ELASTO-PLASTIC DEFORMATION AND FLOWANALYSIS IN OIL SAND MASSESbyTHILLAIKANAGASABAI SRITHARB. Sc (Engineering), University of Peradeniya, Sri Lanka, 1985M. A. Sc. (Civil Engineering) University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment ofCIVIL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril, 1994© THILLAIKANAGASABAI SRITHAR, 1994In . presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_______________________Department of Civil EngineeringThe University of British ColumbiaVancouver, CanadaDate - A?R L 9 LDE-6 (2188)AbstractPrediction of stresses, deformations and fluid flow in oil sand layers are importantin the design of an oil recovery process. In this study, an analytical formulation isdeveloped to predict these responses, and implemented in both 2-dimensional and3-dimensional finite element programs. Modelling of the deformation behaviour ofthe oil sand skeleton and modelling of the three-phase pore fluid behaviour are thekey issues in developing the analytical procedure.The dilative nature of the dense oil sand matrix, stress paths that involve decreasein mean normal stress under constant shear stress, and loading-unloading sequencesare some of the important aspects to be considered in modelling the stress-strainbehaviour of the sand skeleton. Linear and nonlinear elastic models have been foundincapable of handling these aspects, and an elasto-plastic model is postulated tocapture the above aspects realistically. The elasto-plastic model is a double-hardeningtype and consists of cone and cap-type yield surfaces. The model has been verifiedby comparison with laboratory test results on oil sand samples under various stresspaths 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 throughan equivalent fluid that has compressibility and hydraulic conductivity characteristicsrepresentative of the components. Compressibility of the gas phase is obtained usinggas laws and the equivalent compressibility is derived by considering the individualcontributions of the phase components. Equivalent hydraulic conductivity is derivedfrom the knowledge of relative permeabilities and viscosities of the phase components.Effects of temperature changes due to steam injection are also included directly11in the stress-strain relation and in the flow continuity equations. The analyticalequations for the coupled stress, deformation and flow problem are solved by a finiteelement procedure. The finite element programs have been verified by comparing theprogram results with closed form solutions and laboratory test results.The finite element program has been applied to predict the responses of a horizontal well pair in the underground test facility of Alberta Oil Sand Technologyand Research Authority (AOSTRA). The results are discussed and compared withthe measured responses wherever possible, and indicate the analysis gives insightsinto the likely behaviour in terms of stresses, deformations and flow and would beimportant in the successful design and operation of an oil recovery scheme.111Table of ContentsAbstract iiList of Tables xList of Figures xiAcknowledgement xviNomenclature xvii1 Introduction 11.1 Characteristics of Oil Sand 41.2 Scope and Organization of the Thesis 82 Review of Literature 102.1 Stress-Strain Models 102.1.1 Stress-Strain Behaviour of Oil Sands 112.1.2 Stress-Strain Models for Sand 192.1.2.1 Elasto-Plastic Models 202.1.2.2 Constituents of Theory of Plasticity 222.1.3 Stress Dilatancy Relation 232.1.4 Modelling of Stress-Strain Behaviour of Oil Sand 242.2 Modelling of Fluid Flow in Oil Sand 252.3 Coupled Geomechanical-Fluid Flow Models for Oil Sands 272.4 Comments 30iv3 Stress-Strain Model Employed3.1 Introduction3.2 Description of the Model3.3 Plastic Shear Strain by Cone-Type Yielding3.3.1 Background of the Model3.3.2 Yield and Failure Criteria3.3.3 Flow Rule3.3.4 Hardening Rule3.3.5 Development of Constitutive Matrix [C8] .3.4 Plastic Collapse Strain by Cap-Type Yielding3.4.1 Background of the Model3.4.2 Yield Criterion3.4,3 Flow Rule3.4.4 Hardening Rule3.4.5 Development of Constitutive Matrix [Cc]3.5 Elastic Strains by Hooke’s Law3.6 Development of Full Elasto-Plastic Constitutive Matrix3.7 2-Dimensional Formulation of Constitutive Matrix3.8 Inclusion of Temperature Effects3.9 Modelling of Strain Softening by Load Shedding3.9.1 Load Shedding Technique3.10 Discussion4 Stress-Strain Model - Parameter Evaluation and4.1 Introduction4.2 Evaluation of Parameters4.2.1 Elastic Parameters4.2.1.1 Parameters kE and n3232353737424748515555575858596162• 65• 676870727474747575Validationv4.2.1.2 Parameters kB and mEvaluation of Plastic Collapse ParametersEvaluation of Plastic Shear Parameters4.2.3.1 Evaluation of ij and L24.2.3.2 Evaluation of and )4.2.3.3 Evaluation of KG, np and R14.2.4 Evaluation of Strain Softening Parameters4.3 Validation of the Stress-Strain Model4.3.1 Validation against Test Results on Ottawa Sand4.3.1.1 Parameters for Ottawa Sand4.3.1.2 Validation4.3.2 Validation against Test Results on Oil Sand4.3.2.1 Parameters for Oil Sand4.3.2.2 Validation4.4 Sensitivity Analyses of the Parameters4.5 Summary764.2.24.2.379808282838687889196961011071091145 Flow Continuity Equation 1155.1 Introduction 1155.2 Derivation of Governing Flow Equation 1165.3 Permeability of the Porous Medium 1235.4 Evaluation of Relative Permeabilities 1245.5 Viscosity of the Pore Fluid Components 1325.5.1 Viscosity of Oil 1325.5.2 Viscosity of Water 1345.5.3 Viscosity of Gas 1365.6 Compressibility of the Pore Fluid Components 1365.7 Incorporation of Temperature Effects 140vi5.8 Discussion. 1426 Analytical and Finite Element Formulation6.1 Introduction6.2 Analytical Formulation6.2.1 Equilibrium Equation6.2.2 Flow Continuity Equation6.2.3 Boundary Conditions6.3 Drained and Undrained Analyses6.4 Finite Element Formulation6.5 Finite Elements and the Procedure Adopted6.5.1 Selection of Elements6.5.2 Nonlinear Analysis6.5.3 Solution Scheme6.5.4 Finite Element Procedure6.6 Finite Element Programs6.6.1 2-Dimensional Program CONOIL-Il .6.7 3-Dimensional Program CONOIL-Ill7 Verification and Application of the Analytical Procedure7.1 Introduction7.2 Aspects Checked by Previous Researchers . .7.3 Validation of Other Aspects7.4 Verification of the 3-Dimensional Version7.5 Application to an Oil Recovery Problem7.5.1 Analysis with Reduced Permeability .7.6 Other Applications in Geotechnical Engineering144144145146148148149152158158159162164166166167168168168175181183203208vii8 Summary and Conclusions 2168.1 Recommendations for Further Research 219Bibliography 220Appendices 242A Load Shedding Formulation 242A.1 Estimation of {LO}LS 243A.2 Estimation of {F}Ls 245B Relative Permeabilities and Viscosities 247B.1 Calculations of relative permeabilities 247B.1.1 Relevant equations . 247B.1.2 Example data. . 249B.1.3 Sample calculations . 249B.2 Viscosity of water 250B.3 Viscosity of hydrocarbon gases (from Carr et al., 1954) 252B.3.1 Example calculation 254C Subroutines in the Finite Element Codes 258C.1 2-Dimensional Code CONOIL-Il 258C.1.1 Geometry Program 258C.1.2 Main Program 259C.2 3-dimensional code CONOIL-Ill 261D Amounts of Flow of Different Phases 264E User Manual for CONOIL-Il 270E.1 Introduction 270E.2 Geometry Program 272viiiE.3 Main Program.275E.4 Detail Explanations 292E.4.1 Geometry Program 292E.4.2 Main Program 295F User Manual for CONOIL-Ill 304F.1 Introduction 304F.2 Input Data 305F.3 Example Problem 1 319F.3.1 Data File for Example 1 320F.3.2 Output file for Example 1 321ixList of Tables4.1 Summary of Soil Parameters 754.2 Soil Parameters for Ottawa Sand at Dr = 50% 944.3 Details of the Test Samples 1014.4 Soil Parameters for Oil Sand 1075.1 Parameters needed for relative permeability calculations 1337.1 Parameters for Modelling of Triaxial Test in Oil Sand 1787.2 Model Parameters Used for Ottawa Sand 1817.3 Parameters Used for Thermal Consolidation 1847.4 Parameters Used for the Oil Recovery Problem.. 1927.5 Soil Parameters Used for the Example Problem 209B.1 Viscosity of water between 0 and 1000 C 251B.2 Viscosity of water below 00 C 251B.3 Viscosity of water above 1000 C 251D.1 Average Viscosities and Temperatures in Different Zones 266D.2 Initial Saturations and Mobilities of Water and Oil 268D.3 Calculation of Flow and Saturations with Time 269D.4 Saturations and Mobilities of Water and Oil after 300 Days 269E.1 Element Types 294E.2 Time Increment Scheme 300E.3 Load Increments 302xList of Figures1.1 Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978) 21.2 In-situ Structure of Oil Sand (after Dusseault,1980) 61.3 Undrained Equilibrium behaviour of an Element of Soil upon Unloading (after Sobkowicz and Morgenstern, 1984) 72.1 Fabric of Granular Assemblies (after Dusseault and Morgenstetn, 1978) 122.2 Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseaultand Morgenstern, 1978) 132.3 Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987) 142.4 Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agaret al., 1987) 152.5 Effect of Temperature on Stress-Strain Behaviour (after Agar et al.,1987) 162.6 Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et al.,1987) 183.1 A Possible Stress Path During Steam Injection 343.2 Components of Strain Increment 363.3 Mobilized Plane under 2-D Conditions 383.4 Spatial Mobilized Plane under 3-D Conditions 403.5 Yield and Failure Criteria on TSMp— °sMp Space . . 433.6 Matsuoka-Nakai and Mohr-Coulomb Failure Criteria 453.7 Effect of Intermediate Principal Stress (After Salgado (1990)) . . . 4633.8 (TsMp /osMP) Vs— (desMp /d7sMp) for Toyoura Sand (after Matsuoka,1983) 473.9 Flow Rule and The Strain Increments for Conical Yield 493.10 TSMp/o5Mp Vs YsMP for Toyoura Sand (after Matsuoka, 1983) . . . 503.11 Isotropic Compression Test on Loose Sacramento River Sand (afterLade, 1977) 563.12 Conical and Cap Yield Surfaces on the o — o3 Plane 573.13 Possible Loading Conditions 633.14 Modelling of Strain Softening by Frantziskonis and Desai (1987) . . 693.15 Modelling of Strain Softening by Load Shedding 714.1 Evaluation of kE and ii 774.2 Evaluation of kB and m 784.3 Evaluation of C and p 804.4 Evaluation of and L 834.5 Evaluation of ) and it 844.6 Evaluation of G1, and ‘quit 854.7 Evaluation of K0 and np. 864.8 Evaluation of , and q 884.9 Grain Size Distribution Curve for Ottawa Sand (after Neguessy , 1985) 894.10 Stress Paths Investigated on Ottawa Sand 904.11 Variation of Young’s moduli with confining stresses 914.12 Plastic Collapse Parameters for Ottawa Sand 924.13 Failure Parameters for Ottawa Sand 934.14 Flow Rule Parameters for Ottawa Sand 944.15 Hardening Rule Parameters for Ottawa Sand 954.16 Results for Triaxial Compression on Ottawa Sand 974.17 Results for Proportional Loading on Ottawa Sand 98xii4.18 Results for Various Stress Paths on Ottawa Sand 994.19 Grain Size Distribution for Athabasca Oil Sands, (after Edmunds etal., 1987) . .. 1004.20 Determination of kB and m for Oil Sand 1024.21 Plastic Collapse Parameters for Oil Sand 1034.22 Failure Parameters for Oil Sand 1044.23 Determination of and np for Oil Sand 1054.24 Flow Rule Parameters for Oil Sand 1064.25 Results for Isotropic Compression Test on Oil Sand 1084.26 Results for Triaxial Compression Tests on Oil Sand 1104.27 Results for Tests with Various Stress Paths on Oil Sand . . . 1114.28 Sensitivity of Parameters C,p,A and p 1124.29 Sensitivity of Parameters KG, np, R1 and i 1135.1 One dimensional flow of a single phase in an element 1175.2 Typical two-phase relative permeability variations (after Aziz and Settan, 1979) 1255.3 Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979) 1275.4 Comparison of calculated and experimental three-phase oil relative permeability (after Kokal and Maini, 1990) 1305.5 Comparison of calculated and experimental relative permeabilities using power law functions 1315.6 Experimental and predicted values of viscosity (after Puttagunta et al.,1988) 1356.1 Finite Element Types Used in 2-Dimensional Analysis 1606.2 Finite Element Types Used in 3-Dimensional Analysis 1616.3 Flow Chart for the Finite Element Programs 165xiii7.1 Stresses and Displacements Around a Circular Opening for an ElasticMaterial (after Cheung, 1985) 1707.2 Comparison of Observed and Predicted Pore Pressures (after Cheung,1985) 1717.3 Comparison of Observed and Predicted Strains (after Cheung, 1985) 1727.4 Results for a Circular Footing on a Finite Layer (after Vaziri, 1986) 1737.5 Stresses and Displacement in Circular Cylinder (after Srithar, 1989) 1747.6 Pore Pressure Variation with Time for Thermal Consolidation (afterSrithar, 1989) 1767.7 Undrained Volumetric Expansion (after Srithar, 1989) 1777.8 Finite Element Modelling of Triaxial Test 1797.9 Comparison of Measured and Predicted Results in Triaxial Compression Test 1807.10 Comparison of Measured and Predicted Results for a Load-Unload Testin Ottawa Sand 1827.11 Finite Element Mesh for Thermal Consolidation 1847.12 Comparison of Pore pressures for Thermal Consolidation 1857.13 A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991 1877.14 Plan View of the UTF (after Scott et al., 1991) 1887.15 Vertical Cross-Sectional View of the Well Pairs 1897.16 Finite Element Modelling of the Well Pair 1917.17 Temperature Contours in the Oil Sand Layer 1937.18 Pore Pressure Variations in the Oil Sand Layer 1957.19 Comparison of Pore Pressures in the Oil Sand Layer 1967.20 Horizontal Stress Variations in the Oil Sand Layer 1977.21 Vertical Stress Variations in the Oil Sand Layer 1987.22 Stress Ratio Variations in the Oil Sand Layer 199xiv7.23 Comparison of Horizontal Displacements at 7 m from Wells 2007.24 Vertical Displacements at the Injection Well Level 2017.25 Total Amount of Flow with Time 2027.26 Individual Flow Rates of Water and Oil 2047.27 Total Amount of Oil Flow 2057.28 Pore Pressure Variation for Analysis 2 2067.29 Stress Ratio Variation for Analysis 2 2077.30 Details of the Cases Analyzed 2107.31 Variation of Pore Pressure Ratio for Case 1 2127.32 Variation of Pore Pressure Ratio for Case 2 2137.33 Variation of Pore Pressure Ratio for Case 3 214A.1 Strain Softening by Load Shedding 242B.1 Prediction of pseudocritical properties from gas gravity . . 253B.2 Viscosity of hydrocarbon gases at one atmosphere 254B.3 Viscosity ratio vs pseudo-reduced pressure 255B.4 Viscosity ratio vs pseudo-reduced temperature 256D.1 Zones involved in Fluid Flow.. 265E.1 Nodes along element edges . . 290E.2 Element types 293E.3 Plane Strain Condition 296E.4 Axisymmetric Condition . . . 296F.1 Available Element Types . . . 306F.2 Finite element mesh for example problem 1 319xvAcknowledgementThe author is greatly indebted to his supervisor Professor P. M. Byrne for his guidance, valuable suggestions and the encouragement throughout this research.The author wishes to express his appreciation to the members of the supervisorycommittee for reviewing the manuscript and making constructive criticisms. Appreciation is also extended to Mr. Jim Grieg for his valuable helps on the computeraspects.The author expresses his gratitude to his wife, Vasuki, for her support and tolerance of the odd working habits of a graduate student.The author would like to thank his colleagues in Dept. of Civil Engineering , inparticular, Uthayakumar and Hendra for sharing common interest.Finally, the fellowship awarded by the University of British Columbia and theresearch grant provided by Alberta Oil Sand Technology and Research Authority(AO STRA) are gratefully acknowledged.xviNomenclatureB bulk modulusB pore pressure shape function derivativesdisplacement shape function derivativesC plastic collapse modulusCEQ equivalent compressibilityD stress-strain matrixE Young’s moduluse void ratioF body force vectorf plastic collapse yield functioninitial plastic shear parameterGt tangent plastic shear parameterH Henry’s constantI, 12 and 13 stress invariantsK0 plastic shear numberk Darcy’s permeability of the porous mediumkB bulk modulus numberYoung’s modulus numberkEQ equivalent hydraulic conductivitykh permeability in horizontal directionkmi mobility of phase ‘1’kmT total mobilityxviikri relative permeability of phase ‘1’krog relative permeability of oil in oil-gas systemkr relative permeability of oil in oil-water systemrelative permeability of oil at critical water saturationpermeability in vertical directionl, l, and l direction cosines of o to the x, y and z axesM constrained modulusm bulk modulus exponentmz,my and m direction cosines of o-2 to the x, y and z axesN shape functions for pore pressuresN shape functions for displacementsn Young’s modulus exponentn, n, and n2 direction cosines of o3 to the x, y and z axesnp plastic shear exponentP pore pressurePa atmospheric pressurecapillary pressurep plastic collapse exponentq strain softening exponentfailure ratioS saturationnormalized saturationresidual oil saturationS critical water saturationt timeU displacement vectorV volumexviiiW plastic collapse workGreek letterscoefficient of volumetric thermal expansioncEQ equivalent coefficient of thermal expansionshear strainKronecker deltaEl, 62 and 63 principal strainsplastic collapse strains6e elastic strainsplastic shear strainsvolumetric strainstress ratiofailure stress ratio at atmosphere8 temperaturestrain softening constantflow rule slopeproportionality constantp flow rule interceptviscosity of phase ‘1’P30,0 viscosity of oil at 30°C and at 0 gauge pressurev Poisson’s rationormal stress1, 2 and u3 principal stressesmean normal stressr shear stressxix(6m mobilized friction angleSubscriptsf failure stateg gas phasej partial derivative with respect to jMP mobilized planeo oil phaseSMP spatial mobilized planeult ultimate statew water phaseSuperscriptsc plastic collapse conditione elastic conditionplastic shear conditionxxChapter 1IntroductionThe oil contained in oil sand deposits in northern Alberta is one of the major resourcesin Canada. These deposits underlie an area of about 32,000 square kilometres withestimated in-place reserves of 146.5 million cubic meters (Mosscop, 1980). Much of theoil exists as high viscosity bitumen in Arenaceous Cretaceous formations, primarily inthe Athabasca oil sand deposits (see figure 1.1). Approximately 5% of these depositsare found at depths less than 50 m and the rest are encountered at depths from 200to 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 sandformations. In the in-situ extraction procedures some form of heating is often requiredas the very high viscosity of the bitumen makes conventional recovery by pumpingimpractical. In-situ thermal methods such as steam injection through vertical well-bore have been used and are relatively effective for the recovery of heavy oil fromdeep seated formations. There have been, however, numerous well casing failures andinstability problems reported during field injection trials. During steam injection,high pore fluid and stress gradients are created around the well-bore which can leadto the instability and collapse of the well casing. Therefore, to understand the mechanisms involved and to design these oil recovery schemes rationally and economically,analyses which capture the complex engineering characteristics of the oil sand arenecessary.Analyzing the problems related to oil sands is somewhat different from analyzing1Chapter 1. Introduction 2United States of AmencaFigure 1.1: Oil Sand Reserves in Alberta (after Dusseault and Morgenstern, 1978)Northwest TerritoriesChapter 1. Introduction 3a general geotechnical problem because of the nature of the oil sand and the recoveryprocess 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 ofbitumen and gas makes the analytical procedures for oil sands different and difficult.Oil recovery by steam injection will cause changes in temperature and their effectsare also of prime concern. The changes in temperature induce changes in volume andpore 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 voidsin the soil skeleton, there will be an increase in pore pressure and consequently areduction in effective stress. The effective stresses may become zero and liquefactionmay occur, if the oil sand is subjected to rapid increase in temperature and if anundrained 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 porefluids and the behaviour of sand skeleton. An analytical model for the oil sand wasfirst developed by Harris and Sobkowicz (1977); It was later extended by Byrneand Grigg (1980), Byrne and Janzen (1984) and Byrne and Vaziri (1986). However,these analytical models consider a linear or nonlinear elastic behaviour for the sandskeleton. Oil sand is very dense in its natural state and shows significant dilationupon shear. The linear and nonlinear elastic models are not capable of modelling thedilation effectively. Furthermore, steam injection and subsequent recovery will leadto loading and unloading cycles and for realistic modeffing an elasto-plastic modelis necessary. In this study, a double hardening elasto-plastic model is postulated forthe 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 theChapter 1. Introduction 4individual contributions of the pore fluid components in the compressibility but not inthe hydraulic conductivity. In this research work, the relative permeabilities of water,bitumen and gas are considered and an equivalent hydraulic conductivity is derivedto model the pore fluid behaviour appropriately. The equivalent compressibility termas proposed by Byrne and Vaziri (1986) is also included.The effects of temperature changes in stresses and volume changes have been directly included in the governing equilibrium and flow continuity equations. It shouldbe noted that the equation of thermal energy balance is not considered in the analytical model. However, the temperature-time history which is obtained from a separateheat flow analysis or by some other means is considered as an input to the analyticalmodel and, the effects of these temperature changes on the stress-strain behaviourand the fluid flow are evaluated.An analytical procedure considering all these aspects has been developed andincorporated in the 2-dimensional finite element code CONOIL-Il. In order to analyzethe three dimensional effects a new 3-dimensional finite element code CONOIL-Ill isalso developed.1.1 Characteristics of Oil SandSince the oil sand is different form a general soil, it is appropriate to present somebrief descriptions about its unusual characteristics. Oil sand can be considered as afour phase geological material comprising solid, water, bitumen and gas. The twodominant physical characteristics of the oil sand are the quartz mineralogy and thelarge quantity of interstitial bitumen. The quartz grains of the oil sand are 99% waterwet as the water phase forms a continuous film around it. A larger portion of thepore 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, significantquantities of gas can also exist in the dissolved state in the pore fluid. An illustrationChapter 1. Introduction 5of oil sand structure (Dusseault, 1980) is shown in figure 1.2.In its natural state, oil sand is very dense, uncemented, fine to medium grainedand exhibits high shear strength and dilatancy. It shows low compressibility characteristics compared to normal dense sand of similar mineralogy. The extremely highviscosity of bitumen makes the effective hydraulic conductivity very low and causesthe oil sand to behave in an undrained manner.Another unusual characteristic of oil sand is its behaviour upon unloading. Because of the very low effective hydraulic conductivity, oil sand behaves in an undrainedmanner, however, it responds quite differently compared to the undrained behaviourof a normal sand. Above the liquid-gas saturation pressure (U119), oil sand behaveslike a normal sand (path I of figure 1.3). A decrease in confining stress will result in adecrease in pore pressure and the effective stress remains constant. When the level ofconfining stress decreases below the liquid-gas saturation pressure, the dissolved gasin the pore fluid comes out of solution and causes the pore fluid to become very compressible. At this point, the soil matrix commences to take the load and the effectivestress decreases while the pore pressure stays constant (path J). As the effective stressdecreases, the soil skeleton compressibility increases and becomes comparable to thepore fluid compressibility. Then, the pore fluid takes the load and the pore pressurestarts to decrease again (path K). At some stage, the effective stress becomes zeroand the physical consequences of this process are significant increase in volume anda marked reduction in shear strength. Plots of pore pressure versus total stress forsaturated (path M), unsaturated (path L) and gassy soils (path J-K) are shown infigure 1.3. A comprehensive study of the gas exsolution phenomenon upon unloadingcan be found in Sobkowicz and Morgenstern (1984).TjI-. (b I-. U) II-.0CCCC--mrcnoa;-’ -C—m0om-aC-,oo a..C-Chapter 1. Introduction 7UUj/gD(I,C,,w0LAJ00I ..°atmFigure 1.3: Undrained Equilibrium behaviour of an Element of Soil upon Unloading(after Sobkowicz and Morgenstern, 1984)//..—. u=o,____,_,_o-c=o JIN SITUSTRESS/TOTAL STRESSCEGASSEDPORE FLUID0FINE SOILChapter 1. Introduction 81.2 Scope and Organization of the ThesisThe 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. Theanalytical model is developed on the premise that the oil sand is a four phase materialcomprising solid, water, bitumen and gas.In developing the analytical formulation the key issues are; a stress-strain modelfor the sand skeleton, the compressibility and permeability characteristics of the three-phase pore fluid, the effects of temperature, and the overall analytical and finiteelement procedure. Discussions on these issues highlighting the previous researchworks in the literature are given in chapter 2.The main feature in a deformation analysis is the stress-strain model employed. Inthis study, a double-hardening elasto-plastic model is postulated. The fundamentaldetails of the stress-strain model and the development of the constitutive matrixusing plastic theories are described in chapter 3. The parameters required for thestress-strain model, procedures to obtain them, the sensitivity of these parametersand the verification of the stress-strain model against laboratory results are presentedin chapter 4.One of the major concerns in the analytical formulation presented in this studyis the modelling of the multi-phase fluid. Chapter 5 describes the development of theflow continuity equation, considering the contributions from all the fluid phase components, in detail. Inclusion of temperature effects in the flow continuity equation isalso given in this chapter. Inclusion of the temperature effects in stress-strain relationis explained in chapter 3. Details concerning the overall analytical procedure and itsimplementations in 2-dimensional and 3-dimensional finite element formulations aregiven in chapter 6.Verifications and the validations of the developed formulation are presented inchapter 7. Some specific problems where closed form solutions are available and someChapter 1. Introduction 9laboratory experiments are considered and the results are compared. Application toan oil recovery process by steam injection is presented and the results are analyzedin detail. Possible applications of the developed formulation for general geotechnicalproblems are discussed and an example problem is also given.Chapter 8 summarizes the important findings of this research work. Some comments on the aspects which warrant further investigation are also stated in this chapter.Chapter 2Review of LiteratureThe research work carried out in this study can be broadly classified under the following topics; stress-strain model for the oil sand, modelling of flow characteristics ofthe three-phase pore fluid; and the analytical and finite element formulations. Therefore, it is appropriate to present a review on the previous research works under thesesubheadings. The intention of the literature review presented in this chapter is notto critically assess each and every research work but to give an overall picture, andto set the stage to discuss the work carried out in this study.2.1 Stress-Strain ModelsThe stress-strain behaviour of the oil sand skeleton is essentially the stress-strainbehaviour of a dense sand. This conclusion was not widely accepted until the completion of series of research programs at the University of Alberta in the late 1970sand in 1980s. In particular, the perception of bitumen as a cementing material waswidely held until the last decade, as many geologists and petroleum engineers failedto recognize the geomechanical behaviour of the sand skeleton. It is now recognizedthat the oil sands must be considered as a particulate material and its behaviour canbe described by an appropriate stress-strain model. Before going into a detailed review of the stress-strain models, it will be useful to describe the observed stress-strainbehaviour of oil sands. The next subsection summarizes the stress-strain behaviourof oil sands in laboratory experiments.10Chapter 2. Review of Literature 112.1.1 Stress-Strain Behaviour of Oil SandsDusseault (1977) showed that the Athabasca oil sands have an extremely stiff structure in the undisturbed state, accompanied by a large degree of dilation when loadedto failure and subsequent yield. This was attributed to its extreme compactness whichprovides extensive grain-to-grain contact. The grain orientations of the oil sand arecompared schematically to ideal and rounded sand grains in figure 2.1. The angularity of the Athabasca sand grains illustrate why significant dilation can be expectedas the sand is sheared.Dusseault and Morgenstern (1978) studied the shear strength of Athabasca oilsands and stated that the Mohr-Coloumb failure envelope is not a straight line butcurvilinear. The residual and peak shear strengths measured in direct shear tests areshown in figure 2.2. The curvilinear nature is said to be due to the dilatancy and thegrain surface asperity.Agar et al. (1987) carried out extensive testing on Athabasca oil sand to study theeffects of temperature, pressure and stress paths on shear strength and stress-strainbehaviour. 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). Thesecurves illustrate the influence of stress paths on peak deviator stress and stress-strainbehaviour. It can be seen from the figure that the dilatancy is more pronounced oncertain stress paths (see paths B and C), and at lower effective confining stress thanat higher stress levels (compare paths C and D).Figure 2.4 shows the shear strength of Athabasca oil sand compared to denseOttawa sand. The shear strength of oil sand is greater than that of dense Ottawasand at lower effective confining stress levels. However, at higher stress levels, thestrengths 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 sandPoint contacts, with many straight andinterpenetrative contacts (arrows)Figure 2.1: Fabric of Granular Assemblies (after Dusseault and Morgenstern, 1978)Chapter 2. Review of Literature 13a.U,0L0(U0-c(0Figure 2.2: Residual and Peak Shear Strengths of Athabasca Oil Sand (after Dusseaultand Morgenstern, 1978)Three different sampleso o Peak strength• Residual strength•0 200 400 600 800o normal stress, kPa1000 1200(a) Various Stress Pathsb>0(b) Stress-Strain BehaviourFigure 2.3: Effect of Stress Path on Stress-Strain Behaviour (after Agar et al., 1987)I;Chapter 2. Review of Literature 1428242016a2016128400.5120 4 8 12./7O1 (MPa)16—0.50.5 1.0 1.5e (%)Chapter 2. Review of Literature 1560a)aUCDU,ina,40U)CIU-C‘/,— 300a)U)C20Figure 2.4: Shear Strength of Athabasca Oil Sand and Ottawa Sand (after Agar etal., 1987). LEGENDD.Athobasca Oil Sand vATHABASCA OIL SAND (This Study)OTTAWA SAND (This StudyDIJSSEAUT & MORGENSTERN(1978)SOBKOWICZ (1982)DUNCAN & CHANG (1970).1 2 3 4 5 6 7 8Effective Confining Stress c (MPa)Chapter 2. Review of Literature 16The effect of temperature on the stress-strain behaviour does not seem to be significant. For some other stress paths, it appeared that the temperature has considerableinfluence on the stress-strain behaviour. However, Agar et al (1987). concluded thatthe differences in the stress-strain behaviour at various temperatures are small. Theyattributed the observed differences to the disturbances in sampling and the material heterogeneities. The test results appeared to be far more sensitive to sampledisturbances than heating.0 0.5 1.0e (%)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 someessential differences in the geomechanical behaviour. Kosar claimed that in addition20161240041.5 2.0Chapter 2. Review of Literature 17to temperature, pressure and stress paths, the grain mineralogy, geological environment of deposition and the geological history are the major factors affecting thegeomechanical behaviour. The maximum shear strength and the stress-strain moduliof Athabasca oil sands are much greater than those of Cold Lake oil sand reflectingthe grain mineralogy and the geological factors. Athabasca oil sands consist of auniformly graded, predominantly quartz sand, whereas, Cold Lake oil sands containseveral additional minerals which are weaker. Figure 2.6 shows typical drained tn-axial compression test of these two oil sands. Athabasca oil sand exhibits dilatantbehaviour but the Cold Lake oil sand does not. In the Athabasca oil sand, the increasein volume change during shear is also accompanied by strain softening behaviour inthe post peak region. The Cold Lake oil sand shows contractive behaviour and thereason for this is the presence of weaker minerals. The weaker minerals are proneto grain crushing at the applied stress levels. Because of these weaker minerals, thegeomechanical behaviour of Cold Lake oil sand changes with temperature as well.Athabasca oil sands, on the other hand, do not show significant changes in behaviourat different temperatures.Wong et al. (1993) pointed out that testing of oil sand samples should includesome important stress paths which are expected to be encountered in the field. Theycarried out detailed testing on Cold Lake oil sand which includes stress paths withincreasing and decreasing pore pressures under constant total stress. This results inload-unload-reload stress paths in terms of effective stress ratio. They identified fourdifferent modes of granular interactions namely; contact elastic deformation, sheardilation, rolling and grain crushing for the observed geomechanical behaviour. Theyalso noticed grain crushing in Cold Lake oil sand when the effective confining stressincreased above 8 MPa.Chapter 2. Review of Literature 186-Mairjmshearsfl-ength = 16.9 MPa5.Ia—4.OUPa4 / Athabasca (Agar. 1984)03• : Mi,m shaer strength • 6.9 MPaI2. :Athabasca £ - 2200 MPa‘7 CoidLakeSAxial Strain (%)Figure 2.6: Comparison of Athabasca and Cold Lake Oil Sands (after Kosar et aL,1987)Chapter 2. Review of Literature i9Therefore, the modelling of oil sand behaviour should include two significant features; non-recoverable strains and dilatancy. A realistic model must take the deformation history into account, particularly if the stresses are to be cycled through loadingand unloading. The elasto-plastic formulation incorporates these features naturally.There are a number of elasto-plastic stress-strain models available for sands in theliterature and a brief review of those are presented next.2.1.2 Stress-Strain Models for SandA number of models have been proposed in the literature for the stress-strain behaviour of sand. Most of them make use of the well developed classical theories ofelasticity and plasticity either separately or in a combined form. These theories arebased on the observations made on materials that can be described in the context ofcontinuum mechanics. To adopt these theories to model the stress-strain behaviourof sand, they have to be modified. Different modifications are made to capture distinguished features of sand behaviour and thus, different models are proposed bydifferent researchers. One of the difficult features of sand behaviour to model hasbeen the shear induced volume change.Basically, constitutive models can be classified into two categories; linear or incremental elastic models and elasto-plastic models. In the theory of elasticity, thestate of stress is uniquely determined by the state of strain so that the stress-strainresponse of an elastic models is independent of the stress path. The simplest elasticmodel would be the isotropic linear elastic model which requires only two materialparameters. Incremental elastic models (Duncan and Chang (1970), Duncan et al.