Preface ...................................................... xiii
Acknowledgements ............................................... xv
About the Authors ............................................ xvii
List of Symbols ............................................... xxi
PART ONE - FUNDAMENTALS OF SOLID MECHANICS
1 Stress/Strain Definitions and Components ..................... 3
1.1 General Concept ......................................... 3
1.2 Definition of Stress .................................... 4
1.3 Stress Components ....................................... 5
1.4 Definition of Strain .................................... 7
1.5 Strain Components ....................................... 8
Example ..................................................... 10
Problems .................................................... 10
2 Stress and Strain Transformation ............................ 13
2.1 Introduction ........................................... 13
2.2 Transformation Principles .............................. 13
2.3 Two-Dimensional Stress Transformation .................. 15
2.4 Stress Transformation in Space ......................... 18
2.5 Tensor of Stress Components ............................ 21
2.6 Strain Transformation in Space ......................... 22
Example ..................................................... 23
Problems .................................................... 24
3 Principal and Deviatoric Stresses and Strains ............... 27
3.1 Introduction ........................................... 27
3.2 Principal Stresses ..................................... 27
3.3 Average and Deviatoric Stresses ........................ 29
3.4 General Interpretation of Principal Stresses ........... 30
3.5 Two-Dimensional Stress Analysis ........................ 32
3.6 Properties of Strain ................................... 35
Examples .................................................... 36
Problems .................................................... 38
4 Theory of Elasticity ........................................ 41
4.1 Introduction ........................................... 41
4.2 Materials Behavior ..................................... 41
4.3 Hooke's Law ............................................ 43
4.4 Hooke's Law in Shear ................................... 43
4.5 Analysis of Structures ................................. 44
4.6 Theory of Inelasticity ................................. 45
4.7 Constitutive Relation for Rocks ........................ 45
Example ..................................................... 48
Problems .................................................... 49
5 Failure Criteria ............................................ 53
5.1 Introduction ........................................... 53
5.2 Failure Criteria for Rock Materials .................... 53
5.3 The Von-Mises Failure Criterion ........................ 54
5.4 Mohr-Coulomb Failure Criterion ......................... 54
5.5 The Griffith Failure Criterion ......................... 57
5.6 Hoek-Brown Failure Criterion ........................... 57
5.7 Druker-Prager Failure Criterion ........................ 58
5.8 Mogi-Coulomb Failure Criterion ......................... 58
Example ..................................................... 60
Problems .................................................... 61
PART TWO - PETROLEUM ROCK MECHANICS
6 Introduction to Petroleum Rock Mechanics .................... 65
6.1 Introduction ........................................... 65
6.2 Definition and Classification of Rocks ................. 65
6.3 Petroleum Rock Mechanics ............................... 67
6.4 Why Study Stress in Rocks? ............................. 72
6.5 Units of Measurement ................................... 74
Example ..................................................... 76
Problems .................................................... 76
7 Porous Rocks and Effective Stresses ......................... 77
7.1 Introduction ........................................... 77
7.2 Anisotropy and Inhomogeneity ........................... 77
7.3 Anisotropic Rocks, Transversal Isotropy ................ 79
7.3.1 Anisotropic rock properties ..................... 79
7.3.2 Properties of sedimentary rocks ................. 82
7.3.3 Effects of anisotropic rock properties .......... 84
7.3.4 Horizontal wellbore in laminated sedimentary
rocks ........................................... 85
7.4 Porous Rock ............................................ 95
7.5 Formation Pore Pressure ................................ 97
7.6 Effective Stress ....................................... 97
7.7 Formation Porosity and Permeability ................... 100
Example .................................................... 104
Problems ................................................... 104
8 In-situ Stress ............................................. 107
8.1 Introduction .......................................... 107
8.2 Definitions ........................................... 108
8.3 ln-situ Principal Stresses ............................ 108
8.4 Measurement and Estimation of In-Situ Stresses ........ 109
8.5 Probabilistic Analysis of Stress Data ................. 114
8.6 Bounds on ln-situ Stresses ............................ 114
8.6.1 Problem statement .............................. 115
8.6.2 The in-situ stresses ........................... 116
8.6.3 Bounds on the in-situ stresses ................. 117
8.6.4 Application of the model ....................... 118
8.7 Stress Directions from Fracture Traces ................ 120
8.7.1 Traces from fractures ......................... 120
8.7.2 Interpretation of fracture traces ............. 122
Examples ................................................... 123
Problems ................................................... 126
9 Rock Strength and Rock Failure ............................. 131
9.1 Introduction .......................................... 131
9.2 Strength of Rock Material ............................. 131
9.3 Empirical Correlations ................................ 132
9.4 Formation Fracture Gradient ........................... 134
9.4.1 Direct method .................................. 135
9.4.2 Indirect method ................................ 137
9.5 Laboratory Testing of Intact Rocks .................... 139
9.6 Rock Tensile Strength ................................. 142
9.7 Rock Shear Strength ................................... 144
9.7.1 Triaxial test method ........................... 144
9.7.2 Failure criteria ............................... 146
Examples ................................................... 147
