Semi-continuum modeling of unsaturated porous media flow to explain Bauters' paradox

In the gravity-driven free infiltration of a wetting liquid into a homogeneous unsaturated porous medium, the flow pattern is known to depend significantly on the initial saturation. Point source infiltration of a liquid into an initially dry porous medium produces a single finger with an oversaturated tip and an undersaturated tail. In an initially wet medium, a diffusion-like plume is produced with a monotonic saturation profile. We present a semi-continuum model, based on a proper scaling of the retention curve, which is discrete in space and continuous in time. We show that the semi-continuum model is able to describe this transition and to capture the experimentally observed dependence of the saturation overshoot and the finger velocity on the initial saturation.