(1980)) are the most commonly used because they can capture the nonlinearity andare easy to use. Essentially, the incremental elastic models also require only two parameters when analyzing a load increment. However, to update these two materialparameters with stress levels and to model the nonlinearity additional parameters areChapter 2. Review of Literature 20necessary. Generally, in elastic models, the shear and normal stresses and strains areuncoupled from each other. Byrne and Eldrige (1982) incorporated the shear volumecoupling effects in the incremental elastic models using a stress dilatancy equation.Reviews of the existing elastic and elasto-plastic constitutive models are available in the literature as state-of-the-art papers, special workshops and internationalsymposia. Ko and Sture (1980) provided a clear summary of the most importantmodels 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 symposium, Florida, 1980; International Symposium, Deift, 1982) provide better insightsinto the different stress-strain models.Since an elasto-plastic model is proposed in this study, a brief review of elastoplastic models and the related theories are presented next.2.1.2.1 Elasto-Plastic ModelsThe theory of plasticity has been developed on the basis of observed stress-strainbehaviour of metals. Since soils exhibit plastic non-recoverable strains, the theoryof plasticity provides an attractive theoretical framework for the representation ofthe stress-strain behaviour of soils. However, there are major differences such as thepresence of voids and the tendency for volume change during shear that distinguishsoils from metals (Lade, 1987).In the elasto-plastic models, the strain increment is decomposed into an elasticcomponent and a plastic component. The amounts of elastic and plastic strains willvary with the level of loading and unloading. The elastic strain increment is obtainedChapter 2. Review of Literature 21using the theory of elasticity and the plastic strain increment is obtained from thetheory 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 capwhich passes through the isotropic compression axis. Most of the elasto-plastic modelsevolved from this study. The Cam-Clay model (Roscoe et al., 1958) introduced theconcept of critical state and presented an equation for the yield surface consideringenergy dissipation. Prevost and beg (1975) used the critical state line concept intheir model, but defined two yield surfaces, one for volumetric and shear deformationand the other for shear deformation alone. The Cam-Clay model has been used inone 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 ofthe Mohr-Coloumb criterion and incorporate the influence of intermediate principalstress. The yield and failure surfaces are assumed to be described by similar functionsso that both surfaces have similar shapes. Lade (1977) introduced a yielding cap inorder to control the plastic volumetric strain making his model a double hardeningone. Vermeer (1978) also used a double hardening model. He divided the plasticstrain into two parts; one is described by means of a shear surface and the sheardilatancy 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 thepositions, sizes and plastic moduli of each of the yield surfaces be stored for everyintegration point, which is very tedious and therefore not very commonly used.Chapter 2. Review of Literature 222.1.2.2 Constituents of Theory of PlasticityIn the theory of plasticity, existence of a yield function, a potential function anda hardening function are necessary to relate the plastic strain increments to stressincrements mathematically. The yield function defines the stress conditions causingplastic strains. The yield surface represented by the yield function encloses a volumein the stress space, inside of which the strains are fully recoverable. Only stressincrements directed outward form the yield surface cause plastic strains. A stressincrement directed outward from the yield surface causes an expansion or translationof the yield surface. During yielding, the state of stress remains on the yield surfacewhich is known as the consistency condition. A state of stress outside the yield surfaceis not possible.The direction of plastic strain increment is defined by the potential function whichis referred to as flow rule. If the potential function and the yield function are thesame, the flow rule is said to be associative. If these functions are different, then theflow rule is non-associative.The amplitude of the plastic strain increment is specified by the hardening function. In plasticity, two types of hardening have been distinguished; isotropic hardeningand kinematic hardening. In a model undergoing isotropic hardening, the yield surface expands radially about the fixed axes. When the yield surface translates withoutchanging its size, the model undergoes kinematic hardening.Once the constituents of the theory of plasticity are defined, the plastic strainincrement, can be calculated from,=— n (2.1)where,Lo-- applied stress incrementn, - vector defining the unit normal to yield surface at the stress pointChapter 2. Review of Literature 23- vector defining the unit normal to potential surface at the stress pointH - plastic resistance2.1.3 Stress Dilatancy RelationThe stress dilatancy theory derived from theoretical considerations has been usedextensively in stress-strain modeffing of sand. The stress dilatancy theory proposedby Rowe (1962,1971) can be considered a remarkable effort to explain the shear deformation behaviour. After Rowe, a number of other researchers published theoriesto model the dilatancy following different approaches (Murayama (1964), Matsuoka(1974), Oda and Konishi (1974), Nemat-Nasser (1980)). A noticeable difference between Rowe’s theory and the other theories is that Rowe’s theory is independent ofthe spatial distribution of interparticle contacts. Rowe’s theory considers that slidingoccurs on certain favourably oriented contact planes. The orientation of the slidingplanes will be such as to minimize the rate of dissipation of energy in sliding frictionbetween particles with respect to energy supplied.Matsuoka (1974) developed the stress dilatancy relationship through a microscopicpoint of view. He carried out shear tests by using cylindrical rods to model theshearing mechanism of soil particles. From the fundamental measurements of theangle of the interparticle contact, interparticle force and the angle of interparticlefriction, he developed a relationship between the shear resistance and the dilatancy.Lade’s (1977) model incorporates the dilatancy through a empirical relation obtained by curve fitting. The equation relates a dilation parameter to the amount ofplastic work.Nemat-Nasser (1980) presented an equation to describe the volumetric behaviourof soil upon shearing which is based on the mechanics of the relative motion of thegrains at the micro level. The equation was obtained by considering the rate offrictional losses and the energy balance.Chapter 2. Review of Literature 242.1.4 Modelling of Stress-Strain Behaviour of Oil SandModelling of the geomechanical behaviour of oil sand along with the pore fluid behaviour, so as to describe gas exsolution and other related aspects was first presentedby Harris and Sobkowicz (1977). They considered a linear elastic model for the sandskeleton behaviour.A nonlinear elastic model with shear dilation was proposed by Byrne and Griggin 1980 to model the oil sand skeleton behaviour. Their model is based upon anequivalent elastic analysis using a secant modulus and a single step loading. This wassubsequently extended by Byrne and Janzen (1984) who used an incremental tangentmodulus rather than a secant modulus. Vaziri (1986) basically used the same modelas Byrne and Janzen to represent the stress-strain behaviour of oil sand.In the above cited references, the dilative behaviour of the material is incorporatedthrough a procedure borrowed from thermoelasticity. This method involves applyingequivalent nodal loads to predict the correct volume changes. Srithar et al. (1990)pointed out that the thermoelastic approach encounters shortcomings specially in aconsolidation type of analysis. It predicts unrealistic oscillating results when largetime steps are considered. Furthermore, the computer algorithm necessitates twolevels of iterations; one for stress calculations, and the other for shear induced volumechange corrections. Wan et al. (1991) stated that the method of including dilationthrough thermoelastic approach may lead to a decrease in effective mean normal stress0m while in a pressuremeter test, dilation is always accompanied by an increase inTortike (1991) stated that cyclic steam simulation imposes cyclic loads on the oilreservoir. He further suggested that a realistic stress-strain model should have thecapability to model the loading and unloading behaviour. He adopted Hinton andOwen’s (1977) elasto-plastic model which includes a Mohr-Coloumb failure envelopeand an associated flow rule.Chapter 2. Review of Literature 25Wan et al. (1991) also recognized the cyclic loadings caused in the recoveryprocess by steam injection and proposed an elasto-plastic model for oil sand. Theirmodel is based on Vermeer’s (1982) elasto-plastic model. They used Matsuoka andNakai (1982) equation to represent the yield and failure surfaces, and a RambergOsgood type hardening function. The model involves a non-associated flow rule andthe potential function is based upon Rowe’s stress dilatancy equation. However,their model cannot predict the plastic volumetric behaviour for stress paths involvingcompression with constant stress ratio.2.2 Modelling of Fluid Flow in Oil SandIn petroleum reservoir engineering, multiphase fluid flow has been analyzed by anumber of researchers without consideration of the geomechanical behaviour of theoil sand matrix. The first clear attempt to use a finite element method for fluidflow in porous medium that appeared in petroleum engineering was by Javandel andWitherspoon (1968). They considered a single phase isothermal fluid flow throughan isotropic homogeneous porous medium. The numerical solutions were comparedwith the analytical solutions for infinite, bounded and layered radial systems withconstant flow rate or pressure constraints and were found to be in good agreement.Solutions for two-phase isothermal fluid flow problems using variational and finiteelement methods were presented by various researchers (for example: Settari andPrice, 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 formulationfor multi-dimensional, two-phase, immiscible flow using variational method. Theycompared variational and finite difference methods and concluded that the variationalmethod is more efficient than the finite difference method. Galerkin’s procedure wassuccessfully applied to the analytical formulation of the governing equations in thepresence of favourable and unfavourable mobility ratios. Numerical dispersion at theChapter 2. Review of Literature 26front was less in both cases than with the finite difference method. Also, in thevariational 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 weightedform, which was approximated in time by backward Euler differences. The spatialdomain was divided into rectangles and approximated by byliner functions. A sharperfront was noticed when the capillary pressure was not simply a constant function ofsaturation, 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 velocityin miscible and immiscible two-phase reservoir flow problems. Velocity was chosenas the primary variable to ensure that it remains a smooth function throughout thedomain, despite step changes in reservoir properties governing the flow.Faust and Mercer (1976), Huyakorn and Pinder (1977b), Voss(1978) and Lewis etal. (1985) are some of the researchers who analyzed two-phase fluid flow under non-isothermal conditions. Lewis et al. (1985) used the Galerkin method to solve the waterflow and energy equations in two dimensions. Byliner elements were used to modelhot water flooding for thermal oil recovery. Linear and higher order elements wereused to model the heat losses from the reservoir in all directions. Artificial diffusionwas introduced along streamlines to negate any grid orientations. The solutions werefound efficiently at the end of each time step using an alternating direct solutionalgorithm.The solution for multiphase fluid flow problem using finite elements was first presented by McMichael and Thomas (1973). They analyzed a three-phase isothermalflow in a two dimensional domain subdivided into linear finite elements. Reportedly,no difficulties were encountered in finding the solution at each time step. The evaluation of all the reservoir properties at each quadrature point for numerical integrationChapter 2. Review of Literature 27appeared 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 bythe finite element method.Tortike (1991) presented a detailed literature review on modelling of fluid flowunder isothermal and non-isothermal conditions. He solved the three-phase thermal flow problem using finite differences. He also tried to develop a fully coupledgeomechanical 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 flowin oil sand is modelled by two phase system (water and bitumen) with reasonableaccuracy. However, these models solve only the fluid flow problem and do not consider the geomechanical behaviour. Therefore, the effects of stress distribution anddeformation in the oil sand matrix are not included in these models.2.3 Coupled Geomechanical-Fluid Flow Models for Oil SandsSome models in petroleum reservoir engineering include the effects of deformationsin oil sand matrix through poroelasticity. Geertsma (1957) combined the approachesof Biot (1941) and Gassman (1951) to develop the equations of poroelasticity in amore straightforward manner. He clearly defined and related the rock bulk and porecompressibilities, and described the boundary conditions and procedure to determinethe correct parameters defining the compressibilities. Geertsma (1966) reviewed theapplications of poroelasticity in petroleum engineering. An analogy is presented between poroelastic and thermoelastic theories, to take advantage of the many solutionsunder different boundary conditions that have already been published. The conceptof the nucleus of strain for volume elements was described and it has been appliedto predict surface displacements. It should be noted however, the poroelastic theorydoes not consider the effect of stress distribution through a porous medium.Raghavan (1972) derived a one dimensional consolidation equation coupled withChapter 2. Review of Literature 28fluid flow and compared his results with Terzhaghi’s solution. The general solutionwas obtained from the partial differential equations describing the flow of fluid andmaterial displacement using a transform to convert it to an ordinary differential equation. 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 differences which included the effects of compaction on fluid flow and the prediction ofsurface displacements. The problem was formulated by two discretized equations foroil and water flow, and one analytical equation for poroelasticity which was numerically integrated. The variation of permeability and porosity was considered in theanalysis as the effect of compaction on ultimate recoveries. The authors concludedthat the ultimate recoveries of oil increased with compaction.Harris and Sobkowicz (1977) derived an analytical model from a more geotechnical point of view. They presented a coupled mathematical model for the fluid flowand the geomechanical behaviour of oil sand. The model was developed mainly toanalyze excavations, immediate foundation settlements and underground openings inoil sands. Since these scenarios involve short term conditions, and because of thehigh viscosity of the bitumen, their model was only concerned with the undrainedresponse. The authors claimed that the short term conditions govern the design inthe above circumstances.Byrne and Grigg (1980), and Byrne and Janzen (1984) extended Harris andSobkowicz’s formulation. Byrne and Janzen also included the fully drained conditionin their analysis. Their analysis procedure involved an effective stress approach inwhich 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 volumecompatibility between fluid and skeleton phases.Vaziri (1986) coupled the equilibrium equation and the flow continuity equationand analyzed the transient conditions as a consolidation problem. He included theChapter 2. Review of Literature 29thermal effects on stresses, hydraulic conductivity and volume change and presenteda two dimensional finite element formulation. The fluid flow was considered as asingle phase one. The effects of different phase components on compressibility weretaken into account by means of an equivalent compressibility. Vaziri followed thethermoelastic approach to model temperature effects. This approach appeared topredict unrealistic oscillating results. Srithar (1989) incorporated the temperatureinduced stresses and volume changes directly in the governing equilibrium and flowcontinuity equations and presented a better formulation of Vaziri’s model.Dusseault and Rothenberg (1988) reviewed the effect of thermal loading and porepressure changes around a wellbore on dilation and permeability. They describedthe physical process of deformation in terms of particulate media. They concludedthat effective water permeability would increase one or two orders of magnitude withdilation as the thickness of the water film coating the grains would increase by afactor 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 ofa 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-offrates for fracture faces in oil sand. The authors used a nonlinear elastic model and atwo-phase isothermal flow in their analysis. The nonlinear response was shown to givea different pressure distribution than the linear elastic one. Settari (1989) extendedtheir earlier model to thermal flow.Fung (1990) described a control volume finite element approach for coupled isothermal two-phase fluid flow and solid behaviour. He adopted a hyperbolic stress-strainlaw with Rowe’s stress dilatancy theory.Chapter 2. Review of Literature 30Schrefler and Simoni (1991) presented the equations for two-phase flow in a deforming porous medium, which are, a linear momentum balance for the whole multiphase system and continuity equations for solid-water and solid-gas systems. Auxiliary equations included water saturation constraint (S + S9 = 1), and the effective stress equation. Three combinations of solution variables were considered ({ U, F,(,, P}, {U, P,P9}, {U, P, S}). Among these the best convergence was foundwhen using the combination of { U, P, P9}.Tortike (1991) attempted to develop a fully coupled three dimensional formulationfor thermal three-phase fluid flow with geomechanical behaviour of oil sand. He wasnot 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 finiteelements and thermal fluid flow by finite difference and combined the results. Hefound the second approach to be successful and useful.Recently Settari et al. (1993) presented a model to study the geomechanicalresponse of oil sand to fluid injection and to analyze the formation parting in oilsand. They used a generalized form of the hyperbolic model for material behaviour.They also approximated the multiphase fluid flow by means of an effective hydraulicconductivity term. The value of the effective hydraulic conductivity term was foundby matching the results of the single phase model with the rigorous multiphase flowmodel. The authors further examined the behaviour of the constitutive model at loweffective stress ranges and concluded that the frictional properties at low effectivestresses control the development of the failure zone around the injection well and thefractures.2.4 CommentsThe following are some of the important facts that can be extracted from the literaturereview. In the models reviewed, except for Tortike (1991), all other models use elasticChapter 2. Review of Literature 31models. Cyclic loads are more common in the oil recovery procedures such as thecyclic steam simulation. The cyclic loading unloading behaviour cannot be modelledby elastic models. Dilative behaviour is an important feature in oil sands. Modellingof dilation through a thermoelastic approach is inefficient and may lead to unrealisticoscillating results. Temperature effects and the multiphase nature of the pore fluid arevery important aspects to be considered in an analytical model. The multiphase flowmodels with poroelasticity used in petroleum reservoir engineering do not considerthe effect of stress distribution through the porous medium.Chapter 3Stress-Strain Model Employed3.1 IntroductionIn developing a procedure to analyze the geotechnical aspects of oil sands, appropriate 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 ofthe behaviour of pore fluid and modeffing of the behaviour of the sand skeleton. Inthis chapter, modelling of sand skeleton behaviour is described in detail. Modellingof 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 exhibitssignificant shear induced volume expansion or dilation. The dilation in the sandskeleton will increase the pore space and hence increase the permeability and reducethe pore pressure. These changes will have significant effect in the overall deformationand flow predictions. Therefore, realistic modeffing of dilation is important.Generally, oil recovery methods are cyclic processes which will cause the sandskeleton to undergo loading and unloading sequences resulting in irrecoverable plasticstrains. This necessitates the use of an elasto-plastic stress-strain model. There are anumber 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 forthe stress-strain model employed in this study for the following reasons.1. The failure criterion is based on stress ratio rather than shear stress. Thiswould realistically model the behaviour when the soil undergoes a decrease in32Chapter 3. Stress-Strain Model Employed 33mean normal stress with constant shear stress (see figure 3.1) which is a possiblescenario in oil recovery process with steam injection.2. It is based on microscopic analysis of the behaviour of sand grains and not bycurve fitting.3. It considers the effect of the intermediate principal stress.4. It appeared to predict the experimental data best based on the proceedings ofthe Cleveland workshop on constitutive equations for granular materials (Salgado, 1990). A modified version of this model has been extensively used in theUniversity of British Columbia (Salgado (1990), Salgado and Byrne (1991)) andgave very good predictions.The stress-strain model employed in this study is an improved version of the modelused by Salgado (1990). Improvements to Salgado’s model have been made in threeaspects.1. Changes proposed by Nakai and Matsuoka (1983) regarding the strain incrementdirections are implemented.2. A cap type yield criterion is added to model the constant stress ratio typeloadings accurately.3. Modelling of strain softening is added.A detailed description of the stress-strain model, development of the constitutivematrix in a general three dimensional Cartesian coordinate system, its implementationin three dimensional, two dimensional plane strain and axisymmetric conditions arepresented in this chapter. It should be noted that effective stress parameters areimplied throughout this chapter and the prime symbols are omitted for clarity.Chapter 3. Stress-Strain Model Employed 34Cl)Cl)2Failure Envelope(Increasing Steam Injection Pressure)Normal StressFigure 3.1: A Possible Stress Path During Steam InjectionChapter 3. Stress-Strain Model Employed 353.2 Description of the ModelGenerally the total strain increment, de of a soil element can be expressed as a summation of an elastic component, dee and a plastic component, den. In the stress-strainmodel developed in this study, the plastic component is further divided into twoparts; a plastic shear component, de8 (the strain increments caused by the increase instress ratio) and a plastic volumetric or collapse component, dcc (the strain incrementcaused by the increase in mean principal stress). Figure 3.2 schematically illustratesthese elastic, plastic shear and plastic collapse components of the total strain in atypical triaxial compression test.Mathematically, the total strain de can be expressed as,de = dc9 + dcc H- dee (3.1)These different strain components can be calculated separately; the plastic shearstrains by plastic stress-strain theory involving a conical type yield surface, the plasticcollapse strains by plastic stress-strain theory involving a cap type yield surface andthe elastic strains by Hooke’s law.From the stress-strain theories, the strain components can be written as{de8} = [C8] {do}{de} [Ce] {th}{dc6} = [Ce] {d} (3.2)where [C8], [Cc] and [Ce] are the constitutive matrices corresponding to plastic shear,plastic collapse and elastic strains. Combining equations 3.1 and 3.2 a stress-strainrelation for the total strain can be obtained as follows:{de} = [[C8] H- [CC] + [CC]] {do}Chapter 3. Stress-Strain Model Employed 36IcizwwUUC,,C,,UzIC?,ciI-U-J0>Figure 3.2: Components of Strain IncrementChapter 3. Stress-Strain Model Employed 37= [C] {do} (3.3)The theories involved in developing the [C8], [Cc] and [Ce] matrices in generalCartesian coordinate system are explained in the next sections and at the end, thefull elasto-plastic constitutive matrix [C] is formed according to different loadingconditions.In developing a finite element formulation, the stress-strain relation is generallyexpressed asdo = [D] dE (3.4)The above equation is an inverse of equation 3.3. Once the [C] matrix is known, the[D] matrix can be easily obtained as the inverse of [C].3.3 Plastic Shear Strain by Cone-Type Yielding3.3.1 Background of the ModelThe stress-strain relationship for the plastic shear strain is developed based on thecSpatial Mobilized Plane’ concept by Nakai and Matsuoka (1983). Before going intothe three dimensional conditions, a brief description of the concept of mobilized planein two dimensional conditions is given to provide a better insight.The concept of mobilized plane was first developed by Murayama (1964). Theterm ‘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 tooccur. The 2-D representation of this plane is shown in figure 3.3 (a). This planemakes an angle of (45° + m/2) to the major principal stress plane, where q is themobilized friction angle. The Mohr circle for the stress conditions and the mobilizedfriction angle are shown in figure 3.3 (b).Chapter 3. Stress-Strain Model EmployedC,,(I,bJcC’,TMbJC,,Q32-D MobilizedPlane(a)(b)38QNORMAL STRESSFigure 3.3: Mobilized Plane under 2-D ConditionsChapter 3. Stress-Strain Model Employed 39From a large number of tests and from the analysis of the shear mechanism ofgranular material in a microscopic point of view, Murayama and Matsuoka (1973)proposed a relationship between the shear-normal stress ratio (TMp /crMP) and thenormal-shear strain increment ratio (dMp/d7Mp) on the mobilized plane as,rp (_d6MP+ (3.5)MP \ d-yf )where ) and i are constant soil parameters. Equation 3.5 forms the basis for thedevelopments of the constitutive models later by Matsuoka and his co-workers.Under general three dimensional conditions, the stress state of a soil element canbe characterized by the three principal stresses o, 02 and o. Mohr circles for thesethree stresses can be drawn as shown in figure 3.4 (a) and three mobilized frictionangles, ml2423 and ç3 can be obtained. These mobilized friction angles can beexpressed by the following equation:tan(450+) Z (i,j=1,2,3;ucT) (3.6)Using these mobilized friction angles, a 3-D plane ABC can be constructed asshown in figure 3.4 (b). This plane ABC is considered to be the plane where thesoil particles are most mobilized and is called the ‘Spatial Mobilized Plane (SMP)’.Under isotropic stress condition (o = = 03) the mobilized plane will coincide withthe octahedral plane and will vary with possible changes in stresses. The directioncosines 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 expressed by the following equations in terms of principal stresses or the stresses in thegeneral coordinate system.Chapter 3. Stress-Strain Model Employed 40r 13m12o•1(a)1Ia;cI-f-———---———--y.— SpatialMobilizedV’ Plane6 O3—- B7m23450+2A7’ L5•+2(b)Figure 3.4: Spatial Mobilized Plane under 3-D ConditionsChapter 3. Stress-Strain Model Employed 41‘1 = O1+023 =12 = 12 + 0203 + O301 = 0xy + 0yz + 0z — — T2—13 = °y°z +2TTyzTz — OT — — OzTy (3.8)The general stress-strain relationship will be developed basically from the relationship of the stresses on the SMP and the strain components to the SMP. Thenormal stress (oSMP) and the shear stress (TSMp) on the SMP can be obtained fromthe following equations:SMP = o1a + o2a + o3a = 3 (3.9)and\/111213 — 9ITSMp = /(oi —o2)a?a + (o2 —o3)2aa + (o — O1)21= ‘2(3.10)The shear-normal stress ratio, i can be expressed as= TSMp = I1I2 —913 (3.11)SMP 913By assuming that the direction of the principal stresses and the direction of theprincipal strain increments are identical, which is the common assumption in plasticity, the normal and the parallel components of the principal strain increment vectorto the SMP (dcsMp and d7sMp) are given bydEsMp = dea1 + dea2 + dEa3 (3.12)andd7sMp = i,J(dEa2 — d€a1)2+ (deas — dca2)-- (d€ai — d€1a3)2 (3.13)Chapter 3. Stress-Strain Model Employed 42It should be noted that before Nakai and Matsuoka (1983), Matsuoka used thenormal and shear strain increments on the SMP rather than components of the principal strain increments to the SMP. After a thorough investigation of the theoriesinvolved, Nakai and Matsuoka (1983) concluded that the average sliding direction ofthe soil particles coincides with the direction of the principal strain increment vectorand not with the direction of the strain increment vector on the SMP. They denotedtheir earlier model as SMP (Matsuoka and Nakai, 1974, 1977) and the new model asSMP’. The concepts used in this study follow the SMP model.In the theory of plasticity, the stress-strain relation is formulated from a yieldfunction, a plastic potential function (or a flow rule) and a strain hardening function.The model developed by Matsuoka does not explicitly define these functions. However, those can be formulated and the constitutive matrix can be derived easily asexplained in the next subsections.3.3.2 Yield and Failure CriteriaThe yield criterion defines the boundary between the elastic and plastic zones. Afamily of yield surfaces in the TSMp — 0SMP space is shown in figure 3.5. These yieldsurfaces are given by the following equation:77—3\/tanmi2+ tan m23 + tanqm13 = k (3.14)where i TsMp /0sMP, q5m are the mobilized friction angles and k is a constant.The ‘current’ yield surface corresponding to the stress state at a point in a mass ofsoil is defined by the maximum stress ratio mobilized at that point during its historyof loading. For instance, assume the current yield surface is represented by line A andthe stress state of the point is represented by P (see figure 3.5), the shaded area will bethe 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 elasticChapter 3. Stress-Strain Model Employed 43Failure SurfaceBYield SurfacesAP...ElastIc Region°SMPFigure 3.5: Yield and Failure Criteria on TsMp— 05MP SpaceChapter 3. Stress-Strain Model Employed 44strains will occur and it represents an unloading condition. If the stress state movesto FL which is outside the elastic region, there will be elastic and plastic strains. Theyield surface will be dragged along to a new yield surface represented by line B andthe 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 whichis given by the following equation:tan f12 + tan f23 + tan f13 = kf (3.15)where is the failure stress ratio and are the failure friction angles. Salgado(1990) claims that the failure stress ratio is dependent on the normal stress on theSMP at failure, and that a better agreement with the laboratory data will be obtainedif the failure stress ratio is expressed by the following equation:(asMP)f=— log10 (3.16)where- failure stress ratio at (osMp ) = 1 atmosphere- decrement in failure stress ratio for 10 fold increase in (oSMp )The failure surface on the octahedral plane and in the 3-D space is shown infigure 3.6. The Mohr-Coulomb failure surface is also shown in the figure and it canbe seen that the Mohr-Coulomb and Matsuoka-Nakai failure surfaces coincide for thetriaxial conditions (compression and extension) but differ for any other stress path.The Matsuoka-Nakai failure criterion considers the effect of the intermediate principalstress. This effect is shown as the difference between the failure friction angles forMatsuoka-Nakai and Mohr-Coloumb criteria with b-value in figure 3.7. The triaxialcompression condition will correspond to b-value = 0 and triaxial extension conditionwill correspond to b-value = 1.Chapter 3. Stress-Strain Model Employed 4501MOHR-COULOMB\MATSUOKA- NAKAJ(a) Octahedral Plane01/1II\ #\/L\’ “\/ A1/ \%(II ,C/p7C0(b) 3-Dimensional Stress SpaceFigure 3.6: Matsuoka-Nakai and Mohr-Coulomb Failure CriteriaChapter 3. Stress-Strain Model Employed 468-7-6-TX5..7400-a- 3Q0:4- .-a-20° E2I0o1- .00 0.2 0.4 0.6 0.8b-VALUEçb is the failure friction angle in triaxial conditionsis the failure friction angle in Matsuoka-Nakai failure criterionFigure 3.7: Effect of Intermediate Principal Stress (After Salgado (1990))Chapter 3. Stress-Strain Model Employed 47Figure 3.8: (TSMp/oSMp) Vs —(dEsMp/d7sMp) for Toyoura Sand (after Matsuoka,1983)At a particular stress state, the ratio of the normal strain to the shear strain tothe SMP (dEsMp /d7SMp) is given by the following equation:3.3.3 Flow RuleThe flow rule defines the direction of the plastic strain increments at every stressstate. Matsuoka’s model does not explicitly give a plastic potential function definingthe direction of plastic strain increment. Instead, a relationship for the amount ofplastic strain increment components is given, and in fact, this relationship will givethe direction of the plastic strain increment vector. An example of this relationshipobtained from triaxial compression and extension tests for Toyoura sand is shown infigure 3.8 which is essentially a straight line. This straight line relationship holds forall densities.1.002be”a-2080.60.40.2-0.4 -0.2 0 0.2 0.4- ESMp “YSMP0.6Chapter 3. Stress-Strain Model Employed 48[—dESMP’\?7= i , 1+11 (3.17)\a7sMpJwhere A and t are soil parameters and is the stress ratio on the SMP.Rewriting the above equation yields,d6sMp (3.18)d7sMp AEquation 3.18 implies that the plastic strain increment vector will not be perpendicular 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 volumetricstrain 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 figure3.9(b) shows the corresponding results as desMp versus d7sMp.3.3.4 Hardening RuleThe hardening rule defines how the threshold of yielding changes with plastic strain, orin other words how the yield stress state changes with plastic strain. In Matsuoka’smodel, the plastic shear strain to the SMP (7sMp) is considered as the hardener.Therefore, a relationship between i which defines the stress state and the plasticshear strain to the SMP, 7sMP, will form the hardening rule. Matsuoka defines thehardening rule by an empirical equation as follows:7SMP = 7o exp (, — (3.19)\P’ /.