Problems ................................................... 148
10 Stresses Around a Wellbore ................................. 151
10.1 Introduction .......................................... 151
10.2 State of Stresses Around a Wellbore ................... 151
10.3 Properties of Rock Formation Around a Wellbore ........ 153
10.4 Equations Governing Stress Analysis ................... 154
10.4.1 Equations of equilibrium ....................... 154
10.4.2 Equation of compatibility ...................... 155
10.4.3 Constitutive relations ......................... 156
10.4.4 Boundary conditions ............................ 158
10.5 Analysis of Stresses Around a Wellbore ................ 158
10.5.1 Definition of the problem ...................... 158
10.5.2 General assumptions ............................ 159
10.5.3 Analysis methodology ........................... 159
10.5.4 Stress transformation .......................... 160
10.6 Isotropic Solution .................................... 162
10.6.1 Governing Equations ............................ 162
10.6.2 Boundary Conditions ............................ 163
10.7 Anisotropic Solution .................................. 164
10.7.1 Governing Equations ............................ 164
10.7.2 Boundary Conditions ............................ 165
Examples ................................................... 169
Problems ................................................... 170
11 Wellbore Instability Analysis .............................. 173
11.1 Introduction .......................................... 173
11.2 Analysis Procedure .................................... 174
11.3 Wellbore Fracturing Pressure .......................... 177
11.4 Wellbore Collapse Pressure ............................ 180
11.5 Instability Analysis of Multi-lateral Boreholes ....... 183
11.6 Instability Analysis of Adjacent Boreholes ............ 191
11.7 Instability Analysis of Underbalanced Drilling ........ 196
11.8 Shallow Fracturing .................................... 200
11.8.1 Depth-normalized shallow fracture data ......... 200
11.8.2 Estimation of shallow fracture gradient for
a semi-sub and a jack-up rig ................... 203
11.9 General Fracturing Model .............................. 203
11.9.1 Introduction ................................... 203
11.9.2 Development of the model ....................... 204
11.9.3 Field cases .................................... 209
11.10 Compaction Analysis for High-Pressure High-
Temperature Reservoirs ................................ 210
11.11 Breakthrough of a Relief Well into a Blowing Well .... 216
11.11.1 Fracturing at a distance ...................... 217
11.11.2 Collapse when communicating ................... 217
11.11.3 Information from a drillability analysis ...... 220
11.12 Fracture Model for Load History and Temperature ...... 223
11.12.1 The effect of Poisson's ratio ................. 224
11.12.2 The effect of temperature ..................... 225
11.12.3 Initial conditions and history matching ....... 225
11.12.4 Application of the new model .................. 228
11.13 Effects of Flow Induced Stresses ..................... 229
11.14 Sand Production Modeling ............................. 233
11.14.1 Sand production during reservoir depletion .... 234
11.14.2 Sand Production in Elliptical Wellbores ....... 239
Examples ................................................... 245
Problems ................................................... 254
12 Wellbore Instability Analysis Using Inversion Technique .... 257
12.1 Introduction .......................................... 257
12.2 Definitions ........................................... 257
12.3 The Inversion Technique ............................... 257
12.4 Geological Aspects .................................... 260
12.4.1 First scenario - isotropic stress state ........ 260
12.4.2 Second scenario - anisotropic stress state ..... 261
12.5 Analysis Constraints .................................. 261
12.6 Inversion from Fracture Data and Image Logs ........... 263
Examples ................................................... 264
Problems ................................................... 273
13 Wellbore Instability Analysis Using Quantitative Risk
Assessment ................................................. 275
13.1 Introduction .......................................... 275
13.2 Deterministic Analysis Versus Probabilistic
Assessment ............................................ 275
13.3 Why Probabilistic Assessment? ......................... 276
13.4 Quantitative Risk Assessment (QRA) .................... 277
13.4.1 The QRA process ................................ 277
13.4.2 Key physical parameters ........................ 279
13.4.3 Limit state function ........................... 281
13.4.4 Probability failure function ................... 282
13.4.5 Sensitivity analysis ........................... 283
13.5 Quantitative Risk Assessment of Underbalanced
Drilling .............................................. 284
Example .................................................... 288
Problems ................................................... 294
14 The Effect of Mud Lossess on Wellbore Stability ............ 297
14.1 Introduction .......................................... 297
14.2 Mud Losses During Drilling ............................ 297
14.2.1 Experimental work .............................. 297
14.2.2 The fracturing models .......................... 300
14.2.3 Description of the fracturing process .......... 301
14.2.4 Properties of the mud cake ..................... 303
14.2.5 Shallow well field case ........................ 304
14.2.6 Recommended mud recipes ........................ 305
14.3 Interpretation of the Leak-off Tests .................. 306
14.3.1 Experiments with continuous pumping ............ 307
14.3.2 What happens at the fracture failure ........... 309
14.3.3 Leak-off test interpretation ................... 310
14.3.4 Irreversibility of the fracturing process ...... 313
14.3.5 Summary of the key findings .................... 314
14.4 Future Development for Wellbore Stability ............. 314
Examples ................................................... 315
Problems ................................................... 316
References ................................................. 319
Appendix A - Mechanical Properties of Rocks ................... 323
Appendix В - The Poisson's Ratio Effect ....................... 327
Appendix С - Model for the Stress Bridge ...................... 331
Appendix D - Glossary of Terms ................................ 335
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