On Darcy-and Brinkman-Type Models for Two-Phase Flow in Asymptotically Flat Domains

TL;DR

This paper investigates simplified Darcy and Brinkman models for two-phase flow in flat domains, demonstrating their efficiency and equivalence, and introduces a new nonlocal limit model with overshoot behavior.

Contribution

It introduces a new asymptotic limit model for Brinkman regimes and proves its equivalence to multiscale approaches in Darcy regimes, enhancing computational efficiency.

Findings

Asymptotic models are efficient for flat domains in Darcy regimes.

The Brinkman limit model is a nonlocal evolution law with pseudo-parabolic features.

Numerical examples show the model's overshoot behavior, matching observed phenomena.

Abstract

We study two-phase flow for Darcy and Brinkman regimes. To reduce the computational complexity for flow in vertical equilibrium various simplified models have been suggested. Examples are dimensional reduction by vertical integration, the multiscale model approach in [Guo et al., 2014] or the asymptotic approach in [Yortsos, 1995]. The latter approach uses a geometrical scaling. We show the efficiency of the approach in asymptotically flat domains for Darcy regimes. Moreover, we prove that it is vastly equivalent to the multiscale model approach. We apply then asymptotic analysis to the two-phase flow model in Brinkman regimes. The limit model is a single nonlocal evolution law with a pseudo-parabolic extension. Its computational efficiency is demonstrated using numerical examples. Finally, we show that the new limit model exhibits overshoot behaviour as it has been observed for dynamical capillarity laws [Hassanizadeh and Gray, 1993].