‘Jwhere i and i’ are constant soil parameters. The parameter Yo is assumed to be afunction of mean principal stress (crm) and expressed as follows:7o -yo + Cd log10 (--) (3.20)°miChapter 3. Stress-Strain Model Employed 49(a)dEsMpd7SM PDilation71>11Contraction“<It(b)TiDilationContractionA1(dSMp\dy5i,jpFigure 3.9: Flow Rule and The Strain Increments for Conical YieldChapter 3. Stress-Strain Model Employed 50where Cd is a constant, omj is the initial mean principal stress and yoi is the value of7o at 0m = 0mi An example of the hardening rule is shown in figure 3.10, which isobtained from triaxial compression and extension tests on Toyoura sand (Matsuoka,1983).1.0392 kN/mo comp.• ext.•2.0 3.0 4.0Figure 3.10: rsMp/OsMp Vs YsMP for Toyoura Sand (after Matsuoka, 1983)However, the equation 3.19 given by Matsuoka is not used in this study. Instead,the relationship proposed by Salgado (1990) is used because, the parameters in hisrelationship are more meaningful and it is easier to implement in an incremental finiteelement procedure. Salgado (1990) defines the hardening rule using the hyperbolicnature of the relationship and following the procedure by Konder (1963) as7SMP (3.21)+7SMPG,. 1luUwhereChapter 3. Stress-Strain Model Employed 51G,, - initial slope of the i— 7sMP curve- stress ratio (TSMp/JSMp)‘Tlult - asymptotic value of the stress ratioBy differentiating equation 3.21, the plastic shear strain increment /7sMP can beobtained as,dy5Mp = d (3.22)where is the dimensionless tangent plastic shear parameter. This parameter isdependent on both normal stress on SMP (crsMP) and the stress ratio. can beevaluated by a similar procedure as given by Duncan et al. (1980) as follows:= G(1 — Rf __)2 (3.23)1iand= KG (osMP) (3.24)where- plastic shear numbernp - plastic shear exponentPa - atmospheric pressure- stress ratioR1 - failure ratio (7f/ij,zt)3.3.5 Development of Constitutive Matrix [CS]The development of plastic shear constitutive matrix in terms of general Cartesianstress and strain components from the yield criterion, hardening rule and the flowrule 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 52dysMp = —di1 (3.25)IL—?’desMp d-y.9 (3.26)Substituting equation 3.25 in equation 3.26 will give,dEsMp =—(IL ?‘) d (3.27)By assuming that the directions of the principal stresses and the directions of theprincipal strain increments are the same, the direction cosines of desMp are given bya=(i = 1,2,3) (3.28)If it assumed that the direction of d7sMp and the direction of TsMp coincide, thenthe direction cosines of d7sMp are given by— SMP 0jI2 — 3131,: = = (3.29)TSMp /o- ‘2 (I 12 — 913)where 11,12 and 13 are stress invariants as given by equation 3.8. The plastic principalstrain increments due to shear can be obtained from the following equation.de = a desMp H- b d7SMP i = 1,2,3 (3.30)By substituting equation 3.25 and equation 3.27 into equation 3.30,dE=+.i)d?’ (3.31)Equation 3.31 can be written in matrix notation as{defl = {M12} d?’ (3.32)Chapter 3. Stress-Strain Model Employed 53where M1=+The general Cartesian strain increments can be obtained by multiplying the principal strain increment vector by the transformation matrix, as given by the followingmatrix equation:dE l m nde8 12 m2V d€8d8 12 m2 n2z z z z (3.33)d7 2l7,l, 2mm 2nndc2l,l 2mm 2nn2l1 2mm3wherel, l,, and l - direction cosines of o to the x, y and z axesm, m and m - direction cosines of 02 to the x, y and z axes, and n- direction cosines of 03 to the x, y and z axesEquation 3.33 can be written in matrix form as{de8} = [MT] {dc} (3.34)Substitution of equation 3.32 into equation 3.34 yields{de8} = [MT] {M1} di1 (3.35)From equation 3.11 the stress ratio on the SMP, is given by/1112 — 913=91 (3.36)Chapter 3. Stress-Strain Model Employed 54By considering the invariants in terms of Cartesian stresses (equation 3.8) anddifferentiating equation 3.36 with respect to Cartesian stresses the following equationcan be obtained for di7:,I 77 Id={do}T‘213 + 1113 (o, + o) — 1112 (o,o — r) do1213 + 1113 (o + o) — ‘112 (o °•r — T) doy— 1 1213 + I113(0 + o)—IiI2(t717y — r2) do-i— 18iiI dr—2IlI3r— 2IlI2(rr — dr—2IlI3T— 2IiI(rr — or)= {M2}Tdo (3.37)where superscript T denote the transpose of the matrix.Substituting equation 3.37 in equation 3.34 gives{d68} = [MT] {M1} {M2}T {do} (3.38)This can be further written as{d68} = [C8] {dcr} (3.39)where [C8] is the plastic shear constitutive matrix and will be given by[C8] = [MT] {M1} {M2}T (3.40)Chapter 3. Stress-Strain Model Employed 553.4 Plastic Collapse Strain by Cap-Type Yielding3.4.1 Background of the ModelThe plastic stress-strain theory with the conical yield surfaces described in the previous section is not capable of predicting the behaviour of soil under proportionalloading. 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 increasingstresses causes some plastic deformation.An additional yield surface which forms a cap on the earlier conical yield surface isconsidered to circumvent this deficiency as explained in this section. The stress-strainrelationship for predicting the plastic collapse strains was developed by following theconcepts of the cap-type yielding given by Lade (1977).As explained in section 3.2, it is reasonable to assume that the plastic collapsestrains are produced by the increase in mean normal stress and the plastic shearstrains will be associated with the shear stresses. However, under general loading conditions, it is difficult to separate the plastic shear and plastic collapse strains becauseboth will occur simultaneously. Therefore, the development of the cap-type yieldmodel is based on the isotropic compression tests where no plastic shear strains areproduced. Figure 3.11 shows the typical results for loading, unloading and reloadingconditions in an isotropic compression test. The elastic strains which are recoverablecan be calculated using Hooke’s law are also shown in figure 3.11. Then, the collapsestrains 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 acap at the open end of the conical yield surface is used. The yield criterion and thehardening functions for the cap-type yield are explained in the following subsections.The stress-strain relation for the plastic collapse strain is formulated following theChapter 3. Stress-Strain Model EmployedEC‘1,wIC,,00C’)56Figure 3.11: Isotropic Compression Test on Loose Sacramento River Sand (after Lade,1977)VOLUMETRIC STRAIN, eq,, (‘‘Chapter 3. Stress-Strain Model Employed 57general theory of plasticity.3.4.2 Yield CriterionThe yield criterion which defines the onset of plastic collapse strain is given byf = — 212 (3.41)where I and 12 are the first and second stress invariants as given in equation 3.8.The yield criterion which is defined by equation 3.41 represents a sphere with centreat the origin of the principal stress space which forms a cap at the open end of theconical yield surface. Figure 3.12 shows the conical and the cap yield surfaces in01 Hydrostatic AxisConical Yield SurfacePlastic Collapse StrainIncrement /ectorSpherical Yield CapIasti RegiorConical Yield Surface03Figure 3.12: Conical and Cap Yield Surfaces on the o—03 PlaneChapter 3. Stress-Strain Model Employed 58the o-1 — 03 plane. The elastic region at any particular stress state will be boundedby these two yield surfaces. As f increases beyond its current value, the yield capexpands, soil work hardens and collapse strains are produced. It should be noted thatthere are no bounds on the cap yield surface and yielding according to equation 3.41does not result in eventual failure. The failure is entirely controlled by the conicalyield surface.3.4.3 Flow RuleUnder isotropic compression, an isotropic soil shows equal strains in all three principaldirections. Therefore, the direction of strain increment vector should coincide withthe hydrostatic axis pointing outwards from the origin (see figure 3.12). To satisfythis conditioi-i the plastic potential function must be identical to the yield function.This implies the flow rule is associative and will be given by the following equation:de = (3.42)8o.ijwhere is the proportionality constant which gives the magnitude of the plasticcollapse strain and can be determined from the hardening rule.3.4.4 Hardening RuleThe hardening rule gives a relationship between the yield function and the plasticstrain, defining how the yield function changes with plastic strain. For the cap yieldmodel, Lade (1977) developed an empirical relationship between the plastic collapsework (We) and the yield function. The plastic collapse work is a function of plasticcollapse strains and given by= J {}T {dE} (3.43)Chapter 3. Stress-Strain Model Employed 59The relationship between the plastic collapse work and the yield function is givenby= CPa()P(3.44)where C and p are dimensionless constants and called the collapse modulus and thecollapse exponent respectively.The proportionality constant LSX which gives the magnitude of the plastic collapsestrain increment can be obtained as follows. The increment in plastic collapse workcan be expressed asdW = {}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 shownthat= 2f (3.47)From equations 3.46 and 3.47, can be given as= dWC (3.48)3.4.5 Development of Constitutive Matrix [CC]The constitutive matrix relating the plastic collapse strains and the stress incrementscan be developed as described below. Substitution of equation 3.48 in equation 3.42givesChapter 3. Stress-Strain Model Employed 60c dW af (. )Jc O3By differentiating equation 3.43, dW can be obtained as= C p a()121d (3.50)and it can be further written asdW = A df (3.51)where A = (f)P_1df will be obtained by differentiating 3.41 as,df =T2o do2o do2o do= (3.52)4r dr4r dr4Tz drBy combining equations 3.49, 3.51 and 3.52 the following equation can be obtained:A 8f 8fde =—dokj (3.53)2f 8klIn terms of Cartesian components of stress and strain the above equation can bewritten asChapter 3. Stress-Strain Model Employed 61oo- 2or 2OTzm dod 2or 2or2, dodE=o 2o-r 2o-r 2ur do-id79 f 4r dTd-y Symmetry 4r2 4r2r drd7 4r2 dr3,In short matrix notation the constitutive matrix for the plastic collapse strain can bewritten as{Cc]= {8fc}{afc}T3.5 Elastic Strains by Hooke’s LawThe elastic strains which are recoverable upon unloading can be evaluated usingHooke’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 asde 1 —v —v 0 0 0 dode 1—v 0 0 0 do,d 1 1 0 0 0 do2 (3.57)d- 2(1H-v) 0 0 drd72 Symmetry 2(1 + v) 0 dr2d 2(1 + v) dr2Chapter 3. Stress-Strain Model Employed 62where E is the tangential Young’s modulus obtained from the unload-reload portionof a stress-strain curve. i-’ is the Poison ratio which can be calculated from Young’sand bulk moduli asv= (i_&) (3.58)E and B are assumed to be stress dependent and given by the following equations:E = kE Pa()fl(3.59)B = Pa () (3.60)where,kE - Young’s modulus number- bulk modulus numbern - Young’s modulus exponentn - bulk modulus exponent3.6 Development of Full Elasto-Plastic Constitutive MatrixIn the previous sections, the constitutive matrix is formed individually for differentcomponents of strain. One of the major advantages of having the strain componentsseparated is that it is easy to model the different loading conditions. Depending onthe loading condition, the relevant strain components can be included and the corresponding full elasto-plastic constitutive matrix can be formed. The loading conditionscan be classified into four cases which are shown in figure 3.13 on the i — o plane.Case ICase I indicates a loading condition where there is an increase in stress ratio aswell as in mean stress. In this case, all three; the plastic shear, plastic collapse andChapter 3. Stress-Strain Model Employed 63a1 Failure SurfaceIlllastc’Rag ion• 7-.7.Hydrostatic AxisConical Yield SurfaceFailure Surfacea3Figure 3.13: Possible Loading ConditionsChapter 3. Stress-Strain Model Employed 64elastic strains will be present. Then, the full elasto-plastic constitutive matrix will begiven by[C] [[C8] + [CC] + [Ce]] (3.61)Case IIThis case considers a loading condition where there is an increase in stress ratioand a decrease in mean stress. Here, only plastic shear and elastic strains will occur.The full constitutive matrix will comprise those two matrices only, i.e.,[C] = [[C8] + [CC]] (3.62)Case IIICase III considers the loading conditions where there is a decrease in stress ratioand an increase in mean stress. In this case, plastic collapse and elastic strains willoccur and the corresponding full constitutive matrix will be[C] = [[CC] + [CC]] (3.63)Case IVCase IV indicates a complete unloading condition where there will be decrease inboth stress ratio and mean stress. Under these conditions, only elastic strains will berecovered. Therefore, the full elasto-plastic constitutive matrix will be the same asthe constitutive matrix for the elastic strains, i.e.,[C] = [CC] (3.64)Chapter 3. Stress-Strain Model Employed 653.7 2-Dimensional Formulation of Constitutive MatrixGenerally 2-dimensional plane strain and axisymmetric analyses are more often carried out than 3-dimensional analyses because 3-D analysis require tedious work togenerate the relevant input data and more computer time for execution. The constitutive matrix for 2-D plane strain and axisymmetric conditions can be obtained easilyby imposing the appropriate boundary conditions on the 3-D constitutive matrix. Ageneral stress-strain relation under 3-d conditions can be given aswhere C3 arePlane StrainAssume that the horizontal and vertical axes in the 2-D conditions are defined byx and y. Then, all the terms associated with yz and zx and r) willhave no effect in the 2-D plane strain analysis. Hence, equation 3.65 can be reducedtoC11 C12 C13— C21 C22 C23C31 C32 C33C41 C42 C43plane strain boundarydc C11 C12 C13 C14C21 C22 C23 C24= C31 C32 C33 C34C41 C42 C43 C44d’y2 C51 C52 C53 C54C61 C62 C63 C64the components of the constitutiveC15 C16C25 C26C35 C36C45 C46C55 C56C65 C66matrix.drdododr2drza,(3.65)dedcd6d7Now, by imposing theC14 doC24 daC34 doC44 drcondition that = 0, do(3.66)can beChapter 3. Stress-Strain Model Employed 66written asdo2 = — + do-!, + dT) (3.67)Substitution of equation 3.67 in equation 3.66 yields:de C1 C’2 C’3 dode = C;1 C;2 C;3 do-u (3.68)d C;1 C;2 C3 dwhereri — (V C1331 — r_____.C1334‘-‘11 — ‘-‘11—LI12—L112— C33 ‘ ‘—‘13 LI4—_______—— C231 .—(1 C,C32 .— ,-C23‘—‘21 — ‘-‘21 C33 ‘ ‘-‘22 — ‘-‘22— C33 ‘ ‘-‘23 — ‘—‘24— c,3f_I,,— f_I C4331.— f_I C4332 . f_I—f_I C4334— ‘-‘41— C33 ‘ ‘—‘32 — L142— c33 ‘ ‘-‘33— C33In the above 2-D formulation, the 3-D characteristics will not be lost and theeffect of the intermediate principal stress is still considered. The intermediate stresscan be obtained using equation 3.67.AxisymmetricIn case of axisymmetric conditions, the modifications are much simpler. Supposethe 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 willnot have any influence and hence, equation 3.65 can be reduced todEr C11 C12 C13 C64 do.de8 C21 C22 C23 C64 do-8(3.69)de2 C31 C32 C33 C64 do-iC61 C62 C63 C64 drrzChapter 3. Stress-Strain Model Employed 673.8 Inclusion of Temperature EffectsThe effects of temperature changes in oil sand and the works by previous researchersto include these effects in the analytical procedures were described in chapter 2. Theapproach used by Srithar and Byrne (1991) is followed here. This involves additionalterms in the stress-strain relation and in the flow-continuity equation. The changeswhich 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 section5.8.The incremental stress-strain relation can be written as{de} = [C]{do} (3.70)where [C] is the elasto-plastic constitutive matrix. If there is an increase in thetemperature, the sand matrix will expand and there will be additional strains. Then,equation 3.70 will become{d} = [C]{do} — {de8} (3.71)where {dee}T {a d, a8 d6, a8 d6, 0, 0, 0} and a8 is the linear thermal expansioncoefficient of the sand grains and d6 is the change in temperature. It should be notedthat compressive strains are assumed positive.By multiplying equation 3.71 by the inverse of [C] which is referred to as thestress-strain matrix [D] the following equation can be obtained:[D]{dc}={dcr} — [D]{de8} (3.72)Rearranging the terms will give{do} = [D]{dE} + {do8} (3.73)Chapter 3. Stress-Strain Model Employed 68where {do-8} = [D] {de8}, which is the additional term in the stress-strain relationdue to change in temperature. This term will give the induced thermal stresses.3.9 Modelling of Strain Softening by Load SheddingLaboratory tests on oil sand show a decrease in strength after a peak strength isreached which is commonly referred as strain softening. The phenomenon of strainsoftening or loss of strength under progressive straining occurs because of the structural changes in the material such as initiation, propagation and closure of microcracks. Frantziskonis and Desai (1987) stated that strain softening is not a materialproperty of soil when it is treated as a continuum. It is rather a performance ofthe structure composed of micro-cracks and joints that result in an overall loss ofstrength. When the stresses and strains deviate from homogeneity, the behaviour ofa material will no longer be represented by continuum material properties. If strainsoftening is assumed as a true material property, various anomalies may arise withrespect to the solution of boundary and initial value problems. These anomalies canlead to loss of uniqueness in the strain softening part of the stress-strain response andto numerical instabilities as shown by Valanis (1985).A comprehensive review of strain softening is not attempted here as it is beyondthe 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 phenomenonis modelled quantitatively using the ‘load shedding’ or ‘stress transfer’ concept. Inprinciple the load shedding concept is similar to the model presented by Frantziskonisand Desai (1987). They modelled the strain softening behaviour by separating it intotwo parts; a non-softening behaviour of a continuum (topical behaviour) and a damageor stress relieved behaviour with zero stiffness. The true behaviour is estimated asan average of these two (see figure 3.14). In finding the average behaviour, thehydrostatic component is assumed to be the same for both parts and the deviatoricChapter 3. Stress-Strain Model Employed 69ShearStressUltimateTopical Behaviour— Average BehaviourStrainFigure 3.14: Modelling of Strain Softening by Frantziskonis and Desai (1987)Chapter 3. Stress-Strain Model Employed 70stress 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 fromthe 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 givenby Frantziskonis and Desai (1987) for their damage evolution. Thus, in the strainsoftening region the stress-strain relation can be given as= i + (ii, — ‘qr)exp{—k(ysMp—(3.74)where- Residual stress ratio- Peak stress ratio7SMP,p - Peak shear strainIc, q - Constant parameters3.9.1 Load Shedding TechniqueLoad shedding (Zienkiewicz et al. (1968), Byrne and Janzen (1984)) is a techniqueto correct the stress state of an element which has violated the failure criterion, bytaking out the overstress and redistributing to the adjacent unfailed elements. A briefdescription of how the load shedding technique is applied to model strain softening ispresented below. Details of the estimation of overstress and the corresponding loadvector are given in appendix A.Figure 3.15 shows a typical scenario in modelling strain softening by load shedding.The stress state of an element depicted by point P0 in the figure can move to pointChapter 3. Stress-Strain Model Employed‘1]‘ripFigure 3.15: Modelling of Strain Softening by Load Shedding71P1P2F? TI7r71 7Chapter 3. Stress-Strain Model Employed 72P1 in a load increment. But the actual stress state should be point Pia and in orderto bring to this stress state, an overstress of should be removed. The overstresswill then be redistributed to the adjacent stiffer elements. During the redistributionprocess, the modulus of the failed element will be defaulted to a low value so that itwill not take any more load. However, in another load increment the stress state maymove to point P2. Then again the stress state will be brought to point F2a by loadshedding. In the process of load shedding, it is also possible that some other elementsviolate failure criteria and those loads also have to be redistributed. Therefore, severaliterations may be needed to find a solution where failure criteria are satisfied by allthe elements.3.10 DiscussionAlthough the stress-strain model employed in this study is somewhat sophisticated, itwill 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 thedeformations for pure principal stress rotations.In the stress-strain model used in this study, the elastic principal strain incrementdirections are assumed to coincide with the principal stress increment directions andthe plastic principal strain increment directions are assumed to coincide with the principal stress directions. Lade (1977) also stated that the principal strain incrementdirections coincide with the principal stress increment directions at low stress levelswhere elastic strains are predominant and coincide with principal stress directions athigh stress levels where plastic strains are predominant. Salgado (1990) presented acritical review regarding the assumption that the direction of principal strain increments coincide with the direction of principal stresses. He reviewed the results usingthe hollow cylinder device by Symes et al. (1982, 1984, 1988) and Sayao (1989) andconcluded that the assumption is reasonably valid for most of the stress paths exceptChapter 3. Stress-Strain Model Employed 73those that involve significant principal stress rotations.One of the disadvantages of this model is its limited use in the past. Unlike thehyperbolic model, information on the model parameters is very limited. The possiblerange of values for some of the parameters and their physical significance are notwell defined. However, a sensitivity study on the parameters is given in chapter4, 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, itwill result in a non-symmetric stiffness matrix which requires considerable computermemory and time. However, the frontal solution scheme used in this study willcircumvent the requirement for large memory since it does not assemble the fullstiffness matrix and requires only a small memory. Furthermore, these factors of timeand memory requirements may not be considered as disadvantages with the rapidgrowth in computer capabilities.Chapter 4Stress-Strain Model - Parameter Evaluation andValidation4.1 IntroductionThis chapter describes the procedures used to evaluate the soil parameters needed forthe stress-strain model and presents results verifying the stress-strain model againstmeasured responses in laboratory tests. The soil parameters required for the modelcan be classified into four groups; elastic, plastic shear, plastic collapse and strainsoftening. 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 suchas 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 necessaryto obtain a straight line fit. In those cases, it is advisable to have three or more testresults to obtain a better fit. Validations of the stress-strain model against laboratoryresults on Ottawa sand and on oil sand are given in section 4.3. Sensitivity analyseson some of the parameters have been carried out to provide some idea about theirsignificance and these are described in section 4.4.4.2 Evaluation of ParametersIn this section, only the procedures for the evaluation of the parameters are given indetail. Applications of these procedures to actual test data on Ottawa sand and onoil sand can be found in section 4.3.74Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 75Table 4.1: Summary of Soil ParametersL Type Parameter DescriptionElastic kE Young’s modulus numbern Young’s modulus exponentkB Bulk modulus numberm Bulk modulus exponentPlastic Shear Failure stress ratio at one atmosphereLi Decrease in failure stress ratiofor 10 fold increase in 0SMP.\ Flow rule slopei Flow rule interceptKG Plastic shear numbernp Plastic shear exponentR1 Failure ratioPlastic Collapse C Collapse modulus numberp Collapse modulus exponentStrain Softening Strain softening constantq Strain softening exponent4.2.1 Elastic Parameters4.2.1.1 Parameters kE and nThe elastic parameters kE and n can be determined from the unload-reload portion ofa triaxial compression test as explained by Duncan et al. (1980). To determine theseparameters, at least two unload-reload modulus values (see figure 4.1(a)) at differentmean normal stresses are necessary. The unload-reload Young’s modulus is given byE kE Pa ()‘ (4.1)By rearranging and taking the logarithm, the above equation can be written aslog (-) = log kE + n log (i) (4.2)Thus, kE and n can be determined by plotting (E/Pa) against (0m/1Zba) on a log-logChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 76plot as shown in figure 4.1(b).In the standard triaxial compression test, the unload-reload stress path is often notperformed. In the absence of unload-reload results, kE for the unload-reload portioncan be roughly estimated from (kE) for primary loading. The values of (k) can befound in Duncan et al. (1980) and in Byrne et al. (1987) for various soils. Duncan etal. claimed that the ratio of kE/(kE) varies from about 1.2 for stiff soils such as densesands up to about 3 for soft soils such as loose sands. The value of the exponent n forunload-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-reloadE value.4.2.1.2 Parameters kB and mThe best way of evaluating kE and m is from the unload-reload results of an isotropiccompression test. The procedure proposed by Byrne and Eldrige (1982) is followedhere to determine these parameters. The volumetric strain and the mean stress inthe unload-reload path can be related as= a (°m)’ (4.3)where a and b are constants and can be obtained by plotting versus 0m on a log-logscale as shown in figure 4.2.Differentiation of equation 4.3 yieldsck 1 b—i (4.4)Then, the bulk modulus B can be expressed asB = (Om)1 (45)Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 77q-a3 /AE1€(a) Unload-Reload Modulus(E/Pa)1000100‘<E ‘a1 10 (ojP)[log scale] . a(b) Variation of E with a3Figure 4.1: Evaluation of kE and nChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 780.01 - kB = a.b(Pa)6am=1-b100[log scale]Figure 4.2: Evaluation of kB and mChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 79The general expression for B is given byB = kBPa()(4.6)By considering the similarities of equations 4.5 and 4.6, the parameters kB and mcan be obtained from a and b asm=1—b (4.7)kB= ab(Pa)’ (4.8)It should be noted that the parameters kE and kB can be related by the Poisson’sratio v askB= 3(1—2zi) (4.9)Hence, by knowing one parameter, the other one can also be determined fromthe Poisson’s ratio. Lade (1977) stated that the Poisson’s ratio for the unload-reloadpath has often been found to be close to 0.2.4.2.2 Evaluation of Plastic Collapse ParametersOnly two parameters are needed to evaluate the plastic collapse strains. These twoparameters define the hardening law and can be determined from an isotropic compression test. The hardening law is given by= CPa ()‘ (4.10)where W is the plastic collapse work, f defines the yield surface and C and pare constant parameters to be determined. For the isotropic compression loadingcondition, f and W will be given byChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 80f = 3o (4.11)Wc=Jcr3de (4.12)where de = d€,, — d and de is the elastic volumetric strain.By plotting W/P against f/P on a log-log plot, the parameters C and p canbe obtained as shown in figure 4.3.0.01 -[log scale]Figure 4.3: Evaluation of C and p4.2.3 Evaluation of Plastic Shear ParametersIn determining the plastic shear parameters, it is easier to divide them into threegroups as follows:1. Failure parameters i and LSiChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 812. Flow rule parameters i and )3. Hardening rule parameters KG, np and R1The plastic shear parameters can be determined from all types of tests where theprincipal stresses and principal strains can be obtained. By knowing the principalstresses 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 thenbe obtained as explained in the following subsections.The most common laboratory shear tests performed are triaxial compression testsand therefore, special attention is given here to describe how to obtain the plasticshear parameters from those test results.Firstly, the elastic and plastic collapse strains have to be subtracted to obtain theprincipal plastic shear strains:d = de1 — de— (4.13)d€ = de3 — de— d (4.14)Under standard triaxial compression conditions, the elastic and plastic collapsestrains can be given bydo1 (4.15)de = —vde (4.16)de = o do1 (4.17)d = 2A u1o3 do (4.18)where2C (p 1—2p— P\ a)—(o + 2o)2PChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 82It should be noted that if the test samples are preconsolidated to a higher stressand unloaded, then the collapse strains should not be subtracted.By following the equations in section 3.3.1 and imposing the conditions for triaxialcompression loading, the stresses and the strains related to SMP can be obtained asfollows:3o-1oSMP = (4.19)•1 + 03TSMp (4.20)SMP 3— dc/ + 2d4/dEsMp — (4.21)/2o + o2(deW — de/jd7sMp = (4.22)2o1+c34.2.3.1 Evaluation of q’ and zAt least two tests up to failure at different confining stresses are necessary to determinethese parameters. The failure stress ratio on SMP is given by(o-sMP)f= — Li log10 Pa (4.23)The values of and (OsMP)f can be obtained using equations 4.20 and 4.19. Byplotting {(OSMp)f/Pal versus on a semi-log plot, i and ii can be determined asshown in figure 4.4.4.2.3.2 Evaluation ofi and XThe flow rule for the plastic shear is expressed by the following equation.f—c1EsMpi J+/L (4.24)\ u7SMP JChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 83‘if____—_________ ______1 10 100e’ a[log scale]Figure 4.4: Evaluation of 1h and ‘iThe values of i, dEsMp and d7sMp for a triaxial compression test can be obtainedusing equations 4.20, 4.21 and 4.22. The flow rule parameters and ) can be determined by simply plotting versus —(desMp/d7sMp) as shown in figure 4.5.4.2.3.3 Evaluation of KG,rIp and RfAs explained in section 3.3.4, the hardening function is modelled by a hyperbola andis given by7SMP17 = (4.25)+G.The parameters KG, np and R1 which define C and it in the hardening ruleare evaluated following the procedure by Duncan et al. (1980). Basically, there aretwo steps involved in determining these parameters. The first is to determine theChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 84‘17— (dEMp/d4Mp)Figure 4.5: Evaluation of ) and tvalues of G, and the second is to plot those values against °5Mp to determine KGand np. At least two triaxial compression test results are necessary to evaluate theseparameters.Upon rearranging the terms, equation 4.25 becomes7SMP — 1 7SMP 4 261 7u1tNow, by plotting (7sMp/7/) against fsMP the values of G7, and 71,jit can be determined as shown in figure 4.6(b).The failure ratio Rf is defined asRf (4.27)l7ultBy knowing from figure 4.6(b) and i from section 4.2.2.1 Rf can be determined using the above equation.Chapter 4. Stress-Strain Model- Parameter Evaluation and Validation 85“717SMP(a) Hardening Rule7SMP‘1G17SMP(b) Hardening Rule on Transformed PlotFigure 4.6: Evaluation of G and ijChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 86G is expressed as a function of op asG = KG (4.28)The parameters KG and np can be obtained by plotting G,. against (oSMp/Pa)on a log-log plot as shown in figure 4.7.1000‘1)0c-I(I)0U 1001 10 100[log scale]Figure 4.7: Evaluation of K0 and npMP’a4.2.4 Evaluation of Strain Softening ParametersTo determine the strain softening parameters, it is necessary to have experimentalresults which exhibit strain softening phenomenon. As explained in section 3.9, itshould be noted that strain softening is not a fundamental property of soils, rather itis a localized phenomenon. Therefore, it is quite possible that different tests may yielddifferent softening parameters. In those cases, the average value can be consideredappropriate.1npKQChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 87The strain softening region of a stress-strain curve can be given as (see section3.8)= ir + (ip—lir) exp{—i(7sMp— 7SMP,p )} (4.29)The value of the residual stress ratio is i is assumed to be equal to t which isthe flow rule intercept. This assumption is reasonable because, when i = p, theincremental plastic volumetric strain will be zero, which implies a state of shear atconstant 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 obtainedfrom the strain hardening relation (equation 4.25) as7737SMP,p = (4.30)G1— R11where 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 canbe shown thatin () K(7sMp (4.31)Taking natural logarithm of equation 4.31 will giveln [in (ij] ln + qln(7sMp— 7sMp,p) (4.32)Then, the parameters , and q can be determined by plotting {ln [ln ()] }against {ln(7sMp — 7SMP,p )} as shown in figure 4.8.4.3 Validation of the Stress-Strain ModelThe stress-strain model employed in this study has been verified against laboratoryresults on Ottawa sand and oil sand.The triaxial test results reported by NeguessyChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 88in [in (TZr)]Figure 4.8: Evaluation of i and q(1985) on Ottawa sand and by Kosar (1989) on Athabasca McMurray formation interbedded oil sand have been considered. The Ottawa sand is well defined. Uniformtest samples were constituted in the laboratory and the test results were very repeatable. Oil sand samples on the other hand, were obtained from the field andtherefore the samples might not identical. The soil parameters for both sands areobtained as explained in the section 4.2 and then the predicted and measured resultsare compared.4.3.1 Validation against Test Results on Ottawa SandThe Ottawa sand is a naturally occurring uniform, medium silica sand from Ottawa,illinois. Its mineral composition is primarily quartz and the specific gravity is 2.67.The average particle size D50 is 0.4 mm and the particles are rounded. The gradationcurve of the Ottawa sand is shown in figure 4.9.qin(7sMp— 7SMP,p)Chapter 4. Stress-Strain Model- Parameter Evaluation and Validationzp ae 4 48 100‘ II II I89MEDIUM SAND‘II00140 2008060Ct4QLEGE NDX FRESH20• RECYCLEDI0ASTM - C- 109- 69 BAND* MIT CLASSIFICATIONI 0.5 0.1 0.01Diameter (mm)Figure 4.9: Grain Size Distribution Curve for Ottawa Sand (after Neguessy , 1985)Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 90The following test results reported by Negussey (1985) are considered here for thedetermination of the relevant parameters and for the validation:1. Resonant column tests2. Isotropic compression tests3. Triaxial compression tests4. Proportional loading tests (R = o1/o3 = 1.67 and 2)5. Tests along four different stress paths as shown in figure 4.10SP4300 SP3a.200SP2 SP1- = 2.0SP2- (a/u=4.0spi100 SP3 - P = 250 kPa, ConstantSP4- P’ = 350 kPa, Constant100 200 300 400UH(kPa)Figure 4.10: Stress Paths Investigated on Ottawa SandThe test results considered here are for Dr = 50%. The maximum and minimumvoid ratios of the Ottawa sand are 0.82 and 0.50 respectively.Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation4.3.1.1 Parameters for Ottawa Sand91As explained in section 4.2.1, the Young’s modulus values for different confiningstresses are plotted in figure 4.11. The values plotted in the figure are from resonant:-(kE)p= 1180-. Resonant Column --A Tria)_(UnIoad Reload)— •Triaxlal (Primary Loading)I I Ia3 aFigure 4.11: Variation of Young’s moduli with confining stressescolumn tests which yield similar values as are obtained in unload-reload tests. Alsoshown in the figure are one unload-reload modulus and the Young’ modulus valuesfor primary loading from standard triaxial compression tests. It can be seen that theunload-reload value agrees well with the resonant column values. The ratio of theYoung’s modulus for the primary loading condition to the unload-reload condition isabout 2.2 and the exponent for both conditions is 0.46. This agrees with the statementby Duncan et al. (1980) that the ratio of Young’s moduli varies from about 1.2 fordense sands to about 3 for loose sands. From figure 4.11 the values for kE and n can1000050003000200010005000.3 0.5 1 2 3 5 10Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 92be 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 atdifferent confining stress and one value of unload-reload.There are no results of unload-reload conditions available in isotropic compression test to determine kB and m. Therefore, the Poisson ratio is assumed to be0.2 as suggested by Lade (1977). Hence, kB and m are obtained as 1444 and 0.46respectively.The plastic collapse parameters C and p are evaluated as explained in section4.2.2 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 0.89respectively.0.01We/Pa0.0050.0020.00 10.00050.00020.00015E-050.2 0.5 1 2 5 10 20250 100Figure 4.12: Plastic Collapse Parameters for Ottawa SandIn order to obtain the failure parameters, as explained in section 4.2.3.1, the failureChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 93stress ratio i vs usMp for the triaxial compression test results are plotted in figure4.13. The failure parameters () and ZSi7 are determined as 0.49 and 0.0.Figure 4.13: Failure Parameters for Ottawa SandThe four triaxial compression test results are shown as vs. (—desMp/d7sMp) infigure 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 plotsare shown in figure 4.15. The value of R1 is determined as 0.93. From figure 4.15(c),the values of KG and np are obtained as 780 and —0.238 respectively.Table 4.2 summarizes all the parameters for Ottawa sand at Dr = 50%.Ti0.60.50.40.30.20.1—— —71iO49ö::zS7=O.O1000.5 1 2 3 5cTSMP/PChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 94Table 4.2: Soil Parameters for Ottawa Sand at D = 50%Elastic kE 2600n 0.46kB 1444m 0.46Plastic Shear 0.491117 0.0X 0.85u 0.26780np -0.238Rf 0.92Plastic Collapse C 0.00021p 0.89‘70.60.50.4030.20.10-0.3 -0.2 -0.1 0 0.1 0.2 0.3_(dEsMp/d7sMP)0.4Figure 4.14: Flow Rule Parameters for Ottawa SandChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 95(a)0.50.4___________o3=5OkPa0.3 ,‘ ---0---3=150kPa02 —• a_3=50kPa0.1 I a3=45OkPa--.“---.-00 0.2 0.4 0.6 0.87SMP1.87SMP1.6 o1,41.210.80.60.40.2 ‘0 I I0 0.2 0.4 0.6 0.87SMP800G1750.W 735%% ‘UP700650600 Jlr=0.145550500 I I I0.5 1 2 3 5 10Figure 4.15: Hardening Rule Parameters for Ottawa SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 964.3.1.2 ValidationAs a first level of validation, the four triaxial compression tests which were used todetermine the parameters, are modelled. Figure 4.16 shows the experimental resultsand the model predictions and they both agree very well. This implies that the modelsuccessfully represents the test results.The stress-strain model is then used to predict the responses for proportionalloadings and four other stress paths as shown in figure 4.10. Figure 4.17 shows theresults for two proportional loading tests, R = o1/o3 = 1.67 and 2, and it can beseen that the predictions and the measured responses agree very well. Figure 4.18shows the results for four different stress paths and again the predicted and measuredresults are in good agreement.4.3.2 Validation against Test Results on Oil SandThe test results reported by Kosar (1989) on Athabasca McMurray formation oil sandare considered here. Tests were carried out on samples taken form the Alberta OilSands Technology and Research Authority’s (AOSTRA) Underground Test Facility(UTF) at varying depths from 152 m to 161 m. The samples consisted of mediumgrained particles and were uniformly graded. Figure 4.19 shows the gradation curveof the UTF sand and some other oil sands. In UTF sands, pockets and seams of siltyshale were present and their thickness ranged form 1 to several millimetres. The finescontent 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 andVan 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 goodquality samples.The following test results from Kosar (1989) are considered for the determinationChapter 4. Stress-Strain Model- Parameter Evaluation and Validation00097a 3 = 50 kPa----*--a_3• = 5O kPao 3 = 250 kPa-------800600 - _..0400- — .0200 -I Ia_3 =50 kPaSymbols - ExperimentalLines - Analytical(a)0.050.>‘U0.150.20.250 0.2 0.4 0.6 0.8E(%)aFigure 4.16: Results for Triaxial Compression on Ottawa SandChapter 4. Stress-Strain Model- Parameter Evaluation and Validation50040030002001006005004000a-20010000981.67Symbols - ExperImentLines - AnaIytca1I_ —— I I0.1 0.2 0.3 0.4 0.5 0.6Figure 4.17: Results for Proportional Loading on Ottawa SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 99600500400300200100>iLl0.40.60.80.2a0 0.2 0.4 0.6 0.8 1Figure 4.18: Results for Various Stress Paths on Ottawa SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 100>E=10080604020010.0I I I I I 111111 I I-__UTF Sand4• Other McMurray- ——Sands:- coarse I 1- medium-—— I-fine.:::ZE.--——1.0milhimetet 0.1 0.01I I0.1. 0.01 0.001inchesI F F Fl F I F F F F I8 12 18 25 35 45 60 80 120 170 230 325400U.S. meshFigure 4.19: Grain Size Distribution for Athabasca Oil Sands, (after Edmunds et al.,1987)Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation 101of the relevant model parameters and the validation:1. Isotropic compression test2. Standard triaxial compression tests3. i constant compression4. U constant compression0m constant extensionIt should be noted that since the samples tested were undisturbed samples fromthe field, they were not identical. Table 4.3 summarizes the details of the test samplesconsidered.Table 4.3: Details of the Test SamplesTest Sample Bulk Fraction by Weight (%) Void DisturbanceID Density Water Bitumen Solids Fines Ratio Index(kg/rn3) (< 0.074rnrn) (%)Isotropic Comp. UFTOS1 1990 8.3 7.6 84.1 41.2 0.60 12.1Triaxial Comp. 1 UFTOS1 1990 8.3 7.6 84.1 41.2 0.60 12.1Triaxial Comp. 2 UFTOS3 2070 8.5 6.6 84.9 54.0 0.52 6.4Triaxial Comp. 3 UFTOS4 2120 6.4 6.5 87.1 52.9 0.45 10.00.1 Const. Comp. UFTOS1O 2060 6.6 7.3 86.1 71.9 0.50 10.80m Const. Comp. UFTOS9 1960 7.8 8.8 83.4 57.3 0.62 10.90m Const. Ext. UFTOS12 1980 7.0 9.5 83.4 37.7 0.60 9.64.3.2.1 Parameters for Oil SandThe relevant parameters for the oil sand are obtained from an isotropic compressiontest and three standard triaxial compression test results. Since the procedures forobtaining the parameters are discussed in detail in section 4.2 and again briefly insection 4.3.1.1, they are not repeated here.Figure 4.20 shows the data for the unload-reload portion of the isotropic compression test and the elastic parameters kB and m are determined as 1670 and 0.36Chapter 4. Stress-Strain Model- Parameter Evaluation and Validation 1020.03EV0.010.0030.0010.00030.00013E-051 10 100 1000 10000am (kPa)100000Figure 4.20: Determination of kB and m for Oil SandChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 103respectively. The Poisson ratio is assumed to be 0.2 and kE and n are determined as3000 and 0.36. The plastic collapse parameters C and p are obtained from the primaryloading portion of the isotropic compression test as 0.00064 and 0.61 respectively (seefigure 4.21)./ a10.30.10.030.010.0030.0010.0003The failure and hardening rule parameters are obtained from the triaxial compression tests as explained in section 4.2.3. Figure 4.22 shows the relevant graph toobtain the failure parameters. The hardening rule parameters are obtained as shownin figure 4.23.The reduced data to obtain the flow rule parameters are shown in figure 4.24. Theresults from the three triaxial tests do not seem to give a unique set of parameters asobserved 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 obtained1 10 100 1000 10000 100000Figure 4.21: Plastic Collapse Parameters for Oil Sand104Chapter 4. Stress-Strain Model - Parameter Evaluation and Validation0.81f0.750.70.65060.550.5I 2 3 5 10 20 30 50Mp’a100Figure 4.22: Failure Parameters for Oil SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 105200013001000 -500-= -0.662000 0100 -050 I1 2 3 5 10 20 30 50 100MP’aFigure 4.23: Determination of K0 and np for Oil SandChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 1060.80.70.60.50.40.30.20.10.40-0.47,EJA0 --./- .------0.2 0.4o.90.6 0.8(a)110.80.70,60.50—-0.4 -0.2o a_3=1.OMPao_3 = 2.5 MPac,_3 = 4.0 MPa0—(dEsMp/d7sMP)0.30.20.1-0.2 0 0.2 0.4 0.6—(dEsMp/d7sMP)0.8Figure 4.24: Flow Rule Parameters for Oil SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 107if the individual test results are considered. However, an average set of parameterscan be obtained as shown in figure 4.24(b). The flow rule parameters are very muchgoverned by the volumetric strain behaviour and this will be discussed more in section4.3.2.2. The summary of the parameters obtained for oil sand is given in table 4.4.Table 4.4: Soil Parameters for Oil SandElastic kE 3000n 0.36kB 1670m 0.36Plastic Shear 0.75iii 0.13\ 0.53ii 0.31KG 1300rip -0.66R1 0.73Plastic Collapse C 0.00064p 0.614.3.2.2 ValidationFigure 4.25 shows the experimental and predicted results for loading and unloading ofthe 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 compression tests. It can be seen that the predicted and measured deviator stress versus axialstrain agree very well. The volumetric strain versus axial strain agree reasonably wellfor 03 = 1.OMPa and O = 2.5MPa but not for o = 4.OMPa. This is becausethe selected flow rule parameters are the average parameters and they tend to agreeclosely with those two tests. It can be seen from figure 4.24 that for 03 = 4.OMPa, thestraight line relation is much different and steeper, which would have given a higherChapter 4. Stress-Strain Model- Parameter Evaluation and Validation 1081400012000- 010000 -8000- Loadingb 6000 - Unloading4000- 02000 o Line - PredictedSymbols- Measured0 I I I I0 0.5 1 1.5 2 2.5 3LV (%)Figure 4.25: Results for Isotropic Compression Test on Oil SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 109value of ). As the line becomes steeper, there will be less volumetric expansion andthe overall behaviour will be more contractive. If a higher value of ) is selected, thepredictions and observations would agree well for o 4.OMPa. It is also interestingto note that the value of the flow rule parameter i is not much different for the threetests. The value of .t is, in fact, an indication of ultimate stress ratio or a state ofshearing with constant volume.Results for three different stress paths; constant o, compression, constant °mcompression and constant om extension are shown in figure 4.27. The stress pathsare shown in the insert of the figure. It can be seen that the experimental andpredicted results are in good agreement.4.4 Sensitivity Analyses of the ParametersIn order to provide a better understanding about the significance of the parameters,sensitivity analyses on the parameters have been carried out. The parameters obtained for Ottawa sand were chosen as the base parameters and the significance of aparticular parameter was studied by changing only that parameter. A triaxial compression loading condition with the initial confining stress of 500 kPa was consideredand the results in terms of deviator stress and volumetric strain are analyzed. Theresults are shown in figures 4.28 and 4.29The plastic collapse parameters C and p are essentially an indication of isotropiccompressibility. 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 volumetricexpansion for a change in stress ratio. A steeper slope (or higher A) will give lessvolumetric expansion. The parameter 1u is the amount of stress ratio which separatecontraction and dilation (similar to ç5 in general soil mechanics). A smaller value ofwill result in dilation at lower stress ratio.The parameters KG and np define the initial slope of the hardening modulusChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 1100>w10000 -8000 -6000 -4000200000.20.40.60.8--: : MPaa_3 =.5 MPaa_3=,MPaSymbols - ExperimentalLines - Analytical-0.8-0.6-0.4-0.20 0.5 1 1.5 2 2.5 3e(%)aFigure 4.26: Results for Triaxial Compression Tests on Oil SandChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 1115,000—4,000-3,000-2,0001,00006-4I/ SP1-I.1 Const.Comp2 / SP 2- a_v Const Comp./ SP3-i1 Conet Ext.246a_r (MPa)D.C-..C’-,I1‘aa&>WI000or0/0/______o/ spiCSP3Symbols- ExperimentalLines - Analytical-1.4-1.2—1-0.8-0.6-0.4-0.200.20.4zgD Q9OO5/O.-ci - o o-6 -4 -2 0 2 4 6- E_r (%)Figure 4.27: Results for Tests with Various Stress Paths on Oil SandChapter 4. Stress-Strain Model- Parameter Evaluation and ValidationEa (%)(C) Effect of Parameter A00&112c_a (%) c_a (%)(a) Effect of Parameter C (b) Effect of Parameter pI>WI0.4 0.6 0.8 1c_a (%)(d) Effect of Parameter LFigure 4.28: Sensitivity of Parameters C,p, ) and iChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 113(c) Effect of Parameter Rf1.2€_a (%) E_a (%)(a) Effect of Parameter KGp (b) Effect of Parameter rip0&Ea (%) c_a (%)(d) Effect of Parameter flfFigure 4.29: Sensitivity of Parameters KG, np, R1 and iChapter 4. Stress-Strain Model - Parameter Evaluation and Validation 114G,. A higher value of will result in a stiffer deviator stress response and a lowervolumetric strain response. The parameter R1 and i define the shape and the failurestress ratio in the hardening rule. Lower R1 and higher i will give stiffer deviatorstress response.The elastic parameters are not considered here because they have been widelyused and their significance is well understood.4.5 SummaryA double hardening elasto-plastic model has been postulated to model the stress-strain behaviour of oil sands. Procedures for the evaluation of the parameters andthe validation of the proposed model have been presented in this chapter.The model parameters are relatively easy to obtain and can be determined fromconventional isotropic and triaxial compression test results. The parameters havephysical meanings and a sensitivity study has been carried out on the parameters tobetter explain their physical significance. Laboratory test results for various stresspaths have been compared with the model predictions. Measured results and predictions 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 proposedmodel captures the stress-strain behaviour of oil sands very well.Chapter 5Flow Continuity Equation5.1 IntroductionThe pore fluid in the oil sand matrix comprises three phases namely gas, oil and waterand therefore, the fluid flow phenomenon is of multi-phase nature. In petroleumreservoir engineering, the flow in oil sand is often analyzed as multi-phase flow, butsolely as a flow problem without paying much attention to the porous medium. Themost widely used model to analyze the flow in oil sand is called ‘3-model’ or ‘theblack-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 temperaturechanges in the sand skeleton, a flow continuity equation is derived in this chapterfrom the general equation of mass conservation. However, the flow equations are notconsidered separately for individual phases as in petroleum reservoir engineering. Allthree flow equations are combined and a single effective equation is formulated. Inessence, the derived flow continuity equation is similar to a single phase flow equation115Chapter 5. Flow Continuity Equation 116in geomechanics but the permeability and compressibility terms have been changedto include the effects of different phase components. The flow continuity equation willbe combined with the force equilibrium equation and will be solved as a consolidationproblem as explained later in chapter 6.5.2 Derivation of Governing Flow EquationIn this section, the flow continuity equation for a single phase in one dimension is derived first. Later, it is expanded to three phase flow in three dimensions. The amountof flow of one phase component depends on the saturation and the mobility of thatparticular phase. When the fluid is Newtonian and the flow is slow, as it usually is inpetroleum reservoir situations, the volumetric flux of a phase is proportional to thepotential gradient acting on it and inversely proportional to its viscosity. The coefficient of proportionality is the Darcy’s permeability. This is customarily expressedas the product of the relative permeability of phase 1 (krj), and the absolute Darcypermeability (k), of the medium to flow when a single fluid entirely fills the porespace. Mathematically this is expressed asVP1 (5.1)where,v - velocity vector (in m/s)k - permeability matrix of the porous medium (in m2)krt - relative permeability of phase 1 (non dimensional)I’i - viscosity of phase 1 (in kPa.s)P1 - pressure in phase 1 (in kPa)Now, consider a single phase (denoted by 1) flow in one dimension (in z direction)as shown in figure 5.1.Chapter 5. Flow Continuity Equationn - porosityS1 - saturation of phase 1- velocity of phase 1 in z direction- unit weight of phase I1170(vj 71) dzôzS1ndzn dzdzPhase ‘I’ in pore fluidPore fluid— SolidsIFigure 5.1: One dimensional flow of a single phase in an elementChapter 5. Flow Continuity Equation 118Weight of phase 1;wi=nSi7jdz (5.2)Incoming mass flux:v yi (5.3)Outgoing mass flux:= + dz (5.4)Difference between flux coming in and flux going out:QI_Qo_O(vZz_y1)d 55dt 8z (.)Rate of storage:&wlO(nSl7l)d 568t ôtFor conservation of mass, the difference between the incoming and outgoing fluxshould be equal to the rate of storage. Thus,— O(vi 71) — 8(n Si 7i) 5 78z— ôt (.)Expansion of the partial differentials in equation 5.7 givesavzl 871 8n &Y1 8S1—7z—+v2i = iSi+nSi--+n- (5.8)Dividing by 7i yields8v21 v21 871 8m S1 871 8S1——+----—-—=Si——+n—-—+n------ (5.9)8z 71 8z 7’ at atNow, consider all five terms in equation 5.9 separately, starting from the left handside.Chapter 5. Flow Continuity Equation 1190v218zBy Darcy’s law (equation 5.1) v can be written askkr 8P1vz1 =—____—ILl c9z= (5.10)and therefore,92P1= (5.11)wherekmi - mobility of phase 1k - intrinsic permeability of the porous medium[function of void ratio; k = f(e)]k,.1 - relative permeability of phase 1[function of saturation; k,.1 = f(S1)]- viscosity of phase 1function of temperature and pressure; = f(8, F1)]vz - velocity of phase 1 in z directionP1 - pressure in phase 1271 8zThe change in unit weight due to the increase in pressure can be expressed as,871 = 71 (5.12)whereChapter 5. Flow Continuity Equation 120B1 - bulk modulus of phase 1- unit weight of phase 1Therefore,87j—vyi 8z — B1 öz— kmi 6P15 13— B1 8z 8zThis term involves the square of the pressure derivatives and can be neglectedas small compared to the other terms (ERCB, 1975).3. S1--By adopting the usual soil mechanics sign convention as compressive strain andstress positive, it is obvious thatdm =—dEn (5.14)Thus the above term becomes(5.15)wheren - porosityt -time- volumetric strainChapter 5. Flow Continuity Equation 121S1 b-y.4. n—-—7i ôtBy using equation 5.12 this term can be written as,51 87j 51 8P1 (5.16)718t B18tas1.Summation of saturations of all phase components should always be equal tounity. Hence, when combining the equations for all the phases, the summation ofthis term over all the phases will be zero. Mathematically this can be expressedas(5.17)Since the final equation is to be derived by combining all the phases this termneed not to be considered in detail.By making the changes to the terms as explained so far, equation 5.9 can bewritten as82P s1 8P1 8Sikmi-ä—-H-S1 --—n- --—n--O (5.18)Extension of equation 5.18 to three dimensions yieldskmiV2P + 51L — — = 0 (5.19)Chapter 5. Flow Continuity Equation 122where2 b’P ElF t92p (5.20)Hence, the equations of flow for the three phases in oil sand, in three dimensions,will be as follow:for water phase;(5.21)for oil phase;kmo V2P0H-5a—ns0 ap0—n = 0 (5.22)for gas phase;kmg V2P9 + S9 — — n = 0 (5.23)where,km - mobilityS - saturationB - bulk modulusand subscripts o, w and g denote oil, water and gas respectively. It should be notedthat in the formulation the capillary pressure between two phases is assumed to beconstant for the increment and therefore, it will not appear in derivatives.Combining equations 5.21, 5.22 and 5.23 gives(kmo+kmw+kmg) V2p+_n (++) =0 (5.24)This can be written asChapter 5. Flow Continuity Equation 1232 OPkEQ V P + — CEQ -- = 0 (5.25)where,kEQ - equivalent hydraulic conductivity= kmo + kmw + kmgCEQ - equivalent compressibility— (S0 Sw SgEquation 5.25 is similar to the one used by Vaziri (1986) and Srithar (1989), exceptfor the equivalent conductivity term. They considered the contributions from differentfluid phase components in the compressibility but not in the hydraulic conductivity.Recently, Settari et al. (1993) have also used an effective hydraulic conductivity termto model the three-phase fluid which is similar to the equivalent hydraulic conductivityterm derived above. The equivalent hydraulic conductivity is a function of mobilitiesof the phases which in turn depend on their relative permeabilities and viscosities.Evaluations of relative permeabilities and viscosities are described in detail in the nextsections. The equivalent compressibility is a function of saturation and bulk modulusof individual phase components and the details of its evaluation are described insection 5.5.5.3 Permeability of the Porous MediumThe permeability of the porous medium (k) mainly depends on the amount of voidspace. Lambe and Whitman (1969) collected considerable experimental data to studythe variation of k with void ratio. Although there was a considerable scatter in thedata, they found that there is a linear relationship between k and a void ratio functione3/(1 + e) for a wide range of granular materials. It can be argued that various otherrelationships could be established for the varition of k with e. However, without theChapter 5. Flow Continuity Equation 124need for much specific details about the soil, the relationship given by Lambe andWhitman (1969) is quite reasonable for most engineering purposes. Using Lambe andWhitman’s relationship, at a particular void ratio of e, k can be expressed ask =kOe/(l±e) (5.26)e/(1 + eo)where e0 and k0 are the initial void ratio and the initial permeability of the porousmedium respectively.5.4 Evaluation of Relative PermeabilitiesMeasurement of three-phase relative permeability in the laboratory is a difficult andtime consuming task. Due to the complications associated with the three-phase flowexperiments, empirical models have been used extensively in the reservoir simulationstudies. These models use two sets of two-phase experimental data to predict thethree-phase relative permeabilities. Figure 5.2 shows typical results that might beobtained for such two-phase systems. Figure 5.2(a) shows the relative permeabilityvariations for an oil-water system and figure 5.2(b) shows the relative permeabilityvariations for a gas-oil system.Numerous experimental studies on relative permeabilities have been reportedin the petroleum reservoir engineering literature starting from Leverett and Lewis(1941). Many review articles have also appeared in the literature (Saraf and McCaffery (1981), Parameswar and Maerefat (1986), Baker (1988)) and an assessment ofthese studies is beyond the scope of this thesis. However, the general conclusion fromthese studies suggests that the functional dependence of relative permeabilities canbe given by= f(S)krg = f(S9)Chapter 5. Flow Continuity Equation 1250krowk,,,(b) Gas-oil systemFigure 5.2: Typical two-phase relative permeability variations (after Aziz and Settari,1979)Ikr0Ikr0 SwmazSW—,’..(a) Oil-water systemrg_I Sgc SgmozSg 0Chapter 5. Flow Continuity Equation 126k,.0 = f(S0) (5.27)The function for the relative permeability of oil, k,.0, is not readily known and itis estimated from the two-phase data for k,., and k,.09, where, k,.OW is the relativepermeability of oil in an oil-water system and k,.09 is the relative permeability of oilin an oil-gas system. Their functional dependence are given byk,.0 = f(SW)k,.09 = f(S9) (5.28)The simplest way of estimating k,.0 would be,k,.0 = k,.0 k,.09 (5.29)Two more accurate models have been proposed by Stone (1970), only the first ofwhich is considered here. In this model, Stone (1970) defines normalized saturationsasS = — S S (5.30)wc om= SO SomSo Sam (5.31)wc om5; = 15S9 (5.32)wc amWhere, S, is called the critical or connate water saturation at which water startsto 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 displacedsimultaneously by water and gas. If S is less than 5om, k,.0 will be zero.According to Stone (1970), the relative permeability of oil in a three-phase systemis given byk,.0 = S (5.33)Chapter 5. Flow Continuity Equation 127The factors i3 and ,i3 are determined from the end conditions that equation 5.33should match the two-phase data at the extreme points. The two extreme cases of= 0 and SL, = give— 1 — Sk,.09— 1— S’9(5.34)(5.35)The region of mobile oil phase (i.e. k,.0 > 0) predicted by Stone’s model I is shownin figure 5.3 on the ternary diagram assuming increasing 5W and S9. For conditionsdepicted by point outside the hatched area, the relative permeability of oil will bezero.100%WATERFigure 5.3: Zone of mobile oil for three-phase flow (after Aziz and Settari, 1979)100%GASE1S SomI,. 100%OILAziz and Settari (1979) modified Stone’s model because Stone’s model will reduceChapter 5. Flow Continuity Equation 128exactly to two-phase data only if the relative permeabilities at the end points areequal to one, i.e., krow(Swc) = krog(Sg = 0) = 1. They suggest that the oil-gas datahas to be measured in the presence of connate water saturation. In that case, an oil-water system at S, and an oil-gas system at S = 0 are physically identical. Bothsystems will have, S = S and S0 = 1 — S at 59 = 0. At these conditions, therelative permeabilities will bekrow(Swc) = krog(Sg = 0) = krocw (5.36)Then, the modified form of Stone’s equations will bek,.0 = S krocw w /39 (5.37)k— (5.38)—k,.0911 c’ (. ),.0cwI — LI9Kokal and Maini (1990) claim that Aziz and Settari’s method has problems because:1. Measurements of two-phase oil-gas data are not necessarily obtained at connatewater saturation2. The relative permeability at connate water saturation in an oil-water systemgenerally will not be equal to that in an oil-gas systemKokal and Maini (1990) further modified Stone’s model by incorporating anothernormalizing factor. After these modifications, the relevant equations needed to predictthe relative permeability of oil areChapter 5. Flow Continuity Equation 129k,.0 = s(k,?09S;+k,?,&S)(5.40)rOW 4wko(1SIa rog“9k0 (1Srog\ gwhere,k,?0- relative permeability of oil at connate water saturationin a water-oil systemk,?09 - relative permeability of oil at zero gas saturationin an oil-gas systemWhen 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 dataand found very good agreement. The best comparison given in their paper is shownin figure 5.4.From the discussion so far in this section, it can be concluded that the relativepermeabilities in three-phase system can be written as= f(S) (5.43)k,.9 = f(S9) (5.44)k,.0 = f(k,.0,k,.09, Sw, so, S) (5.45)k,.0 = f(5) (5.46)Chapter 5. Flow Continuity Equation 130OILExpenmental— Calculated0.750.700.600.500.400.300200.100.01. . ..WATER “ “ “ “ ‘.‘ “ ‘I’ ‘ GASFigure 5.4: Comparison of calculated and experimental three-phase oil relative permeability (after Kokal and Maini, 1990)Chapter 5. Flow Continuity Equation 131a)Ea,a)>a)— f(S9) (5.47)However, to implement the relative permeability variations in a numerical simulation the variations should be expressed as mathematical functions. Polikar et al.(1989) suggest that these variations can be well represented by power law functions.Thus, mathematically the variations can be given as= C1(S — C2)c3 (5.48)where C1,C2 and C3 are constants. Figure 5.5 shows a comparison of experimentaldata with calculated values using the power law functions.1.2k= 2.769 (0.80 - Sw) 1.996k= 1.820 (Sw - 0.20) 2.735row10.80.60.40.20‘ k rwSymbols- ExperimentalLines - Correlation0 0.2 0.4 0.6 0.8w1 1.2Figure 5.5: Comparison of calculated and experimental relative permeabilities usingpower law functionsChapter 5. Flow Continuity Equation 132In summary, the relevant parameters needed to calculate the relative permeabilities of water, oil and gas phases are given in table 5.1. An example showing thedetails of the calculations of the relative permeabilities and the resulting equivalentpermeability is given in appendix B, to provide a better understanding of the stepsinvolved.5.5 Viscosity of the Pore Fluid Components5.5.1 Viscosity of OilThe mobility of an individual phase in a three-phase system depends on the viscosityof the phase component. Viscosities of the fluid components are generally strongfunctions of temperature and to some extent depend on the pressure as well.Viscosity of oil plays a very important role in reservoir engineering. Crude oilcannot flow at the ambient temperatures because of its high viscosity. The oil recoverymethods require some form of heating to reduce the viscosity and thereby increasemobility. For example, the viscosity of Cold Lake bitumen is 20, 000 mPa.s at 30°Cand 100 mPa.s at 100°C, i.e., a 200-fold reduction at high temperature. There aresome correlations for the viscosity of oil available in the literature. Among thosecorrelations, the one proposed by Puttagunta et al. (1988) has been selected in thisstudy 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 correlationfits the viscosity variation of most bitumens reasonably well.The correlation proposed by Puttaguntta et al. (1988) is expressed by the following equation:Chapter 5. Flow Continuity Equation 133Table 5.1: Parameters needed for relative permeability calculationsParameter DescriptionConnate or critical water saturationSom Residual oil saturationA1,A2,A3 Parameters for variation of k with Si,,in water-oil system = A1(S — A2)A3]B1,B2,B3 Parameters for variation of krow with 5win water-oil system [kro B1(B2 — Sw)B3]C1,C2,C3 Parameters for variation of krg with Sgin oil-gas system [krg = C1(S9 — C2)c3]D1,D2,D3 Parameters for variation of krog with S9in oil-gas system [k.09 = D1(D2— 59)D3]Relative permeability of oil at connate water saturationin water-oil systemk,?09 Relative permeability of oil at zero gas saturationin oil-gas systemChapter 5. Flow Continuity Equation 134lfl(9,p) 2.3026 ( 8-30 — 3.0020] + B0 F exp(d 6) (5.49)+ 30315)where,b log.Lt(3o,o) + 3.0020a = 0.0066940.b + 3.5364B0 = 0.0047424.b + 0.0081709d = —0.0015646.b + 0.00618148 - temperature in degrees CelsiusF - 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 forCold Lake and Wabasca bitumens. The above correlation is implemented in thefinite element program CONOIL. However there is an option in CONOIL to read andinterpolate user specified viscosity-temperature data, in case this correlation does nothold for a particular bitumen.5.5.2 Viscosity of WaterThe viscosity of water does not change as drastically as that of oil. For instance, at30°C the viscosity of water is 0.8 mPa.s and at 100°C, it is 0.28 mFa.s. A changeof 70°C in temperature causes a reduction in viscosity by a factor of 3 as comparedto 200 for oil. The viscosity-temperature data for water are well established andcan be obtained from the international critical tables. The viscosity of water is wellrepresented by the following equation:= (b+8) (5.50)whereChapter 5. Flow Continuity Equation 13550000— empirical equation- * experimental10000-‘‘500o *SC0SI 1000TY 500mPa100S50. *100 20 40 50 80 100 120TEEATUR.E, Ca) Wabasca bitumen50000 -\ — empirical equation* experimental10000VI 5000 -SCo4iooo.500P• 100 -so10 I I23 40 50 80 100 120 140TE’ERATURE, Cb) Cold Lake bitumenFigure 5.6: Experimental and predicted values of viscosity (after Puttagunta et al.,1988)Chapter 5. Flow Continuity Equation 136- viscosity of water- temperaturea, b, n - constantsIt is reasonable to assume the water phase in the oil sand will have the same properties. These data from the International Critical Tables are reproduced in appendixB and built into the computer program CONOIL. There is also an option to read andinterpolate from any other user specified data.5.5.3 Viscosity of GasThere is not much information available about the viscosity of gas in the recentliterature in petroleum engineering. Carr et al. (1954) carried out some work onthe viscosity of hydrocarbon gases as a function of pressure and temperature. Theviscosity of gas appears to be equally dependent on pressure and temperature, but thevariations 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.sand at 200°C it is 0.0105 mPa.s, i.e., increases by only a factor of 1.5. The chartsgiven 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 isvery 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 (forinstance, 0.01 mPa.s). However, there is an option available in CONOIL as for waterand oil, to input any other data at the user’s choice.5.6 Compressibility of the Pore Fluid ComponentsIn the final flow equation derived (equation 5.25), the equivalent compressibility ofthe pore fluid is defined asChapter 5. Flow Continuity Equation 137CEQ =(+±) (5.51)The bulk moduli of the water and oil can be assumed constant, though theydepend slightly on pressure. The important parameter that affects the equivalentcompressibility is the comprssibility of gas. If there is more gas present in the porefluid, it will be more compressible. The compressibility of gas can be determined usingthe gas laws. The basic gas laws governing the volume and pressure relationships areBoyle’s law and Henry’s law. According to Boyle’s law, under constant temperatureconditions,P9V=wRT (5.52)where,P9 - absolute pressure of gasT - absolute temperatureR - universal gas constantV, - volume of gasWg - weight of gasUnder 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’slaw (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 gasabove the solution. Mathematically, this can be written asChapter 5. Flow Continuity Equation 1380 0(5.54)where- weight of dissolved gasand 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 fixedquantity of liquid is constant at a constant temperature and at a confining pressureF, when the volume is measured at F. ThusVd9 = H V0 (5.55)whereH - Henry’s constant, which is temperature dependent and,over a wide range of pressure, is also pressure dependentV0 - volume of oilSince the volume of dissolved gas is constant, free and dissolved gas componentscan be combined. Then application of Boyle’s law to the entire volume yields (Fredlund, 1976)(5.56)where V19 is the volume of free gas.Rearranging the terms yields,ID1IT? in— rgkVdg + VfV19—0 — Vdg9By differentiating equation 5.57,Chapter 5. Flow Continuity Equation 1398V P91(Vdg+Vj)558)8P9° — (Pg0)2By adopting the sign convention that compression is positive,1 - (av1/v)- P(V9+v)559)Bg 8P9°— V°9(P0)2Now,Vd9 = ThiS0TT0V ==nSPg = Pa+F+Pc (5.60)where,- saturation of oilS9 - saturation of gasn - porosity- atmospheric pressureP - pressure in oil- capillary pressureBy substituting these last expressions into equation 5.59iP91(HS0+S)561)B9 S(P9°)2 (Generally, in an incremental procedure the values used are estimated at the beginning of the increment. Therefore, from equation 5.61 the value of (S9/B) at thebeginning of an increment can be given as59 HS0+59562BgPa±P+PcChapter 5. Flow Continuity Equation 140If the capillary effects are neglected (i.e. P = 0), equation 5.62 will be similarto the one derived by Bishop and Henkel (1957). Equation 5.62 is slightly differentfrom the equation derived by Vaziri (1986). In Vaziri’s expression capillary pressurewas assumed to be a function of capillary radius and the capillary radius in turnwas assumed to be a function of saturation. He also included a derivative termof capillary pressure with respect to saturation which is not significant since thechanges in saturation will be very small. In addition, having this derivative termis inconsistent because, in his formulation to derive the flow equation, the capillarypressure was assumed constant over an incremental step. The expression given byequation 5.62 has a practical advantage because, in reservoir engineering, the variationof capillary pressure with saturation is readily available, whereas the capillary radius,critical capillary radius and surface tension values which are needed data for Vaziri’sexpression are not readily available. The capillary pressure P can be well representedby a power function similar to the ones used for relative permeabilities.