgenstern, N.R. (1987), “ Testing to

Determine the Geotechnical Properties Oil Sands”, CIM Paper 87-38-59, 38th

ATM of Pet. Soc. of CIM, Calgary.

[86] Lade, P.V. (1987), “ Behaviour of Plasticity Theory for Metals and Fric

tional Materials “, Constitutive Laws for Engineering Materials; Theory and

Applications.

[87] Lade, P.V. (1977), “Elasto-Plastic Stress-Strain Theory for Cohesionless Soil

with Curved Yield Surfaces”, mt. J of Solids Structures, Vol. 13, pp 1019-1035.

[88] Lade, P.V. and Duncan, J.M. (1975), “Elasto-Plastic Stress-Strain Theory

for Cohesionless Soil “, J. of the Geot. Engr. Div, ASCE, Vol. 101.

[89] Laing, J.M., Graham, J.P., Stokes, A.W. and Collins, P.M. (1992),

Geotechnical Instrumentation and Monitored In-Situ Behaviour of Oil Sands

Bibliography 230

During Steaming at the AOSTRA UTF Phase-A Trial” 44th Can. Geot. Conf.,

Vol. 2, Paper No. 89, Calgary.

[90] Lambe, T.W. and Whitman, RN. (1969), “Soil Mechanics “, John Wiley

& Sons.

[91] Leverett, M.C. and Lewis, W.B. (1941), “ Steady Flow of Gas-Oil-Water

Mixtures Through Unconsolidated Sands “, Trans., AIME, Vol.142, pp 107-

116.

[92] Lewis, R.W., Morgan, K. and Roberts, P.M. (1985), “Finite Element

Simulation of Thermal Recovery Process and Heat Losses to Surrounding Strata

“, 3rd European Meeting on Improved Oil Recovery, Rome, April pp 305-315.

[93] Lewis, R.W., White, I.R. and Wood W.L. (1978), “A Starting Algorithm

for the Numerical Simulation of Two-Phase Flow Problems “, IJNME, Vol. 12,

pp 319-328.

[94] Lewis, R.W., Roberts, G.K. and Zienkiewicz, O.C. (1976), “A Nonlin

ear Flow and Deformation Analysis of Consolidation Problems”, 2nd mt. Conf

on Num. Mtds. in Ceomech., Blacksburg, Virginia, Vol.2, pp 1106-1118.

[95] Light, M.F. and Luxmoore, A.R. (1977), “Application of the Front So

lution to Two and Three Dimensional Elasto-Plastic Crack Problems”, mt. J.

for Num. Mtds. in Engrg., Vol. 11, pp 393-395.

[96] Matsuoka, H. (1983), “ Deformation and Strength of Granular Materials

Based on the Theory of Compounded Mobilized Plane and Spatial Mobilized

Plane “, Advances in the Mech. and the Flow of Granular Materials, Trans.

Tech. Publ., pp 813-836.

Bibliography 231

[97] Matsuoka, H. (1974), “ A Microscopic Study on Shear Mechanism of Gran

ular Materials “, J. Soils and Foundations, Vol. 14, No.1, pp 29-43.

[98] Matsuoka, H. and Nakai, T. (1982), “ A New Failure Criterion for Soils

in Three-Dimensional Stresses “, IUTAM Conf on Deformation and Failure of

Granular Materials, Delfi, pp 253-263.

[99] Matsuoka, H. and Nakai, T. (1977), “ Stress-Strain Relationship of Soil

Based on the SMP “, Proc., Speciality Session 9, 9th ICSMFE, pp 153-162.

[100] Matsuoka, H. and Nakai, T. (1974), “ Stress Deformation and Strength

Characteristics of Soil under Three Different Principal Stresses “, Proc., JSCE,

No. 232., pp 59-70.

[101] Mayer, C. (1973), “ Solution of Linear Equations - State of the Art”, J. of

the Struc. Div., ASCE, Vol. 99, ST?, pp 1507-1526.

[102] McMichael, C.L. and Thomas, G.W. (1973), “ Reservoir Simulation by

Galerkin’s Method “, SPEJ, June, pp 125-138.

[103] McNamee, J. and Gibson, R.E. (1960), “Plane Strain and Axially Sym

metric Problems of Consolidation of A Semi-Infinite Clay Stratum”, Quart. J.

of Mech. and Appi. Maths., Vol. 13, pp 210-227.

[104] Mitchel, J.K., Hooper, O.R. and Campanella, R.J. (1965), “ Perme

ability of Compacted Clay”, J. of SMFD, ASCE, Vol. 91, SM 4, pp 4 1-65.

[105] Mosscop, G.D. (1980), “Geology of the Athabasca Oil Sands”, Science, Vol.

207, No.11, pp 145-152.

[106] Murayama, S. (1964), “ A Theoretical Consideration on the Behaviour of

Sand”, Proc. IUTAM Symp. of Rheology and Soil Mech., Grenoble, pp 65-79.

Bibliography 232

[107] Murayama, S. and Matsuoka, H. (1973), “A Microscopic Study on Shear

ing Mechanism of Soils “, Proc. of the 8th ICSMFE, Vol. 1, pp 293-298.

[108] Nakai T. and Matsuoka H. (1983), “Constitutive Equation for Soils Based

on the Extended Concept of Spatial Mobilized Plane and its Application to

Finite Element Analysis” Soils and Foundations, Vol. 23, No.4, pp 87-105.

[109] Naylor, D.J. (1973), “Discussion”, Proc. of The Symp. on The Role of The

Plasticity in Soil Mech., Cambridge, pp 291-294.

[110] Negussey, D. (1984), “An Experimental Study on the Small Strain Response

of Sand “, Ph.D Thesis, Dept. of Civil Engr., U of British Columbia, Vancouver,

Canada.

[111] Nemat-Nasser, S. (1980), “On the Behaviour of Granular Material in Simple

Shearing”, Soils and Foundations, Vol. 20, No.3, pp 59-73.

[112] Nova, R. and Wood D.M. (1979), “ A Constitutive Model for Sand in

Triaxial Compression “, mt. J. for Num, and Anal. Mtds. in Geomechanics,

Vol.3.

[113] Oda, M. and Konishi, J. (1974), “Microscopic Deformation Mechanism of

Granular Material in Simple Shear”, Soils and Foundations, Vol. 14, No. 4, pp

25-38.

[114] Pande, G., ed (1982), Proc. of the mt. Symposium on Numerical Models in

Geomechanics, Delfi.

[115] Parameswar, R. and Maerefat, N.L. (1986), “A Comparison of Methods

for the Representation of Three-Phase Relative Permeabilities “, SPE Paper

No. 15061, 56th California Regional Meeting, Oakland, California.

Bibliography 233

[116] Pender, M.Z. (1977), “ A Unified Model for Soil Stress-Strain Behaviour “,

IXth ICSMFE, Tokyo, 325-331.

[117] Pina, H.L.G. (1981), “An Algorithm for Frontwidth Reductions” mt. J. for

Num. Mtds. in Engrg., Vol. 17, pp 1539-1546.

[118] Polikar, M., Puttagunta, V.R., DeCastro, V. and Farouq Au, S.M.

(1989), “ Relative Permeability Curves for Bitumen and Water in Oil Sand

Systems “, J. Can. Petro. Tech., Vol.28, No.1, pp 93-99.

[119] Prevost, J.H. (1979), “ Mathematical Modelling of Soil Stress-Strain Be

haviour”, 3rd mt. Conf on Num. Mtds. in Geomechanics, pp 34 7-361.

[120] Prevost, J.H. (1978), “ Plasticity Theory for Soil Stress Behaviour “, J.

Engr. Mech. Div. ASCE, Vol.104, No.5, pp 1177-1194.

[121] Prevost, J.H. and Hoeg. K. (1975), “ Effective Stress-Strain Strength

Model for Soils “, J. of the Soil Mech. and Foun. Div., ASCE. Vol. 101, No.3.

[122] Puttagunta, V.R., Singh, B. and Cooper, E. (1988), “ A Generalized

Viscosity Correlation for Alberta Heavy Oils and Bitumen”, UNITAR/UNDP

mt. Conf. on Heavy Crude and Tar Sands, Edmonton, Alberta, Canada.

[123] Raghavan, R. (1972), “ A Review of the Consolidation and Rebound Pro

cesses in One-Dimensional Porous Columns “, SPE Paper No. 4078, 47th An

nual Fall Meeting of the SPE, San Antonio, Texas.

[124] Read, H.E., Hegemier, G.A. (1984), “ Strain Softening of Rock, Soil and

Concrete - A Review Article “, Mech. Mater., Vol. 3, pp 271-294.

[125] Risnes, R., Bratli, R.K. and Horsrud, P. (1982), “Sand Stresses Around

a Well Bore”, Society of Petroleum Engineers J., pp 883-898.

Bibliography 234

[126] Roscoe, K.H., Schofield, A.N. and Wroth, C.P. (1958), “On the Yield

ing of Soils “, Geotechnique, Vol.9, pp. 22-53.

[127] Rowe, PN. (1971), “ Theoretical Meaning and Observed Values of Defor

mation Parameters for Soils “, Proc. of the Roscoe Memorial Symp. Cambridge

Univ., pp 143-194.

[128] Rowe, P.N. (1962), “ The Stress Dilatancy Relations for Static Equilibrium

of an Assembly of Particles in Contact” , Proc. of the Royal Soc., A 269, pp

500-527.

[129] Saada, A. and Bianchini, G.S. Eds (1987), “ Constitutive Equations for

Granular Non-Cohesive Soils “, Proc. of the mt. Workshop, Case Westeren

Univ., Cleveland.

[130] Salgado, F.M. (1990), “ Analysis Procedures for Caisson-Retained Island

Type Structures”, Ph.D. Thesis, Dept. of Civil Engineering, University of

British Columbia, Vancouver, Canada.

[131] Salgado, F.M. and Byrne, P.M. (1991), “ A Three Dimensional Consti

tutive Elastic-Plastic Model for Sands Following the Spatial Mobilized Plane

Concept”, 7th mt. Conf. of the mt. Assoc. for Comp. Mtds and Advances in

Geomech. Cairns, Australia.

[132] Sandhu, R.S. (1968), “Fluid Flow in Saturated Porous Elastic Media”, Ph.D.

Thesis, Dept. of Civil Engineering, University of California, Berkeley.

[133] Sandhu, R.S. (1972), “Finite Element Analysis of Consolidation and Creep”

State of the Art Lect., Application of FEM in Geotech. Engr., Proc. of Symp.,

Vicksburg, Mississippi, Vol. II, pp 697-738.

Bibliography 235

[134] Sandhu, R.S. and Wilson, E.L. (1969), “Finite Element Analysis of Seep

age in Elastic Media” J. of Engrg. Mech. Div., ASCE, Vol. 95, EMS, pp 641-652.

[135] Sandhu, R.S., Liu, H. and Singh, K.J. (1977), “Numerical Performance of

Some Finite Element Schemes for Analysis of Seepage in Porous Elastic Media”,

mt. J. for Numer. and Anal. Mtds in Geomech., Vol. 1, pp 177-194.

[136] Saraf’, D.N. and McCaffery, F.G. (1981), “Two and Three-Phase Relative

Permeabilities: A Review “, Petroleum Recovery Institute, Report No. 1981-8.

[137] Sayao A.S.F.J (1989), “ Behaviour of Sand Under General Stress Paths in

the Hollow Cylinder Torsional Device “, Ph.D Thesis, U. of British Columbia,

Vancouver, Canada.

[138] Schrefler, B.A. and Simoni, L. (1991), “ Comparison between Different

Finite Element Solutions for Immiscible Two-Phase Flow in Deforming Porous

Media “, Computer Mtds and Advances in Geomech, Beer, C., Booker J.R. and

Carter, J.P. (Eds), Cairns, Australia, pp 1215-1220.

[139] Schuurman, I.E. (1966), “The Compressibility of an Air-Water Mixture and

a Theoretical Relation Between The Air and The Water Pressures”, Geo tech

nique, Vol. 16, pp 269-281.

[140] Scott, R.F. (1985), “Plasticity and Constitutive Relations in Soil Mechanics

“, J. of Geotech. Engr. ASCE, Vol. 111, No. 5, pp 569-605.

[141] Scott, J.D. and Kosar, K.M. (1985), “ Foundation Movements Beneath

Hot Structures”, Proc. of 11th mt. Conf. on Soil Mech. and Foun. Eng., San

Francisco, California.

[142] Scott, J.D. and Kosar, K.M. (1984), “Geotechnical Properties of Oil

Sands”, WRI-DOE Tar Sand Symp., Vail, Colorado.

Bibliography 236

[143] Scott, J.D. and Kosar, K.M. (1982), “ Thermal Expansion of Oil Sands”

Proc. of Forum on Subsidence Due to Fluid Withdrawal U.S. Dept. of Energy

and The Rep. of Venezula Ministry of Energy and Mines, Oklahoma, pp 46-57.

[144] Scott, J.D. and Chalaturnyk, R.J., Stokes, A.W., and Collins, P.M.

(1991), “ The Geotechnical Program at the AOSTRA Underground Test Fa

cility “, 44th Can. Ceotech. Conf, Vol. 2, Paper No. 88, Calgary.

[145] Settari A. (1989), “ Physics and Modelling of Thermal Flow and Soil Me

chanics in Unconsolidated Porous Media “, SPE 18420, 10th Symp. on Reservoir

Simulation, Houston, Texas.

[146] Settari A. (1988), “ Modelling of Fracture and Deformation Processes in Oil

Sands “, Paper No. 43, 4th UNITAR/UNDP mt. Conf on Heavy Crude Tar

Sands.

