Why developing high-resolution schemes for reservoir simulation is challenging

Equations modelling multiphase/multicomponent flow in porous media are widely studied by the hyperbolic community, e.g., as examples of problems with nonconvex and spatially-dependent flux. Various types of high-resolution methods have been developed to solve these equations, but few have managed to permeate into industry use. This is perhaps surprising, given that the numerically predicted outcome of critical engineering decisions can be severely impacted by numerical diffusion and insufficient accuracy. In this talk, I explain some of the challenges and opportunities we have encountered when developing high-resolution methods capable of simulating industry standard reservoir models: 1) Reservoir rocks are typically represented by grids with unstructured topology and polyhedral cell geometries and large aspect ratios. Cells have non-matching interfaces, degenerate geometries, and large variations in sizes. Most methods in the literature are developed for structured topologies or simplices with close to unit aspect ratios. 2) The flow equations are parabolic with a mixed elliptic-hyperbolic character. Strong coupling and large variations in time constants imply that implicit discretizations are necessary also for the hyperbolic part of the problem. Most methods in the literature are explicit.