Inverse Theory for Petroleum Reservoir Characterization and History Matching

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  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2008-06-02
  • Publisher: Cambridge University Press
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Supplemental Materials

What is included with this book?


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.

Table of Contents

Prefacep. xi
Introductionp. 1
The forward problemp. 1
The inverse problemp. 3
Examples of inverse problemsp. 6
Density of the Earthp. 6
Acoustic tomographyp. 7
Steady-state 1D flow in porous mediap. 11
History matching in reservoir simulationp. 18
Summaryp. 22
Estimation for linear inverse problemsp. 24
Characterization of discrete linear inverse problemsp. 25
Solutions of discrete linear inverse problemsp. 33
Singular value decompositionp. 49
Backus and Gilbert methodp. 55
Probability and estimationp. 67
Random variablesp. 69
Expected valuesp. 73
Bayes' rulep. 78
Descriptive geostatisticsp. 86
Geologic constraintsp. 86
Univariate distributionp. 86
Multi-variate distributionp. 91
Gaussian random variablesp. 97
Random processes in function spacesp. 110
Datap. 112
Production datap. 112
Logs and core datap. 119
Seismic datap. 121
The maximum a posteriori estimatep. 127
Conditional probability for linear problemsp. 127
Model resolutionp. 131
Doubly stochastic Gaussian random fieldp. 137
Matrix inversion identitiesp. 141
Optimization for nonlinear problems using sensitivitiesp. 143
Shape of the objective functionp. 143
Minimization problemsp. 146
Newton-like methodsp. 149
Levenberg-Marquardt algorithmp. 157
Convergence criteriap. 163
Scalingp. 167
Line search methodsp. 172
BFGS and LBFGSp. 180
Computational examplesp. 192
Sensitivity coefficientsp. 200
The Frechet derivativep. 200
Discrete parametersp. 206
One-dimensional steady-state flowp. 210
Adjoint methods applied to transient single-phase flowp. 217
Adjoint equationsp. 223
Sensitivity calculation examplep. 228
Adjoint method for multi-phase flowp. 232
Reparameterizationp. 249
Examplesp. 254
Evaluation of uncertainty with a posteriori covariance matrixp. 261
Quantifying uncertaintyp. 269
Introduction to Monte Carlo methodsp. 270
Sampling based on experimental designp. 274
Gaussian simulationp. 286
General sampling algorithmsp. 301
Simulation methods based on minimizationp. 319
Conceptual model uncertaintyp. 334
Other approximate methodsp. 337
Comparison of uncertainty quantification methodsp. 340
Recursive methodsp. 347
Basic concepts of data assimilationp. 347
Theoretical frameworkp. 348
Kalman filter and extended Kalman filterp. 350
The ensemble Kalman filterp. 353
Application of EnKF to strongly nonlinear problemsp. 355
1D example with nonlinear dynamics and observation operatorp. 358
Example - geologic faciesp. 359
Referencesp. 367
Indexp. 378
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