[147] Settari A. and Price H. S. (1976), “Development and Application of Vari

ational Methods for Simulation of Miscible Displacements in Porous Media “,

Paper SPE 5721, 4th SPE-AIME Symp. on Numerical Simulation of Reservoir

Performance, Los Angeles.

[148] Settari A., Ito, N., Fukushima, N. and Vaziri, H. (1993), “Geotechnical

Aspects of Recovery Processes in Oil Sands “, Can. Geotech. J., Vol. 30, No.

1, pp 22-33.

[149] Settari A., Kry, P.R. and Yee, C.T. (1989), “Coupling of Fluid Flow and

Soil Behaviour to Model Injection into Uncemented Oil Sands “, J. Can. Pet.

Tech., Vol. 28, No. 1, pp 81-92.

[150] Settari A., Price, H. S. and Dupont, T. (1977), “ Development and

Application of Variational Methods for Simulation of Miscible Displacement in

Porous Media “, Soc. Pet. Eng. J, Vol. 1?, pp 228-24 6.

Bibliography 237

[151] Singh, R. (1965), “ Unsteady and Saturated Flow in Soils in Two Dimen

sions”, Tech. Report No.54, Dept. of Civil Engr., Stanford Univ.

[152] Sisler, H.H., Vanderwarf, C.A. and Davidson, A.W. (1953), “General

Chemistry - A Systematic Approach”, McMillan Co., New york.

[153] Sobkowicz, J. (1982), “ Mechanics of Gassy Sediments”, Ph.D Thesis, Dept.

of Civil Eng., Univ. of Alberta, Edmonton, Canada.

[154] Sobkowicz, J. and Morgenstern, N.R. (1984), “ The Undrained Equi

librium Behaviour of Gassy Sediments”, Can. Geotech. J., Vol. 21, No. 3, pp

439-448.

[155] Sloan, S.W. (1981), “Numerical Analysis of Incompressible and Plastic Solids

Using Finite Elements”, Ph.D. Thesis, Dept. of Engineering, Univ. of Cam

bridge.

[156] Sloan, S.W. and Randolph, M.F. (1982), “Numerical Prediction of Col

lapse Loads Using Finite Element Methods”, mt. J. for Num. and Anal. Mtds.

in Geomech., Vol. 6, pp 47-76.

[157] Small, J.C., Booker, J.R. and Davis, H.H. (1976), “Elasto-Plastic Con

solidation of Soils”, mt. J. Solids and Structures, Vol. 12, pp 4 31-448.

[158] Sparks, A.D.W. (1963), “Theoretical Consideration of Stress Equations for

Partly Saturated Soils”, Proc. of 3rd Regional Conf. for Africa on Soil Mech.

and Foun. Eng., Salisburg, Southern Rhodesia, Vol. 1, pp 215-218.

[159] Spivak, A., Price, H.S. and Settari, A. (1977), “ Solutions of the Equa

tions for Multidimensional, Two-Phase, Immiscible Flow by Variational Meth

ods “, SPEJ, Feb., pp 27-41.

Bibliography 238

[160] Srithar, T. (1989), “ Stress, Deformation and Flow Analysis of Oil Sand

Masses Under Applied Load and Temperature Changes”, M.A.Sc Thesis, Dept.

of Civil Engineering, Univ. of British Columbia, Vancouver, Canada.

[161] Srithar, T., Byrne, P.M. and Vaziri, H.H (1990), “ Consolidation Anal

ysis of Oil Sand Masses Under Applied Temperature Changes” Can. Geotech.

J., Vol. 27, pp 752-760.

[162] Srithar, T., Byrne, P.M. (1991), “ Stress, Deformation and Flow Analysis

of Oil Sand Masses Under Applied Load and Temperature Changes”, Proc. 7th

liii. Conf on Compu. Mtds and Advances in Geomech., Cairns, pp 1595-1601.

[163] Stone, H.I. (1970), “Probability Model for Estimating Three-Phase Relative

Permeabilities “, J. of Petro. Tech., Vol. 2, pp 214-218.

[164] Symes, M.J. and Hight, D.W. (1988), “Drained Principal Stress Rotation

in Saturated Sand “, Geotechnique 38, No. 1, pp 59-81.

[165] Symes, M.J., Gens, A. and Hight, D.W. (1984), “Undrained Anisotropy

and Principal Stress Rotation in Saturated Sand “, Geotechnique 34, No. 1, pp

11-27.

[166] Symes, M.J., Hight, D.W. and Gens, A. (1982), “ Investigating

Anisotropy and the Effects of Principal Stress Rotation and of the Interme

diate Principal Stress Using Hollow Cylinder Apparatus “, IUTAM Conf. on

Deformation and Failure of Granular Materials.

[167] Taylor, R.L. (1977), Finite Element Method, by Zienkiewicz, O.C., McGraw

Hill, pp 677-757.

[168] Terzaghi, (1923), “Die Beziehungen Zwischen Elastizitat Und Innerdruck”,

Akad. Wiss. wiert. Sitz. Math., Ki, ha, 132, pp 125-138

Bibliography 239

[169] Terzaghi, K. and Peck, R.B. (1948), “Soil Mechanics in Engineering Prac

tice”, John Wiley Sons

[170] Timoshenko, S. (1941), Strength of Materials, Vol. 2, Van Nostrand New

York.

[171] Timoshenko, S. and Goodier, J.N. (1951), Theory of Elasticity, McGraw

Hill, New York.

[172] Tortike, W.S.(1991), “ Numerical Simulation of Thermal, Multiphase Fluid

Flow in an Elastoplastic Deforming Oil Reservoir “, Ph.D Thesis, Dept. of

Mining, Metallurgical and Petroleum Engineering, Univ. of Alberta.

[173] Valanis, K.C. (1985), “ On the Uniqueness of Solution of the Initial Value

Problem in Softening Materials “, J. Appi. Mech., ASME 52, 85-A PM-29.

[174] Vaziri, H.H. (1986), “Nonlinear Temperature and Consolidation Analysis

of Gassy Soils”, Ph.D Thesis, Dept. of Civil Eng., Univ. of British Columbia,

Vancouver, Canada.

[175] Vermeer, P.A. (1978), “A Double Hardening Model for Sand”, Geotechnique,

Vol. 28, No.4, pp 423-433.

[176] Vermeer, P.A. (1982), “A Five-Constant Model Unifying Well-Established

Concepts “, Proc. of the mt. Workshop on Constitutive Relations for Soils,

Grenoble.

[177] Voss, C.I. (1978), “A Finite Element Simulation of Mutiphase Geothermal

Reservoirs”, Ph.D Thesis, Department of Civil Engineering, Princeton Univ.,

New Jersey.

Bibliography 240

[178] Wallace, M.I. (1948), “Experimental Investigation of The Effect of Degree of

Saturation on The Permeability of Sands”, S.M. Thesis, Dept. of Civil Engr.,

M.I.T., Cambridge.

[179] Wan, R., Chan, D.H. and Kosar, K.M. (1991), “A Constitutive Model

for the Effective Stress-Strain Behaviour of Oil Sands “, J. of Can. Petroleum

Tech., Vol. 30, No. 3, pp 22-28.

[180] White, I.R., Lewis, R.W. and Wood, W.L. (1981), “The Numerical

Simulation of Multiphase Flow Through a Poros Medium and Its Application

to Reservoir Engineering”, Applied Mathematical Modelling, Vol.5, pp 165-172.

[181] Wilde, P. (1979), “ Mathematical and Physical Foundations of Elastoplastic

Models for Granular Media “, Colloque Franco-Polonais, Paris.

[182] Wong, R.C.K., Barr, W.E. and Kry P.R. (1993), “Stress-Strain Response

of Cold Lake Oil Sands “, Can. Ceot. J., Vol. 30, No. 2, pp 220-235.

[183] Yan, L. (1986), “ Numerical Studies of Some Aspects With Pressuremeter

Tests and Laterally Loaded Piles”, M.A.Sc Thesis, Dept. of Civil Eng., Univ.

of British Columbia, Vancouver, Canada.

[184] Yokoo, Y., Yamagata, K. and Nagaoka, H. (1971a), “Variational Prin

ciples for Consolidation”, Soils and Foundations, Vol. 11, No. 1, pp 25-35.

[185] Yokoo, Y., Yamagata, K. and Nagaoka, H. (1971b), “ Finite Element

Analysis of Consolidation Following Undrained Deformation”, Soils and Foun

dations, Vol. 11, No. 4, pp 37-58.

[186] Yong, R.N., Ko, H.Y., eds (1980) , “Proceedings of the Workshop on Limit

Equilibrium, Plasticity and Generalized Stress-Strain in Geotechnical Engineer

ing” , ASCE, McGill University.

Bibliography 241

[187] Yong, R.N., Selig, E.T., eds (1980) , “ Proceedings of the Symposium

on the Application of Plasticity and Generalized Stress-Strain in Geotechnical

Engineering “, ASCE, Hollywood, Florida.

[188] Zienkiewicz, O.C., Valliappan, S. and King, I.P. (1968) , “Stress Anal

ysis of Rock as a ‘No Tension Material’ “, Geotechnique 18, pp 56-66.

Appendix A

Load Shedding Formulation

The details of applying the load shedding technique to model strain softening are

described in this appendix. During a load increment it is possible that the stress state

of an element may move from P0 to P as shown in figure A.1 This will violate the

Figure A.1: Strain Softening by Load Shedding

failure criterion and the stress state should be brought back to P1. In load shedding

technique, this is done by taking out the shear stress equivalent to and then

‘r/p

*

1i

P1

1

7SMP,p 7SMP,i

242

Appendix A. Load Shedding Formulation 243

transferring it to the adjacent stiffer elements. The detailed steps of this procedure

will be as follow:

1. Estimate the stress ratio (n’, Figure A.1) in the strain softening region corre

sponding to the shear strain (7sMp,1) using equation 3.63 as

7i 1r + (i— a,,) exp { — K (ysMP, 1 — 7SMP,p

) } (A. 1)

2. Estimate the amount of stress ratio that has to be taken out as

(A.2)

3. Evaluate the changes in the Cartesian stress vector {/.o-}Ls which corresponds

to

4. Evaluate the force vector {/.F}Ls equivalent to {/.o}Ls.

5. Take out {Lo-}Ls from the failed element and set its moduli to low values.

6. Carry out a load step analysis with {F}Ls as the incremental load vector.

7. Check whether any other elements violate the failure criteria and undergo soft

ening, and if so, repeat the load shedding procedure.

A.1 Estimation of {zSJ}Ls

In order to estimate the changes in the Cartesian stress vector, it is easier to first

estimate the changes in principal stresses. By differentiating equation 3.34 in terms

of principal stresses the following equation can be obtained:

T

1213 + 1113(02 + £73) —1112(U2U3) Lo

18I1213+1113(U3+Ui)1112(U3U1) U2 (A.3)

‘213 +1113(ui + £72) —1112(U1t72)


The above equation can be rewritten as

A1/o-1 + A2Io2H- A3z3 (A.4)

To estimate the changes in principal stresses, two more equations are needed, in

addition to equation A.4. The following two conditions are assumed during the load

shedding to obtain the additional two equations:

1. The mean normal stress remains constant during load shedding. This gives

1 + + Lo3 0 (A.5)

2. The b-value [(02 — 03)/(01 — 03)] remains constant. Which implies

(A6)Oi — 03 — (o- + Zoi) — (03 + Lo3) —

By rearranging the terms

(A.7)

By solving equations A.4, A.5 and A.7 the following equations can be obtained

for 02 and o3:

2—b

H- b)(A3 — A2) —(2— b)(A1 — A2)(A.8)

1+b(A.9)

—(Lri + 3) (A.l0)

Now, the changes in the Cartesian stresses can be obtained by simply multiplying

the principal stress vector by the transformation matrix.


l m

12 m2 n2y

12 m2(A.11)

2ll, 2mm 2n,nIO3

2l,l 2mm2 2nn

2mm 2nzna,

where

1a li,, and l - direction cosines of o to the x, y and z axes

m, m and m - direction cosines of O2 to the x, y and z axes

n, n and n - direction cosines of 03 to the x, y and z axes

A.2 Estimation of {F}Ls

The load vector corresponding to the changes in stresses has to be applied at the

nodes of the soil element that failed, to transfer equivalent amount of stresses to the

adjacent stiffer elements. By doing this, the stress equilibrium in the domain will be

maintained. The load vector can be evaluated using the virtual work principle.

By the principle of virtual work, the work done by the virtual displacements (8)

to the system will be equal to the work done by the internal strains caused () within

the system. Mathematically this can be expressed as

{}T{f} = J{}Tfr}dv (A.12)

where

{ f} - Force vector

{o} - Stresses within the system


The virtual strains and the displacements can be related by

{} = [B]{} (A.13)

where [B] is the strain-displacement matrix.

substitution of equation A.13 in equation A.12 will give

{}T{f} = J{}T[B]T{}dv (A.14)

This can be further written as

{f} = J[B]Tfr}dv (A.15)

Following equation A.15 the force vector for load shedding can be obtained as

{IF}Ls J[B]T{U}LSdV (A.16)

Appendix B

Relative Permeabilities and Viscosities

Some detailed explanations which are needed in the evaluation of equivalent per

meability are given in this appendix. To evaluate the equivalent permeability, the

relative permeability and the viscosity values of the pore fluid components are nec

essary. The first section explains how to calculate the relative permeabilities and the

equivalent permeability through an example data set. The viscosity values of water

at different temperatures are given in section 2. Section 3 gives some insights into

the viscosity of hydrocarbon gases and how to evaluate it.

