Rasmus Risnes in Memoriam ....................................... v
Preface to the second edition ................................. vii
Foreword to the 1992 edition ................................... ix
Preface to the 1992 edition .................................... xi
Chapter 1. ELASTICITY ........................................... 1
1.1. Stress ..................................................... 1
1.1.1. The stress tensor ................................... 3
1.1.2. Equations of equilibrium ............................ 5
1.1.3. Principal stresses in two dimensions ................ 7
1.1.4. Mohr's stress circle ................................ 8
1.1.5. Principal stresses in three dimensions .............. 8
1.1.6. Mohr's stress circles in three dimensions .......... 10
1.1.7. Stress invariants .................................. 11
1.1.8. Deviatoric stresses ................................ 11
Geometric interpretation of the deviatoric stress
invariants ......................................... 12
1.1.9. The octahedral stresses ............................ 13
1.2. Strain .................................................... 13
1.2.1. The strain tensor and the strain invariants ........ 17
1.2.2. Compatibility conditions ........................... 17
1.2.3. Principal strains .................................. 18
1.2.4. Plane strain and plane stress ...................... 19
1.3. Elastic moduli ........................................... 20
1.4. Strain energy ............................................. 23
1.5. Thermoelasticity .......................................... 24
1.5.1. Thermal strain ..................................... 24
1.5.2. Thermal stress ..................................... 24
1.5.3. Stress strain relation for linear
thermoelasticity ................................... 25
1.5.4. Isothermal and adiabatic moduli .................... 25
1.5.5. Example: Thermal stresses in a constrained
square plate ....................................... 25
1.6. Poroelasticity ............................................ 26
1.6.1. Suspension of solid particles in a fluid ........... 26
1.6.2. Biot's poroelastic theory for static properties .... 27
1.6.3. The effective stress concept ....................... 32
1.6.4. Pore volume compressibility and related topics ..... 34
1.6.5. The Skempton coefficients .......................... 35
1.6.6. The correspondence to thermoelasticity ............. 36
1.6.7. Other notation conventions ......................... 37
1.7. Anisotropy ................................................ 37
1.7.1. Orthorhombic symmetry .............................. 39
1.7.2. Transverse isotropy ................................ 41
1.8. Nonlinear elasticity ...................................... 42
1.8.1. Stress-strain relations ............................ 42
1.8.2. The impact of cracks ............................... 44
1.9. Time-dependent effects .................................... 46
1.9.1. Consolidation ...................................... 46
1.9.2. Creep .............................................. 50
References ..................................................... 53
Further reading ................................................ 53
Chapter 2 FAILURE MECHANICS .................................... 55
2.1. Basic concepts ............................................ 55
2.1.1. Strength and related concepts ...................... 55
2.1.2. The failure surface ................................ 58
2.2. Tensile failure ........................................... 59
2.3. Shear failure ............................................. 60
2.3.1. The Mohr-Coulomb criterion ......................... 61
2.3.2. The Griffith criterion ............................. 65
2.4. Compaction failure ........................................ 66
2.5. Failure criteria in three dimensions ...................... 68
2.5.1. Criteria independent of the intermediate
principal stress ................................... 69
2.5.2. Criteria depending on the intermediate
principal stress ................................... 70
7r-plane representation ................................... 73
Physical explanations ..................................... 74
2.6. Fluid effects ............................................. 75
2.6.1. Pore pressure ...................................... 75
2.6.2. Partial saturation ................................. 76
2.6.3. Chemical effects ................................... 78
2.7. Presentation and interpretation of data from failure
tests ..................................................... 79
2.8. Beyond the yield point .................................... 80
2.8.1. Plasticity ......................................... 81
Plastic flow ....................................... 82
Associated flow .................................... 84
Non-associated flow ................................ 86
Hardening .......................................... 86
2.8.2. Soil mechanics ..................................... 88
Normally consolidated clays ........................ 90
Overconsolidated clays ............................. 92
2.8.3. Localization ....................................... 93
2.8.4. Liquefaction ....................................... 94
