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Elasto-plastic deformation and flow analysis in oil sand masses Srithar, Thillaikanagasabai 1994

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ELASTO-PLASTIC DEFORMATION AND FLOW ANALYSIS IN OIL SAND MASSES by THILLAIKANAGASABAI SRITHAR B. Sc (Engineering), University of Peradeniya, Sri Lanka, 1985 M. A. Sc. (Civil Engineering) University of British Columbia, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES  Department of CIVIL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  April, 1994  © THILLAIKANAGASABAI SRITHAR, 1994  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my .  department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  Civil Engineering  The University of British Columbia Vancouver, Canada  Date  DE-6 (2188)  -  A?R L  9  L  Abstract  Prediction of stresses, deformations and fluid flow in oil sand layers are important in the design of an oil recovery process. In this study, an analytical formulation is developed to predict these responses, and implemented in both 2-dimensional and 3-dimensional finite element programs. Modelling of the deformation behaviour of the oil sand skeleton and modelling of the three-phase pore fluid behaviour are the key issues in developing the analytical procedure. The dilative nature of the dense oil sand matrix, stress paths that involve decrease in mean normal stress under constant shear stress, and loading-unloading sequences are some of the important aspects to be considered in modelling the stress-strain behaviour of the sand skeleton. Linear and nonlinear elastic models have been found incapable of handling these aspects, and an elasto-plastic model is postulated to capture the above aspects realistically. The elasto-plastic model is a double-hardening type and consists of cone and cap-type yield surfaces. The model has been verified by comparison with laboratory test results on oil sand samples under various stress paths and found to be in very good agreement. The pore fluid in oil sand comprises three phases namely, water, bitumen and gas. The effects of the individual phase components are considered and modelled through an equivalent fluid that has compressibility and hydraulic conductivity characteristics representative of the components. Compressibility of the gas phase is obtained using gas laws and the equivalent compressibility is derived by considering the individual contributions of the phase components. Equivalent hydraulic conductivity is derived from the knowledge of relative permeabilities and viscosities of the phase components. Effects of temperature changes due to steam injection are also included directly  11  in the stress-strain relation and in the flow continuity equations.  The analytical  equations for the coupled stress, deformation and flow problem are solved by a finite element procedure. The finite element programs have been verified by comparing the program results with closed form solutions and laboratory test results. The finite element program has been applied to predict the responses of a hor izontal well pair in the underground test facility of Alberta Oil Sand Technology and Research Authority (AOSTRA). The results are discussed and compared with the measured responses wherever possible, and indicate the analysis gives insights into the likely behaviour in terms of stresses, deformations and flow and would be important in the successful design and operation of an oil recovery scheme.  111  Table of Contents  Abstract  ii  List of Tables  x  List of Figures  xi  Acknowledgement  xvi  Nomenclature 1  2  xvii  Introduction  1  1.1  Characteristics of Oil Sand  4  1.2  Scope and Organization of the Thesis  8  Review of Literature  10  2.1  Stress-Strain Models  10  2.1.1  Stress-Strain Behaviour of Oil Sands  11  2.1.2  Stress-Strain Models for Sand  19  2.1.2.1  Elasto-Plastic Models  20  2.1.2.2  Constituents of Theory of Plasticity  22  2.1.3  Stress Dilatancy Relation  23  2.1.4  Modelling of Stress-Strain Behaviour of Oil Sand  24  2.2  Modelling of Fluid Flow in Oil Sand  25  2.3  Coupled Geomechanical-Fluid Flow Models for Oil Sands  27  2.4  Comments  30  iv  3  Stress-Strain Model Employed  32  3.1  Introduction  32  3.2  Description of the Model  35  3.3  Plastic Shear Strain by Cone-Type Yielding  37  3.4  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 [C ] 8  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 3.9.1  Load Shedding Technique  68 70  3.10 Discussion 4  51  .  72  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  75  Parameters kE and n 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  82  4.2.3.3  Evaluation of KG, np and R 1  4.2.4 4.3  Evaluation of Strain Softening Parameters  Validation of the Stress-Strain Model 4.3.1  4.3.2  5  and )  83 86 87  Validation against Test Results on Ottawa Sand  88  4.3.1.1  Parameters for Ottawa Sand  91  4.3.1.2  Validation  96  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  Flow Continuity Equation  115  5.1  Introduction  115  5.2  Derivation of Governing Flow Equation  116  5.3  Permeability of the Porous Medium  123  5.4  Evaluation of Relative Permeabilities  124  5.5  Viscosity of the Pore Fluid Components  132  5.5.1  Viscosity of Oil  132  5.5.2  Viscosity of Water  134  5.5.3  Viscosity of Gas  136  5.6  Compressibility of the Pore Fluid Components  136  5.7  Incorporation of Temperature Effects  140  vi  5.8 6  Discussion  Analytical and Finite Element Formulation  144  6.1  144  6.2  Introduction 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  6.6  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  Finite Element Programs 6.6.1  6.7 7  142  .  166  2-Dimensional Program CONOIL-Il  166  .  3-Dimensional Program CONOIL-Ill  167  Verification and Application of the Analytical Procedure 7.1  Introduction  7.2  Aspects Checked by Previous Researchers  7.3  Validation of Other Aspects  7.4  Verification of the 3-Dimensional Version  7.5  Application to an Oil Recovery Problem 7.5.1  7.6  .  .  Analysis with Reduced Permeability  Other Applications in Geotechnical Engineering  vii  168  183 .  8  Summary and Conclusions  216  8.1  219  Recommendations for Further Research  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  B.1.3  Sample calculations  .  .  .  249 249  .  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 C.1 2-Dimensional Code CONOIL-Il  258 258  C.1.1  Geometry Program  258  C.1.2  Main Program  259  C.2 3-dimensional code CONOIL-Ill  261  D Amounts of Flow of Different Phases  264  E User Manual for CONOIL-Il  270  E.1 Introduction  270  E.2 Geometry Program  272 viii  E.3 Main Program  .  E.4 Detail Explanations  292  E.4.1  Geometry Program  292  E.4.2  Main Program  295  F User Manual for CONOIL-Ill F.1  275  304  Introduction  304  F.2 Input Data  305  F.3  Example Problem 1  319  F.3.1  Data File for Example 1  320  F.3.2  Output file for Example 1  321  ix  List of Tables  4.1  Summary of Soil Parameters  4.2  Soil Parameters for Ottawa Sand at Dr  4.3  Details of the Test Samples  101  4.4  Soil Parameters for Oil Sand  107  5.1  Parameters needed for relative permeability calculations  133  7.1  Parameters for Modelling of Triaxial Test in Oil Sand  178  7.2  Model Parameters Used for Ottawa Sand  181  7.3  Parameters Used for Thermal Consolidation  184  7.4  Parameters Used for the Oil Recovery Problem.  7.5  Soil Parameters Used for the Example Problem  75 =  50%  94  .  192 209  B.1 Viscosity of water between 0 and 1000 C  251  B.2 Viscosity of water below 00 C  251  B.3 Viscosity of water above 1000 C  251  D.1 Average Viscosities and Temperatures in Different Zones  266  D.2 Initial Saturations and Mobilities of Water and Oil  268  D.3 Calculation of Flow and Saturations with Time  269  D.4 Saturations and Mobilities of Water and Oil after 300 Days  269  E.1 Element Types  294  E.2 Time Increment Scheme  300  E.3 Load Increments  302  x  List of Figures  1.1  Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978)  2  1.2  In-situ Structure of Oil Sand (after Dusseault,1980)  6  1.3  Undrained Equilibrium behaviour of an Element of Soil upon Unload ing (after Sobkowicz and Morgenstern, 1984)  7  2.1  Fabric of Granular Assemblies (after Dusseault and Morgenstetn, 1978) 12  2.2  Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseault and Morgenstern, 1978)  13  2.3  Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987) 14  2.4  Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agar et al., 1987)  2.5  15  Effect of Temperature on Stress-Strain Behaviour (after Agar et al., 1987)  2.6  16  Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et al., 1987)  18  3.1  A Possible Stress Path During Steam Injection  34  3.2  Components of Strain Increment  36  3.3  Mobilized Plane under 2-D Conditions  38  3.4  Spatial Mobilized Plane under 3-D Conditions  40  3.5  Yield and Failure Criteria on  3.6  Matsuoka-Nakai and Mohr-Coulomb Failure Criteria  3.7  Effect of Intermediate Principal Stress (After Salgado (1990))  TSMp  3  —  °sMp  Space  .  43  .  45 .  .  .  46  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  for Toyoura Sand (after Matsuoka, 1983)  YsMP  .  .  .  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  —  3 o  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  77  4.2  Evaluation of kB and m  78  4.3  Evaluation of C and p  80  4.4  Evaluation of  83  4.5  Evaluation of ) and it  4.6  Evaluation of G , and 1  4.7  Evaluation of K 0 and np  4.8  Evaluation of  4.9  Grain Size Distribution Curve for Ottawa Sand (after Neguessy  ii  and L  ,  84 85  ‘quit .  and q  86 88  ,  1985)  89  4.10 Stress Paths Investigated on Ottawa Sand  90  4.11 Variation of Young’s moduli with confining stresses  91  4.12 Plastic Collapse Parameters for Ottawa Sand  92  4.13 Failure Parameters for Ottawa Sand  93  4.14 Flow Rule Parameters for Ottawa Sand  94  4.15 Hardening Rule Parameters for Ottawa Sand  95  4.16 Results for Triaxial Compression on Ottawa Sand  97  4.17 Results for Proportional Loading on Ottawa Sand  98  xii  4.18 Results for Various Stress Paths on Ottawa Sand  99  4.19 Grain Size Distribution for Athabasca Oil Sands, (after Edmunds et al., 1987)  .  .  100  .  4.20 Determination of kB and m for Oil Sand  102  4.21 Plastic Collapse Parameters for Oil Sand  103  4.22 Failure Parameters for Oil Sand  104  4.23 Determination of  105  and np for Oil Sand  4.24 Flow Rule Parameters for Oil Sand  106  4.25 Results for Isotropic Compression Test on Oil Sand  108  4.26 Results for Triaxial Compression Tests on Oil Sand  110  4.27 Results for Tests with Various Stress Paths on Oil Sand  .  .  .  4.28 Sensitivity of Parameters C,p,A and p 4.29 Sensitivity of Parameters KG, np, R 1 and  111 112  i  5.1  One dimensional flow of a single phase in an element  5.2  Typical two-phase relative permeability variations (after Aziz and Set  113 117  tan, 1979)  125  5.3  Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979)  127  5.4  Comparison of calculated and experimental three-phase oil relative per meability (after Kokal and Maini, 1990)  5.5  Comparison of calculated and experimental relative permeabilities us ing power law functions  5.6  130  131  Experimental and predicted values of viscosity (after Puttagunta et al., 1988)  135  6.1  Finite Element Types Used in 2-Dimensional Analysis  160  6.2  Finite Element Types Used in 3-Dimensional Analysis  161  6.3  Flow Chart for the Finite Element Programs  165  xiii  7.1  Stresses and Displacements Around a Circular Opening for an Elastic Material (after Cheung, 1985)  7.2  170  Comparison of Observed and Predicted Pore Pressures (after Cheung, 1985)  171  7.3  Comparison of Observed and Predicted Strains (after Cheung, 1985)  172  7.4  Results for a Circular Footing on a Finite Layer (after Vaziri, 1986)  173  7.5  Stresses and Displacement in Circular Cylinder (after Srithar, 1989)  174  7.6  Pore Pressure Variation with Time for Thermal Consolidation (after Srithar, 1989)  176  7.7  Undrained Volumetric Expansion (after Srithar, 1989)  177  7.8  Finite Element Modelling of Triaxial Test  179  7.9  Comparison of Measured and Predicted Results in Triaxial Compres sion Test  180  7.10 Comparison of Measured and Predicted Results for a Load-Unload Test in Ottawa Sand  182  7.11 Finite Element Mesh for Thermal Consolidation  184  7.12 Comparison of Pore pressures for Thermal Consolidation  185  7.13 A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991  187  7.14 Plan View of the UTF (after Scott et al., 1991)  188  7.15 Vertical Cross-Sectional View of the Well Pairs  189  7.16 Finite Element Modelling of the Well Pair  191  7.17 Temperature Contours in the Oil Sand Layer  193  7.18 Pore Pressure Variations in the Oil Sand Layer  195  7.19 Comparison of Pore Pressures in the Oil Sand Layer  196  7.20 Horizontal Stress Variations in the Oil Sand Layer  197  7.21 Vertical Stress Variations in the Oil Sand Layer  198  7.22 Stress Ratio Variations in the Oil Sand Layer  199  xiv  7.23 Comparison of Horizontal Displacements at 7 m from Wells  200  7.24 Vertical Displacements at the Injection Well Level  201  7.25 Total Amount of Flow with Time  202  7.26 Individual Flow Rates of Water and Oil  204  7.27 Total Amount of Oil Flow  205  7.28 Pore Pressure Variation for Analysis 2  206  7.29 Stress Ratio Variation for Analysis 2  207  7.30 Details of the Cases Analyzed  210  7.31 Variation of Pore Pressure Ratio for Case 1  212  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  .  .  .  306  F.1  Available Element Types  F.2  Finite element mesh for example problem 1  xv  319  Acknowledgement  The author is greatly indebted to his supervisor Professor P. M. Byrne for his guid ance, valuable suggestions and the encouragement throughout this research. The author wishes to express his appreciation to the members of the supervisory committee for reviewing the manuscript and making constructive criticisms. Appre ciation is also extended to Mr. Jim Grieg for his valuable helps on the computer aspects. The author expresses his gratitude to his wife, Vasuki, for her support and toler ance of the odd working habits of a graduate student. The author would like to thank his colleagues in Dept. of Civil Engineering  ,  in  particular, Uthayakumar and Hendra for sharing common interest. Finally, the fellowship awarded by the University of British Columbia and the research grant provided by Alberta Oil Sand Technology and Research Authority (AO STRA) are gratefully acknowledged.  xvi  Nomenclature  B B  bulk modulus pore pressure shape function derivatives displacement shape function derivatives  C CEQ  plastic collapse modulus equivalent compressibility  D  stress-strain matrix  E  Young’s modulus  e  void ratio  F  body force vector  f  plastic collapse yield function initial plastic shear parameter  Gt H I, 12 and 13 0 K k kB  tangent plastic shear parameter Henry’s constant stress invariants plastic shear number Darcy’s permeability of the porous medium bulk modulus number Young’s modulus number  kEQ kh kmi kmT  equivalent hydraulic conductivity permeability in horizontal direction mobility of phase ‘1’ total mobility xvii  kri  relative permeability of phase ‘1’  krog  relative permeability of oil in oil-gas system  kr  relative permeability of oil in oil-water system relative permeability of oil at critical water saturation permeability in vertical direction  l, l, and l  direction cosines of o to the x, y and z axes  M  constrained modulus  m  bulk modulus exponent  mz,my and m  direction cosines of o2 to the x, y and z axes  N  shape functions for pore pressures  N  shape functions for displacements  n n, n, and n 2 np  Young’s modulus exponent direction cosines of o 3 to the x, y and z axes 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 t  critical water saturation time  U  displacement vector  V  volume xviii  W  plastic collapse work  Greek letters  coefficient of volumetric thermal expansion cEQ  equivalent coefficient of thermal expansion shear strain Kronecker delta  El, 62  and  63  principal strains plastic collapse strains  e 6  elastic strains plastic shear strains volumetric strain stress ratio failure stress ratio at atmosphere  8  temperature strain softening constant flow rule slope proportionality constant  p  flow rule intercept viscosity of phase ‘1’  P30,0  v  viscosity of oil at 30°C and at 0 gauge pressure Poisson’s ratio normal stress  1, 2  and u 3  principal stresses mean normal stress  r  shear stress xix  (6m  mobilized friction angle  Subscripts  f  failure state  g  gas phase  j  partial derivative with respect to  MP o SMP ult w  mobilized plane oil phase spatial mobilized plane ultimate state water phase  Superscripts  c  plastic collapse condition  e  elastic condition plastic shear condition  xx  j  Chapter 1  Introduction The oil contained in oil sand deposits in northern Alberta is one of the major resources in Canada. These deposits underlie an area of about 32,000 square kilometres with estimated in-place reserves of 146.5 million cubic meters (Mosscop, 1980). Much of the oil exists as high viscosity bitumen in Arenaceous Cretaceous formations, primarily in the Athabasca oil sand deposits (see figure 1.1). Approximately 5% of these deposits are found at depths less than 50 m and the rest are encountered at depths from 200 to 700 m. Oil recovery schemes involve open pit mining in the shallow oil sand formations, and in-situ extraction techniques such as tunnels and well-bores in the deep oil sand formations. In the in-situ extraction procedures some form of heating is often required as the very high viscosity of the bitumen makes conventional recovery by pumping impractical. In-situ thermal methods such as steam injection through vertical wellbore have been used and are relatively effective for the recovery of heavy oil from deep seated formations. There have been, however, numerous well casing failures and instability problems reported during field injection trials. During steam injection, high pore fluid and stress gradients are created around the well-bore which can lead to the instability and collapse of the well casing. Therefore, to understand the mech anisms involved and to design these oil recovery schemes rationally and economically, analyses which capture the complex engineering characteristics of the oil sand are necessary. Analyzing the problems related to oil sands is somewhat different from analyzing  1  Chapter 1. Introduction  2  Northwest Territories  United States of Amenca  Figure 1.1: Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978)  Chapter 1. Introduction  3  a general geotechnical problem because of the nature of the oil sand and the recovery process involved.  Oil sand comprises four phases; solid, water, bitumen and gas,  whereas, a general soil consists of three phases; solid, water and air. The presence of bitumen and gas makes the analytical procedures for oil sands different and difficult. Oil recovery by steam injection will cause changes in temperature and their effects are also of prime concern. The changes in temperature induce changes in volume and pore fluid pressure which in turn affect the engineering properties such as strength, compressibility and hydraulic conductivity. When there is an increase in temperature, if the volume change of the pore fluid components is greater than that of the voids in the soil skeleton, there will be an increase in pore pressure and consequently a reduction in effective stress. The effective stresses may become zero and liquefaction may occur, if the oil sand is subjected to rapid increase in temperature and if an undrained condition prevails. The deformation and flow behaviour of oil sand is governed by several factors. However, it can be categorized into two major constituents; the behaviour of pore fluids and the behaviour of sand skeleton. An analytical model for the oil sand was first developed by Harris and Sobkowicz (1977);  It was later extended by Byrne  and Grigg (1980), Byrne and Janzen (1984) and Byrne and Vaziri (1986). However, these analytical models consider a linear or nonlinear elastic behaviour for the sand skeleton. Oil sand is very dense in its natural state and shows significant dilation upon shear. The linear and nonlinear elastic models are not capable of modelling the dilation effectively. Furthermore, steam injection and subsequent recovery will lead to loading and unloading cycles and for realistic modeffing an elasto-plastic model is necessary. In this study, a double hardening elasto-plastic model is postulated for the sand skeleton based on the models by Nakai and Matsuoka (1983) and by Lade (1977), and it is very effective in handling the dilation. With regard to the pore fluid behaviour, Byrne and Vaziri (1986) considered the  Chapter 1. Introduction  4  individual contributions of the pore fluid components in the compressibility but not in the hydraulic conductivity. In this research work, the relative permeabilities of water, bitumen and gas are considered and an equivalent hydraulic conductivity is derived to model the pore fluid behaviour appropriately. The equivalent compressibility term as proposed by Byrne and Vaziri (1986) is also included. The effects of temperature changes in stresses and volume changes have been di rectly included in the governing equilibrium and flow continuity equations. It should be noted that the equation of thermal energy balance is not considered in the analyt ical model. However, the temperature-time history which is obtained from a separate heat flow analysis or by some other means is considered as an input to the analytical model and, the effects of these temperature changes on the stress-strain behaviour and the fluid flow are evaluated. An analytical procedure considering all these aspects has been developed and incorporated in the 2-dimensional finite element code CONOIL-Il. In order to analyze the three dimensional effects a new 3-dimensional finite element code CONOIL-Ill is also developed.  1.1  Characteristics of Oil Sand  Since the oil sand is different form a general soil, it is appropriate to present some brief descriptions about its unusual characteristics. Oil sand can be considered as a four phase geological material comprising solid, water, bitumen and gas. The two dominant physical characteristics of the oil sand are the quartz mineralogy and the large quantity of interstitial bitumen. The quartz grains of the oil sand are 99% water wet as the water phase forms a continuous film around it. A larger portion of the pore space is filled with bitumen and since bitumen and water form continuous phases, gas can only exists in the form of discrete bubbles (free gas). However, significant quantities of gas can also exist in the dissolved state in the pore fluid. An illustration  Chapter 1. Introduction  5  of oil sand structure (Dusseault, 1980) is shown in figure 1.2. In its natural state, oil sand is very dense, uncemented, fine to medium grained and exhibits high shear strength and dilatancy. It shows low compressibility charac teristics compared to normal dense sand of similar mineralogy. The extremely high viscosity of bitumen makes the effective hydraulic conductivity very low and causes the oil sand to behave in an undrained manner. Another unusual characteristic of oil sand is its behaviour upon unloading. Be cause of the very low effective hydraulic conductivity, oil sand behaves in an undrained manner, however, it responds quite differently compared to the undrained behaviour of a normal sand. Above the liquid-gas saturation pressure (U ), oil sand behaves 119 like a normal sand (path I of figure 1.3). A decrease in confining stress will result in a decrease in pore pressure and the effective stress remains constant. When the level of confining stress decreases below the liquid-gas saturation pressure, the dissolved gas in the pore fluid comes out of solution and causes the pore fluid to become very com pressible. At this point, the soil matrix commences to take the load and the effective stress decreases while the pore pressure stays constant (path J). As the effective stress decreases, the soil skeleton compressibility increases and becomes comparable to the pore fluid compressibility. Then, the pore fluid takes the load and the pore pressure starts to decrease again (path K). At some stage, the effective stress becomes zero and the physical consequences of this process are significant increase in volume and a marked reduction in shear strength. Plots of pore pressure versus total stress for saturated (path M), unsaturated (path L) and gassy soils (path J-K) are shown in figure 1.3. A comprehensive study of the gas exsolution phenomenon upon unloading can be found in Sobkowicz and Morgenstern (1984).  I  U)  I-.  (b  I-.  Tj  —m 0 o m -  C-  a..  ,oo  m  CC-  CC -  -  a  C  rcn -C  o a;-’  0  I-.  Chapter 1. Introduction  7  IN SITU  U  STRESS  / /..—. u=o, o-c=o Uj/g  ____,_,_  D  J  /  (I, C,,  w 0 LAJ  0  0  I  ..°  TOTAL STRESS  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  1.2  8  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 threephase 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  Chapter 1. Introduction  9  laboratory experiments are considered and the results are compared. Application to an oil recovery process by steam injection is presented and the results are analyzed in detail. Possible applications of the developed formulation for general geotechnical problems are discussed and an example problem is also given. Chapter 8 summarizes the important findings of this research work. Some com ments on the aspects which warrant further investigation are also stated in this chap ter.  Chapter 2  Review of Literature The research work carried out in this study can be broadly classified under the fol lowing topics; stress-strain model for the oil sand, modelling of flow characteristics of the three-phase pore fluid; and the analytical and finite element formulations. There fore, it is appropriate to present a review on the previous research works under these subheadings. The intention of the literature review presented in this chapter is not to critically assess each and every research work but to give an overall picture, and to set the stage to discuss the work carried out in this study.  2.1  Stress-Strain Models  The stress-strain behaviour of the oil sand skeleton is essentially the stress-strain behaviour of a dense sand. This conclusion was not widely accepted until the com pletion of series of research programs at the University of Alberta in the late 1970s and in 1980s. In particular, the perception of bitumen as a cementing material was widely held until the last decade, as many geologists and petroleum engineers failed to recognize the geomechanical behaviour of the sand skeleton. It is now recognized that the oil sands must be considered as a particulate material and its behaviour can be described by an appropriate stress-strain model. Before going into a detailed re view of the stress-strain models, it will be useful to describe the observed stress-strain behaviour of oil sands. The next subsection summarizes the stress-strain behaviour of oil sands in laboratory experiments.  10  Chapter 2. Review of Literature  2.1.1  11  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, with some straight contacts (arrows)  (c) Athabasca oil sand 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 (U 0  -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  ./7O1  12 (MPa)  (a) Various Stress Paths  16  0.5  1.5  1.0  e  (%)  (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  . ATHABASCA OIL SAND (This Study) v OTTAWA SAND (This Study  Athobasca Oil Sand  aU  DIJSSEAUT & MORGENSTERN(1978) SOBKOWICZ (1982) DUNCAN & CHANG (1970)  C D U, in  a,  U)  40  C I  .  D  U -C ‘/, —  0  30  a) U) C  20  1  2  3  4  5  Effective Confining Stress  7  6  c  8  (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  e  1.5  2.0  (%)  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 tnaxial 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•  2.  :  ‘7  Mi,m shaer strength  •  6.9 MPa  I Athabasca £ - 2200 MPa CoidLake  S Axial Strain (%)  Figure 2.6: Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et aL, 1987)  Chapter 2. Review of Literature  i9  Therefore, the modelling of oil sand behaviour should include two significant fea tures; non-recoverable strains and dilatancy. A realistic model must take the deforma tion history into account, particularly if the stresses are to be cycled through loading and unloading. The elasto-plastic formulation incorporates these features naturally. There are a number of elasto-plastic stress-strain models available for sands in the literature and a brief review of those are presented next.  2.1.2  Stress-Strain Models for Sand  A number of models have been proposed in the literature for the stress-strain be haviour of sand. Most of them make use of the well developed classical theories of elasticity and plasticity either separately or in a combined form. These theories are based on the observations made on materials that can be described in the context of continuum mechanics. To adopt these theories to model the stress-strain behaviour of sand, they have to be modified. Different modifications are made to capture dis tinguished features of sand behaviour and thus, different models are proposed by different researchers.  