Inverse Theory for Petroleum Reservoir

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
Introduction	p. 1
The forward problem	p. 1
The inverse problem	p. 3
Examples of inverse problems	p. 6
Density of the Earth	p. 6
Acoustic tomography	p. 7
Steady-state 1D flow in porous media	p. 11
History matching in reservoir simulation	p. 18
Summary	p. 22
Estimation for linear inverse problems	p. 24
Characterization of discrete linear inverse problems	p. 25
Solutions of discrete linear inverse problems	p. 33
Singular value decomposition	p. 49
Backus and Gilbert method	p. 55
Probability and estimation	p. 67
Random variables	p. 69
Expected values	p. 73
Bayes' rule	p. 78
Descriptive geostatistics	p. 86
Geologic constraints	p. 86
Univariate distribution	p. 86
Multi-variate distribution	p. 91
Gaussian random variables	p. 97
Random processes in function spaces	p. 110
Data	p. 112
Production data	p. 112
Logs and core data	p. 119
Seismic data	p. 121
The maximum a posteriori estimate	p. 127
Conditional probability for linear problems	p. 127
Model resolution	p. 131
Doubly stochastic Gaussian random field	p. 137
Matrix inversion identities	p. 141
Optimization for nonlinear problems using sensitivities	p. 143
Shape of the objective function	p. 143
Minimization problems	p. 146
Newton-like methods	p. 149
Levenberg-Marquardt algorithm	p. 157
Convergence criteria	p. 163
Scaling	p. 167
Line search methods	p. 172
BFGS and LBFGS	p. 180
Computational examples	p. 192
Sensitivity coefficients	p. 200
The Frechet derivative	p. 200
Discrete parameters	p. 206
One-dimensional steady-state flow	p. 210
Adjoint methods applied to transient single-phase flow	p. 217
Adjoint equations	p. 223
Sensitivity calculation example	p. 228
Adjoint method for multi-phase flow	p. 232
Reparameterization	p. 249
Examples	p. 254
Evaluation of uncertainty with a posteriori covariance matrix	p. 261
Quantifying uncertainty	p. 269
Introduction to Monte Carlo methods	p. 270
Sampling based on experimental design	p. 274
Gaussian simulation	p. 286
General sampling algorithms	p. 301
Simulation methods based on minimization	p. 319
Conceptual model uncertainty	p. 334
Other approximate methods	p. 337
Comparison of uncertainty quantification methods	p. 340
Recursive methods	p. 347
Basic concepts of data assimilation	p. 347
Theoretical framework	p. 348
Kalman filter and extended Kalman filter	p. 350
The ensemble Kalman filter	p. 353
Application of EnKF to strongly nonlinear problems	p. 355
1D example with nonlinear dynamics and observation operator	p. 358
Example - geologic facies	p. 359
References	p. 367
Index	p. 378
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Inverse Theory for Petroleum Reservoir Characterization and History Matching

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Table of Contents

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