= E1(S9 E2)E3 (5.63)where E1,E2 and E3 are constants.Therefore, by substituting equation 5.62 in equation 5.51, the equivalent compressibility can be written asS, SL (S9+HS0)CEQ=fl (5.64)5.7 Incorporation of Temperature EffectsThe fluid flow model described so far is for isothermal conditions and does not include temperature effects. The final equation obtained for multiphase flow (equation5.25) can be considered as an equation of volume compatibility which is derived fromChapter 5. Flow Continuity Equation 141the basic equation of conservation of mass. If the temperature effects are included,equation 5.25 will become (Srithar (1989), Booker and Savvidou (1985))kEQ V2F +— CEQ + aEQ = 0 (5.65)wherecEQ - equivalent coefficient of thermal expansion- temperatureThe equivalent coefficient of thermal expansion can be obtained by consideringthe coefficients of thermal expansion of the individual soil constituents and theirproportions of the volume, i.e.,0EQ = a8(1 — n) + nSa + flSoto + flSgQg (5.66)where subscripts s, w, o and g denote solid, water, oil and gas respectively.The coefficient of thermal expansion of solids, water and oil can be measured inthe laboratory. The coefficient of thermal expansion of gas can be obtained from theuniversal gas law. According to gas law,Povo Ply1= (5.67)To evaluate the coefficient of thermal expansion, only the volume change due totemperature change has to be considered. Thus, by assuming constant pressurel 80V0Vl—Vo_____= 80(5.68)By adopting the usual notation= (5.69)Chapter 5. Flow Continuity Equation 142Hence,a9 = (5.70)It should be noted that the temperature in the above equation should be absolutetemperature (i.e. in K).5.8 DiscussionIn this chapter, flow continuity equations for individual phases have been derived.Those have been later combined and an equivalent single phase flow continuity equation has been obtained. The effects of individual phases on compressibility and hydraulic conductivity have been modelled by equivalent compressibility and hydraulicconductivity terms. The flow continuity equation will be solved together with theforce equilibrium equation as a consolidation problem. The quantities of flow of individual phases can be estimated from the total amount of flow predicted and fromthe knowledge of the relative permeabilities.In reservoir engineering, only the flow equations for the individual phases (equations 5.21, 5.22 and 5.23) are generally solved and not in combined form as formulatedin 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 therewill be six degrees of freedom per node and the corresponding nodal variables areS, S0,S9, Pt,,, P0 and P9. The solution of the problem therefore requires the following three additional equations:(5.71)P0—P,L? = f(S, S0) (5.72)Pg— Po f(S0,S9) (5.73)Chapter 5. Flow Continuity Equation 143Compared to the flow analysis in reservoir engineering, the major disadvantage ofthe analytical model proposed here is that the treatment of multi-phase fluid as anequivalent single phase fluid. This kind of analytical model is adequate for coupledstress, deformati&n and flow analyses, but may not be effective if a detailed fluid flowanalysis is required. If detailed results about the flow are required, a separate rigorousflow analysis may be necessary. However, the results from the stress-deformationanalysis and the rigorous flow analysis should be looked at together to obtain acomplete picture.There are several advantages in the analytical model suggested in this study. Inreservoir engineering, the stress distribution and the deformation through the porousmedium are generally not considered. But the real problem at hand is a coupledstress, deformation and flow problem and the proposed analytical model in this studyaddresses all these concerns. The combined form of the flow continuity equationmakes the formulation simpler and significantly reduces the number of degrees offreedom, computation time and other such factors.Chapter 6Analytical and Finite Element Formulation6.1 IntroductionOil recovery by steam injection from heavy oil reservoirs is a coupled stress, deformation and fluid flow problem. Therefore, a realistic analytical model should includethe fluid flow behaviour and the mechanical behaviour of the sand matrix. Modellingof the stress-strain behaviour of the sand skeleton and the fluid flow behaviour withmulti-phase fluid has been described in chapters 3 and 5 respectively. This chapterdescribes the development of an analytical model which couples the stress-strain andfluid flow behaviour, and a solution scheme using finite element procedure.Basically, the problem in hand is considered as a consolidation phenomenon. Theanalytical models used in the consolidation analysis are mainly based on theoriesdeveloped by Terzaghi (1923) and Biot (1941). Terzaghi’s theory is restricted to aone dimensional problem under a constant load. Biot extended Terzaghi’s theory tothree dimensions and for any arbitrary load variable with time. Both Terzaghi’s andBiot’s theories assume a linear elastic stress-strain behaviour and an incompressiblepore fluid.Closed form solutions for the consolidation equations have been derived by anumber of researchers, but only for very simplified geometry conditions and for linearelastic material behaviour. For instance, De Josselin de Jong (1957) obtained a solution for consolidation under a uniformly loaded circular area on a semi-infinite soil.MacNamee and Gibson (1960) obtained solutions to plane strain and axisymmetricproblems of strip and circular footings on a consolidating half space. Booker (1974)144Chapter 6. Analytical and Finite Element Formulation 145derived solutions for square, circular and strip footings. A solution for consolidation around a point heat source in a saturated soil mass was derived by Booker andSavvidov (1985).The computer aided techniques such as finite element methods have made theconsolidation analysis possible for more complicated boundary conditions and formore realistic material behaviour. Sandhu (1968) developed the first finite elementformulation for two dimensional consolidation using variational principles. Sandhuand Wilson (1969), Christian and Boehmer (1970) and Hawang et al. (1972) used thefinite element method to solve the general consolidation problem. Ghaboussi and Wilson (1973) took the compressibility of the pore fluid also into account. Ghaboussi andKim (1982) analyzed consolidation in saturated and unsaturated soils with nonlinearskeleton behaviour and nonlinear fluid compressibility. Chang and Duncan (1983)took account of the variation of permeability due to the changes in void ratio andsaturation. Byrne and Vaziri (1986) and Srithar et al. (1990) included the nonlinear skeleton behaviour, nonlinear compressibility, variations in permeability and theeffects of temperature changes in the overall consolidation phenomenon. The analytical 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 sandskeleton, the effects of multi-phase fluid in compressibility and permeability and theeffects of temperature changes. The derived equations are solved by finite elementprocedure using Galerkin’s weighted residual scheme. The details of the formulationof the analytical equations and the finite element procedure are described herein.6.2 Analytical FormulationThe basic equations governing the consolidation problem with changes in temperatureare as follows:Chapter 6. Analytical and Finite Element Formulation 1461. Equilibrium equation.2. Flow continuity equation.3. Thermal energy balance.4. Boundary Conditions.The thermal energy balance will give the temperature profile and its variationwith time over the domain considered. In the analytical formulation presented inthis study, the thermal energy balance is not included. It has been solved separatelywith the heat flow boundary conditions by a separate program. The temperatureprofile and its variation with time is evaluated and considered to be an input to theanalytical model presented in this study. However, the effects of these temperaturechanges on the stress-strain behaviour and the fluid flow are included in the analyticalformulation.6.2.1 Equilibrium EquationUsing the conventional Cartesian tensor notation, the equilibrium of a given body isgiven by— F2 = 0 (6.1)where- total stress tensor- body force vectorsubscript j = -By assuming the geostatic body forces as initial stresses and considering only thechanges in body forces and stresses, the incremental form of the above equation canbe expressed asChapter 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- effective stress tensorP - pore pressure- Kronecker deltaFrom chapter 3, the incremental stress-strain relation including the effects of temperature 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 temperatureThe strains can be expressed in terms of displacements as= (U + (6.5)where- displacement vectorCombining equations 6.3, 6.4 and 6.5 and substituting into equation 6.2 yields,[Dk1 (Uk,I + LU1,k)] + a + ie1]— = 0 (6.6)Chapter 6. Analytical and Finite Element Formulation 1486.2.2 Flow Continuity EquationThe flow continuity equation for a multi-phase fluid including temperature inducedvolume changes was derived in chapter 5. The final equation (see equation 5.78) canbe written in tensor notation as[(kEQ)F]—1 +czEQ = 0 (6.7)wherekEQ - equivalent hydraulic conductivity tensorP- pore pressureU - displacement vectorCEQ - equivalent compressibilityaEQ - equivalent coefficient of thermal expansion6 - temperatureand 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 theconsolidation analysis. In these equations, the fundamental unknowns to be solvedare the displacements, U, and the pore pressure, P. The unknowns are solved byfinite element procedure using Galerkin’s weighted residual scheme.6.2.3 Boundary ConditionsTo define the problem, both the displacement and the flow boundary conditions mustbe specified. For the class of problems considered in this study, the following boundaryconditions can be specified.For the displacement boundary conditions, a part of the surface, can be subjected to known applied traction, while the reminder of the surface, SD, can besubjected to specified displacements, U, which may be zero.Chapter 6. Analytical and Finite Element Formulation 149For 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 simulatea free draining surface. The reminder of the surface, 5q is considered impermeable,i.e. there is no flow across the boundary.Mathematically, these boundary conditions can be expressed asa1n3=t for t0 (6.8)U = (J for t 0 (6.9)P=P for t>0 (6.10)for t0 (6.11)where n is the normal vector to the boundary surface and the bar symbol indicatesa prescribed quantity.To complete the description of the problem, the initial conditions must also bedefined. At t = 0, since there is no time for the fluid to be expelled, the volumechange in the pore fluid and in the soil skeleton must be equal. Thus,tSv = CEQ P at t = 0 (612)6.3 Drained and Undrained AnalysesThe drained and undrained analyses can be easily performed by considering onlythe equilibrium equation (equation 6.6). The flow continuity equation need not beconsidered under drained and undrained conditions. The drained analysis is quitestraight forward as it just involves solving the equilibrium equation. However, toperform an undrained analysis some modifications have to be made.Generally, the undrained response is analyzed with total stress parameters andthe analytical formulation has to be in terms of total stresses. If the pore pressuresare desired, they are commonly computed from the Skempton equation relating totalChapter 6. Analytical and Finite Element Formulation 150stress changes to pore pressure parameters. To use the effective stress formulation forundrained analysis, Byrne and Vaziri (1986) adopted an approach similar to the oneproposed by Naylor (1973). In this approach, the stiffness matrix for a total stressanalysis is obtained from the effective stress parameters and from the compressibilityof the fluid components as described in this section. The solution procedure is thencarried out in the usual manner for a total stress analysis to obtain deformations. Thepore pressures can be evaluated from the computed deformations using the relativecontributions of the pore fluid and the skeleton, without the use of the Skemptonequation.The incremental effective stresses are related to the incremental strains by thefollowing relationship:{o’} = [D’] {L} (6.13)where{ e} - strain vector{ o’} - effective stress vector[D’] - matrix relating effective stress and strainThe volumetric strain can be expressed as= {m}T{e} (6.14)where {m}T = {1 1 1 0 0 0} , is a vector selected such that only direct strains willbe involved in the volumetric strain.For undrained conditions, the volume compatibility requires that the volumechange in the skeleton equals the volume change in the fluid, i.e.,=(6.15)whereChapter 6. Analytical and Finite Element Formulation 151(Lc)j - volume change in the pore fluidn - porosityIn chapter 5, an equivalent compressibility has been obtained by considering allthe fluid components. Based on this approach, the changes in pore pressure can beexpressed as(6.16)where- change in pore pressureCEQ - equivalent compressibilitySubstitution of equations 6.14 and 6.15 into equation 6.16 gives= 1 {m}T{E} (6.17)CEQFrom the definition of effective stress{o-} = {&r’} + {m}i.P (6.18)Substituting equations 6.17 and 6.13 in equation 6.18 yields= [[D’l + 1 {m}{m}T] {e} = [D]{e} (6.19)CEQ ‘Equation 6.19 adds the contributions of both the skeleton and the pore fluid toexpress the stress-strain relation in terms of total stress. Thus, the matrix [D] for thetotal stress analysis is given by[D] = [D’] + 1 {m}{m}T (6.20)CEQChapter 6. Analytical and Finite Element Formulation 152Equation 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 isobtained from equation 6.17, once the deformations are computed.Byrne and Vaziri (1986) claimed that this method has definite advantages suchas 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 zeroand all of the load is carried by the pore fluid. For an incompressible fluid, CEQbecomes zero and the above formulation becomes ill-conditioned. However, this canbe overcome by setting the value of CEQ to a suitably low but finite value (Naylor,1973).6.4 Finite Element FormulationThe equations governing the consolidation with multi-phase fluid and temperatureeffects have been derived and are given by equations 6.6 and 6.7. The best method ofobtaining a solution for these equations is to use a numerical technique such as finiteelement 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 transformations. 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 thefinite element formulation.The choice of the different approaches depends on the type of the problem and theboundary conditions on one hand, and the knowledge of the mathematics involvedon the other hand. In this study, Galerkin’s weighted residual scheme is used todevelop the finite element formulations. The weighted residual scheme is quite straightforward, has relatively less mathematics involved, and is less error prone. In theChapter 6. Analytical and Finite Element Formulation 153Galerkin scheme only a single application of Green’s theorem is needed to obtain aset of integral equations. These equations can be easily turned into matrix form andsolved. However, it should be noted that regardless of the approach used, whetherweighted residual or variational principle, the end results will be the same.From the previous section, the governing differential equations to be solved are,(Uk,j + UZk)j + + — 0 (6.21)[(kEQ) +— CEQ F + aEQ = 0 (6.22)To develop the finite element formulation for these equations, the domain beinganalyzed is subdivided into a finite number of elements. The quantities of the fourindependent variables within each of the elements, U and F, are approximately represented by means of shape functions and their values at the nodes. The equations 6.21and 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, tosolve the resulting integral equations the shape functions for displacements and thepore pressure should be continuous. Hence, the displacement and the pore pressurefields within the element can be written asU = N e (6.23)(6.24)where5eT={S, S2, . . , S} (nodal displacements)q= {qi, q,.. . , q} (nodal pore pressures)U’- displacement field- pore pressure fieldChapter 6. Analytical and Finite Element Formulation 154N - shape functions for displacements- shape functions for pore pressuresU” and F” are approximate solutions and substituting these values into equations6.21 and 6.22 will not exactly satisfy the equations, but will give some residual errorsas follow:[Dkl ± + z1 + [D — (6.25)[(kEQ F], + U”,1 — CEQ F” + aEQ S (6.26)In the weighted residual scheme, these residual errors are minimized in somefashion to give the best approximate solution. Thus, for the best solutionjwrdv=O (6.27)wherer - residual errorw - weighting functionIn Galerkin’s scheme the weighting functions are chosen to be the same as theassumed shape functions. Then the following equations can be obtained to minimizethe residual errors r1 and r2:j Nr1dv = 0 (6.28)jNpr2dvzO (6.29)The strains and the derivative of the pore pressure within an element can bewritten asChapter 6. Analytical and Finite Element Formulation 155(6.30)= m’ (6.31)B Iq6 (6.32)wheremT ={1 1 100 0}B & B - shape function derivativesGreen’s theorem for integration involving two functions, and over the domaincan be expressed asJ ç V dIZ = j c (V) dF— j V V d1 (6.33)where, I’ is the boundary around and i is the normal to the boundary.By substituting equations 6.23 to 6.26 and 6.30 to 6.32 into equations 6.28 and6.29, and by applying Green’s theorem, one obtainsj BDB,4Sdv + / B’mNqdv = / NTds + j NFdv - / B’Dedv(6.34)— / BkEq Bqdv H- / NpTmTBuSdv — / CEQN1’Ndv = — / NpTaeq6dv (6.35)For a time increment t the above set of equations can be written in matrix formas[K] {i6} + [L] {q} = {A} (6.36)Chapter 6. Analytical and Finite Element Formulation 156[L]T {S} - t [E] {q} - [G] {q} = -{C} (6.37)where[K] =fBDBdv[U =fBmNdvrr1l p DTI r,L-’J — Jv -0p nEQ L’p[G] =fCEQNNpdv{LA}= f8 NTds + f NFdv— f B,D6ed{zXC} fvNp0eqMdtEquation 6.37 is considered over a time increment t, and therefore, the term qin that equation has to be expressed as,q = (1—a)qt + aqt (6.38)where a is a parameter corresponding to some integration rule. For example, a = 1/2implies trapezoidal rule, a = 0 implies a fully explicit method and a = 1 gives a fullyimplicit method. Booker (1974) showed that for an unconditionally stable numericalintegration a 1. In the formulation here, the value of a is assumed to be 1,i.e. a fully implicit method. Thus, the term q in equation 6.37 can be given as,q = qt+t = qt + q (6.39)Substitution of equation 6.39 into equation 6.37 yields,[U]T {8}— t [E] {qt + Lq} — [C] {q} = —{zC} (6.40)By rearranging the terms,[U]T {zS}— zSt [[E] — [G]] {q} = —{zC} + [E]qtt (6.41)Chapter 6. Analytical and Finite Element Formulation 157By combining equations 6.36 and 6.41 and writing them in a full matrix formgives,[K] [L] z8 1A ‘1 (6.42)[L]T [[E] Lt— [G]] Lq J [E] /.t{qt} — {tC} JBy changing the notation, equation 6.42 can be written in the usual matrix formas,[K] [L] ILS1 1A1(6.43)[L]T [E’] q J LC’ Jwhere,{LW’} = tt[E}{qt}- {C}[E’] =t[E]-[G]Equation 6.43 gives the matrix equation to be solved for an element. From theelement matrix equations a global matrix equation is formed and solved for displacement and pore pressure unknowns. Stresses and strains are then evaluated from thedisplacements.It should be noted that it may not be possible to use the above consolidationroutine to get the initial condition results, i.e. at t = 0. This is because for Lt = 0{zC’} = 0 and [E’] = [G].If the fluid is incompressible [G] will become zero and equation 6.43 will become illconditioned. For this situation an appropriate solution can be obtained by assuminga small value for t. This will circumvent the ill-conditioning. However, a betterway to get around this problem is to use the undrained routine to obtain the initialcondition and then use the consolidation routine.Chapter 6. Analytical and Finite Element Formulation 1586.5 Finite Elements and the Procedure AdoptedThe principal steps and the details such as obtaining shape functions, its derivatives,formulation of stiffness matrix, numerical integration, etc., can be found in any standard finite element text book. Therefore, only a summary with some discussions oncertain 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 inan existing 2-dimensional finite element code, CONOIL-Il (Srithar (1989)) and alsoin a new 3-dimensional finite element code, CONOIL-Ill. The following subsectionsaddress the key aspects in the development of these finite element codes.6.5.1 Selection of ElementsThe choice of the finite elements has been an important issue when analyzing consolidation problems. Different researchers used different element types. Sandhu andWilson (1969) introduced a composite element, consisting of a six-noded triangle forthe displacements expansion, and only three nodes being used for the pore pressure expansion. The displacements varied quadratically over the element, while the stressesand strains obtained by differentiating the displacements varied linearly. Since thepore 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 componentsand pore pressures. Yooko et al. (1971a) used several different elements, all of whichused the same expansion for the displacements and for the pore pressures. This makesN = 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 axisymmetric triangular ring element an a four noded rectangle. However, Yooko etal. (1971b) had difficulties in obtaining reasonable results for the initial undrainedconditions. Sandhu et al. (1977) also compared several finite elements and concludedChapter 6. Analytical and Finite Element Formulation 159that the elements which had the same expansion for displacement and pore pressuresdo not give satisfactory answers at the initial stages of consolidation. However, theyclaimed that at later stages of consolidation, the differences in the results for differentelement types are insignificant.Ghaboussi and Wilson (1973) used an isoparametric element of four nodes withthe standard expansion for pore pressures and two additional nonconforming degreesof freedom for the displacement expansion. The two additional degrees of freedom areeliminated by static condensation after the element stiffness is completed. However,this procedure does not give the same expansion for pore pressures and stresses, butuses a lower order expansion for pore pressures than for displacement.In the 2-dimensional finite element code employed in this study, element types similar to those proposed by Sandhu and Wilson (1969) are used. Figure 6.1 shows thetwo different triangular elements used for consolidation analysis in the 2-dimensionalcode. Figure 6.2 shows the element types available in the 3-dimensional code. Theeight-noded brick element uses the same expansions for pore pressures and displacements, whereas the 20-noded brick element uses different shape functions for displacements and pore pressures.6.5.2 Nonlinear AnalysisThe solution of the nonlinear problems by the finite element method is usuallyachieved by one of the following techniques:1. Incremental or stepwise procedures2. Iterative or Newton method3. Step-iterative or mixed proceduresThe method employed herein is a form of the mixed procedure which followsthe midpoint Runge-Kutta or modified Euler method. In this scheme, two cyclesChapter 6. Analytical and Finite Element Formulation 160A Displacement nodes (2 d.o.f)Q Pore pressure nodes (1 d.o.f)Linear strain triangle6 displacement nodes3 pore pressure nodes6 nodes and 15 d.o.f.Cubic strain triangle15 displacement nodes10 pore pressure nodes22 nodes and 40 d.o.f.Figure 6,1: Finite Element Types Used in 2-Dimensional AnalysisChapter 6. Analytical and Finite Element Formulation5I161• Corner nodes = 8D.o.f. per node = 4Internal nodes = 0D.o.fper node = 08-Nodded Brick Element• Corner nodes = 8D.o.f per node = 4A Internal nodes = 12D.o.f. per node = 320-Nodded Brick Element8A157,.14A61882044 119Li16s34 A-4-10/11A12Figure 6.2: Finite Element Types Used in 3-Dimensional AnalysisChapter 6. Analytical and Finite Element Formulation 162of analysis are performed for each load increment. In the first cycle of analysis,parameters based on the initial .conditions of the increment are used.. At the end offirst cycle, parameters at the midpoint of the load increment are computed. In thesecond cycle, the midpoint parameters are used to analyze the load increment and thefinal results are evaluated. To obtain more accurate results, this process would haveto be continued until the difference between successive results satisfies the specifiedtolerance. Such an iterative procedure can increase the computer time drasticallyand therefore, was not employed. However, an improvement in the results is madeby estimating the imbalance load at the end of second cycle and adding that to thenext load increment.6.5.3 Solution SchemeSelection of the method for solving the simultaneous algebraic equations is a majorfactor influencing the efficiency of any finite element program, and there are variety ofsolution techniques to choose from. Essentially, there are two classes of methods; oneis the direct solution methods and the other is the iterative solution methods. Thedirect methods use a number of exactly predetermined steps and operations, whereasthe iterative methods make an approximation to solve the equations.The most effective direct solution methods are basically variations of the Gaussianelimination method. Most of the methods take advantage of specific properties of thestiffness matrix, its symmetry, its positive definiteness or its banded nature to reducethe number of operations and the storage requirements to accomplish a solution.Bathe and Wilson (1970) and Meyer (1973) discussed the relative merits of the currentpopular methods, and both of these references contain extensive bibliography.The frontal solution scheme for symmetric matrices (Irons, 1970) and for unsymmetric matrices (Hood, 1976) have been employed in the finite element codes. In thefrontal solution scheme, the element stiffness matrices are assembled and solved byChapter 6. Analytical and Finite Element Formulation 163Gaussian elimination and back substitution process, but the overall global stiffnessmatrix is never formed. The variables are introduced at a later stage and eliminatedearlier than in most of the other direct solution methods. Since the variables areeliminated as soon as conceivably possible, the operations with zero coefficients areminimized and the total arithmetic operations are fewer. As a result, it is faster andrequires less core memory than band routines. In addition, it is not necessary toapply a stringent node numbering scheme. Its efficiency is essentially a function ofelement numbering.Theoretically, the frontal solution scheme will always perform better or at leastas 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 literatureto substantiate this claim (eg: Sloan 1981; Light and Luxmore, 1977; Hood, 1976).The frontal solution scheme is specially attractive for unsymmetric matrices becauseless computer storage is required. The stress-strain model considered in this studydeals with a nonassociated flow rule which results in an unsymmetric stiffness matrixand therefore, using the frontal solution scheme has a definite advantage.The main disadvantage of this method is the complexity of the internal bookkeeping. However, the bookkeeping is a programming problem and does not concernthe user. Another limitation of this technique may be its dependence on the elementnumbering sequence. Although it is rather easier to number the elements in a logicalmanner relative to numbering the nodal sequence, it does place some effort on theuser. However, the difficulty can be easily dealt with, if some form of front widthminimizer is incorporated in the program. There are different front width minimizingschemes 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 the2-dimensional finite element code.Chapter 6. Analytical and Finite Element Formulation 1646.5.4 Finite Element ProcedureA broad overview of the procedures followed in both, the 2-dimensional and 3-dimensional programs is given in the flow chart shown in figure 6.3. The steps involvedin the finite element procedure can be summarized as follows:1. Basic data such as the number of nodes, elements and material types are readand the required storage is allocated for the variables.2. All other data such as nodal coordinates, temperatures, element-nodal information and model parameters are read.3. The initial conditions are read and the initial stresses, strains, pore pressuresand 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 modulibased on the initial stresses.6. The equations are solved using the frontal solution scheme. For linear andnonlinear elastic stress-strain models, the solution scheme for symmetric matrices is used. For the elasto-plastic stress-strain model, the solution scheme forunsymmetric matrices is used.7. Increments in the stresses and strains for the load increment are calculated andif it is the first cycle of analysis, new moduli are evaluated based on the stressesat the mid point of the increment.8. If it is the first cycle of analysis, steps 5 to 7 are repeated once more using newmoduli for step 5.9. The stresses, strains and pore pressures and other relevant results are calculatedand the desired results are printed.Chapter 6. Analytical and Finite Element Formulation 165C Start DRead basic data and allocate storagefor principal arraysRead and set the initial conditionsRead relevant data for theload incrementEvaluate stiffness matrix and load vectorSolve for displacements and pore pressuresEvaluate the changes in stresses,strains and pore pressuresUpdate relevantIs this the last cycle No variables toof analysis’ average valuesYesriJpdate all the results and printNoNoLast increment?YesC Stop DFigure 6.3: Flow Chart for the Finite Element ProgramsChapter 6. Analytical and Finite Element Formulation 16610. If the current stress state exceeds the strength envelope, or if there is strainsoftening, load shedding vector is computed.11. Steps 5 to 9 are repeated until all the elements satisfy the failure criterion or inother words, until the load shedding is converged.12. The imbalance loads at the end of the increment are calculated and added tothe 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 conditionsfor the next load increment.6.6 Finite Element ProgramsThe finite element programs have been written in FORTRAN-77 and are portableto any operating platforms. There are two separate programs; CONOIL-Il whichis 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. Although these finite element programs have been developed with special attention paidto the problems in oil sand, they are capable of doing general drained, undrained andconsolidation analyses effectively. Both programs are capable of analyzing excavationsas well. Brief descriptions of these programs are given in this section. Applicationsof the programs are discussed later in chapter 7.6.6.1 2-Dimensional Program CONOIL-IlThe 2-dimensional program CONOIL-Il was originally developed by Vaziri (1986)based on the program CRISP (University of Cambridge). It was later modified byChapter 6. Analytical and Finite Element Formulation 167Srithar in 1989 with an improved formulation for temperature analysis. CONOILII 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 interiornodes. It also renumbers the elements and nodes to minimize the front width andcreates a input file for the main program, containing the relevant information aboutthe finite element mesh. The program also has some special features. The triplematrix product as suggested by Taylor (1977) is adopted in the formation of stiffnessmatrix, and this will eliminate all the unnecessary arithmetic operations which willresult in zero coefficients. The geometry program consists of 11 subroutines and themain program consists of 58 subroutines. The names of the subroutines and theirfunctions are presented in appendix C.Grieg et al. (1991) developed a pre/post processor package, COPP, for CONOILII to facilitate viewing and plotting the CONOIL-Il input and output data. COPPis menu driven, very user friendly and provides many options for the user.6.7 3-Dimensional Program CONOIL-IllThe 3-dimensional program CONOIL-Ill has been developed from scratch followingthe same sequence of procedures as the 2-dimensional one. However, compared tothe 2-dimensional program, the 3-dimensional program has less special features, andit does not have a post processor yet. The 3-dimensional program comprises 43subroutines. The names of the subroutines and their functions are given in appendixC. A User manual and some example problems are presented in appendix F.Chapter 7Verification and Application of the AnalyticalProcedure7.1 IntroductionThe analytical procedure described in chapter 6 has been incorporated in the finiteelement program, CONOIL. The main intention of this chapter is to verify and validate the finite element program, and to demonstrate its applicability. The programdeals 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 theprogram would be to consider each aspect separately. The program is verified hereby considering some particular problems for which theoretical solutions are available.Once verified, the program is validated by comparing some experimental results withpredictions from the program. Then, the program has been used to predict the responses in a oil recovery problem. A problem concerning pore pressure redistributionafter liquefaction has also been analyzed to show the applicability of the program toother geotechnical problems.7.2 Aspects Checked by Previous ResearchersThe two dimensional version of the finite element program CONOIL has been used atthe University of British Columbia since 1985, with improvements being made fromtime to time. Cheung (1985), Vaziri (1986) and Srithar (1989) have demonstratedthe capability of the program on a number of aspects. Since those aspects are kept168Chapter 7. Verification and Application of the Analytical Procedure 169intact with the improvements made in this study, those verifications and validationsare still valid. These are briefly described herein.The general performance of the program in predicting stresses and strains hasbeen verified by Cheung (1985), by considering a thick wall cylinder under planestrain conditions. Closed form solutions for this problem have been obtained fromTimoshenko (1941). The results from the program and the closed form solutions arein 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 considered. Sobkowicz (1982) carried out triaxial tests to predict the short term undrainedresponse, i.e, no gas exsolution and the long term undrained response, i.e., with complete gas exsolution, The comparisons of the test results with the program results areshown 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 verifiedby Vaziri (1986). The closed form solution developed by Gibson et al. (1976) fora circular footing resting on a layer of fully saturated, elastic material with finitethickness has been considered for the verification. A comparison of the computedresults and the closed form solutions, shown in figure 7.4, demonstrate that they arein very good agreement.Srithar (1989) modified the procedure for thermal analysis in the original CONOILformulation. He verified the new formulation under drained and transient conditions.The closed form solution presented by Timoshenko and Goodier (1951) for a longelastic cylinder subjected to temperature changes has been considered to verify theformulation under drained condition. The closed from solution and the finite elementresults are shown in figure 7.5 and are in remarkably good agreement.To verify the formulation for thermal analysis under transient conditions, a closedform solution was derived by Srithar (1989) for one dimensional thermal consolidationChapter 7. Verification and Application of the Analytical Procedure 1700e000a)closed formo o 0programQca)I I I I I I I I V0Ic’J246810Radii (r/r0)E = 3000 MPaI’— 1/3initial stress : or = o. = 6000 kPafinal stress : o = 2500 kPainside radius : r = 1 inFigure 7.1: Stresses and Displacements Around a Circular Opening for an ElasticMaterial (after Cheung, 1985)Chapter 7. Verification and Application of the Analytical Procedure 171I.’