2.9. Failure of anisotropic and fractured rocks ................ 95
2.9.1. Intrinsic anisotropy and the failure surface ....... 95
2.9.2. The plane of weakness model ........................ 95
2.9.3. Fractured rock ..................................... 97
2.10.Stress history effects .................................... 99
2.10.1.Rate effects and delayed failure ................... 99
2.10.2.Fatigue ........................................... 100
References .................................................... 100
Chapter 3 GEOLOGICAL ASPECTS OF PETROLEUM RELATED ROCK
MECHANICS ........................................... 103
3.1. Underground stresses ..................................... 103
3.2. Pore pressure ............................................ 114
3.3. Sedimentological aspects ................................. 117
3.3.1. Grains and minerals ............................... 117
3.3.2. Pre-deposition and deposition ..................... 120
3.3.3. Post-deposition ................................... 121
3.4. Mechanical properties of sedimentary rocks ............... 123
3.4.1. Sandstone ......................................... 124
3.4.2. Chalk ............................................. 126
3.4.3. Shale ............................................. 128
3.4.4. Rock salt ......................................... 130
References .................................................... 131
Further reading ............................................... 133
Chapter 4 STRESSES AROUND BOREHOLES. BOREHOLE FAILURE CRI
TERIA ............................................... 135
4.1. Stresses and strains in cylindrical coordinates .......... 135
4.2. Stresses in a hollow cylinder ............................ 137
4.2.1. The equilibrium equations ......................... 137
4.2.2. Stress distributions with constant pore
pressure .......................................... 138
4.2.3. Stress distributions with varying pore pressure ... 140
The superposition principle ....................... 142
Radial flow ....................................... 142
4.2.4. Stress distributions with heat flow ............... 143
4.2.5. Stress distributions in nonlinear formations ...... 144
4.3. Elastic stresses around wells—the general solution ....... 145
4.3.1. Transformation formulas ........................... 146
4.3.2. The general elastic solution ...................... 147
4.3.3. Borehole along a principal stress direction ....... 148
4.4. Poroelastic time dependent effects ....................... 150
4.4.1. Wellbore pressure invasion ........................ 151
4.4.2. Drillout induced pore pressure changes ............ 153
4.5. Borehole failure criteria ................................ 154
4.5.1. Vertical hole, isotropic horizontal stresses
and impermeable borehole wall ..................... 155
4.5.2. Vertical hole, isotropic horizontal stresses
and permeable borehole wall ...................... 157
4.5.3. Borehole along a principal stress direction ....... 158
4.5.4. Borehole in a general direction ................... 159
4.6. Beyond failure initiation ................................ 160
4.6.1. A simple plasticity model ......................... 164
The Tresca criterion .............................. 164
The Mohr-Coulomb criterion ........................ 165
Deformation and plastic strain .................... 168
4.7. Spherical coordinates .................................... 170
4.7.1. Basic equations ................................... 170
4.7.2. Stress distribution around a spherical cavity
with no fluid flow ................................ 171
4.7.3. Stress distribution with fluid flow ............... 171
References .................................................... 172
Further reading ............................................... 173
Chapter 5 ELASTIC WAVE PROPAGATION IN ROCKS ................... 175
5.1. The wave equation ........................................ 175
5.2. P- and S-waves ........................................... 177
5.3. Elastic waves in porous materials ........................ 180
5.3.1. Biot's theory of elastic wave propagation ......... 180
5.3.2. Dispersion due to local flow ...................... 184
5.4. Attenuation .............................................. 184
5.5. Anisotropy ............................................... 189
5.5.1. The Christoffel equation .......................... 189
5.5.2. Weak anisotropy ................................... 191
5.6. Rock mechanics and rock acoustics ........................ 192
5.6.1. Static and dynamic moduli ......................... 192
5.6.2. Stress state and stress history ................... 195
5.6.3. Additional effects ................................ 197
Temperature ....................................... 197
Partial saturation ....................................... 197
Chemical effects ......................................... 199
5.7. Reflections and refractions .............................. 200
5.7.1. Interface waves ................................... 203
5.8. Borehole acoustics ....................................... 204
5.8.1. Borehole modes .................................... 206
5.8.2. Borehole alteration ............................... 212