One of the difficult features of sand behaviour to model has  been the shear induced volume change. Basically, constitutive models can be classified into two categories; linear or in cremental elastic models and elasto-plastic models. In the theory of elasticity, the state of stress is uniquely determined by the state of strain so that the stress-strain response of an elastic models is independent of the stress path. The simplest elastic model would be the isotropic linear elastic model which requires only two material parameters. Incremental elastic models (Duncan and Chang (1970), Duncan et al. (1980)) are the most commonly used because they can capture the nonlinearity and are easy to use. Essentially, the incremental elastic models also require only two pa rameters when analyzing a load increment. However, to update these two material parameters with stress levels and to model the nonlinearity additional parameters are  Chapter 2. Review of Literature  20  necessary. Generally, in elastic models, the shear and normal stresses and strains are uncoupled from each other. Byrne and Eldrige (1982) incorporated the shear volume coupling effects in the incremental elastic models using a stress dilatancy equation. Reviews of the existing elastic and elasto-plastic constitutive models are avail able in the literature as state-of-the-art papers, special workshops and international symposia. Ko and Sture (1980) provided a clear summary of the most important models as of 1980 and described the methods needed to obtain their coefficients. Chen (1982) described and analyzed what is meant by different levels of elasticity. He also described some of the elasto-plastic models most commonly used for soils. Scott (1985) presented a very lucid treatise on plasticity and stress-strain relations. A series of workshops held at McGill University (1980), University of Grenoble (1982) and Case Western University (1987) and the international symposia (ASCE sympo sium, Florida, 1980; International Symposium, Deift, 1982) provide better insights into the different stress-strain models. Since an elasto-plastic model is proposed in this study, a brief review of elasto plastic models and the related theories are presented next.  2.1.2.1  Elasto-Plastic Models  The theory of plasticity has been developed on the basis of observed stress-strain behaviour of metals. Since soils exhibit plastic non-recoverable strains, the theory of plasticity provides an attractive theoretical framework for the representation of the stress-strain behaviour of soils. However, there are major differences such as the presence of voids and the tendency for volume change during shear that distinguish soils from metals (Lade, 1987). In the elasto-plastic models, the strain increment is decomposed into an elastic component and a plastic component. The amounts of elastic and plastic strains will vary with the level of loading and unloading. The elastic strain increment is obtained  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  2.1.2.2  22  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, Lon,  -  -  applied stress increment vector defining the unit normal to yield surface at the stress point  Chapter 2. Review of Literature  -  H  2.1.3  -  23  vector defining the unit normal to potential surface at the stress point plastic resistance  Stress Dilatancy Relation  The stress dilatancy theory derived from theoretical considerations has been used extensively in stress-strain modeffing of sand. The stress dilatancy theory proposed by Rowe (1962,1971) can be considered a remarkable effort to explain the shear de formation behaviour. After Rowe, a number of other researchers published theories to model the dilatancy following different approaches (Murayama (1964), Matsuoka (1974), Oda and Konishi (1974), Nemat-Nasser (1980)). A noticeable difference be tween Rowe’s theory and the other theories is that Rowe’s theory is independent of the spatial distribution of interparticle contacts. Rowe’s theory considers that sliding occurs on certain favourably oriented contact planes. The orientation of the sliding planes will be such as to minimize the rate of dissipation of energy in sliding friction between particles with respect to energy supplied. Matsuoka (1974) developed the stress dilatancy relationship through a microscopic point of view.  He carried out shear tests by using cylindrical rods to model the  shearing mechanism of soil particles.  From the fundamental measurements of the  angle of the interparticle contact, interparticle force and the angle of interparticle friction, he developed a relationship between the shear resistance and the dilatancy. Lade’s (1977) model incorporates the dilatancy through a empirical relation ob tained by curve fitting. The equation relates a dilation parameter to the amount of plastic work. Nemat-Nasser (1980) presented an equation to describe the volumetric behaviour of soil upon shearing which is based on the mechanics of the relative motion of the grains at the micro level.  The equation was obtained by considering the rate of  frictional losses and the energy balance.  Chapter 2. Review of Literature  2.1.4  24  Modelling of Stress-Strain Behaviour of Oil Sand  Modelling of the geomechanical behaviour of oil sand along with the pore fluid be haviour, so as to describe gas exsolution and other related aspects was first presented by Harris and Sobkowicz (1977). They considered a linear elastic model for the sand skeleton behaviour. A nonlinear elastic model with shear dilation was proposed by Byrne and Grigg in 1980 to model the oil sand skeleton behaviour. Their model is based upon an equivalent elastic analysis using a secant modulus and a single step loading. This was subsequently extended by Byrne and Janzen (1984) who used an incremental tangent modulus rather than a secant modulus. Vaziri (1986) basically used the same model as Byrne and Janzen to represent the stress-strain behaviour of oil sand. In the above cited references, the dilative behaviour of the material is incorporated through a procedure borrowed from thermoelasticity. This method involves applying equivalent nodal loads to predict the correct volume changes. Srithar et al. (1990) pointed out that the thermoelastic approach encounters shortcomings specially in a consolidation type of analysis. It predicts unrealistic oscillating results when large time steps are considered.  Furthermore, the computer algorithm necessitates two  levels of iterations; one for stress calculations, and the other for shear induced volume change corrections. Wan et al. (1991) stated that the method of including dilation through thermoelastic approach may lead to a decrease in effective mean normal stress m 0  while in a pressuremeter test, dilation is always accompanied by an increase in  Tortike (1991) stated that cyclic steam simulation imposes cyclic loads on the oil reservoir. He further suggested that a realistic stress-strain model should have the capability to model the loading and unloading behaviour. He adopted Hinton and Owen’s (1977) elasto-plastic model which includes a Mohr-Coloumb failure envelope and an associated flow rule.  Chapter 2. Review of Literature  Wan et al.  25  (1991) also recognized the cyclic loadings caused in the recovery  process by steam injection and proposed an elasto-plastic model for oil sand. Their model is based on Vermeer’s (1982) elasto-plastic model. They used Matsuoka and Nakai (1982) equation to represent the yield and failure surfaces, and a Ramberg Osgood type hardening function. The model involves a non-associated flow rule and the potential function is based upon Rowe’s stress dilatancy equation.  However,  their model cannot predict the plastic volumetric behaviour for stress paths involving compression with constant stress ratio.  2.2  Modelling of Fluid Flow in Oil Sand  In petroleum reservoir engineering, multiphase fluid flow has been analyzed by a number of researchers without consideration of the geomechanical behaviour of the oil sand matrix.  The first clear attempt to use a finite element method for fluid  flow in porous medium that appeared in petroleum engineering was by Javandel and Witherspoon (1968). They considered a single phase isothermal fluid flow through an isotropic homogeneous porous medium. The numerical solutions were compared with the analytical solutions for infinite, bounded and layered radial systems with constant flow rate or pressure constraints and were found to be in good agreement. Solutions for two-phase isothermal fluid flow problems using variational and finite element methods were presented by various researchers (for example: Settari and Price, 1976; Huyakorn and Pinder, 1977a; Spivak et al., 1977; Settari et al., 1977; Lewis et al., 1978; White et al., 1981). Spivak et al. (1977) presented a formulation for multi-dimensional, two-phase, immiscible flow using variational method. They compared variational and finite difference methods and concluded that the variational method is more efficient than the finite difference method. Galerkin’s procedure was successfully applied to the analytical formulation of the governing equations in the presence of favourable and unfavourable mobility ratios. Numerical dispersion at the  Chapter 2. Review of Literature  front was less in both cases than with the finite difference method.  26  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 nonisothermal 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  Chapter 2. Review of Literature  appeared  27  to obviate the need for upstream weighting for numerical stability. However,  this result is not in accordance with later studies of the multiphase flow problem by the finite element method. Tortike (1991) presented a detailed literature review on modelling of fluid flow under isothermal and non-isothermal conditions.  He solved the three-phase ther  mal flow problem using finite differences. He also tried to develop a fully coupled geomechanical fluid flow model, but was not successful as the results were unstable. It appears that in most of the research work in petroleum engineering, the flow in oil sand is modelled by two phase system (water and bitumen) with reasonable accuracy. However, these models solve only the fluid flow problem and do not con sider the geomechanical behaviour. Therefore, the effects of stress distribution and deformation in the oil sand matrix are not included in these models.  2.3  Coupled Geomechanical-Fluid Flow Models for Oil Sands  Some models in petroleum reservoir engineering include the effects of deformations in oil sand matrix through poroelasticity. Geertsma (1957) combined the approaches of Biot (1941) and Gassman (1951) to develop the equations of poroelasticity in a more straightforward manner. He clearly defined and related the rock bulk and pore compressibilities, and described the boundary conditions and procedure to determine the correct parameters defining the compressibilities. Geertsma (1966) reviewed the applications of poroelasticity in petroleum engineering. An analogy is presented be tween poroelastic and thermoelastic theories, to take advantage of the many solutions under different boundary conditions that have already been published. The concept of the nucleus of strain for volume elements was described and it has been applied to predict surface displacements. It should be noted however, the poroelastic theory does not consider the effect of stress distribution through a porous medium. Raghavan (1972) derived a one dimensional consolidation equation coupled with  Chapter 2. Review of Literature  28  fluid flow and compared his results with Terzhaghi’s solution. The general solution was obtained from the partial differential equations describing the flow of fluid and material displacement using a transform to convert it to an ordinary differential equa tion. He also presented a significant review of the literature to that time. Finol and Farouq Ali (1975) analyzed a two-phase flow model using finite differ ences which included the effects of compaction on fluid flow and the prediction of surface displacements. The problem was formulated by two discretized equations for oil and water flow, and one analytical equation for poroelasticity which was numer ically integrated. The variation of permeability and porosity was considered in the analysis as the effect of compaction on ultimate recoveries. The authors concluded that the ultimate recoveries of oil increased with compaction. Harris and Sobkowicz (1977) derived an analytical model from a more geotechni cal point of view. They presented a coupled mathematical model for the fluid flow and the geomechanical behaviour of oil sand. The model was developed mainly to analyze excavations, immediate foundation settlements and underground openings in oil sands. Since these scenarios involve short term conditions, and because of the high viscosity of the bitumen, their model was only concerned with the undrained response. The authors claimed that the short term conditions govern the design in the above circumstances. Byrne and Grigg (1980), and Byrne and Janzen (1984) extended Harris and Sobkowicz’s formulation. Byrne and Janzen also included the fully drained condition in their analysis. Their analysis procedure involved an effective stress approach in which the stresses in the sand skeleton were computed using a finite element scheme. The pore fluid pressures were computed from the gas laws together with volume compatibility between fluid and skeleton phases. Vaziri (1986) coupled the equilibrium equation and the flow continuity equation and analyzed the transient conditions as a consolidation problem. He included the  Chapter 2. Review of Literature  29  thermal effects on stresses, hydraulic conductivity and volume change and presented a two dimensional finite element formulation.  The fluid flow was considered as a  single phase one. The effects of different phase components on compressibility were taken into account by means of an equivalent compressibility. thermoelastic approach to model temperature effects.  Vaziri followed the  This approach appeared to  predict unrealistic oscillating results. Srithar (1989) incorporated the temperature induced stresses and volume changes directly in the governing equilibrium and flow continuity equations and presented a better formulation of Vaziri’s model. Dusseault and Rothenberg (1988) reviewed the effect of thermal loading and pore pressure changes around a wellbore on dilation and permeability.  They described  the physical process of deformation in terms of particulate media. They concluded that effective water permeability would increase one or two orders of magnitude with dilation as the thickness of the water film coating the grains would increase by a factor of two. The authors continue to document the changes likely from shear failure, including the localization of shear and the growth of the shear zone from the edge of a hydraulic fracture due to the altered stress state and the increased pore pressures. Settari (1988), Settari et al. (1989) described a model to quantify the leak-off rates for fracture faces in oil sand. The authors used a nonlinear elastic model and a two-phase isothermal flow in their analysis. The nonlinear response was shown to give a different pressure distribution than the linear elastic one. Settari (1989) extended their earlier model to thermal flow. Fung (1990) described a control volume finite element approach for coupled isother mal two-phase fluid flow and solid behaviour. He adopted a hyperbolic stress-strain law with Rowe’s stress dilatancy theory.  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 + S 9  =  1), and the ef  fective stress equation. Three combinations of solution variables were considered  (  }, {U, P, S}). Among these the best convergence was found 9 { U, F,(,, P}, {U, P, P when using the combination of { U, P, P 9 }. 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.  Chapter 3  Stress-Strain Model Employed  3.1  Introduction  In developing a procedure to analyze the geotechnical aspects of oil sands, appropri ate modelling of the deformation behaviour of oil sand is the most important issue. Basically, modeffing of oil sand behaviour can be divided into two parts; modeffing of the behaviour of pore fluid and modeffing of the behaviour of the sand skeleton. In this chapter, modelling of sand skeleton behaviour is described in detail. Modelling of pore fluid behaviour is explained in chapter 5. As explained in section 2.1.1, oil sand is very dense in its natural state and exhibits significant shear induced volume expansion or dilation.  The dilation in the sand  skeleton will increase the pore space and hence increase the permeability and reduce the pore pressure. These changes will have significant effect in the overall deformation and flow predictions. Therefore, realistic modeffing of dilation is important. Generally, oil recovery methods are cyclic processes which will cause the sand skeleton to undergo loading and unloading sequences resulting in irrecoverable plastic strains. This necessitates the use of an elasto-plastic stress-strain model. There are a number of models available in the literature as discussed in chapter 2. Among these, the model proposed by Matsuoka and his co-workers has been chosen as the basis for the stress-strain model employed in this study for the following reasons. 1. The failure criterion is based on stress ratio rather than shear stress.  This  would realistically model the behaviour when the soil undergoes a decrease in 32  Chapter 3. Stress-Strain Model Employed  33  mean normal stress with constant shear stress (see figure 3.1) which is a possible scenario in oil recovery process with steam injection. 2. It is based on microscopic analysis of the behaviour of sand grains and not by curve fitting. 3. It considers the effect of the intermediate principal stress. 4. It appeared to predict the experimental data best based on the proceedings of the Cleveland workshop on constitutive equations for granular materials (Sal gado, 1990). A modified version of this model has been extensively used in the University of British Columbia (Salgado (1990), Salgado and Byrne (1991)) and gave very good predictions. The stress-strain model employed in this study is an improved version of the model used by Salgado (1990). Improvements to Salgado’s model have been made in three aspects. 1. Changes proposed by Nakai and Matsuoka (1983) regarding the strain increment directions are implemented. 2. A cap type yield criterion is added to model the constant stress ratio type loadings accurately. 3. Modelling of strain softening is added. A detailed description of the stress-strain model, development of the constitutive matrix in a general three dimensional Cartesian coordinate system, its implementation in three dimensional, two dimensional plane strain and axisymmetric conditions are presented in this chapter. It should be noted that effective stress parameters are implied throughout this chapter and the prime symbols are omitted for clarity.  Chapter 3. Stress-Strain Model Employed  Cl) Cl)  34  Failure Envelope  2  (Increasing Steam Injection Pressure)  Normal Stress  Figure 3.1: A Possible Stress Path During Steam Injection  Chapter 3. Stress-Strain Model Employed  3.2  35  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, de 8 (the strain increments caused by the increase in stress ratio) and a plastic volumetric or collapse component, dcc (the strain increment caused by the increase in mean principal stress). Figure 3.2 schematically illustrates these elastic, plastic shear and plastic collapse components of the total strain in a typical triaxial compression test. Mathematically, the total strain de can be expressed as,  de = 9 dc + dcc H- dee  (3.1)  These different strain components can be calculated separately; the plastic shear strains by plastic stress-strain theory involving a conical type yield surface, the plastic collapse strains by plastic stress-strain theory involving a cap type yield surface and the elastic strains by Hooke’s law. From the stress-strain theories, the strain components can be written as  } 8 {de  =  [Ce] {th}  {de} } 6 {dc  ] {do} 8 [C  =  [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:  {de}  =  ] H8 [[C  [CC]  + [CC]] {do}  Chapter 3. Stress-Strain Model Employed  I  ci z w w  U U C,, C,,  U  z I  C?,  ci  I-  U  -J  0 >  Figure 3.2: Components of Strain Increment  36  Chapter 3. Stress-Strain Model Employed  =  [C] {do}  37  (3.3)  The theories involved in developing the 8 [C ] , [Cc] and [Ce] matrices in general Cartesian coordinate system are explained in the next sections and at the end, the full elasto-plastic constitutive matrix [C] is formed according to different loading conditions. In developing a finite element formulation, the stress-strain relation is generally expressed as  do = [D] dE  (3.4)  The above equation is an inverse of equation 3.3. Once the [C] matrix is known, the  [D] matrix can be easily obtained as the inverse of [C].  3.3 3.3.1  Plastic Shear Strain by Cone-Type Yielding Background of the Model  The stress-strain relationship for the plastic shear strain is developed based on the cSpatial Mobilized Plane’ concept by Nakai and Matsuoka (1983). Before going into the three dimensional conditions, a brief description of the concept of mobilized plane in two dimensional conditions is given to provide a better insight. The concept of mobilized plane was first developed by Murayama (1964). The term ‘Mobilized Plane (MP)’ refers to the plane where the shear-normal stress ratio  (rMp/crMp) is the maximum. This is the plane on which slip can be considered to occur. The 2-D representation of this plane is shown in figure 3.3 (a). This plane makes an angle of (45° + m/2) to the major principal stress plane, where q is the mobilized friction angle. The Mohr circle for the stress conditions and the mobilized friction angle are shown in figure 3.3 (b).  Chapter 3. Stress-Strain Model Employed  Q3  2-D Mobilized Plane  (a)  C,, (I, bJ  c  C’,  TM bJ C,,  Q  NORMAL STRESS (b) Figure 3.3: Mobilized Plane under 2-D Conditions  38  Chapter 3. Stress-Strain Model Employed  39  From a large number of tests and from the analysis of the shear mechanism of granular material in a microscopic point of view, Murayama and Matsuoka (1973) proposed a relationship between the shear-normal stress ratio  (TMp /crMP)  and the  normal-shear strain increment ratio (dMp/d7Mp) on the mobilized plane as, rp  (_d6MP+  (3.5) \ d-yf ) are constant soil parameters. Equation 3.5 forms the basis for the MP  where ) and  i  developments of the constitutive models later by Matsuoka and his co-workers. Under general three dimensional conditions, the stress state of a soil element can be characterized by the three principal stresses  o, 02  and  o.  Mohr circles for these  three stresses can be drawn as shown in figure 3.4 (a) and three mobilized friction angles,  ml m 4 , 2 23  and ç 3 can be obtained. These mobilized friction angles can be  expressed by the following equation:  Z  tan(450+)  (i,j=1,2,3;ucT)  (3.6)  Using these mobilized friction angles, a 3-D plane ABC can be constructed as shown in figure 3.4 (b). This plane ABC is considered to be the plane where the soil particles are most mobilized and is called the ‘Spatial Mobilized Plane (SMP)’. Under isotropic stress condition  (o =  = 03)  the mobilized plane will coincide with  the octahedral plane and will vary with possible changes in stresses. The direction cosines of the SMP are given by the following equation:  a  (i  =  1,2,3)  (3.7)  =  where 11,12 and 13 are the first, second and third effective stress invariants and ex pressed by the following equations in terms of principal stresses or the stresses in the general coordinate system.  Chapter 3. Stress-Strain Model Employed  r  40  13  12 m  o•1  (a)  1  Ia;  cI -f-———---———--y.— Spatial Mobilized Plane  V’ 6  O3 —  B  -  7 450+  23 m 2  ’ 7 A  •+ 5 L 2 (b)  Figure 3.4: Spatial Mobilized Plane under 3-D Conditions  Chapter 3. Stress-Strain Model Employed  ‘1  =  3 1 O + 2 0  12  =  12  +  0203  41  = +  13 =  =  O301  °y°z +  y x 0  +0 z+0 y z  2 T TyzTz  OT  —  —  —  —  —  2 T  —  (3.8)  OzTy  The general stress-strain relationship will be developed basically from the rela tionship of the stresses on the SMP and the strain components to the SMP. The normal stress (oSMP) and the shear stress  (TSMp)  on the SMP can be obtained from  the following equations: a 1 SMP = o  a+o 2 a 3 +o  =  3  (3.9)  and TSMp =  /(oi  —  2 o a ) 2 ?a + (o  —  o a 2 ) 3 a + (o  \/111213 —  O1)21 =  The shear-normal stress ratio,  i  —  9I  (3.10)  ‘2  can be expressed as  = TSMp  =  SMP  I1I2 —913 913  (3.11)  By assuming that the direction of the principal stresses and the direction of the principal strain increments are identical, which is the common assumption in plastic ity, the normal and the parallel components of the principal strain increment vector to the SMP (dcsMp and d7sMp) are given by  dEsMp  =  1 +2 dea dea + dEa 3  (3.12)  and d7sMp  =  2 i,J(dEa  —  2 + (deas ) 1 d€a  —  ) 2 dca  --  (d€ai  —  2 ) 3 d€1a  (3.13)  Chapter 3. Stress-Strain Model Employed  42  It should be noted that before Nakai and Matsuoka (1983), Matsuoka used the normal and shear strain increments on the SMP rather than components of the prin cipal strain increments to the SMP. After a thorough investigation of the theories involved, Nakai and Matsuoka (1983) concluded that the average sliding direction of the soil particles coincides with the direction of the principal strain increment vector and not with the direction of the strain increment vector on the SMP. They denoted their earlier model as SMP (Matsuoka and Nakai, 1974, 1977) and the new model as SMP’. The concepts used in this study follow the SMP model. In the theory of plasticity, the stress-strain relation is formulated from a yield function, a plastic potential function (or a flow rule) and a strain hardening function. The model developed by Matsuoka does not explicitly define these functions. How ever, those can be formulated and the constitutive matrix can be derived easily as explained in the next subsections.  3.3.2  Yield and Failure Criteria  The yield criterion defines the boundary between the elastic and plastic zones. A family of yield surfaces in the  TSMp  —  0 S MP  space is shown in figure 3.5. These yield  surfaces are given by the following equation:  77  where  i  —  \2 3 /tanmi + tan m23 + tan m q 13  TsMp /0sMP, q m 5  =  k  (3.14)  are the mobilized friction angles and k is a constant.  The ‘current’ yield surface corresponding to the stress state at a point in a mass of soil is defined by the maximum stress ratio mobilized at that point during its history of loading. For instance, assume the current yield surface is represented by line A and the stress state of the point is represented by P (see figure 3.5), the shaded area will be the current elastic region corresponding to that yield surface. In a loading sequence, if the stress state of the point moves to Pu within the elastic region, only elastic  Chapter 3. Stress-Strain Model Employed  43  Failure Surface  B Yield Surfaces  A P...  ElastIc Region  °SMP  Figure 3.5: Yield and Failure Criteria on  TsMp  —  5MP 0  Space  Chapter 3. Stress-Strain Model Employed  44  strains will occur and it represents an unloading condition. If the stress state moves to FL which is outside the elastic region, there will be elastic and plastic strains. The yield surface will be dragged along to a new yield surface represented by line B and the elastic region will expand up to line B. This corresponds to a loading condition. The limit or the boundary of the yield surfaces will be the failure surface which is given by the following equation:  tan f12 + tan f23 + tan f13 where  is the failure stress ratio and  =  kf  (3.15)  are the failure friction angles. Salgado  (1990) claims that the failure stress ratio is dependent on the normal stress on the SMP at failure, and that a better agreement with the laboratory data will be obtained if the failure stress ratio is expressed by the following equation:  =  —  10 log  (asMP)f  (3.16)  where  -  -  failure stress ratio at  (osMp  )  =  1 atmosphere  decrement in failure stress ratio for 10 fold increase in  (oSMp  )  The failure surface on the octahedral plane and in the 3-D space is shown in figure 3.6. The Mohr-Coulomb failure surface is also shown in the figure and it can be seen that the Mohr-Coulomb and Matsuoka-Nakai failure surfaces coincide for the triaxial conditions (compression and extension) but differ for any other stress path. The Matsuoka-Nakai failure criterion considers the effect of the intermediate principal stress. This effect is shown as the difference between the failure friction angles for Matsuoka-Nakai and Mohr-Coloumb criteria with b-value in figure 3.7. The triaxial compression condition will correspond to b-value will correspond to b-value  =  1.  =  0 and triaxial extension condition  Chapter 3. Stress-Strain Model Employed  45  01  MOHR-COULOMB  \  MATSUOKA  -  NAKAJ  (a) Octahedral Plane 01  /1II\  #\ /L\’ “\  / A  1/ \%(  II  ,C/  7 p  0  C (b) 3-Dimensional Stress Space Figure 3.6: Matsuoka-Nakai and Mohr-Coulomb Failure Criteria  Chapter 3. Stress-Strain Model Employed  46  8-  7-  6TX  5..  7400  -a-  3Q0:  4-  .  -a20°  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  3.3.3  47  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 -  Figure 3.8: 1983)  (TSMp/oSMp)  Vs  0.2  0.4  0.6  ESMp “YSMP  —(dEsMp/d7sMp)  for Toyoura Sand (after Matsuoka,  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:  Chapter 3. Stress-Strain Model Employed  48  [—dESMP’\ 1+11 , \a7sMpJ  (3.17)  ?7= i where A and  t  are soil parameters and  is the stress ratio on the SMP.  Rewriting the above equation yields,  d6sMp  (3.18)  A  d7sMp  Equation 3.18 implies that the plastic strain increment vector will not be perpen dicular to the yield surface and therefore the flow rule is nonassociative. For (desMp /d7sMp)  <  will be positive which means there will be an increase in volumetric  strain for an increase in shear strain which implies contractive behaviour. For  i  > u,  (dEsMp/d7sMp) will be negative which indicates dilative behaviour. Figure 3.9(a) shows the flow rule and the regions of dilative and contractive behaviour and figure 3.9(b) shows the corresponding results as desMp versus  3.3.4  d7sMp.  Hardening Rule  The hardening rule defines how the threshold of yielding changes with plastic strain, or in other words how the yield stress state changes with plastic strain. In Matsuoka’s model, the plastic shear strain to the SMP Therefore, a relationship between shear strain to the SMP,  7sMP,  i  (7sMp)  is considered as the hardener.  which defines the stress state and the plastic  will form the hardening rule. Matsuoka defines the  hardening rule by an empirical equation as follows:  where  i  and  i’  7o exp  (,  (3.19) \P’ /.‘J are constant soil parameters. The parameter Yo is assumed to be a 7SMP =  function of mean principal stress 7o  (crm)  —  and expressed as follows:  10 -yo + Cd log  (--) °mi  (3.20)  Chapter 3. Stress-Strain Model Employed  49  Ti  Dilation  A Contraction  1  (dSMp i,jp 5 \ dy  (a)  dEsMp  d7SM P Dilation  71>11  Contraction  “<It  (b)  Figure 3.9: Flow Rule and The Strain Increments for Conical Yield  Chapter 3. Stress-Strain Model Employed  where Cd is a constant, 7o at  mi m = 0 0  omj  50  is the initial mean principal stress and yoi is the value of  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 kN/m 2  o comp. • ext.  •  2.0  Figure 3.10:  rsMp/OsMp  Vs  YsMP  3.0  4.0  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  G,. where  +  7SMP  luU 1  (3.21)  Chapter 3. Stress-Strain Model Employed  G,,  -  -  l‘ T ult  -  initial slope of the stress ratio  i  —  51  7sMP  curve  (TSMp/JSMp)  asymptotic value of the stress ratio  By differentiating equation 3.21, the plastic shear strain increment  /7sMP  can be  obtained as,  dy5Mp  where  d  =  (3.22)  is the dimensionless tangent plastic shear parameter. This parameter is  dependent on both normal stress on SMP (crsMP) and the stress ratio.  