.-Cb000CFigure 7,2: Comparison1985)of Observed and Predicted Pore Pressures (after Cheung,0040 60 80 100 120 140Total Stress (kPa) (X1O’ )Chapter 7. Verification and Application of the Analytical Procedure 1720Cl2.4.)s-Ia)>0___________________.—s-I-I-)0s-I0xc_______0lab data0l.a.4-’Cl)-4-’00__ __ _ _ __ _ __C0 20 40 60 80Effective SigmaP (kPa) (X101 )Figure 7.3: Comparison of Observed and Predicted Strains (after Cheung, 1985)Chapter 7. Verification and Application of the Analytical Procedure 173I I0.25Analytical Solution. Finite Element0.30 - Analysis0.35-________r0.40 -045 xI30y/30DIE—i.5— 0.00.50I I I Iia-4 io io_2 1.0 10CtVTv-a)Amount of settlement0.0 I I IAnalytical Solution0.2-‘%. ‘b%, 0 Finite Element%, %\ Analysis0.4Uy/B—0 vO.3 v—0.00.6— D/E — 10.8 -1.0 I Iia—4 i— 10—2 ia-’ 1.0 10CtV7 -—U2b) Degree of settlementFigure 7.4: Results for a Circular Footing on a Finite Layer (after Vaziri, 1986)Chapter 7. Verification and Application of the Analytical Procedure 1743000 — LaSymbols— CONOIL—IlSolid lines— Closed Form2000 —CVertical Stress1000—Radial Stressci)(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 175with a uniform temperature rise. The closed form solution was obtained by makinganalogy to the closed form solution by Aboshi et al. (1970) for a constant rate ofloading. Figure 7.6 shows the closed form solutions and the program results and theyagree very well. [n the figure, z denotes the depth at which the results are consideredand H denotes the total depth.The performance of the program for undrained thermal analysis has been validatedby comparing the experimental results on oil sand samples in a high temperatureconsolidometer obtained from Kosar (1989). Computed and measured results showgood agreement as illustrated in figure 7.7.7.3 Validation of Other AspectsIn this research work, a new elasto-plastic stress-strain model has been developed andincorporated in the finite element code. This will realistically model the dilation andthe loading-unloading sequences encountered in oil sands. To validate the program’scapability to model the dilation phenomenon, the triaxial test results on oil sandgiven by Kosar (1989) have been considered. The triaxial test has been modelled byfour triangular elements as shown in figure 7.8. An axisymmetric analysis has beencarried out with the relevant boundary conditions as shown in figure 7.8. The modelparameters used are listed in table 7.1. The predicted and the measured results arecompared in figure 7.9. Also shown in that figure are the results using a hyperbolicmodel. It can be seen from the figure that the shear stress versus axial strain responsecan be very well captured by both the elasto-plastic and hyperbolic models. Butthe 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 sandobtained from Negussey (1985) have been considered to validate the loading-unloadingoperation of the program. The triaxial test specimen has been modelled by fourChapter 7. Verification and Application of the Analytical Procedure 176L’z/H = 0.87530 —0 0 0 0 cip 0 00- 2uCl)Cl,Q)0ci)00000 CONOIL—Ilclosed formsolutions0 —,LHz/H = 0.520 —C0ci)C_0 0(1)ci)u-i0000— i I I I I0 1000 2000 3000 4000Time(s)Figure 7.6: Pore Pressure Variation with Time for Thermal Consolidation (afterSrithar, 1989)Chapter 7. Verification and Application of the Analytical Procedure 1778-Test results /00000 CONOIL—Il //06—a)-Cc-- 0C-)a) -E2 4—0>> -D -E0000I I I I I I I I I I20 70 120 170 220Temperature(° C)Figure 7.7: Undrained Volumetric Expansion (after Srithar, 1989)Chapter 7. Verification and Application of the Analytical Procedure 178Table 7.1: Parameters for Modelling of Triaxial Test in Oil Sand(a) Elasto-Plastic ModelElastic kE 3000n 0.36kB 1670m 0.36Plastic Shear 0.72? 0.54! 0.33KG 1300np -0.66Rf 0.80(b) Hyperbolic ModelkE kB m R1100 0.49 700 0.47 0.6 49 13Chapter 7. Verification and Application of the Analytical Procedure 17901.5 cmFigure 7.8: Finite Element Modeffing of Triaxial TestChapter 7. Verification and Application of the Analytical Procedure350030002500‘a 20000S‘015001000500‘IWI180Figure 7.9: Comparison of Measured and Predicted Results in Triaxial CompressionTest.Elasto-PlasticHyperbolicExperimentalEa (%)-0.2-0.11€_a (%)Chapter 7. Verification and Application of the Analytical Procedure 181elements as shown earlier in figure 7.8. The model parameters used are listed in table7.2. The measured and predicted results agree very well as shown in figure 7.10.Table 7.2: Model Parameters Used for Ottawa SandElastic kE 3400m 0.0kB 1888m 0.0Plastic‘i 0.49) 0.85IL 0.26KG 780np -0.238R1 0.70Modelling of the three-phase pore fluid is the other important aspect where majorimprovements have been made in the analytical formulation in this study. There isno theoretical or experimental solutions available to verify or to validate the overallformulation for the modeffing of the three-phase pore fluid. However, validations forthe analytical representation of the relative permeabilities have been made and werepresented in chapter 5.74 Verification of the 3-Dimensional VersionThe 3-dimensional version of CONOIL is newly written following the same operationalframework as the 2-dimensional version. Since the 3-dimensional program is new, itis necessary to check that the performance of the program in all aspects agrees withthe intended theories, as was proven for the 2-dimensional version. The problemsconsidered to verify the 3-dimensional code were similar to those used to verify the2-dimensional code and all gave satisfactory results. Since the verifications are similarto those presented in the previous sections, they are not repeated here. However, theChapter 7. Verification and Application of the Analytical Procedure 182350300250a. 200150100500.00 0.20Figure 7.10: Comparison of Measured and Predicted Results for a Load-Unload Testin Ottawa Sand0.05 0.10 0.15El (%)Chapter 7. Verification and Application of the Analytical Procedure 183verification 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 uniformtemperature increase at a rate of 100°/hr. The boundary and the drainage conditionsare also shown in figure 7.11. The closed form solution for the pore pressure at a depthz under one dimensional thermal consolidation is given by the following equation(Srithar, 1989):16 M n— 1 mrz I (m2ir’\ ‘1pr3 Tm1,3,..—i sin2H — exp — T (7.1)wherep - pore pressure at distance z at time tT - time factorn - porosity- change in temperature at time tM - constrained modulusa1 - coefficient of volumetric thermal expansion of liquida8 - coefficient of volumetric thermal expansion of solidsThe soil properties used for this analysis are given in table 7.3. The soil is assumedto be linear elastic. The predicted pore pressures have been compared with theanalytical solutions at two different depths, at z/H = 0.75 and at z/H 0.5. Theresults agree very well as shown in figure 7.12.7.5 Application to an Oil Recovery ProblemHaving verified the performance of many aspects of the finite element program, it hasbeen 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 SandsChapter 7. Verification and Application of the Analytical Procedure 184imH G__21®22____313 ® 1411 H=lm_1O75 6z 4.. 3.1A BAB, BC, CD, DA - Totally FixedAE, BF, CG, DH - Vertically FreeEF, FG, GH, HE - Drain BoundariesFigure 7.11: Finite Element Mesh for Thermal ConsolidationTable 7.3: Parameters Used for Thermal Consolidationfl V a1 a5 k M Hm3/m°C m3/m°C rn/s MPa rn0.5 0.25 1x103 1x105 2x106 18.3 1Chapter 7. Verification and Application of the Analytical Procedure 18535z/H = 0.75___________________________________________(a)-30-252001.0I-.0010Symbols - Program5 Line- Closed form0 I I0 500 1000 1500 2000 2500 3000 3500Time (s)30z/H = 0.5 (b)4ZSymbols - ProgramLine - Closed form0 I I I I I I0 500 1000 1500 2000 2500 3000 3500Time (s)Figure 7.12: Comparison of Pore pressures for Thermal ConsolidationChapter 7. Verification and Application of the Analytical Procedure 186Technology and Research Authority (AOSTRA) is considered herein for analysis.The UTF uses a steam assisted gravity drainage process with horizontal injectionand production wells. A brief description of the UTF and the problem to be analyzedare 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 currentlybeing used to test the shaft and tunnel access concept for bitumen recovery in deepoil sand formations. The geological stratification at the UTF comprises a number ofsoil layers. However, it can be simplified as consisting of three different soil types, ina broad sense. Devonian Waterway formation limestone exists below a depth of 165m. Overlying the limestone is the McMurray formation oil sand which is about 40 mthick. 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 figures7.13 and 7.14 respectively. There are two shafts accessing the tunnels in limestonebeneath the oil sand layer. The tunnels were constructed in the limestone at a depthof about 178 m with the roof being about 15 m below the limestone-oil sand interface. Three pairs of horizontal injection and production wells were drilled from thetunnels up into the oil sands at about 24 m spacing. A vertical section of the wellpairs was instrumented with thermocouples for measuring temperatures, pneumaticand vibrating wire piezometers for measuring pore pressures and extensometers andincinometers for measuring horizontal and vertical displacements.Figure 7.15 shows a vertical cross-sectional view (section A-A’ in figure 7.14) ofthe three well pairs. Modelling of all three well pairs with their steaming historiesand with the detailed geological stratification would be complex as the steamingof different well pairs started at different times. To illustrate the problem and todemonstrate the applicability of the program in a simple manner, only one well pairis considered here for analysis.Chapter 7. Verification and Application of the Analytical Procedure 187Figure 7.13: A Schematic 3-Dimensional View of the UTF (after Scott et al., 1991Chapter 7. Verification and Application of the Analytical Procedure 188Shaft#1ITObservation Tunnel iInjector/Producer Wellpairs......—Section A-A’ —* Geotechnical A A’Cross Section________4Figure 7.14: Plan View of the UTF (after Scott et al., 1991)—.......................V..:•:••...•.•...•..•.••.•.•.•.•.•.•........CDCD -c0C.11gg0<30CDCD=U)CDCD0=CDCD0.oC;’C;’33300Chapter 7. Verification and Application of the Analytical Procedure 190To analyze the oil recovery with one well pair, the shaded region in figure 7.15 ismodelled by finite elements. The finite element mesh consisted of 240 linear straintriangular elements as shown in figure 7.16. Plane strain boundary conditions areassumed. The injection and production wells are modelled by nodes with known porepressures. 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 tobe at 2000 kPa (500 kPa above the in-situ pressure). The parameters used have beenobtained from laboratory test results reported by Kosar (1989) and from AOSTRAand are listed in table 7.4. The gas saturation is assumed to be zero. i.e., the porefluid is assumed to comprise only water and bitumen. The bitumen saturation isassumed to be 70 %.The temperature-time histories of the nodes have also to be specified as an inputto the program. These were obtained from the field measurements made at theUTF. The temperature contours in the oil sand layer at different times are shown infigure 7.17. The steam chamber which is the region in the oil sand layer where thetemperature is the same the steam temperature, grows with time as shown in thefigure. At time t = 30 days, the steam chamber extends to a distance of about 10 mhorizontally 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. Thepredicted excess pore pressures in the oil sand layer are shown in figure 7.18. Theinjection and production wells are also indicated in the figure. Figure 7.18 (a) showsthe excess pore pressure contours at 10 hours after the steam injection started. Onlya small region adjacent to the injection well experiences significant changes in thepore 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 asshown in figures 7.18 (b) and (c) indicating the growth of the steam chamber which250.0—200.0—-150.0Lii100.0—50.0—0.0—22/7///////22/?/;c0.01I—II100.0II——200.0IIII300.0C-. 0Distance(m)Figure7J6:FiniteElementModellingoftheWellPairI-.Chapter 7. Verification and Application of the Analytical Procedure 192Table 7.4: Parameters Used for the Oil Recovery Problem(a) Soil ParametersElastic 3000n 0.36kB 1670m 0.36Plastic Shear 171 0.75J.:l7 0.13). 0.53t 0.31KG 1300rip -0.66R1 0.73Plastic Collapse C 0.00064p 0.61Other e 0.6k(m2) 1.0 x 10—12a8(3/°C) 3.0 x 1O(b) Pore Fluid ParametersB 5.0 x iOB0 2.5 x iOcx(m3/m°C) 3.0 x i0cz0(/°C) 3.0 x iO1u,o(Pa.s) 20S 0.20.2kro = 2.769(0.8 — S)’996krw = 1.820(S — 0.2)2.735EIevLon(m)EIevLon(m)EIevLon(m)cyq CD CD p CD C)0 0 cn CD 0 (j) p CDp I-’.0NJNJ-01QDQNJNJ-01000NJNJ-010000NJ 0(I) C,D 0 (D 3NJ 00 0 NJ 0 NJ 0(0 C-,f DCJ00(00(0 C-,:3 0 301 0(11 00c,zChapter 7. Verification and Application of the Analytical Procedure 194is also implied by the temperature contours in figure 7.17. The 1000 kPa excess porepressure contour from the field measurements is compared with the predicted contoursin figure 7.19. It can be seen from the figure that the measured zone of 1000 kPa islarger than the predicted zone. However, the shapes of the pore pressure contoursare 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 moreconstrained in the horizontal direction, the horizontal stresses increase. The verticalstresses also increase, but not as much as horizontal stresses. The pattern of the stresscontours 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 thefailure stress state is shown in figure 7.22. It appears that the shape of the steamchamber and the corresponding temperature increases create higher shear stresses inthe region above the steam chamber. This is implied by higher stress ratios and amaximum stress ratio is about 0.45 is predicted in the region about 15 m above theinjection well. Since the predicted stress ratios are well below unity, there would notbe any failure.Figure 7.23 shows the horizontal displacement along a vertical line at 7 m fromthe wells, at time t = 30 days. Also shown in the figure are the field measurementsmade 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 atsome locations, but in the overall picture, the predictions are in reasonable agreementwith the measurements.The variation of vertical displacements with the distance from the wells at theinjection well level is shown in figure 7.24. Maximum displacement of 21 mm ispredicted at a distance 15 m from the well. There is no field measurements availablethat could give the results due to the steaming in a single well pair. The verticalCD I. 0 CD CD Cl) U) I- CD 0 U) CD 0 C,, CDEIevLon(m)-01EIevLonCm)NJNJ-N01NJ 0U Cl) C-, D C) CD 3NJ 0U Co C—, D 0 CD 3NJ 0EIevLon(rn)I.U Cl) C-, 0 CD 301 001 001 &c3 0cJc300CC)01Chapter 7. Verification and Application of the Analytical Procedure 19650S40C01::60Figure 7.19: Comparison of Pore Pressures in the Oil Sand Layerdisplacement measurements were made in bore holes located in between the wellpairs and therefore, those measurements cannot be considered as the result due tothe steaming from a single well pair. Moreover, those measurements were very erraticand 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 isachieved after 20 days. The predicted steady state flow rate is 5.18m3/m/day. In theinitial stages of production, more water will be produced than oil because much ofthe bitumen will be immobile. With time, the temperature will increase, the viscosityof 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, theamount of bitumen produced will become less as it is replaced by water.0 10 20 30 40 50DsLncG (m)oq CD 0 N 0 C,) I- CDEevoLon(m)NJNJ-01EIev3Lon(m)N)NJ-0100C,, E3 C) CD 3NJ 0C,)C-, a, :3 0 CD 3NJ 0ci) C) 0) :3 0 CDEIevLon(m)N)NJ-010000NJ 001 03(ii 0(ii 0c3 00Chapter 7. Verification and Application of the Analytical Procedure 198- () t tlOhrs40C0/6ØO30LU20 IDstance (m)0 10 20 30 40 50 60C030LU40200 10 20 30D9Lance (m)4040EC0>3020•10 20DLnce (m)40 60Figure 7.21: Vertical Stress Variations in the Oil Sand LayerChapter 7. Verification and Application of the Analytical Procedure 19960606040010 20 30 40DsLance (m)40C030Li200 10 20 30DLence (m)4040C030Li200 10 20 30DLncG (m)40Figure 7.22: Stress Ratio Variations in the Oil Sand LayerChapter 7. Verification and Application of the Analytical Procedure 20060Symbols- Field Measurements• Line- Prediction50 ••0••20 I I I I0 5 10 15 20 25Displacement (mm)Figure 7.23: Comparison of Horizontal Displacements at 7 m from WellsChapter 7. Verification and Application of the Analytical Procedure 2012520E151050600 10 20 30 40 50Distance (m)Figure 7.24: Vertical Displacements at the Injection Well LevelChapter 7. Verification and Application of the Analytical Procedure160140120C,,1000U.o 80C0E<6040200Time (days)2020 5 10 15 20 25 30 35Figure 7.25: Total Amount of Flow with TimeChapter 7. Verification and Application of the Analytical Procedure 203The quantity of flow given in figure 7.25 is the total flow of water and oil. Unfortunately, the procedure adopted in the analytical formulation will not give individualamounts of flow directly. However, approximate estimations of the individual amountsof flow of water and oil can be calculated by knowing the area of different temperaturezones and the relative permeabilities. Details of the individual flow calculations aredescribed in appendix D. The individual flow rates of water and oil with time understeady state conditions are given in figure 7.26. The total amount of oil produced withtime in the production well is shown in figure 7.27. It should be noted that the flowpredictions presented here are approximate because of the assumptions made aboutthe 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 PermeabilityTo show the importance of this type of analytical study, the same oil recovery problemis analyzed with reduced permeability. The absolute Darcy’s permeability of theoil sand matrix is reduced from 10’2m to 103m2. The predicted pore pressurecontours 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. Thisis because the pore fluid expands more than the solids and since the permeability islow, there is not enough time for the expanded pore fluid to escape, thus, the porepressure increases. The worst condition occurs after 5 days and a maximum excesspore 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 thosein the earlier analysis. Again, the worst condition is predicted after 5 days and aregion with stress ratio of 0.7 is shown in the figure. The same kind of results wouldChapter 7. Verification and Application of the Analytical Procedure 204•.8-Ia)0UEa)0U-5.554.543.521.510.5012 5 10 20 50 100 200 500Time (days)(a) Flow Rate of Water20Time (days)(b) Flow Rate of Oil500Figure 7.26: Individual Flow Rates of Water and OilChapter 7. Verification and Application of the Analytical ProcedureECE0Li.00E,2205350504030200 50 100 150 200 250 300Time (days)Figure 7.27: Total Amount of Oil FlowCo C-,D D C) CDU Co C-, D C) CD 3EIevton(m)RDRD-01RDRD-R01EIevLon(m)F—(LHevLoncmRDRD-(ii‘SC‘SC‘SC0CD —3co 0 CD fri CD Cl) fri CD 0 ‘-I U) I-. Cl)r\D 0RD 03RD 0U C/) C-, Z3 0 (0 3RD 0I01 001 001 00Chapter 7.C030LUSC030a)LUSC030LUVerification and Application of the Analytical Procedure 207606060Figure 7.29: Stress Ratio Variation for Analysis 240200 10 20 30DtLnce (m)4040200 10 20DsLncG (m)30 4040200 10 20DLance (m)40Chapter 7. Verification and Application of the Analytical Procedure 208have been predicted if the permeability was kept the same and the rate of heatingincreased. The detailed results show that the stress ratio of one of the elements inthe highest stress ratio region reached unity indicating shear failure. Since the regionof 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 deformationsand 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 treatmentfor oil recovery projects. This type of analysis provides important information aboutthe rate of heating, possible failure zones, deformations, stability of the wells etc.,beforehand. Without an analytical treatment, these concerns have to be tested in thefield on a trial and error basis, which would be very costly.7.6 Other Applications in Geotechnical EngineeringEven though the finite element program CONOIL was developed for analyzing problems related to oil sands, it can also be applied to other potential geotechnical problems. An example problem which involves pore pressure migration after liquefactionis described herein.Generally, loose sands are susceptible to liquefaction in the event of an earthquakeand to prevent such liquefaction, loose sand deposits are commonly densified. Thedensified zone in a loose sand deposit will only be stable provided high excess porepressures from the surrounding liquefied sands do not penetrate it during and afterthe earthquake. This concern is examined herein with different densification schemesused in practice.A typical soil profile for Richmond, British Columbia, was considered in the analysis. The soil profile comprised 3 m of clay crust, underlain by 15 m of loose sand andfollowed by 5 m of dense sand as shown in figure 7.30. The earthquake is assumed toChapter 7. Verification and Application of the Analytical Procedure 209generate 100% pore pressure increase in loose sand and 30% pore pressure increase indense sand zones. A hyperbolic stress-strain model was considered and the materialparameters used in the analysis are given in table 7.5. Three cases which representthree different densification schemes were studied as illustrated in figure 7.30Table 7.5: Soil Parameters Used for the Example ProblemSoil Type n kB m R1 k,, kh(m/s) (m/s)Clay 150 0.45 140 0.2 0.7 2.5 x 10 5 x iOLiquefied Sand 300 1.0 180 1.0 0.8 5 X 1O 1 X 10Dense Sand 2000 0.5 1200 0.25 0.6 2.5 x 10 5 x iODense Sand with Drain 2000 0.5 1200 0.25 0.6 1 x 10 1 x iOClay with Drain 150 0.45 140 0.2 0.7 1 x iO 1 x 1OIn case 1, densification is assumed to the full depth of the loose sand withoutany drainage system. This case may represent a field condition where densificationis achieved using timber piles without any drainage provisions. In case 2, the densification is assumed with a perimeter drainage system. This may represent a fieldsituation where densification is achieved using timber piles with a perimeter drainagesystem of vibro-replacement columns. In case 3, the drainage was assumed in thedensified zone. This may represent densification by vibro-replacement. In the analysis, the drains were not considered on an individual basis, instead, the densifiedzone with drains was modelled as a soil with an equivalent permeability. The equivalent permeability can be estimated from the size and spacing of the drains and thepermeabilities of the materials.The excess pore pressures for the three cases considered at various times after theearthquake are shown in figures 7.31, 7.32 and 7.33. The excess pore pressures arer0r0NC)-O0CD00 CD ()0...o. CD CD 00 3Chapter 7. Verification and Application of the Analytical Procedure 211shown in terms of pore pressure ratio u/o0, in which u is the current excess porepressure, and o is the initial vertical effective stress. u/a0 = 0 represents zero porepressure rise and u/o0 = 1 represents 100% pore pressure rise or liquefaction. Thevariations of the excess pore pressure ratios with time and distance from the centreof the densified zone are shown in graphs (a) and (b) in the figures. Graph (a) showsthe variation at a depth of 5 m and graph (b) at a depth of 10 m. Graph (c) showsthe 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 thesurrounding undensified area migrates into the densified zone. The pore pressureratio in the upper part of the densified zone rises to 1 which means liquefactionwill be triggered. However, below a depth of 6 m, liquefaction is not triggered andpiles penetrating below this depth could support vertical load, although significanthorizontal 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 migrationof high pore pressure from the loose zone into the densified zone. A maximum porepressure ratio of 0.5 is predicted 1 mm after the earthquake. The results for case 3where 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 moreeffective in preventing the migration of pore pressure in the densified zone. The porepressure 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 couldbe achieved by driving timber piles will not prevent the high excess pore pressuresfrom the surrounding liquefied zone penetrating the densified zone. Such penetrationwill cause liquefaction to a depth of 6 m for the conditions analyzed. Below this deptheffective stress increases and timber piles would be capable of carrying vertical loadalthough they could be damaged by horizontal movements. Perimeter drains couldChapter 7. Verification and Application of the Analytical Procedure 2121.2d=5m (a)30d=lOm (b)0.8Distance (m)Pore Pressure Ratio0 0.5 1.0 1.52•(c)4- Si, /di’6- / /z’!eI,,10 - QII/i12 - ‘:1 t=lmint=3Ornin14 - t5hrst =lday16Figure 7.31: Variation of Pore Pressure Ratio for Case 1Chapter 7. Verification and Application of the Analytical Procedure 2131.2d=5m (a)d=lOm (b)c05 10 15 20 25 30Distance (m)Pore Pressure Ratio0 0.1 0.2 0.3 0.4 0.L(C)4- //A 0: :: :b-f12- 1 1 t=o4 1t=lmin14- t5rfliflI t=5hr______________I______1€Figure 7.32: Variation of Pore Pressure Ratio for Case 2Chapter 7. Verification and Application of the Analytical Procedure 214d=5m (a)t=rnin15 20 301.2d=lOm (b)__30Distance (m)Pore Pressure Ratio0 0.1 0.2 0.3 04(C)74-4IG)0 /10- 412- 1t 914-0Figure 7.33: Variation Of Pore Pressure Ratio for Case 3Chapter 7. Verification and Application of the Analytical Procedure 215greatly reduce the migration of excess pore pressures into the densified zone. Theprovision of drainage within the densifled zone can be very effective in preventinghigh excess pore pressures in the densifled zone.A more detailed study of this problem including the effect of densification depthis presented in Byrne and Srithar (1992). Some other applications of the programcan be found in Byrne et al. (1991a), Byrne et al. (1991b), Jitno and Byrne (1991)and Crawford et al. (1993).Chapter 8Summary and ConclusionsAn analytical procedure is presented to analyze the geotechnical aspects in an oilrecovery process from oil sand reserves. The key issues in developing an analyticalmodel are: the stress-strain behaviour of the sand skeleton; the behaviour of thethree-phase pore fluid; and the effects of temperature changes associated with steaminjection. A coupled stress-deformation-flow model incorporating these key issues ispresented in this thesis.In modelling the stress-strain behaviour of the oil sand skeleton, shear induceddilation is an important aspect. Such dilation can increase the hydraulic conductivityand hence increase oil recovery. Dilation will also lead to reduced pore fluid pressureand increased stability. The other pertinent aspect is the stress-strain response under stress paths involving a decrease in mean stress under constant shear stress andloading-unloading cycles. The stress-strain models used in the current-state-of-the-practice are linear or nonlinear elastic models which are incapable of modelling theabove mentioned aspects realistically. The major contribution of this thesis is thedevelopment of a suitable elasto-plastic stress-strain model to capture the importantaspects. The stress-strain model postulated in this thesis is a double hardening typeconsisting two yield surfaces. The model has a cone-type yield surface to predictshear induced plastic strains and a cap-type yield surface to predict volumetric plastic strains. The predictions from the stress-strain model have been compared withlaboratory test results under various types of loading and are in good agreement. Thedilation, plastic strains due to cyclic loading, and the response under different stress216Chapter 8. Summary and Conclusions 217paths have been well predicted by the stress-strain model.The pore fluid in oil sand comprises water, bitumen and gas and the three-phasenature 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 elaboratemultiphase thermal simulators. In this study, the effects of multiphase pore fluidare modelled through an equivalent single phase fluid. An effective flow continuityequation is derived from the general equation of mass conservation which is oneof the other contributions of this thesis. An equivalent compressibility term hasbeen derived by considering the individual contributions of the phase components.Compressibility of gas has been obtained from gas laws. An equivalent hydraulicconductivity term has been derived by considering the relative permeabilities andviscosities of the individual phases in the pore fluid. The relative permeabilities havebeen assumed to vary with saturation and the viscosities have been assumed to varywith temperature and pressure. Gas exsolution which would occur when the pore fluidpressure decreases below the gas/liquid saturation pressure has also been modelled.Oil recovery schemes commonly involve some form of heating and therefore, temperature effects on the sand skeleton and pore fluid behaviour are important. Changesin temperature will cause changes in viscosity, stresses and pore pressures and consequently in some of the engineering properties such as strength, compressibility andhydraulic conductivity. In this study, the stress-strain relation and the flow continuity equation have been modified to include the temperature induced effects. Thisapproach of including the temperature effects directly in the governing equations gavevery stable results, compared to the general thermal elastic approach.The final outcome of this research work is a finite element program which incorporates all the above mentioned aspects. The new stress-strain model, flow continuityequation, and other related aspects have been incorporated in the existing two dimensional finite element program CONOIL-Il. This required significant undertakingsChapter 8. Summary and Conclusions 218including a new solution routine as the new stress-strain model results in an unsymmetric stiffness matrix. A frontal solution technique which requires less computermemory has been employed to solve the resulting equations. A new three dimensional 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 comparing the program predictions with closed form solutions and laboratory results.The predicted results agreed very well with the closed form solutions and laboratoryresults.The two dimensional finite element code has been applied to model a horizontalwell pair in the underground test facility of AOSTRA. Results have been presented interms of displacements, stresses, stress ratios and amounts of flow and discussed. Themeasured and predicted results have been compared wherever possible and they agreewell. A method to obtain individual amounts of flow of the pore fluid componentshas also been devised.The type of analytical study presented in this thesis, is very important in oilrecovery projects, since it could give insights into the likely behaviour in terms ofstresses, deformations and flow. For instance, the permeability of the oil sand andthe rate of heating due to steam injection have been examined in some detail. Ithas been revealed that in oil sands with low permeability, higher rates of heatingwould cause shear failure. If the local shear failure zone extends to the wellbore itcould cause significant damage. Information of this kind would be beneficial to thesuccessful operation of an oil recovery scheme.Although the finite element program has been developed to analyze the problemsrelated to oil sand specifically, it can be applied to other geotechnical problems. Todemonstrate its applicability, a problem involving pore pressure redistribution afterliquefaction has been analyzed and the results are discussed.Chapter 8. Summary and Conclusions 2198.1 Recommendations for Further ResearchFollowing the work presented in this study, some aspects can be identified in thisarea which require further study. Application of the finite element codes to moreoil 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 thecode have been verified, it has not been applied to analyze a oil recovery problem ofa three dimensional nature. The three dimensional code needs to be applied to eithera physical model test or a field problem where the responses are measured, in orderto check its capability to model three dimensional effects.Even though the analytical formulation presented in this study includes the effectsof multi-phase fluid through equivalent compressibility and hydraulic conductivityterms, it does not take the flow of thermal energy into account. Incorporation ofan elaborate multi-phase thermal and fluid flow model would be the most desiredenhancement though it may be a very difficult task. Previous researchers concludedthat analyzing the geomechanical behaviour and the thermal and fluid flow behaviourseparately, 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-strainmodel for the sand skeleton. The elasto-plastic stress-strain model described in thisstudy does not consider anisotropy effects. Modelling strain softening by load shedding may also be inefficient since it requires a large number of iterations. A stressstrain model which includes anisotropy and strain softening effects in a realistic manner is worth considering.Fractures in the oil sand layer are sometimes encountered in the oil recoveryprocess by steam injection. Inclusion of modelling of fracture initiation and its propagation will also be beneficial.Bibliography[1] Aboshi, H., Yoshikuni, H. and Maruyama, S. 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(1968) , “Stress Analysis of Rock as a ‘No Tension Material’ “, Geotechnique 18, pp 56-66.Appendix ALoad Shedding FormulationThe details of applying the load shedding technique to model strain softening aredescribed in this appendix. During a load increment it is possible that the stress stateof an element may move from P0 to P as shown in figure A.1 This will violate theFigure A.1: Strain Softening by Load Sheddingfailure criterion and the stress state should be brought back to P1. In load sheddingtechnique, this is done by taking out the shear stress equivalent to and then‘r/p*1iP117SMP,p 7SMP,i242Appendix A. Load Shedding Formulation 243transferring it to the adjacent stiffer elements. The detailed steps of this procedurewill be as follow:1. Estimate the stress ratio (n’, Figure A.1) in the strain softening region corresponding to the shear strain (7sMp,1) using equation 3.63 as7i 1r + (i— a,,) exp { — K (ysMP, 1 — 7SMP,p ) } (A. 1)2. Estimate the amount of stress ratio that has to be taken out as(A.2)3. Evaluate the changes in the Cartesian stress vector {/.o-}Ls which correspondsto4. 