B.1 Calculations of relative permeabilit ies

B.1.1 Relevant equations

The relative permeabilities of water, gas and oil can be obtained from the following

equations:

Jc° S1k° S_qrog 9’ row w a a— Jo

1 t-’w JgI

krw = — A2)A3 (B.2)

krow B1(B2 — Sw)B3 (B.3)

krg = C1(S9 — (B.4)

247

Appendix B. Relative Permeabilities and Viscosities 248

k,.09 = D1(D2 —S9)D3 (B.5)

— krow(Sw)B6/3W

— k0(1 — S,,)

— k,.09(S9)9ko (1—S (

r09\ 9

= 5;— Swc

S S (B.8)Wc om

s S0— Sorn

So > Som (B.9)wc om

9‘9i c’ C’‘-‘uc ‘-‘Orn

where

k,.0 - relative permeability of oil in 3-phase system

k,. - relative permeability of water in 3-phase system

- relative permeability of oil in water-oil system

- relative permeability of oil in oil-gas system

krg - relative permeability of gas in 3-phase system

k - relative permeability of oil at connate water saturation

in a water-oil system

k09 - relative permeability of oil at zero gas saturation

in an oil-gas system

S,,, S, S - Saturation of water, oil and gas respectively

S, S, 5 - Normalized saturation of water, oil and gas respectively

- Critical water saturation

5om - Residual oil saturation

A1,A2... etc. - Constants


= 0.2

k° —107’OW

= 1.0

A1 = 1.820

B1 = 2.769

C1 = 2.201

= 1.640

A2 = 0.20

B2 = 0.80

C2 = 0.05

D2 = 0.80

A3 = 2.375

B3 = 1.996

C3 = 2.704

= 2.547

= 0.5

I-sw = 8 x 104Pa.s

k = 1 x 102m2

S, 0.4

= 2OPa.s

Sg = 0.1

= 2 x 105Pa.s (at 30°C)

B.1.3 Sample calculations

By substituting the data into the relevant equations

0.5 — 0.2=0.5

1 — 0.2 — 0.2

0.15;

= 1 — 0.2 — 0.2= 0.1667

0.4— 0.2= = 0.3333

1 — 0.2 — 0.2

The equivalent permeability is given by

kEQ=kI-o P’g

B.1.2 Example data

(B.11)


= 2.769(0.8 — 0.5)1.996 = 0.25

krog = 1.640(0.8 0.1)2.547 = 0.661

0.25= 1(1 — 0.5)

= 0.5

0.6610 793

— 1(1 — 0.1667) —

krw 1.820(0.4 — 0.2)2.735 = 0.068

k,.9 = 2.201(0.1 — 0.05)2.704 = 0.001

k,.0 = 0.3333(1.0 X 0.1667+1.0 X 0.5)

X 0.5 X 0.793 = 0.132

—12 / 0.068 0.132 0.001‘ m m

kEQ = 1 >< 108 x 10

+20

+2 x 105)

1.350 x 10 —

B.2 Viscosity of water

The viscosity of water at different temperatures are well established and can be ob

tained form the international critical tables. The following tables are given by N.

Ernest Dorsey in the international critical tables and are reproduced here. These

data are also built in the computer program CONOIL.

Appendix B. Relative Permeabiiities and Viscosities

Table B.1: Viscosity of water between 0 and 1000 C

251

Values in rnillipoises (1, 12, 16, 17, 22, 24, 30, 31, 32, 38)

C 0 1 2 3 4 5 6 7 8 9

0 17.93* 17.326 16.74* 16.19a 15.67. 15.18* 14.72* 14.28* 13.872 13.47,10 13.097 12.73s. 12.39o 12.06i 11.748 11.44? 11.15* 10.875 10.60s 10.34o20 10.087 9.843 9.60* 9.38* 9.16i 8.94. 8.74* 8.55i 8.368 8.181

30 8.004 7.834 7.67* 7.511 7.35 7.20* 7.064 6.92 6.791 6.66140 6.536 6.41s 6.29* 6.184 6.075 5.97* 5.86* 5.77* 5.67s 5.58250 5.492 5.40s 5.32* 5.236 5.153 3.07s 4.99* 4.918 4.84a 4.77o60 4.69* 4.62s 4.56i 4.495 4,43i 4.36* 4.30* 4.24s 4.186 4.12s

70 4.07i 4.01* 3.96z 3.909 3.8.5? 3.806 3.756 3.70* 3,66i 3.61s80 3.57. 3.52* 3.483 3.44. 3.39* 3.35i 3.31? 3.27* 3.24* 3.20390 3.16* 3.13* 3.095 3.061 3.027 2.994 2.96a 2.93* 2.89. 2.86*

100 2.83* 2.82 2.79 i 2.76 2.73 2.70 2.67 2.64 2.62 2.59

At a pressure ox 1 atm., = a/(b + t)”.

At a pressure of P kg/cm2, ,7p = ?7i[l + k,(P — 1) X 10’J.‘11 is the value , when P is 1 kg/cm2,which may be taken asThe unit of , is the poise unless otherwise stated.

Table B.2: Viscosity of water below 00 C

H,O ov 100°C (16)

Values as recorded by author accord with I. C. T. values below100°C; the others are given as he has published them. Thepressure is that of the saturated vapor at the temperaturesindicated.

4, °C 110 120 130 140 150 1601000,7 2.56 2.32 2.12 1.96 1.84 1.74

Table B.3: Viscosity of water above 1000 C

H20 BELOW 0°C (39)

Values corrected and adjusted to accord with I. C. T. valuesabove 0°C

FoR,.1Ux.E AND UNITS

the value of,7 at 1 atm.

—2 —4 —5 —6 —8 —10100077 19.1 20.5 21.4 22.2 24.0 26.0


B.3 Viscosity of hydrocarbon gases (from Carr et al., 1954)

The viscosity of hydrocarbon gases can be expressed as a function of reduced pressures

and temperatures, i.e.,

(B.12)IL1

where

- viscosity of gas at reduced temperature TR

and at reduced pressure FR

pi - viscosity of gas at atmospheric pressure and given temperature

TR - temperature/critical temperature (in absolute units)

PR - pressure/critical pressure (in absolute units)

If the gas is a mixture of hydrocarbons, the pseudo-critical concept has to be

applied. Thus, in place of critical temperature and critical pressure, the pseudo-

critical temperature and pressure have to be used. The pseudo-critical temperature

is given by

(B.13)

The pseudo-critical pressure is given by

PPc=XzFci (B.14)

where

- mol fraction of component i in the mixture

- critical temperature of component i in absolute units

- critical pressure of component i in absolute units

a)‘—

4-

)d

0.4

,4

U)

.•-

40

-c

•c:

a)a)—

l

_

a)U)

a)U)

‘—I

;0

to0o

—-d

-c

if)

0a)

Cl)U

)C

)c

;4-

.4

0-4-a

ca)

U)

0Q)

-4to

a)Cl)

)-U

)to

cdto

toa)

o0

--

O-

_-

Cl)H

0-4

a);-1

p-

‘da)

to-0

to

:-

EQ0

a)

4‘-

-4

a)0

-dl-

4a)

0-

a)E

.a

a)0

‘0

Cl)

V

0U

)a,

.4)

;-4a

‘.

0a)4

.44

a)U

)U

)U

)a)

-a)

.-I

_4

--4

-4-a‘-4

(3+

l-40

I—’

-4-4

4)‘-4

2a)

,.D0

:a)

a)a)0-

.a)

-4

--

c3

0

toCl)

r0

oa)

Cl)-4

Cl)a)

0a)

-4

..

.—a)

to-

0Cl)

‘-4

0-d

:0a)

o-0

Cl)d

U)

-4

l-40o00

0

U)

0

:111

if)

-I

:j:==iiI111JW

i1L111FW1

IMIII:.4a.3wvo, an,.!J.L

IiL

IJ_L

LL

_

Ian3L,

tTITFITIITII

—

Ii

IiI1IM1ll1ll12t:

tll1llllillllhIt

r-.l,6A•woIi,,_,

ltav

d’

“llD

c1llllt

4.q_4‘....

,s,wou,,.uo,;i.

Cl)-

II

IIIIc!:

1

U)

VISd

‘UflS

SW

dV

OI1

W3O

OflS

d,d

d—

U.

‘311fl1VU

3dY131

VO

4J4l4OO

OflS

did

Ia,

‘-4.,

oCD

CDCD

.

2U

)

‘-i’

CDCD

,- CD-$

-+

:-

CDo

CD

CD0

.q-

), (

—.

- CDI—

0‘d

c-

I—.

CD

j.

p,U

)0

CD0

0—

.+

0U

)0 p

CD

•-•

CD

pU

)CD

p;;.

U)

o op

0‘—

4.,

U)

,-4.

,

4,_

-’o

o9’

09’

9,

•-

CD,-

,CD

CDCD

‘‘

CDU

)

0•

•P

‘-d

0o

‘-UCD

•C

D9’

4-4—]0

_..

,,.

rJ ‘-U “l

CD0

CD4-

sU

)-

I-s

CD-

r pCD

CDCD

CD—

..

U)

9’:-‘

‘-s

CDU

)CD

C’)

+p

CDCD

--

b0

1E

LJ

P0

‘-UU

)CD 1

4-s 9’

‘-5

4--

CD—

.—

‘0 9, —.‘ .

I—-

4-s CD

C’)

U)

9’

‘-5 CD •

CDo

0 9, 0 -“ CD 0

U)

‘)

oo

4--

U) 0 o 0 4-+

) ,- _4-4 - 0 CD oq

- ‘-UO

u,

9’CD 0

CD

‘-5 CD U) 0 0 U)

‘-4-

0 4-f

,

‘-1 0 0 p I-s 0 9, U) CD ci) 9, 0 CD 9’ 0 U)

‘-U CD ‘-5 CD

0 0 0 ‘-4-

CD U) 0 4-f

,

‘-5 0 0 9, I-s 0 0 CD 9, CD 0 CD U) CD ‘-5

‘-4-

‘-U 0 ‘-4-

U)

O’3

‘-5 CD

VIS

CO

SIT

Y,

AT

IATM

.•AJ, CENTI

PO

IS

8°

80

öb

0b

ION

AD

OE

D-i-

1-ii

F1t4

-iI—

tWttW

ttIt

lilt

I1I

8

FL*J1

i1ir

4’i

‘-U >1 9, ‘-4-

‘-U I..

II+

H+

FfA

ii(r

LjtI

U11

M1W

411

1FF

PII1I

IlIFF

1Fvr1

I14-4a

I4JYL

4AY1

4n’i1

fFIl

CU

UII

ttt1

IIIi

1iV

IlI1

IJ11tI

fl1V

IIIA

FII

,I1

VIl

LIt

IUI

rV

111)

14-I

11

11

11

71

11

5I

1111

1Il

ILU

I.i

ii.n

rrii

iD

IAl

rU1

Er[

I4fl

EC

s.P

111)

11It

’ll

il/l

iV

IKJ

y‘i

iC

OA

CC

TIO

uD

OE

OD

1l1

(—19

ItlA

1-I

II

IA-t

-f-

To

,sc

—G

Pt.

,8

CO

RII

CC

1IO

NA

DO

LDT

I)V

ISC

—G

14

-1 0

r


J )‘‘ =

.-‘. .L. I. .iL. - -

— IH

-- -- —

- .LL.... -- — -

III,

£1 / ‘

:iL.-(-L — -

.k L .t.1- -r ‘--j-•-p.j-r / I—— —--

•- -ir-rr ,4t—— —-- -. 1—-

—— ---r ,

1 “i:0 —. — — I - H- 1! -f-J4 —

1.. — — — -- , -rv i- r;- — -

: ia:: .±Ik— — —- - ‘4- I

1”E

-

___

i!::: .

I ,.,.‘77 I.— - — . .-f ‘ T

—— - - -- r/r7 7 -i,.’_.

_____

i --- :..4 -

__

— . . — -

IC

____

— . —--—-- — — L I . —

2 3 4 .5 6 7.6LD 2 3 4 56 7I9

PSEUCOREDUGED PRESSURC

Figure B.3: Viscosity ratio vs pseudo-reduced pressure

VIS

CO

SIT

YR

AT

IO01

gU

I

__

_

-.---I

H111

HTTD

11I

—-

--

--

--

--

-Iil

lII

lili

liP

I::

E::

:tft

11

1It

t1U

ftI

tumiH

t—

——

—-—- -

AI

III

IIL

iII

IU—

ti±

fliI

JJ-J

kH

lfl-

Ct)—

II

II

i—i-

-i-r

nI

II

IIi

-i-H

I]II

UH

i]CD

—————

-—--

11—

i—1

11

1r

ru1

x4

-rrr

rlL

I-1

-rrt

Trl

]r—

—,——-—-

—- -‘1

I-I

44

II

FL

I+

1H

I-I-

li-I

’1I

III-I

I-I4

J4— ——

1111]J

i—rL

uII

Il-i

-rU

JJJ±

LuII

UL

--—

J-t.I

-[T

u-1-

T[I

TIT

FF

IFrI

TI

1LW

IU

—T

II

III

1—H

ilII

H-I-

flU-

CDi4

———

- -—

IIIT

Iii4

-nT

rLi4

-rn

1-r

Tr

-:

--:

LIJ}

tT[I

WIT

WIT

h[IL

-1J[I-

±tf1±1

±H±ft

H1±th

_:_L

::,

iii[1

4r1±

1±th

tinlll

:I

—-

itiI

lF{*

fl+}1

+HI1

{If

,,

i:

---

rtri

uwiw

rmirr

ni—

L——

—--- -

414-

U4

1414

-I-IW

-1411

1III-

C

3::

::

:•

4I111

II1Ht-1[

1-

I:

7:

::

:41

WhI

I-fll-

UIL[

N-

‘-

EE

HE

E

Appendix B. Relative Permeabilitics and Viscosities 257

temperature was 195°F, and the test pressure was 1800 psig (1815 psia). The gravity

of the liberated gas was determined by the use of tared glass weighing balloon. The

gas gravity was found to be 0.70 18 (air = 1.0). The calculation of viscosity will be as

follows:

1. Molecular weight 0.7018 x 28.95 20.31

2. For which:

Pseudo-critical pressure 667 (figure B.1)

Pseudo-critical temperature = 390 (figure B.1)

If the mole fractions of the hydrocarbon components are known the above values

can be calculated using equations B.13 and B.14.

