Table of contents for Inverse theory for petroleum reservoir characterization and history matching / Dean S. Oliver, Albert C. Reynolds, Ning Liu.


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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 ID 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 Fr6chet 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  Adjoint equations                                                     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 lD example with nonlinear dynamics and observation operator           358
11.7 Example - geologic facies                                             359



Library of Congress subject headings for this publication: Petroleum reserves Mathematical models, Inversion (Geophysics)