Multiscale-stabilized finite element methods for miscible and immiscible flow in porous media

In this paper we study the numerical solution of miscible and immiscible flow in porous media, acknowledging that these phenomena entail a multiplicity of scales. The governing equations are conservation laws, which take the form of a linear advection-diffusion equation and the Buckley-Leverett equation, respectively. We are interested in the case of small diffusion, so that the equations are almost hyperbolic. Here we present a stabilized finite element method, which arises from considering a multiscale decomposition of the variable of interest into resolved and unresolved scales. This approach incorporates the effect of the fine (subgrid) scale onto the coarse (grid) scale. The numerical simulations clearly show the potential of the method for solving multiphase compositional flow in porous media. The results for the Buckley-Leverett problem are particularly remarkable.