A New two-dimensional Second Order Non-oscillatory Central Scheme Applied to multiphase flows in heterogeneous porous media

TL;DR

This paper introduces a new two-dimensional second order non-oscillatory central scheme for simulating multiphase flows in heterogeneous porous media, improving boundary behavior and reducing numerical diffusion compared to existing methods.

Contribution

The paper presents a novel genuinely two-dimensional semi-discrete central scheme that corrects boundary issues found in the dimension-by-dimension approach of the KT scheme.

Findings

The new scheme better handles boundary conditions in complex geometries.

It reduces numerical diffusion in viscous finger simulations.

Numerical examples demonstrate improved accuracy in heterogeneous reservoirs.

Abstract

We compare the Kurganov-Tadmor (KT) two-dimensional second order semi-discrete central scheme in dimension by dimension formulation with a new two-dimensional approach introduced here and applied in numerical simulations for two-phase, two-dimensional flows in heterogeneous formations. This semi-discrete central scheme is based on the ideas of Rusanov's method using a more precise information about the local speeds of wave propagation computed at each Riemann Problem in two-space dimensions. We find the KT dimension by dimension has a much simpler mathematical description than the genuinely two-dimensional one with a little more numerical diffusion, particularly in the presence of viscous fingers. Unfortunately, as one can see, the KT with the dimension by dimension approach might produce incorrect boundary behavior in a typical geometry used in the study of porous media flows: the quarter of a five spot. This problem has been corrected by the authors with the new semi-discrete scheme proposed here. We conclude with numerical examples of two-dimensional, two-phase flow associated with two distinct flooding problems: a two-dimensional flow in a rectangular heterogeneous reservoir (called slab geometry) and a two-dimensional flow in a 5-spot geometry homogeneous reservoir.