can be  evaluated by a similar procedure as given by Duncan et al. (1980) as follows:  =  G(1  —  Rf  __)2  (3.23)  1i  and =  KG  (osMP)  (3.24)  where -  np  Pa  -  -  -  1 R  3.3.5  -  plastic shear number plastic shear exponent atmospheric pressure stress ratio failure ratio  (7f/ij,zt)  Development of Constitutive Matrix  [CS]  The development of plastic shear constitutive matrix in terms of general Cartesian stress and strain components from the yield criterion, hardening rule and the flow rule is described in this section. The hardening rule (equation 3.22) and the flow rule (equation 3.18) give the following:  Chapter 3. Stress-Strain Model Employed  dysMp  =  52  1 —di  IL—?’  desMp  (3.25)  (3.26)  9 d-y.  Substituting equation 3.25 in equation 3.26 will give,  dEsMp  (IL  =  ?‘) d  (3.27)  —  By assuming that the directions of the principal stresses and the directions of the principal strain increments are the same, the direction cosines of desMp are given by  a  (i  =  =  1,2,3)  (3.28)  If it assumed that the direction of d7sMp and the direction of TsMp coincide, then the direction cosines of d7sMp are given by  1,:  —  =  0jI2  SMP  TSMp  where  11,12  —  313  (3.29)  =  /o- ‘2 (I 12  —  913)  and 13 are stress invariants as given by equation 3.8. The plastic principal  strain increments due to shear can be obtained from the following equation.  de  =  a desMp H- b  d7SMP  i  =  1,2,3  (3.30)  By substituting equation 3.25 and equation 3.27 into equation 3.30,  dE=+.i)d?’  (3.31)  Equation 3.31 can be written in matrix notation as  {defl  =  } d?’ 2 {M1  (3.32)  Chapter 3. Stress-Strain Model Employed  where M1  53  + =  The general Cartesian strain increments can be obtained by multiplying the prin cipal strain increment vector by the transformation matrix, as given by the following matrix equation:  dE  l  m  8 de  12  2 m  8z d  12  2z m  7 d  V  n 8 d€  z  2z n  (3.33)  2l7,l, 2mm 2nn dc 2l,l  2 2mm  2 2nn  2l1 2m 3 m 2 where l, l,, and l m, m and m ,  and n  -  -  -  direction cosines of o to the x, y and z axes direction cosines of  02  to the x, y and z axes  direction cosines of  03  to the x, y and z axes  Equation 3.33 can be written in matrix form as  } 8 {de  =  [MT] {dc}  (3.34)  Substitution of equation 3.32 into equation 3.34 yields  } 8 {de  =  [MT] {M1} di 1  From equation 3.11 the stress ratio on the SMP,  =  /1112  —  91  913  (3.35) is given by  (3.36)  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 di : 7  ,  I 77 I  d  {do}  = T  —  —  1 18iiI  ‘213  + 1113 (o, + o)  1213  + 1113 (o + o)  1213 + I113(0 +  —  —  1112 (o,o  —  ‘112 (o °•r  —  IiI ( 2 t717y  —  r)  do  T)  doy  ) 2 r  do-i dr  —2IlI3r — I 2 lI3T =  o)  —  —  —  2IlI2(rr (rr 2 2IiI  dr  —  —  or)  {M2} { T do}  (3.37)  where superscript T denote the transpose of the matrix. Substituting equation 3.37 in equation 3.34 gives  } 8 {d6  }T {do} 2 [MT] {M1} {M  =  (3.38)  This can be further written as  } 8 {d6  =  ] {dcr} 8 [C  (3.39)  where [C ] is the plastic shear constitutive matrix and will be given by 8  ] 8 [C  =  [MT] {M1} {M2} T  (3.40)  Chapter 3. Stress-Strain Model Employed  3.4 3.4.1  55  Plastic Collapse Strain by Cap-Type Yielding Background of the Model  The plastic stress-strain theory with the conical yield surfaces described in the pre vious section is not capable of predicting the behaviour of soil under proportional loading. In that model, the yield surfaces are constant stress ratio lines and therefore, for a stress path having constant stress ratio, only elastic strains will be predicted. However, the laboratory experiments show that proportional loading with increasing stresses causes some plastic deformation. An additional yield surface which forms a cap on the earlier conical yield surface is considered to circumvent this deficiency as explained in this section. The stress-strain relationship for predicting the plastic collapse strains was developed by following the concepts of the cap-type yielding given by Lade (1977). As explained in section 3.2, it is reasonable to assume that the plastic collapse strains are produced by the increase in mean normal stress and the plastic shear strains will be associated with the shear stresses. However, under general loading con ditions, it is difficult to separate the plastic shear and plastic collapse strains because both will occur simultaneously.  Therefore, the development of the cap-type yield  model is based on the isotropic compression tests where no plastic shear strains are produced. Figure 3.11 shows the typical results for loading, unloading and reloading conditions in an isotropic compression test. The elastic strains which are recoverable can be calculated using Hooke’s law are also shown in figure 3.11. Then, the collapse strains can be obtained by subtracting the elastic strains from the total strains. In order to model the plastic collapse behaviour, a yield criterion which forms a cap at the open end of the conical yield surface is used. The yield criterion and the hardening functions for the cap-type yield are explained in the following subsections. The stress-strain relation for the plastic collapse strain is formulated following the  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  =  —  (3.41)  212  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  Hydrostatic Axis  01 Conical Yield Surface  Plastic Collapse Strain Increment /ector Spherical Yield Cap  Iasti Regior Conical Yield Surface  03  Figure 3.12: Conical and Cap Yield Surfaces on the o  —  03  Plane  Chapter 3. Stress-Strain Model Employed  the o1  —  03  58  plane. The elastic region at any particular stress state will be bounded  by these two yield surfaces. As  f  increases beyond its current value, the yield cap  expands, soil work hardens and collapse strains are produced. It should be noted that there are no bounds on the cap yield surface and yielding according to equation 3.41 does not result in eventual failure. The failure is entirely controlled by the conical yield surface.  3.4.3  Flow Rule  Under isotropic compression, an isotropic soil shows equal strains in all three principal directions. Therefore, the direction of strain increment vector should coincide with the hydrostatic axis pointing outwards from the origin (see figure 3.12). To satisfy this conditioi-i the plastic potential function must be identical to the yield function. This implies the flow rule is associative and will be given by the following equation:  de  =  8 o .ij  where  (3.42)  is the proportionality constant which gives the magnitude of the plastic  collapse strain and can be determined from the hardening rule.  3.4.4  Hardening Rule  The hardening rule gives a relationship between the yield function and the plastic strain, defining how the yield function changes with plastic strain. For the cap yield model, Lade (1977) developed an empirical relationship between the plastic collapse work (We) and the yield function. The plastic collapse work is a function of plastic collapse strains and given by  =  J  {}T  {dE}  (3.43)  Chapter 3. Stress-Strain Model Employed  59  The relationship between the plastic collapse work and the yield function is given by ()P  = CPa  (3.44)  where C and p are dimensionless constants and called the collapse modulus and the collapse exponent respectively. The proportionality constant LSX which gives the magnitude of the plastic collapse strain increment can be obtained as follows. The increment in plastic collapse work can be expressed as  dW  =  {}T  {dec}  (3.45)  Substitution of equation 3.42 into equation 3.45 gives  (3.46)  = Since the yield function  f  is a homogeneous function of degree 2, it can be shown  that  = 2f From equations 3.46 and 3.47,  can be given as = dWC  3.4.5  (3.47)  (3.48)  Development of Constitutive Matrix [CC]  The constitutive matrix relating the plastic collapse strains and the stress increments can be developed as described below. Substitution of equation 3.48 in equation 3.42 gives  Chapter 3. Stress-Strain Model Employed  60  af  dW  c  (. )  O3  Jc  By differentiating equation 3.43, dW can be obtained as ()121 =  Cp  d  a  (3.50)  and it can be further written as  dW where  A  =  =  (3.51)  A df  (f)P_1  df will be obtained by differentiating 3.41 as, df  = T  2o  do  2o  do  2o  do  4r  dr  4r  dr  4 T z  dr  =  (3.52)  By combining equations 3.49, 3.51 and 3.52 the following equation can be obtained:  de  =  A  8f 8f —dokj  2f  8 k l  (3.53)  In terms of Cartesian components of stress and strain the above equation can be written as  Chapter 3. Stress-Strain Model Employed  61  oo-  d dE d 7 9  o  =  r 2 2o-  f  2 o r  OTzm 2  do  2or  , 22 2or  do  2o-r  r 2 2u  do-i  4r  d-y  dT  Symmetry  2 4r  7 d  r 2 4r  dr  2 4r  , 3 dr  In short matrix notation the constitutive matrix for the plastic collapse strain can be written as  {Cc]=  3.5  T {8fc}{afc}  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  de d2  —v —v  0  0  0  do  1—v  0  0  0  do,  0  0  0  do  2(1H-v)  0  0  dr  2(1 + v)  0  2 dr  2(1 + v)  2 dr  1  1  dd 7 2 d  Symmetry  (3.57)  Chapter 3. Stress-Strain Model Employed  62  where E is the tangential Young’s modulus obtained from the unload-reload portion of a stress-strain curve.  i-’  is the Poison ratio which can be calculated from Young’s  and bulk moduli as  v= (i_&)  (3.58)  E and B are assumed to be stress dependent and given by the following equations: ()fl  E  =  kE Pa  B =  Pa  (3.59)  ()  (3.60)  where, kE  -  -  n n  3.6  -  -  Young’s modulus number bulk modulus number Young’s modulus exponent bulk modulus exponent  Development of Full Elasto-Plastic Constitutive Matrix  In the previous sections, the constitutive matrix is formed individually for different components of strain. One of the major advantages of having the strain components separated is that it is easy to model the different loading conditions. Depending on the loading condition, the relevant strain components can be included and the corre sponding full elasto-plastic constitutive matrix can be formed. The loading conditions can be classified into four cases which are shown in figure 3.13 on the  i  —  o plane.  Case I Case I indicates a loading condition where there is an increase in stress ratio as well as in mean stress. In this case, all three; the plastic shear, plastic collapse and  Chapter 3. Stress-Strain Model Employed  Failure Surface  1 a  63  Hydrostatic Axis  Ill  Conical Yield Surface  lastc’ Rag ion •  7  Failure Surface  -.7.  3 a  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  ]+ 8 [[C  [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]  =  ] + [CC]] 8 [[C  (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  3.7  65  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  C 1 2 1 C 4 1 C 3 1 C 5 1 C 6  dr  C 2 24 2 1 2 C 2 2 C 3 C C 5 2 C 6  do  C 3 1 3 C 2 3 C 3 3 C 4 3 C 5 3 C 6  do  C 1 1  =  (3.65)  42 4 C 4 1 C 44 C C 45 4 3 C C 6 2 d’y  52 5 C 5 54 5 1 C C 3 C C 5 5 C 6  2 dr  C 6 64 6 1 6 C 2 6 C 3 C C 5 6 C 6  drza,  where C 3 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  C 1 1 1 13 1 C 2 C C 4  do  C 2 1 2 C 2 2 C 3 2 C 4  da  d6  C 3 1 3 C 2 3 C 3 3 C 4  do  7 d  C 4 1  dr  dc  —  C 4 2 4 C 3 4 C 4  Now, by imposing the plane strain boundary condition that  (3.66)  =  0, do can be  Chapter 3. Stress-Strain Model Employed  66  written as 2 do  =  +  —  do-!,  + dT)  (3.67)  Substitution of equation 3.67 in equation 3.66 yields:  de de  =  d  1 C’ C 2 C’ 3  do  1 C; C; 3 2 C;  do-u  1 C; C; 2 C 3  d  (3.68)  where  ri  ‘-‘11  —  (V  —  ‘-‘11  f_I,,  —  —  —  ‘-‘21  f_I  ‘-‘41  —  12 LI  —  —  ‘—‘21  11 C 3 3  —  r L112  —  . —  —  31 C 2 33 C 41 C 3 3 33 C  .  ‘  —  ‘-‘22  —  .  ‘  —  ‘—‘32  —  (1  ‘-‘22  —  f_I  33 C C,C 32 33 C 42 C 3 3  L142 —  33 c  ‘ .  —  —  ‘  ‘-‘23  .  f_I  ‘  ‘-‘33  14 C 3 3  LI4  ‘—‘13  —  —  3 2 C  ,-  ‘—‘24  —  f_I —  3 c,  44 C 3 3 33 C  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  C 1 1 1 C 2 1 C 3 6 C 4  do.  8 de  22 2 C 2 1 C 64 C 3 C  8 do-  2 de  C 3 1  C 3 64 2 3 C 3 C  do-i  C 6 1  C 6 2 6 C 3 6 C 4  drrz  (3.69)  Chapter 3. Stress-Strain Model Employed  3.8  67  Inclusion of Temperature Effects  The effects of temperature changes in oil sand and the works by previous researchers to include these effects in the analytical procedures were described in chapter 2. The approach used by Srithar and Byrne (1991) is followed here. This involves additional terms in the stress-strain relation and in the flow-continuity equation. The changes which have to be made in the stress-strain relation are explained in this section. Inclusion of temperature effects in the flow-continuity equation is described in section 5.8. The incremental stress-strain relation can be written as  {de}  =  [C]{do}  where [C] is the elasto-plastic constitutive matrix.  (3.70) If there is an increase in the  temperature, the sand matrix will expand and there will be additional strains. Then, equation 3.70 will become  {d} where {dee}T  [C]{do}  =  —  } 8 {de  (3.71)  {a d, a 8 d6, a 8 d6, 0, 0, 0} and a 8 is the linear thermal expansion  coefficient of the sand grains and d6 is the change in temperature. It should be noted that compressive strains are assumed positive. By multiplying equation 3.71 by the inverse of [C] which is referred to as the stress-strain matrix [D] the following equation can be obtained:  [D]{dc}  =  {dcr}  —  } 8 [D]{de  (3.72)  Rearranging the terms will give  {do}  =  [D]{dE} + {do } 8  (3.73)  Chapter 3. Stress-Strain Model Employed  where {do} 8  =  68  [D] 8 {de } , which is the additional term in the stress-strain relation  due to change in temperature. This term will give the induced thermal stresses.  3.9  Modelling of Strain Softening by Load Shedding  Laboratory tests on oil sand show a decrease in strength after a peak strength is reached which is commonly referred as strain softening. The phenomenon of strain softening or loss of strength under progressive straining occurs because of the struc tural changes in the material such as initiation, propagation and closure of micro cracks. Frantziskonis and Desai (1987) stated that strain softening is not a material property of soil when it is treated as a continuum. It is rather a performance of the structure composed of micro-cracks and joints that result in an overall loss of strength. When the stresses and strains deviate from homogeneity, the behaviour of a material will no longer be represented by continuum material properties. If strain softening is assumed as a true material property, various anomalies may arise with respect to the solution of boundary and initial value problems. These anomalies can lead to loss of uniqueness in the strain softening part of the stress-strain response and to numerical instabilities as shown by Valanis (1985). A comprehensive review of strain softening is not attempted here as it is beyond the scope of this thesis. Reviews on this subject can be found in Read and Hegemier (1986) and Frantziskonis (1986).  In this study, the strain softening phenomenon  is modelled quantitatively using the ‘load shedding’ or ‘stress transfer’ concept. In principle the load shedding concept is similar to the model presented by Frantziskonis and Desai (1987). They modelled the strain softening behaviour by separating it into two parts; a non-softening behaviour of a continuum (topical behaviour) and a damage or stress relieved behaviour with zero stiffness. The true behaviour is estimated as an average of these two (see figure 3.14).  In finding the average behaviour, the  hydrostatic component is assumed to be the same for both parts and the deviatoric  Chapter 3. Stress-Strain Model Employed  69  Shear Stress Ultimate  Topical Behaviour  —  Average Behaviour  Strain  Figure 3.14: Modelling of Strain Softening by Frantziskonis and Desai (1987)  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  -  -  7SMP,p Ic, q  3.9.1  -  -  Residual stress ratio Peak stress ratio Peak shear strain Constant parameters  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 P 0 in the figure can move to point  Chapter 3. Stress-Strain Model Employed  71  ‘1]  1 P  ‘rip 2 P F?  T I 7r  71  Figure 3.15: Modelling of Strain Softening by Load Shedding  7  Chapter 3. Stress-Strain Model Employed  72  1 in a load increment. But the actual stress state should be point Pia and in order P 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 P . Then again the stress state will be brought to point F 2 a by load 2 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  Chapter 3. Stress-Strain Model Employed  73  those that involve significant principal stress rotations. One of the disadvantages of this model is its limited use in the past. Unlike the hyperbolic model, information on the model parameters is very limited. The possible range of values for some of the parameters and their physical significance are not well defined.  However, a sensitivity study on the parameters is given in chapter  4, which may be helpful to understand the physical significance of the parameters. Another disadvantage of the model is that because of the nonassociated flow rule, it will result in a non-symmetric stiffness matrix which requires considerable computer memory and time.  However, the frontal solution scheme used in this study will  circumvent the requirement for large memory since it does not assemble the full stiffness matrix and requires only a small memory. Furthermore, these factors of time and memory requirements may not be considered as disadvantages with the rapid growth in computer capabilities.  Chapter 4  Stress-Strain Model  -  Parameter Evaluation and  Validation  4.1  Introduction  This chapter describes the procedures used to evaluate the soil parameters needed for the stress-strain model and presents results verifying the stress-strain model against measured responses in laboratory tests. The soil parameters required for the model can be classified into four groups; elastic, plastic shear, plastic collapse and strain softening. A summary of the parameters and their description are given in table 4.1. The procedures used to evaluate these parameters from basic laboratory tests such as isotropic compression and triaxial compression tests are described in section 4.2. For the determination of some of the parameters at least two test results are necessary to obtain a straight line fit. In those cases, it is advisable to have three or more test results to obtain a better fit. Validations of the stress-strain model against laboratory results on Ottawa sand and on oil sand are given in section 4.3. Sensitivity analyses on some of the parameters have been carried out to provide some idea about their significance and these are described in section 4.4.  4.2  Evaluation of Parameters  In this section, only the procedures for the evaluation of the parameters are given in detail. Applications of these procedures to actual test data on Ottawa sand and on oil sand can be found in section 4.3. 74  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  75  Table 4.1: Summary of Soil Parameters  L Type  Parameter  Elastic  kE n kB m  Plastic Shear Li  .\ i  KG  np 1 R C p  Plastic Collapse Strain Softening  q  4.2.1 4.2.1.1  Description Young’s modulus number Young’s modulus exponent Bulk modulus number Bulk modulus exponent Failure stress ratio at one atmosphere Decrease in failure stress ratio for 10 fold increase in 0 SMP Flow rule slope Flow rule intercept Plastic shear number Plastic shear exponent Failure ratio Collapse modulus number Collapse modulus exponent Strain softening constant Strain softening exponent  Elastic Parameters 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  (4.3)  (°m)’  where a and b are constants and can be obtained by plotting  versus  m 0  on a log-log  scale as shown in figure 4.2. Differentiation of equation 4.3 yields 1  ck  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  /  3 q-a  AE 1  €  (a) Unload-Reload Modulus  (E/Pa)  1000  100 ‘<E  ‘a  1 [log scale]  10  (b) Variation of E with a 3  Figure 4.1: Evaluation of kE and n  (ojP) a .  77  Chapter 4. Stress-Strain Model  0.01  -  Parameter Evaluation and Validation  kB = a.b(Pa) 6  -  a  m=1-b 100 [log scale]  Figure 4.2: Evaluation of kB and m  78  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 kB  = ab(Pa)’  (4.7) (4.8)  It should be noted that the parameters kE and kB can be related by the Poisson’s ratio v as  kB  = 3(1—2zi)  (4.9)  Hence, by knowing one parameter, the other one can also be determined from the Poisson’s ratio. Lade (1977) stated that the Poisson’s ratio for the unload-reload path has often been found to be close to 0.2.  4.2.2  Evaluation of Plastic Collapse Parameters  Only two parameters are needed to evaluate the plastic collapse strains. These two parameters define the hardening law and can be determined from an isotropic com pression test. The hardening law is given by  = CPa  where W is the plastic collapse work,  ()‘  (4.10)  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  f  = 3o  Wc=Jcr d 3 e where de = d€,,  —  80  (4.11) (4.12)  d and de is the elastic volumetric strain.  By plotting W/P against f/P on a log-log plot, the parameters C and p can be obtained as shown in figure 4.3.  0.01  -  [log scale]  Figure 4.3: Evaluation of C and p  4.2.3  Evaluation of Plastic Shear Parameters  In determining the plastic shear parameters, it is easier to divide them into three groups as follows: 1. Failure parameters  i  and LSi  Chapter 4. Stress-Strain Model  2. Flow rule parameters  i  Parameter Evaluation and Validation  -  81  and )  3. Hardening rule parameters KG, np and R 1 The plastic shear parameters can be determined from all types of tests where the principal stresses and principal strains can be obtained. By knowing the principal stresses and strains, the stresses and strains on the spatial mobilized plane (SMP) can be evaluated as described in section 3.3. The plastic shear parameters can then be obtained as explained in the following subsections. The most common laboratory shear tests performed are triaxial compression tests and therefore, special attention is given here to describe how to obtain the plastic shear parameters from those test results. Firstly, the elastic and plastic collapse strains have to be subtracted to obtain the principal plastic shear strains:  d  =  1 de  d€  =  3 de  de  —  de  —  (4.13)  —  —  d  (4.14)  Under standard triaxial compression conditions, the elastic and plastic collapse strains can be given by 1 do de de d  =  =  —vde  =  1 o do  3 do o 1 2A u  where —  —  1—2p 2C P\(pa)  (o +  2o)2P  (4.15) (4.16) (4.17) (4.18)  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  82  It should be noted that if the test samples are preconsolidated to a higher stress and unloaded, then the collapse strains should not be subtracted. By following the equations in section 3.3.1 and imposing the conditions for triaxial compression loading, the stresses and the strains related to SMP can be obtained as follows: SMP  =  3 o 1 3o•1  +  (4.19)  03  TSMp  —  dEsMp  d7sMp  4.2.3.1  (4.20)  3  SMP  —  =  dc/ + 2d4/ /2o + o 2(deW  —  (4.21)  de/j (4.22)  3 + 1 2o c  Evaluation of q’ and z  At least two tests up to failure at different confining stresses are necessary to determine these parameters. The failure stress ratio on SMP is given by  = The values of  —  Li  10 log  (o-sMP)f  (4.23)  Pa  and (OsMP)f can be obtained using equations 4.20 and 4.19. By  plotting {(OSMp)f/Pal versus  on a semi-log plot,  i  and  ii  can be determined as  shown in figure 4.4.  4.2.3.2  Evaluation ofi and X  The flow rule for the plastic shear is expressed by the following equation.  f—c1EsMp  i \  J+/L  u7SMP  J  (4.24)  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  83  ‘if  —  1  10 [log scale]  100  e’ a  Figure 4.4: Evaluation of 1 h 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 mined by simply plotting  4.2.3.3  and ) can be deter  versus —(desMp/d7sMp) as shown in figure 4.5.  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  17  7SMP  =  G.  (4.25)  +  The parameters KG, np and R 1 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  —  Figure 4.5: Evaluation of ) and  (dEMp/d4Mp)  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  1 Now, by plotting  (7sMp/7/)  4 26  7u1t  against  fsMP  the values of G , and 7  1,jit 7  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  mined using the above equation.  i  from section 4.2.2.1 Rf can be deter  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  “7  1  7SMP  (a) Hardening Rule  7SMP  ‘1  1 G 7SMP  (b) Hardening Rule on Transformed Plot Figure 4.6: Evaluation of G and  ij  85  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  86  G is expressed as a function of op as  G  =  (4.28)  KG  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.  ‘1) 0  1000  1  np  c-I (I)  0 U  100  KQ  1  10  100  [log scale]  MP’a  Figure 4.7: Evaluation of K 0 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.  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  87  The strain softening region of a stress-strain curve can be given as (see section 3.8)  = ir +  (ip  —  lir) exp{—i(7sMp  The value of the residual stress ratio is the flow rule intercept.  —  7SMP,p  )}  (4.29)  is assumed to be equal to  i  This assumption is reasonable because, when  t i  which is = p, the  incremental plastic volumetric strain will be zero, which implies a state of shear at constant volume. The value of the peak stress ratio, which is the failure stress ratio, can be obtained from equation 4.23. The peak shear strain  7SMp  can be obtained  from the strain hardening relation (equation 4.25) as  7SMP,p  where  =  773  G1  —  (4.30)  R 1 1  is the initial tangent plastic shear parameter and Rf is the failure ratio.  By rearranging the terms in equation 4.29 and taking natural logarithm, it can be shown that  in  ()  (4.31)  K(7sMp  Taking natural logarithm of equation 4.31 will give  ln [in  (ij]  Then, the parameters against {ln(7sMp  4.3  —  7SMP,p  ,  )}  ln + qln(7sMp  —  7sMp,p)  and q can be determined by plotting {ln [ln  (4.32)  ()] }  as shown in figure 4.8.  Validation of the Stress-Strain Model  The stress-strain model employed in this study has been verified against laboratory results on Ottawa sand and oil sand.The triaxial test results reported by Neguessy  Chapter 4. Stress-Strain Model  in [in  -  Parameter Evaluation and Validation  88  (TZr)]  q  in(7sMp  Figure 4.8: Evaluation of  i  —  7SMP,p)  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 5 D 0 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  MEDIUM  -  Parameter Evaluation and Validation  89  SAND  ‘I zp ae  4  48  ‘  I  100  140 200  I00  I  I  80  I  I  60 LEGE ND C  t4Q  20  X  FRESH  •  RECYCLED  I  ASTM  * MIT  0  I  0.5  0.1  -  C  -  109- 69  BAND  CLASSIFICATION  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  =  /o 1 o 3  =  1.67 and 2)  5. Tests along four different stress paths as shown in figure 4.10  SP4  SP3  300  a. 200 SP1-  SP2  2.0  SP2- (a/u=4.0  spi 100  SP3 SP4  100  =  200 300 UH(kPa)  P  -  -  P’  =  250 kPa, Constant  =  350 kPa, Constant  400  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.  Chapter 4. Stress-Strain Model  4.3.1.1  -  Parameter Evaluation and Validation  91  Parameters for Ottawa Sand  As explained in section 4.2.1, the Young’s modulus values for different confining stresses are plotted in figure 4.11. The values plotted in the figure are from resonant  10000  5000  :  3000 -  2000  (kE)p= 1180 1000  . Resonant Column  -  -  Tria)_(UnIoad Reload) •Triaxlal (Primary Loading)  A -  —  500  I  0.3  0.5  2  1  3 a  I  I  3  5  10  a  Figure 4.11: Variation of Young’s moduli with confining stresses  column tests which yield similar values as are obtained in unload-reload tests. Also shown in the figure are one unload-reload modulus and the Young’ modulus values for primary loading from standard triaxial compression tests. It can be seen that the unload-reload value agrees well with the resonant column values. The ratio of the Young’s modulus for the primary loading condition to the unload-reload condition is about 2.2 and the exponent for both conditions is 0.46. This agrees with the statement by Duncan et al. (1980) that the ratio of Young’s moduli varies from about 1.2 for dense sands to about 3 for loose sands. From figure 4.11 the values for kE and n can  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. We/Pa  0.01 0.005  0.002 0.00 1 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  stress ratio  i  vs  usMp  -  Parameter Evaluation and Validation  93  for the triaxial compression test results are plotted in figure  4.13. The failure parameters  ()  and ZSi 7 are determined as 0.49 and 0.0.  0.6 Ti  —71iO49  ——  0.5  ö:  :  0.4  =O.O 7 zS 0.3 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 R 1 is determined as 0.93. From figure 4.15(c), the values of KG and np are obtained as 780 and —0.238 respectively. Table 4.2 summarizes all the parameters for Ottawa sand at Dr  =  50%.  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  0.1  0  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 Elastic  kE n kB m  Plastic Shear 1117  X u  Plastic Collapse  np Rf C p  2600 0.46 1444  0.46 0.49 0.0 0.85 0.26 780 -0.238  0.92 0.00021 0.89  =  50%  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  (a) 0.5 0.4  o3=5OkPa  0.3  ---0---  ,‘  3=150kPa 02  —  a_3=50kPa  •  I  0.1  a3=45OkPa --.“---.-  0  0.2  0  0.4  0.6  0.8 7SMP  1.8 7SMP  o  1.6  1,4 1.2 1 0.8 0.6 0.4 0.2 0  ‘  0  0.4  0.2  I  I  0.6  0.8 7SMP  1 G  800 750  .W %%  ‘UP  735  700 650 Jlr=0.145  600 550 500 0.5  1  I  I  I  2  3  5  10  Figure 4.15: Hardening Rule Parameters for Ottawa Sand  95  Chapter 4. Stress-Strain Model  4.3.1.2  -  Parameter Evaluation and Validation  96  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  =  3 / 1 o 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 600  -  o 3 = 250 kPa -------  _..  a_3 =50 kPa Symbols Experimental Lines Analytical -  400-  .0  —  -  0  0  200  0  -  0  (a) I  I  0.2  0.4  0.05  0. ‘U  >  0.15  0.2  0.25 0  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  a200 1.67  Symbols ExperIment  100  -  Lines AnaIytca1 -  0  I_  0  0.1  0.2  ——  0.3  I  I  0.4  0.5  Figure 4.17: Results for Proportional Loading on Ottawa Sand  0.6  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  99  600  500  400  300  200  100  0.2  iLl  >  0.4  0.6  0.8  0  0.2  0.4  0.6  0.8  a Figure 4.18: Results for Various Stress Paths on Ottawa Sand  1  Chapter 4. Stress-Strain Model  100  -  Parameter Evaluation and Validation  I I I I I  100  111111 I I  -__  UTF Sand 4  80  •  Other McMurray Sands: coarse  -  I  -  medium -fine -  60  —  —  1  I  -——  .:::ZE  >  E=  40  .--——  20  0  10.0  1.0  0.1  milhimetet  0.01  I  I  0.1  .  