3. From figure B.2:

Viscosity at one atmosphere () = 0.01223cp

4. Pseudo-reduced pressure = 1, 815/667 = 2.721

Pseudo-reduced temperature = (460 + 195)/390 = 1.679

5. From figures B.3 and B.4:

IL/IL1 = 1.28

6. Therefore,

The viscosity at 1800 psig and 195°F = 1.28 x 0.01223 = 0.01565cp

Appendix C

Subroutines in the Finite Element Codes

C.1 2-Dimensional Code CONOIL-Il

The 2-dimensional code has been divided into two separate programs; the ‘Geometry

Program’ and the ‘Main Program’. The main reason for having as two separate

programs is to reduce the effort on the user. The geometry program automatically

generates and numbers the midside and interior nodes. It also renumbers the elements

and nodes to minimize the front width and creates a input file for the main program,

containing the relevant information about the finite element mesh. The main program

does the analysis. The geometry program for the 2-dimensional version consists of

11 subroutines and the main program consists of 58 subroutines. The details of the

subroutines are described herein.

C.1.1 Geometry Program

The subroutines in the geometry program and their functions are as follow:

ADDS - forms element-node links.

BCONI - sets up element constants.

FFIN - reads free format input.

MAKENZ - generates an array which contains the number of degrees of freedom

associated with each node.

MIDPOR - generates mid-side pore pressure nodes.

MIDSID - generates mid-side displacement nodes.

258

Appendix C. Subroutines in the Finite Element Codes 259

MLAPZ - marks last appearances of nodes by making them negative.

OPTEL - optimizes and renumbers the elements for frontal solution.

RDELN - reads line data.

SFWZ - calculates the front width for symmetric solution.

SORT2 - changes the element numbers to conform with new ordering.

C.1.2 Main Program

The subroutines in the main program and their functions are given below.

BCON - calculates element constants.

CHANGE - removes/adds elements from geometry mesh and calculates implied

loading.

CHECK - scrutinizes the input data to main program.

CHKLST - checks if there are any changes in fixity for the load increment.

COMP - computes the pore fluid compressibility and permeability.

DATM - reads material property data.

DETJCB - calculates the determinant of the Jacobian matrix.

DHYPER - calculates the stress-strain matrix for elastic model.

DILATE - computes the volume change due to shear deformation (used with hyper

bolic model).

DISTLD - calculates equivalent nodal loads.

DSYMAL - finds the principal stresses and their directions (contains 5 subroutines;

TRED3, TRBAK3, TQLRAT, TQL2, DTRED4).

ELMCH - scrutinizes the list of elements.

EQLBM - calculates unbalanced nodal loads.

EQLIB - calculates nodal forces balancing element stresses.

ERR - records and lists data errors.

FFIN - reads free format input.


FFLOW - calculates amount of flow and updates saturations.

FIXX - updates list of nodal fixities.

FLOWST - calculates vectors for coupled consolidation analysis.

FORMB - forms ‘B’(shape function derivative) matrix.

FRONTZ - frontal solution routine for symmetric matrix.

GETEQN - gets the coefficients of the eliminated equations.

INSIT - sets up in-situ stresses and the equivalent nodal forces.

INSTRS - prints the in-situ stresses before first increment.

INV - inverts a matrix.

LSHED - carries out load shedding operation.

LSTIFA - calculates the element stiffness matrix using fast stiffness formation.

LSTIFF - calculates the element stiffness matrix for elastic model.

MAKENZ - generates an array which contains the number of degrees of freedom

associated with each node.

MBOUND - rearranges the boundary conditions in terms of degrees of freedom.

MLAPZ - marks last appearances of nodes by making them negative.

MODULI - calculates moduli of the soil elements for elastic model.

MSUB - main controlling routine.

PLAS - calculates the stress-strain matrix for elasto-plastic model.

PRINC - calculates principal stresses.

RDN - reads specified range in 1-dimensional array.

REACT - calculates the reactive forces on restrained boundaries.

SCAN - checks for any changes in fixities.

SELF - calculates self weight loads.

SELl - computes nodal forces equivalent to self weight loads.

SFR1 - calculates shape functions and derivatives for 1-dimensional integration along

element edges.


SFWZ - estimates the front width for symmetric matrix solution.

SHAPE - calculates shape functions and derivatives.

SOFT * calculates the overstress for strain softening.

STIF - calculates element stiffness matrix for elasto-plastic model.

STOREQ - writes the terms in a buffer zone when an array becomes saturated.

TEMP - calculates the equivalent force vector terms due to temperature changes.

UFRONT - frontal solution routine for unsymmetric matrix.

UPARAL - allocates storage for subroutine UPOUT.

UPOUT - updates and prints the results.

VISG - calculates viscosity of gas.

VISO - calculates viscosity of oil.

VISW - calculates viscosity of water.

WRTN - writes a specified range in a 1-dimensional array.

ZERO1 - initializes 1-dimensional array.



ZEROI1 - initializes 1-dimensional integer array.

C.2 3-dimensional code CONOIL-Ill

The 3-dimensional code has been developed based on the same sequence of procedures

as the 2-dimensional code. It consists of 43 subroutines and the details of those are

given below.

BOUND - expands the nodal fixity data in terms of degree of freedom.

CHANGE - removes/adds elements from geometry mesh and calculates implied

loading.

COMP - computes the pore fluid compressibility and permeability.

DMAT - reads material property data.

Appendix C. Sn bron tines in the Finite Element Codes 262

DRIVER - main controlling routine.

EPM - calculates stress-strain matrix for elasto-plastic model.

EQLIB - calculates nodal forces balancing element stresses.

FFLOW - calculates amount of flow and updates saturations.

FIXX - updates list of nodal fixities.

FLSD - calculates load vector for load shedding.

FTEMP - calculates force vector terms due to temperature changes.

GETEQN - gets the coefficients of the eliminated equations.

HYPER - calculate moduli values for hyperbolic model.

INSIT - sets up in-situ stresses and the equivalent nodal forces.

JACO - evaluates Jacobian matrix, its determinant and inverse.

LAYOUT - reads nodal geometry data and stores in relevant arrays.

LFIX - sets the load vector for fixed boundaries.

LOAD - evaluates the load vector for applied loads.

LSHED - routine to perform load shedding.

MAKESF - finds last appearance of the nodes, frontwidth and the destination vector.

MFLOW - updates saturations and flow at mid-step.

MINV - inverts a matrix

PRIN - finds the principal stresses and their directions (contains 5 subroutines;

TRED3, TRBAK3, TQLRAT, TQL2, DTRED4).

PRNOUT - calculates, updates and prints the results.

RDN - reads specified range in 1-dimensional array.

SBMATX - calculates B’(shape function derivative) matrix.

SELF - calculates self weight loads.

SELl - calculates self weight loads for gravity changes.

SFRONT - frontal solution routine for symmetric matrix.

SHAPE - calculates shape functions and its derivatives.


SHAPE2 - calculates shape functions and derivatives for 2-dimensional integration.

SMDF - sets up arrays giving nodal degrees of freedom and the first degree of freedom

of the nodes.

STIFF - calculates element stiffness matrix.

STOREQ - writes the terms in a buffer zone when an array becomes saturated.

STRL - calculates and updates stress level.

TEMP - calculates nodal temperature changes.

UFRONT - frontal solution routine for unsymmetric matrix.

UPDATE - updates the results at mid-step for second iteration.

VISG - calculates viscosity of gas.

VISO - calculates viscosity of oil.

VISW - calculates viscosity of water.

WRTN - writes a specified range in a 1-dimensional array.

ZERO 1 - initializes 1-dimensional array.





Appendix D

Amounts of Flow of Different Phases

The formulation for the multi-phase flow presented in chapter 5 considers an equiv

alent conductivity term to model the effects of the different phases in the pore fluid.

This does not give the individual amounts of flow of the fluid phase components.

However, at any time, these individual amounts of flow can be easily estimated by

knowing the total amount of flow, and the relative permeabilities and viscosities of

the phase components. The details of this calculation are presented in this appendix.

To illustrate the steps involved the example problem given in chapter 7 is considered

here.

In the oil sand layer the zone from where the fluid flow occurs, can be obtained from

the temperature contour plot or the pore pressure contour plot (refer to figures 7.17

and 7.18). Such a zone for the example problem is shown in figure D.1.

The fluid flow zone can be divided into a number of zones of different effective mobil

ities. Here, the flow zone is divided into three (zones A, B and C in figure D.1) and

the effective mobilities of the fluid phase components are assumed constant within a

zone. The grater the number of zones the better the results will be.

The mobility of a fluid phase component ‘1’ can be written as

kmikkri

(D.1)

where

kmi - mobility of phase 1

k - intrinsic permeability of the sand matrix (m2)

264

Appendix D. Amounts of Flow of Different Phases 265

50

0 Injection Well

• Production Well

E 40

40 0 60

Distance (m)

Figure D .1: Zones involved in Fluid Flow

k,.1 - relative permeability of phase I

IL1 - viscosity of phase 1

k is a function of void ratio, k,.1 is a function of saturation level and 1u is a function

of temperature. Under steady state conditions, the void ratio and the temperature

are assumed to remain constant. Therefore, the viscosities of the phase components

within a zone can be assumed constant and are summarized in table D.l. The intrinsic

permeability of the sand matrix is assumed to be 1 x 1012 m2.

As the flow continues, the water will replace the oil and therefore, the saturations will

change. Since the relative permeabilities are function of saturation, they will change

as well. The relative permeabilities of water and oil are assumed to be represented

by the following functions:

= 1.820 (S — 0.2)2.375 (D.2)


Table D.1: Average Viscosities and Temperatures in Different Zones

Zone Area (m2) ii(mPa.s) u0(mPa.s) Temp. (°C)

A 96 0.20 8 220

B 252 0.48 40 140

C 312 0.65 1000 50

k,.0 = 2.769 (0.8— S)’996 (D.3)

Now, let us assume that the total flow of water and oil for a time interval /t be

LVT. This total amount of flow will comprise the water and oil flow in all three zones

considered. The effective mobility of water considering all three zones can be given

as,= (kmw)A aA + (kmw)B aB + (kmw) ac

(D.4)aA + a + ac

Where, aA, aB and ac are the areas of zones A, B and C respectively. Similarly, the

effective mobility of oil considering all three zones can be given as,

— (kmo)A aA + (kmo)B aB + (kmo) acmo aA + aB + ac

Then, the amounts of water and oil flow in the total flow can be estimated as,

A TI TI mwL.Vw VT ke j i.e

mw

= LVT e e (D.7)mw mo

Now, because of the flow of oil from the oil sand layer, saturations will change and

those should be updated at the end of the time step. To calculate the new saturations,

the amounts of flow in individual zones should be estimated. This can be done as

follows.


For example, the amount of water flow from zone A can be given by,

fAT? AT? mw A aAiI_1Vw)A = LVw

(kmw)A aA + (kmw)B aJ3 + (kmw) ac

Similarly, all the individual amounts of flow of water and oil in different zones can be

calculated.

Assume that the saturation of oil in zone A at the beginning of a time step be (S0).

Then, the saturation of water in zone A at the beginning of the time step will be,

(S) = 1 — (S0) (D.9)

The volume of oil in zone A at the beginning of the time step will be given by,

(V0) = aA n (S0) (D.1O)

The amount of oil flow from zone A will be,

IAT?\ AT? mo A aAL.1Vo)A =

(kmo)A aA + (kmo)B aB + (kmo) ac

Then, the volume of oil in zone A at the end of the time step will be,

(V (V0) — (V0)A (D.12)

and the new oil saturation will be,

(S0)(V0)

(D.13)fl aA

The new saturation of water in zone A will be given by,

(S) = 1 — (S0) (D.14)

Likewise, the saturations in all the zones can be updated. Then, by knowing the new

saturations, the relative permeabilities of the phase components can be estimated and

subsequently, the new amounts of water and oil flow can be calculated. These steps


of calculations can be continued with time in a step by step manner until the flow of

oil ceases or the amount of oil flow becomes insignificant.

The above described procedure is applied to the example problem considered here.

The initial saturation and the mobilities of water and oil in different zones are given

in table D.2.

Table D.2: Initial Saturations and Mobilities of Water and Oil

Zone Sw S kmw(108m/s) kmo(108m/s)

A 0.3 0.7 37.6 85.1

B 0.3 0.7 19.8 17.0

C 0.3 0.7 11.6 0.68

The stepwise calculations for the amounts of flow and saturations of water and oil

are tabulated in table D.3. The saturations and the mobilities of water and oil at the

end of time t = 300 days, are given in table D.4, which can be compared with table

D.2.


