Upscaling of a Cahn-Hilliard Navier-Stokes Model with Precipitation and Dissolution in a Thin Strip

TL;DR

This paper develops an upscaled multi-scale model for a three-phase flow involving precipitation and dissolution in a thin strip, deriving Darcy-scale equations and analyzing the sharp-interface limit with numerical validation.

Contribution

It introduces a novel upscaling approach for a phase-field model with solid phase evolution in a thin strip geometry, including the sharp-interface limit analysis.

Findings

The upscaled model accurately captures pore-scale phenomena.

The sharp-interface limit simplifies the model to Darcy-scale equations.

Numerical results confirm the validity of the upscaling process.

Abstract

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Peclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip is going to zero. In this way and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell-problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell-problems can be solved explicitly. Notably we find asymptotic consistency, that is the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.