inches I  8  F  12  F  Fl  18 25 35  0.01  F  I  45 60  0.001 F  F  F  F  I  80 120 170 230 325400  U.S. mesh  Figure 4.19: Grain Size Distribution for Athabasca Oil Sands, (after Edmunds et al., 1987)  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  101  of the relevant model parameters and the validation: 1. Isotropic compression test 2. Standard triaxial compression tests constant compression  3.  i  4.  U  constant compression  m 0  constant extension  It should be noted that since the samples tested were undisturbed samples from the field, they were not identical. Table 4.3 summarizes the details of the test samples considered. Table 4.3: Details of the Test Samples Test  Isotropic Comp. Triaxial Comp. 1 Triaxial Comp. 2 Triaxial Comp. 3 0.1 Const. Comp. 0m Const. Comp. 0m Const. Ext.  4.3.2.1  Sample ID  Bulk Density  Water  UFTOS1 UFTOS1 UFTOS3 UFTOS4 UFTOS1O UFTOS9 UFTOS12  (kg/rn ) 3 1990 1990 2070 2120 2060 1960 1980  8.3 8.3 8.5 6.4 6.6 7.8 7.0  Fraction by Weight Bitumen Solids  (%) (<  7.6 7.6 6.6 6.5 7.3 8.8 9.5  84.1 84.1 84.9 87.1 86.1 83.4 83.4  Fines 0.074rnrn) 41.2 41.2 54.0 52.9 71.9 57.3 37.7  Void Ratio  Disturbance Index  0.60 0.60 0.52 0.45 0.50 0.62 0.60  12.1 12.1 6.4 10.0 10.8 10.9 9.6  (%)  Parameters for Oil Sand  The relevant parameters for the oil sand are obtained from an isotropic compression test and three standard triaxial compression test results. Since the procedures for obtaining the parameters are discussed in detail in section 4.2 and again briefly in section 4.3.1.1, they are not repeated here. Figure 4.20 shows the data for the unload-reload portion of the isotropic com pression test and the elastic parameters kB and m are determined as 1670 and 0.36  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  am (kPa)  Figure 4.20: Determination of kB and m for Oil Sand  100000  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 4. Stress-Strain Model  -  Parameter Evaluation and Validation  104  0.8  1f 0.75  0.7  0.65  06  0.55  0.5  I  2  3  5  10  20  30  Mp’a  Figure 4.22: Failure Parameters for Oil Sand  50  100  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  105  2000  1300 1000  -  500  =  -  -0.66  200 0  0  100  -  0  50  I  1  2  3  5  10  20  30  50  MP’a  Figure 4.23: Determination of K 0 and np for Oil Sand  100  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, EJ  0.1 0— -0.4  -0.2  0  0.2  o _3  =  2.5 MPa  c,_3  =  4.0 MPa  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  —(dEsMp/d7sMP)  Figure 4.24: Flow Rule Parameters for Oil Sand  0.8  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 n kB m  Plastic Shear iii  \ ii  Plastic Collapse  4.3.2.2  KG rip 1 R C p  3000 0.36 1670 0.36 0.75 0.13 0.53 0.31 1300 -0.66 0.73 0.00064 0.61  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 10000  -  8000 b  0  -  Loading  -  Unloading  6000  -  4000-  0  o  2000  Line Predicted Symbols Measured -  -  0  0  0.5  I  I  I  I  1  1.5  2  2.5  LV  (%)  Figure 4.25: Results for Isotropic Compression Test on Oil Sand  3  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  109  value of ). As the line becomes steeper, there will be less volumetric expansion and the overall behaviour will be more contractive. If a higher value of ) is selected, the predictions and observations would agree well for o to note that the value of the flow rule parameter tests. The value of  .t  i  4.OMPa. It is also interesting  is not much different for the three  is, in fact, an indication of ultimate stress ratio or a state of  shearing with constant volume. Results for three different stress paths; constant o, compression, constant compression and constant  om  °m  extension are shown in figure 4.27. The stress paths  are shown in the insert of the figure.  It can be seen that the experimental and  predicted results are in good agreement.  4.4  Sensitivity Analyses of the Parameters  In order to provide a better understanding about the significance of the parameters, sensitivity analyses on the parameters have been carried out. The parameters ob tained for Ottawa sand were chosen as the base parameters and the significance of a particular parameter was studied by changing only that parameter. A triaxial com pression loading condition with the initial confining stress of 500 kPa was considered and the results in terms of deviator stress and volumetric strain are analyzed. The results are shown in figures 4.28 and 4.29 The plastic collapse parameters C and p are essentially an indication of isotropic compressibility. The higher the values, the higher the predicted volumetric strains. The parameter ). is the slope of the flow rule and it defines the change in volumetric expansion for a change in stress ratio. A steeper slope (or higher A) will give less volumetric expansion. The parameter 1 u is the amount of stress ratio which separate contraction and dilation (similar to ç 5 in general soil mechanics). A smaller value of will result in dilation at lower stress ratio. The parameters KG and np define the initial slope of the hardening modulus  Chapter 4. Stress-Strain Model  10000  -  Parameter Evaluation and Validation  110  -  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  e(%) a Figure 4.26: Results for Triaxial Compression Tests on Oil Sand  3  Chapter 4. Stress-Strain Model 5,000  Parameter Evaluation and Validation  -  111  —  6 -4  4,000  I  -  3,000  SP1-I.1 Const.Comp SP 2- a_v Const Comp. SP3-i1 Conet Ext. 246 a_r (MPa) /  2 D. C-.. C’-,  /  /  -  000 I1  a  or  0/  &  0/  2,000  o/  spi C  ‘a  SP3  1,000  Experimental Symbols Lines Analytical -  -  0  -1.4  -1.2 —1 > WI  -0.8 -0.6 -0.4  z  -0.2 0 .  0.2 0.4  Q9OO5/O  gD  -6  -4  -ci  -  o o  -2  0 -  E_r  2  4  (%)  Figure 4.27: Results for Tests with Various Stress Paths on Oil Sand  6  Chapter 4. Stress-Strain Model  c_a  Parameter Evaluation and Validation  -  (%)  (%)  c_a  (a) Effect of Parameter  C  112  (b) Effect of Parameter I  0 0  &  > WI  0.4 (C)  0.6 Ea (%)  0.8  Effect of Parameter  1 c_a  A  Figure 4.28: Sensitivity of Parameters  (%)  (d) Effect of Parameter  C,p, )  and  i  L  p  Chapter 4. Stress-Strain Model  -  Parameter Evaluation and Validation  €_a (%)  E_a  (a) Effect of Parameter KGp  113  (%)  (b) Effect of Parameter  rip  0  &  1.2 Ea  (%)  (c) Effect of Parameter  c_a  Rf  (%)  (d) Effect of Parameter flf  Figure 4.29: Sensitivity of Parameters KG, np, R 1 and  i  Chapter 4. Stress-Strain Model  G,. A higher value of  -  Parameter Evaluation and Validation  114  will result in a stiffer deviator stress response and a lower  volumetric strain response. The parameter R 1 and  i  define the shape and the failure  stress ratio in the hardening rule. Lower R 1 and higher  i  will give stiffer deviator  stress response. The elastic parameters are not considered here because they have been widely used and their significance is well understood.  4.5  Summary  A double hardening elasto-plastic model has been postulated to model the stressstrain behaviour of oil sands. Procedures for the evaluation of the parameters and the validation of the proposed model have been presented in this chapter. The model parameters are relatively easy to obtain and can be determined from conventional isotropic and triaxial compression test results.  The parameters have  physical meanings and a sensitivity study has been carried out on the parameters to better explain their physical significance. Laboratory test results for various stress paths have been compared with the model predictions. Measured results and predic tions agree very well and the model predicts the shear induced dilation effectively. From the validations presented in this chapter, it can be concluded that the proposed model captures the stress-strain behaviour of oil sands very well.  Chapter 5  Flow Continuity Equation  5.1  Introduction  The pore fluid in the oil sand matrix comprises three phases namely gas, oil and water and therefore, the fluid flow phenomenon is of multi-phase nature.  In petroleum  reservoir engineering, the flow in oil sand is often analyzed as multi-phase flow, but solely as a flow problem without paying much attention to the porous medium. The most widely used model to analyze the flow in oil sand is called ‘3-model’ or ‘the black-oil model’ (Aziz and Settari, 1979) and it makes the following assumptions. 1. There are three distinct phases; oil, water and gas. 2. Water and oil are immiscible and they do not exchange mass or phases. 3. Gas is assumed to be soluble in oil but not in water. 4. Gas obeys the universal gas law. 5. Gas exsolution occurs instantaneously. With these assumptions, and considering the effects of stresses and temperature changes in the sand skeleton, a flow continuity equation is derived in this chapter from the general equation of mass conservation. However, the flow equations are not considered separately for individual phases as in petroleum reservoir engineering. All three flow equations are combined and a single effective equation is formulated. In essence, the derived flow continuity equation is similar to a single phase flow equation 115  Chapter 5. Flow Continuity Equation  116  in geomechanics but the permeability and compressibility terms have been changed to include the effects of different phase components. The flow continuity equation will be combined with the force equilibrium equation and will be solved as a consolidation problem as explained later in chapter 6.  5.2  Derivation of Governing Flow Equation  In this section, the flow continuity equation for a single phase in one dimension is de rived first. Later, it is expanded to three phase flow in three dimensions. The amount of flow of one phase component depends on the saturation and the mobility of that particular phase. When the fluid is Newtonian and the flow is slow, as it usually is in petroleum reservoir situations, the volumetric flux of a phase is proportional to the potential gradient acting on it and inversely proportional to its viscosity. The coef ficient of proportionality is the Darcy’s permeability. This is customarily expressed as the product of the relative permeability of phase 1 (krj), and the absolute Darcy permeability (k), of the medium to flow when a single fluid entirely fills the pore space. Mathematically this is expressed as  1 VP  (5.1)  where, v k krt I’i  1 P  -  -  -  -  -  velocity vector (in m/s) permeability matrix of the porous medium (in m ) 2 relative permeability of phase 1 (non dimensional) viscosity of phase 1 (in kPa.s) pressure in phase 1 (in kPa)  Now, consider a single phase (denoted by 1) flow in one dimension (in z direction) as shown in figure 5.1.  Chapter 5. Flow Continuity Equation  117  0 ( vj 71) dz ôz S n 1 dz  Phase ‘I’ in pore fluid  n dz  Pore fluid dz  —  Solids  I n S 1  -  -  -  -  porosity saturation of phase 1 velocity of phase 1 in z direction unit weight of phase I  Figure 5.1: One dimensional flow of a single phase in an element  Chapter 5. Flow Continuity Equation  118  Weight of phase 1; wi=nSi7jdz  (5.2)  Incoming mass flux: v  (5.3)  yi  Outgoing mass flux: =  dz  +  (5.4)  Difference between flux coming in and flux going out: QI_Qo_O(vZz_y1)d  dt  8z  55 (.)  Rate of storage: &wlO(nSl7l)d  8t  ôt  56  For conservation of mass, the difference between the incoming and outgoing flux should be equal to the rate of storage. Thus,  —  O(vi 71) 8z  —  —  8(n Si 7i) ôt  5 7 (.)  Expansion of the partial differentials in equation 5.7 gives 8n 1 8S avzl 871 &Y1 7 i 2 —7z—+v i —=7iSi+nSi--+n --  (5.8)  Dividing by 7i yields 8m 1 871 S 1 8S 1 2 8v 1 871 2 v ——+----—-—=Si——+n—-—+n-----8z at 71 8z 7’ at  (5.9)  Now, consider all five terms in equation 5.9 separately, starting from the left hand side.  Chapter 5. Flow Continuity Equation  119  0v 21 8z By Darcy’s law (equation 5.1)  v  vz1  can be written as =  1 kkr 8P —  —  ILl  c9z  =  (5.10)  and therefore,  1 P 2 9  =  (5.11)  where kmi k  -  -  mobility of phase 1 intrinsic permeability of the porous medium [function of void ratio; k = f(e)]  1 k,.  -  relative permeability of phase 1 [function of saturation; k,. 1 = 1 f(S ) ]  -  viscosity of phase 1 function of temperature and pressure;  vz  1 P  2 71  -  -  = f(8, F )] 1  velocity of phase 1 in z direction pressure in phase 1  8z  The change in unit weight due to the increase in pressure can be expressed as,  871 = where  71  (5.12)  Chapter 5. Flow Continuity Equation  1 B  -  -  120  bulk modulus of phase 1 unit weight of phase 1  Therefore, 87j yi 8z  —  —  —  —  v  B öz 1 1 kmi 6P 1 8z 8z B  5 13  This term involves the square of the pressure derivatives and can be neglected as small compared to the other terms (ERCB, 1975).  3.  S 1 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  Chapter 5. Flow Continuity Equation  .4.  121  1 b-y S n—-— 7i ôt By using equation 5.12 this term can be written as,  87j 8t 71 51  8P 1 8t 1 B 51  (5.16)  1 as .  Summation of saturations of all phase components should always be equal to unity. Hence, when combining the equations for all the phases, the summation of this term over all the phases will be zero. Mathematically this can be expressed as  (5.17) Since the final equation is to be derived by combining all the phases this term need not to be considered in detail.  By making the changes to the terms as explained so far, equation 5.9 can be written as P 2 8 1 1 8P s 8Si kmi -ä—-H-S1 --—n- --—n--O  (5.18)  Extension of equation 5.18 to three dimensions yields  L 1 kmiV P 2 i+5  —  —  = 0  (5.19)  Chapter 5. Flow Continuity Equation  122  where 2  b’P  ElF  p 2 9 t  (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;  H-5 P 2 V kmo 0  a  —n  0 ap s 0  —n  = 0  (5.22)  n =0  (5.23)  for gas phase;  V+S 9 P 9 kmg 2  —  —  where, km S B  -  -  -  mobility saturation 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 This can be written as  (++)  =0  (5.24)  Chapter 5. Flow Continuity Equation  kEQ  V2P +  123  —  CEQ  OP --  = 0  (5.25)  where, kEQ  -  equivalent hydraulic conductivity = kmo + kmw + kmg  CEQ  -  equivalent compressibility 0 (S Sg Sw —  Equation 5.25 is similar to the one used by Vaziri (1986) and Srithar (1989), except for the equivalent conductivity term. They considered the contributions from different fluid phase components in the compressibility but not in the hydraulic conductivity. Recently, Settari et al. (1993) have also used an effective hydraulic conductivity term to model the three-phase fluid which is similar to the equivalent hydraulic conductivity term derived above. The equivalent hydraulic conductivity is a function of mobilities of the phases which in turn depend on their relative permeabilities and viscosities. Evaluations of relative permeabilities and viscosities are described in detail in the next sections. The equivalent compressibility is a function of saturation and bulk modulus of individual phase components and the details of its evaluation are described in section 5.5.  5.3  Permeability of the Porous Medium  The permeability of the porous medium (k) mainly depends on the amount of void space. Lambe and Whitman (1969) collected considerable experimental data to study the variation of k with void ratio. Although there was a considerable scatter in the data, they found that there is a linear relationship between k and a void ratio function e / 3 (1 + e) for a wide range of granular materials. It can be argued that various other relationships could be established for the varition of k with e. However, without the  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) e/(1 + eo)  (5.26)  where e 0 and k 0 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  krg  =  f(S)  =  ) 9 f(S  Chapter 5. Flow Continuity Equation  125  I  krow  kr  k,,, 0  Swmaz  0  SW—,’.. (a) Oil-water system  I  kr rg_  0 I  Sgc  Sgmoz  Sg  0  (b) Gas-oil system Figure 5.2: Typical two-phase relative permeability variations (after Aziz and Settari, 1979)  Chapter 5. Flow Continuity Equation  126  0 k,.  =  ) 0 f(S  (5.27)  The function for the relative permeability of oil, k,. , is not readily known and it 0 is estimated from the two-phase data for k,., and k,. , where, k,.OW is the relative 09 permeability of oil in an oil-water system and k,. 09 is the relative permeability of oil in an oil-gas system. Their functional dependence are given by  0 k,.  =  09 k,.  f(SW)  = f(S ) 9  (5.28)  The simplest way of estimating k,. 0 would be,  0 = k,. k,. 09 0 k,.  (5.29)  Two more accurate models have been proposed by Stone (1970), only the first of which is considered here. In this model, Stone (1970) defines normalized saturations as  S  =  —  om  wc  =  SO  Som  om 95 S 15 wc am  S  S  (5.30)  So  Sam  (5.31)  wc  5;  =  (5.32)  Where, S, is called the critical or connate water saturation at which water starts to flow. When S,, is less than S, the relative permeability of water k,., will be zero.  Sam is called the residual oil saturation at which oil ceases to flow when it displaced simultaneously by water and gas. If S is less than o 0 will be zero. 5 m, k,. According to Stone (1970), the relative permeability of oil in a three-phase system is given by  0 k,.  =  S  (5.33)  Chapter 5. Flow Continuity Equation  127  The factors i3 and ,i3 are determined from the end conditions that equation 5.33 should match the two-phase data at the extreme points. The two extreme cases of =  0 and SL,  =  give  —  —  The region of mobile oil phase (i.e. k,. 0  1  —  (5.34)  S  k,. 09 1 S’9  (5.35)  —  >  0) predicted by Stone’s model I is shown  in figure 5.3 on the ternary diagram assuming increasing  W 5  and S . For conditions 9  depicted by point outside the hatched area, the relative permeability of oil will be zero.  100% GAS  E1S Som  100% WATER  I,.  100% OIL  Figure 5.3: Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979)  Aziz and Settari (1979) modified Stone’s model because Stone’s model will reduce  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 oilwater system at S, and an oil-gas system at S = 0 are physically identical. Both systems will have, S = S and S 0 = 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  0 = S krocw w /39 k,.  k  (5.37)  (5.38)  —  —  k,. 09 c’ 11 ,.0cwI —  9 LI  (. )  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  Chapter 5. Flow Continuity Equation  0 k,.  =  129  S;+k,?,&S) 09 (k,?  s  rOW  wko(1SI  (5.40)  4  rog  a  “9k0rog\(1Sg where, 0 k,?  -  relative permeability of oil at connate water saturation in a water-oil system  09 k,?  -  relative permeability of oil at zero gas saturation in an oil-gas system  When k,? 0  =  k,?og, the above model reduces to the one given by Aziz and Settari  (1979). Kokal and Maini (1990) compared model predictions against measured data and found very good agreement. The best comparison given in their paper is shown in figure 5.4. From the discussion so far in this section, it can be concluded that the relative permeabilities in three-phase system can be written as  =  f(S)  (5.43)  =  ) 9 f(S  (5.44)  , Sw, so, S) 09 , k,. 0 f(k,.  (5.45)  9 k,.  0 k,.  =  0 k,.  =  f(5)  (5.46)  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  —  ) 9 f(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  =  (S 1 C  —  C ) 2 c3  (5.48)  where C ,C 1 2 and C 3 are constants. Figure 5.5 shows a comparison of experimental data with calculated values using the power law functions.  1.2  1  =  2.769 (0.80 Sw)  k  =  1.820 (Sw 0.20) 2.735  -  -  row  0.8  a) E a, a)>  1.996  k  k rw  ‘  0.6  a) 0.4 Symbols- Experimental Lines Correlation  0.2  -  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 5. Flow Continuity Equation  133  Table 5.1: Parameters needed for relative permeability calculations Parameter  Som 3 ,A 1 A ,A 2  Description Connate or critical water saturation Residual oil saturation Parameters for variation of k with Si,, )A3] 2 in water-oil system =1 A ( S A —  ,B 1 B ,B 2 3  Parameters for variation of krow with 5 w in water-oil system [kro (B Sw)B3] 1 B 2 —  , 1 C  ,C 2 C 3  Parameters for variation of krg with Sg )c3] 2 in oil-gas system [krg = 9 (S C 1 C —  ,D 1 D 3 ,D 2  Parameters for variation of krog with S 9 ] 3 )D 59 in oil-gas system [k. 09 = 2 (D 1 D —  Relative permeability of oil at connate water saturation in water-oil system  09 k,?  Relative permeability of oil at zero gas saturation in oil-gas system  Chapter 5. Flow Continuity Equation  lfl(9,p)  2.3026  ( +  134  —  8-30 30315)  3.0020] + B 0 F exp(d 6)  (5.49)  where, b  log .Lt(3o,o) + 3.0020  a  =  0.0066940.b + 3.5364  0 B  =  0.0047424.b + 0.0081709  d  =  —0.0015646.b + 0.0061814  8 F  -  -  -  temperature in degrees Celsius pressure in MFa gauge viscosity of oil in Fa.s at 30°C and 1 atmosphere (0 gauge)  Figure 5.6 shows the comparison of this correlation with experimental results for Cold Lake and Wabasca bitumens.  The above correlation is implemented in the  finite element program CONOIL. However there is an option in CONOIL to read and interpolate user specified viscosity-temperature data, in case this correlation does not hold for a particular bitumen.  5.5.2  Viscosity of Water  The viscosity of water does not change as drastically as that of oil. For instance, at 30°C the viscosity of water is 0.8 mPa.s and at 100°C, it is 0.28 mFa.s. A change of 70°C in temperature causes a reduction in viscosity by a factor of 3 as compared to 200 for oil.  The viscosity-temperature data for water are well established and  can be obtained from the international critical tables. The viscosity of water is well represented by the following equation:  =  where  (b+8)  (5.50)  Chapter 5. Flow Continuity Equation  135  50000  —  *  -  10000  empirical equation  experimental  -  ‘‘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  -  \ 10000 V I 5000 S C  —  *  empirical equation experimental  -  o  4iooo. 500  P •  100  -  so  10 23  40  I  I  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  -  -  a, b, n  -  136  viscosity of water temperature 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  Chapter 5. Flow Continuity Equation  CEQ  =(+±)  137  (5.51)  The bulk moduli of the water and oil can be assumed constant, though they depend slightly on pressure. The important parameter that affects the equivalent compressibility is the comprssibility of gas. If there is more gas present in the pore fluid, it will be more compressible. The compressibility of gas can be determined using the gas laws. The basic gas laws governing the volume and pressure relationships are Boyle’s law and Henry’s law. According to Boyle’s law, under constant temperature conditions,  V=w 9 P R T  (5.52)  where, 9 P T R V, Wg  -  -  -  -  -  absolute pressure of gas absolute temperature universal gas constant volume of gas weight of gas  Under undrained conditions, the weight of gas does not change and therefore, equation 5.52 can be written as  (5.53) where K is a constant. Gas can be present in both the dissolved and free states. According to Henry’s law (Sisler et al., 1953); the weight of gas dissolved in a fixed quantity of a liquid, at constant temperature, is directly proportional to the absolute pressure of the gas above the solution. Mathematically, this can be written as  Chapter 5. Flow Continuity Equation  138  0  0  (5.54) where -  weight of dissolved gas  and the superscripts 0 and 1 refer to the initial and final conditions, respectively. In other words, Henry’s law implies that the volume of dissolved gas in a fixed quantity of liquid is constant at a constant temperature and at a confining pressure F, when the volume is measured at F. Thus  9 Vd  =  H V 0  (5.55)  where H  -  Henry’s constant, which is temperature dependent and, over a wide range of pressure, is also pressure dependent  0 V  -  volume of oil  Since the volume of dissolved gas is constant, free and dissolved gas components can be combined. Then application of Boyle’s law to the entire volume yields (Fred lund, 1976)  (5.56) where 1 V 9 is the volume of free gas. Rearranging the terms yields,  —  V 1 9 —  By differentiating equation 5.57,  ID1IT? rgkVdg  +  0 9  in  Vf —  Vdg  Chapter 5. Flow Continuity Equation  8V ° 9 8P  —  139  (Vdg+Vj 1 9 P ) 2 ) 0 (Pg  558)  By adopting the sign convention that compression is positive,  1  -  /v 1 (av )  Bg  ° 9 8P  -  —  P(V +v) 9 (P V° 2 ) 0 9  559)  Now, 9 Vd  =  TT0  =  V  0 ThiS  =nS Pg  =  Pa+F+Pc  (5.60)  where, -  9 S n  -  -  -  P  -  -  saturation of oil saturation of gas porosity atmospheric pressure pressure in oil capillary pressure  By substituting these last expressions into equation 5.59 0 ( 1 9 iP + HS S) 9 B °) 9 S(P 2  ( 561)  Generally, in an incremental procedure the values used are estimated at the be ginning of the increment. Therefore, from equation 5.61 the value of ) /B at the 9 (S beginning of an increment can be given as 9 + 0 HS 5 BgPa±P+Pc 59  562  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.  =  9 ( 1 E S  E ) 2 E3  (5.63)  where E ,E 1 2 and E 3 are constants. Therefore, by substituting equation 5.62 in equation 5.51, the equivalent com pressibility can be written as  CEQ=fl  5.7  S,  SL  0 + 9 (S ) HS  (5.64)  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))  F + 2 kEQ V  —  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.,  0 E Q =  a ( 8 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  1 Ply =  (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  80  l  0 V Vl—Vo  =  80  (5.68)  By adopting the usual notation  =  (5.69)  Chapter 5. Flow Continuity Equation  142  Hence, 9 = a  (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 , Pt,,, P 9 S, S ,S 0 0 and P . The solution of the problem therefore requires the follow 9 ing three additional equations:  (5.71) 0 P Pg  —  —  P,L?  = f(S, S ) 0  (5.72)  Po  9 , 0 f(S ) S  (5.73)  Chapter 5. Flow Continuity Equation  143  Compared to the flow analysis in reservoir engineering, the major disadvantage of the analytical model proposed here is that the treatment of multi-phase fluid as an equivalent single phase fluid. This kind of analytical model is adequate for coupled stress, deformati&n and flow analyses, but may not be effective if a detailed fluid flow analysis is required. If detailed results about the flow are required, a separate rigorous flow analysis may be necessary.  However, the results from the stress-deformation  analysis and the rigorous flow analysis should be looked at together to obtain a complete picture. There are several advantages in the analytical model suggested in this study. In reservoir engineering, the stress distribution and the deformation through the porous medium are generally not considered.  But the real problem at hand is a coupled  stress, deformation and flow problem and the proposed analytical model in this study addresses all these concerns.  The combined form of the flow continuity equation  makes the formulation simpler and significantly reduces the number of degrees of freedom, computation time and other such factors.  Chapter 6  Analytical and Finite Element Formulation  6.1  Introduction  Oil recovery by steam injection from heavy oil reservoirs is a coupled stress, defor mation and fluid flow problem. Therefore, a realistic analytical model should include the fluid flow behaviour and the mechanical behaviour of the sand matrix. Modelling of the stress-strain behaviour of the sand skeleton and the fluid flow behaviour with multi-phase fluid has been described in chapters 3 and 5 respectively. This chapter describes the development of an analytical model which couples the stress-strain and fluid flow behaviour, and a solution scheme using finite element procedure. Basically, the problem in hand is considered as a consolidation phenomenon. The analytical models used in the consolidation analysis are mainly based on theories developed by Terzaghi (1923) and Biot (1941). Terzaghi’s theory is restricted to a one dimensional problem under a constant load. Biot extended Terzaghi’s theory to three dimensions and for any arbitrary load variable with time. Both Terzaghi’s and Biot’s theories assume a linear elastic stress-strain behaviour and an incompressible pore fluid. Closed form solutions for the consolidation equations have been derived by a number of researchers, but only for very simplified geometry conditions and for linear elastic material behaviour. For instance, De Josselin de Jong (1957) obtained a solu tion for consolidation under a uniformly loaded circular area on a semi-infinite soil. MacNamee and Gibson (1960) obtained solutions to plane strain and axisymmetric problems of strip and circular footings on a consolidating half space. Booker (1974) 144  Chapter 6. Analytical and Finite Element Formulation  derived solutions for square, circular and strip footings.  145  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  —  2 F  =  0  (6.1)  where -  -  subscript  j  =  total stress tensor body force vector -  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  Chapter 6. Analytical and Finite Element Formulation  =  —  147  0  (6.2)  The total stresses are the sum of the effective stresses and the pore pressures. Mathematically, this can be written as  =  oj H- P Sj  (6.3)  where -  P  -  -  effective stress tensor pore pressure Kronecker delta  From chapter 3, the incremental stress-strain relation including the effects of tem perature changes can be written as (see equation 3.73)  LE,d  =  H- Dkz  (6.4)  where -  -  -  tensor relating incremental effective stress and strain strain tensor strain due to the change in temperature  The strains can be expressed in terms of displacements as  =  (U +  (6.5)  where -  displacement vector  Combining equations 6.3, 6.4 and 6.5 and substituting into equation 6.2 yields, [Dk1 (Uk,I + LU1,k)]  +  a+  ] 1 ie  —  =  0  (6.6)  Chapter 6. Analytical and Finite Element Formulation  6.2.2  148  Flow Continuity Equation  The flow continuity equation for a multi-phase fluid including temperature induced volume changes was derived in chapter 5. The final equation (see equation 5.78) can be written in tensor notation as  [(kEQ)F]  —  1 +czEQ  =  0  (6.7)  where kEQ P  U CEQ aEQ  6  -  -  -  -  -  -  equivalent hydraulic conductivity tensor pore pressure displacement vector equivalent compressibility equivalent coefficient of thermal expansion temperature  and superscript dot denotes the partial differentiation with respect to time (8 /8t). Equations 6.6 and 6.7 are the resulting equations that have to be solved in the consolidation analysis. In these equations, the fundamental unknowns to be solved are the displacements, U, and the pore pressure, P. The unknowns are solved by finite element procedure using Galerkin’s weighted residual scheme.  6.2.3  Boundary Conditions  To define the problem, both the displacement and the flow boundary conditions must be specified. For the class of problems considered in this study, the following boundary conditions can be specified. For the displacement boundary conditions, a part of the surface, jected to known applied traction,  can be sub  while the reminder of the surface, SD, can be  subjected to specified displacements, U, which may be zero.  