Table D.3: Calculation of Flow and Saturations with Time

Time (S)A (S)B (S)c (S0)A (S0)B (S0)c iW0(days) (m3/day) (m3/day)

0 0.300 0.300 0.300 0.700 0.700 0.700 2.54 2.642 0.394 0.319 0.301 0.606 0.681 0.699 3.89 1.294 0.435 0.330 0.301 0.565 0.670 0.699 4.29 0.896 0.461 0.339 0.302 0.539 0.661 0.698 4.49 0.698 0.479 0.346 0.302 0.521 0.654 0.698 4.61 0.5710 0.494 0.352 0.302 0.506 0.648 0.698 4.69 0.4915 0.525 0.365 0.303 0.475 0.635 0.697 4.82 0.3620 0.546 0.375 0.303 0.454 0.625 0.697 4.89 0.2925 0.562 0.384 0.304 0.438 0.616 0.696 4.94 0.2430 0.574 0.392 0.304 0.426 0.608 0.696 4.97 0.2140 0.595 0.405 0.305 0.405 0.595 0.695 5.01 0.1750 0.610 0.417 0.306 0.390 0.583 0.694 5.04 0.1460 0.622 0.426 0.307 0.378 0.574 0.693 5.06 0.1270 0.632 0.435 0.307 0.368 0.565 0.693 5.07 0.1180 0.640 0.443 0.308 0.360 0.557 0.692 5.08 0.1090 0.647 0.450 0.308 0.353 0.550 0.692 5.09 0.09100 0.653 0.456 0.309 0.347 0.544 0.691 5.10 0.08125 0.667 0.471 0.310 0.333 0.529 0.690 5.11 0.07150 0.678 0.484 0.311 0.322 0.516 0.689 5.12 0.06175 0.686 0.495 0.312 0.314 0.505 0.688 5.13 0.05200 0.693 0.505 0.313 0.307 0.495 0.687 5.13 0.05250 0.704 0.522 0.315 0.296 0.478 0.685 5.14 0.04300 0.713 0.536 0.317 0.287 0.464 0.683 5.15 0.03

Table D.4: Saturations and Mobilities of Water and Oil after 300 Days

Zone S S kmw(108m/s) kmo(108m/s)

A 0.71 0.29 1826 2.61

B 0.54 0.46 352 4.75

C 0.32 0.68 16.7 0.64

Appendix E

User Manual for CONOIL-Il

E.1 Introduction

CONOIL-Il is a finite element program for consolidation analysis in oil sands under

plane strain and axisymmetric conditions. The program includes an elasto-plastic

stress strain model and a formulation to analyze multi-phase fluid flow. It also con

siders temperature effects on stresses and fluid flow in the analysis.

The program can be used to carry out transient, drained or undrained analysis us

ing the same material data base. Elements can be removed to simulate excavation.

Provisions exist for specifying various boundary conditions such as pressure, force,

displacement and pore pressure. Though the program is particularly suited for prob

lems in oil sands, it can be applied for a range of geotechnical problems such as dam

and heavy foundation analyses.

The intention of this manual is to provide sufficient information for an analyst with

a strong geotechnical background to be able to prepare an input file and run the

program. Detailed explanations such as analytical formulation, method of analysis,

formation of stiffness matrix, solving routines etc. can be found in Srithar (1993).

CONOIL-Il has been divided into two separate programs: the ‘Geometry Program’

and the ‘Main Program’. The main purpose of this split is to reduce the effort on the

user. The geometry program automatically generates and numbers the mid-side and

interior nodes. It also renumbers the elements and nodes to minimize the front width

and creates an input file for the main program, containing the relevant information

about the finite element mesh. Therefore, the Geometry Program has to be run first

270

Appendix E. User Manual for CONOIL-Il 271

and the link file has to be submitted to the Main Program.

The data for both the Geometry Program and the Main Program is free format

i.e, particular data items must appear in the correct order on a data record but

they are not restricted to appear only between certain column positions. The data

items are indicated below by mnemonic names, i.e., names which suggest the data

item required by the program. The FORTRAN naming convention is used: names

beginning with the letters I, J, K, L, M and N show that the program is expecting an

INTEGER data item whereas names beginning with any other letter show that the

program is expecting a REAL data item. The only exception is the material property

data where the actual parameter notations are retained to avoid confusions. All the

material property data are real. INTEGER data items must not contain a decimal

point but REAL data items may optionally do so. REAL data items may be entered

in the FORTRAN exponent format if desired. Individual data items must not contain

spaces and are separated from each other by at least one space. Detailed explanations

for some of the records are given in section E.4.

Comments may be included in the input data file in exactly the same way as for the

FORTRAN program. Any line that has the character C in column 1 is ignored by the

programs. This facility enables the user to store information relating to values, units

assumed etc. permanently with the input data rater than separately. The program

only read data from the first 80 columns of each line.


E.2 Geometry Program

Record 1 (one line)

TITLE

TITLE - Title of the problem (up to 80 characters)

Record 2 (one line)

I LINK

LINK - A code number set by the user

Record 3 (one line)

NN NEL ILINK IDEF ISTART SCX SCY

NN - Number of vertex nodes in the mesh

NEL - Number of elements in the mesh

ILINK - Link option:

0 - no link file is created

1 - a link file is created

IDEF - Element default type:

1 - linear strain triangle with displacement unknowns

5 - linear strain triangle with displacement and excess pore

pressure unknowns (linear variation in pore pressure)

7 - cubic strain triangle with displacement unknowns

8 - cubic strain triangle with displacement and excess pore


pressure unknowns (cubic variation in pore pressure)

ISTRAT - Frontal numbering strategy option:

1 - the normal option

2 - only to be used in rare circumstances when the parent’

mesh contains overlapping elements

SCX - Scale factor to be multiplied to all x coordinates

SCY - Scale factor to be multiplied to all y coordinates

Record 4 (NN lines)

N X Y TEMP LCODE VISCO]

N - Node number

X - x coordinate of the node

Y-

y coordinate of the node

TEMP - Initial temperature °C

LCODE - Index for load transfer

o - node can participate in load transfer

1 - node cannot participate in load transfer

VISCO - Initial viscosity factor

(not used in the present formulation, set equal to 1)

Record 5 (NEL lines)

ILN1N2N3MATI

L - Element number

Ni, N2, N3 - Vertex node numbers listed in anticlockwise order


MAT - Material zone, number in range 1 to 10


E.3 Main Program

Record. 1 (one line)

TITLEI


Record 2 (one line)

I LINKI

LINK - Code number set by the user

Record 3 (one line)

I NPLAX NMAT INCJ INC2 IPPJM IUPD ICOR ISELFI

NPLAX - Plane strain/Axisymmetric analysis option:

0 - plane strain

1 - axisymmetric

NMAT - Number of material zones

INC1 - Increment number at start of analysis

INC2 - Increment number at finish of analysis

IPRIM - Number of elements to be removed to from primary mesh

IUPD - Element default type:

1 - linear strain triangle with displacement unknowns

5 - linear strain triangle with displacement and excess pore

pressure unknowns (linear variation in pore pressure)


7 - cubic strain triangle with displacement unknowns

8 - cubic strain triangle with displacement and excess pore

pressure unknowns (cubic variation in pore pressure)

ISTRAT - Frontal numbering strategy option:

1 - the normal option

2 - only to be used in rare circumstances when the ‘parent’

mesh contains overlapping elements

SCX - Scale factor to be multiplied to all x coordinates

SOY - Scale factor to be multiplied to all y coordinates

Record 4 (One line only)

MXITER DIOONV PATM

MXITER - Maximum number of iterations per increment for dilation

and load transfer purposes (zero defaults to 5)

DICONV - Convergence criterion for change in force vector from

dilation calculations (zero defaults to 0.05)

PATM - Atmospheric pressure in user’s units (SI: 101.3 kPa;

Imperial 2116.2 psf (zero defaults to 101.3 kPa)

Record 5 (for HYPERBOLIC stress-strain model)

(Records 5.1 to 5.10 have to repeated NMAT times.

Records 5.5 to 5.10 are necessary only if IMPF = 2.

Records 5.1 to 5.4 are given separately for HYPERBOLIC and ELASTO-PLASTIC

stress-strain models )

Appendix E. User Manual for OONOIL-II 277

Record 5.1

MAT IMODEL e KE n Rf KB m DUO k k

MAT - Material property number. All elements given the same

number in the Geometry Program have the following properties

IMODEL - Stress-strain model number. Use C7 for Hyperbolic model

e - Initial void ratio

KE - Elastic modulus constant

n - Elastic modulus exponent

Rf - Failure ratio

KB - Bulk modulus constant

m - Bulk modulus exponent

DUO - Determines whether Drained/Undrained/Consolidation analysis

i) DUO = 0.0 Drained analysis

ii) DUO = B1 (liquid bulk modulus) - Undrained analysis

NOTE: B1 in the range of 100 to 500 B5k (soil bulk modulus) is

equivalent to using a Poisson’s ratio of 0.495 to 0.499.

If there are temperature changes, use consolidation routine

to do undrained analysis.

iii) DUO = 7i (unit weight of liquid) - Consolidation analysis

- total unit weight of soil

- permeability in x direction

- permeability in y direction

Record 5.2

c - v ot q’cv - B B0


c - Cohesion

- Friction angle at a confining pressure of 1 atmosphere

L4 - Reduction in friction angle for a ten fold increase in

confining pressure—

- 0 (No parameter at present)

- Constant dilation angle. To be specified if the dilation

option is used.

a8t - Coefficient of temperature induced structural reorienta

tion. Only used in temperature analysis.

- Constant volume friction angle. Only used with dilation

option.—


B - Bulk modulus of the water

B0 - Bulk modulus of the oil

Record 5.3

/J’30,0 ‘H H -\U U S S1 cw a0

1’3o,o - Viscosity of oil at 300 C and 1 atmosphere (in Pa.s)

(used in three phase flow, built-in oil viscosity correlation)

- Function to modify Henry’s constant for temperature

H=H+)H*IXT

H - Henry’s coefficient of solubility

- Function to modify bubble pressure for temperature

U - Bubble pressure (Oil/Gas saturation pressure)

S - Initial degree of saturation varying between 0 and 1. (S

= 1 implies 100% saturation)


Sf - Saturation at which fluid begins to move freely. (Used

for modifying permeability. 1 is generally close to zero)

- Coefficient of linear thermal expansion of water

cx0 - Coefficient of linear thermal expansion of oil

- Coefficient of linear thermal expansion of solids

Record 5.4

ISIGE 151GB IMPF IDILAT ILSHD I

ISIGE - Option to calculate Young’s modulus

o - use mean normal stress

1 - use minor principal stress

ISIGB - Option to calculate bulk modulus

o - use mean normal stress


IMPF - Multi phase flow option

o - fully saturated

1 - partially saturated

2 - three phase fluid flow (needs additional parameters)

IDILAT - Dilation option

o - No dilation

1 - Use constant dilation angle

2 - Use Rowe’s stress-dilatancy theory

ILSHD - Load transfer option

o - do not perform load transfer

1 - perform load transfer by keeping o constant

2 - perform load transfer by keeping On constant


Record 5 (for ELASTO-PLASTIC stress-strain model)


Records 5.5 to 5.10 are necessary only if IMPF = 2.

Records 5.1 to 5.4 are given separately for HYPERBOLIC and ELASTO-PLASTIC

stress-strain models )

Record 5.1

MAT IMODEL e KE n (R1) KB m DUO k %

MAT - Material property number. All elements given the same

number in the Geometry Program have the following properties

IMODEL - Stress-strain model number

= 5 Cone type yielding only (single hardening)

= 6 Cone and Cap type yielding (double hardening)




(Rf) - Failure ratio in the hardening rule (cone yield)



DUO - Determines whether Drained/Undrained/Consolidation analysis

i) DUO = 0.0 Drained analysis

ii) DUO = B1 (liquid bulk modulus) - Undrained analysis

NOTE: B1 in the range of 100 to 500 B8,, (soil bulk modulus) is

equivalent to using a Poisson’s ratio of 0.495 to 0.499.


