Inverse Theory for Petroleum Reservoir Characterization and History Matching Book

Preface xi

1 Introduction 1

1.1 The forward problem 1

1.2 The inverse problem 3

2 Examples of inverse problems 6

2.1 Density of the Earth 6

2.2 Acoustic tomography 7

2.3 Steady-state 1D flow in porous media 11

2.4 History matching in reservoir simulation 18

2.5 Summary 22

3 Estimation for linear inverse problems 24

3.1 Characterization of discrete linear inverse problems 25

3.2 Solutions of discrete linear inverse problems 33

3.3 Singular value decomposition 49

3.4 Backus and Gilbert method 55

4 Probability and estimation 67

4.1 Random variables 69

4.2 Expected values 73

4.3 Bayes' rule 78

5 Descriptive geostatistics 86

5.1 Geologic constraints 86

5.2 Univariate distribution 86

5.3 Multi-variate distribution 91

5.4 Gaussian random variables 97

5.5 Random processes in function spaces 110

6 Data 112

6.1 Production data 112

6.2 Logs and core data 119

6.3 Seismic data 121

7 The maximum a posteriori estimate 127

7.1 Conditional probability for linear problems 127

7.2 Model resolution 131

7.3 Doubly stochastic Gaussian random field 137

7.4 Matrix inversion identities 141

8 Optimization for nonlinear problems using sensitivities 143

8.1 Shape of the objective function 143

8.2 Minimization problems 146

8.3 Newton-like methods 149

8.4 Levenberg-Marquardt algorithm 157

8.5 Convergence criteria 163

8.6 Scaling 167

8.7 Line search methods 172

8.8 BFGS and LBFGS 180

8.9 Computational examples 192

9 Sensitivity coefficients 200

9.1 The Frechet derivative 200

9.2 Discrete parameters 206

9.3 One-dimensional steady-state flow 210

9.4 Adjoint methods applied to transient single-phase flow 217

9.5 Adjointequations 223

9.6 Sensitivity calculation example 228

9.7 Adjoint method for multi-phase flow 232

9.8 Reparameterization 249

9.9 Examples 254

9.10 Evaluation of uncertainty with a posteriori covariance matrix 261

10 Quantifying uncertainty 269

10.1 Introduction to Monte Carlo methods 270

10.2 Sampling based on experimental design 274

10.3 Gaussian simulation 286

10.4 General sampling algorithms 301

10.5 Simulation methods based on minimization 319

10.6 Conceptual model uncertainty 334

10.7 Other approximate methods 337

10.8 Comparison of uncertainty quantification methods 340

11 Recursive methods 347

11.1 Basic concepts of data assimilation 347

11.2 Theoretical framework 348

11.3 Kalman filter and extended Kalman filter 350

11.4 The ensemble Kalman filter 353

11.5 Application of EnKF to strongly nonlinear problems 355

11.6 1D example with nonlinear dynamics and observation operator 358

11.7 Example - geologic facies 359

References 367

Index 378