Elasto-Plastic Deformation and Flow Analysis in Oil
ELASTO-PLASTIC
M.
B.
A. Sc
A
Sc.
ANALYSIS (Engineering),
THESIS
© (Civil THE
THE THILLAIKANAGASABAI
THE
THILLAIKANAGASABAI We REQUIREMENTS
SUBMITTED UNIVERSITY Engineering)
FACULTY accept
DOCTOR
to University
CIVIL
the
IN
Department
DEFORMATION
this
April, required OF
ENGINEERING IN OIL OF
OF
thesis
University by
GRADUATE in PARTIAL FOR
PHILOSOPHY
of 1994
BRITISH
SAND
Peradeniya,
standard
as of
THE
conforming
SRITHAR,
of SRITHAR FULFILLMENT COLUMBIA DEGREE
STUDIES
British
MASSES
Sri
AND
Columbia,
1994 OF Lanka,
OF
1985
FLOW 1989 ______
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 Columbia Vancouver, Canada
Date - A?R L 9 L
DE-6 (2188) from gas representative contributions The by type incapable in an paths behaviour capture are key the in 3-dimensional developed Prediction mean equivalent the comparison Effects The The laws issues oil some effects the and and design sand pore the dilative knowledge normal and consists of of in of found of to of of above the fluid skeleton the developing handling temperature fluid the of predict the of stresses, of finite with nature the stress an to sand important the equivalent individual in aspects of that of be oil phase laboratory oil element cone components. and these relative skeleton. in under of these deformations has recovery sand the the very realistically. and modelling components. changes compressibility responses, aspects analytical phase compressibility comprises dense programs. aspects, constant permeabilities good cap-type test Linear process. Compressibility
components Abstract due oil agreement. results to of and sand and The shear procedure. and and three be the to Equivalent yield Modelling 11 fluid In steam and considered elasto-plastic an nonlinear matrix, three-phase implemented on is and stress, phases this surfaces. elasto-plastic derived oil flow hydraulic are viscosities injection study, sand of considered hydraulic stress and of in namely, the elastic in the by oil The samples pore loading-unloading an model conductivity gas modelling in paths considering sand deformation are of analytical water, both models model model phase fluid conductivity the and also is layers that under a phase 2-dimensional behaviour modelled double-hardening bitumen is included has is have the involve obtained characteristics the formulation are postulated various been behaviour components. stress-strain been individual important is sequences and decrease through directly verified derived are found stress using gas. and the to of is
important
into
the izontal
and element in
program
equations
the
The
measured
Research
the
stress-strain
well
finite
procedure.
likely
results
for
in
pair
the
the
element
Authority
responses
behaviour
with
successful
in
coupled
The
relation
the
closed
program
finite
underground
wherever
(AOSTRA).
in
stress,
design
form
terms
and
element
has
deformation
solutions
in
and
possible,
of
been
the
stresses,
The
programs
test operation
111
flow
applied
results
and
facility
and
and
continuity
deformations
laboratory
have
indicate
of
flow
to
are
of
an
predict
been
Alberta
discussed
problem
oil
equations.
the
recovery
test
verified
and
the
analysis
Oil
results.
are
and
responses
flow
by
Sand
scheme.
solved
compared
The
comparing
and
gives
Technology
by
analytical
would
of
insights
a
a
finite
with
hor
the be 2 Nomenclature Acknowledgement 1 Abstract List List 2.2 2.4 2.3 Introduction 2.1 Review 1.2 1.1 of of Figures Tables Modelling 2.1.4 Comments 2.1.3 2.1.2 Coupled 2.1.1 Stress-Strain Scope Characteristics of and Literature Modelling 2.1.2.2 2.1.2.1 Stress Stress-Strain Stress-Strain Geomechanical-Fluid of Organization Fluid Models Dilatancy of Elasto-Plastic Constituents of Oil Table Flow Stress-Strain Models Behaviour Sand Relation of in the Oil for of of Flow Models Thesis Sand of Sand iv Theory Contents Behaviour Oil Models Sands of Plasticity for of Oil Oil Sand Sands xvii xvi xi 10 25 30 27 24 20 23 22 19 11 10 ii x 1 4 8 3 Stress-Strain Model Employed 32
3.1 Introduction 32
3.2 Description of the Model 35
3.3 Plastic Shear Strain by Cone-Type Yielding 37
3.3.1 Background of the Model 37
3.3.2 Yield and Failure Criteria 42
3.3.3 Flow Rule 47
3.3.4 Hardening Rule 48 3.3.5 Development of Constitutive Matrix ]8[C . 51 3.4 Plastic Collapse Strain by Cap-Type Yielding 55
3.4.1 Background of the Model 55
3.4.2 Yield Criterion 57
3.4,3 Flow Rule 58
3.4.4 Hardening Rule 58
3.4.5 Development of Constitutive Matrix [Cc] 59
3.5 Elastic Strains by Hooke’s Law 61
3.6 Development of Full Elasto-Plastic Constitutive Matrix 62
3.7 2-Dimensional Formulation of Constitutive Matrix • 65
3.8 Inclusion of Temperature Effects • 67
3.9 Modelling of Strain Softening by Load Shedding 68
3.9.1 Load Shedding Technique 70
3.10 Discussion 72
4 Stress-Strain Model - Parameter Evaluation and Validation 74
4.1 Introduction 74
4.2 Evaluation of Parameters 74
4.2.1 Elastic Parameters 75
4.2.1.1 Parameters kE and n 75
v 4.2.1.2 Parameters kB and m 76
4.2.2 Evaluation of Plastic Collapse Parameters 79
4.2.3 Evaluation of Plastic Shear Parameters 80
4.2.3.1 Evaluation of ij and L2 82
4.2.3.2 Evaluation of and ) 82
4.2.3.3 Evaluation of KG, np and 1R 83
4.2.4 Evaluation of Strain Softening Parameters 86
4.3 Validation of the Stress-Strain Model 87
4.3.1 Validation against Test Results on Ottawa Sand 88
4.3.1.1 Parameters for Ottawa Sand 91
4.3.1.2 Validation 96
4.3.2 Validation against Test Results on Oil Sand 96
4.3.2.1 Parameters for Oil Sand 101
4.3.2.2 Validation 107
4.4 Sensitivity Analyses of the Parameters 109
4.5 Summary 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 146148149152158168175181203208
5.8 Discussion 142
6 Analytical and Finite Element Formulation 144
6.1 Introduction 144
6.2 Analytical Formulation 145 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 158
6.5.2 Nonlinear Analysis 159
6.5.3 Solution Scheme 162
6.5.4 Finite Element Procedure 164
6.6 Finite Element Programs 166
6.6.1 2-Dimensional Program CONOIL-Il . 166
6.7 3-Dimensional Program CONOIL-Ill 167
7 Verification and Application of the Analytical Procedure 168
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 183
7.5.1 Analysis with Reduced Permeability .
7.6 Other Applications in Geotechnical Engineering
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
F
User
F.3
F.2
F.1
E.4
E.3
F.3.2
F.3.1 Input
Example Introduction
E.4.1
E.4.2
Main
Detail
Manual
Program
Data
Explanations
Data
Output
Main
Geometry
Problem
for
File
Program
CONOIL-Ill file
.
for
Program
1
for
Example
Example
1
1
ix
304
320
321 319
304
305 292
295 275 292
E.3
D.4
E.1 D.3
E.2 D.2 B.3
D.1 B.2
B.1
7.4
4.3
7.5 7.2 4.4
5.1 4.1
7.3 7.1 4.2
Load
Initial Viscosity
Element Viscosity
Time Saturations Viscosity
Average
Calculation
Parameters
Parameters
Model
Parameters Parameters
Soil
Details
Soil Soil
Summary
Parameters
Parameters
Parameters
Increments
Increment
Saturations
Parameters
of
Viscosities
Types
of
of
of
the
of
for
Used needed
of Used
and
water
water
water
Soil
Test
Flow
Modelling
Used for
Mobilities for
Scheme
Parameters
for
for
below
between
above
and
Used
Samples
for
and
Ottawa
Oil
and
the
Thermal
for
List
relative
Mobilities
Sand
Temperatures
for Saturations
00
1000
Oil
the
of
of
0
C
Ottawa
Sand
Triaxial
Recovery
and
Example
Water
of
C
Consolidation
permeability
x
1000
at
of
Tables
Sand
Water
and
Dr
with
Test
Problem.
in
C Problem
=
Oil
Different
Time
in
50%
and
calculations
after
Oil
Oil
Sand
300
Zones Days
.
302
269
294 269 266
300 268
251
251 209
251
192
184
133
178
181
101
107
94 75 3.7 3.6 3.1 3.5 3.3 3.4 3.2 2.6 2.4 2.5 2.3 2.1 2.2 1.2 1.1 1.3 Effect Yield Matsuoka-Nakai A et Mobilized Effect Spatial ing Components Comparison and In-situ Shear Residual Effect Fabric 1987) 1987) Undrained Oil Possible al., (after Sand Morgenstern, and of of Strength 1987) of of Structure Mobilized Intermediate and Stress Temperature Reserves Granular Sobkowicz Failure Plane Stress Equilibrium of of Peak Athabasca Path Strain of under and of Path Criteria Plane 1978) Shear Athabasca in Assemblies Oil on and
Mohr-Coulomb List Alberta Principal Increment 2-D During on behaviour Sand Stress-Strain under Strengths Morgenstern, and on Stress-Strain
Conditions of TSMp (after (after Oil Cold 3-D (after Steam
3 Stress Figures Sand of of Conditions — Lake Dusseault,1980) Dusseault Athabasca Failure an Dusseault Behaviour °sMp Injection (After 1984) and Element Behaviour Oil Space Ottawa Criteria Sands Salgado and and Oil (after of Morgenstern, . (after (after Sand Soil Morgenstetn, Sand . (1990)) Agar upon (after Kosar Agar (after et Unload al., Dusseault et . et Agar 1978) . 1978) 1987) al., al., . 46 43 40 45 38 36 34 16 18 15 14 13 12 7 6 2 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 ,1G and ‘quit 85 4.7 Evaluation of 0K 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
6.3
6.1
5.6
6.2
5.5 5.4
5.3 4.29
5.1 4.28
5.2 4.27
4.26
4.25 4.23
4.24
4.21
4.20
4.22 4.18
4.19
ing
Flow Finite Experimental
Finite meability
tan,
Comparison Comparison
1988) Zone Typical
One Results
Sensitivity Results
Flow
Sensitivity Determination
Results Failure
Plastic
Determination
al., Results
Grain
power
1987)
dimensional
1979)
of
Chart
Rule
Element
Element
Size
Collapse
Parameters
for
for
for
two-phase
mobile
for (after
.
law
Parameters
of
of
Tests Isotropic
Triaxial
Various
Distribution
for
of
of
Parameters
Parameters
and
functions
calculated
calculated
of
of
oil
the .
Types
Types
Kokal
Parameters
flow
with
kB
predicted
for
relative
Finite
for
Stress
Compression
Compression
and
of
three-phase
Used
Used
Various
and
and
for
Oil
a for
and m
KG,
C,p,A
and
Element
Paths
single
Oil
permeability
Sand
np
Maini,
in
in
values
for
Athabasca
for
experimental
experimental
Sand
for
2-Dimensional
3-Dimensional
np,
Stress
xiii
Oil
Oil
and
phase
on
Tests
Oil
flow
R 1
1990)
Test
of
Programs
Sand
Sand
Ottawa
viscosity
p
Paths
Sand
and
(after
in
on
on
variations
Oil an i
.
Oil
Oil
three-phase
Sands,
on
relative
Sand
element
Aziz
Analysis
Analysis
(after
Sand
Sand
Oil
(after
and
Sand
(after
permeabilities
Puttagunta
oil
Settari, Aziz
.
Edmunds relative .
.
and
1979)
et
per
Set
al.,
us
et
165 160
161
135 125
131 130 127 117
112
113 111
110
108 105
106 104
103
102
100 99
7.20
7.22
7.21
7.18
7.19 7.17
7.16 7.14
7.15 7.13
7.12
7.11 7.10
7.9
7.8
7.7
7.6 7.4
7.3
7.5
7.2
7.1
Vertical
Stress Horizontal
Vertical
Pore in
Temperature
Comparison Plan
Finite Finite
A
Comparison sion
Comparison Comparison Undrained
Finite
Pore
Srithar,
Results
Stresses
Comparison Material
Comparison Stresses
1985)
Schematic
Ottawa
Test
Pressure
View
Pressure
Ratio
Element
Element
Element
for
Stress
Cross-Sectional
1989)
and
and
(after
Sand
Stress of
Volumetric
a
of
of
of
Variations
of
of
of
Displacement
Displacements
3-Dimensional
Contours
the Circular
Variations
Measured
Variation
Variations
Pore
Pore
Observed
Measured
Observed
Modelling Mesh
Modelling Cheung,
Variations UTF
Pressures
pressures
for
Footing
Expansion
in
(after
in
View
1985)
and
with
in
Thermal
and
and
in
and
the
of
the
of
in
the
View Around
in
the
the
Triaxial
Predicted
Scott
Predicted
Predicted
Oil Circular
of
for
Oil
Predicted
in
Time
on
the
xiv
Oil
Oil
the
Well
the
Sand
Thermal
of
a
Sand
(after
Consolidation
Oil
et
Sand
Finite
Sand
for
the a
Well
Oil
Test
Pair
al.,
Circular
Cylinder
Sand
Results
Layer
Layer
Pore
Strains
Thermal
Srithar,
UTF
Results
Sand
Layer
Pairs
Layer 1991)
Layer
Consolidation
Layer
Pressures
(after
Layer
for
Opening
(after
(after
in 1989)
(after
Consolidation
a
Triaxial
Load-Unload
Scott
Cheung,
Srithar,
Vaziri,
(after
for
et
an
Compres
al.,
Cheung,
1986)
1989)
Elastic
1985)
(after
1991
Test
199
198
197 195
196 193
191 188
189 187
185
184
182 180
177
179
174
176
173
172
171 170 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
7.32 Variation of Pore Pressure Ratio for Case 2 213
7.33 Variation of Pore Pressure Ratio for Case 3 214
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 particular, research ance (AO aspects. committee The ciation ance, Finally, The The The STRA) author of valuable is the author author author grant also Uthayakumar for odd the is are greatly reviewing suggestions extended expresses provided wishes would gratefully fellowship working indebted like to habits the to by his express and acknowledged. to and awarded Acknowledgement Mr. Alberta gratitude manuscript thank Hendra the to of his Jim a his encouragement his by graduate appreciation Oil for supervisor Grieg to the colleagues xvi and Sand sharing his University for wife, making student. Technology his Professor common to in Vasuki, throughout valuable the Dept. constructive of members British for interest. P. and of helps M. her Civil this Research Byrne Columbia support criticisms. of research. on Engineering the the for supervisory Authority and computer his and Appre toler guid , the in I, 12 and K 0 CEQ kmT kEQ kmi Gt B kB kh H
13 f D E F B C k e mobility total equivalent Young’s permeability bulk Darcy’s initial plastic plastic tangent stress void Henry’s Young’s body equivalent pore stress-strain plastic displacement bulk ratio modulus pressure modulus mobility force invari plastic collapse shear collapse permeability plastic constant modulus modulus of hydraulic compressibility vector phase ants matrix in
number Nomenclature shape shear shape number shear yield modulus horizontal number ‘1’ parameter function function conductivity function of parameter xvii the direction porous derivatives derivatives medium 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 2o- 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 1, El, 2
62
and and P30,0
cEQ
W
u 3 6 e
63 p
r
v
8
Greek
mean
normal flow
principal viscosity
viscosity
shear flow
proportionality
failure
Poisson’s temperature
strain plastic
volumetric
stress plastic
elastic
principal
shear
Kronecker equivalent
coefficient
plastic
letters
rule
rule
stress
normal
strain
ratio
softening
strains
stress
shear
collapse
collapse
stress
of
of
intercept
slope
stresses
strains
ratio
of
delta coefficient
strain
oil
phase
ratio
strains
volumetric
stress
at
constant
strains
work
constant
30°C
‘1’
at
atmosphere
of
xix
and
thermal
thermal
at
0
gauge
expansion
expansion pressure (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 instability formations. necessary. to impractical. high bore in anisms to analyses deep oil as and estimated the are The the 700 Canada. the exists Analyzing Oil found Athabasca in-situ oil pore seated have instability m. very involved recovery contained which as in-place fluid at been problems extraction high formations. high In These In-situ depths the the oil and capture and used schemes viscosity and viscosity in problems sand in-situ reserves deposits stress oil less to thermal reported collapse and techniques sand design deposits There the than involve extraction bitumen gradients are of of related underlie deposits complex methods 146.5 the 50 these of relatively during
have Introduction (see the open m bitumen such in
million Chapter to oil and been, are procedures well in figure an Arenaceous engineering field oil pit such as recovery northern created area the effective casing. sands tunnels 1 however, mining cubic makes injection 1.1). as rest of steam some is about
schemes 1 around meters Alberta Approximately are Therefore, Cretaceous and in for conventional somewhat characteristics numerous the encountered trials. form injection the well-bores 32,000 (Mosscop, the shallow rationally is recovery of one well-bore to During different heating formations, well square understand through of recovery 5% oil in the of at 1980). casing of and the sand of the depths steam is kilometres major from which heavy these often economically, deep vertical oil by primarily failures Much formations, the analyzing injection, resources from pumping sand can required oil deposits oil mech of from well- sand with lead and 200 the are in Chapter 1. Introduction 2
Northwest Territories
United States of Amenca
Figure 1.1: Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978)
the
is
to
first
dilation fluids
(1977), these
upon in
skeleton. and undrained
may
if
However, reduction
pore
compressibility are bitumen
whereas,
process
a
Oil
Chapter
general
necessary.
the
loading
the
With
The
also
sand
recovery
Grigg
developed
occur,
fluid
shear.
analytical
and
volume
soil
and
of
deformation
effectively. involved.
regard
and
1.
skeleton
a
geotechnical
it
in
Oil
the pressure
condition prime
(1980),
and
general
skeleton,
it
if
The
can
Introduction
effective
In
by
gas
is sand
change
the
behaviour
and
unloading
by
this
to
models
very steam
be
linear
concern.
makes
based
Byrne
oil
the
soil
Oil
Harris
which
is
hydraulic
Furthermore,
categorized
study,
prevails.
there
and
effective
stress.
of
very sand
problem
pore
consists
injection
sand
and
consider
the
the
on
of
and
flow
in
The
cycles
and
will
a
dense
is
nonlinear
the
sand
analytical
fluid
pore
The
comprises
turn
double
conductivity.
Janzen
subjected
in
behaviour
changes
because
Sobkowicz
of
be
models
into
will
a
handling
and
skeleton.
effective
behaviour,
fluid
in
steam
three
affect
linear
an
hardening
its
two
cause
(1984)
elastic
for
increase
procedures
components
in
by four
of
phases;
natural
the
to
injection
or
realistic
major
of
temperature
the
stresses
the
(1977);
Nakai
An
When
changes
rapid
nonlinear
engineering
oil
models
phases;
and
Byrne
nature
dilation.
analytical
elasto-plastic
in
solid,
sand
state
constituents;
Byrne
and
there
increase
for
modeffing pore
and
may
and
It
is
in
are
solid,
of
oil
is
water
Matsuoka
and greater
was
elastic
subsequent
induce
temperature
become
is
pressure
the
and
not
Vaziri governed
properties
sands
model
an
in
shows
later
water,
oil
capable
and
Vaziri
increase
an
model
behaviour
temperature
changes
than
the
(1986)
sand
different
zero
elasto-plastic
for
air.
(1983)
and
extended
significant
by
behaviour
recovery
bitumen
(1986).
such
that
is
the
and
and
of
and
The
in
several
consequently
considered
postulated
in
modelling
temperature,
and
and
oil
for
the
of
their
as
liquefaction
volume
presence
by
and
However, sand the
strength,
the
will
and
by
recovery
difficult.
dilation
factors.
of
model
effects
Byrne
voids
Lade
if
sand
pore
lead
gas,
was
and
the
the
for
an
of
a 3
gas wet
pore large four
quantities dominant brief incorporated
also
the
Since model ical
and heat
rectly
be
1.1 individual
to
bitumen
as
the
Chapter
model
proposed
noted
An
The
three
can
hydraulic
model.
as
the
developed.
space phase
flow
descriptions
quantity
the
included
and,
the
Characteristics
analytical
effects
only
fluid
1.
and
that
the
dimensional
analysis
oil
physical
of
is
contributions
water
geological
However,
the
by
Introduction
filled
gas
in
pore
sand
exists
flow
gas
conductivity.
of
the
of
in
the
Byrne
effects
interstitial
can
phase are
temperature
about
equation
with
the
or
procedure
fluid
are
is
characteristics
2-dimensional
in
also
by
different
the
considered
material
governing
evaluated.
effects
and
the
bitumen
of
behaviour
some
forms
its
of
exist
temperature-time
these
form
the
Vaziri
of
bitumen.
In
unusual
a
considering
of
other
thermal
this
form
in
pore
a
changes
new
comprising
and
temperature
equilibrium
of
continuous and
the
Oil
(1986)
finite
of
appropriately.
research
discrete
fluid
3-dimensional
means
a
since
characteristics.
the
dissolved
The
an
general
energy
Sand
in
element
equivalent
is
components
oil
all
bitumen
quartz
stresses
is
also
solid,
work,
history
bubbles
sand
film
and
considered
these
balance
changes
soil,
state
included.
code
flow
grains
The
around
water,
the
are
finite
and
and
it
which
aspects
hydraulic
in
Oil
(free
CONOIL-Il. in
relative
is
the
equivalent
is
continuity
on
water
the
volume
of
the
not
appropriate
element
as
sand
bitumen
it.
quartz
the is
the
gas).
pore
an
compressibility
has
obtained
considered
form
A
permeabilities
oil
can
conductivity
stress-strain
input
changes
larger
fluid.
been
However,
compressibility
code
equations.
sand
mineralogy
and
In
be
continuous
to
to
order
from
considered
are CONOIL-Ill
An
developed
portion
gas.
the
in
have
present
illustration
99%
the
significant
to
a
behaviour
analytical but
is
It
The of
separate
been
and
analyze
phases,
derived
analyt
should
water
of
water,
not
some
term
as
two and
the the
di
in
is
a 4
figure
in
can saturated
a pore
starts
and pressible. like
decreases,
stress
decrease of
confining manner,
viscosity
cause the
teristics
and
of
Chapter
marked
the
a oil
Another
In
be
oil
a
the
exhibits
normal
fluid
normal
of
sand
decreases
1.3.
pore
to
its
found
sand
the
physical
compared
in
however,
1.
of
decrease
stress
natural
reduction
At
(path
compressibility.
the
A
structure
fluid
pore
very
bitumen
unusual
to
sand.
Introduction
comprehensive
high
this
in
sand
soil
behave
while
decreases
Sobkowicz
pressure
M),
comes
low
consequences
point,
state,
it
shear
skeleton
again
(path
Above
to
in
responds
effective
unsaturated
characteristic makes
(Dusseault,
the
normal
in
shear
out
strength
oil
the
I
(path
and
pore
an
below
the
of
Then,
and
of
compressibility
sand
the
soil
study
figure undrained
strength.
hydraulic
the
solution
dense
quite
liquid-gas
pressure
of
K).
Morgenstern
effective
matrix
the
is
1980)
and
the
effective
this
(path
of
1.3).
of
very
At
differently
sand
liquid-gas
the
pore
dilatancy.
oil
process
and
some
stays
conductivity,
is
manner.
Plots
commences
L)
A
dense,
saturation
gas
hydraulic
sand
shown
of
stress
fluid
causes
decrease
increases
and
similar
exsolution
constant
stage,
(1984).
of
are
compared
is
saturation
uncemented,
takes
gassy
pore
remains
It
in
its
the
significant
shows
conductivity
figure
the
to
pressure
in
mineralogy.
oil
behaviour
and
pressure pore
the
(path
take
soils
confining
phenomenon
sand
effective
constant.
becomes
to
low
load
1.2.
pressure,
fluid
the
(path
J).
the
behaves
(U 119 ),
fine
increase
compressibility
and
versus
load
As
upon
to
undrained
stress
very
stress
The
to
J-K)
comparable
the become
When
the
the
oil
and
medium
upon
in
unloading.
low
extremely
total
in
effective
will
becomes
pore
sand
are
an
dissolved
the
volume
the
and
undrained
result
unloading
very behaviour
stress
shown
pressure
effective
behaves
grained
level
charac
causes
to
stress
com
high
zero
in
and
Be
gas
the
for
in
of
a 5 Tj I-. o a;-’ rcn
(b -C I-.
CC CC- I-. m - 0
U)
—m 0 o m
-
I a
C
,oo
a.. -
C- Chapter 1. Introduction 7
U IN SITU / STRESS /..—.u=o, o-c=o J Uj/g ____,_,_ D (I, C,, w / 0
LAJ 0 ..° TOTAL STRESS I atm CEGASSED PORE FLUID
0
FINE SOIL
Figure 1.3: Undrained Equilibrium behaviour of an Element of Soil upon Unloading (after Sobkowicz and Morgenstern, 1984) Chapter 1. Introduction 8
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
ter.
in laboratory
ments problems
an
Chapter
detail.
oil
Chapter
on recovery
1.
the
are
Possible
experiments
8
Introduction
aspects
discussed
summarizes
process
applications
which
and
by
are
the
steam
warrant
considered
an
important
example
of
injection
the
further
developed
and
problem
findings
is
the
investigation
presented
results
formulation
is
of
also
this
are
given.
and
are
research
compared.
the
for
also
general
results
stated
work.
Application
are
in
geotechnical
Some
this
analyzed
chap
com
to 9
of view
be
behaviour widely pletion
that
to fore,
and behaviour to
The to lowing
the
2.1 subheadings.
The
oil
recognize
set
critically
described
three-phase
in
the
research
stress-strain
of
it
sands
the
held
topics;
is
1980s.
Stress-Strain
the
of
oil
appropriate
stage
series
of
of
stress-strain
sands
in
until
assess
the
by
oil
a
work
The
stress-strain
laboratory In
dense
pore
to
of
an
sands.
geomechanical
particular,
the
must behaviour
intention
discuss
each
research
carried
appropriate
fluid;
last
sand.
to
be
models,
The
and
present
Review
experiments.
decade,
Models
the
considered
and
model
out
programs
This
of
the
next
every
of
work
the
the in
behaviour
the
it
stress-strain
a
perception
conclusion
for
subsection
as this
will
review
research
analytical
literature
Chapter
oil
carried
many
the
as
at
of
be
study
sand
a
10
oil
the
useful
on
particulate
of
Literature
out
work
of
sand,
was
the
model.
can
the
summarizes
skeleton
University
and
review
bitumen
in
to
2
previous
sand
not
be
but
finite
modelling
this
describe
broadly
presented
Before
widely
and
material to
is
skeleton.
study.
element
as
of
give
essentially
the
research
a
Alberta
the
going
of
accepted
cementing
classified
an
stress-strain
and
flow
observed in
formulations.
