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

Abstract

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.

Discrete Regularization for Calibration of Geologic Facies Against Dynamic Flow Data

“…Sparse reconstruction algorithms have been developed to search a relatively large number of the ‐SVD expansion functions to identify a subspace (i.e., small subset of expansion functions) that best represents the solution (Khaninezhad et al, ). Sparse reconstruction algorithms use sparsity‐promoting regularization forms that have selection property to identify the best combination of expansion functions based on observed response data (Golmohammadi et al, ; Golmohammadi & Jafarpour, ; Khaninezhad & Jafarpour, ). In our formulation, to preserve geologic connectivity, we use both SVD and ‐SVD methods (Appendix provides a brief overview of the ‐SVD algorithm).…”

“…However, postprocessing must be performed while ensuring that the quality of the data match is not compromised. Parameterization methods that have been used for model calibration include the principal component analysis (Chen et al, ; Gavalas, ) and truncated singular value decomposition (Tavakoli & Reynolds, ), discrete cosine transform (Jafarpour et al, ; Jafarpour & McLaughlin, ), discrete wavelet transform (Golmohammadi et al, ; Jafarpour, ; Sahni & Horne, ), grid‐connectivity transform (Bhark et al, ), and the ‐SVD sparse geologic dictionaries (Khaninezhad et al, ). These parameterization methods have all been used for approximating discrete facies maps with continuous models.…”

Simultaneous geologic scenario identification and flow model calibration with group-sparsity formulations

“…Recently, sparsity-based regularization techniques, inspired by the compressed sensing paradigm [11,20], that induce solution sparsity by minimizing its or -norm have been introduced. In some cases, regularization and parameterization can be combined to effectively exploit certain characteristics of the solution, e.g., transform-domain sparsity [27,38,39,[47][48][49]55].…”

“…An important implication of adopting a geologic scenario prior to model calibration is that flow and monitoring data are not used to constrain it, resulting in an opportunity loss to potentially correct geologic scenarios that are not supported by data. Therefore, an interesting problem is to incorporate the flow data into prior geologic scenario selection [27,49]. When a set of geologic scenarios are proposed as prior knowledge, one could implement a model calibration using each scenario and generate a set of feasible solutions depending on the prior geologic scenario.…”

“…To formulate a model calibration framework to identify consistent prior geologic scenarios, we take advantage of the group-sparsity regularization using mixed norm [4,27,63,66]. The mixed -norm penalty provides an effective regularization form to recover blocksparse signals [22,35,74].…”