An efficient polynomial chaos-based proxy model for history matching and uncertainty quantification of complex geological structures
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A novel polynomial chaos proxy-based history matching and uncertainty quantification method is presented that can be employed for complex geological structures in inverse problems. For complex geological structures, when there are many unknown geological parameters with highly nonlinear correlations, typically more than 106 full reservoir simulation runs might be required to accurately probe the posterior probability space given the production history of reservoir. This is not practical for high-resolution geological models. One solution is to use a "proxy model" that replicates the simulation model for selected input parameters. The main advantage of the polynomial chaos proxy compared to other proxy models and response surfaces is that it is generally applicable and converges systematically as the order of the expansion increases. The Cameron and Martin theorem 2.24 states that the convergence rate of the standard polynomial chaos expansions is exponential for Gaussian random variables. To improve the convergence rate for non-Gaussian random variables, the generalized polynomial chaos is implemented that uses an Askey-scheme to choose the optimal basis for polynomial chaos expansions . Additionally, for the non-Gaussian distributions that can be effectively approximated by a mixture of Gaussian distributions, we use the mixture-modeling based clustering approach where under each cluster the polynomial chaos proxy converges exponentially fast and the overall posterior distribution can be estimated more efficiently using different polynomial chaos proxies. The main disadvantage of the polynomial chaos proxy is that for high-dimensional problems, the number of the polynomial chaos terms increases drastically as the order of the polynomial chaos expansions increases. Although different non-intrusive methods have been developed in the literature to address this issue, still a large number of simulation runs is required to compute high-order terms of the polynomial chaos expansions. This work resolves this issue by proposing the reduced-terms polynomial chaos expansion which preserves only the relevant terms in the polynomial chaos representation. We demonstrated that the sparsity pattern in the polynomial chaos expansion, when used with the Karhunen-Loéve decomposition method or kernel PCA, can be systematically captured. A probabilistic framework based on the polynomial chaos proxy is also suggested in the context of the Bayesian model selection to study the plausibility of different geological interpretations of the sedimentary environments. The proposed surrogate-accelerated Bayesian inverse analysis can be coherently used in practical reservoir optimization workflows and uncertainty assessments.