It
overall
this
into
works
is
its
in
characteristics
the
now
until
the
behaviour
chapter
under
a
material
picture,
stress-strain
stress-strain
under
detailed
recognized
late
behaviour
the
the
There
is
failed
1970s
these
com
was
and
can
not fol
re of 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. Chapter 2. Review of Literature 12
(a) Hexagonal close-packed spheres. Point contacts.
(b) Densely packed rounded sand. Point contacts,withsome straight contacts (arrows)
(c) Athabasca oilsand Point contacts, with many straight and interpenetrative contacts (arrows)
Figure 2.1: Fabric of Granular Assemblies (after Dusseault and Morgenstern, 1978) Chapter 2. Review of Literature 13
Three different samples o o Peak strength • Residual strength
•
a.
U,0 L0 0(U -c (0
0 200 400 600 800 1000 1200 o normal stress, kPa
Figure 2.2: Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseault and Morgenstern, 1978) Chapter 2. Review of Literature 14
20 28
16 24
12 20
8 16 b a
4 12
0 0.5
>0
—0.5 0 4 8 12 16 0.5 1.0 1.5 ./7O1 (MPa) e (%)
(a) Various Stress Paths (b) Stress-Strain Behaviour
Figure 2.3: Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987)
I; Chapter 2. Review of Literature 15
60
a) . LEGEND . ATHABASCAOIL SAND (This Study) a Athobasca Oil Sand v OTTAWA SAND (This Study U C DIJSSEAUT & MORGENSTERN(1978) D U, SOBKOWICZ (1982) in DUNCAN & CHANG (1970) a, 40 U) C I D U . -C ‘/, — 30 0 a) U) C
20 1 2 3 4 5 6 7 8 Effective Confining Stress c (MPa)
Figure 2.4: Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agar et al., 1987) Chapter 2. Review of Literature 16
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.
20
16
12
4
0
04
0 0.5 1.0 1.5 2.0 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 Chapter 2. Review of Literature 17 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. Chapter 2. Review of Literature 18
6-
Mairjmshearsfl-ength = 16.9 MPa
I 5. a—4.OUPa
4 / Athabasca (Agar. 1984) 0
3• : Mi,m shaer strength• 6.9 MPa
I 2. :
Athabasca £ - 2200 MPa CoidLake ‘7
S AxialStrain (%)
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
vary
into
component the
presence of
soils behaviour plastic
The sium,
2.1.2.1 A models
and
He symposia.
Scott coupling necessary.
able
Chen uncoupled
Chapter
series
plasticity
In
Since
Reviews
also
stress-strain
Case
the
with
from
theory
in
(1985)
(1982)
Florida,
the
models
the
as
of
different
described
of 2.
effects
an
the
Western
metals
elasto-plastic
of
workshops
from
Generally,
of
and
Ko
literature
voids
of
Elasto-Plastic
of
Review
elasto-plastic
provides
presented
described
metals.
level
1980
and
plasticity
1980;
the
and
in
a
each
behaviour
(Lade,
stress-strain
and
plastic
the
of
some
University
the
existing
and
Sture
of
loading
International
other.
held in
as
the
an
Since
incremental
related Literature
a
and
elastic
models,
described
state-of-the-art
of
1987).
component.
has
very
attractive
tendency
at
(1980)
model
of
the
Byrne
analyzed elastic
soils
McGill
and
been
soils.
(1987)
models.
lucid
theories
models,
elasto-plastic
Models
the
is
exhibit
unloading.
provided
the and
elastic
Symposium,
developed
and
However,
for
proposed
theoretical
treatise
University
strain
and
The
what
methods
Eldrige
are
volume
the
elasto-plastic
papers,
plastic
the
models
amounts
presented
shear
a
increment
is
on
The
models
international
there
clear
in
on
meant
(1982)
(1980),
change
framework
plasticity
Deift, special
needed
this
non-recoverable
and
elastic
the
using
summary
are
of
study,
next. by
most
normal
basis
incorporated
constitutive
is
elastic
University
during
1982)
major
a
workshops
strain
to
different
decomposed
stress
and
symposia commonly
for
obtain
of
a
provide
of stresses
and
differences
brief
stress-strain
shear
observed
increment
the
dilatancy
the
strains,
of
levels
plastic
models
their and
representation
the
review
Grenoble
that
most
(ASCE
and
better
into
used
shear
international
of
stress-strain
coefficients.
such
strains
is
distinguish
the
equation.
strains
important
an
elasticity.
of are
relations.
obtained
for
insights
sympo
volume
elasto
theory
elastic
(1982)
as
avail
soils.
will
the
are
20 of Chapter 2. Review of Literature 21
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. Chapter 2. Review of Literature 22
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
frictional
tained friction,
grains
of plastic
between
point
shearing tween
angle occurs formation
planes the
to by
(1974), extensively
The
2.1.3
Chapter
soil
model
Rowe
Nemat-Nasser
Lade’s
Matsuoka
spatial
stress
of
of
Rowe’s
upon
by will
at
on
work.
Oda
he
H
particles
the
2.
view.
mechanism
losses (1962,1971)
the
Stress
the
curve
certain
behaviour.
(1977)
distribution
be
developed
dilatancy
in
shearing - Review
-
and
interparticle
(1974)
dilatancy
plastic
vector
theory
micro
such
stress-strain
and
fitting. He
Konishi
model
with
favourably
(1980)
Dilatancy
as
carried
the
developed
of
level.
defining
of
resistance
and
which
theory
can
a
to
After
respect
Literature
of
following
relationship
soil
energy
The
incorporates
minimize
presented
(1974),
interparticle
the
be
contact,
modeffing
out
The
is
particles.
Rowe,
equation
derived
considered
oriented
the
other
based
the
to
balance.
shear
equation
unit
different
Nemat-Nasser
energy
Relation
stress
the
interparticle
an
a
theories
between
on
number
from
the
of
contact
equation
relates normal
tests
rate
contacts.
From
the
a
sand.
dilatancy
supplied.
dilatancy
remarkable
approaches
was
theoretical
of
by
mechanics
is
the
the
of
a
dissipation
to
planes.
that
The
obtained
to
dilation
using
other
force
(1980)).
Rowe’s
potential
shear
fundamental
relationship
describe
stress
through
Rowe’s
cylindrical
effort
and
researchers
(Murayama
The
considerations
resistance
of
parameter
theory
by
of
A
dilatancy
the
surface
the
the
orientation
theory
energy
noticeable
to
a
considering
relative
empirical
through
angle
measurements
volumetric
explain
considers
rods
and
published
at
(1964),
is
to
in
theory
the
of
independent
has
sliding
the
the
motion
to
difference
a
of
the
interparticle
relation
microscopic
that
stress
the
the
model
amount
dilatancy.
been
behaviour
Matsuoka
shear
proposed
theories
friction
rate
sliding
sliding
of
of
point
used
the the
ob the
be
de
23
of of of
levels
reservoir.
and through
0 m
capability equivalent
Owen’s change time pointed
through in
modulus
consolidation
as equivalent
subsequently
skeleton haviour,
by
Modelling
2.1.4
Chapter
Byrne
1980
Harris
Tortike
In
A
an while
steps
of
the nonlinear
associated
corrections.
(1977)
out
thermoelastic iterations;
to
a
behaviour.
so
2.
rather
and
above
Modelling
and
in
procedure
He
to
elastic
of
nodal
(1991)
are
model
as
that
Review
a
extended
model
the
Janzen
type
further
to
Sobkowicz
elasto-plastic
pressuremeter
considered.
than
elastic
cited
describe
loads
the
analysis
geomechanical
flow
stated
the
one
of
Wan
the
of
borrowed
a
thermoelastic
references,
to
suggested
analysis.
approach
oil
by
rule. for
model
Literature
to
secant
represent
loading
of
that
(1977).
et
gas
using
Byrne
predict
sand
stress
Furthermore,
al.
model
Stress-Strain
exsolution
test,
with
cyclic
modulus.
from
may
skeleton
a
It
(1991)
that
the
calculations,
and
and
behaviour
They
secant
the
the
predicts
dilation
shear
which
approach
thermoelasticity.
dilative
steam
lead
Janzen
unloading
a
correct
stress-strain
considered
stated
realistic
Vaziri
and
modulus
behaviour.
to
dilation
includes
the
simulation
is
of
unrealistic
a
behaviour
(1984)
other
decrease
and
encounters
volume
always
oil
that
computer
(1986)
Behaviour
behaviour.
stress-strain
sand
and
was
the
a
related
a
behaviour
the
who
linear
Mohr-Coloumb
Their
accompanied
other
This
changes.
imposes
a
basically
in
proposed
of
oscillating
along
method
single
used
effective
the
shortcomings
algorithm
aspects
elastic
method
for
model
He
material
model
with
an
of
step
cyclic
shear
of
used
Srithar
adopted
of
oil by
incremental
mean
model
results
was
including
the
is
loading.
by
involves
Oil
sand.
Byrne
necessitates
should
failure
the
induced
loads
based
is
first
an
pore
specially
normal
et
incorporated
for
Sand
same
Hinton
when
increase
al.
and
on
presented
have
This
envelope
applying
upon fluid
the
dilation
tangent
volume
the
(1990)
model
stress
Grigg
large
sand
in
and
two was
the
be
oil
an
24
in a
for
presence
successfully
method
compared element
Lewis
Price, with
constant flow
Witherspoon oil In
number
an
their
2.2
the
compression model
Nakai process
Osgood
Chapter
isotropic
petroleum
sand
Solutions
multi-dimensional,
Wan
potential
in
the
model
et
1976;
(1982)
is
porous
Modelling
is
by
type
of
methods
of
al.,
flow matrix.
2.
et
based
analytical
variational
more
researchers
favourable
steam
cannot
applied
al.
homogeneous
Review
Huyakorn
with
1978;
for
hardening
(1968).
rate
equation
function
reservoir
medium
on
efficient
two-phase
(1991)
injection
were
constant
The
or
White
Vermeer’s
predict
solutions
to
of
and
pressure
They
and
of
without
two-phase,
the first
Literature
that
and
to
presented
is
engineering,
function.
also
than
finite
et
Fluid based
represent
porous
unfavourable
isothermal
the
stress
analytical
and
Pinder,
clear
considered
appeared
al.,
recognized
for
the
(1982)
constraints
plastic
difference
consideration
proposed
1981).
upon
infinite,
ratio.
finite
attempt
The
by
medium.
immiscible
Flow
1977a;
the
elasto-plastic
multiphase
various
formulation
volumetric
fluid
in
model
Rowe’s
difference Spivak
a
mobility
yield
petroleum the
methods
single
an
and
bounded
in
to
flow
Spivak
The
elasto-plastic
cyclic
involves
of
researchers
use
flow
were
Oil
and
et
stress
problems
phase
the
fluid
numerical
behaviour
ratios.
al.
method.
and
of
a
model.
failure
using
et
found
Sand
engineering
loadings
and
geomechanical
finite
the
(1977)
dilatancy
a
al.,
concluded flow
isothermal
non-associated
layered
Numerical
governing
variational
using
(for
to
1977;
surfaces,
element
model
They
Galerkin’s
solutions
for
has
presented
be
caused
example:
stress
variational
was
in
equation.
been
radial
Settari
used
that
for
fluid
good
method
dispersion
behaviour
equations
and
by
in
method.
paths
were
oil
the
analyzed
procedure
Matsuoka
a
systems
Javandel
flow
flow
the
et
agreement.
a
sand.
formulation
Settari
variational
compared
Ramberg
and
al.,
involving
However,
for
rule
recovery
through
in
at
of
Their
They 1977; finite
by
fluid
with
was and
and
and
and
the
the
the
25 a Chapter 2. Review of Literature 26
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
the
does of tween
to under in
more
applications of
compressibilities, in
Some 2.3 geomechanical
sider mal
deformation accuracy.
under
the
this
appeared
Chapter
oil
the
predict
Biot
oil
Raghavan
It
Tortike
correct
finite
result
flow
not
the
straightforward
sand
appears
models
poroelastic
sand
different
nucleus
isothermal
Coupled
(1941)
consider 2.
geomechanical
problem to
element
surface
However,
is
matrix
parameters
is
(1991)
obviate
of
in Review
not
in
(1972)
modelled
that
of
fluid
and
poroelasticity
the
boundary
petroleum
in
and
strain and
the
displacements.
method.
and
through
using
presented
oil
in
Geomechanical-Fluid
Gassman
accordance
the
flow
these
of
derived
manner.
effect
described
thermoelastic
sand most
non-isothermal
Literature
defining
need
by
for
behaviour.
finite
model,
conditions
models
poroelasticity.
two
reservoir
matrix
of volume
of
for
a
a
in
(1951)
stress
He
the
differences.
one
detailed
with
the
the
phase
petroleum
upstream
but
It
solve
clearly
research
are
dimensional
boundary
theories,
elements
compressibilities.
should
was
that Therefore,
distribution
later
to
engineering
system
conditions.
not
only
develop
literature
not
defined
have
Geertsma
weighting
studies
included
engineering.
be
He
work
the
to
successful
was
conditions
(water
noted
already
also
take
the
consolidation
the
fluid
through
include
and
in
described Flow
of
review
He
effects
tried
for
(1957) in
advantage petroleum
equations
the
however,
and
related
Geertsma flow
these
been
as
numerical
solved
An
multiphase
and
Models
the
to
a
on
the
bitumen)
of
problem
combined
porous
analogy
and
published.
models. develop
procedure
the
modelling
effects
stress
equation
results
the
the
of
of
engineering,
(1966)
it
rock
the
stability.
poroelasticity
poroelastic
three-phase
medium. has
distribution
and
flow
for
is
with
of
a
many
the
were
bulk
presented
fully
coupled
deformations
been
reviewed
to
The
of
do
Oil
problem
approaches
reasonable
determine
unstable.
fluid
However,
and
solutions
not
the
concept
coupled
applied
theory
Sands
ther
with
pore
con
in
flow
flow
and
the
be
by
27 a
in
and which
compatibility response.
The the high
oil
Sobkowicz’s
and that ically
analyze
cal
analysis oil
ences
surface material fluid
tion.
was
Chapter
their
Vaziri
and
Byrne sands.
Harris
Finol
point
above
analyzed
the
obtained
pore
the
viscosity
flow
He
integrated.
the
which
displacements.
water
excavations,
analysis.
geomechanical
as
ultimate
displacement
2.
also
and
of
(1986)
stresses
fluid
circumstances.
The and and
and
Since
the
formulation.
view.
Review
included
between
presented
the
Farouq
from
flow,
of
Sobkowicz
compared
authors
Grigg
effect
pressures
coupled
these
the
recoveries
transient
in
The
Their
They
the
and
of
immediate
the
bitumen,
Ali
of
(1980),
the
using
fluid
variation
The
Literature
scenarios
claimed
partial
behaviour
a
one
compaction
analysis
sand
presented
his
Byrne
the
(1975)
(1977)
significant
were
effects
conditions
problem
and
a
of
analytical
results
equilibrium
transform
skeleton
and
oil
differential
their
computed
foundation
that
and
skeleton
analyzed
of
derived
involve
procedure
of
increased
of
Byrne
a
permeability
with
compaction
Janzen
was
review
on
model
the
oil
coupled
as were
equation
ultimate
to
sand.
an
phases.
formulated
short
short
Terzhaghi’s
a
equation
a
from
and
equations
convert
settlements
consolidation
of
with
was
analytical
computed
two-phase
also
involved
mathematical
the
term
term
Janzen
The
the
for
only
included
on
and
compaction.
recoveries.
literature
it
and
poroelasticity
model
fluid
gas
by
conditions
describing
conditions,
to
an
porosity
concerned
solution.
model
using
flow
and
(1984)
an
the two
laws
effective
flow
problem. the
ordinary
was
underground
model
flow
discretized
model
a
to
from
The
fully
together
finite
was
and
extended
that
govern
developed The
the
with
continuity
and
stress
authors
which
using
a
drained
considered
for
the
differential
flow
element
He
time.
more
general
because
the
the
the
equations
with
prediction
included
approach
finite
of
was
openings
Harris
undrained
geotechni
concluded
mainly
fluid
condition
design
fluid
equation
scheme.
solution
volume
numer
of
in
differ
equa
flow
and
and
the
the
the
for
to
in
28
in
in of
law
mal
their two-phase
including
factor
a rates
a that
dilation pressure
the induced
thermoelastic
continuity predict
taken
single
thermal
a
Chapter
hydraulic
different
two
Fung
Settari
Dusseault
with
physical
two-phase
effective
earlier
for
of
phase
into
dimensional
unrealistic
as
effects
two. stresses
(1990)
fracture
2.
Rowe’s changes
the
isothermal
equations
pressure
(1988),
account
the
fracture
model
Review
process
one.
water
localization
The
and
approach
fluid
on
described
thickness
stress
and
faces
around
Rothenberg
The
authors
stresses,
to
Settari
oscillating
finite
permeability
distribution
by
flow
due
of
of
and
flow
volume
thermal
in
effects
dilatancy
Literature
means
deformation
to
to
and
a presented
oil
of
element
a
of
in
continue
et
control
hydraulic
the
model
wellbore
shear
their
the
sand.
changes
solid
al.
of
flow.
results.
(1988)
of
altered
than
different
theory. would
water
an
analysis. (1989)
and
volume
formulation.
temperature
behaviour.
The
to
a
in
equivalent
on
reviewed
conductivity
better
the
directly
document
the
stress
Srithar
terms
film
increase
authors
dilation
described
linear
phase
finite
growth
The
formulation
coating
state
of
in
He
the
(1989)
nonlinear
element
used
effects.
one
elastic
compressibility.
components
the
The
particulate
the
and
adopted
of
and
effect
and
a
changes
or
the
the
a
governing
model fluid
permeability.
incorporated
nonlinear
two
volume
one.
the
approach
This
of
of
shear grains
response
flow
a
increased thermal
orders
Vaziri’s
media.
likely
to
Settari
on
hyperbolic
approach
zone
change
equilibrium
quantify
would
was
compressibility
elastic
Vaziri
for
of
from
was
model.
loading
from
the
(1989)
They
magnitude
coupled
considered
They
pore
and
increase
shown
shear
model
temperature
followed
stress-strain
appeared
the
the
pressures.
concluded
presented
described
extended
and
and
isother
leak-off
edge
failure,
to
and
by
with
were
pore
give
as
flow
the
29
to
of
a
a a Chapter 2. Review of Literature 30
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 + 9S = 1), and the ef fective stress equation. Three combinations of solution variables were considered ( {U,F,(,,P}, {U, P, 9P}, {U,P, S}). Among these the best convergence was found when using the combination of {U,P, 9P }. 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 Chapter 2. Review of Literature 31
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.
the number
the
skeleton and
the
strains.
of the
significant this
skeleton
ate
Basically, In
3.1
pore
1.
developing
Generally,
As
pore
model
stress-strain
modelling
behaviour
flow
chapter,
would
The
explained
Introduction
fluid
of
This
to
pressure.
will
predictions.
modeffing
failure proposed
models
shear
undergo
realistically
behaviour
increase
necessitates
modelling
a
oil
of
of
procedure
model
in
Stress-Strain
the
pore
induced
criterion recovery
available
These
section
loading
by
of
deformation
the
Therefore,
fluid
oil
employed
Matsuoka
is
of
model
changes
the
pore
explained
sand
volume
2.1.1,
to
sand
methods
is
in
and
and
use
analyze
based
the
space
behaviour
the
unloading
skeleton
modeffing
oil
realistic
will
of
in
and
behaviour
literature
expansion
Chapter
behaviour
sand
an
in
this
on
are
and
have
the
his
chapter
elasto-plastic
Model
stress
is
study
32
cyclic
behaviour
hence
modeffing
co-workers
geotechnical
can
significant
sequences
very
of
as
of
or
the
be
when
ratio
for
5.
discussed
oil
processes
increase dense
3
dilation.
divided
behaviour
the
sand
Employed
is
of
the
rather
stress-strain
has
resulting
effect
in
described
following
dilation
aspects
its
is
the
soil
been
into
in
which
the
The
in
natural
than chapter
permeability
of
undergoes
the
two
in
chosen
of
most
is
the
dilation
reasons.
in
will
irrecoverable
important.
oil
overall
model.
shear
parts;
state
sand
detail.
2.
important
sands,
cause
as
Among
a
stress.
the
modeffing
and
deformation
in skeleton.
There
decrease
and
Modelling
appropri
the
the
basis
exhibits
reduce
plastic
these,
issue.
are
sand
This sand
for
in
In
of a 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.
Chapter
3.
2
Cl)
Cl)
Stress-Strain
Figure
3.1:
A
Model
Possible
Employed
Stress
Path
(Increasing
Normal
During
Failure
Stress
Steam
Steam
Envelope
Injection
Injection
Pressure) 34 Chapter 3. Stress-Strain Model Employed 35
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, 8de (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,
dcc dee de = 9dc + H- (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
}8{de = ]8[C {do} {de} [Ce] {th} }6{dc = [Ce] {d} (3.2) where 8],[C [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:
[CC] [CC]] {de} = ]8[[C H- + {do}
Chapter
3.
U
C,, w
> w
0 z U z C?, U ci
C,,
U ci
-J I- I
I
Stress-Strain
Figure
Model
3.2:
Components
Employed
of
Strain
Increment 36
friction
makes
mobilized occur.
in
the
term
(rMp/crMp)
The
cSpatial
3.3.1 3.3
full
The expressed
conditions.
[D]
Cartesian
Chapter
two
The
In
The
three
matrix
elasto-plastic
above
stress-strain
‘Mobilized
developing
an
dimensional
Plastic
The
angle
concept
Mobilized
theories
3.
dimensional
Background
angle
friction
as
coordinate
equation
can
is
2-D
Stress-Strain
are
the
be
of
Plane
of
representation
involved
shown
relationship
a
Shear
angle.
(45°
constitutive
easily
maximum.
Plane’
conditions
mobilized
finite
is
conditions, system
an
(MP)’
+
in
The
obtained
m/2)
inverse
concept
element
Model
in
Strain
figure
of
developing
Mohr are
refers
plane
for
is
This
the
matrix =
of
to
given
Employed
a
of explained
3.3
the
by
as formulation,
this
brief
the
[C]
circle
to
is equation
do
was
by
Model
the
(b).
Nakai
plastic
the
to
the
{do}
major
[C]
plane
=
description
first
inverse
the
Cone-Type
provide
for
[D]
plane
plane
is
in
and
the
shear
3.3.
[C 8 ],
is
principal
developed
formed
dE
the
the
shown
Matsuoka
of
where
on
stress
a
Once
[Cc]
next
better
strain [C].
stress-strain
of
which
according
the
in
and
conditions
stress
the
the
sections
by
Yielding
figure
is
concept
insight.
slip
(1983).
shear-normal
[C]
Murayama
[Ce]
developed
plane,
can
matrix
relation
and
3.3
to
matrices
of
and
Before
be
different
where
(a).
mobilized
at
the
considered is
based
(1964).
the
is
known,
stress
This
going
in
mobilized
q
generally
end,
loading
general
on
is
plane plane
ratio
(3.4)
(3.3)
into
The
the
the
the
the
to 37 Chapter 3. Stress-Strain Model Employed 38
Q3
2-D Mobilized Plane
(a)
C,, (I, cbJ C’, TM bJ C,,
Q NORMALSTRESS
(b)
Figure 3.3: Mobilized Plane under 2-D Conditions
general
pressed
where
the
soil
cosines Under
shown
expressed three
angles,
be
where
developments
normal-shear
proposed
granular
Chapter
characterized
Using
Under
From
octahedral particles
tan(450+)
stresses
11,12
isotropic
)
in of
ml 2 , 4 m23
coordinate
by
and
3.
material
a
figure
these
the
a
by
general
the
large
relationship
and
Stress-Strain
are the
i
strain
SMP
of
can
following
plane
stress
mobilized
are
3.4
13 the
by
most
following
number
three
and
in
be
system.
are
are
constant
increment
the
(b).
constitutive
and
a
condition drawn
ç 3
mobilized
the
given
microscopic
dimensional
three
equations
between
a
This
will
of
Model
friction
equation: first,
=
can
Z
tests
as
by soil MP
rp
principal
vary
ratio plane
(o
shown
be
the
second
Employed
models
and
parameters.
the
and
in
angles,
obtained. with
=
point
(dMp/d7Mp)
following
conditions,
terms
ABC
is
shear-normal
in from
stresses
called
(_d6MP+
\
and later
=
possible
figure
of
d-yf a
03)
is
of
the
3-D
third
view,
These
principal
considered
by
equation:
the
the Equation
o,
(i
the
3.4
analysis
)
Matsuoka
plane changes
=
on
effective
‘Spatial mobilized
02
Murayama
stress
stress
(a)
mobilized
1,2,3)
(i,j=1,2,3;ucT)
the
and
stresses
ABC
and
3.5
to
of
mobilized
in
state ratio
o.
Mobilized
stress
and
the
be
three
plane
forms
stresses.
can
Mohr
friction
and
or
the of
(TMp
his
shear
invariants
be
the
a
mobilized
will
Matsuoka
the
co-workers.
plane
plane
soil circles
/crMP)
constructed
Plane
stresses
The
mechanism
angles
coincide
basis
element
where
as,
direction
for
(SMP)’.
and
and
friction
for
can
(1973)
in
these
(3.7)
(3.6)
(3.5)
with
can
the
the
the
ex
the
be
as
39 of Chapter 3. Stress-Strain Model Employed 40
r 13 m12
o•1
(a)
1
Ia; cI -f-———---———--y.—Spatial Mobilized V’ Plane 6 O3 — - B 7 450+ m23 ’7A 52L•+ 2 (b)
Figure 3.4: Spatial Mobilized Plane under 3-D Conditions
ity,
and principal
to
The
and
normal
the tionship Chapter
TSMp
the
By
The
the
following
shear-normal
d7sMp
SMP assuming
=
normal
stress
general
13 12
‘1
of
3.
strain
/(oi
the
(dcsMp
=
=
=
Stress-Strain equations:
= (oSMP)
—
and increments 12
O 1 +0 2 +0 3
stresses
i,J(dEa 2
stress-strain
that
o 2 ) 2 a?a
stress
the
and
+
the and
0203
parallel
ratio,
on d7sMp) —
SMP
direction
Model
+
the
dEsMp
d€a 1 ) 2
are
the
+
(o 2
relationship
°y°z
= shear i
O301 identical, = =
—
components
SMP
can
are Employed SMP o 1 a =
TSMp
o 3 ) 2 aa
+
+
of
dea 1
stress
=
given
be (deas
2 TTyzTz
and
the +
0 x 0 y expressed
=
which
o 2 a
+
will
principal by
(TSMp)
I1I2
the +
—
dea 2
of
+
(o
dca 2 ) 2
+
be
0 y 0 z
is strain
—
the
o 3 a 913
—
—913 the
OT
+
developed
as
on principal
O1)21
stresses
dEa 3 +
=
-- common
the
components
0 z 0
—
3
(d€ai SMP
— and
=
basically strain
—
assumption
can \/111213
—
d€1a 3 ) 2 the —
OzTy
to
be increment
T 2 direction
‘2
the
from obtained —
—
9I
SMP.
in
the
plastic
vector
of
(3.13)
(3.12)
(3.11)
(3.10)
from
(3.9)
(3.8)
rela
The
the 41
if
the
of
the
where family
soil
surfaces
The ever, function,
explained
3.3.2
their the
The involved,
SMP’. and
cipal
normal
Chapter
the
loading.
