An arbitrary-order scheme on generic meshes for miscible displacements in porous media

Abstract

We design, analyze, and implement an arbitrary-order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretization is based on several adaptations of the hybrid-high-order (HHO) method due to Di Pietro, Ern, and Lemaire [Comput. Methods Appl. Math., 14 (2014), pp. 461–472]. The equation governing the pressure is discretized using an adaptation of the HHO method for variable diffusion, while the discrete concentration equation is based on the HHO method for advection-diffusion-reaction problems combined with numerically stable flux reconstructions for the advective velocity that we have derived using the results of Cockburn, Di Pietro, and Ern [ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 635–650]. We perform some rigorous analysis of the method to demonstrate its L2 stability under the irregular data often presented by reservoir engineering problems and present several numerical tests to demonstrate the quality of the results that are produced by the proposed scheme.