A phase field model of unsaturated flow

We present a phase field model of infiltration that explains the formation of gravity fingers during water infiltration in soil. The model is an extension of the traditional Richards equation, and it introduces a new term, a fourth-order derivative in space, but not a new parameter. We propose a scaling that links the magnitude of the new term to the relative strength of gravity-to-capillary forces already present in Richards' equation. We exploit the thermodynamic framework to design a flow potential that constrains the water saturation to be between 0 and 1, its physically admissible values. The model predicts a saturation overshoot at the wetting front, which is in good agreement with experimental measurements. Two-dimensional numerical simulations predict gravity fingers with the appearance and characteristics observed in visual laboratory experiments. A linear stability analysis of the model shows that there is a direct relation between saturation overshoot and the strength of the front instability. Therefore our theory supports the conjecture that saturation overshoot, a pileup of water at the wetting front, is a prerequisite for gravity fingering.