The
In
It
stress
current
is
soil
not
model
yield
earlier
those
strain
defined
should stress
i
the
of
The
and
‘current’
particles
with
are
3.
yield
a
state
Yield Nakai
in
criterion
TsMp theory
For
plastic
elastic developed
can
shear
given
concepts
model
increments
Stress-Strain
the
state
be
the
by
of surfaces instance,
/0sMP,
be
and
next
yield
noted
the
coincides
the
direction
of
strain
by
and
potential
region
of
as
formulated
defines Matsuoka
77
maximum
plasticity,
point
the
the used
subsections.
SMP
surface by
—
that in
q 5 m
to
increments
assume
Failure
following
3 \/tanmi 2
corresponding
Matsuoka
point
the
Model
in
with
is
of
the
the
(Matsuoka
function
before
represented
are
this
the corresponding
TSMp
and (1983)
SMP.
stress
the
boundary
the
moves
the
the
Employed
strain
study
equation:
Criteria
Nakai the
—
direction
stress-strain
on
current
does
mobilized
(or
+
ratio After
0 SMP
concluded
and
to
the
tan
constitutive
increment
follow
to
a
by
not
Pu and
flow
between
that
SMP
mobilized Nakai,
m23
yield
P
a
space
to
of
explicitly
within
thorough
(see
the
Matsuoka
rule)
friction
the
the
yield
+
rather
that
relation
surface
is
1974,
figure
vector
tan
SMP
stress
the
principal
matrix and
shown
the
at
surface.
the
q m13
angles
than
elastic
define
investigation
that
1977)
is
a
3.5),
state
model.
is
(1983),
on
elastic
average
strain
represented in
=
formulated
can
the
components
point
strain
the
and
k
figure
In
these
at
and
and
be
SMP.
region,
hardening
a
a
Matsuoka
shaded
sliding
k
loading
point
the
during
plastic
derived increment
functions.
is
3.5.
of
a
They
new
by
from
the
constant.
in
only
These
area
direction
of
line
its
sequence,
zones.
a
function.
model
the
easily
used
denoted
theories
mass
history
a
will
elastic
A
vector
(3.14)
How
yield
yield
prin
and
the
be
42
as
of as
of A Chapter 3. Stress-Strain Model Employed 43
Failure Surface
B
YieldSurfaces
A
P...
ElastIc Region
°SMP
Figure 3.5: Yield and Failure Criteria on TsMp — 05MP Space
will
figure
Matsuoka-Nakai
triaxial
compression stress. be
The
if
where
is
where
(1990)
yield
SMP
the
to
strains
Chapter
the
given
FL
seen
The
The
elastic
correspond
Matsuoka-Nakai
at
surface
failure
which
3.6.
claims
This
will
that
conditions
failure,
limit
failure
by
3.
is
region
The
the
occur
the
condition
the is
Stress-Strain
effect
stress
will
or
that
outside
following
and
surface to -
-
Mohr-Coulomb
failure
Mohr-Coulomb
the
and
failure
decrement
be
will
and
b-value
is (compression
ratio
the
that
boundary
dragged
failure
Mohr-Coloumb
shown
expand
will
it
the
failure
on
stress
stress
a
is
represents
equation:
Model
better
tan
= elastic
correspond
expressed
the
criterion
in
as
1. along
=
up
ratio
stress
ratio
failure
of
octahedral
the
failure
and
and
agreement
Employed
to region,
f12 the
—
an
to
difference
and
line
at
Matsuoka-Nakai
considers
ratio
criteria by
extension)
yield
+
stress
to
a
unloading surface
(osMp
tan
new
the
B.
b-value
there
log 10
is
plane
surfaces with
f23
This
following
ratio
are
dependent
yield
with
)
between
the
will is (asMP)f
=
+ but
=
the the
also
corresponds
condition.
and
for
effect
b-value
tan
surface
be
1
0
will
laboratory
atmosphere
differ
and
failure
failure
10
shown
elastic
equation: in
f13
the
of
on
fold
be
the
triaxial
in
represented
for the
=
failure
the
the
friction
If
surfaces
in
and
increase
kf
figure
3-D
to
any
intermediate
the
the
normal
data failure
a
extension
plastic
space
friction
loading
stress
other
figure
3.7.
angles.
will
coincide
in
by
surface stress
(oSMp
strains.
The
is
stress
be
state
line
and
angles
condition.
condition
shown
principal
obtained
Salgado
triaxial
for
on
B
it moves
(3.16)
which
(3.15)
)
path.
The
and
can
the
the
for
44 in Chapter 3. Stress-Strain Model Employed 45
01 MOHR-COULOMB \
MATSUOKA - NAKAJ
(a) Octahedral Plane
01
/1II\ #\ /L\’ “\ / A\%( II1/ ,C/
0
C p7 (b) 3-Dimensional Stress Space
Figure 3.6: Matsuoka-Nakai and Mohr-Coulomb Failure Criteria Chapter 3. Stress-Strain Model Employed 46
8-
7-
6-
TX 5.. 7400
-a- 3Q0: 4- .
-a- 20° E
2 I0o
1- .
0 0 0.2 0.4 0.6 0.8 b-VALUE
çb is the failure friction angle in triaxial conditions is the failure friction angle in Matsuoka-Nakai failure criterion
Figure 3.7: Effect of Intermediate Principal Stress (After Salgado (1990)) Chapter 3. Stress-Strain Model Employed 47
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 08 0 2 be”0.6 a- 2 0.4
0.2
-0.4 -0.2 0 0.2 0.4 0.6 - ESMp “YSMP
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:
function
where
hardening in
shear model,
Therefore,
The
3.9(b)
shows
3.3.4 strain
(desMp
(dEsMp/d7sMp) dicular
where
Chapter
other
Equation
Rewriting
hardening
strain for
i
the
A shows
/d7sMp)
the
to
words
of
and
and
3.
Hardening
an
rule
the
flow
a
mean
plastic
to
Stress-Strain increase
t the
i’
3.18
relationship
the
yield
rule
by
how
the are
rule
are
will
will
corresponding
principal
implies
above
an
defines
soil
shear
SMP,
constant the
and
surface
be
be
empirical
in
parameters
positive
yield
shear
equation
the negative
7sMP,
strain
Rule
that
between
Model
how
stress
and
regions 7o
7SMP
soil
stress
strain
the
?7=
the equation d7sMp
d6sMp
results
to
therefore
which
will Employed
(crm)
parameters.
yields,
-yo
which
plastic
threshold and
i the
=
state
of
which
form
+
which
7o i [—dESMP’\
\a7sMpJ
and
means
dilative
as
Cd SMP
indicates
as exp
,
is
changes
strain
desMp
the
the
implies
expressed
log 10
follows:
the
defines
of
(, \P’
(7sMp)
flow
there
hardening
The
yielding
and
A
stress increment
1+11 (--)
— versus
°mi
with
dilative
rule
contractive
parameter
/.‘J
contractive
will
the
as
is
ratio
changes
plastic is
d7sMp.
follows:
considered
be
stress
rule.
nonassociative.
vector
behaviour.
an
on
Yo
increase
behaviour.
the
strain.
state
Matsuoka
with
behaviour
is
will
SMP.
as
assumed
plastic
and
not
In
the
Figure
in
Matsuoka’s
the
defines
be
volumetric
For
For
and
hardener.
strain,
perpen
to plastic
i
(3.20)
(3.19)
3.9(a)
(3.18)
figure
(3.17) be <
>
the
48
or u, a
Chapter
Figure
3.
dEsMp
Stress-Strain 3.9: Contraction
Contraction
Flow
“ Model Rule and Ti Employed The Strain (b) (a) Dilation Increments Dilation 71>11 1 for Conical A d7SM \ (dSMp dy 5 i,jp Yield P 49 Chapter 3. Stress-Strain Model Employed 50 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). 1.0 392 2kN/m 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,. 1luU where rule stress The (equation 3.3.5 where and evaluated where obtained dependent Chapter By is development and described differentiating 3. as, Development 3.18) by ‘ T lult strain is G,, R 1 on np Pa Stress-Strain a the both similar give ------ - in failure stress plastic plastic atmospheric components initial dimensionless asymptotic stress of this normal the plastic equation procedure section. ratio ratio slope following: ratio shear shear Model stress shear of value (TSMp/JSMp) (7f/ij,zt) of pressure from number exponent 3.21, tangent The the Constitutive Employed as dy5Mp on constitutive = of = the given the i hardening G(1 KG SMP the — yield plastic plastic 7sMP = stress by (osMP) — (crsMP) Duncan Rf criterion, curve matrix rule shear shear ratio d __)2 1i Matrix and (equation strain et parameter. in hardening the al. terms increment (1980) stress 3.22) [CS] of This general rule ratio. as and follows: /7sMP and parameter the Cartesian flow the can can (3.24) (3.23) (3.22) flow rule be be 51 is strain where the principal Chapter Equation By If By Substituting direction it increments 11,12 substituting assuming assumed 3. strain and Stress-Strain 3.31 cosines increments 13 equation that that due can de are equation of the the be to = 1,: stress d7sMp a = shear a written direction directions 3.25 Model dE=+.i)d?’ desMp are = invariants 3.25 TSMp dEsMp — in desMp can are the SMP {defl in Employed equation and H- dysMp given of be same, matrix of b = d7sMp equation obtained the d7SMP as = — IL—?’ by {M1 2 } given the = = principal 3.26 notation (IL —di 1 /o- and direction ?‘) d-y. 9 by will from 3.27 0jI2 (i ‘2 the d?’ equation (I = give, stresses into d as — direction 1,2,3) the 12 cosines 313 — i following equation = 913) 3.8. and 1,2,3 of of The the TsMp desMp 3.30, equation. directions plastic coincide, are principal given (3.32) (3.31) (3.30) (3.29) (3.28) of (3.27) (3.26) (3.25) then the by 52 m, where matrix cipal where Chapter m l, , From Substitution Equation The l,, strain and M1 and equation: and general 3. equation m n = increment l Stress-Strain 3.33 - - - Cartesian of direction direction direction can 3.11 equation d 7 de 8 dE d 8 be vector the z written + Model strain cosines cosines cosines stress 3.32 by {de 8 } {de 8 } the increments Employed into in 2l,l 2l1 2l7,l, 12 12 l ratio V z of of of = matrix transformation = 02 03 o equation = [MT] on /1112 m 2 m 2 2mm 2 m 2m 2 m 3 2mm to to to [MT] z the the the the form 91 can — {M1} SMP, x, {dc} x, x, 913 3.34 be as n n 2 2nn 2 2nn y y y z obtained and and and di 1 matrix, yields is z z z given axes axes axes as dc d€ 8 by by given multiplying by the following the (3.36) (3.34) (3.35) (3.33) prin 53 Chapter 3. Stress-Strain Model Employed 54 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 :7di , I 77 I d {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) — IiI(t717y — r)2 do-i — 2 18iiI dr —2IlI3r — 2IlI2(rr — dr 2—IlI3T — (rr2IiI — or) = 2 T{do}{M2} (3.37) where superscript T denote the transpose of the matrix. Substituting equation 3.37 in equation 3.34 gives }8{d6 = [MT] {M1} 2}T{M {do} (3.38) This can be further written as }8{d6 = ]8[C {dcr} (3.39) where ]8[C is the plastic shear constitutive matrix and will be given by ]8[C = [MT] {M1} T{M2} (3.40) hardening cap The produced. both model strains can conditions relationship strains for ditions, strains concepts loading. considered stresses vious However, The 3.4.1 3.4 Chapter In As An a be at stress-strain plastic will stress order section is the explained additional will calculated can Plastic are it causes In based 3. of the Background is occur functions in Figure open to be be that produced the path to stress-strain for difficult Stress-Strain an circumvent laboratory is associated obtained model some on predicting cap-type end model, isotropic not simultaneously. in having relation yield using 3.11 Collapse the section for to of capable plastic by the isotropic shows surface the separate the the Hooke’s by with experiments theory constant yielding this the compression plastic Model for the subtracting conical yield cap-type of 3.2, deformation. the of deficiency increase the the plastic which Strain the law predicting compression with the it surfaces Employed typical collapse given Therefore, shear plastic stress is yield plastic are forms yield Model reasonable the collapse show test. in the stresses. by as also by ratio, surface results are collapse mean conical behaviour, are explained Lade a elastic the that shear The shown cap the tests constant Cap-Type explained strains only behaviour normal for proportional is (1977). elastic However, to on yield development and strains strain where used. in loading, assume the elastic in a was plastic figure stress surfaces this yield strains earlier stress in is The no from developed of under the strains section. formulated that Yielding unloading ratio 3.11. plastic collapse criterion soil loading yield conical and following of which the described general the under lines the Then, will total the criterion shear The by plastic with strains are yield cap-type which and and following be following plastic proportional loading stress-strain strains. subsections. the recoverable in strains increasing predicted. therefore, reloading surface the collapse collapse because and forms shear yield con pre the the the are 55 is a Chapter 3. Stress-Strain Model Employed 56 E C ‘1, w I C,, 0 0 C’) VOLUMETRIC STRAIN, eq,,(‘‘ Figure 3.11: Isotropic Compression Test on Loose Sacramento River Sand (after Lade, 1977) Chapter 3. Stress-Strain Model Employed 57 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 YieldSurface Plastic Collapse Strain Increment /ector Spherical Yield Cap Iasti Regior Conical YieldSurface 03 Figure 3.12: Conical and Cap Yield Surfaces on the o — 03 Plane model, work collapse strain, The where collapse 3.4.4 This the this directions. yield Under expands, there the does 3.4.3 by Chapter these hydrostatic o- 1 conditioi-i hardening implies not surface. (We) are isotropic — defining Lade strains strain 3. 03 two soil result no Hardening Flow and is Therefore, plane. Stress-Strain bounds the (1977) work yield the the and the how rule and compression, in axis flow Rule proportionality plastic hardens eventual can surfaces. yield The developed given gives the on pointing rule the be the yield elastic function. Rule potential is by Model a determined direction and cap failure. relationship associative an As function outwards an region collapse yield isotropic f Employed empirical constant de = The increases function The of J surface at from = changes strain {}T and strains from plastic between soil failure any which relationship the will must and shows beyond 8 o.ij the particular increment {dE} are with collapse hardening is be yielding origin gives the be produced. entirely equal given plastic its identical yield the work stress (see between vector current strains by according rule. controlled magnitude function strain. figure the It is state to should should a in value, following the the function all For 3.12). to will and plastic yield three by coincide be equation of the the be the the the noted equation: To of cap yield function. principal bounded collapse conical plastic plastic plastic satisfy (3.43) (3.42) yield with that 3.41 cap 58 gives can The that 3.4.5 can where strain by collapse Chapter From Since Substitution The The be be constitutive increment C developed expressed proportionality relationship exponent 3. the equations and Development Stress-Strain yield p are of matrix can as as respectively. function equation 3.46 dimensionless between described be constant relating and obtained Model f 3.42 3.47, of the is below. dW a the LSX Constitutive Employed into plastic constants homogeneous as plastic = = follows. which = equation Substitution can {}T CPa collapse = be = gives collapse dWC and ()P The 2f {dec} given 3.45 function called the work increment of Matrix strains gives as magnitude equation and the of the collapse and degree in [CC] yield 3.48 the of plastic the 2, stress modulus in function it plastic equation collapse can increments be collapse and is shown (3.47) (3.48) (3.46) (3.45) (3.44) given work 3.42 the 59 Chapter 3. Stress-Strain Model Employed 60 c dW af ) Jc O3 (. By differentiating equation 3.43, dW can be obtained as ()121 = C p a 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 = 4r dr (3.52) 4r dr 4Tz dr By combining equations 3.49, 3.51 and 3.52 the following equation can be obtained: A 8f de = 8f (3.53) 2f —dokj8kl In terms of Cartesian components of stress and strain the above equation can be written as Chapter 3. Stress-Strain Model Employed 61 oo- 2or 2OTzm do d 2or 222or do dE , = o r22o- 2o-r r22u do-i d79 f 4r dT d-y Symmetry 24r r24r dr d7 24r ,3dr In short matrix notation the constitutive matrix for the plastic collapse strain can be written as {Cc]= T{8fc}{afc} 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 2 1 1 0 0 0 do (3.57) d- 2(1H-v) 0 0 dr d72 Symmetry 2(1 + v) 0 2dr d 2(1 + v) 2dr well can the In Case sponding components separated 3.6 where, of and where Chapter the a Case E loading be as bulk stress-strain and I E previous classified Development in I 3. is full moduli mean is B indicates kE the condition, of n that n are Stress-Strain elasto-plastic strain. tangential - - - - sections, stress. assumed into as bulk bulk Young’s Young’s it curve. is a four loading easy the One modulus modulus In i-’ to the cases modulus modulus Model relevant Young’s of this constitutive is of to be the constitutive the Full model condition stress case, which exponent number B E Employed Poison major v= modulus strain exponent = number = Elasto-Plastic all dependent the kE are matrix three; where advantages (i_&) ratio Pa different Pa components shown matrix obtained () ()fl which can the there and in is loading be plastic figure formed of given can is formed. from can having an Constitutive be 3.13 shear, be by conditions. increase the individually calculated the included The the on unload-reload plastic following the strain loading in i and stress Depending from collapse — for components conditions equations: o Matrix the different Young’s ratio portion plane. corre (3.60) (3.59) (3.58) and on 62 as Chapter 3. Stress-Strain Model Employed 63 Failure Surface a1 Hydrostatic Axis Ill Conical YieldSurface lastc’ Ragion • 7 Failure Surface -.7. a3 Figure 3.13: Possible Loading Conditions Chapter 3. Stress-Strain Model Employed 64 elastic strains will be present. Then, the full elasto-plastic constitutive matrix will be given by [C] ]8[[C + [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] = ]8[[C + [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) Chapter 3. Stress-Strain Model Employed 65 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 dc C11 C12 C13 C14 C15 C16 dr C21 C22 C23 C24 C25 C26 do = C31 C32 C33 C34 C35 C36 do (3.65) C41 C42 C43 C44 C45 C46 2d’y C51 C52 C53 C54 C55 C56 2dr C61 C62 C63 C64 C65 C66 drza, where 3C are the components of the constitutive matrix. 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 de C11 C12 C13 C14 do dc — C21 C22 C23 C24 da (3.66) d6 C31 C32 C33 C34 do d7 C41 C42 C43 C44 dr Now, by imposing the plane strain boundary condition that = 0, do can be ______ Chapter 3. Stress-Strain Model Employed 66 written as 2do = — + do-!, + dT) (3.67) Substitution of equation 3.67 in equation 3.66 yields: de 1C 2C’ 3C’ do de = 1C; 2C; 3C; do-u (3.68) d 1C; 2C; 3C d where — (V — ri C1331 — L112r . 1334 — ‘-‘11 LI12 LI4 ‘-‘11 ‘ C — — C33 ‘—‘13 — — — 31 . — (1 32C,C . — ,- — ‘-‘21 C ‘ ‘—‘21 ‘-‘22 — ‘-‘22 ‘ — ‘—‘24 C 2 C33 — C33 ‘-‘23 — 23 3c, f_I,, — f_I — f_I . 4331 . 4332 f_I — f_I 4334 — ‘-‘41 C ‘ — L142 C ‘ ‘—‘32 C — C33 — 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. 8de C21 C22 C23 C64 8do- (3.69) 2de C31 C32 C33 C64 do-i C61 C62 C63 C64 drrz stress-strain that where temperature, coefficient equation where Inclusion terms which 5.8. to approach The 3.8 Chapter include Rearranging By The effects compressive in {dee}T have [C] multiplying Inclusion incremental 3. the 3.70 of used of these is of temperature to matrix Stress-Strain the stress-strain the the will temperature by be the effects {a sand strains sand elasto-plastic Srithar become made equation terms [D] stress-strain d, of grains matrix in the a 8 are Temperature effects in and the relation Model will changes d6, following the [D]{dc} assumed and 3.71 {do} analytical will {d} Byrne give a 8 in constitutive stress-strain d6 Employed relation d6, expand by the {de} and = = in is = (1991) equation 0, the positive. [D]{dE} [C]{do} flow-continuity oil the in {dcr} = 0, procedures sand can the 0} inverse [C]{do} and change is Effects — matrix. and relation flow-continuity be + followed — there and can [D]{de 8 } {do 8 } {de 8 } written in a 8 of the be were temperature. will is [C] equation If obtained: are here. works the be which there as described explained linear additional This by equation. is is is previous described an thermal involves It referred in should increase in chapter strains. The this researchers expansion additional to in be changes section. section 2. as in (3.73) (3.72) Then, (3.71) (3.70) noted The the the 67 is hydrostatic an two principle or lead and the to respect softening (1986) a the strength. property reached tural Laboratory cracks. softening where 3.9 due Chapter material modelled numerical stress A average scope parts; structure Desai to to comprehensive changes {do- 8 } loss change and Modelling to Frantziskonis which the 3. relieved of is or of When a (1987). the will of component non-softening of Frantziskonis quantitatively tests assumed this soil loss Stress-Strain load instabilities = uniqueness composed in solution in these is no [D] the when thesis. of the temperature. commonly behaviour shedding on They longer strength {de 8 }, stresses review two oil material as and of it of is modelled Reviews a sand of in as be is boundary behaviour (see assumed Model Desai Strain (1986). true using concept which micro-cracks the treated shown of with represented referred under and show figure such strain This material strain (1987) on the Employed strains the zero is by to progressive is In as Softening this and of a as the term as ‘load softening 3.14). similar strain be softening Valanis this decrease a stiffness. initiation, a stated strain continuum initial additional subject property, deviate by the and continuum. will shedding’ study, softening continuum to In same joints give (1985). that softening. straining value is the in part finding can from The not propagation by the various strength for the model term (topical strain that of be problems. or true behaviour attempted homogeneity, both It Load strain induced the material found ‘stress the occurs in is result The softening presented behaviour anomalies stress-strain behaviour) rather the parts after average softening in phenomenon and Shedding transfer’ thermal because These stress-strain in by properties. here Read a and an a separating closure is peak the by is performance may behaviour, as not the anomalies overall and and Frantziskonis response estimated phenomenon behaviour stresses. of it concept. strength a arise deviatoric the is Hegemier a of of material If damage relation beyond loss micro it strain struc strain with into and can the In 68 as of of of is Chapter Figure Stress Shear 3. 3.14: Stress-Strain Ultimate Modelling Model of Strain Employed Softening by Frantziskonis — Topical Average Behaviour Behaviour and Desai Strain (1987) 69 Chapter 3. Stress-Strain Model Employed 70 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 0P in the figure can move to point Chapter 3. ‘rip Figure ‘1] Stress-Strain 3.15: Modelling Model Employed of Strain 71 Softening F? P 1 by P 2 Load I T Shedding 7r 7 71 Chapter 3. Stress-Strain Model Employed 72 1P 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 .2P Then again the stress state will be brought to point a2F 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 growth and stiffness memory circumvent will Another well 4, range hyperbolic those Chapter which One result memory defined. that of in matrix of disadvantage 3. and values may computer in involve the model, the Stress-Strain a requirements time. be However, disadvantages non-symmetric and requirement for helpful information significant requires some capabilities. However, of a the to Model of sensitivity may only understand the model for principal of on stiffness the this not Employed large parameters a the small frontal is model be model that study stress memory matrix considered the memory. because solution is parameters physical on rotations. and its which since the limited their Furthermore, of as scheme parameters requires significance it the disadvantages physical is does use nonassociated very used in considerable not limited. the these significance is of in assemble given past. the this with factors The parameters. flow in Unlike study the computer possible the chapter are of rule, rapid time will full not the 73 it oil In detail. results 4.2 results on significance to For as softening. can the measured This 4.1 Stress-Strain this obtain isotropic sand some The the stress-strain be chapter to section, on Applications Evaluation Introduction determination classified procedures can of obtain a responses Ottawa A straight the and compression be summary describes only found parameters these a model into better sand used line the in of in of four are the of laboratory these of and and Model to procedures fit. some section fit. and the described procedures evaluate Parameters groups; have presents on In Validations procedures parameters triaxial of those oil 4.3. the been Validation tests. sand Chapter elastic, these for in - parameters results cases, compression used Parameter carried section the are of to 74 The and parameters the to evaluation plastic it actual given verifying their soil is evaluate 4.4. out stress-strain 4 at advisable in parameters least test tests to shear, description section from the provide of the data two are the stress-strain plastic Evaluation to basic soil model test described 4.3. on parameters have required some parameters are laboratory Ottawa results Sensitivity collapse against three given idea in model for are sand or in are about section laboratory and needed the tests more necessary table analyses given and against and model strain their such test 4.2. 4.1. for on in Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 75 Table 4.1: Summary of Soil Parameters Type Parameter Description L Elastic kE Young’s modulus number n Young’s modulus exponent kB Bulk modulus number m Bulk modulus exponent Plastic Shear Failure stress ratio at one atmosphere Li Decrease in failure stress ratio for 10 fold increase in 0SMP .\ Flow rule slope i Flow rule intercept KG Plastic shear number np Plastic shear exponent 1R Failure ratio Plastic Collapse C Collapse modulus number p Collapse modulus exponent Strain Softening Strain softening constant q 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 76 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) Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 77 3q-a / AE 1 € (a) Unload-Reload Modulus (E/Pa) 1000 100 ‘ 1 10 [log scale] (ojP). a (b) Variation of E with a3 Figure 4.1: Evaluation of kE and n Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 78 kB = a.b(Pa) 0.01 - 6 a m=1-b 100 [log scale] Figure 4.2: Evaluation of kB and m Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 79 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 (4.8) = ab(Pa)’ It should be noted that the parameters kE and kB can be related by the Poisson’s ratio v as kB (4.9) = 3(1—2zi) 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 80 f = 3o (4.11) deWc=Jcr (4.12) where d€,, — and is the3 elastic volumetric de = d de 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 where strains principal shear be and principal stresses can Chapter 3. 