Inverse theory for petroleum reservoir characterization and history matching

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1 The forward problem......Page 15
1.2 The inverse problem......Page 17
2.1 Density of the Earth......Page 20
2.2 Acoustic tomography......Page 21
2.3 Steady-state 1D flow in porous media......Page 25
2.4 History matching in reservoir simulation......Page 32
2.5 Summary......Page 36
3 Estimation for linear inverse problems......Page 38
3.1 Characterization of discrete linear inverse problems......Page 39
3.1.1 The null space and range......Page 40
3.1.2 Underdetermined problems......Page 43
3.1.3 Overdetermined and mixed determined problems......Page 44
3.2.1 Gradient operators......Page 47
3.2.2 Solution of purely underdetermined problems......Page 50
3.2.3 Solution of purely overdetermined problems......Page 55
3.2.4 Regularized least-squares solution......Page 56
3.2.5 General regularization......Page 59
3.3 Singular value decomposition......Page 63
3.4 Backus and Gilbert method......Page 69
3.4.1 Least-squares estimation for accurate data......Page 72
3.4.2 Example: estimation of permeability from well test data......Page 75
4 Probability and estimation......Page 81
4.1 Random variables......Page 83
4.1.1 Example: permeability and porosity......Page 84
4.2 Expected values......Page 87
4.2.1 Variance, covariance, and correlation......Page 89
4.2.2 Random vectors......Page 90
4.3.1 Example: total depth of a well......Page 92
4.3.2 Example: location of a barrier from well test......Page 96
5.2 Univariate distribution......Page 100
5.2.2 Cumulative distribution plot......Page 101
5.2.3 Box plot......Page 102
5.2.4 Representative values......Page 104
5.3.1 Stationarity......Page 105
5.3.2 Transformation of variables......Page 107
5.3.3 Experimental covariance......Page 109
5.4 Gaussian random variables......Page 111
5.4.1 Covariance models......Page 112
5.4.2 Covariance matrix......Page 115
5.4.3 Covariance model selection......Page 116
5.4.4 Anisotropic covariance......Page 122
5.5 Random processes in function spaces......Page 124
6.1 Production data......Page 126
6.1.1 Drawdown tests......Page 127
6.1.2 Interference tests......Page 128
6.1.3 Tracer tests......Page 129
6.1.4 Water–oil ratio......Page 131
6.1.6 Sources of errors in production data......Page 132
6.2 Logs and core data......Page 133
6.3 Seismic data......Page 135
6.3.1 Seismic data acquisition......Page 137
6.3.2 Seismic data processing......Page 138
6.3.4 Seismic data inversion......Page 139
7.1 Conditional probability for linear problems......Page 141
7.1.1 Posteriori mean......Page 142
7.1.2 Maximum of the posterior PDF......Page 143
7.1.3 The posteriori covariance and variance......Page 144
7.2 Model resolution......Page 145
7.2.2 1D example – well-logs......Page 146
7.2.3 Well-testing example......Page 148
7.2.4 The resolution of permeability......Page 150
7.3 Doubly stochastic Gaussian random field......Page 151
7.3.1 Doubly stochastic model – example......Page 153
7.4 Matrix inversion identities......Page 155
8.1 Shape of the objective function......Page 157
8.1.1 Evaluation of quality of estimate......Page 159
8.2 Minimization problems......Page 160
8.2.1 Maximum likelihood estimate......Page 161
8.2.2 MAP estimate......Page 162
8.3 Newton-like methods......Page 163
8.3.1 Newton’s method for minimization......Page 164
8.3.2 Gauss–Newton method......Page 165
8.3.3 Gauss–Newton for generating the MAP estimate......Page 167
8.3.4 Restricted-step method for calculation of step size......Page 168
8.4 Levenberg–Marquardt algorithm......Page 171
8.4.1 Spectral analysis of Levenberg–Marquardt algorithm......Page 173
8.4.2 Effects of ill-conditioning......Page 175
8.5 Convergence criteria......Page 177
8.5.2 Convergence of algorithms......Page 179
8.6 Scaling......Page 181