If there are temperature changes, use consolidation routine

iii) DUO = 71 (unit weight of liquid) - Consolidation analysis

to do undrained analysis.



k - permeability in y direction

Record 5.2

(r/o)1,i (r/o-) q — — B B0

—


(T/o-)f,i - Failure stress ratio at 1 atmosphere

(r/o) - Reduction in failure stress ratio for a ten fold increase

in confining pressure

- Strain softening number

q - Strain softening exponent

a8t - Coefficient of temperature induced structural reorienta

tion. Only used in temperature analysis.—


—




Record 5.3

f-3O,O H H u U S S a a0

Appendix K User Manual for CONOIL-Il 282

1130,0 - Viscosity of oil at 300 C and 1 atmosphere (in Pa.s)



H =H+\H*T


- Function to modify bubble pressure for temperature


S - Initial degree of saturation varying between 0 and 1. (S

= 1 implies 100% saturation)

S, - Saturation at which fluid begins to move freely. (Used

for modifying permeability. S is generally close to zero)


- Coefficient of linear thermal expansion of oil

a5 - Coefficient of linear thermal expansion of solids

Record 5.4

ISIGE ISIGB IMPF ILSHD F F KGp GP 11


0 - use mean normal stress


ISIGB - Option to calculate bulk modulus




0 - fully saturated


Appendix E. User Manual for GONOIL-Il 283



0 - do not perform load transfer



- Collapse modulus number (cap yield)

F - Collapse modulus exponent (cap yield)

KGp - Plastic shear parameter (cone yield, hardening rule)

GP - Plastic shear exponent (cone yield, hardening rule)

- Flow rule intercept (cone yield)

- Flow rule slope (cone yield)

Record 5.5 (necessary only if IMPF = 2, all are real variables except IV)

Sw So Sg S S k0g IVL, IVO IV9

S - Initial water saturation

S, - Initial oil saturation

S9 - Initial gas saturation

(S + S0 + S must be equal to 1)

5om - Residual oil saturation

S - Connate water saturation (irreducible water saturation)

- Relative permeability of oil at connate water saturation

(oil-water)

- Relative permeability of oil at zero gas saturation (oil-gas)

IV, - Options to estimate viscosity of water

0 - use a given constant value (in Pa.s)


1 - use the built-in feature in the program (International

critical tables)

>1 - interpolate using given temperature-viscosity profile

(IV data pairs, maximum 10)

IV, - Options to estimate viscosity of oil


1 - use the built-in feature in the program (Correlation by

Puttangunta et.al (1988), to,o should be given in record

6.4)


(1V0 data pairs, maximum 10)

IVg - Options to estimate viscosity of gas


1 - use the built-in feature in the program (a constant value

2.E-5 Pa.s)


(I17 data pairs, maximum 10)

Record 5.6 (necessary only if IMPF = 2)

Al A2 A3 Bi B2 B3 Cl C2 03 Dl D2 D3

Al...A3 - Parameters for relative permeability of water (oil-water)

krw = A1(S — A2)A3

Bl...B3 - Parameters for relative permeability of oil (oil-water)

= B1(B2 — S)B3

Cl... 03 - Parameters for relative permeability of gas (oil-gas)

k,.9 = C1(S9 — C2)c3

Dl...D3 - Parameters for relative permeability of oil (oil-gas)


k,.09 = D1(D2


I Fl F2 F31

Fi...F3 - Parameters for oil-gas capillary pressure

of gas (oil-gas)

Pc = Fl Pa(S9 — F2)’3

Record 5.8 (necessary only if IMPF = 2 and IV,,, = 0 or >1)

V,,, (ifIV=0)

Vi Ti V2 T2•.• I (if IV, , 1, IV, data pairs, maximum 10)

V - Constant viscosity value of water (in Pa.s)

Vi,... - Viscosity values in the given profile (in Pa.s)

Ti,... - Temperature values in the given profile (in °C)

Record 5.9 (necessary only if IMPF = 2 and 1V0 = 0 or >1)

V01 (ifIV0=0)

Vi Ti V2 T2•.. I (if 1V0> 1, 1V0 data pairs, maximum 10)

V0 - Constant viscosity value of oil (in Pa.s)





(ifIV=0)

I Vi Ti V2 T2... I (if IVg> 1, 1V9 data pairs, maximum 10)

- Constant viscosity value of gas (in Pa.s)



Record 6 ((IPRIM-1)/10 + 1 lines, only if IPRIM> 0)

I Li L2

Li,... - List of element numbers to be removed to form mesh at

the beginning of the analysis (LPPJM element numbers)

There must be 10 data per line, except the last line

Record 7 (one line only)

INSIT NNI NELl NO UT I

INSIT - In-situ stress option:

0 - Set in-situ stresses to zero

1 - Direct specification of in-situ stresses

NNI - Number of nodes in-situ mesh

NELl - Number of elements in-situ mesh

NOUT - In-situ stress printing option:

0 - Do not print the in-situ stresses

1 - Print the variables at the centroids of each element

2 - Print the variables at each integration point per element

and print the equilibrium loads for in-situ stresses.


Record 8 (NNI lines)

NI XI Yl o- o, o- r u

NI - In-situ mesh node number

XI - x coordinate

Y1-

y coordinate

o, o, o - Normal components of the effective stress vector

- Shear stress component

ii - Pore fluid pressure

(Note that effective stress parameters are assumed)

Record 9 (NELl lines)

LI NIl N12 NI3]

LI - In-situ mesh element number

NIl, N12, N13 - In-situ mesh node numbers (anticlockwise order)

Record 10 (one line only, but records 10 to 14 are repeated for each analysis incre

ment)

INC ICHEL NLOD IFIX lOUT DTIME DGRAV NSINC NTEMP NPTSI

INC - Increment number

ICHEL - Number of elements to be removed

NLOD - Number of CHANGES to incremental nodal loads or (if

NLOD is negative) the number of element sides which

have their increment loading changed.


IFIX - Number of changes to nodal fixities

lOUT - Output option for this increment - a four digit number

abcd where:a - out of balance loads and reactions

o - no out of balance loads

1 - out of balance loads at vertex nodes

2 - out of balance loads at all nodes

b - option for prescribed boundary conditions (e.g. fixity

condition or equivalent nodal loads at specified nodes)

o no information printed

I - data printed for each relevant d.o.f

c - option for general stresses

o - no stresses printed

1 - stresses at element centroids

2 - stresses at integration points

d - option for nodal displacements

o - no displacements printed

1 - displacements at vertex nodes

2 - displacements at all nodes

DTIME - Time increment for consolidation analysis

DGRAV - Increment in gravity level

(change in number of gravities)

NSINC - The number of sub increments (this is presently equal to 1)

NTEMP - Number of changes to nodal temperature

DGRAV - Number of data pairs in the temperature-time history profile

Record 11 ((ICHEL-1)/1O + 1 lines, only if ICHEL > 0)


rLi

Li,... - List of element numbers to be removed in this increment


Record 12 (NLOD lines)

(a) For .1\TLOD > 0

NDFX DFY1

N - Node number

DFX - Increment of x force

DFY - Increment of y force

For NLOD < 0

(b.1) For linear strain triangle

LNJN2TJS1 T3S3T2S200001

(b.2) For cubic strain triangle

I L Ni N2 Ti Si T3 S3 T4 S4 T5 55 T2 S

L - Element number

Ni, N2 - Node numbers at the end of the loaded element side

Ti - Increment of shear stress at Ni (see the following figure E.1

Si - Increment of normal stress at Ni

Ti - Increment of shear stress at Ni


Si - Increment of normal stress at Ni etc.

Sign convention for stresses:

Shear - which act in an anticlockwise direction about element

centroid are positive

Normal - compressive stresses are positive

Ni

N2

Linear Strain Triangle Cubic Strain Triangle

Figure E.1: Nodes along element edges

Record 13 (one line only, but record from 10 to 15 are repeated for each analysis

increment)

N ICODE DX DY DPI

NSN4

N - Node number


ICODE - A three digit code abc which specifies the degrees of

freedom associated with this node that are fixed to par

ticular valuesa - fix for x direction

o - node is free in x direction

1 - node is to have a prescribed incremental displacement

DXb - fix for y direction

o - node is free in y direction

1 - node is to have a prescribed incremental displacement

DYc - fix for excess pore pressure

o - no prescribed excess pore pressure

1 - the increment of excess pore pressure at this node is to

have a prescribed value DP

2 - the absolute excess pore pressure at this node is to have

a zero value for this and all subsequent increments of

analysis

DX - Prescribed displacement in x direction

DY - Prescribed displacement in y direction

DP - Prescribed pore pressure

Record 14 (NTEMP lines, only if NTEMP > 0)

N TEM1 TIMEJ TEM2 TIME2 .J (NPTS data pairs, maximum 15)

N - Node number

TEMJ,... - Temperature in the given temperature time profile

TIMEJ,... - Time in the temperature time profile


E.4 Detail Explanations

Detailed explanations for some of the records are given in this section to provide a

better understanding.

E.4.1 Geometry Program

Record 2

The geometry program stores basic information describing the finite element mesh on

a computer disk file (the ‘Link’ file) which is subsequently read by the Main Program.

A user of CONOIL will often set up several (different) finite element meshes and run

the Main Program several times for each of these meshes. In order to ensure that a

particular Main Program run accesses the correct Link file the LINK number is stored

on the Link file by the Geometry program and must be quoted correctly in the input

for the Main Program. Hence LINK should be set to a different integer number for

each finite element mesh that the user specifies.

Record 3

LDEF (Element Types)

The element type is defined by LDEF which at present can take one of four values

associated with the elements shown in Figure E.2. The variation of displacements

(and consequently strains) and where appropriate, the excess pore pressures are sum

marized in table E.1. All elements are basically standard displacement finite elements

which are described in most texts on the finite element method.

Although CONOIL allows the user complete freedom in the choice of element type,

the following recommendations should lead to the selection of an appropriate element

type:

(i) Plane Strain Analysis

For drained or undrained analysis use element type 1 (LST) and for consolidation


0 u,v — displacement unknowrs

A p — pore pressure unknowns

a.1.

6

22

S2

(a) Element type 1 (LST) (b) Element type 5 (LST)6 nodes, 12 d.o.f. 6 nodes, 15 d.o.f.

(consolidation)

412

—._. 1216

216 11

- / ,‘ 112

/ 106

/ 1//

102S - /

.188 19 9 1

(c) Element type 7 (CuST) (d) Element type 8 (CuST)15 nodes, 30 d.o.f. 22 nodes, 40 d.o.f.

(consolidation)

Figure E.2: Element types


Table E.1: Element Types

Variation ofLEDF Element Name Displacement Strain Pore Pressure

1 Linear strain triangle (LST) Quadratic Linear N/A5 LST with linearly varying Quadratic Linear Linear

pore pressures7 Cubic strain triangle (CST) Quartic Cubic N/A5 CST with cubic variation of Quartic Cubic Cubic

pore_pressures

analysis use element type 5.

(ii) Axisymmetric Analysis

For drained analysis or consolidation analysis where collapse is not expected then

element types 1 and 5 will probably be adequate (i.e. the same as (i) above). For

undrained analysis or in a situation where collapse is expected then element types 7

and 8 are recommended. Recent research has shown that in axisymmetric analysis

the constraint of no volume change (which occurs in undrained situations) leads to

finite element meshes ‘locking up’ if low order finite elements (such as the LST) are

used.

NN (Number of Vertex Nodes)

It should be noted that NN refers to the number of vertex nodes in the finite element

mesh. The geometry program automatically generates node numbers and coordinates

for any nodes lying on element sides or within elements.

Records 4 and 5

ulElement and Nodal Numbering

The program user must assign each element and each vertex node in the finite element

mesh unique integer numbers in the following ranges:

1 < node number 750

1 < element number < 500


It is not necessary for either the node numbers or the element numbers to form

a complete set of consecutive integers, i.e., there may be ‘gaps’ in the numbering

scheme adopted. This facility means that users may modify existing finite element

meshes by removing elements without the need for renumbering the whole mesh. The

Geometry Program assigns numbers in the range 751 upwards to nodes on element

sides and in element interiors.

MAT Material Zone Numbers

The user must assign a zone number (in the range 1 to 10) to each finite element.

The zone number associates each element with a particular set of material properties

(Record 5 of Main Program input). Thus, if there are three zones of soil with different

material properties, they can be modelled by different stress-strain relations. (Note:

the material zone numbers have to consecutive).

E.4.2 Main Program

Record 2

The link number must be the same as that specified in the Geometry Program input

data for the appropriate finite element mesh (see Record 2 in section E.4.1).

Record 3

NPLAX Plane strain/Axisymmetric

The selection of axes and the strain conditions under plane strain and axisymmetric

conditions are shown in figures E.3 and E.4 respectively.

NMAT Number of Materials

sl NMAT must be equal to the number of different material zones specified in the

geometry program.

IPRIM

CONOIL allows excavations to be modelled in an analysis via the removal of elements

as the analysis proceeds. All the elements that appear at any stage in the analysis


KZZ.

Figure E.3: Plane Strain Condition

xis the’adia1 direcSon

z is the circwnferentiai direction

Figure E.4: Axisymmetric Condition


must have been included in the input data for the Geometry Program. IPRIM is the

number of finite elements that must be removed to form the initial (or primary) finite

element mesh before the analysis is started.

IUPD

IUPD = 0: This corresponds to the normal assumption that is made in linear elas

tic finite element programs and also in most finite element programs with nonlinear

material behaviour. External loads and internal stresses are assumed to be in equi

librium in relation to the original (i.e., undeformed) geometry of the finite element

mesh. This is usually known as the ‘small displacement’ assumption.

IUPD = 1: When this option is used the nodal coordinates are updated after each

increment of the analysis by adding the displacements undergone by the nodes during

the increment to the coordinates. The stiffness matrix of the continuum is then

calculated with respect to these new coordinates during the next analysis increment.

The intension of this process is that at the end of the analysis equilibrium will be

satisfied in the final (deformed) configuration. Although this approach would seem to

be intuitively more appropriate when there are significant deformations it should be

noted that it does not constitute a rigorous treatment of the large strain/displacement

behaviour for which new definitions of strains and stresses are required. Various

research workers have examined the influence of a large strain formulation on the load

deformation response calculated by the finite element method using elastic perfectly

plastic models of soil behaviour. The general conclusion seems to be that the influence

of large strain effects is not very significant for the range of material parameters

associated with most soils. In most situations, the inclusion of large strain effects

leads to a stiffer load deformation response near failure and some enhancement of

the load carrying capacity of the soil. If a program user is mainly interested in the

estimation of a collapse load using an elastic perfectly plastic soil model then it is

probably best to use the small displacement approach (i.e., sl IUPD = 0). Collapse


loads can then be compared (and should correspond) with those obtained from a

classical theory of plasticity approach.

ISELF

In many analyses the stresses included in the soil by earth’s gravity will be insignificant

compared to the stresses induced by boundary loads (e.g., in a laboratory triaxial

test). For this type of analysis it is convenient to set ISELF = 0 and correspondingly

7 set to zero in Record 5.

When the stresses due to the self weight of the soil do have a significant effect in

the analysis then ISELF should be set to 1 and 7should be set to the appropriate

(non zero) value. If the program simulates an excavation by removing elements then

the assumption is made that the original in-situ stresses were in equilibrium with the

various densities (-y) in the Records 5.

