Two-phase flow through porous media in the fixed-contact-line regime

The complete set of equations controlling immiscible two-phase flow through porous media are derived from first principles under the sole restriction that contact lines between the two fluids and the grain surfaces are not allowed to migrate irreversibly. Because rough grain surfaces have the ability to trap contact lines over significant ranges of capillary-pressure variation, such laws are of practical interest. As distinct from previous coarse-graining work, we explicitly allow for the stretching of the fluid interface, which results in considerable nonlinearity at the macroscopic scale. The laws are obtained through an asymptotic analysis and have several new features compared to the standard laws conventionally used in two-phase flow modeling. These include the need to (i) distinguish between measurable fluxes and the volume-averaged flow; (ii) allow for flow induced by the time rate of change of the capillary pressure; and (iii) include quadratic-force terms in the generalized Darcy laws when macroscopic-pressure diffusion is slow (as defined herein).