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

摘要

We design, analyse and implement an arbitrary order scheme applicable togeneric meshes for a coupled elliptic-parabolic PDE system describing miscibledisplacement in porous media. The discretisation is based on severaladaptations of the Hybrid-High-Order (HHO) method due to Di Pietro et al.[Computational Methods in Applied Mathematics, 14(4), (2014)]. The equationgoverning the pressure is discretised using an adaptation of the HHO method forvariable diffusion, while the discrete concentration equation is based on theHHO method for advection-diffusion-reaction problems combined with numericallystable flux reconstructions for the advective velocity that we have derivedusing the results of Cockburn et al. [ESAIM: Mathematical Modelling andNumerical Analysis, 50(3), (2016)]. We perform some rigorous analysis of themethod to demonstrate its L 2 stability under the irregular data oftenpresented by reservoir engineering problems and present several numerical teststo demonstrate the quality of the results that are produced by the proposedscheme.