Monotone Multi-dimensional Upstream Weighting on General Grids

The governing equations for multi-phase flow in porous media often have a mixed elliptic and (nearly) hyperbolic character. The total flux for each phase consists of two parts: a geometry and rock dependent term that resembles a single-phase flux and a mobility term representing fluid properties and rock-fluid interactions. The geometric term is commonly discretized by two or multi point flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, e.g. gas injection, SPU yields grid orientation effects. For these physical processes, the governing equations are unstable on the scale at which they are discretized, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results and to improve the overall performance prediction of enhanced oil recovery processes. In this work, we present a framework for multi-dimensional upstream weighting for multi-phase flow with gravity on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multi-dimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for treatment of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multi-dimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multi-dimensional upstream weighting is used.