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,  q 5  is considered impermeable,  i.e. there is no flow across the boundary. Mathematically, these boundary conditions can be expressed as a = 3 n 1 t U  =  (J  P=P  for t0 for t  (6.8)  0  (6.9)  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 =  6.3  CEQ P  at  t  =  0  (612)  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}  -  { o’}  -  [D’]  -  strain vector effective stress vector matrix relating effective stress and strain  The volumetric strain can be expressed as  = {m}T{e}  where  {m}T =  {1 1 1 0 0 0}  ,  (6.14)  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.,  =  where  (6.15)  Chapter 6. Analytical and Finite Element Formulation  (Lc)j  n  -  -  151  volume change in the pore fluid porosity  In chapter 5, an equivalent compressibility has been obtained by considering all the fluid components. Based on this approach, the changes in pore pressure can be expressed as  (6.16) where  -  CEQ  -  change in pore pressure equivalent compressibility  Substitution of equations 6.14 and 6.15 into equation 6.16 gives 1  =  {m}T{E}  (6.17)  CEQ  From the definition of effective stress  {o-} =  {&r’} + {m}i.P  (6.18)  Substituting equations 6.17 and 6.13 in equation 6.18 yields  =  [[D’l +  1 CEQ  {m}{m}T] {e} =  [D]{e}  (6.19)  ‘  Equation 6.19 adds the contributions of both the skeleton and the pore fluid to express the stress-strain relation in terms of total stress. Thus, the matrix [D] for the total stress analysis is given by  [D]  =  [D’] +  1 CEQ  {m}{m}T  (6.20)  Chapter 6. Analytical and Finite Element Formulation  152  Equation 6.19 is used in the finite element formulation for undrained conditions. The pore pressure is not an unknown in the resulting system of equations, but is obtained from equation 6.17, once the deformations are computed. Byrne and Vaziri (1986) claimed that this method has definite advantages such as adaptability for saturated or unsaturated soils and for any stress-strain relation. In particular, this method gives stable solutions when the effective stresses go to zero and all of the load is carried by the pore fluid. For an incompressible fluid, CEQ becomes zero and the above formulation becomes ill-conditioned. However, this can be overcome by setting the value of CEQ to a suitably low but finite value (Naylor, 1973).  6.4  Finite Element Formulation  The equations governing the consolidation with multi-phase fluid and temperature effects have been derived and are given by equations 6.6 and 6.7. The best method of obtaining a solution for these equations is to use a numerical technique such as finite element method. The finite element procedure can be formulated in a number of ways. For instance, Sandhu and Wilson (1969) used a Gurtin type variational principle. Booker and Small (1975) employed a variational theorem involving Laplace trans formations. Christian and Boehmer (1970), Carter (1977) and Small et al. (1976) obtained the solutions through the principle of virtual work. Hwang et al. (1972) and Chang and Duncan (1983) used the weighted residual technique to develop the finite element formulation. The choice of the different approaches depends on the type of the problem and the boundary conditions on one hand, and the knowledge of the mathematics involved on the other hand. In this study, Galerkin’s weighted residual scheme is used to develop the finite element formulations. The weighted residual scheme is quite straight forward, has relatively less mathematics involved, and is less error prone. In the  Chapter 6. Analytical and Finite Element Formulation  153  Galerkin scheme only a single application of Green’s theorem is needed to obtain a set of integral equations. These equations can be easily turned into matrix form and solved. However, it should be noted that regardless of the approach used, whether weighted residual or variational principle, the end results will be the same. From the previous section, the governing differential equations to be solved are,  (Uk,j + UZk)j  +  +  +  [(kEQ)  —  CEQ F + aEQ  0  —  =  0  (6.21)  (6.22)  To develop the finite element formulation for these equations, the domain being analyzed is subdivided into a finite number of elements. The quantities of the four independent variables within each of the elements, U and F, are approximately repre sented by means of shape functions and their values at the nodes. The equations 6.21 and 6.22 have at most second order derivatives of displacements and pore pressures. However, by applying Green’s theorem, it can be reduced to first order. Therefore, to solve the resulting integral equations the shape functions for displacements and the pore pressure should be continuous. Hence, the displacement and the pore pressure fields within the element can be written as  U  =  N  e  (6.23) (6.24)  where 5eT  q  U’  =  {S,  , 2 S  .  .  =  {qi,  q,..  .  -  -  ,  S}  , q}  displacement field pore pressure field  (nodal displacements) (nodal pore pressures)  Chapter 6. Analytical and Finite Element Formulation  N  -  -  154  shape functions for displacements shape functions for pore pressures  U” and F” are approximate solutions and substituting these values into equations  6.21 and 6.22 will not exactly satisfy the equations, but will give some residual errors as follow: + z1  ±  [Dkl  [(kEQ  F]  ,  1 + U”,  + [D  —  CEQ F” +  —  aEQ  S  (6.25) (6.26)  In the weighted residual scheme, these residual errors are minimized in some fashion to give the best approximate solution. Thus, for the best solution  jwrdv=O  (6.27)  where r w  -  -  residual error weighting function  In Galerkin’s scheme the weighting functions are chosen to be the same as the assumed shape functions. Then the following equations can be obtained to minimize the residual errors r 1 and r : 2  dv 1 jNr  0  (6.28)  jNpr2dvzO  (6.29)  =  The strains and the derivative of the pore pressure within an element can be written as  Chapter 6. Analytical and Finite Element Formulation  155  (6.30) =  m’  (6.31)  B Iq 6  (6.32)  where mT  B & B  ={1 1 100 0} -  shape function derivatives  Green’s theorem for integration involving two functions,  and  over the domain  j V V d1  (6.33)  can be expressed as  J  ç V dIZ =  j  c  (V) dF —  where, I’ is the boundary around  and i is the normal to the boundary.  By substituting equations 6.23 to 6.26 and 6.30 to 6.32 into equations 6.28 and 6.29, and by applying Green’s theorem, one obtains  j BDB,4Sdv + / B’mNqdv  =  /  NTds +  j NFdv  -  /  B’Dedv  (6.34)  —  /  BkEq Bqdv H-  /  NpTmTBuSdv  —  /  ’Ndv 1 CEQ N  =  —  /  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  T {S} [L]  -  t [E] {q}  -  [G] {q}  =  156  -{C}  (6.37)  where [K] =fBDBdv [U rr1l L-’J  =fBmNdv Jv  r,  DTI  p  —  p 0 -  nEQ L’p  [G] =fCEQNNpdv {LA} {zXC}  =  f8 NTds  +  f NFdv  —  ed 6 f B,D  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 implies trapezoidal rule, a  0 implies a fully explicit method and a  =  =  =  1/2  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 {8} [U]  t [E] {qt + Lq}  —  —  [C] {q}  =  —{zC}  (6.40)  By rearranging the terms,  T {zS} [U]  —  zSt [[E]  —  [G]] {q}  =  —{zC} + [E]qtt  (6.41)  Chapter 6. Analytical and Finite Element Formulation  157  By combining equations 6.36 and 6.41 and writing them in a full matrix form gives,  [K]  [L]  T [L]  [[E] Lt  1A  z8 —  Lq  [G]]  J  ‘1  [E] /.t{qt}  —  {tC}  (6.42)  J  By changing the notation, equation 6.42 can be written in the usual matrix form as,  [K]  [L]  ILS1  T [L]  [E’]  q  J  1A1 LC’ J  (6.43)  where,  {LW’}  =  tt[E}{qt}  -  {C}  [E’] =t[E]-[G] Equation 6.43 gives the matrix equation to be solved for an element. From the element matrix equations a global matrix equation is formed and solved for displace ment and pore pressure unknowns. Stresses and strains are then evaluated from the displacements. It should be noted that it may not be possible to use the above consolidation routine to get the initial condition results, i.e. at t {zC’}  =  0 and [E’]  =  =  0. This is because for Lt  =  0  [G].  If the fluid is incompressible [G] will become zero and equation 6.43 will become ill conditioned. For this situation an appropriate solution can be obtained by assuming a small value for t. This will circumvent the ill-conditioning. However, a better way to get around this problem is to use the undrained routine to obtain the initial condition and then use the consolidation routine.  Chapter 6. Analytical and Finite Element Formulation  6.5  158  Finite Elements and the Procedure Adopted  The principal steps and the details such as obtaining shape functions, its derivatives, formulation of stiffness matrix, numerical integration, etc., can be found in any stan dard finite element text book. Therefore, only a summary with some discussions on certain issues which are important for the class of problems considered in this study, are given in this section. The developed analytical model has been incorporated in an existing 2-dimensional finite element code, CONOIL-Il (Srithar (1989)) and also in a new 3-dimensional finite element code, CONOIL-Ill. The following subsections address the key aspects in the development of these finite element codes.  6.5.1  Selection of Elements  The choice of the finite elements has been an important issue when analyzing con solidation problems. Different researchers used different element types. Sandhu and Wilson (1969) introduced a composite element, consisting of a six-noded triangle for the displacements expansion, and only three nodes being used for the pore pressure ex pansion. The displacements varied quadratically over the element, while the stresses and strains obtained by differentiating the displacements varied linearly. Since the pore pressures are expressed in terms of three nodal values, they vary linearly too. Therefore, the element has the same order of expansion for both stress components and pore pressures. Yooko et al. (1971a) used several different elements, all of which used the same expansion for the displacements and for the pore pressures. This makes N  =  N for any choice of element and the relevant matrices can be derived easily.  The examples they presented include a two noded bar element, a three noded ax isymmetric triangular ring element an a four noded rectangle. However, Yooko et al. (1971b) had difficulties in obtaining reasonable results for the initial undrained conditions. Sandhu et al. (1977) also compared several finite elements and concluded  Chapter 6. Analytical and Finite Element Formulation  159  that the elements which had the same expansion for displacement and pore pressures do not give satisfactory answers at the initial stages of consolidation. However, they claimed that at later stages of consolidation, the differences in the results for different element types are insignificant. Ghaboussi and Wilson (1973) used an isoparametric element of four nodes with the standard expansion for pore pressures and two additional nonconforming degrees of freedom for the displacement expansion. The two additional degrees of freedom are eliminated by static condensation after the element stiffness is completed. However, this procedure does not give the same expansion for pore pressures and stresses, but uses a lower order expansion for pore pressures than for displacement. In the 2-dimensional finite element code employed in this study, element types sim ilar to those proposed by Sandhu and Wilson (1969) are used. Figure 6.1 shows the two different triangular elements used for consolidation analysis in the 2-dimensional code. Figure 6.2 shows the element types available in the 3-dimensional code. The eight-noded brick element uses the same expansions for pore pressures and displace ments, whereas the 20-noded brick element uses different shape functions for displace ments and pore pressures.  6.5.2  Nonlinear Analysis  The solution of the nonlinear problems by the finite element method is usually achieved by one of the following techniques: 1. Incremental or stepwise procedures 2. Iterative or Newton method 3. Step-iterative or mixed procedures The method employed herein is a form of the mixed procedure which follows the midpoint Runge-Kutta or modified Euler method. In this scheme, two cycles  Chapter 6. Analytical and Finite Element Formulation  A  Displacement nodes (2 d.o.f)  Q  Pore pressure nodes (1 d.o.f)  Linear strain triangle  Cubic strain triangle  6 displacement nodes 3 pore pressure nodes 6 nodes and 15 d.o.f.  15 displacement nodes 10 pore pressure nodes 22 nodes and 40 d.o.f.  Figure 6,1: Finite Element Types Used in 2-Dimensional Analysis  160  161  Chapter 6. Analytical and Finite Element Formulation  A  157 A  8  8  ,.  20  s34  18  4 A10  4  1 •  5I  19Li16  /11 4  6  14  A  12 •  Corner nodes  =  8  4  D.o.f per node  =  4  =  0  A Internal nodes  =  12  =  0  D.o.f. per node  =  3  Corner nodes  =  8  D.o.f. per node  =  Internal nodes D.o.fper node  8-Nodded Brick Element  20-Nodded Brick Element  Figure 6.2: Finite Element Types Used in 3-Dimensional Analysis  Chapter 6. Analytical and Finite Element Formulation  of analysis are performed for each load increment.  162  In the first cycle of analysis,  parameters based on the initial .conditions of the increment are used.. At the end of first cycle, parameters at the midpoint of the load increment are computed. In the second cycle, the midpoint parameters are used to analyze the load increment and the final results are evaluated. To obtain more accurate results, this process would have to be continued until the difference between successive results satisfies the specified tolerance.  Such an iterative procedure can increase the computer time drastically  and therefore, was not employed. However, an improvement in the results is made by estimating the imbalance load at the end of second cycle and adding that to the next load increment.  6.5.3  Solution Scheme  Selection of the method for solving the simultaneous algebraic equations is a major factor influencing the efficiency of any finite element program, and there are variety of solution techniques to choose from. Essentially, there are two classes of methods; one is the direct solution methods and the other is the iterative solution methods. The direct methods use a number of exactly predetermined steps and operations, whereas the iterative methods make an approximation to solve the equations. The most effective direct solution methods are basically variations of the Gaussian elimination method. Most of the methods take advantage of specific properties of the stiffness matrix, its symmetry, its positive definiteness or its banded nature to reduce the number of operations and the storage requirements to accomplish a solution. Bathe and Wilson (1970) and Meyer (1973) discussed the relative merits of the current popular methods, and both of these references contain extensive bibliography. The frontal solution scheme for symmetric matrices (Irons, 1970) and for unsym metric matrices (Hood, 1976) have been employed in the finite element codes. In the frontal solution scheme, the element stiffness matrices are assembled and solved by  Chapter 6. Analytical and Finite Element Formulation  163  Gaussian elimination and back substitution process, but the overall global stiffness matrix is never formed. The variables are introduced at a later stage and eliminated earlier than in most of the other direct solution methods.  Since the variables are  eliminated as soon as conceivably possible, the operations with zero coefficients are minimized and the total arithmetic operations are fewer. As a result, it is faster and requires less core memory than band routines.  In addition, it is not necessary to  apply a stringent node numbering scheme. Its efficiency is essentially a function of element numbering. Theoretically, the frontal solution scheme will always perform better or at least as well as the bandwidth solving routines in terms of accuracy and efficiency (Irons, 1970; Irons and Ahmad, 1980). Some comparisons have already given in the literature to substantiate this claim (eg: Sloan 1981; Light and Luxmore, 1977; Hood, 1976). The frontal solution scheme is specially attractive for unsymmetric matrices because less computer storage is required. The stress-strain model considered in this study deals with a nonassociated flow rule which results in an unsymmetric stiffness matrix and therefore, using the frontal solution scheme has a definite advantage. The main disadvantage of this method is the complexity of the internal book keeping. However, the bookkeeping is a programming problem and does not concern the user. Another limitation of this technique may be its dependence on the element numbering sequence. Although it is rather easier to number the elements in a logical manner relative to numbering the nodal sequence, it does place some effort on the user. However, the difficulty can be easily dealt with, if some form of front width minimizer is incorporated in the program. There are different front width minimizing schemes available such as by Sloan and Randolph (1981), Akin and Pardue (1975) and Pina (1981).  The procedure by Sloan and Randolph (1981) is built into the  2-dimensional finite element code.  Chapter 6. Analytical and Finite Element Formulation  6.5.4  164  Finite Element Procedure  A broad overview of the procedures followed in both, the 2-dimensional and 3dimensional 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 and Finite Element Formulation  C  Start  165  D  Read basic data and allocate storage for principal arrays  Read and set the initial conditions Read relevant data for the load increment Evaluate stiffness matrix and load vector Solve for displacements and pore pressures Evaluate the changes in stresses, strains and pore pressures  Is this the last cycle of analysis’  No  Update relevant variables to average values  Yes  riJpdate all the results and print  No  No  Last increment? Yes  C  Stop  D  Figure 6.3: Flow Chart for the Finite Element Programs  Chapter 6. Analytical and Finite Element Formulation  166  10. If the current stress state exceeds the strength envelope, or if there is strain softening, load shedding vector is computed. 11. Steps 5 to 9 are repeated until all the elements satisfy the failure criterion or in other words, until the load shedding is converged. 12. The imbalance loads at the end of the increment are calculated and added to the next load increment, if any. 13. Steps 4 to 12 are repeated until all the load increment data have been analyzed. The final states of the previous load increment are used as the initial conditions for the next load increment.  6.6  Finite Element Programs  The finite element programs have been written in FORTRAN-77 and are portable to any operating platforms.  There are two separate programs; CONOIL-Il which  is a 2-dimensional program to perform axisymmetric and plane strain analyses and, CONOIL-Ill which is a 3-dimensional program to perform 3-dimensional analysis. Al though these finite element programs have been developed with special attention paid to the problems in oil sand, they are capable of doing general drained, undrained and consolidation analyses effectively. Both programs are capable of analyzing excavations as well. Brief descriptions of these programs are given in this section. Applications of the programs are discussed later in chapter 7.  6.6.1  2-Dimensional Program CONOIL-Il  The 2-dimensional program CONOIL-Il was originally developed by Vaziri (1986) based on the program CRISP (University of Cambridge). It was later modified by  Chapter 6. Analytical and Finite Element Formulation  167  Srithar in 1989 with an improved formulation for temperature analysis. CONOIL II has been divided into two separate programs; the ‘Geometry Program’ and the ‘Main Program’. The main purpose of this split is to reduce the effort for the user. The geometry program automatically generates and numbers the midside and interior nodes. It also renumbers the elements and nodes to minimize the front width and creates a input file for the main program, containing the relevant information about the finite element mesh. The program also has some special features.  The triple  matrix product as suggested by Taylor (1977) is adopted in the formation of stiffness matrix, and this will eliminate all the unnecessary arithmetic operations which will result in zero coefficients. The geometry program consists of 11 subroutines and the main program consists of 58 subroutines. The names of the subroutines and their functions are presented in appendix C. Grieg et al. (1991) developed a pre/post processor package, COPP, for CONOIL II to facilitate viewing and plotting the CONOIL-Il input and output data. COPP is menu driven, very user friendly and provides many options for the user.  6.7  3-Dimensional Program CONOIL-Ill  The 3-dimensional program CONOIL-Ill has been developed from scratch following the same sequence of procedures as the 2-dimensional one. However, compared to the 2-dimensional program, the 3-dimensional program has less special features, and it does not have a post processor yet.  The 3-dimensional program comprises 43  subroutines. The names of the subroutines and their functions are given in appendix C. A User manual and some example problems are presented in appendix F.  Chapter 7  Verification and Application of the Analytical Procedure  7.1  Introduction  The analytical procedure described in chapter 6 has been incorporated in the finite element program, CONOIL. The main intention of this chapter is to verify and val idate the finite element program, and to demonstrate its applicability. The program deals with a number of aspects such as, dilative nature of sand, three-phase pore fluid, gas exsolution, effects of temperature changes, etc.. and the best way of verifying the program would be to consider each aspect separately. The program is verified here by considering some particular problems for which theoretical solutions are available. Once verified, the program is validated by comparing some experimental results with predictions from the program. Then, the program has been used to predict the re sponses in a oil recovery problem. A problem concerning pore pressure redistribution after liquefaction has also been analyzed to show the applicability of the program to other geotechnical problems.  7.2  Aspects Checked by Previous Researchers  The two dimensional version of the finite element program CONOIL has been used at the University of British Columbia since 1985, with improvements being made from time to time. Cheung (1985), Vaziri (1986) and Srithar (1989) have demonstrated the capability of the program on a number of aspects. Since those aspects are kept 168  Chapter 7. Verification and Application of the Analytical Procedure  169  intact with the improvements made in this study, those verifications and validations are still valid. These are briefly described herein. The general performance of the program in predicting stresses and strains has been verified by Cheung (1985), by considering a thick wall cylinder under plane strain conditions. Closed form solutions for this problem have been obtained from Timoshenko (1941). The results from the program and the closed form solutions are in excellent agreement and are shown in figure 7.1. Cheung (1985) also validated the gas exsolution phenomenon in the program. Laboratory test results by Sobkowicz (1982) on gassy soil samples have been consid ered. Sobkowicz (1982) carried out triaxial tests to predict the short term undrained response, i.e, no gas exsolution and the long term undrained response, i.e., with com plete gas exsolution, The comparisons of the test results with the program results are shown in figures 7.2 and 7.3. The measured and predicted results agree very well. The overall structure of operations for a consolidation analysis has been verified by Vaziri (1986). The closed form solution developed by Gibson et al. (1976) for a circular footing resting on a layer of fully saturated, elastic material with finite thickness has been considered for the verification.  A comparison of the computed  results and the closed form solutions, shown in figure 7.4, demonstrate that they are in very good agreement. Srithar (1989) modified the procedure for thermal analysis in the original CONOIL formulation. He verified the new formulation under drained and transient conditions. The closed form solution presented by Timoshenko and Goodier (1951) for a long elastic cylinder subjected to temperature changes has been considered to verify the formulation under drained condition. The closed from solution and the finite element results are shown in figure 7.5 and are in remarkably good agreement. To verify the formulation for thermal analysis under transient conditions, a closed form solution was derived by Srithar (1989) for one dimensional thermal consolidation  Chapter 7.  Verification and Application of the Analytical Procedure  170  0 e 0  0  0  a) closed form o  o  I  I  program 0  Qc  a)  I  0  I  I  I  I  I  V  I c’J  246810  ) 0 Radii (r/r 3000 MPa 1/3 initial stress : or final stress : o inside radius : r  E  =  I’  —  = = =  o. = 6000 kPa 2500 kPa 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  0 b  0 0 C  I.’ .-  40  60  80  100  120  Total Stress (kPa) (X1O’  140  )  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 a)  >0  .— s-I  -I-)  0 s-I  0 xc  0  lab data 0 l.a .4-’  Cl)  -4-’  0 0 C  0  20  40  60  80  1 Effective SigmaP (kPa) (X10  )  Figure 7.3: Comparison of Observed and Predicted Strains (after Cheung, 1985)  Chapter 7.  Verification and Application of the Analytical Procedure  I  I  0.25  0.30  Analytical Solution . Finite Element Analysis  -  0.35 -  0.40  173  r  -  xI30  045  y/30  DIE—i  .5 —  0.50 I  4 ia-  io  I  0.0  I 1.0  I  io_2  10  Ct V  v T  -  a)Amount of settlement 0.0  I  I  I  Analytical Solution  0.2‘b%,  ‘%. %,  0  %\  Finite Element Analysis  0.4 U 0.6  —  0.8  -  1.0  y/B—0 D/E — 1  I  4 ia—  i—  vO.3  v—0.0  I  10—2  ia-’  1.0 Ct  7  V  -—  2 U  b) Degree of settlement Figure 7.4: Results for a Circular Footing on a Finite Layer (after Vaziri, 1986)  10  Chapter 7.  3000  Verification and Application of the Analytical Procedure  —  Symbols Solid lines  —  2000  CONOIL—Il Closed Form  174  La  —  —  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)  Chapter 7.  Verification and Application of the Analytical Procedure  175  with a uniform temperature rise. The closed form solution was obtained by making analogy to the closed form solution by Aboshi et al. (1970) for a constant rate of loading. Figure 7.6 shows the closed form solutions and the program results and they agree very well. [n the figure, z denotes the depth at which the results are considered and H denotes the total depth. The performance of the program for undrained thermal analysis has been validated by comparing the experimental results on oil sand samples in a high temperature consolidometer obtained from Kosar (1989). Computed and measured results show good agreement as illustrated in figure 7.7.  7.3  Validation of Other Aspects  In this research work, a new elasto-plastic stress-strain model has been developed and incorporated in the finite element code. This will realistically model the dilation and the loading-unloading sequences encountered in oil sands. To validate the program’s capability to model the dilation phenomenon, the triaxial test results on oil sand given by Kosar (1989) have been considered. The triaxial test has been modelled by four triangular elements as shown in figure 7.8. An axisymmetric analysis has been carried out with the relevant boundary conditions as shown in figure 7.8. The model parameters used are listed in table 7.1. The predicted and the measured results are compared in figure 7.9. Also shown in that figure are the results using a hyperbolic model. It can be seen from the figure that the shear stress versus axial strain response can be very well captured by both the elasto-plastic and hyperbolic models. But the hyperbolic model does not predict the volumetric strain behaviour as measured, whereas, the elasto-plastic model predicts results that match the measured values. The triaxial test results for a load-unload-reload type loading on Ottawa sand obtained from Negussey (1985) have been considered to validate the loading-unloading operation of the program.  The triaxial test specimen has been modelled by four  Chapter 7.  Verification and Application of the Analytical Procedure  z/H 30  L’  0.875  —  0  -  =  176  0  0  0  0  ci  0  p  0  2u  Cl) Cl, Q)  0 ci)  00000  0  —,  z/H 20  CONOIL—Il closed form solutions  =  LH  0.5  —  0  0  C  0 ci) C (1)  ci)  0 u-i 0 0  0— 0  i  I  I  1000  2000 Time(s)  I  I  3000  4000  Figure 7.6: Pore Pressure Variation with Time for Thermal Consolidation (after Srithar, 1989)  Chapter 7. Verification and Application of the Analytical Procedure  177  800000  /  Test results CONOIL—Il  / /0  a) C  6— -  c  -  C-)  a) E  0  -  -  2 4— 0 > > D  -  -  E 0 0 0 0 I  20  I  I  70  I  I  I  I  I  120 170 Temperature(° C)  I  I  220  Figure 7.7: Undrained Volumetric Expansion (after Srithar, 1989)  Chapter 7.  Verification and Application of the Analytical Procedure  Table 7.1: Parameters for Modelling of Triaxial Test in Oil Sand (a) Elasto-Plastic Model  Elastic  kE n kB m  Plastic Shear ? ! KG np Rf  3000 0.36 1670 0.36 0.72 0.54 0.33 1300 -0.66 0.80  (b) Hyperbolic Model  kE 1100  0.49  kB  m  R  700  0.47  0.6  49  13  178  Chapter 7.  Verification and Application of the Analytical Procedure  0 1.5  cm  Figure 7.8: Finite Element Modeffing of Triaxial Test  179  Chapter 7. Verification and Application of the Analytical Procedure  180  3500  3000  .  2500  ‘a 0  2000  S  ‘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  Plastic  kE m kB m ‘i  ) IL KG np 1 R  3400 0.0 1888 0.0 0.49 0.85 0.26 780 -0.238 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  Verification and Application of the Analytical Procedure  Chapter 7.  182  350  300  250  a .  200  150  100  50  0.00  0.05  0.10  0.15  0.20  El (%)  Figure 7.10: Comparison of Measured and Predicted Results for a Load-Unload Test in Ottawa Sand  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):  p  16 M n  —  3 T r  m1,3,..  1 mrz —i sin 2H  I —  exp  ir 2 (m ’ \ —  T  ‘1  (7.1)  where p T n  -  -  -  -  M 1 a 8 a  -  -  -  pore pressure at distance z at time t time factor porosity change in temperature at time t constrained modulus coefficient of volumetric thermal expansion of liquid 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  Chapter 7.  