2. obtained Under Firstly, The The be therefore, Flow Hardening parameters can evaluated most plastic and 4. plastic stresses standard the rule be as Stress-Strain strains, common given elastic special explained shear parameters shear rule from as and triaxial by described the parameters parameters and laboratory attention strains: principal those stresses in Model plastic i the compression test and in d€ d following d strains is - can KG, section collapse results. shear and de ) Parameter given = = — — de = be = de 3 de 1 (o np strains 2A 2C tests = can determined 3.3. here — conditions, — P\ and subsections. + strains —vde (p u 1 o 3 do 1 o de de 2o)2P be performed a) Evaluation The on R 1 1—2p to do 1 — — obtained. do the have d describe plastic from the spatial to are be elastic and all shear how By triaxial subtracted types mobilized Validation knowing to parameters and obtain of compression plastic tests plane to the the obtain where principal can collapse (SMP) plastic (4.18) (4.17) (4.15) (4.16) (4.14) (4.13) tests then the the 81 The 4.2.3.2 plotting shown these 4.2.3.1 At follows: compression and Chapter least The It By flow unloaded, parameters. should in following values two {(OSMp)f/Pal 4. rule figure tests Evaluation Evaluation Stress-Strain loading, be for of then 4.4. the noted up the The the equations to and plastic the versus failure that failure collapse (OsMP)f d7sMp stresses SMP TSMp Model if dEsMp ofi of shear = at the stress in q’ different on strains section = - SMP and can test is — Parameter and — — and a ratio Li expressed 2(deW semi-log i f—c1EsMp dc/ \ be the samples = u7SMP should 3.3.1 3 X obtained log 10 confining /2o z on strains •1 3o- 1 o 3 2o 1 +c 3 SMP + and Evaluation + plot, by are not — (o-sMP)f J + 2d4/ J+/L 03 related de/j imposing o the using Pa stresses is preconsolidated be i given following subtracted. and equations to and are ii by the SMP Validation necessary can conditions equation. can 4.20 be to be a determined and to higher obtained for determine 4.19. triaxial (4.24) (4.23) (4.22) (4.19) (4.21) (4.20) stress By 82 as as ______ Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 83 ‘if — 1 10 100 [log scale] e’ a 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 7SMP 17= (4.25) G. + The parameters KG, np and 1R 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 84 ‘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 7SMP 4 26 1 7u1t Now, by plotting (7sMp/7/) against fsMP the values of ,7G 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. Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 85 “7 1 7SMP (a) Hardening Rule 7SMP ‘1 1G 7SMP (b) Hardening Rule on Transformed Plot Figure 4.6: Evaluation of G and ij Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 86 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) 1 0 c-I np (I) 0 U 100 KQ 1 10 100 MP’a [log scale] Figure 4.7: Evaluation of 0K and np 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. results The 4.3 against where be from incremental constant can the 3.8) Chapter shown Then, Taking By The The flow be stress-strain the rearranging Validation on obtained {ln(7sMp value strain 4. volume. rule strain that the Ottawa is natural plastic Stress-Strain the intercept. parameters of softening hardening initial from the model — The ln sand logarithm the 7SMP,p volumetric residual [in = equation value of tangent terms ir employed and in (ij] region This Model the , relation )} + () of and oil (ip of as stress in assumption 7SMP,p the strain plastic 4.23. equation of Stress-Strain shown sand.The — equation - q in Parameter lir) a (equation peak can ratio stress-strain this will The exp{—i(7sMp = ln in shear be K(7sMp stress G1 4.31 is figure study triaxial 4.29 be peak + is determined i parameter 4.25) qln(7sMp zero, Evaluation 773 reasonable will ratio, — is and 4.8. has shear R 11 assumed curve test give as which Model taking been — which strain results 7SMP,p by and — can because, and verified implies 7sMp,p) to plotting natural is Rf be 7SMp be )} the Validation reported given is equal failure a against the when logarithm, state {ln can as failure to by [ln be stress i (see of laboratory t ()] Neguessy = obtained which shear ratio. section p, it (4.32) (4.31) (4.30) (4.29) ratio, can the 87 at is } Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 88 in [in (TZr)] q in(7sMp — 7SMP,p) 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. Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 89 MEDIUM SAND ‘I zp ae 4 48 100 140 200 I00 ‘ I I I 80 I I 60 LEGE ND C t4Q X FRESH • RECYCLED 20 I ASTM - C - 109- 69 BAND * MIT CLASSIFICATION 0 I 0.5 0.1 0.01 Diameter (mm) Figure 4.9: Grain Size Distribution Curve for Ottawa Sand (after Neguessy , 1985) Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 90 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 = o13/o = 1.67 and 2) 5. Tests along four different stress paths as shown in figure 4.10 SP4 300 SP3 a. 200 SP1- = 2.0 SP2 SP2- spi (a/u=4.0 100 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. for by unload-reload Young’s about shown dense column stresses As 4.3.1.1 Chapter Duncan primary explained 10000 2000 3000 5000 1000 500 sands 2.2 in tests are modulus 4. 0.3 and the Figure et loading plotted to Parameters Stress-Strain which figure the value al. in about - 4.11: exponent section for (1980) from in yield 0.5 are agrees the I 3 figure Variation for one primary that standard 4.2.1, similar Model loose for well unload-reload 4.11. for the both the with sands. of values - loading ratio The Parameter Ottawa triaxial 1 Young’s conditions Young’s the values From of as (kE)p= condition resonant : Young’s compression are modulus moduli modulus a 3 figure Evaluation Sand plotted is obtained 1180 0.46. a 2 — - - moduli column to with 4.11 and in This the values tests. in the the the confining and 3 unload-reload I A . unload-reload varies •Triaxlal agrees values. Tria)_(UnIoad figure values Resonant Young’ It Validation for can from with different stresses are (Primary for The be modulus 5 I Column from the kE about seen ratio condition tests. and statement Reload) Loading) confining resonant that 1.2 - values of n Also can the the for 91 is 10 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 92 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. 0.01 We/Pa 0.005 0.002 0.001 0.0005 0.0002 0.0001 5E-05 0.2 0.5 1 2 5 10 20 50 100 2 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 93 stress ratio i vs usMp for the triaxial compression test results are plotted in figure 4.13. The failure parameters () and 7ZSi are determined as 0.49 and 0.0. 0.6 Ti —— —71iO49 0.5 ö: : 0.4 zS=O.O 0.3 7 0.2 0.1 0 0.5 1 2 3 5 10 cTSMP/P 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 1R is determined as 0.93. From figure 4.15(c), the values of KG and np are obtained as 780 and —0.238respectively. Table 4.2 summarizes all the parameters for Ottawa sand at Dr = 50%. Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 94 0.6 ‘7 0.5 0.4 03 0.2 0.1 0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 _(dEsMp/d7sMP) Figure 4.14: Flow Rule Parameters for Ottawa Sand Table 4.2: Soil Parameters for Ottawa Sand at D = 50% Elastic kE 2600 n 0.46 kB 1444 m 0.46 Plastic Shear 0.49 1117 0.0 X 0.85 u 0.26 780 np -0.238 Rf 0.92 Plastic Collapse C 0.00021 p 0.89 ______ Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 95 (a) 0.5 0.4 o3=5OkPa 0.3 ---0--- ,‘ 3=150kPa 02 — • a_3=50kPa 0.1 I a3=45OkPa --.“---.- 0 0 0.2 0.4 0.6 0.8 7SMP 1.8 7SMP 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 0.8 7SMP 1G 800 750 .W 735 %% ‘UP 700 650 600 Jlr=0.145 550 I I 500 I 0.5 1 2 3 5 10 Figure 4.15: Hardening Rule Parameters for Ottawa Sand Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 96 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 = o13/o = 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 97 800 a----*--3 = 50 kPa a_3• = 5O kPa o 3 = 250 kPa 600 - _.. ------a_3 =50 kPa Symbols - Experimental 0 Lines - Analytical 400- — .0 0 0 200 - 0 (a) I I 0.05 0. > ‘U 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 E(%) a Figure 4.16: Results for Triaxial Compression on Ottawa Sand Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 98 500 400 300 0 200 100 600 500 400 0 a- 200 1.67 100 Symbols - ExperIment Lines - AnaIytca1 0 I_ —— I I 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 4.17: Results for Proportional Loading on Ottawa Sand Chapter iLl > 0.8 200 300 100 0.6 400 0.4 500 0.2 600 4. 0 Figure Stress-Strain 4.18: 0.2 Results Model for - Parameter Various 0.4 a Stress Evaluation Paths 0.6 on and Ottawa Validation 0.8 Sand 1 99 Figure 1987) Chapter E = > 4.19: 4. 100 20 40 60 80 0 10.0 Stress-Strain Grain .--—— .:::ZE • I Size Sands: Other UTF I I Distribution 0.1 Sand I Model 8 McMurray I I -fine - - I 12 medium coarse F - Parameter 18 1.0 F 111111 for 25 Fl U.S. 35 milhimetet Athabasca -—— inches . mesh 45 Evaluation I F 0.01 60 I I Oil 80 F I 120 Sands, F I 1 and 4 0.1 170 F 230 Validation (after F - 325400 I — 0.001 Edmunds I — -__ et 0.01 100 al., pression obtaining test section The 4.3.2.1 considered. the of Chapter 0m 0m 0.1 Isotropic Triaxial Triaxial Triaxial Test the 4. 3. 2. 1. Figure Const. It field, Const. Const. relevant and i U 0 m Isotropic Standard should relevant Comp. Comp. Comp. 4.3.1.1, Comp. test constant Comp. three 4. they Ext. Comp. constant constant the 4.20 Parameters Stress-Strain 3 2 1 parameters and be were parameters standard they model compression shows triaxial ID UFTOS12 UFTOS9 UFTOS1O UFTOS3 UFTOS1 UFTOS1 noted UFTOS4 Sample the compression extension compression not are elastic parameters the that Table identical. compression not triaxial for Model are (kg/rn 3 ) Density data Bulk 1980 2070 1960 2060 2120 1990 1990 since test repeated the parameters 4.3: for discussed for oil compression - Table the Details and Oil Water Parameter sand 8.5 8.3 8.3 6.6 6.4 the 7.0 7.8 tests samples here. the 4.3 Sand unload-reload in kB are Bitumen of Fraction validation: summarizes detail 9.5 8.8 6.5 6.6 7.3 7.6 7.6 and the obtained Evaluation test tested Test by m in Solids 83.4 83.4 84.9 84.1 86.1 84.1 87.1 results. Weight are section were Samples from portion the determined (%) (< and undisturbed 0.074rnrn) details Fines Since an 41.2 37.7 41.2 54.0 57.3 71.9 52.9 4.2 Validation isotropic of and the the of as the Ratio again Void 0.60 0.52 0.60 0.62 0.50 0.60 0.45 procedures isotropic 1670 samples compression test Disturbance briefly and samples Index 10.9 12.1 12.1 10.8 10.0 (%) 9.6 6.4 com from 0.36 101 for in Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 102 0.03 EV 0.01 0.003 0.001 0.0003 0.0001 3E-05 1 10 100 1000 10000 100000 am (kPa) Figure 4.20: Determination of kB and m for Oil Sand Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 103 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). 1 / a 0.3 0.1 0.03 0.01 0.003 0.001 0.0003 1 10 100 1000 10000 100000 Figure 4.21: Plastic Collapse Parameters for Oil Sand 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 Chapter 1f 0.55 0.65 0.75 0.5 06 0.7 0.8 4. I Stress-Strain Figure 2 Model 3 4.22: Failure - 5 Parameter Mp’a Parameters 10 Evaluation for 20 Oil and Sand 30 Validation 50 100 104 Chapter 2000 1000 200 500 100 50 4. 1 Stress-Strain - - - Figure 4.23: 2 Model Determination 3 1300 - Parameter 5 of K 0 Evaluation 10 MP’a = 0 and -0.66 np 20 and for Oil Validation 30 0 Sand 50 0 I 100 105 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 106 0.8 11 (a) 0.7 o.9 0.6 A 0.5 0 -- 0.4 ./- .------ 0.3 o a_3=1.OMPa 0.2 7, o EJ _3 = 2.5 MPa 0.1 c,_3= 4.0 MPa 0— -0.4 -0.2 0 0.2 0.4 0.6 0.8 —(dEsMp/d7sMP) 0.8 0.7 0,6 0.5 0.4 0.3 0.2 0.1 0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 —(dEsMp/d7sMP) Figure 4.24: Flow Rule Parameters for Oil Sand Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 107 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 3000 n 0.36 kB 1670 m 0.36 Plastic Shear 0.75 iii 0.13 \ 0.53 ii 0.31 KG 1300 rip -0.66 1R 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 Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 108 14000 12000 - 0 10000 - 8000 - Loading Unloading b 6000 - 4000- 0 2000 o Line - Predicted Symbols - Measured 0 I 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 volumetric contraction expansion The particular compressibility. results pression and tained In sensitivity predicted 4.4 are tests. shearing to compression predictions the value Chapter will order note The The Results overall shown the parameter of result The for are Sensitivity that plastic parameters results loading ). to with 4. results parameter Ottawa for shown analyses expansion. in behaviour value provide and and for As in the Stress-Strain and constant a the the dilation collapse ). three in observations dilation change value The condition are is of constant in sand terms insert line the a on .t KG figures in higher was better different will of is, The Analyses volume. the becomes parameters at good slope were in (similar the of of and in studied Model be lower with om stress parameter parameters 4.28 deviator the fact, the understanding flow would chosen agreement. of more np extension stress the the stress values, and figure. steeper, to - rule ratio. define an by Parameter contractive. ç 5 initial C agree flow as of stress indication 4.29 changing paths; parameter and 1 u ratio. in the have the the is A the rule It are there well general confining p the base steeper and about can higher are initial shown constant been and Evaluation Parameters for amount only will volumetric of If parameters essentially be i o soil it the a ultimate carried is slope the be slope seen that stress defines in higher not mechanics). significance less o, 4.OMPa. figure of predicted parameter. that much (or and of stress compression, of out. volumetric value strain an the stress and the 500 higher 4.27. Validation indication the different change ratio The the hardening It kPa of volumetric are of ratio A experimental is ) The A) significance the smaller A parameters which was also analyzed. is expansion in will triaxial or constant for selected, stress parameters, of volumetric considered interesting a the isotropic modulus give separate value state strains. paths three com The of and less and ob 109 the °m of of a Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 110 10000 - 8000 - 6000 - 0 4000 MPa -: a_3 =.5 MPa : a_3=,MPa 2000 - Symbols - Experimental Lines - Analytical 0 -0.8 -0.6 -0.4 > w -0.2 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 e(%) a Figure 4.26: Results for Triaxial Compression Tests on Oil Sand ______ Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 111 5,000 — 6 -4 4,000 - I 2 / SP1-I.1 Const.Comp D. / SP 2- a_v Const Comp. C-.. / SP3-i1 Conet Ext. C’-, 246 a_r (MPa) 3,000 - 000 a I1 0/or & 0/ 2,000 o/ spi C ‘a 1,000 SP3 - Experimental Symbols Lines - Analytical 0 -1.4 -1.2 —1 > WI -0.8 -0.6 -0.4 -0.2 0 gD Q9OO5/O z . -ci - o 0.2 o 0.4 -6 -4 -2 0 2 4 6 - E_r (%) Figure 4.27: Results for Tests with Various Stress Paths on Oil Sand Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 112 c_a (%) c_a (%) (a) Effect of Parameter C (b) Effect of Parameter p I 0 0 & > WI 0.4 0.6 0.8 1 Ea (%) c_a (%) (C) Effect of Parameter A (d) Effect of Parameter L Figure 4.28: Sensitivity of Parameters C,p,) and i Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 113 €_a(%) E_a (%) (a) Effect of Parameter KGp (b) Effect of Parameter rip 0 & 1.2 Ea (%) c_a (%) (c) Effect of Parameter Rf (d) Effect of Parameter flf Figure 4.29: Sensitivity of Parameters KG, np, 1R and i model physical tions better From paths conventional the strain A 4.5 used stress volumetric stress G,. Chapter double The The validation A and agree the have response. ratio behaviour captures explain Summary higher model elastic meanings 4. validations their hardening strain been very in Stress-Strain isotropic value of the their parameters significance parameters the compared well the of response. hardening and stress-strain oil of physical presented proposed elasto-plastic and and a sands. sensitivity Model the are will with is triaxial The are rule. well significance. model relatively model Procedures in result not parameter the behaviour - this understood. Parameter model Lower compression study considered model have in predicts chapter, a easy R 1 stiffer has has been predictions. for Laboratory R 1 of Evaluation oil to and been the and the been it here presented deviator obtain sands can test higher shear i evaluation carried postulated because be define results. test very Measured and concluded induced and i stress in out results the well. can will this they of Validation The on response shape to be the give dilation chapter. results the that have model for parameters determined parameters stiffer and parameters various the been and and the effectively. the proposed deviator predic a widely failure stress- stress lower from have and 114 to 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 where, as permeability ficient as petroleum space. potential of In particular rived in problem be to 5.2 Chapter shown flow include the this geomechanics combined Now, first. of product Mathematically section, Derivation of as consider 5. in proportionality gradient one phase. the reservoir Later, krt explained P 1 I’i figure Flow k v with (k), effects phase the of - - - - - but relative viscosity permeability velocity pressure When it a of the the Continuity 5.1. acting flow single is situations, component of the the later force expanded relative this of different continuity the medium is permeability permeability on vector phase in of Governing in equilibrium is the fluid it phase phase Equation expressed chapter permeability matrix and the Darcy’s to (denoted phase depends (in is to equation three 1 volumetric Newtonian 1 inversely m/s) (in (in flow of 6. components. of and equation permeability. kPa) as phase kPa.s) the phase on by when Flow compressibility for of porous 1) the proportional phase flow flux and flow a 1 VP 1 a and saturation single (non Equation single the in of in The medium will 1 three a one dimensional) This (krj), flow phase phase flow be fluid to dimension terms is dimensions. is and solved and (in continuity its in slow, customarily is entirely one m 2 ) the viscosity. proportional the have as as dimension mobility absolute (in a it been fills consolidation equation The usually z direction) expressed The the changed amount of to Darcy is (5.1) coef pore is that will the 116 de in Chapter dz n 5. Figure dz Flow S 1 ndz 5.1: Continuity S 1 One n - - - - dimensional velocity unit porosity saturation Equation weight of of phase of flow I phase phase of 1 in a 1 I z single 0 (vj direction ôz — phase 71) Solids Pore Phase dz in an fluid ‘I’ element in pore fluid 117 Chapter 5. Flow Continuity Equation 118 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 55 dt 8z (.) Rate of storage: &wlO(nSl7l)d 56 8t 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 7 8z — ôt (.) Expansion of the partial differentials in equation 5.7 gives avzl 871 8n &Y1 18S —7z—+vi—=7iSi+nSi--+n (5.8) Dividing byi--7i yields 8v 871 8m 1S 871 8S ——+----—-—=Si——+n—-—+n------21 v21 1 (5.9) 8z 71 8z 7’ at at Now, consider27 all five terms in equation 5.9 separately, starting from the left hand side. ____ Chapter 5. Flow Continuity Equation 119 0v21 8z By Darcy’s law (equation 5.1) v can be written as kkr 18P vz1 = — — ILl c9z = (5.10) and therefore, = 921P (5.11) where kmi - mobility of phase 1 k - intrinsic permeability of the porous medium [function of void ratio; k = f(e)] 1k,. - relative permeability of phase 1 [function of saturation; 1k,. = 1)]f(S - viscosity of phase 1 function of temperature and pressure; = f(8, 1F)] vz - velocity of phase 1 in z direction 1P - pressure in phase 1 2 71 8z The change in unit weight due to the increase in pressure can be expressed as, 871 = 71 (5.12) where Chapter 5. Flow Continuity Equation 120 1B - bulk modulus of phase 1 - unit weight of phase 1 Therefore, 87j — v yi 8z — 1B öz — kmi 16P 5 13 — 1B 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. 1S-- 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 written Chapter .4. . Extension By need unity. this Since as Summation By n—-— making as 5. as 1 7i S 1 using term not the Hence, b-y Flow ôt of to over equation final the equation be of Continuity when changes all kmi equation saturations considered kmiV 2 Pi the -ä—-H-S1 5.12 8 2 P combining 5.18 phases to this Equation is to + the in to of 5 1 L three term will detail. --—n- be all terms the 71 8t 51 be derived phase dimensions — can equations 87j zero. as be s 1 components explained by written Mathematically --—n--O 8P 1 B 1 8t 51 for combining — yields 8P 1 all as, 8Si the so should = far, phases, all 0 this equation the always the phases can summation be be 5.9 expressed this equal can (5.19) (5.18) (5.17) (5.16) term 121 be to of Chapter 5. Flow Continuity Equation 122 where b’P ElF 2 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; a s0 0ap kmo20PVH-5 —n —n = 0 (5.22) for gas phase; kmg29PV + 9S — — 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 relationships e 3 /(1 the The of data, space. section of fluid Evaluations term for to 5.3 sections. Recently, Chapter model the individual the Equation where, variation permeability phase derived + they phases Lambe equivalent Permeability e) 5.5. the 5. The Settari CEQ for kEQ found components of Flow three-phase 5.25 could above. of a phase equivalent and which relative wide - - k that et equivalent equivalent = is — conductivity of with Whitman Continuity be al. similar kmo components the The range in there (S 0 established permeabilities (1993) void turn in + fluid porous compressibility equivalent of kmw kEQ of the is to hydraulic compressibility ratio. (1969) depend a granular Sw which the Equation the have term. compressibility + linear V medium 2 kmg one and for P Although also collected Porous Sg is hydraulic + They and on relationship used the similar the materials. conductivity used is their viscosities (k) varition by a details considered — function considerable an mainly CEQ Vaziri to there relative conductivity Medium but the effective It between of of -- OP not was equivalent are can (1986) k depends its of the permeabilities = with in described saturation a be evaluation hydraulic 0 contributions experimental the considerable k and argued is e. and a hydraulic on hydraulic However, Srithar function a the in void that and conductivity detail are amount and (1989), ratio scatter bulk from data various conductivity. conductivity described of without in viscosities. mobilities modulus function to different the of except in study (5.25) other term next void the the 123 in Chapter 5. Flow Continuity Equation 124 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 = )9f(S Chapter 5. Flow Continuity Equation 125 I krow kr k,,, 0 Swmaz 0 SW—,’.. (a) Oil-water system I kr rg_ 0 Sgc Sgmoz I Sg 0 (b) Gas-oil system Figure 5.2: Typical two-phase relative permeability variations (after Aziz and Settari, 1979) is to Sam simultaneously which as in is permeability Chapter given flow. estimated an According Where, Two The The is oil-gas is called by When simplest function more considered 5. S, Flow system. from the of to accurate S,, is by oil Stone way residual for called is Continuity water the S 5; in here. less the Their of an two-phase (1970), models = = = the estimating than relative and oil-water oil In functional critical 15 S 95 this saturation gas. S, Equation SO the have wc permeability wc wc model, — data If the relative k,. 0 k,. 0 k,. 09 system k,. 0 or k,. 0 Som k,. 0 been S connate relative = = dependence for would am om om is at = = = Stone S k,. 0 less proposed permeability and which k,., f(S 0 ) f(SW) f(S 9 ) k,. 09 be, than permeability of water k,. 09 (1970) and oil, oil are 5 om, is by ceases k,. 09 , saturation k,. 0 , the defines given Stone of So S k,. 0 is relative where, oil of to not will by in (1970), Sam normalized water S flow at readily a be three-phase k,.OW permeability which when zero. k,., only is known will water it the saturations the displaced be relative system first and (5.33) (5.31) (5.32) (5.30) (5.29) (5.28) starts (5.27) of zero. 126 oil of it zero. in depicted should Chapter Figure figure = Aziz The The 0 and match region factors and 5.3: 5. 5.3 by SL, WATER point Flow Settari on 100% Zone the of = i3 the mobile Continuity two-phase outside and of ternary (1979) mobile give ,i3 oil are the modified phase diagram oil data Equation determined hatched for (i.e. at three-phase I,. Stone’s assuming the k,. 0 — — area, GAS 100% extreme 1 from 1 > E1S k,. 09 — — model 0) the S’ S 9 increasing the predicted flow relative points. because end Som (after conditions by 5 W permeability The Stone’s Aziz Stone’s and two and S 9 . that model model extreme 100% Settari, OIL For equation of will oil conditions I is cases 1979) will reduce shown (5.35) (5.34) 5.33 127 be of Chapter 5. Flow Continuity Equation 128 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 0S = 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 0k,. = S krocww /39 (5.37) k — (5.38) — k,.09 c’ ) ,.0cwI 11 — 9LI (. 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 permeabilities in and (1979). Chapter figure From When where, found Kokal 5.4. 5. the k,? 0 k,? 0 k,? 09 very Flow discussion in and = good - - three-phase k,?og, in in relative relative Continuity Maini an a agreement. water-oil the oil-gas so k,. 0 permeability (1990) permeability above far k,. 0 = system Equation system in = s system compared model “9k0 The a f(k,. 0 , this wko(1SI (k,? 09 S;+k,?,&S) k,. 0 k,. 9 can best section, of of reduces = = rog\ = k,. 09 , be oil oil f(S 9 ) f(S) f(5) (1S comparison model rog rOW written at at Sw, it zero connate to g can so, predictions the gas as S) be one given water saturation concluded given in against saturation their by that Aziz paper measured the and is relative Settari shown (5.46) (5.45) (5.44) (5.43) (5.40) data 129 4 Chapter 5. Flow Continuity Equation 130 OIL Expenmental — Calculated 0.75 0.70 0.60 0.50 0.40 0.30 020 0.10 0.01 . . .. WATER “ “ “ “ ‘.‘ “ ‘I’ ‘ GAS Figure 5.4: Comparison of calculated and experimental three-phase oil relative per meability (after Kokal and Maini, 1990) Chapter 5. Flow Continuity Equation 131 — )9f(S (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 = 1C(S — 2C)c3 (5.48) where ,1C 2C and 3C are constants. Figure 5.5 shows a comparison of experimental data with calculated values using the power law functions. 1.2 1.996 k = 2.769 (0.80 - Sw) 2.735 k = 1.820 (Sw - 0.20) 1 0.8 row k a) ‘ rw E a, 0.6 a)> a) 0.4 Symbols- Experimental 0.2 Lines - Correlation 0 0 0.2 0.4 0.6 0.8 1 1.2 w Figure 5.5: Comparison of calculated and experimental relative permeabilities using power law functions Chapter 5. Flow Continuity Equation 132 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: Chapter D 1 , Parameter B 1 , A 1 , C 1 , Table 5. D 2 , B 2 , A 2 , C 2 , Flow k,? 