Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media

TL;DR

This paper derives a macroscopic phase field model for immiscible fluid flows in porous media using multiscale analysis, enabling better interface tracking and understanding of dispersion effects in complex porous structures.

Contribution

It introduces a rigorous derivation of effective macroscopic equations for two-fluid flows in porous media, incorporating dispersion and interface dynamics, based on thermodynamic and variational principles.

Findings

Derivation of diffusion-dispersion relations including Taylor-Aris dispersion.

Development of a computational framework for interface tracking in porous media.

Validation of the model's applicability to multiphase flow scenarios.

Abstract

Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations [Schmuck et al., Proc. R. Soc. A 468:3705-3724, 2012.], we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large P\'eclet number. As for classical convection-diffusion problems, we obtain systematically diffusion-dispersion relations (including Taylor-Aris-dispersion). Our results also provide a convenient analytical and computational framework to macroscopically track interfaces in porous media. In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells.