Elasto-Plastic Deformation and Flow Analysis in Oil
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ELASTO-PLASTIC DEFORMATION AND FLOW ANALYSIS IN OIL SAND MASSES by THILLAIKANAGASABAI SRITHAR B. Sc (Engineering), University of Peradeniya, Sri Lanka, 1985 M. A. Sc. (Civil Engineering) University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1994 © THILLAIKANAGASABAI SRITHAR, 1994 _______________________ In . presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of Civil Engineering The University of British Columbia Vancouver, Canada Date - A?R L 9 L DE-6 (2188) Abstract Prediction of stresses, deformations and fluid flow in oil sand layers are important in the design of an oil recovery process. In this study, an analytical formulation is developed to predict these responses, and implemented in both 2-dimensional and 3-dimensional finite element programs. Modelling of the deformation behaviour of the oil sand skeleton and modelling of the three-phase pore fluid behaviour are the key issues in developing the analytical procedure. The dilative nature of the dense oil sand matrix, stress paths that involve decrease in mean normal stress under constant shear stress, and loading-unloading sequences are some of the important aspects to be considered in modelling the stress-strain behaviour of the sand skeleton. Linear and nonlinear elastic models have been found incapable of handling these aspects, and an elasto-plastic model is postulated to capture the above aspects realistically. The elasto-plastic model is a double-hardening type and consists of cone and cap-type yield surfaces. The model has been verified by comparison with laboratory test results on oil sand samples under various stress paths and found to be in very good agreement. The pore fluid in oil sand comprises three phases namely, water, bitumen and gas. The effects of the individual phase components are considered and modelled through an equivalent fluid that has compressibility and hydraulic conductivity characteristics representative of the components. Compressibility of the gas phase is obtained using gas laws and the equivalent compressibility is derived by considering the individual contributions of the phase components. Equivalent hydraulic conductivity is derived from the knowledge of relative permeabilities and viscosities of the phase components. Effects of temperature changes due to steam injection are also included directly 11 in the stress-strain relation and in the flow continuity equations. The analytical equations for the coupled stress, deformation and flow problem are solved by a finite element procedure. The finite element programs have been verified by comparing the program results with closed form solutions and laboratory test results. The finite element program has been applied to predict the responses of a hor izontal well pair in the underground test facility of Alberta Oil Sand Technology and Research Authority (AOSTRA). The results are discussed and compared with the measured responses wherever possible, and indicate the analysis gives insights into the likely behaviour in terms of stresses, deformations and flow and would be important in the successful design and operation of an oil recovery scheme. 111 Table of Contents Abstract ii List of Tables x List of Figures xi Acknowledgement xvi Nomenclature xvii 1 Introduction 1 1.1 Characteristics of Oil Sand 4 1.2 Scope and Organization of the Thesis 8 2 Review of Literature 10 2.1 Stress-Strain Models 10 2.1.1 Stress-Strain Behaviour of Oil Sands 11 2.1.2 Stress-Strain Models for Sand 19 2.1.2.1 Elasto-Plastic Models 20 2.1.2.2 Constituents of Theory of Plasticity 22 2.1.3 Stress Dilatancy Relation 23 2.1.4 Modelling of Stress-Strain Behaviour of Oil Sand 24 2.2 Modelling of Fluid Flow in Oil Sand 25 2.3 Coupled Geomechanical-Fluid Flow Models for Oil Sands 27 2.4 Comments 30 iv 3 Stress-Strain Model Employed 32 3.1 Introduction 32 3.2 Description of the Model 35 3.3 Plastic Shear Strain by Cone-Type Yielding 37 3.3.1 Background of the Model 37 3.3.2 Yield and Failure Criteria 42 3.3.3 Flow Rule 47 3.3.4 Hardening Rule 48 3.3.5 Development of Constitutive Matrix ] . [C8 51 3.4 Plastic Collapse Strain by Cap-Type Yielding 55 3.4.1 Background of the Model 55 3.4.2 Yield Criterion 57 3.4,3 Flow Rule 58 3.4.4 Hardening Rule 58 3.4.5 Development of Constitutive Matrix [Cc] 59 3.5 Elastic Strains by Hooke’s Law 61 3.6 Development of Full Elasto-Plastic Constitutive Matrix 62 3.7 2-Dimensional Formulation of Constitutive Matrix • 65 3.8 Inclusion of Temperature Effects • 67 3.9 Modelling of Strain Softening by Load Shedding 68 3.9.1 Load Shedding Technique 70 3.10 Discussion 72 4 Stress-Strain Model - Parameter Evaluation and Validation 74 4.1 Introduction 74 4.2 Evaluation of Parameters 74 4.2.1 Elastic Parameters 75 4.2.1.1 Parameters kE and n 75 v 4.2.1.2 Parameters kB and m 76 4.2.2 Evaluation of Plastic Collapse Parameters 79 4.2.3 Evaluation of Plastic Shear Parameters 80 4.2.3.1 Evaluation of ij and L2 82 4.2.3.2 Evaluation of and ) 82 4.2.3.3 Evaluation of KG, np and R1 83 4.2.4 Evaluation of Strain Softening Parameters 86 4.3 Validation of the Stress-Strain Model 87 4.3.1 Validation against Test Results on Ottawa Sand 88 4.3.1.1 Parameters for Ottawa Sand 91 4.3.1.2 Validation 96 4.3.2 Validation against Test Results on Oil Sand 96 4.3.2.1 Parameters for Oil Sand 101 4.3.2.2 Validation 107 4.4 Sensitivity Analyses of the Parameters 109 4.5 Summary 114 5 Flow Continuity Equation 115 5.1 Introduction 115 5.2 Derivation of Governing Flow Equation 116 5.3 Permeability of the Porous Medium 123 5.4 Evaluation of Relative Permeabilities 124 5.5 Viscosity of the Pore Fluid Components 132 5.5.1 Viscosity of Oil 132 5.5.2 Viscosity of Water 134 5.5.3 Viscosity of Gas 136 5.6 Compressibility of the Pore Fluid Components 136 5.7 Incorporation of Temperature Effects 140 vi 175 146 148 149 152 158 168 181 203 208 . 5.8 Discussion 142 6 Analytical and Finite Element Formulation 144 6.1 Introduction 144 6.2 Analytical Formulation 145 6.2.1 Equilibrium Equation 6.2.2 Flow Continuity Equation 6.2.3 Boundary Conditions 6.3 Drained and Undrained Analyses 6.4 Finite Element Formulation 6.5 Finite Elements and the Procedure Adopted 6.5.1 Selection of Elements 158 6.5.2 Nonlinear Analysis 159 6.5.3 Solution Scheme 162 6.5.4 Finite Element Procedure 164 6.6 Finite Element Programs 166 6.6.1 2-Dimensional Program CONOIL-Il . 166 6.7 3-Dimensional Program CONOIL-Ill 167 7 Verification and Application of the Analytical Procedure 168 7.1 Introduction 7.2 Aspects Checked by Previous Researchers . 7.3 Validation of Other Aspects 7.4 Verification of the 3-Dimensional Version 7.5 Application to an Oil Recovery Problem 183 7.5.1 Analysis with Reduced Permeability . 7.6 Other Applications in Geotechnical Engineering vii 8 Summary and Conclusions 216 8.1 Recommendations for Further Research 219 Bibliography 220 Appendices 242 A Load Shedding Formulation 242 A.1 Estimation of {LO}LS 243 A.2 Estimation of {F}Ls 245 B Relative Permeabilities and Viscosities 247 B.1 Calculations of relative permeabilities 247 B.1.1 Relevant equations . 247 B.1.2 Example data . 249 B.1.3 Sample calculations . 249 B.2 Viscosity of water 250 B.3 Viscosity of hydrocarbon gases (from Carr et al., 1954) 252 B.3.1 Example calculation 254 C Subroutines in the Finite Element Codes 258 C.1 2-Dimensional Code CONOIL-Il 258 C.1.1 Geometry Program 258 C.1.2 Main Program 259 C.2 3-dimensional code CONOIL-Ill 261 D Amounts of Flow of Different Phases 264 E User Manual for CONOIL-Il 270 E.1 Introduction 270 E.2 Geometry Program 272 viii . E.3 Main Program 275 E.4 Detail Explanations 292 E.4.1 Geometry Program 292 E.4.2 Main Program 295 F User Manual for CONOIL-Ill 304 F.1 Introduction 304 F.2 Input Data 305 F.3 Example Problem 1 319 F.3.1 Data File for Example 1 320 F.3.2 Output file for Example 1 321 ix List of Tables 4.1 Summary of Soil Parameters 75 4.2 Soil Parameters for Ottawa Sand at Dr = 50% 94 4.3 Details of the Test Samples 101 4.4 Soil Parameters for Oil Sand 107 5.1 Parameters needed for relative permeability calculations 133 7.1 Parameters for Modelling of Triaxial Test in Oil Sand 178 7.2 Model Parameters Used for Ottawa Sand 181 7.3 Parameters Used for Thermal Consolidation 184 7.4 Parameters Used for the Oil Recovery Problem.