A variational multiscale method for the numerical simulation of multiphase flow in porous media

We present a stabilized finite element method for the numerical solution of multiphase flow in porous media, based on a multiscale decomposition of pressures and fluid saturations into resolved (or grid) scales and unresolved (or subgrid) scales. The multiscale split is invoked in a variational setting, which leads to a rigorous definition of a grid scale problem and a subgrid scale problem. The subgrid problem is modeled using an algebraic approximation. This model requires the definition of a matrix of intrinsic time scales, which we design carefully after a stability analysis. We illustrate the performance of the method with simulations of a waterflood in a heterogeneous oil reservoir. The proposed method yields stable, highly accurate solutions on very coarse grids, which we compare with those obtained by the classical Galerkin method or the upstream finite difference method. Although this paper is restricted to multiphase flow in porous media, the formulation is general and can be applied to any nonlinear system of conservation laws, like the shallow water equations.