MHM as preconditioner

Abstract

This work is a continuation of the PhD thesis of Omar Duran Triana [DURAN] who developed the computational tools to apply the Multiscale Hybrid-Mixed (MHM) [PAREDES] and Reduced Order Modeling (ROM) of geomechanic deformation [DURAN], to simulate two phase reservoir flow. In a first project associated with a fellowship request (the selected candidate will assume the fellowship in October 2020) a Multiscale Solver will be developed as a prototype solver simulating flow through porous media. In this proposal the candidate will assert the robustness of the Multiscale Solver by applying it todatasets available in the OPM repository. Different strategies to improve the quality of MHM approximations will be evaluated: a) an a-priori evaluation of the permeability contrast; b) bya spectral analysis of the boundary matrix; and c) by iteratively improving the approximate solution. In a PhD thesis recently concluded in São Carlos, Franciane Fracalossi Rocha demonstrates that a considerable loss in accuracy is observed when applying either the MHM or MMM method to problems where strong contrast of permeability is observed between macro domains. She demonstrated that adaptively switching between both methods can mitigate this problem. In this research will show that the same quality improvement can be obtained by adaptive lyadding flux functions between the domains. The advantage of MHM over MMM is that MMM approximations require a (costly) post-processing step in order to yield locally conservative solutions. Studying work on multiscale solvers by Efendiev the author noted a similarity between the "mixed generalized multiscale finite element method" and MHM. In his publications, Efendiev documents different procedures to compute specific macro-fluxes between domains that could be applied to improve MHM approximations. The quality of MHM approximation will also be assessed by iteratively improving the fluxes between macro domains. These fluxes will be computed based on residual computations. The fine scale fluxes can then be used to progressively improve the MHM during time stepping. This approach can also be view as an iterative method where the MHM approximation is a preconditioner for the full scale approximation. The focus of this contribution will be: apply MHM to datasets available in the OPM repository and/or suggested by the Equinor reservoir teams; improve MHM approximations based on either a-priori evaluation of permeability contrasts or local spectral decomposition as documented by Efendiev in "multiscale finiteelement methods for high-contrast problems using local spectral basis functions"; estimate the error induced by MHM macro fluxes by computing fine scale flux reconstructions between two domains and comparing them to macro fluxes; use the interaction between fine-scale fluxes and macro fluxes as a pre-conditioner for afine scale solver. (AU)