Bayesian History Matching and Uncertainty Quantification under Sparse Priors: A Randomized Maximum Likelihood Approach

Abstract

Abstract Despite their apparent high dimensionality, spatially distributed reservoir properties can often be compactly (sparsely) described in a properly designed basis. Hence estimation of high-dimensional reservoir flow properties from dynamic performance data, a.k.a. history matching, can be formulated and solved as a sparse reconstruction inverse problem. Recent advances in statistical signal processing, formalized under the compressed sensing paradigm, provide important guidelines on formulating and solving sparse inverse problems, primarily for linear models. Sparse inverse problems have been mainly formulated as a deterministic inverse problem. Given the uncertainty in describing the reservoir properties, even after integration of the dynamic data, it is important to develop a practical sparse Bayesian inversion framework to enable uncertainty quantification. We use sparse geologic dictionaries to compactly represent uncertain reservoir properties and develop a sparse Bayesian approach for effective data integration and uncertainty quantification. In formulating the Bayesian inversion, commonly used Gaussian distributions are not appropriate for representing sparse prior models. Following the results presented by the compressed sensing paradigm, Laplace (or double exponential) probability distributions are appropriate for representing sparse parameters. However, combining Laplace priors with frequently used Gaussian likelihood functions leads to neither Laplace nor Gaussian posterior distributions, thereby complicating the analytical characterization of the posterior distribution. Here, we first express the form of the Maximum A-Posteriori (MAP) estimate for Laplace priors, and then, as a practical approximation, use the Monte-Carlo-based Randomize Maximum Likelihood (RML) method to sample from the derived posterior distribution for uncertainty quantification. The Sparse RML (SpRML) realizations are approximate samples from the posterior distribution and provide a mechanism for assessing the uncertainty in the calibrated reservoir model with a relatively modest computational complexity. We demonstrate the suitability and effectiveness of the proposed SpRML formulation with a synthetic dataset and the model adapted from the PUNQ-S3 benchmark reservoir.