Records 7, 8 and 9

In the nonlinear analyses performed by CONOIL, the stiffness matrix of a finite el

ement is dependent on the stress state within the element. In general, the stress

state will vary across an element and the stiffness terms are calculated by integrat

ing expressions dependent on these varying stresses over the volume of each element.

CONOIL integrates these expressions numerically by ‘sampling’ the stresses at par

ticular points within the element and then using standard numerical integration rules

for triangular areas.

The purpose of Records 7, 8 and 9 is to enable the program to calculate the stresses

before the analysis starts. Although the in-situ mesh elements are specified in exactly

the same way as finite elements in the Geometry Program input, it should be noted

that they are not finite elements. The specification of the ‘in-situ mesh’ is simply a

device to allow stresses to be calculated at all integration points by a process of linear

interpolation over triangular regions. Thus, if the initial stresses vary linearly over

the finite element mesh, it is usually possible to use an in-situ mesh with one or two


triangular elements.

Records 10

When a nonlinear or consolidation analysis is performed using CONOIL, it is neces

sary to divide either the loading or the time span off the analysis into a number of

increments. Thus, if a total stress of 20 kN/m2 is applied to part of the boundary of

the finite element mesh it might be divided into ten equal increments of 2 kN/m2 each

of which is applied in turn. CONOIL calculates the incremental displacements for

each increment using a tangent stiffness approach, i.e., the current stiffness properties

are based on the stress state at the start of each increment. While it is desirable to use

as many increments as possible to obtain accurate results, the escalating computer

costs that this entails will inevitably mean that some compromise is made between

accuracy and cost. The recommended way of reviewing the results to determine

whether enough increments have been used in an analysis is to examine the values

of shear stress level at each integration point. \Talues less than 1.10 are generally

regarded as leading to sufficiently accurate calculations. If values greater than 1.1 are

seen then the size of the load increments should be reduced. Alternatively, the stress

transfer option can be invoked.

The time intervals for consolidation analysis (DTIME) should be chosen after giving

consideration to the following factors:

1. Excess pore pressures are assumed to vary linearly with time during each incre

ment.

2. In a nonlinear analysis the increments of effective stress must not be too large

(i.e., the same criteria apply as for a drained or undrained analysis)

3. It is a good idea to use the same number of time increments in each log cycle of

time (thus for linear elastic analysis the same number of time increments would

be used in carrying the analysis forwarded from one day to ten days as from


ten days to one hundred days). Not less than three time steps should be used

per log cycle off time (for a log base of ten). Thus a suitable scheme may be as

shown in table E.2

Table E.2: Time Increment Scheme

Increment No. DTIME Total Time1 1 12 1 23 3 54 5 105 10 206 30 507 50 1008 100 2009 300 50010 500 1000

This scheme would be modified slightly near the start and end of an analysis

(see below).

4. If a very small time increment is used near the start of the analysis then the

finite element equations will be ill conditioned.

5. When a change in pore pressure boundary condition is applied, the associated

time step should be large enough to allow the effect of consolidation to be

experienced by those nodes in the mesh with excess pore pressure variables

that are close to the boundary. If this is not done then the solution will predict

excess pore pressures that show oscillations (both in time and space).

The application of item 5 will often mean that the true undrained response will

not be captured in the solution The following procedure, however, usually leads to

satisfactory results.

Appendix E. User Manual for GONOIL-Il 301

1. Apply loads in the first increment (or first few increments for a nonlinear anal

ysis) but do not introduce any pore pressure boundary conditions.

2. Introduce the excess pore pressure boundary conditions in the increment fol

lowing the application of the loads.

NLOD and IFIX

It is important to note that NLOD and sl IFIX refer to the number of changes in

loading and nodal fixities in a particular increment. CONOIL maintains a list of

loads and nodal fixities which the user may update by providing the program with

appropriate data. Thus, if NLOD 0 and IFIX = 0, the program assumes that the

same incremental loads and fixities will be applied in the current increment as were

applied in the previous increment. Another point to note is that loads applied are

incremental, thus the total loads acting at any particular time are given by adding

together all the previous incremental loads. The following example is intended to

clarify these points for a consolidation analysis:

1. Part of the boundary of a soil mass is loaded with a load of ten units (this is

applied in ten equal increments).

2. Consolidation takes place for some period of time (over ten increments)

3. The load is removed from boundary of the soil mass in five equal increments.

4. Consolidation takes place with no total load acting.

This loading history requires the data shown in table E.3.

Note that in increments 11 and 26 it is necessary to apply a zero load to cancel the

incremental loads which CONOIL would otherwise assume.

DGRAV


Table E.3: Load Increments

LoadsIncrement No. Input to Incremental load Total load

program applied acting12345678910111213

2122232425262728

etc.

1

0

-2

0

1111111111000

-2-2-2-2-2000

123456789

10101010

86420000


DGRAV is used in problems in which the material’s self weight is increased during

an analysis (e.g. in the ‘wind-up’ stage of a centrifuge test increasing centrifugal

acceleration can be regarded as having this effect).

Appendix F

User Manual for CONOIL-Ill

F.1 Introduction

CONOIL-Ill is a three dimensional finite element program developed to analyze the

stresses, deformations and flow in oil sands. Though CONOIL-Ill is specifically writ

ten for oil sands, it can be used for general geotechnical problems. CONOIL-Ill can

perform drained, undrained and consolidation analyses and has the following special

features.

1. Elasto-Plastic stress strain model. Modified form of Matsuoka’s model is im

plemented.

2. Three phase fluid flow. This is a special feature required to analyze the problems

in oil sands where the pore fluid contains three phases; water, bitumen and gas.

3. Temperature effects on stresses and strains.

This manual provides neither detail information about the program nor the theories

behind its development. Only the input parameters needed, their format and some

brief descriptions are given here. For detail explanations such as, method of analysis,

derivation of differential equations, formation of stiffness matrix, solving routines etc.,

please refer Srithar (1993). A sample data file and the corresponding output file are

given at the end of this manual.

The source code is written in FORTRAN-77. Input parameter names are given ac

cording to the standard FORTRAN naming convention. Names begin with the letters

304

Appendix F. User Manual for CONOIL-III 305

1 J, L, M and N implies that the program expects integer data. Integer data should

not contain a decimal point. There are exceptions to this naming convention in record

6 where the material property data are read. Actual material parameter notations

are retained to avoid confusions.

F.2 Input Data

Record 1 (one line)

TITLEI


Record 2 (one line)

NCNOD, NINOD, NTEL, ITYPE, NINT, IPRN

NCNOD - Total number of corner nodes

NINOD - Total number of internal nodes (0 for ITYPE 1 and 3)

NTEL - Total number of elements

ITYPE - Element type (see fig. F.l)

= 1 for drained/undrained analysis

= 3 for consolidation analysis

NINT - Number of integration points

= 8 or 27 (generally 8 is good enough)

IPRN - Index to print nodal and element information

0 - Do not print the information

1 - Print the information

Appendix F. User Manual for CONOIL-Ill

TYPE 1 TYPE 3

306

o Corner nodes = 8

D.o.f. per node = 3

Internal nodes = 0


D.o.f. per node = 4

Internal nodes = 0

Figure F.1: Available Element Types

Appendix F. User Manual for CONOIL-IlI 307

Record 3 (NCNOD+NINOD lines)

NN, X(NN), Y(NN), Z(NN), T(NN)

I\TN- Node number

X(NN) - X coordinate of the node NN

Y(NN) - Y coordinate of the node NN

Z(NN) - Z coordinate of the node NN

T(NN) - Initial temperature of the node NN

Repeat record 3 for all nodes.

Record 4 (NTEL lines)

NE, Ni, N2, N3, N4, N5, N6, N7, N8, MAT

NE - Element number

N1...N8 - Corner node numbers of the element in anticlockwise

order (see fig.F.1)

MAT - Material type of the element (maximum 10)

Record 4 has to be repeated for all elements. H elements cards are omitted, the

element data for a series of elements are generated by increasing the preceding nodal

numbers by one. The material number for the generated elements are set equal to

the material number for the previous element. The first and the last elements must

be specified.

Record 5 (one line)

PATM, GAMW, IDUC, INCi, INC2, NMAT, NTEMP, NPTS, IPRIM, ISELF


PATM - Atmospheric pressure

GAMW - Unit weight of water

ID UC - Index for Drained/Undrained/Consolidation analysis

0 - Drained analysis

1 - Undrained analysis

2 - Consolidation analysis

If there are temperature changes, use consolidation

routine with no flow boundary conditions to perform

undrained analysis.

INCJ - First increment number of the analysis

1N02 - Last increment number of the analysis

NMAT - Number of material types (maximum 10)

NTEMP - Number of nodes where temperature changes

NPTS - Number of data pairs in the temperature-time profile (max. 15)

IPRIM - Number of elements to be removed to form the primary mesh

ISELF - Option to specify self weight load as in-situ stresses

0 - in-situ stresses do not include self weight

1 - in-situ stresses include self weight

Record 6


Record 6.5 is necessary only if MODEL 2 or 3.

Records 6.6 to 6.11 are necessary only if IMPF = 2.)

Record 6.1

MAT, MODEL, ISICE, 151GB, ILSHD, IMPF


MAT - Material number

MODEL - Stress-Strain model type

1 - hyperbolic model

2 - modified Matsuoka’s model

3 - modified Matsuoka’s model with Cap-type yield




181GB - Option to calculate bulk modulus




0 - do not perform load transfer

1 - perform load transfer


0 - fully saturated



Record 6.2 (all are real variables)

e,KE,n,Rf,KB,m,7,k,k,k2





- Failure ratio





- permeability in y direction

- permeability in z direction

if IMPF = 0 or 1 give the absolute permeability values (rn/s)

if IMPF = 2 give intrinsic permeability values (m2)


c - Cohesion

- Friction angle at a confining pressure of 1 atmosphere

- Reduction in friction angle for a ten fold increase in

confining pressure

- strain softening constant

q - strain softening exponent

S - Initial degree of saturation (between 0 and 1, not in %)

- Saturation at which fluid begins to move freely. (used

to modify permeability for partially saturated soils. S1

generally close to zero)

B8 - Bulk modulus of the solids





,U30,0, )H, H, Au, U, —, ant, c.z8, a, a0j

fL3o,o - Viscosity of oil at 300 C and 1 atmosphere (in Pa.s)



H=H+AH*T


Au - Function to modify bubble pressure for temperature


—


a8t - Coefficient of volume change due to temperature in

duced structural reorientation- Coefficient of linear thermal expansion of solids


a0 - Coefficient of linear thermal expansion of oil

Record 6.5 (necessary only if MODEL = 2 or 3, all are real variables)

C, p, K, rip, R1p, i, A, (r/), (r/o),—

C - Cap-yield collapse modulus number

p - Cap-yield collapse modulus exponent

K - Plastic shear number

lip - Plastic shear exponent

R1 - Plastic shear failure ratio

- flow rule intercept


A - flow rule slope

r/o- - Failure stress ratio at 1 atmosphere

- Reduction in failure ratio for a ten fold increase in con

fining pressure—


Record 6.6 (necessary only if IMPF = 2, all are real variables except IV)

Sw, So, S, Sam, Swc, 1ow, ‘og IVj

S,, - Initial water saturation

S0 - Initial oil saturation

S9 - Initial gas saturation

(S + S0 H- S9 must be equal to 1)

S - Residual oil saturation

S - Connate water saturation (irreducible water saturation)

- Relative permeability of oil at connate water saturation

(oil-water)

k,?09 - Relative permeability of oil at zero gas saturation (oil-gas)

IV,, - Options to estimate viscosity of water

0 - use a given constant value (in Fa.s)

1 - use the built-in feature in the program (International

critical tables)


(IV data pairs, maximum 10)

1V0 - Options to estimate viscosity of oil



1 - use the built-in feature in the program (Correlation by

Puttangunta et.al (1988), to,o should be given in record

6.4)



1V9 - Options to estimate viscosity of gas

0 - use a given constant value (in Pa..s)

1 - use the built-in feature in the program (a constant value

2.E-5 Pa.s)




Al, A2, A3, Bi, B2, B3, Cl, 02, 03, Dl, D2, D3

Al.. .A3 - Parameters for relative permeability of water (oil-water)

= A1(SL, — A2)-3

Bl...B3 - Parameters for relative permeability of oil (oil-water)

= B1(B2 —

Cl... C3 - Parameters for relative permeability of gas (oil-gas)

ICrg = C1(Sg —

Dl...D3 - Parameters for relative permeability of oil (oil-gas)

k,.09 = D1(D2 S9)”3


Fl, F2, F3

Fl...F3 - Parameters for oil-gas capillary pressure


of gas (oil-gas)

Pc = Fl Pa(Sg — F2)F3

Record 6.9 (necessary only if IMPF = 2 and IV,, = 0 or >1)

V (ifIV=0)

Vi, Ti, V2, T2,... (if IV 1, IV, data pairs, maximum 10)

Vi,, - Constant viscosity value of water (in Pa.s)




___

(ifIV0=0)

Vi, Ti, V2, T2,... (if 1V0 > 1, IV,, data pairs, maximum 10)

V0 - Constant viscosity value of oil (in Pa.s)

Vi,... - Viscosity values in the given profile (in Fa.s)


Record 6.11 (necessary only if IMPF = 2 and IVg = 0 or >1)

___

(ifIV=0)

Vi, Ti, V2, T2,... (if 1V9> 1, 1V9 data pairs, maximum 10)

- Constant viscosity value of gas (in Pa.s)




Record 7 (NTEMP lines, only if NTEMP > 0)

TEM1, TIME1, TEM2, TIMEj (NPTS data pairs, maximum 15)

N - Node number

TEM1,... - Temperature in the given temperature time profile

TIMEJ,... - Time in the temperature time profile

Record 8 (one line)

LINSIT, PINSIT

LINSIT - Option to specify in-situ stresses

o - set the in-situ stresses to zero

1 - read the in-situ stresses from data

PINSIT - Option to print in-situ stress data

o - do not print

1 - print in-situ stress data

Record 9 (NTEL lines, only if LINSIT = 1)

M, SIGX, SIGY, SIGZ, SIGXY, SIGYZ, SIGZX, PP1

M - Element number

SIGX - Stress in x direction

SIGY - Stress in y direction

SIGZ - Stress in z direction

SIGXY - Stress in xy direction


SIGYZ - Stress in yz direction

SIGZX - Stress in zx direction

PP - Pore pressure

Record 9 has to be repeated for all elements. If elements cards are omitted, the

stresses for a series of elements are generated by assigning the same stresses as the

previous element. Stresses for the first and the last elements must be specified.

