# An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media

## TL;DR

This paper presents a novel numerical scheme combining characteristic and finite volume methods for simulating miscible incompressible flows in porous media on complex polygonal meshes, improving accuracy and conservation.

## Contribution

It introduces a Darcy velocity reconstruction with local consistency, addresses implementation challenges with distorted cells, and enhances treatment near injection wells for better fluid conservation.

## Key findings

- Applicable to generic polygonal meshes, including non-conforming ones.
- Ensures local consistency in Darcy velocity reconstruction.
- Improves fluid conservation near injection wells.

## Abstract

We design a numerical approximation of a system of partial differential equations modelling the miscible displacement of a fluid by another in a porous medium. The advective part of the system is discretised using a characteristic method, and the diffusive parts by a finite volume method. The scheme is applicable on generic (possibly non-conforming) meshes as encountered in applications. The main features of our work are the reconstruction of a Darcy velocity, from the discrete pressure fluxes, that enjoys a local consistency property, an analysis of implementation issues faced when tracking, via the characteristic method, distorted cells, and a new treatment of cells near the injection well that accounts better for the conservativity of the injected fluid.