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

TL;DR

This paper introduces a high-order numerical scheme for simulating miscible displacement in porous media, capable of handling complex meshes and irregular data with proven stability and demonstrated accuracy through numerical tests.

Contribution

It develops and analyzes an arbitrary order Hybrid-High-Order scheme tailored for coupled elliptic-parabolic PDEs on generic meshes, extending existing methods for improved stability and flexibility.

Findings

The scheme achieves $L^2$ stability with irregular data.

Numerical tests confirm high accuracy and robustness.

Applicable to complex reservoir engineering problems.

Abstract

We design, analyse and implement an arbitrary order scheme applicable to generic meshes for a coupled elliptic-parabolic PDE system describing miscible displacement in porous media. The discretisation is based on several adaptations of the Hybrid-High-Order (HHO) method due to Di Pietro et al. [Computational Methods in Applied Mathematics, 14(4), (2014)]. The equation governing the pressure is discretised 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 et al. [ESAIM: Mathematical Modelling and Numerical Analysis, 50(3), (2016)]. We perform some rigorous analysis of the method to demonstrate its $L^2$ 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.