5.9. Seismics ................................................. 214
References .................................................... 217
Further reading ............................................... 218
Chapter 6 ROCK MODELS ......................................... 219
6.1. Layered media ............................................ 220
6.2. Models involving porosity only ........................... 222
6.3. Grain pack models ........................................ 225
6.4. Models for cracks and other inclusions ................... 230
6.4.1. Linear, isotropic models .......................... 231
6.4.2. Anisotropic models ................................ 233
6.4.3. Models accounting for interactions ................ 235
6.4.4. Crack development in stressed rocks ............... 240
6.5. Fractured rocks .......................................... 243
6.5.1. Single fractures .................................. 243
6.5.2. Rocks with many fractures ......................... 246
References .................................................... 249
Chapter 7 MECHANICAL PROPERTIES AND STRESS DATA FROM LABO
RATORY ANALYSIS ..................................... 251
7.1. Core samples for rock mechanical laboratory analysis ..... 252
7.1.1. Core representativeness and size effects .......... 252
7.1.2. Core alteration ................................... 252
7.1.3. Core handling ..................................... 254
7.1.4. Preparation of test samples ....................... 255
7.2. Laboratory equipment ..................................... 257
7.2.1. The load frame .................................... 258
7.2.2. The triaxial cell ................................. 259
7.2.3. Measurements of stresses and strains .............. 261
7.2.4. Acoustic measurements ............................. 261
7.3. Laboratory tests for rock mechanical property
determination ............................................ 263
7.3.1. Stresses on cylindrical samples ................... 263
7.3.2. Drained and undrained test conditions ............. 264
7.3.3. Standard triaxial compression tests ............... 265
7.3.4. Interpretation of elastic moduli from triaxial
tests ............................................. 266
7.3.5. Unconfined (uniaxial) compression tests ........... 267
7.3.6. Hydrostatic tests ................................. 268
7.3.7. Triaxial testing of shales ........................ 269
7.3.8. Oedometer (K0) test ............................... 270
7.3.9. Stress path tests ................................. 272
Constant stress ratio tests ....................... 272
Constant Mean Stress (CMS) tests .................. 272
7.3.10.Other triaxial failure tests ...................... 273
Extension tests ................................... 273
Multiple and continuous failure state triaxial
tests ............................................. 273
True triaxial tests ............................... 274
7.3.11.Hollow cylinder tests ............................. 275
7.3.12.Measurements on small samples ..................... 276
7.4. Laboratory tests for stress determination ................ 277
7.4.1. Differential strain curve analysis ................ 277
7.4.2. Anelastic strain recovery ......................... 279
7.4.3. Acoustic techniques ............................... 279
Differential wave velocity analysis ............... 279
Acoustic emission ................................. 280
7.5. Index tests .............................................. 280
7.5.1. Tensile strength indicators ....................... 281
7.5.2. Hardness measurements ............................. 282
7.5.3. Indentation and scratch tests ..................... 283
7.5.4. Specific shale characterization tests ............. 283
References .................................................... 284
Chapter 8 MECHANICAL PROPERTIES AND IN SITU STRESSES FROM
FIELD DATA .......................................... 289
8.1. Estimation of elastic parameters ......................... 289
8.1.1. Acoustic wireline logs ............................ 290
Full waveform sonic tools ................................ 290
Multipole sonic tools .................................... 291
8.1.2. Acoustic logging while drilling ................... 292
8.1.3. Acoustic measurements on drill cuttings ........... 292
8.2. Estimation of strength parameters ........................ 293
8.2.1. Log data (wireline and MWD) ....................... 293
8.2.2. Drill cuttings measurements ....................... 293
8.2.3. Empirical correlations ............................ 294
8.2.4. Drilling data ..................................... 295
8.3. Estimation of in situ stresses ........................... 295
8.3.1. The density log (overburden stress) ............... 296
8.3.2. Borehole logs (horizontal stress directions) ...... 296
Caliper logs ............................................. 297
Image logs ............................................... 298
8.3.3. Fracture tests (horizontal stress magnitudes) ..... 298
Leak-off tests and extended leak-off tests ........ 299
Mini-frac tests ................................... 304
Wireline tools .................................... 305
Empirical relations ............................... 305