Record 10 ((IPRIM-1)/10 + 1 lines, only if IPRIM> 0)

Li, L2,...

Li,... - List of element numbers to be removed to form mesh at

the beginning of the analysis (LPPJM element numbers)


Record 11

(one line, records 11 to 14 have to be repeated for incre

ments from INC1 to INC2)

INC, ICHEL, NLOAD, NFIX, 10 UT, DTIME, DGRAV

INC - Increment number

ICHEL - Number of elements to be removed from primary mesh

NLOAD - Number of nodes where loads are applied

NFIX - Number of nodes where nodal fixities are changed

lOUT - Option for printing results (5 digit code ‘ abcde’)

a = 1 print nodal displacements

b = 1 print moduli values and saturations


c = 1 print strains and coordinates of the integration point

where results are printed

d = 1 print stresses and pore pressure

e = 1 print velocity vectors

DTIME - Time increment

DGRAV - Increase in gravity

Record 12 ((ICH.EL-1)/10 + 1 lines, only if ICHEL > 0)

Li, L2

Li,... - List of element numbers to be removed in this increment


Record 13 (NLOAD lines, only if NLOAD> 0)

N, DFX, DFY, DFZI

N - Node number

DFX - Increment in x force

DFY - Increment in y force

DFZ - Increment in z force

Record 14 (NFIX lines, only if NFIX> 0)

N, NFCODE, DX, DY, DZ, DP

N - Node number


NFCODE - Four digit code ‘abcd’ which specifies the fixity condi

tions associated with the nodea = 0 free in x direction

= 1 will have prescribed incremental displacement DX

b = 0 free in y direction

= 1 will have prescribed incremental displacement DY

c = 0 free in z direction

= 1 will have prescribed incremental displacement DZ

d = 0 pore pressure can have any value (undrained boundary)

= 1 will have prescribed incremental pore pressure DP

= 2 will have zero absolute pore pressure for this and all

subsequent increments

DX - Prescribed displacement in x direction

DY - Prescribed displacement in y direction

DZ - Prescribed displacement in z direction

DP - Prescribed pore pressure


F.3 Example Problem 1

An example of a general stress analysis under one dimensional loading is illustrated

here. The material is assumed to be linear elastic. The finite element mesh consists

of two brick elements as shown in figure F.2. The data file and the corresponding

output file from the program are given in subsections.

25 kN

H G

Ei‘12

6...I

0ZL

ol...

AB, BC, CD, DA - Totally Fixed

AE, BF, CG, DH - Vertically Free

Figure F.2: Finite element mesh for example problem 1

“O”O”OO’OOLLL“O”O”O”O’OOLLLL“O”O”O”000LL01“O”O”O”OOOLL6

0”O”O”OOOLL’8“O”O”O”OOOU.L

OOO”OOOL.V9O”O”O”OOOLL9

“OOO0’OLLLV•.0000’OLLLE“O”O”O”O’OILL“O”O”O”O’OLLLL

S-”O”O’LS-O”OLL

S-”O”O6Q’‘LLLLLIV’OL

“o”o”o”o”oog”oog”oog’••O••O••O••O•OOS•OOS’OOS’L

LL‘•o••o••o’•o•o’•o••o•o•o

SL3L’SL3LS3LOOO”O”O”sOOO”O”OOOOOO”O”OOLOL

0000LL‘O’000Lt096’OOL

‘L’LLLOL68L99‘L8L9S’V’Li

0LOL‘OL’L’LL

O”O”LOL“O”O”O6o”VL”o9“O”L”I.”L1“O”L”O”L9“O•L009“O”O”L”0V“O”O”L”LE

“o”o”o”oH•LLLOL

ONIOVO71VNOISN3YUO3N0SISA1VNVSS3UISVH3N35

jIdmxaiojuir

111710N00io;nuvJjisfljxrpudd

EL

EM

EN

T-N

OD

AL

INFO

RM

AT

ION

NO

DE

SE

LE

.N

O.

12

34

56

78

o CD ‘•l

—C

)U

11

23

45

67

82

56

78

910

1112

MA

TE

RIA

LP

RO

PE

RT

IES

MA

TE

RIA

LI

=

MO

DEL

NIS

IGE

ISIG

BIL

SH

DI

MPF

LIN

EA

R/N

ON

LIN

EA

RE

LA

ST

ICM

OD

ELU

SEM

EAN

NO

RM

AL

ST

RE

SS

USE

MEA

NN

OR

MA

LS

TR

ES

SN

OLO

AD

SHE

DD

ING

FUL

LY

SAT

UR

AT

ED

SO

IL

O.1

00E

+O

10.1

50E

+04

O.0

00E

÷O

OO

.000

E+

OO

0.1

00

E+

04

O.0

00E

÷O

OO

.200E

+O

2O

.000

E+

OO

O.0

00E

+O

OO

.00

0E

+O

OO

.000E

÷O

OO

.350E

+02

O.0

00E

÷O

OO

.000

E+

OO

O.0

00E

+O

OO

.000E

+O

OO

.000

E+

OO

O.1

00

E+

16

0,1

00

E+

16

O.1

00E

+16

O.0

00

E+

OO

O.0

00E

+O

OO

.000

E+

OO

O.0

00E

+O

OO

.OO

OE

OO

O.0

00E

+O

OO

.000

E+

OO

O.0

00E

+O

OO

.00

0E

+O

OO

.000E

+O

0

•A

OFO

IL

(A)N

AL

YS

ISO

F(D

)EF

OR

MA

TIO

NA

ND

(F)L

OW

IN(O

IL)

SAN

DS

GE

NE

RA

LS

TR

ES

SA

NA

LY

SIS

,O

NE

DIM

EN

SIO

NA

LL

OA

DIN

G

NO

DA

LC

OO

RD

INA

TE

SA

ND

TE

MPE

RA

TU

RE

NO

DE

XC

OO

RD

Y-C

OO

RD

Z-C

OO

RD

TEM

P

10

.00

00000

0.0

00

0.0

00

21

.00

00.0

00

0.0

00

0.0

00

31

.00

01.0

00

0.0

00

0.0

00

40

.00

01

.00

00.0

00

0.0

00

50

.00

00

.00

01.0

00

0.0

00

61

.00

00

.00

01.0

00

0.0

00

71

.00

01

.00

01.0

00

0.0

00

80

.00

01.0

00

1.0

00

0.0

00

90

.00

00

.00

02

.00

00.0

00

101

.00

00

.00

02

.00

00.0

00

111

.00

01

.00

02

.00

00.0

00

120.0

00

1.0

00

2.0

00

0.0

00

=0

=0

=0

=0

L’3

INIT

IAL

ST

RE

SS

ES X

EL

EM

ST

RE

SS

1O

.5000E

+03

2O

.50

00

E+

03

INC

RE

ME

NT

NU

MB

ER=

VZ

SHE

AR

-XY

SHE

AR

-YZ

SHE

AR

-ZX

POR

ES

TR

ES

SS

TR

ES

SS

TR

ES

SS

TR

ES

SS

TR

ES

SPR

ESS

UR

E

O.5

000E

+03

O.5

000E

+03

O.0

000E

+O

OO

.0000E

+O

OO

.0000E

+O

OO

.000

0E+

OO

O.5

OO

OE

O3

O.5

00

0E

+03

O.0

000E

+O

OO

.0000E

+O

OO

.0000E

+O

OO

.0000E

+O

O

INC

H.

ING

RA

VIT

Y=

0.0

00

0E

÷O

OTO

TAL

GR

AV

ITY

=O

.00

00

E+

O0

TIM

EIN

CR

EM

EN

T=

O.1

000E

+O

1TO

TAL

TIM

E=

O.1

000E

+O

1

NO

DA

LD

ISPL

AC

EM

EN

TS

I-s

NO

DE

XI

ZI

XA

1-0

.8333E

-16

-O.8

333E

-16

02

O.8

333E

-16

O.8

33

3E

-16

3O

.8333E

-16

O.8

33

3E

-16

4-O

.8333E

-16

-O.8

333E

-16

5-O

.16

67

E-1

5-O

.1667E

-15

6O

.1667E

-15

O.1

66

7E

-15

7O

.1667E

-15

O.1

66

7E

-15

8-O

.16

67

E-1

5-O

.1667E

-15

9-O

.8333E

-16

-O.8

333E

-16

10O

.8333E

-16

O.8

33

3E

-16

11O

.8333E

-16

O.8

33

3E

-16

12-O

.8333E

-16

-O.8

333E

-16

MO

DU

LIV

AL

UE

S

PL

AS

TIC

PAR

AM

ET

ER

0.0

000E

+O

O0.

00

00

E+

0O

INC

RE

ME

NT

AL VI

-O.8

333E

-16

-0.8

333E

-16

0.B

333E

-16

0.8

333E

-16

-0.

1667E

-15

-0.

1667E

-15

0.

16

67

E-

150

.1

66

7E

-15

-O.8

333E

-16

-O.8

333E

-16

O.8

333E

-16

O.8

333E

-16

BU

LKM

OD

ULU

S

0.

1000E

+06

0.

1000E

+06 V

ST

RA

IN

-O.2

981E

-15

-D.2

019E

-15

AB

SOL

UT

E VA

-O.8

333E

-16

-O.8

33

3E

-16

0.8

333E

-16

O.8

333E

-16

-0.

1667E

-15

-0.

1667E

-15

0.

1667E

-15

0.

1667E

-15

-O.8

333E

-16

-O.8

333E

-16

O.8

33

3E

-16

O.8

33

3E

-16

VO

IDR

AT

IO

0.

I00

0E

+0

10.

1000E

+01

ELEM

2

ZA

-O.2

500E

-15

-0.

2500E

-15

-O.2

500E

-15

-O.2

500E

-15

-0.

55

56

E-0

3O

.55

56

E-0

3-0

.5

55

6E

-03

•O.5

556E

-O3

-0.

1111E

-O2

-0.

1111E

-02

-0.1

11

IE-0

2-0

.11

11

E-0

2

WA

TER

SAT

UR

AN

0.

I000

E+

O1

0.

1000

E+

O1

EL

AS

TIC

MO

DU

LUS

0.

1500E

+06

0.

1500

E+

O6

-O.2

500E

-15

0.2

500E

-15

-0.

2500E

-15

-O.2

500E

-15

-0.

5556E

-03

-O.5

55

6E

-O3

-0.5

55

6E

-03

-O.5

556E

-O3

-0.

1111E

-O2

-0.

11

11

E-0

2-0

.11

11E

-O2

-0.1

11

1E

-02

PO

ISS

ION

RA

TIO

0.2

500E

+O

00.

25

00

E+

00 Z

ST

RA

IN

0.5

55

6E

-O3

O.5

55

6E

-03

ST

RA

INS

EL

EM 2

XS

TR

AIN

-O.2

981E

-15

-0.

20lY

E-

15

OIL

SAT

UR

AN

0.0

000E

+O

0O

.0000E

+0

0

XVY

ZS

TR

AIN

ST

RA

IN

GA

SSA

TU

RA

N

0.

0000

E+

OO

0.

00

00

E+

00

TN

T.

PO

INT

CO

OR

DIN

AT

ES

XV

ZZX

VO

L.

ST

RA

INS

TR

AIN

0.1

23

3E

-31

O.4

82

5E

-16

-O.4

816E

-16

O.5

55

6E

-03

O.7

9E

+0O

0.2

1E

+00

O.7

9E

+0O

-0.3

69

8E

-31

-0.4

77

9E

-16

O.4

78

0E

-16

0.5

55

6E

-03

O.7

9E+

OO

0.2

1E

+00

0.1

8E

+0

lC

3

I.3

ST

RE

SS

AN

DPO

RE

PR

ES

SU

RE

S

XV

ZXV

YZ

ZXPO

RE

ST

RE

SS

EL

EM

ST

RE

SS

ST

RE

SS

ST

RE

SS

ST

RE

SS

ST

RE

SS

ST

RE

SS

PRE

SSU

RE

LE

VE

L

1O

.53

33

E-O

3O

.5333E

+03

O.6

000E

+03

O.7

396E

-27

O.8

95E

-11

-O.2

88

9E

-11

O.0

00

0E

+O

OO

.23

98

E-O

1O

.53

33

E+

O3

O.5

33

3E

+03

O.6

000E

+03

-O.2

219E

-26

-O.2

86

7E

-11

O.2

868E

-I1

O.0

00

0E

+O

OO

.23

g8

E-O

rj I

ELASTO-PLASTIC DEFORMATION AND FLOW ANALYSIS IN ...

Documents

Transcript of ELASTO-PLASTIC DEFORMATION AND FLOW ANALYSIS IN ...