The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media

We consider the one-dimensional two-phase flow including capillary effects through a heterogeneous porous medium. The heterogeneity is due to the spatial variation of the absolute permeability and the porosity. Both these quantities are assumed to be piecewise constant. At interfaces where the rock properties are discontinuous, we derive, by a regularisation technique, conditions to match the values of the saturation on both sides. There are two conditions: a flux condition and an extended pressure condition. Applying these conditions we show that trapping of the wetting phase may occur near heterogeneities. To illustrate the behaviour of the saturation we consider a time-dependent diffusion problem without convection, a stationary convection-diffusion problem, and the full time-dependent convection-diffusion problem (numerically). In particular the last two problems explicitly show the trapping behaviour.