Kernel Principal Component Analysis for Efficient,Differentiable Parameterization of Multipoint Geostatistics |
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Authors: | Pallav Sarma Louis J Durlofsky Khalid Aziz |
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Institution: | (1) Chevron Energy Technology Company, San Ramon, CA 94583, USA;(2) Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA |
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Abstract: | This paper describes a novel approach for creating an efficient, general, and differentiable parameterization of large-scale
non-Gaussian, non-stationary random fields (represented by multipoint geostatistics) that is capable of reproducing complex
geological structures such as channels. Such parameterizations are appropriate for use with gradient-based algorithms applied
to, for example, history-matching or uncertainty propagation. It is known that the standard Karhunen–Loeve (K–L) expansion,
also called linear principal component analysis or PCA, can be used as a differentiable parameterization of input random fields
defining the geological model. The standard K–L model is, however, limited in two respects. It requires an eigen-decomposition
of the covariance matrix of the random field, which is prohibitively expensive for large models. In addition, it preserves
only the two-point statistics of a random field, which is insufficient for reproducing complex structures.
In this work, kernel PCA is applied to address the limitations associated with the standard K–L expansion. Although widely
used in machine learning applications, it does not appear to have found any application for geological model parameterization.
With kernel PCA, an eigen-decomposition of a small matrix called the kernel matrix is performed instead of the full covariance
matrix. The method is much more efficient than the standard K–L procedure. Through use of higher order polynomial kernels,
which implicitly define a high-dimensionality feature space, kernel PCA further enables the preservation of high-order statistics
of the random field, instead of just two-point statistics as in the K–L method. The kernel PCA eigen-decomposition proceeds
using a set of realizations created by geostatistical simulation (honoring two-point or multipoint statistics) rather than
the analytical covariance function. We demonstrate that kernel PCA is capable of generating differentiable parameterizations
that reproduce the essential features of complex geological structures represented by multipoint geostatistics. The kernel
PCA representation is then applied to history match a water flooding problem. This example demonstrates that kernel PCA can
be used with gradient-based history matching to provide models that match production history while maintaining multipoint
geostatistics consistent with the underlying training image. |
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Keywords: | Kernel Geostatistics Principal component Karhunen– Loeve History-matching Reservoir characterization |
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