A Model for Three-Phase Flow in Porous Media with Rate-Dependent Capillary Pressure

Abstract

As a contaminant, such as oil, travels through a porous media, such as soil, there is contact between the contaminant, groundwater, and the intermediate gas, air. At this interface there is a pressure difference, capillary pressure, which impacts the flow of the contaminant through the porous media. We derive a model for three-phase flow in porous media with the inclusion of capillary pressure, as given by thermodynamically constrained averaging theory (TCAT). Starting with conservation of mass, an incompressibility condition, and Darcy’s law, we include constitutive equations that extend a strictly hyperbolic system analyzed by Juanes and Patzek. In the absence of gravity and capillarity, they show that solutions include rarefaction waves and shocks which satisfy the Liu entropy criterion. By incorporating capillary pressure, we show that the model gains dissipation and dispersion terms, the latter of which is rate-dependent. This extends the framework developed by Hayes and LeFloch in which there are solutions involving shocks which do not satisfy the Liu entropy criterion.