Modeling stable and unstable flow in unsaturated porous media for different infiltration rates

The gravity-driven flow in unsaturated porous medium is still one of the biggest unsolved problems in multiphase flow. Sometimes a stable flow with an uniform wetting front is observed, but at other times it is unstable with distinct preferential pathways even if the porous material is homogeneous. The formation of an unstable wetting front in a porous medium depends on many factors such as the type of the porous medium, the initial saturation or the applied infiltration rate. As the infiltration rate increases, the wetting front first transitions from stable to unstable for low infiltration rates, and then from unstable to stable for high infiltration rates. We propose a governing equation and its discretized form, the semi-continuum model, to describe this significant non-monotonic transition. We show that the semi-continuum model is able to capture the influx dependence together with the correct finger width and spacing. We also present that the instability of the wetting front is closely related to the saturation overshoot in 1D. Finally, we demonstrate that the flow can be still preferential even when the porous medium is completely wetted.