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 softening, and if so, repeat the load shedding procedure.A.1 Estimation of {zSJ}LsIn order to estimate the changes in the Cartesian stress vector, it is easier to firstestimate the changes in principal stresses. By differentiating equation 3.34 in termsof principal stresses the following equation can be obtained:T1213 + 1113(02 + £73) —12(U3) Lo18I 1(U+ i)1) U2 (A.3)‘213 +13(ui + £72) —12(Ut7)Appendix A. Load Shedding Formulation 244The above equation can be rewritten asA1/o- + A2Io H- A3z (A.4)To estimate the changes in principal stresses, two more equations are needed, inaddition to equation A.4. The following two conditions are assumed during the loadshedding to obtain the additional two equations:1. The mean normal stress remains constant during load shedding. This gives1 + + Lo3 0 (A.5)2. The b-value [(02 — 03)/(01 — 03)] remains constant. Which implies(A6)Oi — 03 — (o- + Zoi) — (03 + Lo3) —By rearranging the terms(A.7)By solving equations A.4, A.5 and A.7 the following equations can be obtainedfor 02 and o3:2—bH- b)(A3 — A2) —(2— b)(A1 — A2) (A.8)1+b (A.9)—(Lri + 3) (A.l0)Now, the changes in the Cartesian stresses can be obtained by simply multiplyingthe principal stress vector by the transformation matrix.Appendix A. Load Shedding Formulation 245l m12 m2 n2y12 m2(A.11)2ll, 2mm 2n,nIO32l,l 2mm 2nn2mm 2nzna,where1a li,, and l - direction cosines of o to the x, y and z axesm, m and m - direction cosines of O2 to the x, y and z axesn, n and n - direction cosines of 03 to the x, y and z axesA.2 Estimation of {F}LsThe load vector corresponding to the changes in stresses has to be applied at thenodes of the soil element that failed, to transfer equivalent amount of stresses to theadjacent stiffer elements. By doing this, the stress equilibrium in the domain will bemaintained. 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 () withinthe system. Mathematically this can be expressed as{}Tf} = J{}Tfr}dv (A.12)where{ f} - Force vector{o}- Stresses within the systemAppendix A. Load Shedding Formulation 246The 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{}Tf} = J{}T[B]T{}dv (A.14)This can be further written as{f} = J[B]Tfr}dv (A.15)Following equation A.15 the force vector for load shedding can be obtained as{IF}Ls J[B]T{U}LSdV (A.16)Appendix BRelative Permeabilities and ViscositiesSome detailed explanations which are needed in the evaluation of equivalent permeability are given in this appendix. To evaluate the equivalent permeability, therelative permeability and the viscosity values of the pore fluid components are necessary. The first section explains how to calculate the relative permeabilities and theequivalent permeability through an example data set. The viscosity values of waterat different temperatures are given in section 2. Section 3 gives some insights intothe viscosity of hydrocarbon gases and how to evaluate it.B.1 Calculations of relative permeabilit iesB.1.1 Relevant equationsThe relative permeabilities of water, gas and oil can be obtained from the followingequations:Jc° S1k° S_qrog 9’ row w a a— Jo1 t-’w JgIkrw = — A2)A3 (B.2)krow B1(B2 — Sw)B3 (B.3)krg = C1(S9 — (B.4)247Appendix B. Relative Permeabilities and Viscosities 248k,.09 = D1(D2 — S9)D3 (B.5)— krow(Sw)B6/3W— k0(1 — S,,)—k,.09(S9)9ko (1—S (r09\ 9= 5;— Swc S S (B.8)Wc oms S0 — Sorn So > Som (B.9)wc om9‘9i c’ C’‘-‘uc ‘-‘Ornwherek,.0 - relative permeability of oil in 3-phase systemk,. - relative permeability of water in 3-phase system- relative permeability of oil in water-oil system- relative permeability of oil in oil-gas systemkrg - relative permeability of gas in 3-phase systemk - relative permeability of oil at connate water saturationin a water-oil systemk09 - relative permeability of oil at zero gas saturationin an oil-gas systemS,,, S, S - Saturation of water, oil and gas respectivelyS, S, 5 - Normalized saturation of water, oil and gas respectively- Critical water saturation5om - Residual oil saturationA1,A2... etc. - ConstantsAppendix B. Relative Permeabilities and Viscosities 249= 0.2k° —107’OW= 1.0A1 = 1.820B1 = 2.769C1 = 2.201= 1.640A2 = 0.20B2 = 0.80C2 = 0.05D2 = 0.80A3 = 2.375B3 = 1.996C3 = 2.704= 2.547= 0.5I-sw = 8 x 104Pa.sk = 1 x 102mS, 0.4= 2OPa.sSg = 0.1= 2 x 105Pa.s (at 30°C)B.1.3 Sample calculationsBy substituting the data into the relevant equations0.5 — 0.2=0.51 — 0.2 — 0.20.15;= 1 — 0.2 — 0.2 = 0.16670.4— 0.2== 0.33331 — 0.2— 0.2The equivalent permeability is given bykEQ=kI-o P’gB.1.2 Example data(B.11)Appendix B. Relative Permeabilities and Viscosities 250= 2.769(0.8 — 0.5)1.996 = 0.25krog = 1.640(0.8 0.1)2.547 = 0.6610.25= 1(1 — 0.5) = 0.50.6610 793— 1(1 — 0.1667) —krw 1.820(0.4 — 0.2)2.735 = 0.068k,.9 = 2.201(0.1 — 0.05)2.704 = 0.001k,.0 = 0.3333(1.0 X 0.1667+1.0 X 0.5)X 0.5 X 0.793 = 0.132—12 / 0.068 0.132 0.001 ‘ m mkEQ = 1 >< 10 8 x 10 + 20 + 2 x 105) 1.350 x 10 —B.2 Viscosity of waterThe viscosity of water at different temperatures are well established and can be obtained form the international critical tables. The following tables are given by N.Ernest Dorsey in the international critical tables and are reproduced here. Thesedata are also built in the computer program CONOIL.Appendix B. Relative Permeabiiities and ViscositiesTable B.1: Viscosity of water between 0 and 1000 C251Values in rnillipoises (1, 12, 16, 17, 22, 24, 30, 31, 32, 38)C 0 1 2 3 4 5 6 7 8 90 17.93* 17.326 16.74* 16.19a 15.67. 15.18* 14.72* 14.28* 13.872 13.47,10 13.097 12.73s. 12.39o 12.06i 11.748 11.44? 11.15* 10.875 10.60s 10.34o20 10.087 9.843 9.60* 9.38* 9.16i 8.94. 8.74* 8.55i 8.368 8.18130 8.004 7.834 7.67* 7.511 7.35 7.20* 7.064 6.92 6.791 6.66140 6.536 6.41s 6.29* 6.184 6.075 5.97* 5.86* 5.77* 5.67s 5.58250 5.492 5.40s 5.32* 5.236 5.153 3.07s 4.99* 4.918 4.84a 4.77o60 4.69* 4.62s 4.56i 4.495 4,43i 4.36* 4.30* 4.24s 4.186 4.12s70 4.07i 4.01* 3.96z 3.909 3.8.5? 3.806 3.756 3.70* 3,66i 3.61s80 3.57. 3.52* 3.483 3.44. 3.39* 3.35i 3.31? 3.27* 3.24* 3.20390 3.16* 3.13* 3.095 3.061 3.027 2.994 2.96a 2.93* 2.89. 2.86*100 2.83* 2.82 2.79 i 2.76 2.73 2.70 2.67 2.64 2.62 2.59At a pressure ox 1 atm., = a/(b + t)”.At a pressure of P kg/cm2, ,7p = ?7i[l + k,(P — 1) X 10’J.‘11 is the value , when P is 1 kg/cm2 which may be taken asThe unit of , is the poise unless otherwise stated.Table B.2: Viscosity of water below 00 CH,O ov 100°C (16)Values as recorded by author accord with I. C. T. values below100°C; the others are given as he has published them. Thepressure is that of the saturated vapor at the temperaturesindicated.4, °C 110 120 130 140 150 1601000,7 2.56 2.32 2.12 1.96 1.84 1.74Table B.3: Viscosity of water above 1000 CH20 BELOW 0°C (39)Values corrected and adjusted to accord with I. C. T. valuesabove 0°CFoR,.1Ux.E AND UNITSthe value of,7 at 1 atm.—2 —4 —5 —6 —8 —10100077 19.1 20.5 21.4 22.2 24.0 26.0Appendix B. Relative Permeabilities and Viscosities 252B.3 Viscosity of hydrocarbon gases (from Carr et al., 1954)The viscosity of hydrocarbon gases can be expressed as a function of reduced pressuresand temperatures, i.e.,(B.12)IL1where- viscosity of gas at reduced temperature TRand at reduced pressure FRpi - viscosity of gas at atmospheric pressure and given temperatureTR - temperature/critical temperature (in absolute units)PR - pressure/critical pressure (in absolute units)If the gas is a mixture of hydrocarbons, the pseudo-critical concept has to beapplied. Thus, in place of critical temperature and critical pressure, the pseudo-critical temperature and pressure have to be used. The pseudo-critical temperatureis given by(B.13)The pseudo-critical pressure is given byPPc=XzFci (B.14)where- mol fraction of component i in the mixture- critical temperature of component i in absolute units- critical pressure of component i in absolute unitsa)‘—4-)d0.4,4U).•-40-c•c:a)a)—l_a)U)a)U)‘—I;0to0o—-d-cif)0a)Cl)U)C)c;4-.40-4-aca)U)0Q)-4toa)Cl))-U)tocdtotoa)o0--O-_-Cl)H0-4a);-1p-‘da)to-0to:-EQ0a)4‘--4a)0-dl-4a)0-a)E.aa)0‘0Cl)V0U)a,.4);-4a‘.0a)4.44a)U)U)U)a)-a).-I_4--4-4-a‘-4(3+l-40I—’-4-44)‘-42a),.D0:a)a)a)0-.a)-4--c30toCl)r0oa)Cl)-4Cl)a)0a)-4...—a)to-0Cl)‘-40-d:0a)o-0Cl)dU)-4l-40o000U)0:111if)-I:j:==iiI111JWi1L111FW1IMIII:.4a.3wvo,an,.!J.LIiLIJ_LLL_Ian3L,tTITFITIITII—IiIiI1IM1ll1ll12t:tll1llllillllhItr-.l,6A•woIi,,_,ltavd’“llDc1llllt4 .q_4‘....,s,wou,,.uo,;i.Cl)-IIIIIIc!:1U)VISd‘UflSSWdVOI1W3OOflSd,dd—U.‘311fl1VU3dY131VO4J4l4OOOflSdidIa,‘-4.,oCDCDCD.2U)‘-i’CDCD,- CD-$-+:-CDoCDCD0.q-),(—.-CDI—0‘dc-I—.CDj.p,U)0CD00—.+0U)0 pCD•-•CDpU)CDp;;.U)o op0‘—4.,U),-4.,4,_-’oo9’09’9,•-CD,-,CDCDCD‘‘CDU)0••P‘-d0o‘-UCD•CD9’4- 4—]0 _..,,.rJ‘-U“lCD0CD4-sU)-I-sCD-r pCDCDCDCD—..U)9’:-‘‘-sCDU)CDC’) +pCDCD--b01ELJP0‘-UU)CD 14-s 9’‘-54--CD—.—‘0 9, —.‘ .I—-4-s CDC’)U)9’‘-5 CD •CDo0 9, 0 -“ CD 0 U) ‘) o o 4-- U) 0 o 0 4-+) ,- _4-4 - 0 CD oq - ‘-U O u, 9’ CD 0 CD‘-5 CD U) 0 0 U) ‘-4-0 4-f,‘-1 0 0 p I-s 0 9, U) CD ci) 9, 0 CD 9’ 0 U)‘-U CD ‘-5 CD0 0 0 ‘-4-CD U) 0 4-f,‘-5 0 0 9, I-s 0 0 CD 9, CD 0 CD U) CD ‘-5 ‘-4-‘-U 0 ‘-4-U)O’3‘-5 CDVISCOSITY,ATIATM.•AJ,CENTIPOIS8°80öb0bIONADOED-i-1-iiF1t4-iI—tWttWttItliltI1I8FL*J1i1ir4’i‘-U >1 9, ‘-4-‘-U I..II+H+FfAii(rLjtIU11M1W4111FFPII1IIlIFF1Fvr1I14-4aI4JYL4AY14n’i1fFIlCUUIIttt1IIIi1iVIlI1IJ11tIfl1VIIIAFII,I1VIlLItIUIrV111)14-I11111171115I11111IlILUI.iii.nrriii DIAlrU1Er[I4flECs.P111)11It’llil/liVIKJy‘iiCOACCTIOuDOEOD1l1(—19ItlA1-IIIIA-t-f-To,sc—GPt.,8CORIICC1IONADOLDTI)VISC—G14-1 0rAppendix B. Relative Permeabilities and Viscosities 255J )‘‘ =.-‘. .L. I. .iL. - -— IH---- —- .LL.... -- — -III,£1 /‘:iL.-(-L — -.k L .t.1- -r ‘--j-•-p.j-r / I—— —--•- -ir-rr ,4t—— —---. 1—-—— ---r ,1 “i:0 —. — — I - H- 1! -f-J4 —1.. — — — -- , -rv i- r;- — -: ia:: .±Ik— — —- - ‘4- I1”E-___i!::: .I ,.,.‘77 I.— - — ..-f ‘ T—— -- -- r/r7 7 -i,.’_._____i --- :..4-__— . . — -IC_ __— . —--—-- — —L I . —2 3 4 .5 6 7.6LD 2 3 4 56 7I9PSEUCOREDUGED PRESSURCFigure B.3: Viscosity ratio vs pseudo-reduced pressureVISCOSITYRATIO01gUI___-.---IH111HTTD11I—----------IillIIlililiPI::E:::tft111Itt1UftItumiHt————-—--AIIIIIILiIIIU—ti±fliIJJ-JkHlfl-Ct)—IIIIi—i--i-rnIIIIIi-i-HI]IIUHi]CD—————-—--11—i—1111rru1x4-rrrrlLI-1-rrtTrl]r——,——-—-—--‘1I-I44IIFLI+1HI-I-li-I’1IIII-II-I4J4———1111]Ji—rLuIIIl-i-rUJJJ±LuIIUL--—J-t.I-[Tu-1-T[ITITFFIFrITI1LWIU—TIIIII1—HilIIH-I-flU-CDi4———--—IIITIii4-nTrLi4-rn1-rTr-:--:LIJ}tT[IWITWITh[IL-1J[I-±tf1±1±H±ftH1±th_:_L::,iii[14r1±1±thtinlll:I—-itiIlF{*fl+}1+HI1{If,,i:---rtriuwiwrmirrni—L———----414-U41414-I-IW-14111III-C3:::::•4I111II1Ht-1[1-I:7::::41WhII-fll-UIL[N-‘-EEHEEAppendix B. Relative Permeabilitics and Viscosities 257temperature was 195°F, and the test pressure was 1800 psig (1815 psia). The gravityof the liberated gas was determined by the use of tared glass weighing balloon. Thegas gravity was found to be 0.70 18 (air = 1.0). The calculation of viscosity will be asfollows:1. Molecular weight 0.7018 x 28.95 20.312. 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 valuescan be calculated using equations B.13 and B.14.3. From figure B.2:Viscosity at one atmosphere () = 0.01223cp4. Pseudo-reduced pressure = 1, 815/667 = 2.721Pseudo-reduced temperature = (460 + 195)/390 = 1.6795. From figures B.3 and B.4:IL/IL1 = 1.286. Therefore,The viscosity at 1800 psig and 195°F = 1.28 x 0.01223 = 0.01565cpAppendix CSubroutines in the Finite Element CodesC.1 2-Dimensional Code CONOIL-IlThe 2-dimensional code has been divided into two separate programs; the ‘GeometryProgram’ and the ‘Main Program’. The main reason for having as two separateprograms is to reduce the effort on the user. The geometry program automaticallygenerates and numbers the midside and interior nodes. It also renumbers the elementsand 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 programdoes the analysis. The geometry program for the 2-dimensional version consists of11 subroutines and the main program consists of 58 subroutines. The details of thesubroutines are described herein.C.1.1 Geometry ProgramThe subroutines in the geometry program and their functions are as follow:ADDS - forms element-node links.BCONI - sets up element constants.FFIN - reads free format input.MAKENZ - generates an array which contains the number of degrees of freedomassociated with each node.MIDPOR- generates mid-side pore pressure nodes.MIDSID - generates mid-side displacement nodes.258Appendix C. Subroutines in the Finite Element Codes 259MLAPZ- marks last appearances of nodes by making them negative.OPTEL - optimizes and renumbers the elements for frontal solution.RDELN - reads line data.SFWZ - calculates the front width for symmetric solution.SORT2 - changes the element numbers to conform with new ordering.C.1.2 Main ProgramThe subroutines in the main program and their functions are given below.BCON - calculates element constants.CHANGE- removes/adds elements from geometry mesh and calculates impliedloading.CHECK- scrutinizes the input data to main program.CHKLST- checks if there are any changes in fixity for the load increment.COMP - computes the pore fluid compressibility and permeability.DATM - reads material property data.DETJCB- calculates the determinant of the Jacobian matrix.DHYPER- calculates the stress-strain matrix for elastic model.DILATE - computes the volume change due to shear deformation (used with hyperbolic model).DISTLD - calculates equivalent nodal loads.DSYMAL- finds the principal stresses and their directions (contains 5 subroutines;TRED3, TRBAK3, TQLRAT, TQL2, DTRED4).ELMCH- scrutinizes the list of elements.EQLBM- calculates unbalanced nodal loads.EQLIB - calculates nodal forces balancing element stresses.ERR - records and lists data errors.FFIN - reads free format input.Appendix C. Subroutines in the Finite Element Codes 260FFLOW - calculates amount of flow and updates saturations.FIXX - updates list of nodal fixities.FLOWST - calculates vectors for coupled consolidation analysis.FORMB - forms ‘B’(shape function derivative) matrix.FRONTZ - frontal solution routine for symmetric matrix.GETEQN - gets the coefficients of the eliminated equations.INSIT - sets up in-situ stresses and the equivalent nodal forces.INSTRS - prints the in-situ stresses before first increment.INV - inverts a matrix.LSHED - carries out load shedding operation.LSTIFA - calculates the element stiffness matrix using fast stiffness formation.LSTIFF - calculates the element stiffness matrix for elastic model.MAKENZ - generates an array which contains the number of degrees of freedomassociated with each node.MBOUND - rearranges the boundary conditions in terms of degrees of freedom.MLAPZ- marks last appearances of nodes by making them negative.MODULI - calculates moduli of the soil elements for elastic model.MSUB - main controlling routine.PLAS - calculates the stress-strain matrix for elasto-plastic model.PRINC - calculates principal stresses.RDN - reads specified range in 1-dimensional array.REACT- calculates the reactive forces on restrained boundaries.SCAN - checks for any changes in fixities.SELF - calculates self weight loads.SELl - computes nodal forces equivalent to self weight loads.SFR1 - calculates shape functions and derivatives for 1-dimensional integration alongelement edges.Appendix C. Subroutines in the Finite Element Codes 261SFWZ - estimates the front width for symmetric matrix solution.SHAPE - calculates shape functions and derivatives.SOFT * calculates the overstress for strain softening.STIF - calculates element stiffness matrix for elasto-plastic model.STOREQ - writes the terms in a buffer zone when an array becomes saturated.TEMP - calculates the equivalent force vector terms due to temperature changes.UFRONT- frontal solution routine for unsymmetric matrix.UPARAL - allocates storage for subroutine UPOUT.UPOUT - updates and prints the results.VISG- calculates viscosity of gas.VISO - calculates viscosity of oil.VISW - calculates viscosity of water.WRTN- writes a specified range in a 1-dimensional array.ZERO1 - initializes 1-dimensional array.ZERO2 - initializes 2-dimensional array.ZERO3 - initializes 3-dimensional array.ZEROI1 - initializes 1-dimensional integer array.C.2 3-dimensional code CONOIL-IllThe 3-dimensional code has been developed based on the same sequence of proceduresas the 2-dimensional code. It consists of 43 subroutines and the details of those aregiven below.BOUND- expands the nodal fixity data in terms of degree of freedom.CHANGE- removes/adds elements from geometry mesh and calculates impliedloading.COMP- computes the pore fluid compressibility and permeability.DMAT - reads material property data.Appendix C. Sn bron tines in the Finite Element Codes 262DRIVER - main controlling routine.EPM - calculates stress-strain matrix for elasto-plastic model.EQLIB - calculates nodal forces balancing element stresses.FFLOW - calculates amount of flow and updates saturations.FIXX- updates list of nodal fixities.FLSD- calculates load vector for load shedding.FTEMP - calculates force vector terms due to temperature changes.GETEQN- gets the coefficients of the eliminated equations.HYPER - calculate moduli values for hyperbolic model.INSIT- sets up in-situ stresses and the equivalent nodal forces.JACO- evaluates Jacobian matrix, its determinant and inverse.LAYOUT- reads nodal geometry data and stores in relevant arrays.LFIX - sets the load vector for fixed boundaries.LOAD- evaluates the load vector for applied loads.LSHED- routine to perform load shedding.MAKESF - finds last appearance of the nodes, frontwidth and the destination vector.MFLOW- updates saturations and flow at mid-step.MINV- inverts a matrixPRIN- finds the principal stresses and their directions (contains 5 subroutines;TRED3, TRBAK3, TQLRAT, TQL2, DTRED4).PRNOUT- calculates, updates and prints the results.RDN- reads specified range in 1-dimensional array.SBMATX- calculates B’(shape function derivative) matrix.SELF- calculates self weight loads.SELl - calculates self weight loads for gravity changes.SFRONT- frontal solution routine for symmetric matrix.SHAPE- calculates shape functions and its derivatives.Appendix C. Subroutines in the Finite Element Codes 263SHAPE2 - calculates shape functions and derivatives for 2-dimensional integration.SMDF - sets up arrays giving nodal degrees of freedom and the first degree of freedomof the nodes.STIFF - calculates element stiffness matrix.STOREQ - writes the terms in a buffer zone when an array becomes saturated.STRL - calculates and updates stress level.TEMP - calculates nodal temperature changes.UFRONT - frontal solution routine for unsymmetric matrix.UPDATE - updates the results at mid-step for second iteration.VISG - calculates viscosity of gas.VISO - calculates viscosity of oil.VISW - calculates viscosity of water.WRTN - writes a specified range in a 1-dimensional array.ZERO 1 - initializes 1-dimensional array.ZERO2 - initializes 2-dimensional array.ZERO3 - initializes 3-dimensional array.ZEROI1 - initializes 1-dimensional integer array.ZEROI2 - initializes 2-dimensional integer array.Appendix DAmounts of Flow of Different PhasesThe formulation for the multi-phase flow presented in chapter 5 considers an equivalent 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 byknowing the total amount of flow, and the relative permeabilities and viscosities ofthe 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 consideredhere.In the oil sand layer the zone from where the fluid flow occurs, can be obtained fromthe temperature contour plot or the pore pressure contour plot (refer to figures 7.17and 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 mobilities. Here, the flow zone is divided into three (zones A, B and C in figure D.1) andthe effective mobilities of the fluid phase components are assumed constant within azone. The grater the number of zones the better the results will be.The mobility of a fluid phase component ‘1’ can be written askmi kkri (D.1)wherekmi - mobility of phase 1k - intrinsic permeability of the sand matrix (m2)264Appendix D. Amounts of Flow of Different Phases 265500 Injection Well• Production WellE 4040 0 60Distance (m)Figure D .1: Zones involved in Fluid Flowk,.1 - relative permeability of phase IIL1 - viscosity of phase 1k is a function of void ratio, k,.1 is a function of saturation level and 1u is a functionof temperature. Under steady state conditions, the void ratio and the temperatureare assumed to remain constant. Therefore, the viscosities of the phase componentswithin a zone can be assumed constant and are summarized in table D.l. The intrinsicpermeability of the sand matrix is assumed to be 1 x 1012 m2.As the flow continues, the water will replace the oil and therefore, the saturations willchange. Since the relative permeabilities are function of saturation, they will changeas well. The relative permeabilities of water and oil are assumed to be representedby the following functions:= 1.820 (S — 0.2)2.375 (D.2)Appendix D. Amounts of Flow of Different Phases 266Table D.1: Average Viscosities and Temperatures in Different ZonesZone Area (m2) ii(mPa.s) u0(mPa.s) Temp. (°C)A 96 0.20 8 220B 252 0.48 40 140C 312 0.65 1000 50k,.0 = 2.769 (0.8— S)’996 (D.3)Now, let us assume that the total flow of water and oil for a time interval /t beLVT. This total amount of flow will comprise the water and oil flow in all three zonesconsidered. The effective mobility of water considering all three zones can be givenas,= (kmw)A aA + (kmw)B aB + (kmw) ac (D.4)aA + a + acWhere, aA, aB and ac are the areas of zones A, B and C respectively. Similarly, theeffective mobility of oil considering all three zones can be given as,— (kmo)A aA + (kmo)B aB + (kmo) acmoaA + aB + acThen, the amounts of water and oil flow in the total flow can be estimated as,A TI TI mwL.Vw VT ke j i.emw= LVT e e (D.7)mw moNow, because of the flow of oil from the oil sand layer, saturations will change andthose 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 asfollows.Appendix D. Amounts of Flow of Different Phases 267For example, the amount of water flow from zone A can be given by,fAT? AT? mw A aAiI_1Vw)A = LVw (kmw)A aA + (kmw)B aJ3 + (kmw) acSimilarly, all the individual amounts of flow of water and oil in different zones can becalculated.Assume that the saturation of oil in zone A at the beginning of a time step be (S0).Then, the saturation of water in zone A at the beginning of the time step will be,(S) = 1 — (S0) (D.9)The volume of oil in zone A at the beginning of the time step will be given by,(V0) = aA n (S0) (D.1O)The amount of oil flow from zone A will be,IAT?\ AT? mo A aAL.1Vo)A = (kmo)A aA + (kmo)B aB + (kmo) acThen, the volume of oil in zone A at the end of the time step will be,(V (V0)— (V0)A (D.12)and the new oil saturation will be,(S0) (V0) (D.13)fl aAThe new saturation of water in zone A will be given by,(S) = 1 — (S0) (D.14)Likewise, the saturations in all the zones can be updated. Then, by knowing the newsaturations, the relative permeabilities of the phase components can be estimated andsubsequently, the new amounts of water and oil flow can be calculated. These stepsAppendix D. Amounts of Flow of Different Phases 268of calculations can be continued with time in a step by step manner until the flow ofoil 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 givenin table D.2.Table D.2: Initial Saturations and Mobilities of Water and OilZone Sw S kmw(108m/s) kmo(108m/s)A 0.3 0.7 37.6 85.1B 0.3 0.7 19.8 17.0C 0.3 0.7 11.6 0.68The stepwise calculations for the amounts of flow and saturations of water and oilare tabulated in table D.3. The saturations and the mobilities of water and oil at theend of time t = 300 days, are given in table D.4, which can be compared with tableD.2.Appendix D. Amounts of Flow of Different Phases 269Table D.3: Calculation of Flow and Saturations with TimeTime (S)A (S)B (S)c (S0)A (S0)B (S0)c iW0(days) (m3/day) (m3/day)0 0.300 0.300 0.300 0.700 0.700 0.700 2.54 2.642 0.394 0.319 0.301 0.606 0.681 0.699 3.89 1.294 0.435 0.330 0.301 0.565 0.670 0.699 4.29 0.896 0.461 0.339 0.302 0.539 0.661 0.698 4.49 0.698 0.479 0.346 0.302 0.521 0.654 0.698 4.61 0.5710 0.494 0.352 0.302 0.506 0.648 0.698 4.69 0.4915 0.525 0.365 0.303 0.475 0.635 0.697 4.82 0.3620 0.546 0.375 0.303 0.454 0.625 0.697 4.89 0.2925 0.562 0.384 0.304 0.438 0.616 0.696 4.94 0.2430 0.574 0.392 0.304 0.426 0.608 0.696 4.97 0.2140 0.595 0.405 0.305 0.405 0.595 0.695 5.01 0.1750 0.610 0.417 0.306 0.390 0.583 0.694 5.04 0.1460 0.622 0.426 0.307 0.378 0.574 0.693 5.06 0.1270 0.632 0.435 0.307 0.368 0.565 0.693 5.07 0.1180 0.640 0.443 0.308 0.360 0.557 0.692 5.08 0.1090 0.647 0.450 0.308 0.353 0.550 0.692 5.09 0.09100 0.653 0.456 0.309 0.347 0.544 0.691 5.10 0.08125 0.667 0.471 0.310 0.333 0.529 0.690 5.11 0.07150 0.678 0.484 0.311 0.322 0.516 0.689 5.12 0.06175 0.686 0.495 0.312 0.314 0.505 0.688 5.13 0.05200 0.693 0.505 0.313 0.307 0.495 0.687 5.13 0.05250 0.704 0.522 0.315 0.296 0.478 0.685 5.14 0.04300 0.713 0.536 0.317 0.287 0.464 0.683 5.15 0.03Table D.4: Saturations and Mobilities of Water and Oil after 300 DaysZone S S kmw(108m/s) kmo(108m/s)A 0.71 0.29 1826 2.61B 0.54 0.46 352 4.75C 0.32 0.68 16.7 0.64Appendix EUser Manual for CONOIL-IlE.1 IntroductionCONOIL-Il is a finite element program for consolidation analysis in oil sands underplane strain and axisymmetric conditions. The program includes an elasto-plasticstress strain model and a formulation to analyze multi-phase fluid flow. It also considers temperature effects on stresses and fluid flow in the analysis.The program can be used to carry out transient, drained or undrained analysis using 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 problems in oil sands, it can be applied for a range of geotechnical problems such as damand heavy foundation analyses.The intention of this manual is to provide sufficient information for an analyst witha strong geotechnical background to be able to prepare an input file and run theprogram. 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 theuser. The geometry program automatically generates and numbers the mid-side andinterior nodes. It also renumbers the elements and nodes to minimize the front widthand creates an input file for the main program, containing the relevant informationabout the finite element mesh. Therefore, the Geometry Program has to be run first270Appendix E. User Manual for CONOIL-Il 271and the link file has to be submitted to the Main Program.The data for both the Geometry Program and the Main Program is free formati.e, particular data items must appear in the correct order on a data record butthey are not restricted to appear only between certain column positions. The dataitems are indicated below by mnemonic names, i.e., names which suggest the dataitem required by the program. The FORTRAN naming convention is used: namesbeginning with the letters I, J, K, L, M and N show that the program is expecting anINTEGER data item whereas names beginning with any other letter show that theprogram is expecting a REAL data item. The only exception is the material propertydata where the actual parameter notations are retained to avoid confusions. All thematerial property data are real. INTEGER data items must not contain a decimalpoint but REAL data items may optionally do so. REAL data items may be enteredin the FORTRAN exponent format if desired. Individual data items must not containspaces and are separated from each other by at least one space. Detailed explanationsfor 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 theFORTRAN program. Any line that has the character C in column 1 is ignored by theprograms. This facility enables the user to store information relating to values, unitsassumed etc. permanently with the input data rater than separately. The programonly read data from the first 80 columns of each line.Appendix E. User Manual for CONOIL-Il 272E.2 Geometry ProgramRecord 1 (one line)TITLETITLE- Title of the problem (up to 80 characters)Record 2 (one line)I LINKLINK- A code number set by the userRecord 3 (one line)NN NEL ILINK IDEF ISTART SCX SCYNN - Number of vertex nodes in the meshNEL - Number of elements in the meshILINK- Link option:0 - no link file is created1 - a link file is createdIDEF - Element default type:1 - linear strain triangle with displacement unknowns5 - linear strain triangle with displacement and excess porepressure unknowns (linear variation in pore pressure)7 - cubic strain triangle with displacement unknowns8 - cubic strain triangle with displacement and excess poreAppendix E. User Manual for CONOIL-Il 273pressure unknowns (cubic variation in pore pressure)ISTRAT - Frontal numbering strategy option:1 - the normal option2 - only to be used in rare circumstances when the parent’mesh contains overlapping elementsSCX- Scale factor to be multiplied to all x coordinatesSCY - Scale factor to be multiplied to all y coordinatesRecord 4 (NN lines)N X Y TEMP LCODE VISCO]N - Node numberX - x coordinate of the nodeY-y coordinate of the nodeTEMP- Initial temperature °CLCODE- Index for load transfero - node can participate in load transfer1 - node cannot participate in load transferVISCO- Initial viscosity factor(not used in the present formulation, set equal to 1)Record 5 (NEL lines)ILN1N2N3MATIL - Element numberNi, N2, N3 - Vertex node numbers listed in anticlockwise orderAppendix E. User Manual for CONOIL-Il 274MAT- Material zone, number in range 1 to 10Appendix E. User Manual for CONOIL-Il 275E.3 Main ProgramRecord. 1 (one line)TITLEITITLE - Title of the problem (up to 80 characters)Record 2 (one line)I LINKILINK - Code number set by the userRecord 3 (one line)I NPLAX NMAT INCJ INC2 IPPJM IUPD ICOR ISELFINPLAX- Plane strain/Axisymmetric analysis option:0 - plane strain1 - axisymmetricNMAT - Number of material zonesINC1 - Increment number at start of analysisINC2 - Increment number at finish of analysisIPRIM- Number of elements to be removed to from primary meshIUPD - Element default type:1 - linear strain triangle with displacement unknowns5 - linear strain triangle with displacement and excess porepressure unknowns (linear variation in pore pressure)Appendix E. User Manual for CONOIL-Il 2767 - cubic strain triangle with displacement unknowns8 - cubic strain triangle with displacement and excess porepressure unknowns (cubic variation in pore pressure)ISTRAT - Frontal numbering strategy option:1 - the normal option2 - only to be used in rare circumstances when the ‘parent’mesh contains overlapping elementsSCX- Scale factor to be multiplied to all x coordinatesSOY- Scale factor to be multiplied to all y coordinatesRecord 4 (One line only)MXITER DIOONV PATMMXITER - Maximum number of iterations per increment for dilationand load transfer purposes (zero defaults to 5)DICONV- Convergence criterion for change in force vector fromdilation 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-PLASTICstress-strain models )Appendix E. User Manual for OONOIL-II 277Record 5.1MAT IMODEL e KE n Rf KB m DUO k kMAT - Material property number. All elements given the samenumber in the Geometry Program have the following propertiesIMODEL - Stress-strain model number. Use C7 for Hyperbolic modele - Initial void ratioKE - Elastic modulus constantn - Elastic modulus exponentRf - Failure ratioKB - Bulk modulus constantm - Bulk modulus exponentDUO - Determines whether Drained/Undrained/Consolidation analysisi) DUO = 0.0 Drained analysisii) DUO = B1 (liquid bulk modulus)- Undrained analysisNOTE: B1 in the range of 100 to 500 B5k (soil bulk modulus) isequivalent to using a Poisson’s ratio of 0.495 to 0.499.If there are temperature changes, use consolidation routineto do undrained analysis.iii) DUO= 7i (unit weight of liquid) - Consolidation analysis- total unit weight of soil- permeability in x direction- permeability in y directionRecord 5.2c - v ot q’cv - B B0Appendix E. User Manual for CONOIL-Il 278c - Cohesion- Friction angle at a confining pressure of 1 atmosphereL4 - Reduction in friction angle for a ten fold increase inconfining pressure—- 0 (No parameter at present)- Constant dilation angle. To be specified if the dilationoption is used.a8t - Coefficient of temperature induced structural reorientation. Only used in temperature analysis.- Constant volume friction angle. Only used with dilationoption.—- 0 (No parameter at present)B - Bulk modulus of the waterB0 - Bulk modulus of the oilRecord 5.3/J’30,0 ‘H H -\U U S S1 cw a01’3o,o - Viscosity of oil at 300 C and 1 atmosphere (in Pa.s)(used in three phase flow, built-in oil viscosity correlation)- Function to modify Henry’s constant for temperatureH=H+)H*IXTH - Henry’s coefficient of solubility- Function to modify bubble pressure for temperatureU - Bubble pressure (Oil/Gas saturation pressure)S - Initial degree of saturation varying between 0 and 1. (S= 1 implies 100% saturation)Appendix E. User Manual for CONOIL-Il 279Sf - Saturation at which fluid begins to move freely. (Usedfor modifying permeability. 1 is generally close to zero)- Coefficient of linear thermal expansion of watercx0 - Coefficient of linear thermal expansion of oil- Coefficient of linear thermal expansion of solidsRecord 5.4ISIGE 151GB IMPF IDILAT ILSHD IISIGE- Option to calculate Young’s moduluso - use mean normal stress1 - use minor principal stressISIGB- Option to calculate bulk moduluso - use mean normal stress1 - use minor principal stressIMPF - Multi phase flow optiono - fully saturated1 - partially saturated2 - three phase fluid flow (needs additional parameters)IDILAT- Dilation optiono - No dilation1 - Use constant dilation angle2 - Use Rowe’s stress-dilatancy theoryILSHD- Load transfer optiono - do not perform load transfer1 - perform load transfer by keeping o constant2 - perform load transfer by keeping On constantAppendix E. User Manual for CONOIL-Il 280Record 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-PLASTICstress-strain models )Record 5.1MAT IMODEL e KE n (R1) KB m DUO k %MAT- Material property number. All elements given the samenumber in the Geometry Program have the following propertiesIMODEL- Stress-strain model number= 5 Cone type yielding only (single hardening)= 6 Cone and Cap type yielding (double hardening)e - Initial void ratioKE - Elastic modulus constantn - Elastic modulus exponent(Rf) - Failure ratio in the hardening rule (cone yield)KB - Bulk modulus constantm - Bulk modulus exponentDUO - Determines whether Drained/Undrained/Consolidation analysisi) DUO = 0.0 Drained analysisii) DUO = B1 (liquid bulk modulus)- Undrained analysisNOTE: B1 in the range of 100 to 500 B8,, (soil bulk modulus) isequivalent to using a Poisson’s ratio of 0.495 to 0.499.Appendix E. User Manual for CONOIL-Il 281If there are temperature changes, use consolidation routineiii) DUO= 71 (unit weight of liquid) - Consolidation analysisto do undrained analysis.- total unit weight of soil- permeability in x directionk - permeability in y directionRecord 5.2(r/o)1,i (r/o-) q — — B B0—- 0 (No parameter at present)(T/o-)f,i- Failure stress ratio at 1 atmosphere(r/o)- Reduction in failure stress ratio for a ten fold increasein confining pressure- Strain softening numberq - Strain softening exponenta8t - Coefficient of temperature induced structural reorientation. Only used in temperature analysis.—- 0 (No parameter at present)—- 0 (No parameter at present)B - Bulk modulus of the waterB0 - Bulk modulus of the oilRecord 5.3f-3O,O H H u U S S a a0Appendix K User Manual for CONOIL-Il 2821130,0 - 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 temperatureH =H+\H*TH - Henry’s coefficient of solubility- Function to modify bubble pressure for temperatureU - Bubble pressure (Oil/Gas saturation pressure)S - Initial degree of saturation varying between 0 and 1. (S= 1 implies 100% saturation)S, - Saturation at which fluid begins to move freely. (Usedfor modifying permeability. S is generally close to zero)- Coefficient of linear thermal expansion of water- Coefficient of linear thermal expansion of oila5 - Coefficient of linear thermal expansion of solidsRecord 5.4ISIGE ISIGB IMPF ILSHD F F KGp GP 11ISIGE - Option to calculate Young’s modulus0 - use mean normal stress1 - use minor principal stressISIGB - Option to calculate bulk modulus0 - use mean normal stress1 - use minor principal stressIMPF - Multi phase flow option0 - fully saturated1 - partially saturatedAppendix E. User Manual for GONOIL-Il 2832 - three phase fluid flow (needs additional parameters)ILSHD- Load transfer option0 - do not perform load transfer1 - perform load transfer by keeping o constant2 - perform load transfer by keeping o constant- Collapse modulus number (cap yield)F - Collapse modulus exponent (cap yield)KGp - Plastic shear parameter (cone yield, hardening rule)GP - Plastic shear exponent (cone yield, hardening rule)- Flow rule intercept (cone yield)- Flow rule slope (cone yield)Record 5.5 (necessary only if IMPF = 2, all are real variables except IV)Sw So Sg S S k0g IVL, IVO IV9S - Initial water saturationS, - Initial oil saturationS9 - Initial gas saturation(S + S0 + S must be equal to 1)5om - Residual oil saturationS - Connate water saturation (irreducible water saturation)- Relative permeability of oil at connate water saturation(oil-water)- Relative permeability of oil at zero gas saturation (oil-gas)IV,- Options to estimate viscosity of water0 - use a given constant value (in Pa.s)Appendix E. User Manual for CONOIL-Il 2841 - use the built-in feature in the program (Internationalcritical tables)>1 - interpolate using given temperature-viscosity profile(IV data pairs, maximum 10)IV, - Options to estimate viscosity of oil0 - use a given constant value (in Pa.s)1 - use the built-in feature in the program (Correlation byPuttangunta et.al (1988), to,o should be given in record6.4)>1 - interpolate using given temperature-viscosity profile(1V0 data pairs, maximum 10)IVg - Options to estimate viscosity of gas0 - use a given constant value (in Pa.s)1 - use the built-in feature in the program (a constant value2.