Mary F. Wheeler

Saumik Dana, Mary F Wheeler

We perform a convergence analysis of a two-grid staggered solution algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The algorithm first solves the flow subproblem on a fine grid using a mixed finite element method (by freezing a certain measure of the mean stress) followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method. Restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The analysis is based on studying the equations satisfied by the difference of iterates to show that the two-grid scheme is a contraction map under certain conditions. Those conditions are used to construct the restriction and prolongation operators as well as arrive at coarse scale elastic properties in terms of the fine scale data. We show that the adjustable parameter in the measure of the mean stress is linked to the Voigt and Reuss bounds frequently encountered in computational homogenization of multiphase composites.

Gurpreet Singh, Mary F. Wheeler

We present an adaptive space-time mesh refinement approach based a domain decomposition approach (Singh and Wheeler, 2018) that allows different time-step sizes and mesh refinements in different subdomains. Our numerical experiments indicate that non-linear solvers fail to converge, to the desired tolerance, due to large non-linear residuals in a smaller subdomain. We exploit this feature to identify subdomains where smaller time-step sizes are necessary while using large time-step sizes in the rest of the reservoir domain. The three key components of our approach are: (1) a space-time, enhanced velocity, domain decomposition approach that allows different mesh refinements and time-step sizes in different subdomains while preserving local mass conservation, (2) a residual based error estimator to identify or mark regions (or subdomains) that pose non-linear convergence issues, and (3) a fully coupled monolithic solver is also presented that solves the coarse and fine subdomain problems, both in space and time, simultaneously. This solution scheme is fully implicit and is therefore unconditionally stable. The proposed space-time domain decomposition approach, with smaller time-step sizes in a subdomain and large time-step sizes everywhere else, circumvents the non-linear convergence issue without adding computational costs. Additionally, a space-time monolithic solver renders a massively parallel, time concurrent framework for solving flow and transport problems in subsurface porous media. Since the proposed approach is similar to the widely used finite difference scheme, it can be easily integrated in any existing legacy reservoir simulator.