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

Reservoir simulations for complex multiphase flow and transport problems often suffer from non-linear solver convergence issues. These manifest in the form of restrictively small time-step sizes even while using unconditionally stable fully implicit schemes. These problems are further compounded when a local mesh refinement is used to accurately represent reservoir parameters available such as permeability, porosity, etc., at multiple spatial scales. We discuss a domain decomposition approach that allows different time-step sizes and mesh refinements in different subdomains (Singh and Wheeler (2018)) of the reservoir that circumvents these issues without compromising computational efficiency and prediction accuracy. This approach extends the well-known methodology of local mesh refinement in space (Wheeler et al. (2002)) to time. 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 results indicate that using large time-step sizes for the entire reservoir domain poses serious non-linear solver convergence issues. Although using a smaller time step size for the entire domain reduces the convergence issues, it also results in substantial computational overheads. 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.