Verification and Application of the Analytical Procedure  184  im  H  G  21®22____3  13  ®  14  H=lm  11  7 5  6  4..  z  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  1 a /°C 3 m / m  5 a /°C 3 m / m  k rn/s  M MPa  H rn  0.5  0.25  3 1x10  5 1x10  6 2x10  18.3  1  Chapter 7.  Verification and Application of the Analytical Procedure  185  35 z/H  =  (a)  0.75 -  30 -25 20 0 1. 0 I-. 0 0  10 Symbols Program Line Closed form -  5  -  0  I  0  500  I  1000  1500  2000  2500  3000  3500  Time (s)  4Z 30  z/H  =  (b)  0.5  Symbols Program Line Closed form -  -  0  0  500  I  I  1000  1500  I  I  I  I  2000  2500  3000  Time (s)  Figure 7.12: Comparison of Pore pressures for Thermal Consolidation  3500  Chapter 7.  Verification and Application of the Analytical Procedure  186  Technology and Research Authority (AOSTRA) is considered herein for analysis. The UTF uses a steam assisted gravity drainage process with horizontal injection and production wells. A brief description of the UTF and the problem to be analyzed are presented here. Further details about the UTF can be found in Scott et al. (1992), Laing et al. (1992) and in AOSTRA reports on UTF. The UTF of AOSTRA is located near Fort McMurray, Alberta, and is currently being used to test the shaft and tunnel access concept for bitumen recovery in deep oil sand formations. The geological stratification at the UTF comprises a number of soil layers. However, it can be simplified as consisting of three different soil types, in a broad sense. Devonian Waterway formation limestone exists below a depth of 165 m. Overlying the limestone is the McMurray formation oil sand which is about 40 m thick. The top 125 m overburden can be classified as Clearwater formation shale. A schematic 3-dimensional view and a plan view of the UTF are shown in figures 7.13 and 7.14 respectively. There are two shafts accessing the tunnels in limestone beneath the oil sand layer. The tunnels were constructed in the limestone at a depth of about 178 m with the roof being about 15 m below the limestone-oil sand inter face. Three pairs of horizontal injection and production wells were drilled from the tunnels up into the oil sands at about 24 m spacing. A vertical section of the well pairs was instrumented with thermocouples for measuring temperatures, pneumatic and vibrating wire piezometers for measuring pore pressures and extensometers and incinometers for measuring horizontal and vertical displacements. Figure 7.15 shows a vertical cross-sectional view (section A-A’ in figure 7.14) of the three well pairs. Modelling of all three well pairs with their steaming histories and with the detailed geological stratification would be complex as the steaming of different well pairs started at different times. To illustrate the problem and to demonstrate the applicability of the program in a simple manner, only one well pair is considered here for analysis.  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 7. Verification and Application of the Analytical Procedure  188  Shaft#1  I Observation Tunnel  i  T Injector/Producer Wellpairs...... Section A-A’  —*  Geotechnical  —  A  Cross Section  Figure 7.14: Plan View of the UTF (after Scott et al., 1991)  A’  4  U)  C.11  -c  CD  —.  =  CD  CD  0<  gg  .... ..  CD  0  3  CD  C;’  3 3  ..  C;’  0.  =  ......  0  ......  .  .  V..  :•:••  .  . .•  .•.  .  .•.•  CD  .•.  •.•  .•.• .  •.•.•.  .......  o 3 00  CD  0  CD  Chapter 7.  Verification and Application of the Analytical Procedure  190  To analyze the oil recovery with one well pair, the shaded region in figure 7.15 is modelled by finite elements. The finite element mesh consisted of 240 linear strain triangular elements as shown in figure 7.16. Plane strain boundary conditions are assumed. The injection and production wells are modelled by nodes with known pore pressures. The steam injection pressure is assumed to be maintained at 2800 kPa (1300 kPa above the in-situ pore pressure) and the production pressure is assumed to be at 2000 kPa (500 kPa above the in-situ pressure). The parameters used have been obtained from laboratory test results reported by Kosar (1989) and from AOSTRA and are listed in table 7.4. The gas saturation is assumed to be zero. i.e., the pore fluid is assumed to comprise only water and bitumen. The bitumen saturation is assumed to be 70  %.  The temperature-time histories of the nodes have also to be specified as an input to the program.  These were obtained from the field measurements made at the  UTF. The temperature contours in the oil sand layer at different times are shown in figure 7.17. The steam chamber which is the region in the oil sand layer where the temperature is the same the steam temperature, grows with time as shown in the figure. At time t  =  30 days, the steam chamber extends to a distance of about 10 m  horizontally and vertically from the injection well. Even though a larger domain is analyzed as shown in the finite element mesh, the results are plotted only for the oil sand layer which is of primary interest. The predicted excess pore pressures in the oil sand layer are shown in figure 7.18. The injection and production wells are also indicated in the figure. Figure 7.18 (a) shows the excess pore pressure contours at 10 hours after the steam injection started. Only a small region adjacent to the injection well experiences significant changes in the pore pressures. This correlates very well with the temperature contours at that time, as shown in figure 7.17 (a). With time, the region of higher pore pressure expands as shown in figures 7.18 (b) and (c) indicating the growth of the steam chamber which  Lii  0.0  50.0  100.0  -150.0  200.0  250.0  —  —  —  —  —  0.0  ;  c 1  I  I  100.0  I  I  Distance (m)  I  200.0  ——  I  Figure 7J6: Finite Element Modelling of the Well Pair  —  I  2 2/?/  22/7// ///// I  I  300.0  I-.  0  C-.  Chapter 7.  Verification and Application of the Analytical Procedure  Table 7.4: Parameters Used for the Oil Recovery Problem (a) Soil Parameters  Elastic n kB m Plastic Shear  171 7 J.:l  ). t  Plastic Collapse Other  KG rip 1 R C p e ) 2 k(m ( 8 a / / 3 °C) m m  3000 0.36 1670 0.36 0.75 0.13 0.53 0.31 1300 -0.66 0.73 0.00064 0.61 0.6 1.0 x 10—12 3.0 x 1O  (b) Pore Fluid Parameters  B 0 B / cx(m / 3 °C) m ( 0 cz / / 3 °C) m m u,o(Pa.s) 1 S  kro = 2.769(0.8 krw = 1.820(S  5.0 x iO 2.5 x iO 3.0 x i0 3.0 x iO 20 0.2 0.2  —  —  996 S)’ 0.2)2.735  192  CD  p  (j)  0  CD  cn  0  0  C)  CD  p  CD  CD  cyq  -  3  0 (D  D  0  01 0  0  (0  00  DCJ  f  C-,  01 Q  (0  D  C,  NJ  (m)  (I)  NJ 0  NJ 0  NJ Q  EIevLon NJ  NJ 0  EIevLon -  0  (m) 01 0  3  0  :3  C-,  (0  (11 0  NJ 0  NJ 0  0  0  NJ 0 NJ 0  EIevLon 0  -  (m) 01 0  c,z  0  I-’.  p  Chapter 7.  Verification and Application of the Analytical Procedure  194  is also implied by the temperature contours in figure 7.17. The 1000 kPa excess pore pressure contour from the field measurements is compared with the predicted contours in figure 7.19. It can be seen from the figure that the measured zone of 1000 kPa is larger than the predicted zone. However, the shapes of the pore pressure contours are similar to the measured ones. The predicted horizontal and vertical stresses are shown in figures 7.20 and 7.21. As the steam chamber grows the soil matrix expands and since the soil is more constrained in the horizontal direction, the horizontal stresses increase. The vertical stresses also increase, but not as much as horizontal stresses. The pattern of the stress contours also indicates the movement of soil and the shape of the steam chamber. The stress ratio which is an index giving the current stress state relative to the failure stress state is shown in figure 7.22. It appears that the shape of the steam chamber and the corresponding temperature increases create higher shear stresses in the region above the steam chamber. This is implied by higher stress ratios and a maximum stress ratio is about 0.45 is predicted in the region about 15 m above the injection well. Since the predicted stress ratios are well below unity, there would not be any failure. Figure 7.23 shows the horizontal displacement along a vertical line at 7 m from the wells, at time t  =  30 days. Also shown in the figure are the field measurements  made in a instrumented bore hole at about the same distance away from the wells. It can be seen that the field measurements are slightly larger than the predictions at some locations, but in the overall picture, the predictions are in reasonable agreement with the measurements. The variation of vertical displacements with the distance from the wells at the injection well level is shown in figure 7.24.  Maximum displacement of 21 mm is  predicted at a distance 15 m from the well. There is no field measurements available that could give the results due to the steaming in a single well pair. The vertical  CD  C,,  0  0  c3  01 0  3  3  CD  0  c3  cJ  0  01 &  3  CD  0  01 0  NJ 0  CD  0  D  NJ 0  D  C) CD  0  U)  C-,  Cl)  01  C—,  -N  Cm)  C-,  NJ  EIevLon  U  NJ 0  NJ  Co  01  U  (m)  Cl)  -  U  CD  I-  Cl) U)  CD  0 CD  I.  CD  EIevLon  EIevLon(rn)  01  CC)  I.  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 3 initial stages of production, more water will be produced than oil because much of the bitumen will be immobile. With time, the temperature will increase, the viscosity of bitumen will reduce, it will become mobile and more bitumen will be produced. As the oil recovery process continues at the steady state conditions, eventually, the amount of bitumen produced will become less as it is replaced by water.  CD  I-  C,)  0  N  0  CD  oq  3  C) CD  E3  C,,  0  c3  01 0  NJ 0  NJ  NJ  EevoLon -  (m) 01  3  0 CD  :3  a,  C-,  C,)  (ii 0  NJ 0  0  N)  NJ  EIev3Lon -  (m) 01  3  0 CD  :3  C) 0)  ci)  0  (ii 0  NJ 0  0  N) 0 NJ 0  EIevLon 0  -  (m) 01 0  Chapter 7.  Verification and Application of the Analytical Procedure  ()  -  t  198  tlOhrs  40 C  0  30  /6ØO  LU I  20 0  10  20  30 Dstance  40  50  60  (m)  40 C  0  30 LU  20  0  10  20  30 D9Lance  40 (m)  E  40 C  0  >30  20 •10  20  40 DLnce  (m)  Figure 7.21: Vertical Stress Variations in the Oil Sand Layer  60  Chapter 7.  Verification and Application of the Analytical Procedure  199  40 0  10  20  30 DsLance  40  60  40  60  40  60  (m)  40 C  0  30 Li 20 0  10  20  30 DLence  (m)  40 C  0  30 Li 20 0  10  20  30 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  ••  20  I  I  0  5  I  I  10  15  20  Displacement (mm)  Figure 7.23: Comparison of Horizontal Displacements at 7 m from Wells  25  Chapter 7. Verification and Application of the Analytical Procedure  201  25  20 E 15  10  5  0 0  10  20  30  40  50  Distance (m)  Figure 7.24: Vertical Displacements at the Injection Well Level  60  Chapter 7.  Verification and Application of the Analytical Procedure  202  160  140  120 C,,  100 0 U.  o  80  C 0  E  <60  40  20  0 0  5  10  15  20  25  Time (days)  Figure 7.25: Total Amount of Flow with Time  30  35  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 10’ m to 10 2 . The predicted pore pressure 2 m 13 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  Chapter 7.  Verification and Application of the Analytical Procedure  204  5.5  I  5 •.8 -  4.5  a)  0 U  4 3.5  2  5  10  20  50  100  200  500  Time (days) (a) Flow Rate of Water  2  1.5  E a)  1  0.5 0 U-  01  20  Time (days) (b) Flow Rate of Oil  Figure 7.26: Individual Flow Rates of Water and Oil  500  Chapter 7.  Verification and Application of the Analytical Procedure  205  50  E  40  C  E 0 Li.  30  0  0  E  20  ,2  0  50  100  150  200  Time (days)  Figure 7.27: Total Amount of Oil Flow  250  300  350  CD  Cl)  U)  ‘-I  0  I-.  CD  fri  Cl)  CD  fri  0 CD  co  —3  3  C) CD  D  D  C-,  Co  0  01 0  RD 0  RD  RD  EIevton -  (m) 01  3  C) CD  D  C-,  Co  U  01 0  RD 0  0  r\D  0  RD RD  EIevLon -R  (m) 01  3  (0  0  Z3  C-,  C/)  U  01 0  RD 0  RD RD ‘SC  LHevLon  F—  ‘SC  -  ( cm  (ii ‘SC  I  Chapter 7.  Verification and Application of the Analytical Procedure  207  40 C  0  30 LU  20 0  10  20  30 DtLnce  40  60  40  60  40  60  (m)  S 40 C  0  30  a) LU  20 0  10  20  30 DsLncG  (m)  S  40 C  0  30 LU  20 0  10  20  DLance (m) Figure 7.29: Stress Ratio Variation for Analysis 2  Chapter 7. Verification and Application of the Analytical Procedure  208  have been predicted if the permeability was kept the same and the rate of heating increased. The detailed results show that the stress ratio of one of the elements in the highest stress ratio region reached unity indicating shear failure. Since the region of shear failure is small and away from the wells, it will not cause any problems. However, if the region of shear failure is large, there will be significant deformations and if the region extends to the wells, it may cause significant damage to the wells. To avoid this kind of situation, the rate of heating should be reduced. The above example illustrates the usefulness of this type of analytical treatment for oil recovery projects. This type of analysis provides important information about the rate of heating, possible failure zones, deformations, stability of the wells etc., beforehand. Without an analytical treatment, these concerns have to be tested in the field on a trial and error basis, which would be very costly.  7.6  Other Applications in Geotechnical Engineering  Even though the finite element program CONOIL was developed for analyzing prob lems related to oil sands, it can also be applied to other potential geotechnical prob lems. An example problem which involves pore pressure migration after liquefaction is described herein. Generally, loose sands are susceptible to liquefaction in the event of an earthquake and to prevent such liquefaction, loose sand deposits are commonly densified. The densified zone in a loose sand deposit will only be stable provided high excess pore pressures from the surrounding liquefied sands do not penetrate it during and after the earthquake. This concern is examined herein with different densification schemes used in practice. A typical soil profile for Richmond, British Columbia, was considered in the anal ysis. The soil profile comprised 3 m of clay crust, underlain by 15 m of loose sand and followed by 5 m of dense sand as shown in figure 7.30. The earthquake is assumed to  Chapter 7.  Verification and Application of the Analytical Procedure  209  generate 100% pore pressure increase in loose sand and 30% pore pressure increase in dense sand zones. A hyperbolic stress-strain model was considered and the material parameters used in the analysis are given in table 7.5. Three cases which represent three different densification schemes were studied as illustrated in figure 7.30 Table 7.5: Soil Parameters Used for the Example Problem Soil Type  n  kB  m  1 R  k,, (m/s) 2.5 x 10  kh (m/s) 5 x iO  Clay  150  0.45  140  0.2  0.7  Liquefied Sand  300  1.0  180  1.0  0.8  Dense Sand  2000  0.5  1200  0.25  0.6  2.5 x 10  5 x iO  Dense Sand with Drain  2000  0.5  1200  0.25  0.6  1 x 10  1 x iO  Clay with Drain  150  0.45  140  0.2  0.7  1 x iO  1 x 1O  5  X  1O  1  X  10  In case 1, densification is assumed to the full depth of the loose sand without any drainage system. This case may represent a field condition where densification is achieved using timber piles without any drainage provisions. In case 2, the den sification is assumed with a perimeter drainage system. This may represent a field situation where densification is achieved using timber piles with a perimeter drainage system of vibro-replacement columns. In case 3, the drainage was assumed in the densified zone. This may represent densification by vibro-replacement. In the anal ysis, the drains were not considered on an individual basis, instead, the densified zone with drains was modelled as a soil with an equivalent permeability. The equiv alent permeability can be estimated from the size and spacing of the drains and the permeabilities of the materials. The excess pore pressures for the three cases considered at various times after the earthquake are shown in figures 7.31, 7.32 and 7.33. The excess pore pressures are  CD ()  0  r0  0  0.  CD  NC)  -O  CD 0  CD  .o.  .  0  3  0  r0  Chapter 7.  Verification and Application of the Analytical Procedure  211  shown in terms of pore pressure ratio u/o , in which u is the current excess pore 0 pressure, and o is the initial vertical effective stress. pressure rise and u/o 0  =  0 u/a  =  0 represents zero pore  1 represents 100% pore pressure rise or liquefaction. The  variations of the excess pore pressure ratios with time and distance from the centre of the densified zone are shown in graphs (a) and (b) in the figures. Graph (a) shows the variation at a depth of 5 m and graph (b) at a depth of 10 m. Graph (c) shows the excess pore pressure ratio with depth along the centre line. The results for case 1. (figure 7.31) show that the excess pore pressure in the surrounding undensified area migrates into the densified zone.  The pore pressure  ratio in the upper part of the densified zone rises to 1 which means liquefaction will be triggered. However, below a depth of 6 m, liquefaction is not triggered and piles penetrating below this depth could support vertical load, although significant horizontal displacements are likely to occur.  The results for case 2 (figure 7.32)  indicate that a perimeter drainage system is quite effective in preventing the migration of high pore pressure from the loose zone into the densified zone. A maximum pore pressure ratio of 0.5 is predicted 1  mm after the earthquake. The results for case 3  where the drainage is assumed throughout the densified zone are shown in figure 7.33. It can be seen from the figure that the drains in the densified zone are much more effective in preventing the migration of pore pressure in the densified zone. The pore pressure ratio in the densified zone increases from an initial value of 0.3 at time t  =  0, to 0.4 after 10 seconds and then reduces. The conclusions from the analyses are as follows. Densification alone such as could be achieved by driving timber piles will not prevent the high excess pore pressures from the surrounding liquefied zone penetrating the densified zone. Such penetration will cause liquefaction to a depth of 6 m for the conditions analyzed. Below this depth effective stress increases and timber piles would be capable of carrying vertical load although they could be damaged by horizontal movements. Perimeter drains could  Chapter 7.  Verification and Application of the Analytical Procedure 1.2  d=5m  (a)  30  d=lOm  (b)  0.8  Distance (m) Pore Pressure Ratio 2  0  1.5  1.0  0.5  (c) 4-  Si,  •  di’ 6 -  ! e  / /z’  / I,,  10  12  -  -  QI  I/i ‘:1  t=lmin t=3Ornin  14  -  t5hrs  t =lday 16  Figure 7.31: Variation of Pore Pressure Ratio for Case 1  212  Chapter 7.  Verification and Application of the Analytical Procedure 1.2  d=5m  (a)  d=lOm  (b)  0 c  10  5  15  25  20  30  Distance (m) Pore Pressure Ratio 0.1  0  0.2  0.3  0.4  0.  L  (C)  4-  /  / A  ::  0  :  b-f 1  12- 1  4  t=o  1  t=lmin t5rflifl  14-  I 1€  t=5hr  I  Figure 7.32: Variation of Pore Pressure Ratio for Case 2  213  Chapter 7.  Verification and Application of the Analytical Procedure d=5m  (a)  t=rnin  15  20  30  1.2  d=lOm  (b)  30  Distance (m) Pore Pressure Ratio 0.2 0.3  0  0.1  04 (C)  7 4-  4  I  G)  0  10-  12-  /  4  1  t  9  140  Figure 7.33: Variation Of Pore Pressure Ratio for Case 3  214  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).  Chapter 8  Summary and Conclusions An analytical procedure is presented to analyze the geotechnical aspects in an oil recovery process from oil sand reserves. The key issues in developing an analytical model are: the stress-strain behaviour of the sand skeleton; the behaviour of the three-phase pore fluid; and the effects of temperature changes associated with steam injection. A coupled stress-deformation-flow model incorporating these key issues is presented in this thesis. In modelling the stress-strain behaviour of the oil sand skeleton, shear induced dilation is an important aspect. Such dilation can increase the hydraulic conductivity and hence increase oil recovery. Dilation will also lead to reduced pore fluid pressure and increased stability. The other pertinent aspect is the stress-strain response un der stress paths involving a decrease in mean stress under constant shear stress and loading-unloading cycles. The stress-strain models used in the current-state-of-thepractice are linear or nonlinear elastic models which are incapable of modelling the above mentioned aspects realistically.  The major contribution of this thesis is the  development of a suitable elasto-plastic stress-strain model to capture the important aspects. The stress-strain model postulated in this thesis is a double hardening type consisting two yield surfaces.  The model has a cone-type yield surface to predict  shear induced plastic strains and a cap-type yield surface to predict volumetric plas tic strains. The predictions from the stress-strain model have been compared with laboratory test results under various types of loading and are in good agreement. The dilation, plastic strains due to cyclic loading, and the response under different stress  216  Chapter 8. Summary and Conclusions  217  paths have been well predicted by the stress-strain model. The pore fluid in oil sand comprises water, bitumen and gas and the three-phase nature of the pore fluid has to be recognized in modelling the behaviour of pore fluid. In petroleum reservoir engineering, multiphase fluid flow is modelled by elaborate multiphase thermal simulators.  In this study, the effects of multiphase pore fluid  are modelled through an equivalent single phase fluid. An effective flow continuity equation is derived from the general equation of mass conservation which is one of the other contributions of this thesis.  An equivalent compressibility term has  been derived by considering the individual contributions of the phase components. Compressibility of gas has been obtained from gas laws. An equivalent hydraulic conductivity term has been derived by considering the relative permeabilities and viscosities of the individual phases in the pore fluid. The relative permeabilities have been assumed to vary with saturation and the viscosities have been assumed to vary with temperature and pressure. Gas exsolution which would occur when the pore fluid pressure decreases below the gas/liquid saturation pressure has also been modelled. Oil recovery schemes commonly involve some form of heating and therefore, tem perature effects on the sand skeleton and pore fluid behaviour are important. Changes in temperature will cause changes in viscosity, stresses and pore pressures and con sequently in some of the engineering properties such as strength, compressibility and hydraulic conductivity. In this study, the stress-strain relation and the flow conti nuity equation have been modified to include the temperature induced effects. This approach of including the temperature effects directly in the governing equations gave very stable results, compared to the general thermal elastic approach. The final outcome of this research work is a finite element program which incorpo rates all the above mentioned aspects. The new stress-strain model, flow continuity equation, and other related aspects have been incorporated in the existing two di mensional finite element program CONOIL-Il. This required significant undertakings  Chapter 8. Summary and Conclusions  218  including a new solution routine as the new stress-strain model results in an unsym metric stiffness matrix. A frontal solution technique which requires less computer memory has been employed to solve the resulting equations. A new three dimen sional finite element program has also been developed following the same concepts. The validity of the finite element codes has been checked for various aspects by com paring the program predictions with closed form solutions and laboratory results. The predicted results agreed very well with the closed form solutions and laboratory results. The two dimensional finite element code has been applied to model a horizontal well pair in the underground test facility of AOSTRA. Results have been presented in terms of displacements, stresses, stress ratios and amounts of flow and discussed. The measured and predicted results have been compared wherever possible and they agree well. A method to obtain individual amounts of flow of the pore fluid components has also been devised. The type of analytical study presented in this thesis, is very important in oil recovery projects, since it could give insights into the likely behaviour in terms of stresses, deformations and flow. For instance, the permeability of the oil sand and the rate of heating due to steam injection have been examined in some detail. It has been revealed that in oil sands with low permeability, higher rates of heating would cause shear failure. If the local shear failure zone extends to the wellbore it could cause significant damage. Information of this kind would be beneficial to the successful operation of an oil recovery scheme. Although the finite element program has been developed to analyze the problems related to oil sand specifically, it can be applied to other geotechnical problems. To demonstrate its applicability, a problem involving pore pressure redistribution after liquefaction has been analyzed and the results are discussed.  Chapter 8. Summary and Conclusions  8.1  219  Recommendations for Further Research  Following the work presented in this study, some aspects can be identified in this area which require further study. Application of the finite element codes to more oil recovery problems should be carried out to increase the credibility of the models. The three dimensional code is newly written and even though various aspects of the code have been verified, it has not been applied to analyze a oil recovery problem of a three dimensional nature. The three dimensional code needs to be applied to either a physical model test or a field problem where the responses are measured, in order to check its capability to model three dimensional effects. Even though the analytical formulation presented in this study includes the effects of multi-phase fluid through equivalent compressibility and hydraulic conductivity terms, it does not take the flow of thermal energy into account. Incorporation of an elaborate multi-phase thermal and fluid flow model would be the most desired enhancement though it may be a very difficult task. Previous researchers concluded that analyzing the geomechanical behaviour and the thermal and fluid flow behaviour separately, and combining the results by partial coupling is useful and successful. However, a fully integrated analytical formulation may be more efficient. Perhaps another aspect which require further study would be the stress-strain model for the sand skeleton. The elasto-plastic stress-strain model described in this study does not consider anisotropy effects. Modelling strain softening by load shed ding may also be inefficient since it requires a large number of iterations. A stress strain model which includes anisotropy and strain softening effects in a realistic man ner is worth considering. Fractures in the oil sand layer are sometimes encountered in the oil recovery process by steam injection. Inclusion of modelling of fracture initiation and its prop agation will also be beneficial.  Bibliography  [1] Aboshi, H., Yoshikuni, H. and Maruyama, S. (1970), “Constant Load ing Rate Consolidation Test”, Soils and Foundation, vol. X, No. 1, pp 43-56. [2] Aboustit, B.L., Advani, S.H., Lee, J.K. and Sandu, R.S. (1982), Finite Element Evaluation of Thermo-Elastic Consolidation”, Issues on Rock Mech., 23rd Sym. on Rock Mech., Univ. of California, Berkeley, pp 587-595. [3] Aboustit, B.L., Advani, S.H. and Lee, J.K. 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(1971b),  “  Finite Element  Analysis of Consolidation Following Undrained Deformation”, Soils and Foun dations, Vol. 11, No.  4, pp 37-58.  [186] Yong, R.N., Ko, H.Y., eds (1980)  ,  “Proceedings of the Workshop on Limit  Equilibrium, Plasticity and Generalized Stress-Strain in Geotechnical Engineer ing”  ,  ASCE, McGill University.  Bibliography  241  [187] Yong, R.N., Selig, E.T., eds (1980)  ,  “  Proceedings of the Symposium  on the Application of Plasticity and Generalized Stress-Strain in Geotechnical Engineering  “,  ASCE, Hollywood, Florida.  [188] Zienkiewicz, O.C., Valliappan, S. and King, I.P. (1968) ysis of Rock as a ‘No Tension Material’  “,  ,  “Stress Anal  Geotechnique 18, pp 56-66.  Appendix A  Load Shedding Formulation The details of applying the load shedding technique to model strain softening are described in this appendix. During a load increment it is possible that the stress state of an element may move from P 0 to P as shown in figure A.1 This will violate the  ‘r/p  1 P  *  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 P . In load shedding 1 technique, this is done by taking out the shear stress equivalent to 242  and then  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 7i  1r +  (7sMp,1)  (i —  using equation 3.63 as  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  (U 2 1 ) 3 U  Lo  + + 3 ( 2 ) 1 U Ui)1 U 1  2 U  1213 + 1113(02 +  18I  ‘213  (ui + 3 +1  £73)  £72)  —  —  (U 2 t 1 ) 7  (A.3)  Appendix A. Load Shedding Formulation  244  The above equation can be rewritten as  /o- + 2 1 A Io H- A A z 3  (A.4)  To estimate the changes in principal stresses, two more equations are needed, in addition to equation A.4. The following two conditions are assumed during the load shedding to obtain the additional two equations: 1. The mean normal stress remains constant during load shedding. This gives +  1  2. The b-value  [(02  —  03)/(01  Oi  —  03  —  —  03)]  +  3 Lo  0  (A.5)  remains constant. Which implies  (o- + Zoi)  —  (03  ) 3 + Lo  (A6) —  By rearranging the terms (A.7) By solving equations A.4, A.5 and A.7 the following equations can be obtained for  02  and  o3:  3 H- b)(A  —  2—b ) —(2— b)(A 2 A 1 1+b  —(Lri  +  —  ) 2 A  (A.8)  (A.9)  3)  (A.l0)  Now, the changes in the Cartesian stresses can be obtained by simply multiplying the principal stress vector by the transformation matrix.  