09 Som 5.1: D 3 B 3 A 3 C 3 Continuity Parameters Relative Relative Parameters Parameters Parameters Parameters Residual Connate in in in in in in water-oil oil-gas oil-gas oil-gas water-oil water-oil permeability permeability Equation or oil needed for for for for system system critical system saturation system system system variation variation variation variation for water [k. 09 [krg relative of of Description [kro = oil oil = of of of of saturation C 1 (S 9 = D 1 (D 2 krog krg krow k at at A 1 (S B 1 (B 2 permeability zero connate with with with with — — C 2 )c3] gas — Sg 59 )D 3 ] — Si,, S 9 5 w A 2 )A3] Sw)B3] water saturation calculations saturation 133 where represented to of interpolate 30°C hold finite The can 5.5.2 Cold Chapter 70°C 200 Figure where, be viscosity for element the Lake obtained for in a 5. viscosity Viscosity particular 5.6 temperature oil. user B 0 and by F lfl(9,p) Flow d a 8 b of program shows the - - - specified = = = The Wabasca water from pressure viscosity temperature Continuity log 0.0066940.b 0.0047424.b —0.0015646.b of following the bitumen. viscosity-temperature water .Lt(3o,o) the does CONOIL. 2.3026 of causes comparison viscosity-temperature international bitumens. in of Water is not + equation: MFa oil Equation 0.8 3.0020 in + + a ( change + in 3.5364 0.0081709 reduction degrees However mPa.s + 0.0061814 Fa.s gauge of 30315) 8-30 The = this critical as at (b+8) Celsius and there drastically above in 30°C data correlation — viscosity at 3.0020] data, tables. is for 100°C, and correlation an water in option 1 as The + by with atmosphere case it that B 0 a is are viscosity this in factor experimental F 0.28 is of CONOIL well exp(d implemented oil. correlation mFa.s. of (0 For established of 3 6) gauge) water as instance, to results compared A read does change is in (5.50) (5.49) well and and not the 134 for at Chapter 5. Flow Continuity Equation 135 50000 — empirical equation - * experimental 10000 - ‘‘500o * S C 0 S I 1000 T Y 500 m P a 100 S 50. * 10 0 20 40 50 80 100 120 TEEATUR.E, C a) Wabasca bitumen 50000 - — empirical equation \ * experimental 10000 V I 5000 - S C o 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) Chapter 5. Flow Continuity Equation 136 - 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 law above where at equation the fluid, conditions, Boyle’s compressibility depend Chapter constant Gas Under where, The gas (Sisler it the K laws. law will can slightly bulk 5. 5.52 is solution. undrained et Wg P 9 a V, be temperature, and R be T Flow constant. The moduli can al., more present is - - - - - on Henry’s weight volume universal absolute absolute the 1953); be basic Continuity Mathematically, pressure. compressible. written conditions, comprssibility of in gas of law. the the of is both pressure temperature gas gas laws gas directly CEQ water weight as Equation According The constant the governing =(+±) the P 9 V=w 9 RT The important of and dissolved proportional this of of weight gas compressibility gas gas. oil to can the dissolved Boyle’s can If be of and parameter volume there gas written be to free law, assumed the does is in of and states. more a under absolute as gas that not fixed pressure can gas constant, change According affects constant quantity be present pressure determined relationships the and though temperature of in to equivalent of therefore, a the Henry’s the liquid, (5.53) (5.52) (5.51) using pore they gas 137 are where lund, can where F, and quantity where Chapter when By Rearranging Since In be the 1976) other V 19 differentiating combined. superscripts the 5. of the is liquid V 0 H words, Flow the volume volume - - - the volume volume over Henry’s weight is Then Continuity Henry’s terms constant is 0 of equation a and measured dissolved wide application of of of constant, yields, free dissolved 1 oil law V 19 refer range Equation at 5.57, gas. — — implies a at gas to constant rgkVdg ID1IT? which of of F. Vd 9 gas the is pressure, Boyle’s 0 Thus constant, = that 0 initial 9 is + H temperature temperature in Vf V 0 the 0 law is and — free volume also to final Vdg the and pressure and entire conditions, dependent of dissolved dissolved at dependent volume a confining gas and, respectively. gas yields components in pressure a (Fred (5.56) (5.55) (5.54) fixed 138 beginning ginning Chapter Generally, By where, Now, By substituting adopting of 5. of the S 9 P n Flow an in increment. - - - - - - increment the an pressure capillary porosity atmospheric saturation saturation Continuity these incremental sign Bg 1 convention last Therefore, in pressure can - of of oil Equation expressions pressure 8V 8P 9 ° be oil gas TT0 9 B Vd 9 V Pg (av 1 /v 1 ) procedure BgPa±P+Pc iP 9 1 (HS 0 +S) 59 given 8P 9 ° — that = =nS = = from as HS 0 +5 9 P 9 1 (Vdg+Vj 9 ) Pa+F+Pc ThiS 0 S(P 9 °) 2 into compression equation the — - (Pg 0 ) 2 equation P(V 9 values V° 9 (P 9 0 ) 2 5.61 +v) used is 5.59 positive, the are value estimated of (S 9 /B 9 ) at the at 562 ( (5.60) 561) 559) 558) the be 139 Chapter 5. Flow Continuity Equation 140 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. = 19E(S 2E)E3 (5.63) where ,1E 2E and 3E are constants. Therefore, by substituting equation 5.62 in equation 5.51, the equivalent com pressibility can be written as S, SL +HS CEQ=fl )90(S (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 _____ Chapter 5. Flow Continuity Equation 141 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 2FV + — 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 = 1Ply (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 0V Vl—Vo (5.68) = 80 By adopting the usual notation = (5.69) Chapter 5. Flow Continuity Equation 142 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, ,0S ,9S Pt,,,0P and .9P The solution of the problem therefore requires the follow ing three additional equations: (5.71) 0P — P,L? = f(S, )0S (5.72) Pg — Po )09,Sf(S (5.73) freedom, makes addresses reservoir medium stress, flow complete analysis analysis equivalent stress, the Chapter There Compared analytical analysis deformation deformati&n the is and are engineering, 5. computation picture. are all required. single formulation Flow generally these the may several to model the phase rigorous Continuity be and and concerns. If flow proposed advantages the time detailed necessary. not flow simpler flow fluid. stress analysis flow considered. and problem analyses, Equation This results The here distribution other and analysis in in However, kind the combined is reservoir significantly and about but such that analytical But of the should may factors. analytical the and the the the proposed engineering, form not treatment flow the results be real reduces model be of looked deformation are problem effective model analytical the from required, suggested the the of flow at is multi-phase the major number if at together adequate continuity a model through a hand stress-deformation detailed separate in disadvantage this of in is to for the fluid degrees this a study. fluid equation obtain rigorous coupled coupled porous study as flow 143 an In of of a problems elastic tion number one MacNamee pore three Biot’s fluid multi-phase describes of developed analytical mation the Oil 6.1 the Closed Basically, fluid recovery dimensional for fluid. flow dimensions Analytical theories stress-strain material Introduction and of consolidation flow the of behaviour, researchers, by models form and fluid strip fluid by the development behaviour Terzaghi assume behaviour. Gibson steam problem solutions flow problem and and has used behaviour and for problem. been under circular but a injection (1923) in and (1960) and linear any under a in for of only the For described solution the hand a an arbitrary of footings the uniformly and instance, obtained elastic consolidation Therefore, for Finite a mechanical analytical from the constant is consolidation very Chapter Biot scheme sand considered in heavy stress-strain load on simplified 144 chapters (1941). De solutions loaded skeleton a a model Element load. using behaviour variable Josselin consolidating realistic oil analysis as 6 reservoirs circular Terzaghi’s equations Biot which 3 finite geometry to a and behaviour and consolidation de with analytical plane extended of are the Jong element couples 5 area the time. respectively. half Formulation is mainly fluid strain have theory conditions sand a (1957) on and space. coupled model the Both Terzaghi’s procedure. flow a been phenomenon. matrix. and based an semi-infinite is stress-strain obtained behaviour incompressible Terzaghi’s restricted Booker should axisymmetric and derived This stress, on Modelling theory for theories chapter include a (1974) linear defor by solu with to soil. The and and to a a 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: Chapter 6. Analytical and Finite Element Formulation 146 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 — 2F = 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 where where perature where Mathematically, Chapter Combining The From The strains total changes 6. chapter [Dk1 Analytical stresses equations can P this (Uk,I 3, can be the ------ - can be displacement strain tensor strain effective pore Kronecker expressed are incremental and written 6.3, + be the pressure LU1,k)] Finite due tensor written relating 6.4 sum stress = to in as delta and Element = vector the terms (see stress-strain of as + incremental tensor = 6.5 the change oj (U — LE,d equation and of effective H- Formulation a displacements + H- P substituting in = Dkz + Sj relation 0 effective temperature 3.73) stresses ie 1 ] including into stress as and equation — and the the strain pore effects = 6.2 0 pressures. yields, of (6.6) (6.5) (6.4) tem (6.3) (6.2) 147 jected subjected be finite conditions To are 6.2.3 consolidation and where be volume The 6.2.2 Chapter specified. define written For Equations the superscript flow element to the changes displacements, 6. known the Boundary continuity to Flow can in displacement Analytical For specified problem, tensor analysis. procedure 6.6 CEQ aEQ kEQ be dot was the applied P U 6 Continuity specified. and denotes class - - - - - - equation notation derived [(kEQ)F] temperature equivalent equivalent equivalent displacements, pore displacement 6.7 both U, In and using Conditions traction, boundary of are these pressure and problems the Finite the in as for the Galerkin’s chapter the displacement partial equations, coefficient compressibility hydraulic a Equation resulting Element conditions, vector multi-phase pore U, considered while 5. differentiation which pressure, — weighted The conductivity of the Formulation the equations and thermal a final may fundamental reminder fluid in 1 part the this +czEQ P. residual equation be including flow of with that expansion study, The zero. tensor the of boundary have scheme. respect unknowns = surface, the unknowns (see the 0 temperature surface, to following equation to be conditions time are solved to can SD, boundary solved be 5.78) (8 induced be can solved in /8t). must (6.7) sub can the 148 by be Chapter 6. Analytical and Finite Element Formulation 149 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 Chapter 6. Analytical and Finite Element Formulation 150 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 total express where expressed the Chapter Equation From Substituting Substitution In fluid stress chapter the 6. the components. as analysis stress-strain (Lc)j Analytical 6.19 definition CEQ 5, equations of n an adds equations is - - - - equivalent equivalent change porosity volume given = Based of the and relation [[D’l effective 6.17 [D] contributions Finite by {o-} 6.14 in on change + = and compressibility pore in compressibility CEQ this [D’] and = Element stress terms = 1 6.13 CEQ pressure in approach, + {&r’} ‘ 6.15 1 {m}{m}T] the CEQ in of of 1 into equation {m}T{E} Formulation total pore both + {m}i.P {m}{m}T has equation the stress. fluid the {e} been changes 6.18 skeleton Thus, = obtained 6.16 yields [D]{e} in gives the and pore matrix by the pressure considering pore [D] fluid for can (6.20) (6.18) (6.19) (6.17) (6.16) the 151 be all to forward, develop boundary finite formations. on obtained Booker and obtaining element For effects The 6.4 be becomes In and 1973). obtained as The Chapter particular, adaptability overcome the The Byrne Equation instance, Chang equations all pore element have Finite other the choice and method. of has zero 6. the from a conditions pressure and the finite and been solution Small Analytical Christian by 6.19 relatively solutions hand. this formulation. and Sandhu of equation Vaziri load governing for Duncan setting Element the element derived The is method the (1975) saturated is is for used different In on finite (1986) not above and and carried through less and these the 6.17, this one (1983) and formulations. the in an employed gives Wilson element Boehmer value Finite mathematics the formulation hand, claimed study, Formulation are approaches once equations or unknown consolidation by used the stable finite unsaturated given of the Element the (1969) and procedure principle Galerkin’s a CEQ the (1970), that pore element solutions deformations variational by The in is the depends weighted becomes involved, to to equations used this the fluid. weighted Formulation with knowledge use a Carter soils of can formulation suitably resulting method weighted when a virtual a multi-phase on be Gurtin ill-conditioned. theorem numerical For and residual and 6.6 are (1977) formulated the residual the an for of low has and work. computed. system is type residual effective type the incompressible any for less technique involving definite but and 6.7. technique scheme of fluid mathematics undrained Hwang stress-strain variational error finite the in of Small The scheme However, stresses a equations, and problem advantages number is best Laplace prone. to value quite et et such temperature develop conditions. fluid, al. al. is method go principle. involved relation. (Naylor, this as of straight used and to In (1972) (1976) trans but ways. finite such CEQ zero can the the the 152 to of is where fields pore independent solve However, and sented analyzed weighted solved. set Galerkin Chapter To From of 6.22 pressure the within integral develop by However, have 6. is the resulting scheme residual by means subdivided Analytical variables applying the previous should (Uk,j at equations. the 5eT q U’ element of most only or it integral finite shape - - = = should variational be [(kEQ) pore displacement within Green’s section, a + {S, {qi, second into and continuous. single UZk)j element can functions These S 2 , q,.. pressure equations be Finite a each be theorem, order finite the . application noted . . equations principle, written , , formulation S} q} of + governing Element + and field field the number derivatives U Hence, the that it their = elements, as (nodal — (nodal can shape N the can regardless of CEQ Formulation the of + values differential e Green’s be for end be elements. of pore functions displacements) F displacement reduced these easily U displacements results + at and aEQ pressures) of theorem the equations, turned the equations F, to will The for nodes. first = are approach — displacements be 0 and quantities is into approximately order. and the needed The the to the matrix same. pore be equations 0 used, domain pore Therefore, solved to of pressures. form obtain pressure the whether and repre (6.23) (6.24) (6.21) (6.22) being are, four 6.21 and the 153 to a written the where fashion assumed as 6.21 Chapter follow: In The In U” residual and Galerkin’s the and [Dkl to as strains shape 6. 6.22 give weighted F” errors Analytical will are the functions. w N and r scheme not approximate r 1 best - - [(kEQ - - the residual residual weighting and exactly shape shape ± and approximate derivative the r 2 : Then Finite F] functions functions error weighting scheme, satisfy function solutions , the + + Element j jNpr2dvzO jwrdv=O of following U”, 1 z1 the N solution. for for the these functions equations, r 1 dv — and pore displacements pore CEQ Formulation + residual substituting equations = Thus, [D pressures F” 0 pressure are + but for aEQ errors chosen will can the within these S give be — are best to obtained values some be minimized an solution the element residual into same to equations minimize in can as (6.29) (6.28) (6.27) (6.26) (6.25) errors some the 154 be Chapter 6. Analytical and Finite Element Formulation 155 (6.30) = m’ (6.31) B 6Iq (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 ç V dIZ c (V) dF V V d1 (6.33) J = j — j 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) — NpTmTBuSdv — / BkEq Bqdv H-/ / CEQ1N’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) Chapter 6. Analytical and Finite Element Formulation 156 T[L] {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} NTds + NFdv = f8 f — f 6edB,D {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, T[U] {8} — t [E] {qt + Lq} — [C]{q} = —{zC} (6.40) By rearranging the terms, T[U] {zS} — zSt[[E] — [G]]{q} = —{zC} + [E]qtt (6.41) way condition a conditioned. {zC’} routine ment displacements. element as, gives, Chapter small If It Equation where, By By to the and should changing = get combining to value matrix fluid 6. 0 and pore get [L] T around and [K] {LW’} Analytical For 6.43 be is for then the [E’] pressure equations [E’] incompressible the noted this [[E] [L] t. gives initial equations this use = notation, = =t[E]-[G] situation Lt [G]. This tt[E}{qt} the problem that and the unknowns. condition — a [L] T [K] consolidation [G]] global matrix Finite will it 6.36 equation an may [G] is circumvent [E’] [L] - and appropriate matrix to will Element results, equation {C} Stresses not Lq z8 use become 6.41 ILS1 6.42 be J routine. the q equation i.e. Formulation and possible and can the to undrained J 1A solution zero at be [E] be writing ill-conditioning. strains t is 1A1 solved and /.t{qt} written = to LC’ formed 0. can equation use routine are them This for J — be in the then and {tC} an the obtained is in to above 6.43 because element. evaluated solved a However, usual obtain full J ‘1 will consolidation by for matrix matrix for become the From assuming from displace a Lt better initial (6.43) (6.42) form form = the the 157 ill 0 isymmetric al. used conditions. N The and pore pansion. the Therefore, Wilson and solidation in The formulation 6.5.1 address an are dard certain The 6.5 Chapter a existing = displacements (1971b) given pore strains examples principal new the choice pressures finite N Finite issues (1969) the same The 6. pressures. 3-dimensional for Selection in problems. the element Sandhu triangular of 2-dimensional obtained key had of Analytical this any displacements which expansion steps the introduced they stiffness element are aspects difficulties Elements section. choice expansion, finite expressed et text and Yooko presented are Different by al. ring matrix, has of and finite in the important elements for book. of differentiating (1977) a The finite the et Elements element element the in the composite Finite details varied and in al. element include obtaining researchers development and developed Therefore, numerical displacements same terms element also (1971a) only has Element for quadratically such and an the compared order three been the code, a of element, a the as the two code, reasonable used three four analytical integration, only used class Procedure obtaining of nodes an Formulation displacements of relevant CONOIL-Ill. noded and expansion several these noded CONOIL-Il important a nodal several different of consisting over summary being for problems results bar model finite shape matrices the rectangle. different values, the etc., finite used element, element for pore element, issue Adopted element of varied The can has with (Srithar functions, for considered both elements for a they pressures. can elements, six-noded be the following been when the some However, types. linearly. a stress found vary be codes. while initial pore (1989)) three incorporated analyzing and its derived discussions in linearly Sandhu components in pressure all the This triangle subsections derivatives, this undrained noded concluded Since Yooko any of and stresses makes easily. which study, stan con too. and also the ax 158 ex for on et in the ilar The ments, two achieved ments eight-noded code. 6.5.2 uses this of eliminated the element that claimed do Chapter freedom 2. 3. 1. not The In Ghaboussi standard midpoint to different procedure a the solution Iterative Incremental Step-iterative the Figure and lower those whereas give method that types elements by 6. Nonlinear 2-dimensional for pore by satisfactory one brick Analytical order proposed at triangular expansion 6.2 Runge-Kutta of the static are or and does the later pressures. of employed the shows Newton which element or displacement insignificant. expansion the Wilson 20-noded or not condensation stepwise stages nonlinear mixed by following elements and for finite the give answers had Analysis Sandhu method uses (1973) pore herein of Finite or element the the brick for procedures element procedures consolidation, modified the expansion. problems pressures techniques: pore at same same used and used is element after Element same the types a pressures code for expansion expansion Wilson initial form an the Euler expansions consolidation and by uses isoparametric employed available The Formulation element the of the stages (1969) two method. different two than the differences for for finite additional additional displacement in stiffness mixed for pore in of for are the this consolidation. analysis pore shape displacement. element In element used. pressures in 3-dimensional study, procedure this is pressures nonconforming the degrees functions completed. Figure in scheme, method results of and element the and four However, pore of which 2-dimensional 6.1 and stresses, for freedom for nodes two code. types is shows pressures However, displace displace different degrees usually follows cycles with they sim The but 159 the are Chapter 6. Analytical and Finite Element Formulation 160 A Displacement nodes (2 d.o.f) Q Pore pressure nodes (1 d.o.f) Linear strain triangle Cubic strain triangle 6 displacement nodes 15 displacement nodes 3 pore pressure nodes 10 pore pressure nodes 6 nodes and 15 d.o.f. 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 161 A 157 14 6 A 8 8 19Li16 5I 18 ,. 20 s34 -4A- 10 /11 A 4 4 1 12 • Corner nodes = 8 • Corner nodes = 8 D.o.f. per node = 4 D.o.f per node = 4 Internal nodes = 0 A Internal nodes = 12 D.o.fper node = 0 D.o.f. per node = 3 8-Nodded Brick Element 20-Nodded Brick Element Figure 6.2: Finite Element Types Used in 3-Dimensional Analysis frontal metric popular the elimination the is factor Bathe stiffness direct next solution Selection by tolerance. final 6.5.3 first and to second parameters of Chapter the be analysis estimating The The iterative number therefore, cycle, load results continued influencing methods and direct matrices solution cycle, most frontal methods, matrix, techniques 6. of Solution increment. Wilson parameters Such method. based are the are of Analytical methods solution effective the the was solution use operations (Hood, scheme, performed evaluated. its method until an midpoint the and (1970) imbalance on a symmetry, not to iterative Most number efficiency methods the direct make the choose both Scheme at employed. and 1976) scheme the for and initial the difference of parameters and for To of an Finite solution solving element the load Meyer of from. have procedure midpoint its these obtain and of approximation each for exactly the .conditions methods any positive at Element been However, symmetric the Essentially, (1973) the storage references between load methods the stiffness finite more are other predetermined of simultaneous employed can end increment. take definiteness used the discussed of element accurate Formulation requirements increase an successive of is are the matrices matrices to load advantage contain to there the second improvement solve basically analyze increment in program, increment iterative the results, the are algebraic the or In steps the extensive are (Irons, cycle results relative its finite two of the the computer to variations equations. assembled banded specific and are this solution and classes and accomplish load first in are element 1970) satisfies equations merits used.. operations, bibliography. the process there adding increment computed. cycle properties nature of time of and results methods. and At of methods; codes. are the the of would the that for a is drastically the solved to variety Gaussian solution. specified analysis, whereas a is unsym and current reduce In end In to major of made have The one the the the 162 the the by of of minimizer 2-dimensional user. manner less schemes numbering keeping. and the to and deals The as minimized element requires apply earlier eliminated 1970; matrix Gaussian Chapter well substantiate The Theoretically, user. computer therefore, frontal Pina with Irons However, a than as is relative main numbering. stringent available less However, 6. Another never elimination (1981). the is a and sequence. as and solution incorporated nonassociated Analytical in core finite bandwidth disadvantage soon using storage the Ahmad, this most formed. the to the node such limitation memory the The total numbering as element difficulty claim the scheme Although of frontal and is conceivably bookkeeping as and numbering procedure 1980). frontal the The arithmetic required. solving in by (eg: back flow than of Finite is the code. other of variables solution Sloan can specially this the Some it this Sloan rule solution program. substitution band routines is be possible, Element direct by nodal The scheme. method operations is technique rather which and comparisons easily 1981; a scheme Sloan are routines. attractive programming stress-strain scheme Randolph solution sequence, in There introduced results easier Formulation Light the dealt is Its terms and process, may will the are efficiency operations has are to In Randolph in have with, and for methods. fewer. always complexity be of (1981), it number different an a addition, model at but unsymmetric accuracy its problem definite does Luxmore, already unsymmetric if a dependence As is perform the later some with Akin place considered the essentially (1981) a front Since overall advantage. it result, of and given and stage elements form zero is 1977; and some the width better matrices the does not is efficiency in stiffness on and global coefficients it internal Pardue of built in the Hood, a effort variables necessary is not front the minimizing in function or eliminated this faster literature a into element because stiffness concern at matrix on logical (Irons, (1975) 1976). width book study least and the the 163 are are to of Chapter 6. Analytical and Finite Element Formulation 164 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. Chapter 6. Analytical Figure Solve Evaluate Read 6.3: Read riJpdate for and No Evaluate Flow basic displacements strains Read Finite stiffness and for Chart Is Last data the load C this all principal set relevant C Element and of the changes the Start increment? for the analysis’ matrix and Stop increment pore results the initial No Yes Yes last and data allocate arrays Formulation D pressures and D Finite cycle in pore conditions for and stresses, load the storage Element print pressures vector No Programs Update average variables relevant values to 165 based The of though is to consolidation 6.6.1 to as The CONOIL-Ill 6.6 Chapter 13. 