A Domain Decomposition Approach for Local Mesh Refinement in Space and Time

TL;DR

This paper introduces a novel adaptive space-time mesh refinement method using domain decomposition that enables different time-step sizes in subdomains, improving convergence and efficiency in reservoir simulations.

Contribution

It presents a new space-time domain decomposition approach with residual-based error estimation and a fully coupled solver, enhancing stability and parallelism in flow simulations.

Findings

Non-linear solvers fail to converge with large residuals in small subdomains.

The method allows smaller time-steps in critical regions while using larger steps elsewhere.

The approach is fully implicit, unconditionally stable, and easily integrable into existing reservoir simulators.

Abstract

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.