Mohammadreza M. Khaninezhad

Prior model identification during subsurface flow data integration with adaptive sparse representation techniques

Construction of predictive reservoir models invariably involves interpretation and interpolation between limited available data and adoption of imperfect modeling assumptions that introduce significant subjectivity and uncertainty into the modeling process. In particular, uncertainty in the geologic continuity model can significantly degrade the quality of fluid displacement patterns and predictive modeling outcomes. Here, we address a standing challenge in flow model calibration under uncertainty in geologic continuity by developing an adaptive sparse representation formulation for prior model identification (PMI) during model calibration. We develop a flow-data-driven sparsity-promoting inversion to discriminate against distinct prior geologic continuity models (e.g., variograms). Realizations of reservoir properties from each geologic continuity model are used to generate sparse geologic dictionaries that compactly represent models from each respective prior. For inversion initially the same number of elements from each prior dictionary is used to construct a diverse geologic dictionary that reflects a wide range of variability and uncertainty in the prior continuity. The inversion is formulated as a sparse reconstruction problem that inverts the flow data to identify and linearly combine the relevant elements from the large and diverse set of geologic dictionary elements to reconstruct the solution. We develop an adaptive sparse reconstruction algorithm in which, at every iteration, the contribution of each dictionary to the solution is monitored to replace irrelevant (insignificant) elements with more geologically relevant (significant) elements to improve the solution quality. Several numerical examples are used to illustrate the effectiveness of the proposed approach for identification of geologic continuity in practical model calibration problems where the uncertainty in the prior geologic continuity model can lead to biased inversion results and prediction.

Group‐sparsity regularization for ill‐posed subsurface flow inverse problems

Sparse representations provide a flexible and parsimonious description of high-dimensional model parameters for reconstructing subsurface flow property distributions from limited data. To further constrain ill-posed inverse problems, group-sparsity regularization can take advantage of possible relations among the entries of unknown sparse parameters when: (i) groups of sparse elements are either collectively active or inactive and (ii) only a small subset of the groups is needed to approximate the parameters of interest. Since subsurface properties exhibit strong spatial connectivity patterns they may lead to sparse descriptions that satisfy the above conditions. When these conditions are established, a group-sparsity regularization can be invoked to facilitate the solution of the resulting inverse problem by promoting sparsity across the groups. The proposed regularization penalizes the number of groups that are active without promoting sparsity within each group. Two implementations are presented in this paper: one based on the multiresolution tree structure of Wavelet decomposition, without a need for explicit prior models, and another learned from explicit prior model realizations using sparse principal component analysis (SPCA). In each case, the approach first classifies the parameters of the inverse problem into groups with specific connectivity features, and then takes advantage of the grouped structure to recover the relevant patterns in the solution from the flow data. Several numerical experiments are presented to demonstrate the advantages of additional constraining power of group-sparsity in solving ill-posed subsurface model calibration problems.