8.7.1 Strong Wolfe conditions......Page 186
8.7.2 Quadratic and cubic line search algorithms......Page 189
8.8 BFGS and LBFGS......Page 194
8.8.1 Theoretical overview of BFGS......Page 197
8.8.2 Convergence rate of BFGS......Page 198
8.8.3 Scaling of BFGS......Page 200
8.8.4 Efficient implementation of BFGS......Page 203
8.8.5 Limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm......Page 205
8.9 Computational examples......Page 206
8.9.1 2D three-phase example......Page 207
8.9.2 3D three-phase example......Page 212
9.1 The Fréchet´ derivative......Page 214
9.1.2 Frechet´ derivative approach......Page 216
9.1.3 Adjoint approach to calculate Frechét derivative......Page 217
9.2.1 Generic finite-dimensional problem......Page 220
9.2.3 Sensitivity by the adjoint method......Page 222
9.2.4 Sensitivity by finite-difference method......Page 223
9.3 One-dimensional steady-state flow......Page 224
9.3.1 Direct method......Page 227
9.3.2 The adjoint method......Page 229
9.3.3 Finite-difference method......Page 230
9.4 Adjoint methods applied to transient single-phase flow......Page 231
9.4.1 Discrete and semidiscrete flow equations......Page 232
9.5 Adjoint equations......Page 237
9.5.1 Solution of the adjoint equations......Page 241
9.6 Sensitivity calculation example......Page 242
9.6.2 Sensitivities from the direct method......Page 243
9.6.3 Sensitivities from the adjoint method......Page 244
9.7.1 The reservoir simulator......Page 246
9.7.2 Adjoint equations and sensitivities......Page 250
9.7.3 Compact derivation of sensitivities......Page 258
9.7.4 G times a vector......Page 261
9.7.6 Comments on sensitivities......Page 262
9.8 Reparameterization......Page 263
9.8.1 Pilot- and master-point methods......Page 265
9.8.2 The master-point method......Page 267
9.9.1 Sensitivity examples......Page 268
9.9.2 History-matching example......Page 271
9.10.1 Approximate a posteriori covariance matrix......Page 275
9.10.2 Normalized variance......Page 276
9.10.3 Dimensionless sensitivity matrix......Page 277
9.10.4 Example......Page 278
10 Quantifying uncertainty......Page 283
10.1 Introduction to Monte Carlo methods......Page 284
10.1.1 Calculating expectations......Page 285
10.2 Sampling based on experimental design......Page 288
10.2.1 Screening designs......Page 290
10.2.2 Response surface modeling......Page 296
10.3 Gaussian simulation......Page 300
10.3.1 Generating (pseudo) random numbers......Page 301
10.3.2 Cholesky or square-root method......Page 302
10.3.3 Moving average......Page 304
10.3.4 Truncated Gaussian simulation......Page 309
10.4.1 Rejection method......Page 315
10.4.2 Markov chain Monte Carlo......Page 320
10.4.3 Markov random fields......Page 326
10.5 Simulation methods based on minimization......Page 333
10.5.2 RML for linear inverse problems......Page 337
10.5.3 RML for nonlinear inverse problems......Page 341
10.5.4 RML as Markov chain Monte Carlo......Page 343
10.6 Conceptual model uncertainty......Page 348
10.7.1 Pilot point......Page 351
10.7.2 Gradual deformation......Page 352
10.8 Comparison of uncertainty quantification methods......Page 354
10.8.1 Test problem......Page 355
10.8.2 Results analysis......Page 356
10.8.3 Discussion......Page 360
11.1 Basic concepts of data assimilation......Page 361
11.2 Theoretical framework......Page 362
11.3 Kalman filter and extended Kalman filter......Page 364
11.4 The ensemble Kalman filter......Page 367
11.5 Application of EnKF to strongly nonlinear problems......Page 369
11.5.1 Nonlinear dynamic system......Page 370
11.5.2 Implementation of the EnRML......Page 371
11.6 1D example with nonlinear dynamics and observation operator......Page 372
11.7 Example – geologic facies......Page 373
11.7.1 Matching facies observations at wells......Page 375
11.7.2 Matching production data......Page 377
References......Page 381
Index......Page 392