On upscaling pore-scale models for two-phase flow with evolving interfaces

Abstract

The modelling and simulation of the unsaturated flow or the flow of two immiscible fluid phases in a porous medium is challenging as this flow takes place through the pores of the medium, which form a highly complex domain. Next to the complexity of the domain, a major challenge is to account for the interface separating the fluids, or the unsaturated fluid from the inert filling part, as the location of this interface is not known a-priori. The evolution of this interface depends on the flow of both fluids and of the surface tension. Moreover, the surface tension may depend on the concentration of a surfactant dissolved in one fluid phase. In this work, such aspects are taken into account, and effective, Darcy-scale models are derived based on the known physics at the pore scale. In this sense a thin strip is used as the representation of a single pore in the porous medium. The Darcy-scale models are derived for various regimes, accounting for different pore-scale processes. Numerical examples show that the upscaled models are a good approximation of the transversal average of the solution to the pore-scale models, as the ratio of the width and the length of the pore approaches zero.