8.3.4. Other methods ..................................... 306
References .................................................... 306
Further reading ............................................... 308
Chapter 9 STABILITY DURING DRILLING ........................... 309
9.1. Unstable boreholes: Symptoms, reasons and consequences ... 310
9.1.1. Tight hole/stuck pipe ............................. 311
9.1.2. Lost circulation .................................. 312
9.2. Rock mechanics analysis of borehole stability during
drilling ................................................. 314
9.3. Time-delayed borehole failure ............................ 320
9.3.1. Establishment of pore pressure equilibrium ........ 320
9.3.2. Temperature effects ............................... 321
9.3.3. Creep ............................................. 322
9.4. Interaction between shale and drilling fluid ............. 323
9.5. Borehole stability analysis for well design .............. 325
9.6. Use of pressure gradients ................................ 329
9.6.1. Introduction ...................................... 329
9.6.2. Depth reference and depth corrections ............. 330
9.7. Beyond simple stability analysis ......................... 331
9.7.1. Field cases: The borehole stability problem in
complex geology ................................... 331
Case 1: Drilling-induced lateral shifts
along pre-existing fractures (Meillon
St. Faust Field, France) .................. 331
Case 2: Drilling in a complex geological
setting with high tectonic stresses
(Cusiana Field, Colombia) ................. 331
Case 3: The Heidrun Field, Norwegian Sea .......... 332
9.7.2. Drilling in depleted reservoirs ................... 333
9.7.3. Drilling below deep water ......................... 333
9.7.4. Surge and swab effects ............................ 334
9.7.5. Hole cleaning ..................................... 335
9.7.6. Amount and quality of input data .................. 335
References .................................................... 336
Further reading ............................................... 338
Chapter 10 SOLIDS PRODUCTION .................................. 341
10.1.Operational aspects of solids production ................. 341
10.1.1.Consequences of solids production ................. 341
10.1.2.Well completion and solids control ................ 342
10.2.Sand ..................................................... 343
10.2.1.Necessary and sufficient conditions for sand
production ........................................ 344
10.2.2.Forces on a sand grain ............................ 344
10.2.3.Critical drawdown for cylindrical cavities ........ 346
Shear failure ..................................... 347
Tensile failure ................................... 351
10.2.4.Stability and collapse of sand arches ............. 354
The impact of water ...................................... 357
10.2.5.Rate of produced sand ............................. 357
10.2.6.Sand transport .................................... 362
10.2.7.Sand prediction ................................... 364
10.3.Chalk .................................................... 365
References .................................................... 366
Further reading ............................................... 367
Chapter 11 MECHANICS OF HYDRAULIC FRACTURING .................. 369
11.1.Conditions for tensile failure ........................... 370
11.2.Fracture initiation and formation breakdown .............. 372
11.3.Fracture orientation, growth and confinement ............. 376
11.4.Fracture size and shape .................................. 380
11.5.Fracture closure ......................................... 382
11.5.1.Estimation of 03 from shut-in/decline tests ....... 383
11.5.2.Estimation of стз from flowback tests ............. 385
11.6.Thermal effects on hydraulic fracturing .................. 387
References .................................................... 389
Further reading ............................................... 390
Chapter l2 RESERVOIR GEOMECHANICS ............................. 391
12.1.Compaction and subsidence ................................ 391
12.2.Modelling of reservoir compaction ........................ 392
12.2.1.Uniaxial reservoir compaction ..................... 392
12.2.2.The depleting sphere .............................. 394
12.2.3.Reservoir stress path ............................. 395
The ellipsoidal reservoir ......................... 396
Elastic contrast .................................. 398
Non-ellipsoidal reservoirs ........................ 399
12.2.4.Beyond simple elastic theory ...................... 399
12.2.5.Time delayed reservoir compaction ................. 401
12.3.From compaction to subsidence ............................ 402
12.3.1.Geertsma's nucleus of strain model ................ 402
The effect of the free surface .................... 403
The size of the subsidence bowl ................... 404
Subsidence above a disk shaped reservoir .......... 405
Some example results .............................. 406
12.3.2.Stress alteration in the overburden ............... 409
12.4.Geomechanical effects on reservoir performance ........... 414
