A vertex scheme for two-phase flow in heterogeneous media

TL;DR

This paper introduces a finite element method for simulating two-phase flow in heterogeneous porous media, emphasizing accuracy, convergence, and conservation properties, suitable for reservoir simulation applications.

Contribution

The paper presents a novel finite element scheme with mass-lumping and flux up-winding that ensures high accuracy, optimal convergence, and local mass conservation in heterogeneous media.

Findings

Method converges optimally for manufactured solutions.

High-accuracy saturation profiles with minimal numerical diffusion.

Effectively handles heterogeneities in permeability fields.

Abstract

This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux up-winding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions. For both structured and unstructured meshes, we observe the high-accuracy wetting saturation profile that ensures minimal numerical diffusion at the front. Performing several examples of quarter-five spot problems in two and three dimensions, we show that the method can easily handle heterogeneities in the permeability field. Two distinct features that make the method appealing to reservoir simulators are: (i) maximum principle is satisfied, and (ii) mass balance is locally conserved.