Two-phase flow: Travelling waves in a reacting porous medium

There are many situations where a fluid flows through a porous medium and reacts with it to alter the porosity and permeability of the medium. Important examples include flows in the Earth's mantle and crust, and formation of karst topography. An everyday example is the flow of water poured through a bed of dissolving sugar. Instabilities in these systems can create remarkable fingering patterns, arising from the localisation of flow in high permeability pathways.



There remains a question of how to model these systems in a continuum model, allowing for a smooth transition between fully reacted and non-reacted components. I develop a continuum model describing such systems with a two-phase fluid approach. In this framework, both the porous medium and fluid occupy the whole domain, with their relative proportions described by a spatially variable porosity.

In the configuration of a 1D transition from reacted to non-reacted regions of the porous solid, there exist travelling wave solutions, and we use the model to predict the spatial scales on which a transition between two such regions occurs. This work provides basic solutions to lay the footings for future numerical work on the non-linear development of the system (over long time scales) in a computationally efficient manner.