Multiscale mixed methods for two-phase flows in high-contrast porous media

Abstract

The Multiscale Robin Coupled Method (MRCM) is a domain decomposition procedure that has been developed to efficiently approximate velocity and pressure fields for single-phase flows in highly heterogeneous porous media. It generalizes other well-established multiscale domain decomposition mixed methods and it adds great flexibility to the choice of interface spaces as well as in the boundary conditions for subdomain coupling. We investigate the approximation of two phase flows in porous media using the MRCM to compute velocity fields. We consider an operator splitting strategy, where a scalar conservation law for the saturation of one of the phases and the velocity field are updated sequentially. We find that the coupling of nearest neighbor subdomains through the imposition of a continuous pressure (respectively, normal fluxes) is the best strategy to approximate two-phase flows in the presence of high (resp., low) permeability channels (resp., regions). A new adaptivity strategy for setting an algorithmic parameter of the MRCM, that controls the relative importance of Dirichlet and Neumann boundary conditions in the coupling of subdomains, is proposed and tested in challenging, high-contrast permeability fields. Our numerical simulations of two-phase flows show that by switching between existing multiscale procedures we can observe unprecedented accuracy, in that we produce better solutions for problems with high-contrast permeability coefficients when compared to solutions obtained with some standard multiscale mixed methods.