12. 11. 10. a the the well. any 2-dimensional finite 2-dimensional for The the other The Steps If softening, Steps on programs problems Finite these operating the Brief 6. the the next 2-Dimensional final imbalance element 4 words, 5 current Analytical which finite next analyses to to program descriptions load states in load 12 9 are Element platforms. load are oil program element until are is programs program increment, discussed stress shedding loads a of sand, repeated effectively. repeated CRISP increment. 3-dimensional and the the of state programs at they to previous load Finite CONOIL-Il Programs these have There later the vector perform Program if (University until until are exceeds any. shedding Both end Element been in programs capable all are load have all program is chapter of programs axisymmetric the computed. the written two the the was increment been is of elements Formulation load CONOIL-Il of converged. increment strength separate are Cambridge). originally to 7. doing developed are in increment perform given FORTRAN-77 are capable satisfy general and envelope, programs; in used are developed 3-dimensional with plane this data It the calculated of drained, as was analyzing special section. failure the strain have or CONOIL-Il later and by initial if undrained been there attention criterion Vaziri analyses and Applications analysis. are modified excavations conditions analyzed. added portable is (1986) which strain or paid and, and 166 Al by to in it the is the subroutines. II functions C. The 6.7 result main the nodes. matrix matrix, creates II The ‘Main Srithar Chapter menu to does has A Grieg 2-dimensional same finite geometry 3-dimensional facilitate User program in Program’. been 3-Dimensional product a It in not and driven, zero 6. et input are sequence element also manual 1989 al. have Analytical this divided The presented coefficients. program viewing consists renumbers (1991) file as very with will names The a program, suggested and mesh. program for of post user into eliminate an procedures developed main the some and automatically in and of of improved processor friendly two The the appendix The 58 main the plotting the Finite Program purpose CONOIL-Ill by example subroutines. separate elements subroutines geometry program all 3-dimensional Taylor program, a as and Element pre/post the formulation yet. C. the of the generates problems provides unnecessary (1977) programs; this and CONOIL-Il also program 2-dimensional has The CONOIL-Ill containing and The Formulation processor split nodes has been is program 3-dimensional their and are for many adopted names is some consists the to temperature presented developed arithmetic numbers to functions input the options package, minimize reduce ‘Geometry has of one. special in relevant the of and less the the 11 However, in program for from the subroutines are operations COPP, formation features. output special the appendix subroutines analysis. midside the effort information Program’ given scratch front user. for comprises features, data. compared for in and of width The which F. CONOIL CONOIL and appendix following the stiffness and and interior COPP about triple their user. and and will the 167 the 43 to the time the idate The other predictions by sponses program after gas Once element 7.2 deals The 7.1 considering Verification exsolution, University capability two analytical liquefaction to with verified, the geotechnical Aspects Introduction in time. program, dimensional would finite a a from number oil the of Cheung of some effects procedure recovery element be the the British has program CONOIL. problems. to Checked of particular version program program. also of consider aspects (1985), program, and problem. temperature Columbia described been is of The validated on such problems each the Vaziri analyzed Application Then, by Procedure main a and A finite Chapter since number as, in aspect problem Previous changes, (1986) to the dilative chapter by intention 168 element for demonstrate to 1985, comparing program separately. show of which and concerning etc.. aspects. nature with 6 7 program of has the Srithar Researchers theoretical has and this improvements of some been applicability of its The been Since the chapter sand, pore the applicability. (1989) CONOIL incorporated experimental best program used pressure solutions those three-phase Analytical is way have of to to being has aspects the of is predict verify demonstrated redistribution The are verifying been verified in results program made pore available. the program are and used the fluid, finite from with kept here val the re to at 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 Chapter 7. Verification and Application of the Analytical Procedure 170 0 0e 0 0 a) closed form o program Qc o a) 0 I 0 I I I I I I I I V c’J 246810 Radii 0(r/r E = 3000 MPa ) I’ — 1/3 initial stress : or = o. = 6000 kPa final stress : o = 2500 kPa inside radius : r = 1 in Figure 7.1: Stresses and Displacements Around a Circular Opening for an Elastic Material (after Cheung, 1985) Chapter 7. Verification and Application of the Analytical Procedure 171 0 0 C b0 0 0 C I.’ .- 40 60 80 100 120 140 Total Stress (kPa) (X1O’ ) Figure 7,2: Comparison of Observed and Predicted Pore Pressures (after Cheung, 1985) ______ Chapter 7. Verification and Application of the Analytical Procedure 172 0 Cl2 .4.) s-I >0a) .— s-I -I-) 0 0s-I xc 0 lab data 0 l.a .4-’ Cl) -4-’0 0 C 0 20 40 60 80 Effective SigmaP (kPa) 1(X10 ) Figure 7.3: Comparison of Observed and Predicted Strains (after Cheung, 1985) ______ Chapter 7. Verification and Application of the Analytical Procedure 173 I I 0.25 Analytical Solution . Finite Element 0.30 - Analysis 0.35 - r 0.40 - xI30 045 y/30 DIE—i .5 — 0.0 0.50 I I I I 4ia- io io_2 1.0 10 CtV vT - a)Amount of settlement 0.0 I I I 0.2- Analytical Solution ‘%. ‘b%, 0 Finite Element %, %\ Analysis 0.4 U y/B—0 vO.3 v—0.0 0.6 — D/E — 1 0.8 - 1.0 I I ia— i— 10—2 ia-’ 1.0 10 4 Ct 7 -— V 2U b) Degree of settlement Figure 7.4: Results for a Circular Footing on a Finite Layer (after Vaziri, 1986) Chapter 7. Verification and Application of the Analytical Procedure 174 3000 — La Symbols — CONOIL—Il Solid lines — Closed Form 2000 — C Vertical Stress 1000— Radial Stress ci) (1) 0— Hoop Stress —1000— Radial Distance(m) Figure 7.5: Stresses and Displacement in Circular Cylinder (after Srithar, 1989) obtained whereas, operation the model. parameters four incorporated compared can given the carried capability In good by loading. consolidometer and 7.3 with agree analogy Chapter this The comparing The loading-unloading hyperbolic be H triangular a agreement by very research Validation denotes uniform out It very performance triaxial to from Figure the Kosar 7. can in of to well. with the used Verification well elasto-plastic figure in the model be Negussey the model elements the work, (1989) obtained the closed temperature 7.6 test [n as the seen are program. captured experimental the illustrated total 7.9. shows finite relevant of listed the results from a does sequences form have of figure, the new (1985) Also and as depth. dilation from element the Other model the program in not by shown been elasto-plastic solution The for rise. boundary Application shown table z closed in figure have Kosar both predict denotes a encountered figure results triaxial considered. predicts phenomenon, code. The load-unload-reload in 7.1. been for Aspects in form the that by figure (1989). the undrained closed that 7.7. conditions the This The on Aboshi considered elasto-plastic the of test stress-strain solutions results volumetric depth oil 7.8. figure the predicted will shear in The form Computed specimen sand the Analytical oil et An that realistically thermal at triaxial as are stress al. and solution sands. to triaxial axisymmetric which samples shown type strain model validate match the and and (1970) the has and versus analysis test results To Procedure the program loading hyperbolic the in was test behaviour been has the model in measured validate for figure has the results measured axial obtained a been results measured using has a loading-unloading modelled been high analysis on constant the results 7.8. strain been developed are the models. Ottawa as a results temperature modelled dilation on The hyperbolic results by considered measured, program’s validated has values. and response oil by making rate model show been sand sand they four But and and 175 are by of _ Chapter 7. Verification and Application of the Analytical Procedure 176 L’ z/H = 0.875 30 — 0 0 0 ci p 0 0 0 0 - 2u Cl) Cl, Q) 0 ci) 00000 CONOIL—Il closed form solutions 0 —, LH z/H = 0.5 20 — 0 0 0C ci) C (1) 0u-ici) 00 I 0— i I I I 0 1000 2000 3000 4000 Time(s) Figure 7.6: Pore Pressure Variation with Time for Thermal Consolidation (after Srithar, 1989) Chapter 7. Verification and Application of the Analytical Procedure 177 8- Test results / 00000 CONOIL—Il / /0 6— a) - C - c C-) - 0 a) - E 2 4— 0 > > - D - E 0 0 0 0 I I I I I I I I I I 20 70 120 170 220 Temperature(° C) Figure 7.7: Undrained Volumetric Expansion (after Srithar, 1989) Chapter 7. Verification and Application of the Analytical Procedure 178 Table 7.1: Parameters for Modelling of Triaxial Test in Oil Sand (a) Elasto-Plastic Model Elastic kE 3000 n 0.36 kB 1670 m 0.36 Plastic Shear 0.72 ? 0.54 ! 0.33 KG 1300 np -0.66 Rf 0.80 (b) Hyperbolic Model kE kB m R 1100 0.49 700 0.47 0.6 49 13 Chapter 7. Verification and Application of the Analytical Procedure 179 0 1.5 cm Figure 7.8: Finite Element Modeffing of Triaxial Test Hyperbolic Chapter 7. Verification and Application of the Analytical Procedure 180 3500 3000 . 2500 ‘a 2000 0S ‘0 1500 1000 Elasto-Plastic 500 Experimental Ea (%) -0.2 -0.1 ‘I WI 1 €_a (%) Figure 7.9: Comparison of Measured and Predicted Results in Triaxial Compression Test Chapter 7. Verification and Application of the Analytical Procedure 181 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 3400 m 0.0 kB 1888 m 0.0 Plastic ‘i 0.49 ) 0.85 IL 0.26 KG 780 np -0.238 1R 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 in Figure Chapter Ottawa . a 350 300 250 200 7.10: 150 100 50 7. 0.00 Sand Verification Comparison and of 0.05 Measured Application and of Predicted El the 0.10 (%) Analytical Results Procedure for 0.15 a Load-Unload 0.20 Test 182 Chapter 7. Verification and Application of the Analytical Procedure 183 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 (m ‘1 p —i sin — exp — 2ir’\ T (7.1) r3 T m1,3,.. 2H 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 _1O __ Chapter 7. Verification and Application of the Analytical Procedure 184 im H G 21®22____3 ® 13 14 11 H=lm 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 H 3m/m/°C 3m/m/°C rn/s MPa rn 0.5 0.25 31x10 51x10 62x10 18.3 1 ______ Chapter 7. Verification and Application of the Analytical Procedure 185 35 z/H = 0.75 - (a) 30 -25 20 1.0 0I-. 00 10 Symbols - Program 5 Line - Closed form I 0 I 0 500 1000 1500 2000 2500 3000 3500 Time (s) 30 4Zz/H = 0.5 (b) Symbols - Program Line - Closed form I I I 0 I I I 0 500 1000 1500 2000 2500 3000 3500 Time (s) Figure 7.12: Comparison of Pore pressures for Thermal Consolidation is incinometers pairs of the tunnels demonstrate face. thick. and and of beneath m. 7.13 a soil being oil Laing and The are Technology Chapter broad considered about different sand Figure A The Overlying presented three layers. vibrating production with and UTF was Three schematic used et The up UTF sense. formations. the al. 7. 178 7.14 well instrumented the 7.15 uses top into However, to well pairs and oil (1992) Verification the here m here. for of the wire test respectively. detailed pairs. Devonian 125 shows sand wells. a the 3-dimensional with AOSTRA pairs applicability Research measuring limestone of steam for the piezometers m Further and oil layer. horizontal The it the Modelling analysis. overburden a A started shaft can geological sands in with vertical brief and assisted Waterway geological roof AOSTRA is Authority The be details is There horizontal and Application thermocouples located description at of the being at view simplified injection for tunnels of cross-sectional the about tunnel gravity different can McMurray stratification about are measuring all and stratification formation program about reports near (AOSTRA) be three two and 24 were access a the as and of classified of drainage plan Fort m times. shafts vertical 15 the consisting the for well UTF on constructed formation spacing. production in pore m limestone concept view view McMurray, UTF would Analytical measuring UTF. a below pairs at accessing can simple is To as process pressures displacements. the of (section and considered Clearwater be illustrate of A the be with for oil the UTF exists three found in wells the manner, vertical complex bitumen sand UTF Procedure temperatures, with Alberta, the limestone-oil the their and problem A-A’ comprises different were in below limestone tunnels the which are horizontal herein formation extensometers Scott section steaming only in as recovery shown problem drilled and a to figure is the one et soil depth in for a be is about sand of pneumatic at al. number limestone in currently steaming from well analyzed types, injection shale. histories analysis. the 7.14) a in (1992), and figures of depth inter 40 deep pair well and 165 186 the to in of m of Chapter 7. Verification and Application of the Analytical Procedure 187 Figure 7.13: A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991 Chapter Shaft#1 7. I Section T Figure Verification Injector/Producer 7.14: A-A’ Observation Plan and —* Cross Geotechnical Application View Wellpairs...... Section of Tunnel the of UTF the A (after Analytical ____ Scott — Procedure et al., 1991) ____ i A’ 4 188 —...... V.. :•:•• . . .• .•. . .•. .•.• •.• .•.• . •.•.•...... CD CD -c 0 C.11 gg CD 0< 3 0 CD = U) CD CD 0 = CD CD 0. o C;’ C;’ 3 3 3 00 injection shown pore the as the horizontally a figure. figure predicted temperature fluid to UTF. assumed and be obtained pressures. triangular (1300 modelled assumed. Chapter small shown the at Even The To excess results are pressures. is 2000 kPa The 7.17. in analyze At program. temperature-time assumed region listed to and though 7. in figures from by excess The time pore above temperature elements kPa are The figure be The and Verification is finite production in injection laboratory the plotted 70 adjacent This pressure t the (500 steam 7.18 vertically the pore steam a = table to %. 7.17 These oil larger elements. same 30 comprise kPa as correlates in-situ (b) recovery pressures days, injection (a). only 7.4. shown contours chamber and contours to and above wells and were the domain histories test from the With pore The for production the Application steam The (c) only in are results very obtained with the injection the in the steam pressure gas pressure) in which indicating time, figure at is also finite the of in-situ water the well oil injection one 10 temperature, saturation analyzed the reported indicated oil chamber sand the hours oil wells is 7.16. with well element from well of is nodes and the pressure). sand and sand region the assumed the layer are pair, the well. after experiences region Plane as the bitumen. the by is layer have growth layer Analytical extends modelled in mesh temperature shown of assumed which grows the Kosar production the field the higher The strain in also to are at shaded steam consisted figure. of the measurements different be in to is with The (1989) parameters shown to by significant the Procedure of pore to a the boundary maintained oil be injection distance nodes primary region pressure bitumen be contours steam time Figure sand finite specified pressure and in of times zero. figure with 240 as layer in from chamber used changes element 7.18 of conditions started. interest. is shown made i.e., at saturation figure are at linear about known as assumed expands 7.18. that where have AOSTRA (a) 2800 shown an the at in in 7.15 which mesh, shows strain input time, Only 10 been pore pore The The kPa the the the the 190 are as in m to is is 250.0 — 200.0 — C-. 22/7// 0 -150.0 ///// Lii 100.0 — 2 2/?/ 50.0 — ; c 0.0 — —— I 1 I — I I I I I I I 0.0 100.0 200.0 300.0 Distance (m) Figure 7J6: Finite Element Modelling of the Well Pair I-. Chapter 7. Verification and Application of the Analytical Procedure 192 Table 7.4: Parameters Used for the Oil Recovery Problem (a) Soil Parameters Elastic 3000 n 0.36 kB 1670 m 0.36 Plastic Shear 171 0.75 7J.:l 0.13 ). 0.53 t 0.31 KG 1300 rip -0.66 1R 0.73 Plastic Collapse C 0.00064 p 0.61 Other e 0.6 2k(m 1.0 x 10—12 a83/m(m/°C) ) 3.0 x 1O (b) Pore Fluid Parameters B 5.0 x iO 0B 2.5 x iO /m/°C)cx(m 3.0 x i0 /m(m/°C) 3.0 x iO 3czu,o(Pa.s) 20 03 0.2 1 S 0.2 kro = 2.769(0.8 — S)’996 krw = 1.820(S — 0.2)2.735 EIevLon (m) EIevLon (m) EIevLon (m) NJ NJ - 01 NJ NJ - 01 NJ NJ - 01 Q D Q 0 0 0 0 0 0 0 0 cyq CD p I-’. 0 0 CD p NJ NJ 0 0 CD C) (I) (0 0 (0 C, C-, C-, f 0 D NJ DCJ :3 NJ cn 0 0 00 0 0 (D (0 3 CD 3 0 0 (j) p CD 01 (11 0 0 0 c,z injection predicted that with injection It the be some made the maximum failure chamber stresses larger contours constrained in As are is pressure Chapter can also figure any The Figure The The the wells, similar region could the locations, in be than implied stress failure. also steam variation predicted stress contour a measurements. and well. also 7.19. 7. well seen at at stress 7.23 instrumented above give to in the increase, a Verification time the state indicates the level distance that ratio Since the by It chamber but shows the predicted from ratio corresponding can the of measured the t horizontal horizontal is the in is = which results the vertical but steam be shown temperature the the the is 30 shown field the 15 predicted grows bore about and seen not days. field overall horizontal m zone. is movement due ones. measurements chamber. in from and displacements Application an as direction, in from hole measurements 0.45 the figure Also much temperature to index figure picture, However, vertical the stress contours at the the soil is shown displacement about as of well. 7.22. predicted giving This figure steaming 7.24. the matrix horizontal soil ratios the stresses of are the horizontal There with in in the is It the and increases is predictions that the Maximum figure implied slightly the appears shapes are compared expands Analytical in same in the the are current is the figure along well stresses. the a no 7.17. shape distance shown single distance stresses of create measured larger by region below field that are a and are with displacement the stress higher The vertical Procedure of The in in measurements well the the than higher pore since unity, about the the from increase. reasonable figures away 1000 state pattern zone shape field stress pair. predicted steam the pressure line the the shear there kPa 15 from relative measurements of predictions 7.20 of of m at The ratios wells soil The of chamber. 1000 excess agreement would stresses the 21 above the 7 the and available contours contours is vertical m vertical mm at to kPa steam and wells. stress more from 7.21. pore not the the the 194 at in is is a EIevLon (m) EIevLon Cm) EIevLon(rn) - 01 NJ NJ -N 01 CD I. 0 CD CD Cl) U) NJ I- 0 CD U U U Cl) Co Cl) C-, C—, C-, 0 D NJ D NJ U) C) 0 0 0 0 CD CD CD CD 3 3 3 0 C,, CD 01 01 01 0 0 & I. c3 cJ c3 CC) 0 0 0 01 Chapter 7. Verification and Application of the Analytical Procedure 196 50 S 40 C 0 1:: 0 10 20 30 40 50 60 DsLncG (m) 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 5.18m/m/day. In the initial stages of production, more water will be produced than3 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. EevoLon (m) EIev3Lon (m) EIevLon (m) - NJ NJ 01 N) NJ - 01 N) NJ - 01 0 0 0 0 0 oq CD 0 0 N 0 C,) I- CD C,, C,) ci) C-, C) a, 0) E3 NJ :3 NJ :3 NJ C) 0 0 0 0 0 CD CD CD 3 3 3 01 (ii (ii 0 0 0 c3 0 0 Chapter 7. Verification and Application of the Analytical Procedure 198 - () t tlOhrs 40 C 0 30 LU /6ØO 20 I 0 10 20 30 40 50 60 Dstance (m) 40 C 0 30 LU 20 0 10 20 30 40 D9Lance (m) E 40 C 0 >30 20 •10 20 40 60 DLnce (m) Figure 7.21: Vertical Stress Variations in the Oil Sand Layer Chapter 7. Verification and Application of the Analytical Procedure 199 40 0 10 20 30 40 60 DsLance (m) 40 C 0 30 Li 20 0 10 20 30 40 60 DLence (m) 40 C 0 30 Li 20 0 10 20 30 40 60 DLncG (m) Figure 7.22: Stress Ratio Variations in the Oil Sand Layer Chapter 7. Verification and Application of the Analytical Procedure 200 60 Symbols - Field Measurements • Line - Prediction 50 • • 0 •• I I I 20 I 0 5 10 15 20 25 Displacement (mm) Figure 7.23: Comparison of Horizontal Displacements at 7 m from Wells Chapter 10 15 E 20 25 0 5 7. 0 Figure Verification 7.24: 10 Vertical and Application Displacements 20 Distance of (m) the 30 at Analytical the Injection 40 Procedure Well Level 50 60 201 Chapter C,, U. <60 0 o E 0 C 160 120 140 100 80 40 20 0 7. 0 Verification Figure 5 and 7.25: Application 10 Total Time Amount 15 of (days) the of Analytical Flow 20 with Procedure 25 Time 30 35 202 Chapter 7. Verification and Application of the Analytical Procedure 203 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 2m10’ to 2m1013 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 U U- I Chapter 0 a) 0 a) E 0.5 1.5 3.5 4.5 5.5 01 2 1 4 5 7. Verification 2 Figure 7.26: and 5 Individual Application - (a) 10 (b) Time Flow •.8 Time Flow Flow of 20 20 (days) Rate Rate (days) the Rates of Analytical of Water Oil of 50 Water Procedure 100 and Oil 200 500 500 204 Chapter 7. Verification and Application of the Analytical Procedure 205 50 40 E C E 0 30 Li. 0 0 20 E ,2 0 50 100 150 200 250 300 350 Time (days) Figure 7.27: Total Amount of Oil Flow EIevton (m) EIevLon (m) LHevLonF— cm( RD RD - 01 RD RD -R 01 RD RD - (ii ‘SC ‘SC ‘SC CD —3 0 co 0 CD r\D 0 fri CD U U Cl) Co Co C/) fri CD C-, C-, C-, D D RD D RD Z3 RD C) 0 C) 0 0 0 CD CD (0 0 3 3 3 ‘-I U) I I-.Cl) 01 01 01 0 0 0 0 Chapter 7. Verification and Application of the Analytical Procedure 207 40 C 0 30 LU 20 0 10 20 30 40 60 DtLnce (m) S 40 C 0 30 a) LU 20 0 10 20 30 40 60 DsLncG (m) S 40 C 0 30 LU 20 0 10 20 40 60 DLance (m) Figure 7.29: Stress Ratio Variation for Analysis 2 followed ysis. used the lems. pressures is lems and densified field beforehand. for Even the 7.6 of increased. To However, and the have Chapter described shear A Generally, oil avoid The earthquake. highest rate to if on in The related been typical though An recovery the prevent Other practice. a above by of failure zone 7. this soil from example if trial region predicted The 5 stress heating, the herein. Without to Verification soil the m profile kind loose in example and the oil projects. such region of detailed This is Applications profile finite a extends ratio sands, dense problem of small surrounding error loose sands comprised possible liquefaction, if concern situation, an of element region the illustrates for analytical sand results and basis, This sand and it shear are to Richmond, which permeability can the Application susceptible failure is reached away as type deposit 3 which program failure the examined liquefied also show m shown wells, loose involves the treatment, in of of rate from be zones, would clay analysis that unity usefulness is will British it Geotechnical in applied sand of CONOIL to large, was sands may the of herein figure crust, pore heating only the liquefaction deformations, indicating the be deposits kept wells, these Columbia, provides cause there stress very do to pressure Analytical be underlain 7.30. with of was other the not should stable this concerns it costly. significant will ratio different The are developed shear will same penetrate important in type potential migration was be the provided Engineering stability be commonly by earthquake of Procedure not significant failure. and have of reduced. considered event one 15 densification damage cause analytical m it for the geotechnical information of to of after of high during of Since analyzing be the rate densified. loose any the an is deformations tested to liquefaction in excess assumed elements earthquake of wells the treatment problems. the and sand the schemes heating region in about prob prob wells. anal after pore etc., The and 208 the to in earthquake permeabilities ysis, zone alent is densified situation system sification any three parameters generate dense Chapter achieved Dense The In drainage with the permeability different sand Clay case Liquefied of excess Dense zone. where 100% 7. drains is Sand Soil drains vibro-replacement are using 1, with zones. used assumed Clay Verification Table system. Type densification of pore densification pore with Sand shown This densification were Drain the Sand was in timber can A 7.5: pressures the pressure Drain may materials. hyperbolic with modelled not This in be Soil analysis and piles figures represent estimated considered a is case schemes Parameters 2000 2000 perimeter is increase 300 150 150 for Application columns. assumed without achieved as stress-strain the are may 7.31, a 0.45 0.45 densification 0.5 0.5 soil 1.0 from three n given were in represent on 7.32 any drainage In to loose with using Used an of 1200 1200 the studied cases 140 180 140 in case the kB and drainage the individual model an table sand for size timber full 3, a considered 7.33. Analytical equivalent 0.25 system. 