Optimization with the Gradual Deformation Method

Building reservoir models consistent with production data and prior geological knowledge is usually carried out through the minimization of an objective function. Such optimization problems are nonlinear and may be difficult to solve because they tend to be ill-posed and to involve many parameters. The gradual deformation technique was introduced recently to simplify these problems. Its main feature is the preservation of the spatial structure: perturbed realizations exhibit the same spatial variability as the starting ones. It is shown that optimizations based on gradual deformation converge exponentially to the global minimum, at least for linear problems. In addition, it appears that combining the gradual deformation parameterization with optimizations may remove step by step the structure preservation capability of the gradual deformation method. This bias is negligible when deformation is restricted to a few realization chains, but grows increasingly when the chain number tends to infinity. As in practice, optimization of reservoir models is limited to a small number of iterations with respect to the number of gridblocks, the spatial variability is preserved. Last, the optimization processes are implemented on the basis of the Levenberg–Marquardt method. Although the objective functions, written in terms of Gaussian white noises, are reduced to the data mismatch term, the conditional realization space can be properly sampled.

[1]  A. Tarantola Inverse problem theory : methods for data fitting and model parameter estimation , 1987 .

[2]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[3]  Dean S. Oliver,et al.  Moving averages for Gaussian simulation in two and three dimensions , 1995 .

[4]  M. Boucher,et al.  Interpretation of Interference Tests in a Well Field Using Geostatistical Techniques to Fit the Permeability Distribution in a Reservoir Model , 1984 .

[5]  L. Hu Gradual Deformation and Iterative Calibration of Gaussian-Related Stochastic Models , 2000 .

[6]  L. Hu,et al.  Gradual Deformation of Continuous Geostatistical Models for History Matching , 1998 .

[7]  Benoit Noetinger,et al.  Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations , 2001 .

[8]  Dean S. Oliver,et al.  Reparameterization Techniques for Generating Reservoir Descriptions Conditioned to Variograms and Well-Test Pressure Data , 1996 .

[9]  Benoit Noetinger,et al.  The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations , 2000 .

[10]  L. Hu,et al.  Conditioning to dynamic data: an improved zonation approach , 2001, Petroleum Geoscience.

[11]  S. P. Neuman Calibration of distributed parameter groundwater flow models viewed as a multiple‐objective decision process under uncertainty , 1973 .

[12]  N. Sun Inverse problems in groundwater modeling , 1994 .