Upscaling and Effective Behavior for Two-Phase Porous-Medium Flow using a Diffuse Interface Model

TL;DR

This paper develops a two-scale model for two-phase flow in porous media using a diffuse interface approach, capturing pore-scale effects and surface tension to improve macro-scale predictions.

Contribution

It introduces a novel two-scale model derived via homogenization that incorporates pore-scale phase distribution and surface tension effects into Darcy-scale flow equations.

Findings

The model produces non-monotone relative permeability-saturation relations.

Pore-scale fluid distribution significantly influences effective parameters.

Incorporating pore-scale information improves macro-scale flow predictions.

Abstract

We investigate two-phase flow in porous media and derive a two-scale model, which incorporates pore-scale phase distribution and surface tension into the effective behavior at the larger Darcy scale. The free-boundary problem at the pore scale is modeled using a diffuse interface approach in the form of a coupled Allen-Cahn Navier-Stokes system with an additional momentum flux due to surface tension forces. Using periodic homogenization and formal asymptotic expansions, a two-scale model with cell problems for phase evolution and velocity contributions is derived. We investigate the computed effective parameters and their relation to the saturation for different fluid distributions, in comparison to commonly used relative permeability saturation curves. The two-scale model yields non-monotone relations for relative permeability and saturation. The strong dependence on local fluid distribution and effects captured by the cell problems highlights the importance of incorporating pore-scale information into the macro-scale equations.