Inverse theory for petroleum reservoir characterization and history matching

This book is a guide to the use of inverse theory for estimation and conditional simulation of flow and transport parameters in porous media. It describes the theory and practice of estimating properties of underground petroleum reservoirs from measurements of flow in wells, and it explains how to characterize the uncertainty in such estimates. Early chapters present the reader with the necessary background in inverse theory, probability and spatial statistics. The book demonstrates how to calculate sensitivity coefficients and the linearized relationship between models and production data. It also shows how to develop iterative methods for generating estimates and conditional realizations. The text is written for researchers and graduates in petroleum engineering and groundwater hydrology, and can be used as a textbook for advanced courses on inverse theory in petroleum engineering. It includes many worked examples to demonstrate the methodologies and a selection of exercises.

Preface	p. xi
1 Introduction	p. 1
1.1 The forward problem	p. 1
1.2 The inverse problem	p. 3
2 Examples of inverse problems	p. 6
2.1 Density of the Earth	p. 6
2.2 Acoustic tomography	p. 7
2.3 Steady-state 1D flow in porous media	p. 11
2.4 History matching in reservoir simulation	p. 18
2.5 Summary	p. 22
3 Estimation for linear inverse problems	p. 24
3.1 Characterization of discrete linear inverse problems	p. 25
3.2 Solutions of discrete linear inverse problems	p. 33
3.3 Singular value decomposition	p. 49
3.4 Backus and Gilbert method	p. 55
4 Probability and estimation	p. 67
4.1 Random variables	p. 69
4.2 Expected values	p. 73
4.3 Bayes' rule	p. 78
5 Descriptive geostatistics	p. 86
5.1 Geologic constraints	p. 86
5.2 Univariate distribution	p. 86
5.3 Multi-variate distribution	p. 91
5.4 Gaussian random variables	p. 97
5.5 Random processes in function spaces	p. 110
6 Data	p. 112
6.1 Production data	p. 112
6.2 Logs and core data	p. 119
6.3 Seismic data	p. 121
7 The maximum a posteriori estimate	p. 127
7.1 Conditional probability for linear problems	p. 127
7.2 Model resolution	p. 131
7.3 Doubly stochastic Gaussian random field	p. 137
7.4 Matrix inversion identities	p. 141
8 Optimization for nonlinear problems using sensitivities	p. 143
8.1 Shape of the objective function	p. 143
8.2 Minimization problems	p. 146
8.3 Newton-like methods	p. 149
8.4 Levenberg-Marquardt algorithm	p. 157
8.5 Convergence criteria	p. 163
8.6 Scaling	p. 167
8.7 Line search methods	p. 172
8.8 BFGS and LBFGS	p. 180
8.9 Computational examples	p. 192
9 Sensitivity coefficients	p. 200
9.1 The Frechet derivative	p. 200
9.2 Discrete parameters	p. 206
9.3 One-dimensional steady-state flow	p. 210
9.4 Adjoint methods applied to transient single-phase flow	p. 217
9.5 Adjoint equations	p. 223
9.6 Sensitivity calculation example	p. 228
9.7 Adjoint method for multi-phase flow	p. 232
9.8 Reparameterization	p. 249
9.9 Examples	p. 254
9.10 Evaluation of uncertainty with a posteriori covariance matrix	p. 261
10 Quantifying uncertainty	p. 269
10.1 Introduction to Monte Carlo methods	p. 270
10.2 Sampling based on experimental design	p. 274
10.3 Gaussian simulation	p. 286
10.4 General sampling algorithms	p. 301
10.5 Simulation methods based on minimization	p. 319
10.6 Conceptual model uncertainty	p. 334
10.7 Other approximate methods	p. 337
10.8 Comparison of uncertainty quantification methods	p. 340
11 Recursive methods	p. 347
11.1 Basic concepts of data assimilation	p. 347
11.2 Theoretical framework	p. 348
11.3 Kalman filter and extended Kalman filter	p. 350
11.4 The ensemble Kalman filter	p. 353
11.5 Application of EnKF to strongly nonlinear problems	p. 355
11.6 1D example with nonlinear dynamics and observation operator	p. 358
11.7 Example - geologic facies	p. 359
References	p. 367
Index	p. 378

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