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 D3Al...A3 - Parameters for relative permeability of water (oil-water)krw = A1(S — A2)A3Bl...B3- Parameters for relative permeability of oil (oil-water)= B1(B2 — S)B3Cl... 03 - Parameters for relative permeability of gas (oil-gas)k,.9 = C1(S9 — C2)c3Dl...D3 - Parameters for relative permeability of oil (oil-gas)Appendix E. User Manual for CONOIL-Il 285k,.09 = D1(D2Record 5.7 (necessary only if IMPF = 2)I Fl F2 F31Fi...F3 - Parameters for oil-gas capillary pressureof gas (oil-gas)Pc = Fl Pa(S9 — F2)’3Record 5.8 (necessary only if IMPF = 2 and IV,,, = 0 or >1)V,,, (ifIV=0)Vi Ti V2 T2•.• I (if IV, , 1, IV, data pairs, maximum 10)V - Constant viscosity value of water (in Pa.s)Vi,...- Viscosity values in the given profile (in Pa.s)Ti,...- Temperature values in the given profile (in °C)Record 5.9 (necessary only if IMPF = 2 and 1V0 = 0 or >1)V01 (ifIV0=0)Vi Ti V2 T2•.. I (if 1V0> 1, 1V0 data pairs, maximum 10)V0 - Constant viscosity value of oil (in Pa.s)Vi,...- Viscosity values in the given profile (in Pa.s)Ti,...- Temperature values in the given profile (in °C)Record 5.10 (necessary only if IMPF = 2 and 1V9 = 0 or >1)Appendix E. User Manual for CONOIL-Il 286(ifIV=0)I Vi Ti V2 T2 ... I (if IVg> 1, 1V9 data pairs, maximum 10)- Constant viscosity value of gas (in Pa.s)Vi,...- Viscosity values in the given profile (in Pa.s)Ti,...- Temperature values in the given profile (in °C)Record 6 ((IPRIM-1)/10 + 1 lines, only if IPRIM> 0)I Li L2Li,... - List of element numbers to be removed to form mesh atthe beginning of the analysis (LPPJM element numbers)There must be 10 data per line, except the last lineRecord 7 (one line only)INSIT NNI NELl NO UT IINSIT - In-situ stress option:0 - Set in-situ stresses to zero1 - Direct specification of in-situ stressesNNI - Number of nodes in-situ meshNELl - Number of elements in-situ meshNOUT- In-situ stress printing option:0 - Do not print the in-situ stresses1 - Print the variables at the centroids of each element2 - Print the variables at each integration point per elementand print the equilibrium loads for in-situ stresses.Appendix E. User Manual for CONOIL-Il 287Record 8 (NNI lines)NI XI Yl o- o, o- r uNI- In-situ mesh node numberXI - x coordinateY1-y coordinateo, o, o - Normal components of the effective stress vector- Shear stress componentii - Pore fluid pressure(Note that effective stress parameters are assumed)Record 9 (NELl lines)LI NIl N12 NI3]LI - In-situ mesh element numberNIl, N12, N13 - In-situ mesh node numbers (anticlockwise order)Record 10 (one line only, but records 10 to 14 are repeated for each analysis increment)INC ICHEL NLOD IFIX lOUT DTIME DGRAV NSINC NTEMP NPTSIINC - Increment numberICHEL - Number of elements to be removedNLOD - Number of CHANGES to incremental nodal loads or (ifNLOD is negative) the number of element sides whichhave their increment loading changed.Appendix E. User Manual for CONOIL-Il 288IFIX - Number of changes to nodal fixitieslOUT - Output option for this increment- a four digit numberabcd where:a - out of balance loads and reactionso - no out of balance loads1 - out of balance loads at vertex nodes2- out of balance loads at all nodesb - option for prescribed boundary conditions (e.g. fixitycondition or equivalent nodal loads at specified nodes)o no information printedI - data printed for each relevant d.o.fc - option for general stresseso - no stresses printed1 - stresses at element centroids2 - stresses at integration pointsd - option for nodal displacementso - no displacements printed1 - displacements at vertex nodes2 - displacements at all nodesDTIME- Time increment for consolidation analysisDGRAV- Increment in gravity level(change in number of gravities)NSINC - The number of sub increments (this is presently equal to 1)NTEMP- Number of changes to nodal temperatureDGRAV- Number of data pairs in the temperature-time history profileRecord 11 ((ICHEL-1)/1O + 1 lines, only if ICHEL > 0)Appendix E. User Manual for CONOIL-Il 289rLiLi,... - List of element numbers to be removed in this incrementThere must be 10 data per line, except the last lineRecord 12 (NLOD lines)(a) For .1\TLOD > 0NDFX DFY1N - Node numberDFX - Increment of x forceDFY - Increment of y forceFor NLOD < 0(b.1) For linear strain triangleLNJN2TJS1 T3S3T2S200001(b.2) For cubic strain triangleI L Ni N2 Ti Si T3 S3 T4 S4 T5 55 T2 SL - Element numberNi, N2 - Node numbers at the end of the loaded element sideTi - Increment of shear stress at Ni (see the following figure E.1Si - Increment of normal stress at NiTi - Increment of shear stress at NiAppendix E. User Manual for CONOIL-Il 290Si - Increment of normal stress at Ni etc.Sign convention for stresses:Shear - which act in an anticlockwise direction about elementcentroid are positiveNormal- compressive stresses are positiveNiN2Linear Strain Triangle Cubic Strain TriangleFigure E.1: Nodes along element edgesRecord 13 (one line only, but record from 10 to 15 are repeated for each analysisincrement)N ICODE DX DY DPINSN4N - Node numberAppendix E. User Manual for CONOIL-Il 291ICODE - A three digit code abc which specifies the degrees offreedom associated with this node that are fixed to particular valuesa - fix for x directiono - node is free in x direction1 - node is to have a prescribed incremental displacementDXb- fix for y directiono - node is free in y direction1- node is to have a prescribed incremental displacementDYc - fix for excess pore pressureo - no prescribed excess pore pressure1- the increment of excess pore pressure at this node is tohave a prescribed value DP2 - the absolute excess pore pressure at this node is to havea zero value for this and all subsequent increments ofanalysisDX - Prescribed displacement in x directionDY - Prescribed displacement in y directionDP - Prescribed pore pressureRecord 14 (NTEMP lines, only if NTEMP > 0)N TEM1 TIMEJ TEM2 TIME2.J (NPTS data pairs, maximum 15)N - Node numberTEMJ,...- Temperature in the given temperature time profileTIMEJ,...- Time in the temperature time profileAppendix E. User Manual for CONOIL-Il 292E.4 Detail ExplanationsDetailed explanations for some of the records are given in this section to provide abetter understanding.E.4.1 Geometry ProgramRecord 2The geometry program stores basic information describing the finite element mesh ona 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 runthe Main Program several times for each of these meshes. In order to ensure that aparticular Main Program run accesses the correct Link file the LINK number is storedon the Link file by the Geometry program and must be quoted correctly in the inputfor the Main Program. Hence LINK should be set to a different integer number foreach finite element mesh that the user specifies.Record 3LDEF (Element Types)The element type is defined by LDEF which at present can take one of four valuesassociated with the elements shown in Figure E.2. The variation of displacements(and consequently strains) and where appropriate, the excess pore pressures are summarized in table E.1. All elements are basically standard displacement finite elementswhich 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 elementtype:(i) Plane Strain AnalysisFor drained or undrained analysis use element type 1 (LST) and for consolidationAppendix E. User Manual for CONOIL-Il 2930 u,v— displacement unknowrsA p — pore pressure unknownsa.1.622S2(a) Element type 1 (LST) (b) Element type 5 (LST)6 nodes, 12 d.o.f. 6 nodes, 15 d.o.f.(consolidation)412—._. 1216216 11- / ,‘ 112/ 106/ 1// 102S - /.188 19 9 1(c) Element type 7 (CuST) (d) Element type 8 (CuST)15 nodes, 30 d.o.f. 22 nodes, 40 d.o.f.(consolidation)Figure E.2: Element typesAppendix E. User Manual for CONOIL-Il 294Table E.1: Element TypesVariation ofLEDF Element Name Displacement Strain Pore Pressure1 Linear strain triangle (LST) Quadratic Linear N/A5 LST with linearly varying Quadratic Linear Linearpore pressures7 Cubic strain triangle (CST) Quartic Cubic N/A5 CST with cubic variation of Quartic Cubic Cubicpore_pressuresanalysis use element type 5.(ii) Axisymmetric AnalysisFor drained analysis or consolidation analysis where collapse is not expected thenelement types 1 and 5 will probably be adequate (i.e. the same as (i) above). Forundrained analysis or in a situation where collapse is expected then element types 7and 8 are recommended. Recent research has shown that in axisymmetric analysisthe constraint of no volume change (which occurs in undrained situations) leads tofinite element meshes ‘locking up’ if low order finite elements (such as the LST) areused.NN (Number of Vertex Nodes)It should be noted that NN refers to the number of vertex nodes in the finite elementmesh. The geometry program automatically generates node numbers and coordinatesfor any nodes lying on element sides or within elements.Records 4 and 5ulElement and Nodal NumberingThe program user must assign each element and each vertex node in the finite elementmesh unique integer numbers in the following ranges:1 < node number 7501 < element number < 500Appendix E. User Manual for CONOIL-Il 295It is not necessary for either the node numbers or the element numbers to forma complete set of consecutive integers, i.e., there may be ‘gaps’ in the numberingscheme adopted. This facility means that users may modify existing finite elementmeshes by removing elements without the need for renumbering the whole mesh. TheGeometry Program assigns numbers in the range 751 upwards to nodes on elementsides and in element interiors.MAT Material Zone NumbersThe 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 differentmaterial properties, they can be modelled by different stress-strain relations. (Note:the material zone numbers have to consecutive).E.4.2 Main ProgramRecord 2The link number must be the same as that specified in the Geometry Program inputdata for the appropriate finite element mesh (see Record 2 in section E.4.1).Record 3NPLAX Plane strain/AxisymmetricThe selection of axes and the strain conditions under plane strain and axisymmetricconditions are shown in figures E.3 and E.4 respectively.NMAT Number of Materialssl NMAT must be equal to the number of different material zones specified in thegeometry program.IPRIMCONOIL allows excavations to be modelled in an analysis via the removal of elementsas the analysis proceeds. All the elements that appear at any stage in the analysisAppendix E. User Manual for CONOIL-Il 296KZZ.Figure E.3: Plane Strain Conditionxis the’adia1 direcSonz is the circwnferentiai directionFigure E.4: Axisymmetric ConditionAppendix E. User Manual for CONOIL-Il 297must have been included in the input data for the Geometry Program. IPRIM is thenumber of finite elements that must be removed to form the initial (or primary) finiteelement mesh before the analysis is started.IUPDIUPD = 0: This corresponds to the normal assumption that is made in linear elastic finite element programs and also in most finite element programs with nonlinearmaterial behaviour. External loads and internal stresses are assumed to be in equilibrium in relation to the original (i.e., undeformed) geometry of the finite elementmesh. This is usually known as the ‘small displacement’ assumption.IUPD = 1: When this option is used the nodal coordinates are updated after eachincrement of the analysis by adding the displacements undergone by the nodes duringthe increment to the coordinates. The stiffness matrix of the continuum is thencalculated 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 besatisfied in the final (deformed) configuration. Although this approach would seem tobe intuitively more appropriate when there are significant deformations it should benoted that it does not constitute a rigorous treatment of the large strain/displacementbehaviour for which new definitions of strains and stresses are required. Variousresearch workers have examined the influence of a large strain formulation on the loaddeformation response calculated by the finite element method using elastic perfectlyplastic models of soil behaviour. The general conclusion seems to be that the influenceof large strain effects is not very significant for the range of material parametersassociated with most soils. In most situations, the inclusion of large strain effectsleads to a stiffer load deformation response near failure and some enhancement ofthe load carrying capacity of the soil. If a program user is mainly interested in theestimation of a collapse load using an elastic perfectly plastic soil model then it isprobably best to use the small displacement approach (i.e., sl IUPD = 0). CollapseAppendix E. User Manual for CONOIL-Il 298loads can then be compared (and should correspond) with those obtained from aclassical theory of plasticity approach.ISELFIn many analyses the stresses included in the soil by earth’s gravity will be insignificantcompared to the stresses induced by boundary loads (e.g., in a laboratory triaxialtest). For this type of analysis it is convenient to set ISELF = 0 and correspondingly7 set to zero in Record 5.When the stresses due to the self weight of the soil do have a significant effect inthe 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 thenthe assumption is made that the original in-situ stresses were in equilibrium with thevarious densities (-y) in the Records 5.Records 7, 8 and 9In the nonlinear analyses performed by CONOIL, the stiffness matrix of a finite element is dependent on the stress state within the element. In general, the stressstate will vary across an element and the stiffness terms are calculated by integrating expressions dependent on these varying stresses over the volume of each element.CONOIL integrates these expressions numerically by ‘sampling’ the stresses at particular points within the element and then using standard numerical integration rulesfor triangular areas.The purpose of Records 7, 8 and 9 is to enable the program to calculate the stressesbefore the analysis starts. Although the in-situ mesh elements are specified in exactlythe same way as finite elements in the Geometry Program input, it should be notedthat they are not finite elements. The specification of the ‘in-situ mesh’ is simply adevice to allow stresses to be calculated at all integration points by a process of linearinterpolation over triangular regions. Thus, if the initial stresses vary linearly overthe finite element mesh, it is usually possible to use an in-situ mesh with one or twoAppendix E. User Manual for CONOIL-Il 299triangular elements.Records 10When a nonlinear or consolidation analysis is performed using CONOIL, it is necessary to divide either the loading or the time span off the analysis into a number ofincrements. Thus, if a total stress of 20 kN/m2 is applied to part of the boundary ofthe finite element mesh it might be divided into ten equal increments of 2 kN/m eachof which is applied in turn. CONOIL calculates the incremental displacements foreach increment using a tangent stiffness approach, i.e., the current stiffness propertiesare based on the stress state at the start of each increment. While it is desirable to useas many increments as possible to obtain accurate results, the escalating computercosts that this entails will inevitably mean that some compromise is made betweenaccuracy and cost. The recommended way of reviewing the results to determinewhether enough increments have been used in an analysis is to examine the valuesof shear stress level at each integration point. \Talues less than 1.10 are generallyregarded as leading to sufficiently accurate calculations. If values greater than 1.1 areseen then the size of the load increments should be reduced. Alternatively, the stresstransfer option can be invoked.The time intervals for consolidation analysis (DTIME) should be chosen after givingconsideration to the following factors:1. Excess pore pressures are assumed to vary linearly with time during each increment.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 oftime (thus for linear elastic analysis the same number of time increments wouldbe used in carrying the analysis forwarded from one day to ten days as fromAppendix E. User Manual for CONOIL-Il 300ten days to one hundred days). Not less than three time steps should be usedper log cycle off time (for a log base of ten). Thus a suitable scheme may be asshown in table E.2Table E.2: Time Increment SchemeIncrement No. DTIME Total Time1 1 12 1 23 3 54 5 105 10 206 30 507 50 1008 100 2009 300 50010 500 1000This 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 thefinite element equations will be ill conditioned.5. When a change in pore pressure boundary condition is applied, the associatedtime step should be large enough to allow the effect of consolidation to beexperienced by those nodes in the mesh with excess pore pressure variablesthat are close to the boundary. If this is not done then the solution will predictexcess pore pressures that show oscillations (both in time and space).The application of item 5 will often mean that the true undrained response willnot be captured in the solution The following procedure, however, usually leads tosatisfactory results.Appendix E. User Manual for GONOIL-Il 3011. Apply loads in the first increment (or first few increments for a nonlinear analysis) but do not introduce any pore pressure boundary conditions.2. Introduce the excess pore pressure boundary conditions in the increment following the application of the loads.NLOD and IFIXIt is important to note that NLOD and sl IFIX refer to the number of changes inloading and nodal fixities in a particular increment. CONOIL maintains a list ofloads and nodal fixities which the user may update by providing the program withappropriate data. Thus, if NLOD 0 and IFIX = 0, the program assumes that thesame incremental loads and fixities will be applied in the current increment as wereapplied in the previous increment. Another point to note is that loads applied areincremental, thus the total loads acting at any particular time are given by addingtogether all the previous incremental loads. The following example is intended toclarify these points for a consolidation analysis:1. Part of the boundary of a soil mass is loaded with a load of ten units (this isapplied 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 theincremental loads which CONOIL would otherwise assume.DGRAVAppendix E. User Manual for CONOIL-Il 302Table E.3: Load IncrementsLoadsIncrement No. Input to Incremental load Total loadprogram applied acting123456789101112132122232425262728etc.10-201111111111000-2-2-2-2-20001234567891010101086420000Appendix E. User Manual for CONOIL-Il 303DGRAV is used in problems in which the material’s self weight is increased duringan analysis (e.g. in the ‘wind-up’ stage of a centrifuge test increasing centrifugalacceleration can be regarded as having this effect).Appendix FUser Manual for CONOIL-IllF.1 IntroductionCONOIL-Ill is a three dimensional finite element program developed to analyze thestresses, deformations and flow in oil sands. Though CONOIL-Ill is specifically written for oil sands, it can be used for general geotechnical problems. CONOIL-Ill canperform drained, undrained and consolidation analyses and has the following specialfeatures.1. Elasto-Plastic stress strain model. Modified form of Matsuoka’s model is implemented.2. Three phase fluid flow. This is a special feature required to analyze the problemsin 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 theoriesbehind its development. Only the input parameters needed, their format and somebrief 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 aregiven at the end of this manual.The source code is written in FORTRAN-77. Input parameter names are given according to the standard FORTRAN naming convention. Names begin with the letters304Appendix F. User Manual for CONOIL-III 3051 J, L, M and N implies that the program expects integer data. Integer data shouldnot contain a decimal point. There are exceptions to this naming convention in record6 where the material property data are read. Actual material parameter notationsare retained to avoid confusions.F.2 Input DataRecord 1 (one line)TITLEITITLE - Title of the problem (up to 80 characters)Record 2 (one line)NCNOD, NINOD, NTEL, ITYPE, NINT, IPRNNCNOD - Total number of corner nodesNINOD - Total number of internal nodes (0 for ITYPE 1 and 3)NTEL - Total number of elementsITYPE - Element type (see fig. F.l)= 1 for drained/undrained analysis= 3 for consolidation analysisNINT - Number of integration points= 8 or 27 (generally 8 is good enough)IPRN - Index to print nodal and element information0 - Do not print the information1 - Print the informationAppendix F. User Manual for CONOIL-IllTYPE 1 TYPE 3306o Corner nodes = 8D.o.f. per node = 3Internal nodes = 0• Corner nodes = 8D.o.f. per node = 4Internal nodes = 0Figure F.1: Available Element TypesAppendix F. User Manual for CONOIL-IlI 307Record 3 (NCNOD+NINOD lines)NN, X(NN), Y(NN), Z(NN), T(NN)I\TN- Node numberX(NN)- X coordinate of the node NNY(NN)- Y coordinate of the node NNZ(NN) - Z coordinate of the node NNT(NN)- Initial temperature of the node NNRepeat record 3 for all nodes.Record 4 (NTEL lines)NE, Ni, N2, N3, N4, N5, N6, N7, N8, MATNE - Element numberN1...N8 - Corner node numbers of the element in anticlockwiseorder (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, theelement data for a series of elements are generated by increasing the preceding nodalnumbers by one. The material number for the generated elements are set equal tothe material number for the previous element. The first and the last elements mustbe specified.Record 5 (one line)PATM, GAMW, IDUC, INCi, INC2, NMAT, NTEMP, NPTS, IPRIM, ISELFAppendix F. User Manual for CONOIL-IlI 308PATM - Atmospheric pressureGAMW- Unit weight of waterID UC - Index for Drained/Undrained/Consolidation analysis0 - Drained analysis1 - Undrained analysis2- Consolidation analysisIf there are temperature changes, use consolidationroutine with no flow boundary conditions to performundrained analysis.INCJ - First increment number of the analysis1N02- Last increment number of the analysisNMAT- Number of material types (maximum 10)NTEMP- Number of nodes where temperature changesNPTS- Number of data pairs in the temperature-time profile (max. 15)IPRIM- Number of elements to be removed to form the primary meshISELF- Option to specify self weight load as in-situ stresses0 - in-situ stresses do not include self weight1 - in-situ stresses include self weightRecord 6(Records 6.1 to 6.11 have to repeated NMAT times.Record 6.5 is necessary only if MODEL 2 or 3.Records 6.6 to 6.11 are necessary only if IMPF = 2.)Record 6.1MAT, MODEL, ISICE, 151GB, ILSHD, IMPFAppendix F. User Manual for CONOIL-III 309MAT - Material numberMODEL - Stress-Strain model type1 - hyperbolic model2 - modified Matsuoka’s model3 - modified Matsuoka’s model with Cap-type yieldISIGE - Option to calculate Young’s modulus0 - use mean normal stress1 - use minor principal stress181GB - Option to calculate bulk modulus0 - use mean normal stress1 - use minor principal stressILSHD - Load transfer option0 - do not perform load transfer1 - perform load transferIMPF - Multi phase flow option0 - fully saturated1 - partially saturated2 - three phase fluid flow (needs additional parameters)Record 6.2 (all are real variables)e,KE,n,Rf,KB,m,7,k,k,ke - Initial void ratioKE - Elastic modulus constantn - Elastic modulus exponentAppendix F. User Manual for CONOIL-III 310- Failure ratioKB - Bulk modulus constantm- Bulk modulus exponent- total unit weight of soil- permeability in x direction- permeability in y direction- permeability in z directionif IMPF = 0 or 1 give the absolute permeability values (rn/s)if IMPF = 2 give intrinsic permeability values (m2)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 inconfining pressure- strain softening constantq - strain softening exponentS - Initial degree of saturation (between 0 and 1, not in %)- Saturation at which fluid begins to move freely. (usedto modify permeability for partially saturated soils. S1generally close to zero)B8- Bulk modulus of the solidsB - Bulk modulus of the waterB0 - Bulk modulus of the oilAppendix F. User Manual for CONOIL-III 311Record 6.4 (all are real variables),U30,0, )H, H, Au, U, —, ant, c.z8, a, a0jfL3o,o- Viscosity of oil at 300 C and 1 atmosphere (in Pa.s)(used in three phase flow, built-in oil viscosity correlation)- Function to modify Henry’s constant for temperatureH=H+AH*TH - Henry’s coefficient of solubilityAu - Function to modify bubble pressure for temperatureU - Bubble pressure (Oil/Gas saturation pressure)—- 0 (No parameter at present)a8t - Coefficient of volume change due to temperature induced structural reorientation- Coefficient of linear thermal expansion of solids- Coefficient of linear thermal expansion of watera0 - Coefficient of linear thermal expansion of oilRecord 6.5 (necessary only if MODEL = 2 or 3, all are real variables)C, p, K, rip, R1p, i, A, (r/), (r/o),—C - Cap-yield collapse modulus numberp - Cap-yield collapse modulus exponentK - Plastic shear numberlip - Plastic shear exponentR1 - Plastic shear failure ratio- flow rule interceptAppendix F. User Manual for CONOIL-III 312A - flow rule sloper/o-- Failure stress ratio at 1 atmosphere- Reduction in failure ratio for a ten fold increase in confining 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 IVjS,, - Initial water saturationS0 - Initial oil saturationS9 - Initial gas saturation(S + S0 H- S9 must be equal to 1)S - Residual oil saturationS - Connate water saturation (irreducible water saturation)- Relative permeability of oil at connate water saturation(oil-water)k,?09- Relative permeability of oil at zero gas saturation (oil-gas)IV,, - Options to estimate viscosity of water0 - use a given constant value (in Fa.s)1 - use the built-in feature in the program (Internationalcritical tables)>1 - interpolate using given temperature-viscosity profile(IV data pairs, maximum 10)1V0 - Options to estimate viscosity of oil0 - use a given constant value (in Pa.s)Appendix F. User Manual for CONOIL-III 3131- use the built-in feature in the program (Correlation byPuttangunta et.al (1988), to,o should be given in record6.4)>1 - interpolate using given temperature-viscosity profile(1V0 data pairs, maximum 10)1V9 - Options to estimate viscosity of gas0 - use a given constant value (in Pa..s)1 - use the built-in feature in the program (a constant value2.E-5 Pa.s)>1 - interpolate using given temperature-viscosity profile(1V9 data pairs, maximum 10)Record 6.7 (necessary only if IMPF = 2)Al, A2, A3, Bi, B2, B3, Cl, 02, 03, Dl, D2, D3Al.. .A3 - Parameters for relative permeability of water (oil-water)= A1(SL, — A2)-3Bl...B3 - Parameters for relative permeability of oil (oil-water)= B1(B2 —Cl... C3 - Parameters for relative permeability of gas (oil-gas)ICrg = C1(Sg —Dl...D3 - Parameters for relative permeability of oil (oil-gas)k,.09 = D1(D2 S9)”3Record 6.8 (necessary only if IMPF = 2)Fl, F2, F3Fl...F3 - Parameters for oil-gas capillary pressureAppendix F. User Manual for CONOIL-III 314of gas (oil-gas)Pc = Fl Pa(Sg — F2)F3Record 6.9 (necessary only if IMPF = 2 and IV,, = 0 or >1)V (ifIV=0)Vi, Ti, V2, T2,... (if IV 1, IV, data pairs, maximum 10)Vi,, - Constant viscosity value of water (in Pa.s)Vi,... - Viscosity values in the given profile (in Pa.s)Ti,... - Temperature values in the given profile (in °C)Record 6.10 (necessary only if IMPF = 2 and 1V0 = 0 or >1)___(ifIV0=0)Vi, Ti, V2, T2,... (if 1V0 > 1, IV,, data pairs, maximum 10)V0 - Constant viscosity value of oil (in Pa.s)Vi,... - Viscosity values in the given profile (in Fa.s)Ti,... - 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 1V9> 1, 1V9 data pairs, maximum 10)- Constant viscosity value of gas (in Pa.s)Vi,... - Viscosity values in the given profile (in Pa.s)Ti,... - Temperature values in the given profile (in °C)Appendix F. User Manual for CONOIL-III 315Record 7 (NTEMP lines, only if NTEMP > 0)TEM1, TIME1, TEM2, TIMEj (NPTS data pairs, maximum 15)N - Node numberTEM1,... - Temperature in the given temperature time profileTIMEJ,... - Time in the temperature time profileRecord 8 (one line)LINSIT, PINSITLINSIT - Option to specify in-situ stresseso - set the in-situ stresses to zero1 - read the in-situ stresses from dataPINSIT - Option to print in-situ stress datao - do not print1 - print in-situ stress dataRecord 9 (NTEL lines, only if LINSIT = 1)M, SIGX, SIGY, SIGZ, SIGXY, SIGYZ, SIGZX, PP1M - Element numberSIGX - Stress in x directionSIGY - Stress in y directionSIGZ- Stress in z directionSIGXY - Stress in xy directionAppendix F. User Manual for CONOIL-III 316SIGYZ- Stress in yz directionSIGZX- Stress in zx directionPP - Pore pressureRecord 9 has to be repeated for all elements. If elements cards are omitted, thestresses for a series of elements are generated by assigning the same stresses as theprevious 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 atthe beginning of the analysis (LPPJM element numbers)There must be 10 data per line, except the last lineRecord 11(one line, records 11 to 14 have to be repeated for increments from INC1 to INC2)INC, ICHEL, NLOAD, NFIX, 10 UT, DTIME, DGRAVINC - Increment numberICHEL- Number of elements to be removed from primary meshNLOAD- Number of nodes where loads are appliedNFIX - Number of nodes where nodal fixities are changedlOUT- Option for printing results (5 digit code ‘ abcde’)a = 1 print nodal displacementsb = 1 print moduli values and saturationsAppendix F. User Manual for CONOIL-IlI 317c = 1 print strains and coordinates of the integration pointwhere results are printedd = 1 print stresses and pore pressuree = 1 print velocity vectorsDTIME - Time incrementDGRAV - Increase in gravityRecord 12 ((ICH.EL-1)/10 + 1 lines, only if ICHEL > 0)Li, L2Li,... - List of element numbers to be removed in this incrementThere must be 10 data per line, except the last lineRecord 13 (NLOAD lines, only if NLOAD> 0)N, DFX, DFY, DFZIN - Node numberDFX - Increment in x forceDFY - Increment in y forceDFZ - Increment in z forceRecord 14 (NFIX lines, only if NFIX> 0)N, NFCODE, DX, DY, DZ, DPN - Node numberAppendix F. User Manual for CONOIL-III 318NFCODE - Four digit code ‘abcd’ which specifies the fixity conditions associated with the nodea = 0 free in x direction= 1 will have prescribed incremental displacement DXb = 0 free in y direction= 1 will have prescribed incremental displacement DYc = 0 free in z direction= 1 will have prescribed incremental displacement DZd = 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 allsubsequent incrementsDX - Prescribed displacement in x directionDY - Prescribed displacement in y directionDZ - Prescribed displacement in z directionDP - Prescribed pore pressureAppendix F. User Manual for CONOIL-IlI 319F.3 Example Problem 1An example of a general stress analysis under one dimensional loading is illustratedhere. The material is assumed to be linear elastic. The finite element mesh consistsof two brick elements as shown in figure F.2. The data file and the correspondingoutput file from the program are given in subsections.25 kNH GEi‘126...I0ZLol...AB, BC, CD, DA- Totally FixedAE, BF, CG, DH - Vertically FreeFigure 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”OOOLL60”O”O”OOOLL’8“O”O”O”OOOU.LOOO”OOOL.V9O”O”O”OOOLL9“OOO0’OLLLV•.0000’OLLLE“O”O”O”O’OILL“O”O”O”O’OLLLLS-”O”O’LS-O”OLLS-”O”O6Q’‘LLLLLIV’OL“o”o”o”o”oog”oog”oog’••O••O••O••O•OOS•OOS’OOS’LLL‘•o••o••o’•o•o’•o••o•o•oSL3L’SL3LS3LOOO”O”O”sOOO”O”OOOOOO”O”OOLOL0000LL‘O’000Lt096’OOL‘L’LLLOL68L99‘L8L9S’V’Li0LOL‘OL’L’LLO”O”LOL“O”O”O6o”VL”o9“O”L”I.”L1“O”L”O”L9“O•L009“O”O”L”0V“O”O”L”LE“o”o”o”oH•LLLOLONIOVO71VNOISN3YUO3N0SISA1VNVSS3UISVH3N35jIdmxaiojuir111710N00io;nuvJjisfljxrpuddELEMENT-NODALINFORMATIONNODESELE.NO.12345678o CD ‘•l —C)U112345678256789101112MATERIALPROPERTIESMATERIALI=MODELNISIGEISIGBILSHDIMPFLINEAR/NONLINEARELASTICMODELUSEMEANNORMALSTRESSUSEMEANNORMALSTRESSNOLOADSHEDDINGFULLYSATURATEDSOILO.100E+O10.150E+04O.000E÷OOO.000E+OO0.100E+04O.000E÷OOO.200E+O2O.000E+OOO.000E+OOO.000E+OOO.000E÷OOO.350E+02O.000E÷OOO.000E+OOO.000E+OOO.000E+OOO.000E+OOO.100E+160,100E+16O.100E+16O.000E+OOO.000E+OOO.000E+OOO.000E+OOO.OOOEOOO.000E+OOO.000E+OOO.000E+OOO.000E+OOO.000E+O0•AOFOIL(A)NALYSISOF(D)EFORMATIONAND(F)LOWIN(OIL)SANDSGENERALSTRESSANALYSIS,ONEDIMENSIONALLOADINGNODALCOORDINATESANDTEMPERATURENODEXCOORDY-COORDZ-COORDTEMP10.00000000.0000.00021.0000.0000.0000.00031.0001.0000.0000.00040.0001.0000.0000.00050.0000.0001.0000.00061.0000.0001.0000.00071.0001.0001.0000.00080.0001.0001.0000.00090.0000.0002.0000.000101.0000.0002.0000.000111.0001.0002.0000.000120.0001.0002.0000.000=0=0=0=0L’3INITIALSTRESSES XELEMSTRESS1O.5000E+032O.5000E+03INCREMENTNUMBER=VZSHEAR-XYSHEAR-YZSHEAR-ZXPORESTRESSSTRESSSTRESSSTRESSSTRESSPRESSUREO.5000E+03O.5000E+03O.0000E+OOO.0000E+OOO.0000E+OOO.0000E+OOO.5OOOEO3O.5000E+03O.0000E+OOO.0000E+OOO.0000E+OOO.0000E+OOINCH.INGRAVITY=0.0000E÷OOTOTALGRAVITY=O.0000E+O0TIMEINCREMENT=O.1000E+O1TOTALTIME=O.1000E+O1NODALDISPLACEMENTSI-sNODEXIZIXA1-0.8333E-16-O.8333E-1602O.8333E-16O.8333E-163O.8333E-16O.8333E-164-O.8333E-16-O.8333E-165-O.1667E-15-O.1667E-156O.1667E-15O.1667E-157O.1667E-15O.1667E-158-O.1667E-15-O.1667E-159-O.8333E-16-O.8333E-1610O.8333E-16O.8333E-1611O.8333E-16O.8333E-1612-O.8333E-16-O.8333E-16MODULIVALUESPLASTICPARAMETER0.0000E+OO0.0000E+0OINCREMENTAL VI-O.8333E-16-0.8333E-160.B333E-160.8333E-16-0.1667E-15-0.1667E-150.1667E-150.1667E-15-O.8333E-16-O.8333E-16O.8333E-16O.8333E-16BULKMODULUS0.1000E+060.1000E+06 VSTRAIN-O.2981E-15-D.2019E-15ABSOLUTE VA-O.8333E-16-O.8333E-160.8333E-16O.8333E-16-0.1667E-15-0.1667E-150.1667E-150.1667E-15-O.8333E-16-O.8333E-16O.8333E-16O.8333E-16VOIDRATIO0.I000E+010.1000E+01ELEM 2ZA-O.2500E-15-0.2500E-15-O.2500E-15-O.2500E-15-0.5556E-03O.5556E-03-0.5556E-03•O.5556E-O3-0.1111E-O2-0.1111E-02-0.111IE-02-0.1111E-02WATERSATURAN0.I000E+O10.1000E+O1ELASTICMODULUS0.1500E+060.1500E+O6-O.2500E-150.2500E-15-0.2500E-15-O.2500E-15-0.5556E-03-O.5556E-O3-0.5556E-03-O.5556E-O3-0.1111E-O2-0.1111E-02-0.1111E-O2-0.1111E-02POISSIONRATIO0.2500E+O00.2500E+00 ZSTRAIN0.5556E-O3O.5556E-03STRAINSELEM2XSTRAIN-O.2981E-15-0.20lYE-15OILSATURAN0.0000E+O0O.0000E+00XVYZSTRAINSTRAINGASSATURAN0.0000E+OO0.0000E+00TNT.POINTCOORDINATESXVZZXVOL.STRAINSTRAIN0.1233E-31O.4825E-16-O.4816E-16O.5556E-03O.79E+0O0.21E+00O.79E+0O-0.3698E-31-0.4779E-16O.4780E-160.5556E-03O.79E+OO0.21E+000.18E+0lC3I.3STRESSANDPOREPRESSURESXVZXVYZZXPORESTRESSELEMSTRESSSTRESSSTRESSSTRESSSTRESSSTRESSPRESSURELEVEL1O.5333E-O3O.5333E+03O.6000E+03O.7396E-27O.895E-11-O.2889E-11O.0000E+OOO.2398E-O1O.5333E+O3O.5333E+03O.6000E+03-O.2219E-26-O.2867E-11O.2868E-I1O.0000E+OOO.23g8E-Orj I
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Elasto-plastic deformation and flow analysis in oil sand masses Srithar, Thillaikanagasabai 1994
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Title | Elasto-plastic deformation and flow analysis in oil sand masses |
Creator |
Srithar, Thillaikanagasabai |
Date Issued | 1994 |
Description | 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 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. |
Extent | 7514983 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050410 |
URI | http://hdl.handle.net/2429/6941 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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