12.4.1.Compaction drive .................................. 414
12.4.2.Stress effects on porosity ........................ 416
Porosity change during depletion .................. 416
Overburden correction of laboratory measured
porosity .......................................... 417
12.4.3.Stress effects on permeability .................... 418
Flood directionality in the field ................. 419
Permeability changes under isotropic stress
conditions ........................................ 419
Permeability changes under anisotropic stress
conditions ........................................ 422
12.4.4.Geomechanics in reservoir simulation .............. 424
Weakly coupled technique .......................... 424
Fully coupled simulation .......................... 424
12.4.5.Seismic reservoir monitoring ...................... 425
Fluid substitution ................................ 425
Changes in temperature ............................ 426
Changes in pore pressure and reservoir stresses ... 426
12.5.Well problems and reservoir geomechanics ................. 427
12.5.1.Casing damage ..................................... 428
12.5.2.Reservoir geomechanics as a tool to optimize
drilling and production strategies ................ 430
References .................................................... 430
Further reading ............................................... 433
Appendix A. ROCK PROPERTIES ................................... 435
Appendix В. SI METRIC CONVERSION FACTORS ...................... 443
Appendix C. MATHEMATICAL BACKGROUND ........................... 445
С.1. Introduction ...................................... 445
C.2. Matrices .......................................... 445
C.2.1. The transpose of a matrix .................. 445
C.2.2. Symmetric matrix ........................... 445
C.2.3. Diagonal matrix ............................ 446
C.2.4. Matrix addition ............................ 446
C.2.5. Multiplication by a scalar ................. 446
C.2.6. Matrix multiplication ...................... 446
C.2.7. The identity matrix ........................ 447
C.2.8. The inverse matrix ......................... 447
C.2.9. The trace of a matrix ...................... 447
C.2.10.Determinants ............................... 448
C.2.11.Systems of linear equations ................ 448
Homogeneous systems ........................ 449
C.2.12.Eigenvalues and eigenvectors ............... 449
C.2.13.Similarity transforms and orthogonal
transforms ................................. 450
C.3. Vectors and coordinate transforms ................. 451
C.4. Tensors and coordinate transforms ................. 452
C.5. Eigenvalues, eigenvectors and diagonalization ..... 452
C.6. Rotation of the coordinate system: The Euler
angles ............................................ 453
C.7. Examples .......................................... 454
C.7.1. Rotation of the stress tensor .............. 455
C.7.2. Inversion of an axis ....................... 455
C.8. Matrix invariants ................................. 455
C.9. Some trigonometric formulas ....................... 457
C.10.The Voigt notation spelled out .................... 457
С.10.1.The Voigt mapping for the stress tensor .... 458
С.10.2.The Voigt mapping for the strain tensor .... 458
С.10.3.The Voigt mapping for the stiffness
tensor ..................................... 458
C.10.4.The Voigt mapping for the compliance
tensor ..................................... 459
С.11.The Einstein summing convention and other
notation conventions .............................. 459
C.11.1.The Einstein summing convention ............ 459
С.11.2.Kronecker's delta .......................... 460
C.11.3.Comma notation for partial derivatives ..... 460
C.11.4.Operator notation for partial derivatives .. 460
References ............................................. 461
Appendix D. SOME RELEVANT FORMULAS ......................... 463
D.1. Elasticity ........................................ 463
D.1.1. Stress invariants .......................... 463
D.1.2. Strain in spherical coordinates ............ 464
D.1.3. Isotropic linear elastic stiffness
tensor ..................................... 464
D.1.4. Isotropic linear poro-thermo-elastic
stress strain law .......................... 464
D.1.5. The force balance equation ................. 465
D.2. Elastic wave propagation in rocks ................. 466
D.2.1. Correction for non-laminar flow ............ 466
D.2.2. Reflection, transmission and conversion
coefficients at non-normal in cidence ...... 466
D.3. Rock models ....................................... 467
D.3.1. Stresses at a crack tip .................... 467
D.3.2. Self-consistent model for composite
media ...................................... 468
D.4. Solids production ................................. 469
D.4.1. Critical drawdown for turbulent flow ....... 469
D.5. Subsidence ........................................ 470
D.6. Vector operators in cylindrical coordinates ....... 471
References ............................................. 473
Appendix E. LIST OF SYMBOLS ................................... 475
Index ......................................................... 483
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