Appendix A. Load Shedding Formulation  l  m  12  y  2 m  12  2 m  245  2 n (A.11)  2ll, 2mm  2n,n  2l,l  2 2mm  2nn  2mm  2 n zna,  IO3  where a 1  li,, and l  m, m and m n, n and n  A.2  -  -  -  direction cosines of o to the x, y and z axes direction cosines of  O2  to the x, y and z axes  direction cosines of  03  to the x, y and z axes  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  where  { f} {o}  -  -  Force vector Stresses within the system  (A.12)  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)  =  J[B]Tfr}dv  (A.15)  This can be further written as  {f}  Following equation A.15 the force vector for load shedding can be obtained as  {IF}Ls  J[B]T{U}LSdV  (A.16)  Appendix B  Relative Permeabilities and Viscosities Some detailed explanations which are needed in the evaluation of equivalent per meability are given in this appendix. To evaluate the equivalent permeability, the relative permeability and the viscosity values of the pore fluid components are nec essary. The first section explains how to calculate the relative permeabilities and the equivalent permeability through an example data set. The viscosity values of water at different temperatures are given in section 2. Section 3 gives some insights into the viscosity of hydrocarbon gases and how to evaluate it.  B.1  Calculations of relative permeabilit ies  B.1.1  Relevant equations  The relative permeabilities of water, gas and oil can be obtained from the following equations:  Jc° S1k° Sw _qrog row 9’ —  Jo  1  I  krw =  krow  —  2 ( 1 B B  (S 1 C krg = 9 247  —  —  a Jg a t-’w  A ) 2 A3  (B.2)  Sw)B3  (B.3)  (B.4)  Appendix B. Relative Permeabilities and Viscosities  248  3 ) 9 S D  (B.5)  09 = D k,. 2 ( 1 D  —  krow(Sw) (1 S,,) 0 k  —  /3W  B6  —  —  k,. ) 9 (S 09 9kor09\(1—S 9 —  =  5;  —  Swc  s  0 S  —  S  om  Wc  Sorn  wc  (  om  S  (B.8)  So > Som  (B.9)  9 ‘9i  c’ ‘-‘uc  C’  ‘-‘Orn  where 0 k,. k,.  -  -  -  -  krg k  -  -  relative permeability of oil in 3-phase system relative permeability of water in 3-phase system relative permeability of oil in water-oil system relative permeability of oil in oil-gas system relative permeability of gas in 3-phase system relative permeability of oil at connate water saturation in a water-oil system  09 k  -  relative permeability of oil at zero gas saturation in an oil-gas system  S,,, S, S S, S, 5  -  -  -  om 5  2 , 1 A . A .. etc.  -  -  Saturation of water, oil and gas respectively Normalized saturation of water, oil and gas respectively Critical water saturation Residual oil saturation Constants  Appendix B. Relative Permeabilities and Viscosities  249  The equivalent permeability is given by  kEQ=k  B.1.2  (B.11)  P’g  I-o  Example data  = 0.2 k°7’OW —10 1.0  =  1 A  =  1.820  2 A  =  0.20  3 A  =  2.375  1 B  =  2.769  2 B  =  0.80  3 B  =  1.996  1 C  =  2.201  2 C  =  0.05  3 C  =  2.704  =  1.640  2 D  =  0.80  =  2.547  =  0.5  S,  I-sw =  k  =  B.1.3  Pa.s 4 8 x 10  0.4 =  Sg = 0.1  2OPa.s  m 2 1 x 10  Sample calculations  By substituting the data into the relevant equations 0.5 0.2 =0.5 1 0.2 0.2 —  —  5;  —  0.1 0.2  =  1  =  0.4— 0.2 1 0.2 0.2  —  —  —  —  0.2  =  0.1667  =  0.3333  =  2  x 10 Pa.s 5  (at 30°C)  Appendix B. Relative Permeabilities and Viscosities  = 2.769(0.8  —  krog = 1.640(0.8  0.5)1.996  = 0.25  0.1)2.547  = 0.661  250  0.25 = 0.5 = 1(1 0.5) —  —  krw  0.661 1(1 0.1667) —  1.820(0.4  9 = 2.201(0.1 k,.  0 = 0.3333 k,.  kEQ  B.2  = 1  ><  0.2)2.735 = 0.068  —  —  (1.0  0 793 —  0.05)2.704  X  = 0.001  0.1667+1.0  X  0.5) X  0.5  X  0.793 = 0.132  0.068 0.132 0.001 ‘ 10 —12 / + + 8 x 10 20 2 x 105)  1.350 x 10  m  m  —  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 C 0 10 20 30 40 50 60 70 80 90 100  0 17.93* 13.097 10.087 8.004 6.536 5.492 4.69* 4.07i 3.57. 3.16* 2.83*  1 17.326 12.73s. 9.843 7.834 6.41s  5.40s 4.62s 4.01* 3.52* 3.13* 2.82  2 16.74* 12.39o 9.60* 7.67* 6.29* 5.32* 4.56i 3.96z 3.483 3.095 2.79  (1, 12, 16, 17, 22, 24, 30, 31, 32, 38)  3 16.19a 12.06i 9.38*  7.511  i  6.184 5.236 4.495 3.909 3.44. 3.061 2.76  4 15.67. 11.748 9.16i 7.35 6.075 5.153 4,43i 3.8.5? 3.39* 3.027 2.73  5 15.18* 11.44? 8.94. 7.20* 5.97* 3.07s 4.36* 3.806 3.35i 2.994 2.70  6 14.72* 11.15* 8.74* 7.064 5.86* 4.99* 4.30* 3.756 3.31? 2.96a 2.67  7 14.28* 10.875 8.55i 6.92 5.77* 4.918 4.24s 3.70* 3.27* 2.93* 2.64  FoR,.1Ux.E AND UNITS At a pressure ox 1 atm., = a/(b + t)”. 1) X 10’J. , ,7p = ?7i[l + k,(P 2 At a pressure of P kg/cm , which may be taken as the value of,7 at 1 atm. 2 ‘11 is the value , when P is 1 kg/cm The unit of , is the poise unless otherwise stated. —  Table B.2: Viscosity of water below  00  C  H,O ov 100°C (16) Values as recorded by author accord with I. C. T. values 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 120 110 140 160 150 130 1000,7 2.32 1.84 2.56 1.96 1.74 2.12  Table B.3: Viscosity of water above 1000 C 0 BELOW 0°C (39) 2 H Values corrected and adjusted to accord with I. C. T. values above 0°C —2 —4 —6 —5 —8 —10 100077 19.1 20.5 21.4 22.2 24.0 26.0  8 13.872 10.60s 8.368 6.791 5.67s 4.84a 4.186 3,66i 3.24* 2.89. 2.62  9 13.47, 10.34o 8.181  6.661  5.582 4.77o 4.12s 3.61s 3.203 2.86*  2.59  Appendix B. Relative Permeabilities and Viscosities  252  Viscosity of hydrocarbon gases (from Carr et al., 1954)  B.3  The viscosity of hydrocarbon gases can be expressed as a function of reduced pressures and temperatures, i.e.,  1 IL  (B.12)  where  -  viscosity of gas at reduced temperature TR and at reduced pressure FR  pi  -  TR  -  PR  -  viscosity of gas at atmospheric pressure and given temperature temperature/critical temperature (in absolute units) pressure/critical pressure (in absolute units)  If the gas is a mixture of hydrocarbons, the pseudo-critical concept has to be applied.  Thus, in place of critical temperature and critical pressure, the pseudo-  critical temperature and pressure have to be used. The pseudo-critical temperature is given by  (B.13) The pseudo-critical pressure is given by  PPc=XzFci where  -  -  -  mol fraction of component i in the mixture critical temperature of component i in absolute units critical pressure of component i in absolute units  (B.14)  o O  : a)  .  a) .4  c  .•-  a) a)  0  0  ‘—4  4 •  a) U)  a)  U)  Cl)  -  o  a)  _-  0 a)  0  -d U) .4  a)  0 -  U)  )  ,4 -  0  d  —l  to  —  ;4-  -4-a  c  H  0 4) ‘-4  +  a)  to  -  to a)  U)  to  4-4  (3  a  a) Cl) V  -  0  0  Cl)  to  ) cd  -4  c  if)  ‘—I  0  c:  ; c  0 C) U)  Q) Cl)  to -  ‘d  ;-1  p-  0  -  E -4  ‘-  0  l-4  0  a  ‘  U)  U) .-I -4-a ‘-4  a)  4.44  a, ;-4  0  0 4)  .  -d a)  4  Q  to  a)  -4  a)  -  a)  0  E 0  ‘.  a) U) U)  a)  a) -  -4 -  _4-  l-4 I—’  2a) ,.D 0  U)  -I  if)  t  Ii t:  r-.l  ‘....  “  c .q_4 4  Cl)  ;i  .  II  c!:  ,  —  :  :j: ==  .4  a. 3wvo, an,.!  Ia,  IMIII: J.LI iLIJ_LLL_  tTITF ITIITI I  1  Ian3L,  iiI1 11 JWi 1L111FW 1  IiI1IM1ll1l 12 ltavd’  6A•ll1llllillllhIt woIi, _,  llD 1llllt ,s,wou,,.uo, -  VOI1W3OOflSd ,dd—U. ‘311fl1VU3dY131 VO4J4l4OOOflSd id  IIII VISd ‘UflSSWd  -4  c3  to  o  Cl)  a)  -4  0  ‘-4  0  :  o  -  U)  a)  0  -.  -  a)  a)  a) -  0  Cl)  0  Cl)  r  Cl)  0  Cl)  -  a)  0  a)  -4  a)  ..  .—  to  -d  a)  0 d  0  -4  0  0  l-4  o  0  U)  0  :1  11  ‘-4.,  ‘-s  —.  CD  4-s  CD  ,,.  o  •  CD  ,-,  9,  o  o‘—4.,  o  p  CD  I—  o  +  CD  2  o  .  U)  0  U)  p •-•  0 0  U)  c-  CD  0  ,-4.,  CD  CD  U)  —.  0  (  CD  U)  .  CD  CD  ‘‘  U)  CD  9’  •-  0  4,_-’ o 9’  CD  0 0  p,  ‘d I—. CD  )  CD  :-  CD  ‘-i’  CD  -,  -$-  CD  CD  ‘-U “l 0  U)  CD  CD  -  U)  CD  0  9’  CD  0 ‘-d  U)  9’  CD  -  _..  •  0  •  CD  CD  p  I-s  CD  4r—J] 4r :-‘  ‘-U  P  CD  p  p;;.  U)  +  j.  0  —.  .q-  CD  CD  ,-  CD  --  o  I—  —‘  1ELJ  + CD  C’)  4-s  U)  0 0  CD  ,-  CD  9’  O  oq  CD  -  4-+)  0  o  U)  0  4--  o  U)  0  CD  u,  ‘-U  -  0  _4-4  o  ‘)  0  9, CD  0  CD  C’) 9’  0 -“  CD •  ‘-5  U)  4-s  CD  .  0  —.  4--  9’  9, —.- ‘  CD  ‘-5  1  CD  b P ‘-U  p  0  ‘-5 CD  CD  0 U) ‘-U  CD 9’  0  9,  ci)  CD  9,  U)  0  I-s  p  0  ‘-1  4-f,  0  ‘-4-  U)  0 0  U)  CD  ‘-5  C r  ION  ADOED  8 0  b  rU1Er[I4flE  I I+H+FfAii(r  r  P 111)11 It’ll il/li VIKJ y ‘ii D1 l1(—19ItlA 1-II I IA-t-f-  .iii.nrriii DIAl  I  1  8  CORIICC1ION ADOLD TI) VISC—G14 -  DOEO To ,sc —G P t. , COACCTIOu  V 111)14-I 11111171115I 11111 Il ILUI  UUIIttt1IIIi1iVIlI1IJ11tIfl1VIIIAFII,I1VIlLItIU  .  0  -i-1-ii F1t4-iI—tWttW ttItliltI 1  ö  AJ, CENTIPOIS  *J1i1ir4’i Lj  8  °  •  tIU1 1M1W411 FFPII1IIlIFF1Fvr1I14-4aI4JYL4AY14n’i1fFIl  FL  I  8  VISCOSITY, AT I ATM.  0  1  Cs.  b  0  U)  ‘-5  CD  O’3  U)  ‘-U 0 ‘-4-  ‘-4-  ‘-5  CD  CD  0  CD  9,  CD  9, I-s 0 0  0  0  ‘-5  0  4-f,  U)  CD  ‘-4-  0  0  I.  ‘-U  9,  ‘-4-  >1  ‘-U  Appendix B. Relative Permeabilities and Viscosities  255  )  J  =  ‘‘ .-‘.  —  I.  .L. IH  .iL.  --  .LL.... III,  -  £1 / :i  r  ——  —--  r  L  —--  1..  —.  —  —  —  —  —  I  -  -  -  : —  —  -  —  -  -  /  ,  1—-  I  ,  1  0  —  .t.1-  -ir-rr  -.  ---r  ——  --  ‘--j-•-p.j-  •-  4t——  -  —  ‘  L.-(-L  .k  -  --  “i:  -f-J4 r;ia:: .±Ik ,  1! H-rv i-  ‘4-  -  —-  — —  -  I  E 1 ” -  i!  :::  I —  .  ,.,.‘77  —  -  r/r7  -  -  ——  i  4 IC  2  -  7  ---  -  3  4  .5  —  I.  .-f -i,.’_.  .  .  —--—--  —  6 7.6LD 2 PSEUCOREDUGED  — —  .  :..  .  I 4 56 7I9 3 PRESSURC  L  T  ‘  .  — —  -  Figure B.3: Viscosity ratio vs pseudo-reduced pressure  i4  CD  N  —  :  —  •  —L  i  I  -  -  -  -  -  -  :  —  -  -  -  -  :  7  EEHEE  4  : : :  3 : : : : :  :  g UI  -  —  -  :  -  J}tT[IWITWITh[IL  41  -  41WhII-fll-UIL[  I111II1Ht-1[1  r triuwiwrmirrni 4- U4 1414-I-I W-14111 I II-  ii i[14r1±1±thtinlll: i tiIlF{*fl+}1+HI1{If  -1  LI  I U—TI I I II 1—Hil I I H-I-fl U—I I IT I ii4-nTrLi4-rn1-rTr  -  1111]Ji—rLuIIIl-i-rUJJJ±LuIIUL J-t.I [T u-1-T[ITITFFIFrITI 1 LW  J[I-±tf1±1±H±ftH1±th  I I i—i--i-rn I I I I I i-i-HI] I IUHi]  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  I I  :: E:::tft111Itt1Uft I tumiHt A I I II II Li IIIU—ti±fliIJJ-JkHlfl-  -  _:_L::,  ‘  C  ,,  —  —  —  -  — ——, — ———- —— -- —— -- --— — —- -—— — - — - - - -  —  RATIO  -.---IH111HTTD11 I -I ill I I lililiP I  VISCOSITY 01  I  CD  Ct)  Appendix B. Relative Permeabilitics and Viscosities  257  temperature was 195°F, and the test pressure was 1800 psig (1815 psia). The gravity of the liberated gas was determined by the use of tared glass weighing balloon. The gas gravity was found to be 0.70 18 (air  =  1.0). The calculation of viscosity will be as  follows: 1. Molecular weight  0.7018 x 28.95  20.31  2. For which: Pseudo-critical pressure  667 (figure B.1)  Pseudo-critical temperature  =  390 (figure B.1)  If the mole fractions of the hydrocarbon components are known the above values can be calculated using equations B.13 and B.14. 3. From figure B.2:  ()  Viscosity at one atmosphere 4. Pseudo-reduced pressure  =  =  0.01223cp  1, 815/667  Pseudo-reduced temperature  =  =  2.721  (460 + 195)/390  =  1.679  5. From figures B.3 and B.4: 1 IL/IL  =  1.28  6. Therefore, The viscosity at 1800 psig and 195°F  =  1.28 x 0.01223  =  0.01565cp  Appendix C  Subroutines in the Finite Element Codes  2-Dimensional Code CONOIL-Il  C.1  The 2-dimensional code has been divided into two separate programs; the ‘Geometry Program’ and the ‘Main Program’.  The main reason for having as two separate  programs is to reduce the effort on the user. The geometry program automatically generates and numbers the midside and interior nodes. It also renumbers the elements and nodes to minimize the front width and creates a input file for the main program, containing the relevant information about the finite element mesh. The main program does the analysis. The geometry program for the 2-dimensional version consists of 11 subroutines and the main program consists of 58 subroutines. The details of the subroutines are described herein.  Geometry Program  C.1.1  The subroutines in the geometry program and their functions are as follow: ADDS  -  BCONI FFIN  -  forms element-node links. sets up element constants.  -  reads free format input.  MAKENZ  -  generates an array which contains the number of degrees of freedom  associated with each node. MIDPOR MIDSID  -  -  generates mid-side pore pressure nodes.  generates mid-side displacement nodes.  258  Appendix C. Subroutines in the Finite Element Codes  MLAPZ OPTEL  SFWZ  marks last appearances of nodes by making them negative.  -  optimizes and renumbers the elements for frontal solution.  -  RDELN  259  reads line data.  -  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  CHKLST  COMP DATM  -  -  checks if there are any changes in fixity for the load increment.  -  computes the pore fluid compressibility and permeability. reads material property data.  DETJCB  calculates the determinant of the Jacobian matrix.  -  DHYPER DILATE  scrutinizes the input data to main program.  -  -  -  calculates the stress-strain matrix for elastic model.  computes the volume change due to shear deformation (used with hyper  bolic model). DISTLD  -  DSYMAL  calculates equivalent nodal loads. -  finds the principal stresses and their directions (contains 5 subroutines;  TRED3, TRBAK3, TQLRAT, TQL2, DTRED4). ELMCH EQLBM EQLIB ERR FFIN  -  -  -  scrutinizes the list of elements. calculates unbalanced nodal loads.  calculates nodal forces balancing element stresses.  records and lists data errors.  -  -  reads free format input.  Appendix C. Subroutines in the Finite Element Codes  FFLOW FIXX  calculates amount of flow and updates saturations.  -  updates list of nodal fixities.  -  FLOWST FORMB  forms ‘B’(shape function derivative) matrix.  -  frontal solution routine for symmetric matrix.  -  GETEQN  gets the coefficients of the eliminated equations.  -  sets up in-situ stresses and the equivalent nodal forces.  -  INSTRS INV  calculates vectors for coupled consolidation analysis.  -  FRONTZ  INSIT  260  prints the in-situ stresses before first increment.  -  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 MLAPZ  -  MODULI MSUB PLAS  SELF SELl SFR1  -  -  calculates moduli of the soil elements for elastic model.  reads specified range in 1-dimensional array.  -  -  -  calculates principal stresses.  -  REACT SCAN  marks last appearances of nodes by making them negative.  calculates the stress-strain matrix for elasto-plastic model.  -  -  rearranges the boundary conditions in terms of degrees of freedom.  main controlling routine.  -  PRINC RDN  -  -  calculates the reactive forces on restrained boundaries.  checks for any changes in fixities. calculates self weight loads.  computes nodal forces equivalent to self weight loads. calculates shape functions and derivatives for 1-dimensional integration along  element edges.  Appendix C. Subroutines in the Finite Element Codes  SFWZ  estimates the front width for symmetric matrix solution.  -  SHAPE SOFT STIF  calculates shape functions and derivatives.  -  calculates the overstress for strain softening.  *  calculates element stiffness matrix for elasto-plastic model.  -  STOREQ TEMP  calculates the equivalent force vector terms due to temperature changes.  -  UPARAL UPOUT  VISO  allocates storage for subroutine UPOUT.  -  updates and prints the results.  -  calculates viscosity of oil.  -  calculates viscosity of water.  -  WRTN  -  ZERO1  writes a specified range in a 1-dimensional array. initializes 1-dimensional array.  -  ZERO2  initializes 2-dimensional array.  -  ZERO3  initializes 3-dimensional array.  -  ZEROI1  C.2  frontal solution routine for unsymmetric matrix.  -  calculates viscosity of gas.  -  VISW  writes the terms in a buffer zone when an array becomes saturated.  -  UFRONT  VISG  261  -  initializes 1-dimensional integer array.  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  -  CHANGE  expands the nodal fixity data in terms of degree of freedom. -  removes/adds elements from geometry mesh and calculates implied  loading. COMP DMAT  -  -  computes the pore fluid compressibility and permeability. reads material property data.  Appendix C. Sn bron tines in the Finite Element Codes  DRIVER EPM  EQLIB  calculates nodal forces balancing element stresses.  -  FFLOW  calculates amount of flow and updates saturations.  -  updates list of nodal fixities.  -  FLSD  calculates load vector for load shedding.  -  FTEMP  calculates force vector terms due to temperature changes.  -  GETEQN HYPER INSIT  -  calculate moduli values for hyperbolic model.  -  evaluates Jacobian matrix, its determinant and inverse.  -  LAYOUT  reads nodal geometry data and stores in relevant arrays.  -  sets the load vector for fixed boundaries.  -  LOAD  gets the coefficients of the eliminated equations.  sets up in-situ stresses and the equivalent nodal forces.  -  JACO  LFIX  main controlling routine.  -  calculates stress-strain matrix for elasto-plastic model.  -  FIXX  262  -  LSHED  evaluates the load vector for applied loads. -  routine to perform load shedding.  MAKESF finds last appearance of the nodes, frontwidth and the destination vector. -  MFLOW MINV PRIN  -  updates saturations and flow at mid-step.  -  inverts a matrix finds the principal stresses and their directions (contains 5 subroutines;  -  TRED3, TRBAK3, TQLRAT, TQL2, DTRED4). PRNOUT RDN  -  -  reads specified range in 1-dimensional array.  SBMATX SELF SELl  -  calculates B’(shape function derivative) matrix.  calculates self weight loads.  -  -  calculates, updates and prints the results.  calculates self weight loads for gravity changes.  SFRONT SHAPE  -  -  frontal solution routine for symmetric matrix.  calculates shape functions and its derivatives.  Appendix C. Subroutines in the Finite Element Codes  SHAPE2  263  calculates shape functions and derivatives for 2-dimensional integration.  -  SMDF sets up arrays giving nodal degrees of freedom and the first degree of freedom -  of the nodes. STIFF  calculates element stiffness matrix.  -  STOREQ STRL  calculates nodal temperature changes.  -  UFRONT UPDATE VISG VISO VISW  writes the terms in a buffer zone when an array becomes saturated.  calculates and updates stress level.  -  TEMP  -  -  -  frontal solution routine for unsymmetric matrix. updates the results at mid-step for second iteration.  calculates viscosity of gas.  -  calculates viscosity of oil.  -  -  WRTN ZERO 1 ZERO2 ZERO3 ZEROI1 ZEROI2  calculates viscosity of water. -  writes a specified range in a 1-dimensional array. initializes 1-dimensional array.  -  initializes 2-dimensional array.  -  initializes 3-dimensional array.  -  -  -  initializes 1-dimensional integer array. initializes 2-dimensional integer array.  Appendix D  Amounts of Flow of Different Phases The formulation for the multi-phase flow presented in chapter 5 considers an equiv alent conductivity term to model the effects of the different phases in the pore fluid. This does not give the individual amounts of flow of the fluid phase components. However, at any time, these individual amounts of flow can be easily estimated by knowing the total amount of flow, and the relative permeabilities and viscosities of the phase components. The details of this calculation are presented in this appendix. To illustrate the steps involved the example problem given in chapter 7 is considered here. In the oil sand layer the zone from where the fluid flow occurs, can be obtained from the temperature contour plot or the pore pressure contour plot (refer to figures 7.17 and 7.18). Such a zone for the example problem is shown in figure D.1. The fluid flow zone can be divided into a number of zones of different effective mobil ities. Here, the flow zone is divided into three (zones A, B and C in figure D.1) and the effective mobilities of the fluid phase components are assumed constant within a zone. The grater the number of zones the better the results will be. The mobility of a fluid phase component ‘1’ can be written as kmi  kkri  where kmi k  -  -  mobility of phase 1 intrinsic permeability of the sand matrix (m ) 2 264  (D.1)  Appendix D. Amounts of Flow of Different Phases  265  50  0 Injection Well • E  Production Well  40  40  0  60  Distance (m)  Figure D .1: Zones involved in Fluid Flow  1 k,. IL1  -  -  relative permeability of phase I viscosity of phase 1  k is a function of void ratio, k,. 1 is a function of saturation level and 1 u 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 m . 2 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  ) 2 Area (m  ii(mPa.s)  (u 0 mPa.s)  Temp. (°C)  A  96  0.20  8  220  B  252  0.48  40  140  C  312  0.65  1000  50  0 = 2.769 (0.8 k,.  —  996 S)’  (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  Where,  aA, aB  and  ac  aA + (kmw)B aB + (kmw) ac aA + a + ac  (D.4)  are the areas of zones A, B and C respectively. Similarly, the  effective mobility of oil considering all three zones can be given as, —  mo  (kmo)A aA + (kmo)B  aA +  aB  + (kmo) ac + ac aB  Then, the amounts of water and oil flow in the total flow can be estimated as, A TI  L.Vw  mw  TI  VT  = LVT  kemw e  mw  j  i.e  e  (D.7)  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?  iI_1Vw)A  =  mw A aA  AT?  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 (S ) 0  (D.1O)  The amount of oil flow from zone A will be, IAT?\  L.1Vo)A =  mo A aA  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  —  (V ) 0 A  (D.12)  and the new oil saturation will be, ) 0 (S  ) 0 (V fl aA  (D.13)  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  kmw(10 m 8 /s)  kmo(10 m 8 /s)  A  0.3  0.7  37.6  85.1  B  0.3  0.7  19.8  17.0  C  0.3  0.7  11.6  0.68  The stepwise calculations for the amounts of flow and saturations of water and oil are tabulated in table D.3. The saturations and the mobilities of water and oil at the end of time t D.2.  =  300 days, are given in table D.4, which can be compared with table  Appendix D. Amounts of Flow of Different Phases  269  Table D.3: Calculation of Flow and Saturations with Time Time (days)  (S)A  0 2 4 6 8 10 15 20 25 30 40 50 60 70 80 90 100 125 150 175 200 250 300  0.300 0.394 0.435 0.461 0.479 0.494 0.525 0.546 0.562 0.574 0.595 0.610 0.622 0.632 0.640 0.647 0.653 0.667 0.678 0.686 0.693 0.704 0.713  (S)B  0.300 0.319 0.330 0.339 0.346 0.352 0.365 0.375 0.384 0.392 0.405 0.417 0.426 0.435 0.443 0.450 0.456 0.471 0.484 0.495 0.505 0.522 0.536  (S)c  0.300 0.301 0.301 0.302 0.302 0.302 0.303 0.303 0.304 0.304 0.305 0.306 0.307 0.307 0.308 0.308 0.309 0.310 0.311 0.312 0.313 0.315 0.317  (S ) 0 A  0.700 0.606 0.565 0.539 0.521 0.506 0.475 0.454 0.438 0.426 0.405 0.390 0.378 0.368 0.360 0.353 0.347 0.333 0.322 0.314 0.307 0.296 0.287  (S ) 0 B  0.700 0.681 0.670 0.661 0.654 0.648 0.635 0.625 0.616 0.608 0.595 0.583 0.574 0.565 0.557 0.550 0.544 0.529 0.516 0.505 0.495 0.478 0.464  (m / 3 day)  0 iW (m / 3 day)  2.54 3.89 4.29 4.49 4.61 4.69 4.82 4.89 4.94 4.97 5.01 5.04 5.06 5.07 5.08 5.09 5.10 5.11 5.12 5.13 5.13 5.14 5.15  2.64 1.29 0.89 0.69 0.57 0.49 0.36 0.29 0.24 0.21 0.17 0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.04 0.03  (S ) 0 c  0.700 0.699 0.699 0.698 0.698 0.698 0.697 0.697 0.696 0.696 0.695 0.694 0.693 0.693 0.692 0.692 0.691 0.690 0.689 0.688 0.687 0.685 0.683  Table D.4: Saturations and Mobilities of Water and Oil after 300 Days Zone  S  S  kmw(10 m 8 /s)  kmo(10 m 8 /s)  A  0.71  0.29  1826  2.61  B  0.54  0.46  352  4.75  C  0.32  0.68  16.7  0.64  Appendix E  User Manual for CONOIL-Il  E.1  Introduction  CONOIL-Il is a finite element program for consolidation analysis in oil sands under plane strain and axisymmetric conditions. The program includes an elasto-plastic stress strain model and a formulation to analyze multi-phase fluid flow. It also con siders temperature effects on stresses and fluid flow in the analysis. The program can be used to carry out transient, drained or undrained analysis us ing the same material data base. Elements can be removed to simulate excavation. Provisions exist for specifying various boundary conditions such as pressure, force, displacement and pore pressure. Though the program is particularly suited for prob lems in oil sands, it can be applied for a range of geotechnical problems such as dam and heavy foundation analyses. The intention of this manual is to provide sufficient information for an analyst with a strong geotechnical background to be able to prepare an input file and run the program. Detailed explanations such as analytical formulation, method of analysis, formation of stiffness matrix, solving routines etc. can be found in Srithar (1993). CONOIL-Il has been divided into two separate programs: the ‘Geometry Program’ and the ‘Main Program’. The main purpose of this split is to reduce the effort on the user. The geometry program automatically generates and numbers the mid-side and interior nodes. It also renumbers the elements and nodes to minimize the front width and creates an input file for the main program, containing the relevant information about the finite element mesh. Therefore, the Geometry Program has to be run first 270  Appendix E. User Manual for CONOIL-Il  271  and the link file has to be submitted to the Main Program. The data for both the Geometry Program and the Main Program is free format i.e, particular data items must appear in the correct order on a data record but they are not restricted to appear only between certain column positions. The data items are indicated below by mnemonic names, i.e., names which suggest the data item required by the program. The FORTRAN naming convention is used: names beginning with the letters I, J, K, L, M and N show that the program is expecting an INTEGER data item whereas names beginning with any other letter show that the program is expecting a REAL data item. The only exception is the material property data where the actual parameter notations are retained to avoid confusions. All the material property data are real. INTEGER data items must not contain a decimal point but REAL data items may optionally do so. REAL data items may be entered in the FORTRAN exponent format if desired. Individual data items must not contain spaces and are separated from each other by at least one space. Detailed explanations for some of the records are given in section E.4. Comments may be included in the input data file in exactly the same way as for the FORTRAN program. Any line that has the character C in column 1 is ignored by the programs. This facility enables the user to store information relating to values, units assumed etc. permanently with the input data rater than separately. The program only read data from the first 80 columns of each line.  Appendix E. User Manual for CONOIL-Il  Geometry Program  E.2  Record 1 (one line) TITLE  TITLE  -  Title of the problem (up to 80 characters)  Record 2 (one line)  I  LINK  LINK  -  A code number set by the user  Record 3 (one line) NN NEL ILINK IDEF IS TART SCX SCY  NN NEL ILINK  -  -  -  Number of vertex nodes in the mesh Number of elements in the mesh Link option: 0 1  IDEF  -  -  -  no link file is created a link file is created  Element default type: 1 5  -  -  linear strain triangle with displacement unknowns linear strain triangle with displacement and excess pore  pressure unknowns (linear variation in pore pressure) 7 8  -  -  cubic strain triangle with displacement unknowns cubic strain triangle with displacement and excess pore  272  Appendix E. User Manual for CONOIL-Il  pressure unknowns (cubic variation in pore pressure)  ISTRAT  -  Frontal numbering strategy option: 1 2  -  -  the normal option only to be used in rare circumstances when the parent’  mesh contains overlapping elements SCX SCY  -  -  Scale factor to be multiplied to all x coordinates Scale factor to be multiplied to all y coordinates  Record 4 (NN lines) N X Y TEMP LCODE VISCO]  N X Y TEMP LCODE  -  -  -  -  -  Node number x coordinate of the node y coordinate of the node Initial temperature °C Index for load transfer  o 1 VISCO  -  -  -  node can participate in load transfer node cannot participate in load transfer  Initial viscosity factor (not used in the present formulation, set equal to 1)  Record 5 (NEL lines)  ILN1N2N3MATI  L  Ni, N2, N3  -  -  Element number Vertex node numbers listed in anticlockwise order  273  Appendix E. User Manual for CONOIL-Il  MAT  -  Material zone, number in range 1 to 10  274  Appendix E. User Manual for CONOIL-Il  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)  I  NPLAX NMAT INCJ INC2 IPPJM IUPD ICOR ISELFI  NPLAX  -  Plane strain/Axisymmetric analysis option: 0 1  NMAT INC1 INC2 IPRIM IUPD  -  -  -  -  -  -  -  plane strain axisymmetric  Number of material zones Increment number at start of analysis Increment number at finish of analysis Number of elements to be removed to from primary mesh Element default type: 1 5  -  -  linear strain triangle with displacement unknowns linear strain triangle with displacement and excess pore  pressure unknowns (linear variation in pore pressure)  275  Appendix E. User Manual for CONOIL-Il  7 8  -  -  276  cubic strain triangle with displacement unknowns cubic strain triangle with displacement and excess pore  pressure unknowns (cubic variation in pore pressure) ISTRAT  -  Frontal numbering strategy option: 1 2  -  -  the normal option only to be used in rare circumstances when the ‘parent’  mesh contains overlapping elements SCX SOY  -  -  Scale factor to be multiplied to all x coordinates 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  MAT  -  k k  Material property number. All elements given the same number in the Geometry Program have the following properties  IMODEL e KE n Rf KB m DUO  -  -  -  -  -  -  -  -  Stress-strain model number. Use  C7  for Hyperbolic model  Initial void ratio Elastic modulus constant Elastic modulus exponent Failure ratio Bulk modulus constant Bulk modulus exponent Determines whether Drained/Undrained/Consolidation analysis i) DUO  =  ii) DUO  0.0 Drained analysis  =  1 (liquid bulk modulus) B  -  Undrained analysis  NOTE: B 1 in the range of 100 to 500 B k (soil bulk modulus) is 5 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)  total unit weight of soil permeability in x direction permeability in y direction  Record 5.2 c  -  v ot  q’cv  -  0 B B  -  Consolidation analysis  Appendix E. User Manual for CONOIL-Il  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. t 8 a  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 0 B  Bulk modulus of the water  -  Bulk modulus of the oil  -  Record 5.3 /J’30,0  ‘H  1’3o,o  H  -  -\U  U S S 1 cw a 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*IXT  H  -  -  U S  -  -  Henry’s coefficient of solubility Function to modify bubble pressure for temperature Bubble pressure (Oil/Gas saturation pressure) Initial degree of saturation varying between 0 and 1. (S =  1 implies 100% saturation)  278  Appendix E. User Manual for CONOIL-Il  Sf  -  -  0 cx  -  -  279  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 Coefficient of linear thermal expansion of oil Coefficient of linear thermal expansion of solids  Record 5.4 ISIGE 151GB IMPF IDILAT ILSHD  ISIGE  -  Option to calculate Young’s modulus  o 1 ISIGB  -  1 -  1 2 -  1 2 -  use minor principal stress  -  -  use mean normal stress use minor principal stress  -  -  -  fully saturated partially saturated three phase fluid flow (needs additional parameters)  Dilation option  o  ILSHD  -  use mean normal stress  Multi phase flow option  o  IDILAT  -  Option to calculate bulk modulus  o IMPF  I  -  -  -  No dilation Use constant dilation angle Use Rowe’s stress-dilatancy theory  Load transfer option  o 1 2  -  -  -  do not perform load transfer perform load transfer by keeping o constant 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 1  MAT  -  m DUO  k  %  Material property number. All elements given the same number in the Geometry Program have the following properties  IMODEL  e KE n (Rf) KB m DUO  -  -  -  -  -  -  -  -  Stress-strain model number =  5 Cone type yielding only (single hardening)  =  6 Cone and Cap type yielding (double hardening)  Initial void ratio Elastic modulus constant Elastic modulus exponent Failure ratio in the hardening rule (cone yield) Bulk modulus constant Bulk modulus exponent Determines whether Drained/Undrained/Consolidation analysis i) DUO ii) DUO  =  =  0.0 Drained analysis 1 (liquid bulk modulus) B  -  Undrained analysis  NOTE: B 1 in the range of 100 to 500 B ,, (soil bulk modulus) is 8 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. -  -  k  -  total unit weight of soil permeability in x direction permeability in y direction  Record 5.2 (r/o) , 1 i (r/o-)  —  -  (T/o-)f,i  (r/o)  -  -  q  —  —  0 B B  0 (No parameter at present) Failure stress ratio at 1 atmosphere Reduction in failure stress ratio for a ten fold increase in confining pressure  -  q t 8 a  -  -  Strain softening number Strain softening exponent Coefficient of temperature induced structural reorienta tion. Only used in temperature analysis.  — -  — -  B 0 B  -  -  0 (No parameter at present) 0 (No parameter at present) Bulk modulus of the water Bulk modulus of the oil  Record 5.3  f-3O,O  H  H u U S S a a 0  Appendix K User Manual for CONOIL-Il  1130,0  -  282  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  -  -  U S  -  -  Henry’s coefficient of solubility Function to modify bubble pressure for temperature Bubble pressure (Oil/Gas saturation pressure) 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)  -  -  5 a  -  Coefficient of linear thermal expansion of water Coefficient of linear thermal expansion of oil Coefficient of linear thermal expansion of solids  Record 5.4  ISIGE ISIGB IMPF ILSHD F F KGp  ISIGE  -  Option to calculate Young’s modulus 0 1  ISIGB  -  1 -  -  -  use mean normal stress use minor principal stress  Option to calculate bulk modulus 0  IMPF  GP  -  -  use mean normal stress use minor principal stress  Multi phase flow option 0 1  -  -  fully saturated partially saturated  11  Appendix E. User Manual for GONOIL-Il  2 ILSHD  -  1 2  F KGp GP  -  -  -  -  -  three phase fluid flow (needs additional parameters)  Load transfer option 0  -  -  -  -  -  do not perform load transfer perform load transfer by keeping o constant perform load transfer by keeping o constant  Collapse modulus number (cap yield) Collapse modulus exponent (cap yield) Plastic shear parameter (cone yield, hardening rule) Plastic shear exponent (cone yield, hardening rule) Flow rule intercept (cone yield) Flow rule slope (cone yield)  Record 5.5 (necessary only if IMPF  Sw So Sg S S  S S, 9 S  -  -  -  283  =  2, all are real variables except IV)  g IVL, 0 k  IVO  9 IV  Initial water saturation Initial oil saturation Initial gas saturation 0 + S must be equal to 1) (S + S  om 5  S  -  -  -  Residual oil saturation Connate water saturation (irreducible water saturation) Relative permeability of oil at connate water saturation (oil-water)  -  IV,  -  Relative permeability of oil at zero gas saturation (oil-gas) Options to estimate viscosity of water 0  -  use a given constant value (in Pa.s)  Appendix E. User Manual for CONOIL-Il  1  -  284  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 1  -  -  use a given constant value (in Pa.s) 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 data pairs, maximum 10) (1V  IVg  -  Options to estimate viscosity of gas 0 1  -  -  use a given constant value (in Pa.s) 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  Bl...B3  -  A1(S  =  -  -  B1(B2  —  S)B3  Parameters for relative permeability of gas (oil-gas) 9 k,.  Dl...D3  3 A2)A  Parameters for relative permeability of oil (oil-water) =  Cl... 03  —  =  9 C1(S  —  C2)c3  Parameters for relative permeability of oil (oil-gas)  Appendix E. User Manual for CONOIL-Il  09 = D1(D2 k,.  Record 5.7 (necessary only if IMPF = 2)  I  Fl F2 F31 Fi...F3  -  Parameters for oil-gas capillary pressure  of gas (oil-gas) Pc = Fl Pa(S 9  —  3 F2)’  Record 5.8 (necessary only if IMPF = 2 and IV,,, = 0 or >1) V,,,  (ifIV=0)  Vi Ti V2 T2  I  •.•  V  1, IV, data pairs, maximum 10)  Viscosity values in the given profile (in Pa.s)  -  Ti,...  ,  Constant viscosity value of water (in Pa.s)  -  Vi,...  (if IV,  Temperature values in the given profile (in °C)  -  Record 5.9 (necessary only if IMPF = 2 and 1V 0 = 0 or >1) (ifIV = 0 0)  1 0 V Vi Ti V2 T2  0 V Vi,... Ti,...  I  •..  -  -  -  > 1, 1V 0 (if 1V 0 data pairs, maximum 10)  Constant viscosity value of oil (in Pa.s) Viscosity values in the given profile (in Pa.s) Temperature values in the given profile (in °C)  Record 5.10 (necessary only if IMPF = 2 and 1V 9 = 0 or >1)  285  Appendix E. User Manual for CONOIL-Il  (ifIV=0)  I  Vi Ti V2 T2  I  ...  -  Vi,... Ti,...  -  -  (if IVg> 1, 1V 9 data pairs, maximum 10)  Constant viscosity value of gas (in Pa.s) Viscosity values in the given profile (in Pa.s) Temperature values in the given profile (in °C)  Record 6 ((IPRIM-1)/10 + 1 lines, only if IPRIM> 0)  I  Li L2  Li,...  -  List of element numbers to be removed to form mesh at the beginning of the analysis (LPPJM element numbers)  There must be 10 data per line, except the last line Record 7 (one line only) INSIT NNI NELl NO UT I  INSIT  -  In-situ stress option: 0 1  NNI NELl NOUT  -  -  -  -  -  Set in-situ stresses to zero Direct specification of in-situ stresses  Number of nodes in-situ mesh Number of elements in-situ mesh In-situ stress printing option: 0 1 2  -  -  -  Do not print the in-situ stresses Print the variables at the centroids of each element Print the variables at each integration point per element and print the equilibrium loads for in-situ stresses.  286  Appendix E. User Manual for CONOIL-Il  287  Record 8 (NNI lines) NI XI Yl o- o, o- r u  NI XI Y1 o, o, o  -  -  -  -  -  ii  -  In-situ mesh node number x coordinate y coordinate Normal components of the effective stress vector Shear stress component Pore fluid pressure (Note that effective stress parameters are assumed)  Record 9 (NELl lines) LI NIl N12 NI3]  LI NIl, N12, N13  -  -  In-situ mesh element number 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 ICHEL NLOD  -  -  -  Increment number Number of elements to be removed 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  IFIX lOUT  -  -  288  Number of changes to nodal fixities Output option for this increment  -  a four digit number  abcd where: a out of balance loads and reactions -  o 1 2 b  -  -  -  -  no out of balance loads out of balance loads at vertex nodes out of balance loads at all nodes  option for prescribed boundary conditions (e.g. fixity condition or equivalent nodal loads at specified nodes)  o I c  -  o 1 2 d  -  o 1 2 DTIME DGRAV  -  -  no information printed -  data printed for each relevant d.o.f  option for general stresses -  -  -  no stresses printed stresses at element centroids stresses at integration points  option for nodal displacements -  -  -  no displacements printed displacements at vertex nodes displacements at all nodes  Time increment for consolidation analysis Increment in gravity level (change in number of gravities)  NSINC NTEMP DGRAV  -  -  -  The number of sub increments (this is presently equal to 1) Number of changes to nodal temperature Number of data pairs in the temperature-time history profile  Record 11 ((ICHEL-1)/1O + 1 lines, only if ICHEL > 0)  Appendix E. User Manual for CONOIL-Il  rLi  Li,...  List of element numbers to be removed in this increment  -  There must be 10 data per line, except the last line Record 12 (NLOD lines) (a) For .1\TLOD > 0 NDFX DFY1  N DFX DFY  Node number  -  -  -  Increment of x force Increment of y force  For NLOD < 0 (b.1) For linear strain triangle  LNJN2TJS1 T3S3T2S200001  (b.2) For cubic strain triangle  I  L Ni N2 Ti Si T3 S3 T4 S4 T5 55 T2 S  L  Ni, N2 Ti Si Ti  -  -  -  -  -  Element number Node numbers at the end of the loaded element side Increment of shear stress at Ni (see the following figure E.1 Increment of normal stress at Ni Increment of shear stress at Ni  289  Appendix E. User Manual for CONOIL-Il  Si  -  290  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  N  -  DPI  Node number  Appendix E. User Manual for CONOIL-Il  ICODE  -  291  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 1 b  -  o 1  -  -  node is free in x direction node is to have a prescribed incremental displacement  DX fix for y direction -  -  node is free in y direction node is to have a prescribed incremental displacement  DY c fix for excess pore pressure -  o 1  -  -  no prescribed excess pore pressure 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 DY DP  -  -  -  Prescribed displacement in x direction Prescribed displacement in y direction Prescribed pore pressure  Record 14 (NTEMP lines, only if NTEMP > 0) N TEM1 TIMEJ TEM2 TIME2  N TEMJ,... TIMEJ,...  -  -  -  .J  (NPTS data pairs, maximum 15)  Node number Temperature in the given temperature time profile Time in the temperature time profile  Appendix E. User Manual for CONOIL-Il  E.4  292  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  0  u,v  A  p  —  —  293  displacement unknowrs  pore pressure unknowns  a. 1.  6  2 2  S (a)  2  Element type 1 (LST) 6 nodes, 12 d.o.f.  (b)  Element type 5 (LST) 6 nodes, 15 d.o.f. (consolidation)  4 12 —._.  16  12 21  11  6  /  -  2  /  6  10  /  11  ,‘  1// 2  10  S  /  -  .18  (c) Element type 7 (CuST) 15 nodes, 30 d.o.f.  Figure  (d)  E.2: Element  8  19  9  Element type 8 (CuST) 22 nodes, 40 d.o.f. (consolidation)  types  1  Appendix E. User Manual for CONOIL-Il  294  Table E.1: Element Types LEDF 1 5 7 5  Element Name Linear strain triangle (LST) LST with linearly varying pore pressures Cubic strain triangle (CST) CST with cubic variation of pore_pressures  Displacement Quadratic Quadratic Quartic Quartic  Variation of Strain Pore Pressure Linear N/A Linear Linear Cubic Cubic  N/A Cubic  analysis use element type 5. (ii) Axisymmetric Analysis For drained analysis or consolidation analysis where collapse is not expected then element types 1 and 5 will probably be adequate (i.e. the same as (i) above). For undrained analysis or in a situation where collapse is expected then element types 7 and 8 are recommended. Recent research has shown that in axisymmetric analysis the constraint of no volume change (which occurs in undrained situations) leads to finite element meshes ‘locking up’ if low order finite elements (such as the LST) are used. NN (Number of Vertex Nodes) It should be noted that NN refers to the number of vertex nodes in the finite element mesh. The geometry program automatically generates node numbers and coordinates for any nodes lying on element sides or within elements. Records 4 and 5 ulElement and Nodal Numbering The program user must assign each element and each vertex node in the finite element mesh unique integer numbers in the following ranges: 1 < node number  750  1 < element number < 500  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 E. User Manual for CONOIL-Il  296  KZZ. Figure E.3: Plane Strain Condition  xis the’adia1 direcSon is the circwnferentiai direction z  Figure E.4: Axisymmetric Condition  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 kN/m 2 is applied to part of the boundary of the finite element mesh it might be divided into ten equal increments of 2 kN/m 2 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. T \ a lues 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  Appendix E. User Manual for CONOIL-Il  300  ten days to one hundred days). Not less than three time steps should be used per log cycle off time (for a log base of ten). Thus a suitable scheme may be as shown in table E.2 Table E.2: Time Increment Scheme Increment No. 1 2 3 4 5 6 7 8 9 10  DTIME 1 1 3 5 10 30 50 100 300 500  Total Time 1 2 5 10 20 50 100 200 500 1000  This scheme would be modified slightly near the start and end of an analysis (see below). 4. If a very small time increment is used near the start of the analysis then the finite element equations will be ill conditioned. 5. When a change in pore pressure boundary condition is applied, the associated time step should be large enough to allow the effect of consolidation to be experienced by those nodes in the mesh with excess pore pressure variables that are close to the boundary. If this is not done then the solution will predict excess pore pressures that show oscillations (both in time and space). The application of item 5 will often mean that the true undrained response will not be captured in the solution The following procedure, however, usually leads to satisfactory results.  Appendix E. User Manual for GONOIL-Il  301  1. Apply loads in the first increment (or first few increments for a nonlinear anal ysis) but do not introduce any pore pressure boundary conditions. 2. Introduce the excess pore pressure boundary conditions in the increment fol lowing the application of the loads. NLOD and IFIX It is important to note that NLOD and sl IFIX refer to the number of changes in loading and nodal fixities in a particular increment.  CONOIL maintains a list of  loads and nodal fixities which the user may update by providing the program with appropriate data. Thus, if NLOD  0 and IFIX  =  0, the program assumes that the  same incremental loads and fixities will be applied in the current increment as were applied in the previous increment. Another point to note is that loads applied are incremental, thus the total loads acting at any particular time are given by adding together all the previous incremental loads. The following example is intended to clarify these points for a consolidation analysis: 1. Part of the boundary of a soil mass is loaded with a load of ten units (this is applied in ten equal increments). 2. Consolidation takes place for some period of time (over ten increments) 3. The load is removed from boundary of the soil mass in five equal increments. 4. Consolidation takes place with no total load acting. This loading history requires the data shown in table E.3. Note that in increments 11 and 26 it is necessary to apply a zero load to cancel the incremental loads which CONOIL would otherwise assume. DGRAV  Appendix E. User Manual for CONOIL-Il  302  Table E.3: Load Increments Increment No. 1 2 3 4 5 6 7 8 9 10 11 12 13  21 22 23 24 25 26 27 28 etc.  Input to program 1  0  -2  0  Loads Incremental load applied 1 1 1 1 1 1 1 1 1 1 0 0 0  Total load acting 1 2 3 4 5 6 7 8 9 10 10 10 10  -2 -2 -2 -2 -2 0 0 0  8 6 4 2 0 0 0 0  Appendix E. User Manual for CONOIL-Il  303  DGRAV is used in problems in which the material’s self weight is increased during an analysis (e.g. in the ‘wind-up’ stage of a centrifuge test increasing centrifugal acceleration can be regarded as having this effect).  Appendix F  User Manual for CONOIL-Ill  F.1  Introduction  CONOIL-Ill is a three dimensional finite element program developed to analyze the stresses, deformations and flow in oil sands. Though CONOIL-Ill is specifically writ ten for oil sands, it can be used for general geotechnical problems. CONOIL-Ill can perform drained, undrained and consolidation analyses and has the following special features. 1. Elasto-Plastic stress strain model. Modified form of Matsuoka’s model is im plemented. 2. Three phase fluid flow. This is a special feature required to analyze the problems in oil sands where the pore fluid contains three phases; water, bitumen and gas. 3. Temperature effects on stresses and strains. This manual provides neither detail information about the program nor the theories behind its development. Only the input parameters needed, their format and some brief descriptions are given here. For detail explanations such as, method of analysis, derivation of differential equations, formation of stiffness matrix, solving routines etc., please refer Srithar (1993). A sample data file and the corresponding output file are given at the end of this manual. The source code is written in FORTRAN-77. Input parameter names are given ac cording to the standard FORTRAN naming convention. Names begin with the letters 304  Appendix F. User Manual for CONOIL-III  305  1 J, L, M and N implies that the program expects integer data. Integer data should not contain a decimal point. There are exceptions to this naming convention in record 6 where the material property data are read. Actual material parameter notations are retained to avoid confusions.  F.2  Input Data  Record 1 (one line) TITLEI  TITLE  -  Title of the problem (up to 80 characters)  Record 2 (one line) NCNOD, NINOD, NTEL, ITYPE, NINT, IPRN  NCNOD NINOD NTEL ITYPE  NINT  -  -  -  -  -  Total number of corner nodes Total number of internal nodes (0 for ITYPE 1 and 3) Total number of elements Element type (see fig. F.l) =  1 for drained/undrained analysis  =  3 for consolidation analysis  Number of integration points 8 or 27 (generally 8 is good enough)  =  IPRN  -  Index to print nodal and element information 0 1  -  -  Do not print the information Print the information  Appendix F. User Manual for CONOIL-Ill  306  TYPE 3  TYPE 1  o  Corner nodes  =  8  D.o.f. per node  =  Internal nodes  =  •  Corner nodes  =  8  3  D.o.f. per node  =  4  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  X(NN) Y(NN) Z(NN) T(NN)  -  -  -  -  -  Node number X coordinate of the node NN Y coordinate of the node NN Z coordinate of the node 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 N1...N8  -  -  Element number 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  PATM GAMW ID UC  -  -  -  308  Atmospheric pressure Unit weight of water Index for Drained/Undrained/Consolidation analysis 0 1 2  -  -  -  Drained analysis Undrained analysis Consolidation analysis  If there are temperature changes, use consolidation routine with no flow boundary conditions to perform undrained analysis.  INCJ 1N02 NMAT NTEMP NPTS IPRIM ISELF  -  -  -  -  -  -  -  First increment number of the analysis Last increment number of the analysis Number of material types (maximum 10) Number of nodes where temperature changes Number of data pairs in the temperature-time profile (max. 15) Number of elements to be removed to form the primary mesh Option to specify self weight load as in-situ stresses 0 1  -  -  in-situ stresses do not include self weight 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  =  Record 6.1  MAT, MODEL, ISICE, 151GB, ILSHD, IMPF  2.)  Appendix F. User Manual for CONOIL -III  MAT MODEL  -  -  Material number Stress-Strain model type 1 2 3  ISIGE  -  1 -  1 -  1 -  modified Matsuoka’s model with Cap-type yield  -  -  use mean normal stress use minor principal stress  -  -  use mean normal stress use minor principal stress  Load transfer option 0  IMPF  -  modified Matsuoka’s model  Option to calculate bulk modulus 0  ILSHD  -  hyperbolic model  Option to calculate Young’s modulus 0  181GB  -  -  -  do not perform load transfer perform load transfer  Multi phase flow option 0 1 2  -  -  -  fully saturated partially saturated three phase fluid flow (needs additional parameters)  Record 6.2 (all are real variables) 2 e,KE,n,Rf,KB ,m,7,k,k,k  e KE n  -  -  -  Initial void ratio Elastic modulus constant Elastic modulus exponent  309  Appendix F. User Manual for CONOIL-III  -  KB m  -  -  -  -  -  -  310  Failure ratio Bulk modulus constant 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 (m ) 2  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  -  q S  -  -  -  strain softening constant strain softening exponent 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. S 1 generally close to zero)  8 B B 0 B  -  -  -  Bulk modulus of the solids Bulk modulus of the water Bulk modulus of the oil  Appendix F. User Manual for CONOIL-III  311  Record 6.4 (all are real variables) ,U30,0, )H,  fL o 3 ,o  H, Au, U,  -  —,  , a, a 8 j 0 ant, c.z  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 Au U  -  -  -  — -  t 8 a  -  -  -  0 a  -  Henry’s coefficient of solubility Function to modify bubble pressure for temperature Bubble pressure (Oil/Gas saturation pressure) 0 (No parameter at present) 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 Coefficient of linear thermal expansion of oil  Record 6.5 (necessary only if MODEL C, p, K,  rip,  R p 1 ,  i,  =  2 or 3, all are real variables)  A, (r/), (r/o), —  C p K lip  1 R  -  -  -  -  -  -  Cap-yield collapse modulus number Cap-yield collapse modulus exponent Plastic shear number Plastic shear exponent Plastic shear failure ratio flow rule intercept  Appendix F. User Manual for CONOIL -III  A r/o-  -  -  -  312  flow rule slope 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  S,, 0 S 9 S  -  -  -  IVj  Initial water saturation Initial oil saturation Initial gas saturation 0 H- S 9 must be equal to 1) (S + S  S S  -  -  -  Residual oil saturation Connate water saturation (irreducible water saturation) Relative permeability of oil at connate water saturation (oil-water)  09 k,? IV,,  -  -  Relative permeability of oil at zero gas saturation (oil-gas) Options to estimate viscosity of water 0 1  -  -  use a given constant value (in Fa.s) use the built-in feature in the program (International critical tables)  >1  -  interpolate using given temperature-viscosity (IV data pairs, maximum 10)  0 1V  -  Options to estimate viscosity of oil 0  -  use a given constant value (in Pa.s)  profile  Appendix F. User Manual for CONOIL -III  1  -  313  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 data pairs, maximum 10) (1V 9 1V  -  Options to estimate viscosity of gas 0 1  -  -  use a given constant value (in Pa..s) use the built-in feature in the program (a constant value 2.E-5 Pa.s)  >1  -  interpolate using given temperature-viscosity profile 9 data pairs, maximum 10) (1V  Record 6.7 (necessary only if IMPF  =  2)  Al, A2, A3, Bi, B2, B3, Cl, 02, 03, Dl, D2, D3  Al.. .A3  -  Parameters for relative permeability of water (oil-water) A1(SL,  =  Bl...B3  -  -  -  B1(B2  —  Parameters for relative permeability of gas (oil-gas) ICrg  Dl...D3  3 A2)-  Parameters for relative permeability of oil (oil-water) =  Cl... C3  —  =  C1(Sg  —  Parameters for relative permeability of oil (oil-gas) 09 k,.  =  D1(D2  3 ) 9 S ”  Record 6.8 (necessary only if IMPF  =  2)  Fl, F2, F3 Fl...F3  -  Parameters for oil-gas capillary pressure  Appendix F. User Manual for CONOIL-III  of gas (oil-gas) Pc = Fl Pa(Sg  —  F2)F3  Record 6.9 (necessary only if IMPF = 2 and IV,, = 0 or >1) V  (ifIV=0)  Vi, Ti, V2, T2,... (if IV  Vi,, Vi,... Ti,...  -  -  -  1, IV, data pairs, maximum 10)  Constant viscosity value of water (in Pa.s) Viscosity values in the given profile (in Pa.s) Temperature values in the given profile (in °C)  Record 6.10 (necessary only if IMPF = 2 and 1V 0 = 0 or >1) (ifIV = 0 0) Vi, Ti, V2, T2,... (if 1V 0 > 1, IV,, data pairs, maximum 10)  0 V Vi,... Ti,...  -  -  -  Constant viscosity value of oil (in Pa.s) Viscosity values in the given profile (in Fa.s) Temperature values in the given profile (in °C)  Record 6.11 (necessary only if IMPF = 2 and IVg = 0 or >1) (ifIV=0) Vi, Ti, V2, T2,... (if 1V 9 data pairs, maximum 10) > 1, 1V 9  -  Vi,... Ti,...  -  -  Constant viscosity value of gas (in Pa.s) Viscosity values in the given profile (in Pa.s) Temperature values in the given profile (in °C)  314  Appendix F. User Manual for CONOIL-III  Record 7 (NTEMP lines, only if NTEMP  315  >  0)  TEM1, TIME1, TEM2, TIMEj (NPTS data pairs, maximum 15)  N TEM1,... TIMEJ,...  -  -  -  Node number Temperature in the given temperature time profile Time in the temperature time profile  Record 8 (one line) LINSIT, PINSIT  LINSIT  -  Option to specify in-situ stresses  o 1 PINSIT  -  -  -  set the in-situ stresses to zero read the in-situ stresses from data  Option to print in-situ stress data  o 1  -  -  do not print print in-situ stress data  Record 9 (NTEL lines, only if LINSIT  =  1)  M, SIGX, SIGY, SIGZ, SIGXY, SIGYZ, SIGZX, PP1  M SIGX SIGY SIGZ SIGXY  -  -  -  -  -  Element number Stress in x direction Stress in y direction Stress in z direction Stress in xy direction  Appendix F. User Manual for CONOIL -III  SIGYZ SIGZX PP  -  -  -  316  Stress in yz direction Stress in zx direction 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 ICHEL NLOAD NFIX lOUT  -  -  -  -  -  Increment number Number of elements to be removed from primary mesh Number of nodes where loads are applied Number of nodes where nodal fixities are changed Option for printing results (5 digit code  ‘  a  =  1 print nodal displacements  b  =  1 print moduli values and saturations  abcde’)  Appendix F. User Manual for CONOIL-IlI  c  =  1 print strains and coordinates of the integration point where results are printed  DTIME DGRAV  -  -  d  =  1 print stresses and pore pressure  e  =  1 print velocity vectors  Time increment Increase in gravity  Record 12 ((ICH.EL-1)/10 + 1 lines, only if ICHEL > 0)  Li, L2  Li,...  -  List of element numbers to be removed in this increment  There must be 10 data per line, except the last line Record 13 (NLOAD lines, only if NLOAD> 0)  N, DFX, DFY,  N DFX DFY DFZ  -  -  -  -  DFZI  Node number Increment in x force Increment in y force Increment in z force  Record 14 (NFIX lines, only if NFIX> 0)  N, NFCODE, DX, DY, DZ, DP  N  -  Node number  317  Appendix F. User Manual for CONOIL-III  NFCODE  -  Four digit code ‘abcd’ which specifies the fixity condi tions associated with the node a = 0 free in x direction =  b  c  d  =  1 will have prescribed incremental displacement DX 0 free in y direction  =  1 will have prescribed incremental displacement DY  =  0 free in z direction  =  1 will have prescribed incremental displacement DZ  =  0 pore pressure can have any value (undrained boundary)  =  1 will have prescribed incremental pore pressure DP  =  2 will have zero absolute pore pressure for this and all subsequent increments  DX DY DZ DP  -  -  -  -  Prescribed displacement in x direction Prescribed displacement in y direction Prescribed displacement in z direction Prescribed pore pressure  318  Appendix F. User Manual for CONOIL-IlI  F.3  319  Example Problem 1  An example of a general stress analysis under one dimensional loading is illustrated here. The material is assumed to be linear elastic. The finite element mesh consists of two brick elements as shown in figure F.2. The data file and the corresponding output file from the program are given in subsections.  25 kN  H  G  Ei ‘12  6  ...I  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  “O”O”OO’OOLLL “O”O”O”O’OOLLLL “O”O”O”000LL01 “O”O”O”OOOLL6 0”O”O”OOOLL’8 “O”O”O”OOOU.L OOO”OOOL.V9 O”O”O”OOOLL9 “OOO 0’OLLLV • .0 0 0 0’OLLLE “O”O”O”O’OILL “O”O”O”O’OLLLL S-”O”O’L S-O”OLL S-”O”O6 Q’ ‘LLLLLIV’OL “o”o”o”o”oog”oog”oog’ ••O••O••O••O•OOS•OOS’OOS’L LL ‘•o• •o••o’•o •o’•o••o •o •o SL3L’SL3LS3LOOO”O”O”sO OO”O”OOOOOO”O”OOLOL 0000 L L ‘O’000 L t 0 96’ OOL ‘L’LLLOL68L99 ‘L 8L9S’V’Li L 0 OL ‘OL’L’LL O”O”LOL “O”O”O6 o”VL”o9 “O”L”I.”L1 “O”L”O”L9 “O• L 0 09 “O”O”L”0V “O”O”L”LE  ONIOVO7 1VNOISN3YUO 3N0  j  “o”o”o”oH • L L L OL SISA1VNV SS3UIS VH3N35  Idmxa ioj  u  ir  111710N00 io; nuvJj isfl j xrpudd  AOFOIL  0.000 1.000 1.000 0.000 0.000 1.000 1.000 0.000 0.000 1.000 1.000 0.000  1 2 3 4 5 6 7 8 9 10 11 12  2 6  1 5  1 2  =0 =0 =0 =0  =  O.100E+O1 O.000E÷OO O.000E+OO  MATERIAL I MODEL N IS I GE ISIGB ILSHD I MPF  3 7  3 4 8  NODES 4 5 9  5 6 10  6 7 7 11  0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 2.000 2.000 2.000 2.000  Z-COORD  0.150E+04 O.350E+02 O.000E+OO O.000E÷OO O.000E÷OO O.000E+OO  8 12  8  O.000E+OO O.000E+OO O.000E+OO  LINEAR/NONLINEAR ELASTIC MODEL USE MEAN NORMAL STRESS USE MEAN NORMAL STRESS NO LOAD SHEDDING FULLY SATURATED SOIL  MATERIAL PROPERTIES  2  1  ELE. NO.  0000 0.000 1.000 1.000 0.000 0.000 1.000 1.000 0.000 0.000 1.000 1.000  Y-COORD  ELEMENT-NODAL INFORMATION  XCOORD  NODE  NODAL COORDINATES AND TEMPERATURE  GENERAL STRESS ANALYSIS, ONE DIMENSIONAL LOADING  0.100E+04 O.000E+OO O.OOOEOO  0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000  TEMP  (A)NALYSIS OF (D)EFORMATION AND (F)LOW IN (OIL) SANDS  •  O.000E÷OO O.000E+OO O.000E+OO  O.200E+O2 O.000E+OO O.000E+OO  O.000E+OO O.100E+16 O.000E+OO  O.000E+OO 0,100E+16 O.000E+OO  O.000E+OO O.100E+16 O.000E+O0  U  ‘•l —  CD  o  L’3  C)  X STRESS V STRESS  = =  IN GRAVITY  INCH.  TIME INCREMENT  -0.8333E-16 O.8333E-16 O.8333E-16 -O.8333E-16 -O.1667E-15 O.1667E-15 O.1667E-15 -O.1667E-15 -O.8333E-16 O.8333E-16 O.8333E-16 -O.8333E-16  1 2 3 4 5 6 7 8 9 10 11 12  V STRAIN  STRESS AND PORE PRESSURES  -O.2981E- 15 -O.2981E- 15 2 -0. 20 lYE- 15 -D.2019E- 15  X STRAIN  0. 1000E+06 0. 1000E+06  0. 1500E+06 0. 1500E+O6  2  STRAINS  BULK MODULUS  16 16 16 16 15 15 15 15 16 16 16 16  ELASTIC MODULUS  -O.8333E-0.8333E0.B333E0.8333E-0. 1667E-0. 1667E0. 1667E0. 1667E-O.8333E-O.8333EO.8333EO.8333E-  INCREMENTAL VI  SHEAR-XY STRESS  =  Z STRAIN  0.2500E+O0 0. 2500E+00  POISSION RATIO  -O.2500E-15 0.2500E-15 -0. 2500E- 15 -O.2500E- 15 -0. 5556E-03 -O.5556E-O3 -0.5556E-03 -O.5556E-O3 -0. 1111E-O2 -0. 1111E-02 -0. 111 1E-O2 -0.1 111E-02  ZI  SHEAR-YZ STRESS  =  XA  SHEAR-ZX STRESS  XV STRAIN  0 .0000E+OO 0. 0000E+0O  PLASTIC PARAMETER  PORE PRESSURE  ZX STRAIN  0. I000E+O1 0. 1000E+O1  WATER SATURAN  -O.2500E- 15 -0. 2500E- 15 -O.2500E- 15 -O.2500E- 15 -0. 5556E-03 O.5556E-03 -0. 5556E-03 •O.5556E-O3 -0. 1111E-O2 -0. 1111E-02 -0.111 IE-02 -0.111 1E-02  ZA  O.4825E-16 -O.4816E-16 -0.4779E-16 O.4780E-16  YZ STRAIN  0. I000E+01 0. 1000E+01  VOID RATIO  -O.8333E- 16 -O.8333E- 16 0.8333E- 16 O.8333E- 16 -0. 1667E-15 -0. 1667E- 15 0. 1667E- 15 0. 1667E- 15 -O.8333E- 16 -O.8333E- 16 O.8333E- 16 O.8333E- 16  ABSOLUTE VA  O.1000E+O1  O.0000E+O0  -O.8333E-16 O.8333E-16 O.8333E-16 -O.8333E-16 -O.1667E-15 O.1667E-15 O.1667E-15 -O.1667E-15 -O.8333E-16 O.8333E-16 O.8333E-16 -O.8333E-16  TOTAL TIME  TOTAL GRAVITY  0.5556E-O3 0.1233E-31 O.5556E-03 -0.3698E-31  O.1000E+O1  0.0000E÷OO  EL EM  MODULI VALUES  XI  NODE  NODAL DISPLACEMENTS  =  INCREMENT NUMBER  ELEM  Z STRESS  O.5556E-03 0.5556E-03  VOL. STRAIN  0 .0000E+O0 O.0000E+00  OIL SATURAN  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  ELEM  INITIAL STRESSES  O.79E+0O O.79E+OO  0.21E+00 0.21E+00  O.79E+0O 0.18E+0l  TNT. POINT COORDINATES X V Z  0. 0000E+OO 0. 0000E+00  GAS SATURAN  I.3  C3  0  I-s  1  ELEM  O.5333E-O3 O.5333E+O3  X STRESS O.5333E+03 O.5333E+03  V STRESS  STRESS LEVEL O.2398E-O1 O.23g8E-O  O.0000E+OO O.0000E+OO  O.6000E+03 O.7396E-27 O.895E-11 -O.2889E-11 O.6000E+03 -O.2219E-26 -O.2867E-11 O.2868E-I1  YZ STRESS  PORE PRESSURE  XV STRESS  ZX STRESS  Z STRESS  I  rj  

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