0.25 by 0.2 0.2 1.0 the m field as and the and depth 7.5. was vibro-replacement. illustrated provisions. piles Example The drainage spacing condition 0.7 0.6 0.6 0.8 0.7 R 1 30% basis, Three considered This of with excess at Procedure permeability. pore 2.5 2.5 the 1 5 1 various instead, (m/s) cases may Problem of x x X a in x x was k,, loose where In pressure iO 10 1O perimeter the pore 10 10 figure and represent case which assumed times drains sand pressures the densification the In 7.30 5 5 1 1 1 2, The increase (m/s) x x x X x the represent densified after drainage the kh material without and 1O iO iO iO 10 a equiv in anal den field the the the 209 are in r0 r0 0 3 NC) -O 0 CD 0 0 CD () 0. . .o. CD CD 0 from although will be effective indicate pressure It effective where pressure of horizontal 0, piles ratio will the the of surrounding variations pressure pressure, shown Chapter to can high the achieved The The excess cause variation be the 0.4 penetrating in the densified be in triggered. pore conclusions that results in stress the ratio ratio after rise surrounding 7. they and liquefaction terms seen drainage of pore displacements preventing by Verification pressure undensified upper a the and at o in 10 of perimeter could increases from driving zone pressure for a of the 0.5 excess seconds is below u/o 0 depth However, is pore from part the case the is be densified are assumed liquefied from to the predicted timber initial damaged = pore this pressure and a figure shown and the ratio drainage of 1. of area and are migration depth 1 the (figure 5 the below analyses represents depth timber Application pressure m then likely with vertical zone migrates throughout zone that piles loose in and densified of 1 ratio system by graphs 7.31) a 6 reduces. mm depth increases could penetrating the to piles will graph of m zone depth horizontal are effective ratios pore u/o 0 , for 100% occur. into after drains show as not is (a) of support would along zone the into the (b) quite follows. of the pressure with the from and prevent the pore densified 6 in conditions that the at stress. in rises The the the movements. Analytical m, be effective densified which earthquake. (b) a time the vertical an pressure densified liquefaction capable depth densified Densification centre the results in to in the initial u/a 0 densified zone and the the u excess 1 analyzed. in high is of which load, line. zone. Procedure figures. densified preventing rise distance of zone. for are = value the 10 zone. Perimeter The 0 carrying excess pore zone m. shown or is case although represents current means alone The of not A liquefaction. results Such Below Graph Graph from pressure 0.3 zone. are maximum 2 pore pore in triggered the such vertical (figure drains penetration liquefaction at excess much figure this significant the for migration (c) (a) pressures The zero time pressure as in case centre depth shows shows could could more 7.32) 7.33. pore pore load pore pore The and t 211 the = 3 Chapter 7. Verification and Application of the Analytical Procedure 212 1.2 d=5m (a) 30 d=lOm (b) 0.8 Distance (m) Pore Pressure Ratio 0 0.5 1.0 1.5 2 (c) 4- Si, • di’ / 6 - / /z’ e! I,, 10 - QI I/i 12 - ‘:1 t=lmin t=3Ornin 14 - t5hrs t =lday 16 Figure 7.31: Variation of Pore Pressure Ratio for Case 1 ______ Chapter 7. Verification and Application of the Analytical Procedure 213 1.2 d=5m (a) d=lOm (b) c0 5 10 15 20 25 30 Distance (m) Pore Pressure Ratio 0 0.1 0.2 0.3 0.4 0. L (C) 4- / / A 0 ::: b-f : 1 12- 1 t=o 4 1 t=lmin 14- t5rflifl I t=5hr I 1€ Figure 7.32: Variation of Pore Pressure Ratio for Case 2 __ Chapter 7. Verification and Application of the Analytical Procedure 214 d=5m (a) t=rnin 15 20 30 1.2 d=lOm (b) 30 Distance (m) Pore Pressure Ratio 0 0.1 0.2 0.3 04 (C) 7 4- 4 I G) 0 / 10- 4 12- 1 t 9 14- 0 Figure 7.33: Variation OfPore Pressure Ratio for Case 3 Chapter 7. Verification and Application of the Analytical Procedure 215 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). laboratory loading-unloading practice tic injection. dilation, presented aspects. recovery shear development above three-phase model consisting and der dilation and An In strains. stress analytical hence increased induced modelling mentioned are: is are The plastic process in paths A an increase two test The pore the linear this coupled of important stress-strain plastic stability. procedure results yield a strains involving stress-strain predictions the from fluid; thesis. suitable aspects cycles. or oil Summary surfaces. stress-strain stress-deformation-flow strains nonlinear under recovery. oil and due aspect. The is elasto-plastic realistically. sand model a The the to presented decrease various from and other behaviour cyclic stress-strain effects The reserves. elastic Such Dilation postulated a behaviour the Chapter pertinent cap-type types model loading, and in dilation of stress-strain to models The stress-strain mean 216 of temperature will The analyze of the has models in major yield loading Conclusions of model aspect also can and stress key this which sand the a 8 increase lead cone-type the the surface issues thesis model contribution oil incorporating used is under model and skeleton; are changes response geotechnical to the sand are in in reduced incapable is to the have stress-strain constant to developing a the in yield predict skeleton, double hydraulic capture associated good the under been current-state-of-the- of pore these surface behaviour this aspects of volumetric shear agreement. hardening compared different the shear modelling fluid an response conductivity thesis key with important to stress analytical in pressure issues induced predict of steam an is stress plas with type The and un the the the oil is 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 liquefaction related successful would the has demonstrate recovery has could terms stresses, well. measured well results. paring The The memory including sional metric Chapter Although The The rate also been pair predicted validity A cause of finite cause the to stiffness two type method been displacements, deformations of in projects, has 8. revealed and a operation oil the program heating new has significant dimensional element Summary of shear its the been sand of devised. predicted results the underground been matrix. applicability, solution analytical to finite since failure. that specifically, employed finite obtain due of predictions program analyzed agreed and element and an damage. stresses, in results it to element finite routine A oil individual could oil flow. If study Conclusions steam frontal test very to recovery the a has sands it and element have program problem solve stress Information facility with as give can For codes local presented well injection also the solution the been with instance, be amounts the insights scheme. ratios closed with been shear results has code new has of applied involving compared resulting low AOSTRA. been the have technique been stress-strain in and developed has failure of form are into permeability, of the this closed this to checked amounts been been developed flow discussed. pore solutions permeability other the wherever equations. thesis, zone kind Results form applied which of examined following likely pressure model for the geotechnical would extends of is solutions to flow various higher and possible requires pore have behaviour very to analyze A results be of redistribution and the model in laboratory new to been fluid the important beneficial rates aspects some same the and discussed. and less problems. in the three oil presented a components in wellbore laboratory an of they horizontal detail. computer sand problems concepts. terms by heating unsym results. dimen to in agree com after The and 218 the To oil in of It it agation process ner study ding strain model However, enhancement that of separately, terms, an to a a The oil code Following area 8.1 Chapter three physical multi-phase check Fractures recovery Perhaps elaborate Even is analyzing three may which have worth model does for it Recommendations by dimensional will its though 8. a does also the the been dimensional model steam not fully and require capability also another problems considering. Summary which multi-phase in though sand be the work not fluid consider verified, combining the the integrated be injection. test inefficient geomechanical includes take skeleton. beneficial. further nature. analytical presented oil aspect through it or should to and code may anisotropy sand the it a model thermal field has Inclusion the Conclusions analytical study. anisotropy The which since is flow be be The equivalent layer formulation newly in not results problem a three carried three behaviour of very this it elasto-plastic for and require been effects. Application are requires thermal of written formulation dimensional dimensional study, difficult and by Further fluid modelling out sometimes applied where compressibility partial presented further strain Modelling and to flow a and energy some increase large task. the stress-strain the of to even softening model of may coupling effects. study the code Research analyze responses aspects encountered thermal fracture number in into Previous strain though finite this be the needs and would would account. more effects credibility a study is can and model softening of element initiation oil hydraulic are to useful researchers various iterations. be efficient. be be recovery fluid in be measured, includes in the identified described Incorporation the the applied a and flow codes of aspects realistic by and most conductivity stress-strain oil the problem load successful. the concluded behaviour A its recovery to in to models. desired in in stress effects of either prop shed man order more this this 219 the of of [7] [6] [5] [4] [3] [2] [1] Akin, man, Agar, Solutions”, pansion and Agar, 23, and Adachi, ciples ing Aboustit, Consolidated Mech., Aboustit, Finite J. 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Prin 240 of Bibliography [188] [187] ysis Yong, Zienkiewicz, on Engineering the of Rock Application R.N., “, as O.C., ASCE, Selig, a ‘No of Valliappan, Tension Plasticity E.T., Hollywood, eds Material’ and Florida. (1980) S. Generalized and “, Geotechnique , King, “ Proceedings Stress-Strain I.P. (1968) 18, of pp the in , 56-66. “Stress Geotechnical Symposium Anal 241 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 0P to P as shown in figure A.1 This will violate the ‘r/p 1P * 1i 1 7SMP,p 7SMP,i Figure A.1: Strain Softening by Load Shedding failure criterion and the stress state should be brought back to .1P In load shedding technique, this is done by taking out the shear stress equivalent to and then 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) — 123U(U Lo 18I +Ui)1 ) 2U (A.3) (U+1‘213 + (ui + £72) — (U )123U 13 )12t7 the for addition shedding Appendix 2. 1. Now, By To The principal By The The estimate solving above 02 the to rearranging to b-value mean A. equation obtain changes stress and Load equation equations the normal o3: [(02 vector the changes Shedding in A.4. the — additional the can stress 03)/(01 Oi A.4, terms by The Cartesian — be in the H- 03 Formulation A.5 remains following rewritten principal b)(A 3 A 1 /o- 1 — — transformation 1 two and 03)] (o- — stresses —(Lri equations: + constant A.7 remains + + A 2 ) two stresses, as A 2 Io 2 Zoi) 1+b 2—b the —(2— conditions + + can — following Lo 3 3) constant. during matrix. H- (03 two be b)(A 1 A 3 z 3 obtained + more 0 load Lo 3 ) are — equations Which A 2 ) assumed equations shedding. — by implies simply can during are This be multiplying needed, obtained gives the (A.l0) (A.9) (A.8) (A6) (A.7) (A.5) (A.4) load 244 in Appendix A. Load Shedding Formulation 245 l m 12 y 2m n2 12 2m (A.11) 2ll, 2mm 2n,n IO3 2l,l 22mm 2nn 2mm 2nzna, where a1 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 Appendix A. Load Shedding Formulation 246 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) equations: the equivalent meability The essary. at relative B.1 B.1.1 Some different viscosity relative detailed The Calculations permeability Relative are Relevant permeability temperatures first permeabilities of given explanations hydrocarbon section in and this equations through explains Permeabilities the are of of appendix. — _qrog which gases relative given viscosity water, krow Jo krw krg an how Appendix Jc° and = = are in example gas to C 1 (S 9 B 1 (B 2 section S1k° how To values needed 247 I 1 9’ calculate and permeabilit evaluate to — — — data row oil of 2. evaluate A 2 )A3 Sw)B3 in B S can the and w the Section set. the the pore t-’w be a relative evaluation The it. equivalent obtained Jg a Viscosities ies 3 fluid viscosity gives permeabilities components from some of permeability, equivalent values the insights following of are and water (B.3) (B.2) (B.4) nec per into the the Appendix B. Relative Permeabilities and Viscosities 248 — )D k,.09 = 12D(D 93S (B.5) — krow(Sw) /3W B6 — k0(1 — S,,) — (Sk,.09 9 ( 9ko (1—S r09\) 9 = 5; — Swc S S (B.8) Wc om 0S — Sorn s So > Som (B.9) wc om 9 ‘9i c’ C’ ‘-‘uc ‘-‘Orn where 0k,. - 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 12A...,A etc. - Constants By B.1.3 B.1.2 Appendix substituting The I-sw 5; k B 1 C 1 A 1 k° equivalent 7’OW = Sample Example B. ====== = = 1 = —10 2.201 8 1 0.5 2.769 1.820 1.640 1 1 x Relative 0.2 1.0 — — — x 0.4— 0.5 the 10 2 m 2 0.2 10 4 Pa.s 0.2 0.2 0.1 permeability — data — — — 0.2 0.2 calculations Permeabilities 0.2 0.2 0.2 data into = = =0.5 kEQ=k D 2 A 2 C 2 0.1667 S, B 2 0.3333 the is = = = = = relevant given 2OPa.s 0.4 0.05 0.20 0.80 0.80 and by equations Viscosities I-o A 3 Sg B 3 C 3 P’g = = = = = = 2 0.1 2.704 2.375 1.996 2.547 x 10 5 Pa.s (at 30°C) (B.11) 249 Appendix B. Relative Permeabilities and Viscosities 250 = 2.769(0.8 — 0.5)1.996 = 0.25 krog= 1.640(0.8 0.1)2.547 = 0.661 0.25 = 0.5 = 1(1 — 0.5) 0.661 0 793 — 1(1 — 0.1667) — krw 1.820(0.4 — 0.2)2.735= 0.068 9k,. = 2.201(0.1 — 0.05)2.704 = 0.001 (1.0 X 0.1667+1.0 X 0.5) 0k,. = 0.3333 X 0.5 X 0.793 = 0.132 —12 / 0.068 0.132 0.001 ‘ m m kEQ = 1 >< 10 + + 1.350 x 10 — 8 x 10 20 2 x 105) 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 251 Table B.1: Viscosity of water between 0 and 1000 C 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 10.34o 20 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 6.661 40 6.536 6.41s 6.29* 6.184 6.075 5.97* 5.86* 5.77* 5.67s 5.582 50 5.492 5.40s 5.32* 5.236 5.153 3.07s 4.99* 4.918 4.84a 4.77o 60 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 3.61s 80 3.57. 3.52* 3.483 3.44. 3.39* 3.35i 3.31? 3.27* 3.24* 3.203 90 3.16* 3.13* 3.095 3.061 3.027 2.994 2.96a 2.93* 2.89. 2.86* 100 2.83* 2.76 2.73 2.70 2.67 2.64 2.62 2.59 2.82 2.79 i FoR,.1Ux.E AND UNITS At a pressure ox 1 atm., = a/(b + t)”. At a pressure of P kg/cm ,7p = ?7i[l + k,(P — 1) X 10’J. of,7 at 1 atm. ‘11 is the value when2 P is 1 kg/cm which may be taken as the value , The unit of is the poise unless 2otherwise 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 below 100°C; the others are given as he has published them. The pressure is that of the saturated vapor at the temperatures indicated. 4, °C 110 120 130 140 150 160 1000,7 2.56 2.32 2.12 1.96 1.84 1.74 Table B.3: Viscosity of water above 1000 C 20H BELOW 0°C (39) Values corrected and adjusted to accord with I. C. T. values above 0°C —2 —4 —5 —6 —8 —10 100077 19.1 20.5 21.4 22.2 24.0 26.0 where is critical applied. where and The B.3 Appendix given The If temperatures, viscosity the PR TR pi temperature pseudo-critical Viscosity by Thus, gas B. ------ - critical critical pressure/critical viscosity temperature/critical viscosity and of is Relative hydrocarbon in a fraction at i.e., mixture place reduced pressure temperature and of of of pressure Permeabilities gas gas pressure hydrocarbon of of of component critical pressure at at gases of pressure hydrocarbons, reduced atmospheric component is of temperature have given PPc=XzFci IL 1 can component temperature FR and be (in to i temperature by in expressed absolute be Viscosities i gases the pressure in used. the (in i absolute mixture in and pseudo-critical absolute units) as The absolute (from TR and a critical function pseudo-critical units given units) Carr units pressure, temperature of concept reduced et al., temperature the has pressures pseudo- 1954) (B.13) (B.14) (B.12) to 252 be _ a) ‘—4 - ) d a) a) .4 ,4 U) if) 0 -I : .•- 4 0 - c • c: :j: = = a) a) iiI1 11 JWi 1L111FW1 0 -. -4 - -a) a) a) —l U) IMIII:.4 a. 3wvo, an,.! c3 0 a) U) ‘—I J.LI iLIJ_LLL_ ; 0 to to 0 Cl) o if) — r 0 -d - c o 0 — a) Cl) a) Cl) ;4- -4 U) C) c Cl) .4 Ii IiI1IM1ll1ll12 a) 0 -4-a c Ian3L, 0 U) t: t -4 ..a) a) 0 Q) -4 to tTITF ITIITI I Cl) .— a) a) ) ll1llllillllhIt - - U) to r-.l , to cd 6A•woIi,,_, ltavd’ 0 to ‘-4 Cl) 0 to a) -d o 0 - - O Cl) H “ - _- 0 : -4 llD 1 a) ;-1 0 a) p- ‘d o a) c 1llllt - to - 4.q_4 ‘.... ,s,wou,,.uo, 0 Cl) d 0 to U) - E : Q 0 l-4 -4 .;iCl) - 0 a) 4 ‘- -4 a) 0 -d l-4 o 0 - a) 0 a) II 0 0 E . a a) 0 ‘ 0 Cl) V c!: IIII U) 0 a,U) . 4) ;-4 a 0 ‘. 0 a) 4.44 Ia, U) U) a) U) a) - a) .-I-4-a VISd ‘UflSSWd VOI1W3OOflSd ,dd—U. ‘311fl1VU3dY131 VO4J4l4OOOflSd id :1 _4- -4 ‘-4 (3 +0 l-4 I—’ - 4) 4-4 ‘-4 2a) U) ,.D 0 11 o‘-4., C’) 0 CD CD CD . 2 U) ‘-U ‘-i’ CD CD VISCOSITY, AT I ATM. AJ, CENTIPOIS ,- + p CD CD ° -$- CD 8 8 0 • ö b 0 b 0 + :- CD 0 >1 o CD -- b 0 ‘-4- 1ELJ P 0 CD CD ‘-U U) ION ADOED 0 CD -i-1-ii F1t4-iI—tWttW .q- ) ( 1 ttItliltI 1 —. I 8 U) CD ‘-5 I— 0 , CD 4-s 0 ‘d- c- 9’ 4-f, 9, I—. ‘-5 4-- ‘-4- CD CD —. —‘ 0 FL j. U) p, 0 . *J1i1ir4’i ‘-5 CD 9, —.‘ 0 ‘-U 0 0 0 —. U) + 0 U) I— 0 9, 0 4-s 0 I-s CD U) 0 -C’) ‘-4- 0 p U) 9’ CD •-• ‘-5 Lj CD CD 0 • CD 4-f, U) o 0 p p;;. U) CD o ‘-1 CD 9, 0 1 CD 0 tIU11M1W411 o p 0 p ‘—4., 0 FFPII1IIlIFF1Fvr1I14-4aI4JYL4AY14n’i1fFIl 9, -“ I-s U) ,-4., C UUIIttt1IIIi1iVIlI1IJ11tIfl1VIIIAFII,I1VIlLItIU r V 111)14-I 11111171115I 11111 Il ILUI I CD 0 .iii.nrriii DIAl rU1Er[I4flE Cs. 0 P 111)11 It’ll il/li VIKJ y ‘ii COACCTIOu DOEO D1 l1(—19ItlA 1-II I IA-t-f- To ,sc —G P t. 0 4,_-’o , 8 o U) ‘) 9, . I I+H+FfAii(r 9’ 0 9’ o U) 4-- o CD ci) 9, I. •- CD CD ,-, CD 9, CD CD ‘‘ CD U) 0 U) U) CD 0 • 0 CD ‘-5 • P ‘-d 0 o ‘-4- o ‘-U CD • 9’ CD ‘-U 9’ 0 CORIICC1ION ‘-4- 0 ADOLD U) 0 _.. U) TI) VISC—G14 - 1 ,,. 0 4—] 4- 4-+) ‘-U 0 rJ _4-4 - 0 CD CD,- ‘-5 ‘-U CD O’3 “l oq - CD 0 CD ‘-5 CD 4-s U) - I-s r CD - p ‘-U CD O u, r CD 9’ CD CD CD —. . U) 9’ 0 ‘-s CD U) CD:-‘ CD ______ Appendix B. Relative Permeabilities and Viscosities 255 J ) = .L. ‘‘ I. .iL. - - — .-‘. IH -- -- — - .LL....III, -- — - :i£1 / ‘ .k L.-(-L — - r ‘--j-•-p.j-L .t.1- - —— —-- r / I •- -ir-rr , 4t—— —-- -. 1—- —— ---r , 1 —. — — - “i: — 0 I H- 1! -f-J4 1.. — — - — - , -rv i- r;- — - : ia:: .±Ik — —- - — ”E‘4- I - 1 ::: . I ,.,.‘77i! I. — - — . .-f ‘ T - r/r7 7 -i,.’_. —— ------:.. 4 - — . . — - I — L . IC i — . —--—-- — — 2 3 4 .5 6 7.6LD 2 3 4 56 7I9 PSEUCOREDUGED PRESSURC Figure B.3: Viscosity ratio vs pseudo-reduced pressure ___ VISCOSITY RATIO 01 g UI -.---IH111HTTD11 I — ------I ill I I lililiP I :: E:::tft111Itt1Uft I tumiHt — — — A I I II II Li IIIU—ti±fliIJJ-JkHlfl- Ct) — — - — - - I I I I i—i--i-rn I I I I I i-i-HI] I IUHi] - CD — - 11—i—1 1 1 1 r ru 1x4-rrrrlLI-1-rrtTrl]r — ‘1I- I 4 4 I I F LI +1H I-I-li-I’1I II I-I I-I4J4 — —,— — —-— — - — - - 1111]Ji—rLuIIIl-i-rUJJJ±LuIIUL - — —— - — J-t.I [T u-1-T[ITITFFIFrITI 1 LW - I U—TI I I II 1—Hil I I H-I-fl U- CD i4 —I I IT I ii4-nTrLi4-rn1-rTr — — —- - - - : : LI J}tT[IWITWITh[IL - -1 J[I-±tf1±1±H±ftH1±th _:_L::, ii i[14r1±1±thtinlll: I — - i tiIlF{*fl+}1+HI1{If ,, i : - r triuwiwrmirrni —L - - 41 4- U4 1414-I-I W-14111 I II- C — — — - - - - : : : : : • 3 4 I111II1Ht-1[1 - : : : : : I N 41WhII-fll-UIL[ 7 - ‘ - EEHEE 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 0.7018 (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: 1IL/IL = 1.28 6. Therefore, The viscosity at 1800 psig and 195°F = 1.28 x 0.01223 = 0.01565cp MIDSID MAKENZ MIDPOR FFIN ADDS BCONI associated The generates containing does and subroutines programs Program’ The C.1.1 11 C.1 subroutines nodes 2-dimensional subroutines the - Subroutines - 2-Dimensional reads analysis. - forms and is and Geometry - to with the sets - generates are to - generates minimize numbers generates free relevant the and up reduce element-node each described in element format code The the ‘Main the node. mid-side the information the the mid-side geometry main geometry has an Program herein. input. Program’. midside effort front constants. array been in Code program links. displacement width pore on the program Appendix divided which program and about the CONOIL-Il pressure The consists and interior Finite 258 user. contains the into and main creates for nodes. finite two The nodes. their nodes. the of C reason the 58 separate a element geometry 2-dimensional Element functions input subroutines. number It for also file mesh. programs; having renumbers program for are of degrees the The version as The Codes as follow: main the main automatically details two the of consists ‘Geometry program, elements program separate freedom of the of 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. Appendix C. Subroutines in the Finite Element Codes 260 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. Appendix C. Subroutines in the Finite Element Codes 261 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. ZERO2 - initializes 2-dimensional array. ZERO3 - initializes 3-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. Snbrontines 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. WRTN VISW ZEROI2 VISO ZEROI1 VISG ZERO3 UPDATE TEMP ZERO2 UFRONT ZERO STOREQ STRL STIFF of SHAPE2 SMDF Appendix the nodes. 1 ------ - calculates calculates - - calculates - - calculates sets calculates - calculates - writes initializes initializes initializes C. - initializes initializes - - - calculates writes updates frontal up Subroutines arrays a viscosity viscosity specified and viscosity the element nodal 2-dimensional 3-dimensional 1-dimensional solution 2-dimensional the 1-dimensional shape giving terms updates results temperature in of range of stiffness of functions routine the nodal in oil. gas. water. stress a at Finite in buffer array. array. array. mid-step degrees integer integer a matrix. for level. and 1-dimensional changes. Element unsymmetric zone derivatives of array. array. for freedom when second Codes an array. matrix. and for array iteration. 2-dimensional the becomes first degree saturated. integration. of freedom 263 ities. zone. where the here. the In the The knowing The To and However, alent This The the illustrate effective phase temperature 7.18). fluid mobility formulation does conductivity Here, The oil the flow sand at components. grater kmi not Amounts Such the mobilities k the any total of zone layer give flow - - a a steps contour for intrinsic mobility time, the fluid zone term amount can the zone the the number involved of phase these be for The to individual zone multi-phase plot the is permeability of divided model the of of divided details phase individual fluid from of component flow, or example the Flow zones the Appendix the into phase where example of 1 amounts and into kmi pore flow effects this the a of problem the amounts 264 three number components of ‘1’ the presented pressure the calculation better kkri relative can problem of of fluid Different sand (zones the flow be D is the of of flow contour shown different matrix written zones in permeabilities flow of given results are are A, chapter occurs, the B can presented assumed of in in (m 2 ) plot and as fluid phases different will figure chapter be Phases can 5 C (refer be. easily phase considers in constant D.1. and be in in figure effective 7 to this the obtained estimated is viscosities components. figures considered pore appendix. an D.1) within mobil equiv (D.1) fluid. from 7.17 and by of a 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 1k,. - relative permeability of phase I IL1 - viscosity of phase 1 k is a function of void ratio, 1k,. 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 .2m 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) Appendix D. Amounts of Flow of Different Phases 266 Table D.1: Average Viscosities and Temperatures in Different Zones Zone Area )2(m 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 0k,. = 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) ac mo aA + aB + ac Then, the amounts of water and oil flow in the total flow can be estimated as, A TI TI mw L.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. Appendix D. Amounts of Flow of Different Phases 267 For example, the amount of water flow from zone A can be given by, fAT? AT? mw A aA iI_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 0).(S Then, the saturation of water in zone A at the beginning of the time step will be, (S) = 1 — )0(S (D.9) The volume of oil in zone A at the beginning of the time step will be given by, )0(V = aA n )0(S (D.1O) The amount of oil flow from zone A will be, mo A aA L.1Vo)AIAT?\ = AT? (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 )0(V — 0(V)A (D.12) and the new oil saturation will be, (V (S )0 (D.13) 0 fl ) aA The new saturation of water in zone A will be given by, (S) = 1 — )0(S (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 Appendix D. Amounts of Flow of Different Phases 268 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 m/s)kmw(10 m/s)kmo(10 A 0.7 8 0.3 8 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. Appendix D. Amounts of Flow of Different Phases 269 Table D.3: Calculation of Flow and Saturations with Time Time (S)A (S)B (S)c 0(S)A 0(S)B 0)c(S 0iW (days) 3(m/day) 3(m/day) 0 0.300 0.300 0.300 0.700 0.700 0.700 2.54 2.64 2 0.394 0.319 0.301 0.606 0.681 0.699 3.89 1.29 4 0.435 0.330 0.301 0.565 0.670 0.699 4.29 0.89 6 0.461 0.339 0.302 0.539 0.661 0.698 4.49 0.69 8 0.479 0.346 0.302 0.521 0.654 0.698 4.61 0.57 10 0.494 0.352 0.302 0.506 0.648 0.698 4.69 0.49 15 0.525 0.365 0.303 0.475 0.635 0.697 4.82 0.36 20 0.546 0.375 0.303 0.454 0.625 0.697 4.89 0.29 25 0.562 0.384 0.304 0.438 0.616 0.696 4.94 0.24 30 0.574 0.392 0.304 0.426 0.608 0.696 4.97 0.21 40 0.595 0.405 0.305 0.405 0.595 0.695 5.01 0.17 50 0.610 0.417 0.306 0.390 0.583 0.694 5.04 0.14 60 0.622 0.426 0.307 0.378 0.574 0.693 5.06 0.12 70 0.632 0.435 0.307 0.368 0.565 0.693 5.07 0.11 80 0.640 0.443 0.308 0.360 0.557 0.692 5.08 0.10 90 0.647 0.450 0.308 0.353 0.550 0.692 5.09 0.09 100 0.653 0.456 0.309 0.347 0.544 0.691 5.10 0.08 125 0.667 0.471 0.310 0.333 0.529 0.690 5.11 0.07 150 0.678 0.484 0.311 0.322 0.516 0.689 5.12 0.06 175 0.686 0.495 0.312 0.314 0.505 0.688 5.13 0.05 200 0.693 0.505 0.313 0.307 0.495 0.687 5.13 0.05 250 0.704 0.522 0.315 0.296 0.478 0.685 5.14 0.04 300 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 m/s)kmw(10 m/s)kmo(10 A 8 0.71 0.29 8 1826 2.61 B 0.54 0.46 352 4.75 C 0.32 0.68 16.7 0.64 interior formation and user. about program. lems ing and The CONOIL-Il and a Provisions siders plane displacement The stress CONOIL-Il E.1 strong the the heavy creates intention program in The temperature strain strain the nodes. oil Introduction same ‘Main geotechnical finite geometry Detailed of foundation sands, exist an has is model stiffness and and material can of Program’. a It input element been for finite this also it axisymmetric User pore be effects explanations and can program specifying manual file matrix, used renumbers divided background analyses. element data pressure. a be mesh. for The formulation on to Manual applied base. the automatically stresses is solving carry main into program various Therefore, to conditions. such Though main the Appendix Elements provide two to for purpose out elements routines and be as to program, a boundary separate transient, for 270 range for analytical the analyze able fluid the sufficient generates consolidation of can The program etc. and Geometry to this of CONOIL-Il flow programs: containing be E geotechnical program multi-phase prepare conditions nodes drained can split formulation, removed in information and is the be is particularly Program to analysis found numbers an to analysis. includes or minimize the the such reduce to input undrained fluid problems ‘Geometry relevant simulate in method for has as in Srithar flow. the file an the an suited pressure, oil the to elasto-plastic mid-side analyst and effort such sands information analysis be front It of excavation. (1993). Program’ for also analysis, run run as on width under force, prob with dam con first and the the us 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. Appendix E. User Manual for CONOIL-Il 272 E.2 Geometry Program Record 1 (one line) TITLE TITLE - Title of the problem (up to 80 characters) Record 2 (one line) LINK I 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 Appendix E. User Manual for CONOIL-Il 273 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 Appendix E. User Manual for CONOIL-Il 274 MAT - Material zone, number in range 1 to 10 Appendix E. User Manual for CONOIL-Il 275 E.3 Main Program Record. 1 (one line) TITLEI TITLE - Title of the problem (up to 80 characters) Record 2 (one line) I LINKI LINK - Code number set by the user Record 3 (one line) NPLAX NMAT INC2 I INCJ 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) Appendix E. User Manual for CONOIL-Il 276 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 C7 IMODEL - Stress-strain model number. Use 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 = 1B (liquid bulk modulus) - Undrained analysis NOTE: 1B in the range of 100 to 500 5kB (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 0B Appendix E. User Manual for CONOIL-Il 278 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. ta8 - Coefficient of temperature induced structural reorienta tion. Only used in temperature analysis. - Constant volume friction angle. Only used with dilation option. — - 0 (No parameter at present) B - Bulk modulus of the water 0B - Bulk modulus of the oil Record 5.3 /J’30,0 ‘H H -\U U S 1S 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) Appendix E. User Manual for CONOIL-Il 279 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 0cx - 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 1 - use minor principal 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 Appendix E. User Manual for CONOIL-Il 280 Record 5 (for ELASTO-PLASTIC 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 ) Record 5.1 MAT IMODEL e KE n (R KB m DUO k )1 % 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) e - Initial void ratio KE - Elastic modulus constant n - Elastic modulus exponent (Rf) - Failure ratio in the hardening rule (cone yield) KB - Bulk modulus constant m - Bulk modulus exponent DUO - Determines whether Drained/Undrained/Consolidation analysis i) DUO = 0.0 Drained analysis ii) DUO = 1B (liquid bulk modulus) - Undrained analysis NOTE: 1B in the range of 100 to 500 8,,B (soil bulk modulus) is equivalent to using a Poisson’s ratio of 0.495 to 0.499. Appendix E. User Manual for CONOIL-Il 281 If there are temperature changes, use consolidation routine iii) DUO = 71 (unit weight of liquid) - Consolidation analysis to do undrained analysis. - total unit weight of soil - permeability in x direction k - permeability in y direction Record 5.2 1,i(r/o) (r/o-) q — — B 0B — - 0 (No parameter at present) (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 ta8 - Coefficient of temperature induced structural reorienta tion. Only used in temperature analysis. — - 0 (No parameter at present) — - 0 (No parameter at present) B - Bulk modulus of the water 0B - Bulk modulus of the oil 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) (used in three phase flow, built-in oil viscosity correlation) - Function to modify Henry’s constant for temperature H =H+\H*T 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) 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 water - 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 ISIGE - Option to calculate Young’s modulus 0 - use mean normal stress 1 - use minor principal stress ISIGB - Option to calculate bulk modulus 0 - use mean normal stress 1 - use minor principal stress IMPF - Multi phase flow option 0 - fully saturated 1 - partially saturated Appendix E. User Manual for GONOIL-Il 283 2 - three phase fluid flow (needs additional parameters) ILSHD - Load transfer option 0 - do not perform load transfer 1 - perform load transfer by keeping o constant 2 - perform load transfer by keeping o constant - 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 9IV S - Initial water saturation S, - Initial oil saturation 9S - Initial gas saturation (S + 0S + 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) Appendix E. User Manual for CONOIL-Il 284 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 0 - use a given constant value (in Pa.s) 1 - use the built-in feature in the program (Correlation by Puttangunta et.al (1988), to,o should be given in record 6.4) >1 - interpolate using given temperature-viscosity profile 0(1V data pairs, maximum 10) IVg - 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) >1 - interpolate using given temperature-viscosity profile (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 — 3A2)A Bl...B3 - Parameters for relative permeability of oil (oil-water) = B1(B2 — S)B3 Cl... 03 - Parameters for relative permeability of gas (oil-gas) 9k,. = 9C1(S — C2)c3 Dl...D3 - Parameters for relative permeability of oil (oil-gas) Appendix E. User Manual for CONOIL-Il 285 k,.09 = D1(D2 Record 5.7 (necessary only if IMPF = 2) Fl F2 I F31 Fi...F3 - Parameters for oil-gas capillary pressure of gas (oil-gas) Pc = Fl 9Pa(S — 3F2)’ Record 5.8 (necessary only if IMPF = 2 and IV,,, = 0 or >1) V,,, (ifIV=0) Vi Ti V2 T2 (if IV, 1, •.• I , 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 01V = 0 or >1) 01V =0)(ifIV Vi V2 T2 0 1, Ti (if •.. I 0>1V 01V data pairs, maximum 10) 0V - Constant viscosity value of oil (in Pa.s) Vi,... - Viscosity values in the given profile (in Pa.s) Ti,... - Temperature values in the given profile (in °C) Record 5.10 (necessary only if IMPF = 2 and 91V = 0 or >1) Appendix E. User Manual for CONOIL-Il 286 (ifIV=0) Vi Ti V2 T2 (if 1, I ... I IVg> 91V data pairs, maximum 10) - Constant viscosity value of gas (in Pa.s) Vi,... - Viscosity values in the given profile (in Pa.s) Ti,... - Temperature values in the given profile (in °C) Record 6 ((IPRIM-1)/10 + 1 lines, only if IPRIM> 0) Li L2 I 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. Appendix E. User Manual for CONOIL-Il 287 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. Appendix E. User Manual for CONOIL-Il 288 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) Record (a) There Appendix (b.2) (b.1) For For I rLi L LNJN2TJS1 NDFX must For For .1\TLOD NLOD Ni, Ni 12 Li,... DFY DFX E. linear cubic N2 be Ti (NLOD Si Ti N2 N L User > < 10 DFY1 ------ - 0 Ti 0 strain strain Increment Increment Increment Element Node data Increment Increment Node List Manual lines) Si of per triangle numbers number triangle element T3 T3S3T2S200001 number line, for of of of of of S3 CONOIL-Il normal shear x shear y except at force force T4 numbers the stress stress S4 stress the end T5 to last at at of at be 55 Ni Ni the line Ni removed (see T2 loaded S the in element following this increment side figure E.1 289 Appendix E. User Manual for CONOIL-Il 290 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 N2 NS N4 Ni 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 N - Node number Appendix E. User Manual for CONOIL-Il 291 ICODE - A three digit code abc which specifies the degrees of freedom associated with this node that are fixed to par ticular values a - fix for x direction o - node is free in x direction 1 - node is to have a prescribed incremental displacement DX b - fix for y direction o - node is free in y direction 1 - node is to have a prescribed incremental displacement DY c - 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 Appendix E. User Manual for CONOIL-Il 292 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 Appendix E. User Manual for CONOIL-Il 293 0 u,v — displacement unknowrs A p — pore pressure unknowns a. 1. 6 2 2 S 2 (a) Element type 1 (LST) (b) Element type 5 (LST) 6 nodes, 12 d.o.f. 6 nodes, 15 d.o.f. (consolidation) 4 12 —._. 12 16 21 6 11 - ,‘ 11 2 / 6 / 10 1// / 2 10 S - / .18 8 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 mesh for Records ulElement It finite The mesh. used. NN the undrained and element For analysis (ii) Appendix should any LEDF Axisymmetric constraint (Number drained program 8 unique element 5 5 7 1 are The nodes types use be 4 recommended. E. and analysis geometry and Linear CST noted Cubic element analysis integer user LST of lying meshes User of 1 Nodal Vertex and 5 no with must with Analysis that Element strain pore_pressures pore or strain Manual on numbers volume program type 5 or Numbering in ‘locking cubic element will assign linearly Nodes) NN pressures consolidation a triangle triangle 5. Recent situation refers for probably Name 1 variation change Table in 1 automatically each < up’ sides CONOIL-Il < varying the element to (CST) research node (LST) if element E.1: where following the (which low or be of analysis within number number Element adequate order number Displacement has collapse and occurs generates Quadratic Quadratic Quartic Quartic ranges: elements. finite shown each where of Types < 750 (i.e. vertex in is 500 vertex elements expected undrained node that collapse Variation the nodes Linear Linear Strain Cubic Cubic node in numbers same axisymmetric (such is then in in situations) not of as the the Pore as and element (i) expected finite Linear finite Cubic the N/A N/A above). Pressure coordinates LST) leads analysis element element types then For 294 are to 7 Appendix E. User Manual for CONOIL-Il 295 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 Appendix KZZ. E. User Manual Figure Figure for E.4: xis z CONOIL-Il E.3: is Axisymmetric the’adia1 the Plane circwnferentiai Strain direcSon Condition Condition direction 296 Appendix E. User Manual for CONOIL-Il 297 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 Appendix E. User Manual for CONOIL-Il 298 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 Appendix E. User Manual for CONOIL-Il 299 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 2kN/m is applied to part of the boundary of the finite element mesh it might be divided into ten equal increments of 2 2kN/m 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 not satisfactory The Appendix 4. 5. be application excess finite experienced When that time If This (see per shown ten captured a log days very below). are E. scheme step element results. a pore in cycle close change User to small table should in of pressures one by would off to Manual the equations item E.2 time those in hundred time the Increment solution be Table pore 5 be boundary. increment that large (for for will nodes modified 10 9 8 4 5 3 2 6 pressure 7 1 will E.2: days). CONOIL-Il a show often The enough log No. be in Time If base is following ill the oscillations Not slightly mean boundary this DTIME used conditioned. to Increment 500 300 mesh 100 of 50 30 10 less 3 5 1 1 is allow ten). that near not near than procedure, with condition (both Total done the Thus the the the three Scheme 1000 excess 200 500 100 20 50 10 start true 5 2 1 effect then start in Time a suitable time time however, is undrained of the pore and applied, of the steps and solution consolidation end pressure scheme analysis space). usually should the of response will an associated may variables then leads analysis be predict to be used will 300 the be to as 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 Appendix E. User Manual for CONOIL-Il 302 Table E.3: Load Increments Loads Increment No. Input to Incremental load Total load program applied acting 1 1 1 1 2 1 2 3 1 3 4 1 4 5 1 5 6 1 6 7 1 7 8 1 8 9 1 9 10 1 10 11 0 0 10 12 0 10 13 0 10 21 -2 -2 8 22 -2 6 23 -2 4 24 -2 2 25 -2 0 26 0 0 0 27 0 0 28 0 0 etc. acceleration an DGRAV Appendix analysis is E. used (e.g. can User be in in Manual problems regarded the ‘wind-up’ for as in CONOIL-Il having which stage the this of material’s effect). a centrifuge self weight test increasing is increased centrifugal during 303 cording please given The brief derivation behind features. This ten perform stresses, CONOIL-Ill F.1 3. 2. 1. for source descriptions manual in Temperature plemented. Three Elasto-Plastic at refer oil its to Introduction oil the drained, deformations the of sands, development. sands code Srithar phase end differential provides is standard a of is User where undrained it three are fluid effects this written (1993). can stress given neither and flow. dimensional FORTRAN manual. the equations, be Only on Manual flow strain here. used in A pore and This stresses detail sample FORTRAN-77. the in for fluid consolidation For model. is Appendix oil formation input naming a general finite information sands. detail and special data contains for 304 parameters Modified element strains. file explanations convention. geotechnical Though feature of CONOIL-Ill analyses three Input and stiffness F about program form the required phases; CONOIL-Ill parameter needed, corresponding the and Names matrix, such of problems. program Matsuoka’s developed has water, to their as, begin analyze solving the names method is bitumen format CONOIL-Ill following specifically nor with output to are routines the model the analyze of the and given problems analysis, and theories file special letters is some writ etc., gas. can im are the ac Record Record not 6 1 are F.2 Appendix where J, retained contain L, NCNOD, TITLEI NCNOD NINOD M Input ITYPE TITLE NTEL the 2 1 NINT IPRN and F. (one (one a to material decimal User N avoid NINOD, line) line) Data ------ - implies Index 0 Element Number Total = = Total = Total 1 Title Manual - - 8 3 1 confusions. point. Do Print property for for or of number number number to NTEL, that not 27 consolidation drained/undrained the of type print for the There (generally print integration the problem CONOIL data information (see ITYPE, of of of nodal program are the internal elements corner are fig. exceptions information 8 and (up -III analysis read. is F.l) NINT, points nodes expects good to nodes element analysis 80 Actual IPRN enough) to characters) (0 integer this information for material naming ITYPE data. parameter convention 1 Integer and 3) data notations in should record 305 Appendix F. User Manual for CONOIL-Ill 306 TYPE1 TYPE3 o Corner nodes = 8 • Corner nodes = 8 D.o.f. per node = 3 D.o.f. per node = 4 Internal nodes = 0 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 Appendix F. User Manual for CONOIL-IlI 308 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 (Records 6.1 to 6.11 have to repeated NMAT times. 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 Record Appendix e,KE,n,Rf,KB,m,7,k,k,k 2 MODEL ILSHD 6.2 181GB ISIGE IMPF MAT F. KE (all n e User - - - - - - are - - - Elastic Initial Elastic 0 2 Multi Load 0 0 1 0 1 Option 3 Option 1 2 Stress-Strain 1 Material 1 Manual ------ - real fully perform partially three use use use do use modified modified hyperbolic transfer phase not void variables) mean minor mean modulus minor modulus to to saturated for phase number calculate perform calculate ratio load saturated flow Matsuoka’s CONOIL Matsuoka’s normal normal model principal principal option model fluid exponent constant transfer option load bulk Young’s type flow stress stress -III stress stress transfer model model modulus (needs modulus with additional Cap-type parameters) yield 309 Appendix F. User Manual for CONOIL-III 310 - Failure ratio KB - Bulk modulus constant m - Bulk modulus exponent - total unit weight of soil - permeability in x direction - 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 )2(m Record 6.3 (all are real variables) 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. 1S generally close to zero) 8B - Bulk modulus of the solids B - Bulk modulus of the water 0B - Bulk modulus of the oil Appendix F. User Manual for CONOIL-III 311 Record 6.4 (all are real variables) ,U30,0, )H, H, Au, U, —, ant, ,8c.z a, ja0 3fLo,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+AH*T H - Henry’s coefficient of solubility Au - Function to modify bubble pressure for temperature U - Bubble pressure (Oil/Gas saturation pressure) — - 0 (No parameter at present) ta8 - Coefficient of volume change due to temperature in duced structural reorientation - Coefficient of linear thermal expansion of solids - Coefficient of linear thermal expansion of water 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, Rp, i, A, (r/), (r/o), 1 — C - Cap-yield collapse modulus number p - Cap-yield collapse modulus exponent K - Plastic shear number lip - Plastic shear exponent 1R - Plastic shear failure ratio - flow rule intercept Appendix F. User Manual for CONOIL-III 312 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 — - 0 (No parameter at present) 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 0S - Initial oil saturation 9S - Initial gas saturation (S + 0S H-9S 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) 09k,? - 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) >1 - interpolate using given temperature-viscosity profile (IV data pairs, maximum 10) 01V - Options to estimate viscosity of oil 0 - use a given constant value (in Pa.s) Record Record Appendix Fl, Al, Dl...D3 Al.. Bl...B3 Cl... Fl...F3 6.8 6.7 F2, A2, F. .A3 1V 9 C3 (necessary (necessary F3 A3, User - - - - - - >1 >1 Parameters k,. 09 Parameters ICrg Parameters Parameters Parameters Bi, 0 Options 1 1 Manual - - - - - use use 2.E-5 = 6.4) use interpolate Puttangunta interpolate (1V 9 (1V 0 = B2, = = only only C1(Sg A1(SL, D1(D2 B1(B2 the a the B3, to data data for given Pa.s) if if built-in estimate built-in for for for for for IMPF IMPF — Cl, CONOIL — pairs, pairs, — oil-gas constant relative relative relative relative A2)- 3 using using S 9 )” 3 et.al 02, feature = = feature maximum maximum viscosity 03, -III 2) (1988), 2) capillary given given permeability permeability permeability permeability value Dl, in in the to,o D2, temperature-viscosity of temperature-viscosity the (in 10) 10) pressure gas program D3 Pa..s) program should of of of of oil gas oil water be (a (oil-gas) (oil-water) (oil-gas) (Correlation given constant (oil-water) in record profile profile value by 313 Record Record Record of Appendix gas ___ ___ Vi, V Vi, Vi, (oil-gas) 6.11 6.10 6.9 Ti, Ti, Ti, F. (necessary (necessary V2, (necessary V2, User V2, Vi,... Ti,... Vi,... Ti,... Vi,... Ti,... T2,... T2,... Pc T2,... Manual Vi,, V 0 = Fl only - - - - - (if - (ifIV=0) - - (if - (ifIV 0 =0) (ifIV=0) (if only only Viscosity Temperature Temperature Viscosity Constant Viscosity Constant Temperature Constant for Pa(Sg 1V 9 > 1V 0 IV if if if IMPF CONOIL > IMPF IMPF — 1, 1, 1, values values viscosity values F2)F3 viscosity viscosity IV,, 1V 9 IV, = = = values values values -III 2 data data 2 data 2 and in in in and and value the value the value the pairs, pairs, in in pairs, IV,, in IVg 1V 0 the the given given the given of of of = = = maximum maximum maximum given given given gas 0 oil water 0 0 profile profile profile or or or (in (in >1) profile profile profile >1) >1) (in Pa.s) Pa.s) (in (in (in 10) 10) 10) Pa.s) Pa.s) Fa.s) Pa.s) (in (in (in °C) °C) °C) 314 Record Record Record Appendix TIMEJ,... M, LINSIT, TEM1,... PINSIT LINSIT SIGXY 9 8 7 TEM1, SIGX, SIGY SIGX SIGZ F. (NTEL (one (NTEMP N M User PINSIT line) SIGY, ------ - TIME1, Element Stress Stress Stress Stress o Option o Option 1 1 Temperature Time Node lines, Manual - - - - lines, print read do set SIGZ, in number only in in in in not the TEM2, to to the only the number in-situ xy x y z for print specify print in-situ direction if direction direction SIGXY, in-situ direction temperature CONOIL LINSIT in if TIMEj NTEMP in-situ stress the in-situ stresses stresses SIGYZ, given -III = data stress 1) > stresses time temperature to (NPTS from 0) SIGZX, zero data profile data data PP1 time pairs, profile maximum 15) 315 Appendix F. User Manual for CONOIL-III 316 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) There must be 10 data per line, except the last line 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 Record Record There Record Appendix N, N, Li, must DGRAV DTIME 14 NFCODE, 13 DFX, 12 Li,... L2 DFY DFX DFZ F. be (NFIX (NLOAD ((ICH.EL-1)/10 N N User DFY, 10 ------ - data Node Increment Increment Increment Node List Increase Time e d c lines, Manual DX, = = = DFZI lines, 1 of 1 1 per number number increment print print where DY, print only element in line, for only + in in in DZ, gravity if velocity stresses strains 1 CONOIL-IlI results x y z except NFIX> lines, if force force force numbers DP NLOAD> only and are vectors and the 0) printed pore to coordinates last if ICHEL be 0) line pressure removed > of 0) in the this integration increment point 317 Appendix NFCODE F. DY DX DP DZ User - - - - - Prescribed Prescribed Prescribed Prescribed tions Four d c a b Manual ====== = 2 1 1 1 1 0 0 0 0 digit associated will subsequent will will will will free free pore free for have have have have have in in in code pore displacement displacement displacement pressure CONOIL z x y direction direction prescribed prescribed direction prescribed prescribed zero pressure with ‘abcd’ increments can absolute -III the which have in in in node incremental incremental incremental incremental x y z direction direction direction any specifies pore value pressure pore displacement displacement displacement the (undrained fixity pressure for this condi boundary) DZ DY DX DP and all 318 Appendix F. User Manual for CONOIL-IlI 319 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 ...I 6 0 ZL ol... AB, BC, CD, DA - Totally Fixed AE, BF, CG, DH - Vertically Free Figure F.2: Finite element mesh for example problem 1 L •o L LL 09 OOL OL •o ir 0 L OL VH3N35 xrpudd 0’OLLLV 0’OLLLE L 96’ L j 0 0000 0 L S-”O”O’L S-O”OLL S-”O”O6 t u L 0 O”O”LOL “O”O”O6 “O”O”L”LE o”VL”o9 “O”L”I.”L1 “O”L”O”L9 “O• “O”O”L”0V “o”o”o”oH • 0 SS3UIS ‘OL’L’LL isfl ‘LLLLLIV’OL •o’•o••o L 0”O”O”OOOLL’8 OOO”OOOL.V9 O”O”O”OOOLL9 8L9S’V’Li .0 “O”O”O”OOOU.L “OOO “O”O”O”OOOLL6 • “O”O”O”O’OILL “O”O”O”O’OLLLL Q’ “O”O”O”O’OOLLLL “O”O”OO’OOLLL “O”O”O”000LL01 ‘L ‘L’LLLOL68L99 SISA1VNV nuvJj •o••o’•o ‘O’000 ioj io; 3N0 “o”o”o”o”oog”oog”oog’ ••O••O••O••O•OOS•OOS’OOS’L ‘•o• OO”O”OOOOOO”O”OOLOL Idmxa SL3L’SL3LS3LOOO”O”O”sO 1VNOISN3YUO 111710N00 j ONIOVO7 • AOFOIL (A)NALYSIS OF (D)EFORMATION AND (F)LOW IN (OIL) SANDS o GENERAL STRESS ANALYSIS, ONE DIMENSIONAL LOADING NODAL COORDINATES AND TEMPERATURE CD NODE XCOORD Y-COORD Z-COORD TEMP 1 0.000 0000 0.000 0.000 ‘•l 2 1.000 0.000 0.000 0.000 3 1.000 1.000 — C) 0.000 0.000 U 4 0.000 1.000 0.000 0.000 5 0.000 0.000 1.000 0.000 6 1.000 0.000 1.000 0.000 7 1.000 1.000 1.000 0.000 8 0.000 1.000 1.000 0.000 9 0.000 0.000 2.000 0.000 10 1.000 0.000 2.000 0.000 11 1.000 1.000 2.000 0.000 12 0.000 1.000 2.000 0.000 ELEMENT-NODAL INFORMATION NODES ELE. NO. 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 5 6 7 8 9 10 11 12 MATERIAL PROPERTIES MATERIAL I = MODEL N LINEAR/NONLINEAR ELASTIC MODEL IS I GE =0 USE MEAN NORMAL STRESS ISIGB =0 USE MEAN NORMAL STRESS ILSHD =0 NO LOAD SHEDDING I MPF =0 FULLY SATURATED SOIL L’3 O.100E+O1 0.150E+04 O.000E÷OO O.000E+OO 0.100E+04 O.000E÷OO O.200E+O2 O.000E+OO O.000E+OO O.000E+OO O.000E÷OO O.350E+02 O.000E÷OO O.000E+OO O.000E+OO O.000E+OO O.000E+OO O.100E+16 0,100E+16 O.100E+16 O.000E+OO O.000E+OO O.000E+OO O.000E+OO O.OOOEOO O.000E+OO O.000E+OO O.000E+OO O.000E+OO O.000E+O0 INITIAL STRESSES X V Z SHEAR-XY SHEAR-YZ SHEAR-ZX PORE ELEM STRESS STRESS STRESS STRESS STRESS STRESS PRESSURE 1 O.5000E+03 O.5000E+03 O.5000E+03 O.0000E+OO O.0000E+OO O.0000E+OO O.0000E+OO 2 O.5000E+03 O.5OOOEO3 O.5000E+03 O.0000E+OO O.0000E+OO O.0000E+OO O.0000E+OO INCREMENT NUMBER = INCH. IN GRAVITY = 0.0000E÷OO TOTAL GRAVITY = O.0000E+O0 TIME INCREMENT = O.1000E+O1 TOTAL TIME = O.1000E+O1 NODAL DISPLACEMENTS INCREMENTAL ABSOLUTE I-s NODE XI VI ZI XA VA ZA 1 -0.8333E-16 -O.8333E- 16 -O.2500E-15 -O.8333E-16 -O.8333E- 16 -O.2500E- 15 0 2 O.8333E-16 -0.8333E- 16 0.2500E-15 O.8333E-16 -O.8333E- 16 -0. 2500E- 15 3 O.8333E-16 0.B333E- 16 -0. 2500E- 15 O.8333E-16 0.8333E- 16 -O.2500E- 15 4 -O.8333E-16 0.8333E- 16 -O.2500E- 15 -O.8333E-16 O.8333E- 16 -O.2500E- 15 5 -O.1667E-15 -0. 1667E- 15 -0. 5556E-03 -O.1667E-15 -0. 1667E-15 -0. 5556E-03 6 O.1667E-15 -0. 1667E- 15 -O.5556E-O3 O.1667E-15 -0. 1667E- 15 O.5556E-03 7 O.1667E-15 0. 1667E- 15 -0.5556E-03 O.1667E-15 0. 1667E- 15 -0. 5556E-03 8 -O.1667E-15 0. 1667E- 15 -O.5556E-O3 -O.1667E-15 0. 1667E- 15 •O.5556E-O3 9 -O.8333E-16 -O.8333E- 16 -0. 1111E-O2 -O.8333E-16 -O.8333E- 16 -0. 1111E-O2 10 O.8333E-16 -O.8333E- 16 -0. 1111E-02 O.8333E-16 -O.8333E- 16 -0. 1111E-02 11 O.8333E-16 O.8333E- 16 -0. 111 1E-O2 O.8333E-16 O.8333E- 16 -0.111 IE-02 12 -O.8333E-16 O.8333E- 16 -0.1 111E-02 -O.8333E-16 O.8333E- 16 -0.111 1E-02 MODULI VALUES ELASTIC BULK POISSION PLASTIC VOID WATER OIL GAS EL EM MODULUS MODULUS RATIO PARAMETER RATIO SATURAN SATURAN SATURAN 0. 1500E+06 0. 1000E+06 0.2500E+O0 0 .0000E+OO 0. I000E+01 0. I000E+O1 0 .0000E+O0 0. 0000E+OO 2 0. 1500E+O6 0. 1000E+06 0. 2500E+00 0. 0000E+0O 0. 1000E+01 0. 1000E+O1 O.0000E+00 0. 0000E+00 STRAINS X V Z XV YZ ZX VOL. TNT. POINT COORDINATES ELEM STRAIN STRAIN STRAIN STRAIN STRAIN STRAIN STRAIN X V Z -O.2981E- 15 -O.2981E- 15 0.5556E-O3 0.1233E-31 O.4825E-16 -O.4816E-16 O.5556E-03 O.79E+0O 0.21E+00 O.79E+0O C3 2 -0. 20 lYE- 15 -D.2019E- 15 O.5556E-03 -0.3698E-31 -0.4779E-16 O.4780E-16 0.5556E-03 O.79E+OO 0.21E+00 0.18E+0l I.3 STRESS AND PORE PRESSURES X V Z XV YZ ZX PORE STRESS ELEM STRESS STRESS STRESS STRESS STRESS STRESS PRESSURE LEVEL 1 O.5333E-O3 O.5333E+03 O.6000E+03 O.7396E-27 O.895E-11 -O.2889E-11 O.0000E+OO O.2398E-O1 O.5333E+O3 O.5333E+03 O.6000E+03 -O.2219E-26 -O.2867E-11 O.2868E-I1 O.0000E+OO O.23g8E-O rj I