On the continuum‐scale modeling of gravity‐driven fingers in unsaturated porous media: The inadequacy of the Richards Equation with standard monotonic constitutive relations and hysteretic equations of state

We consider the ability of the continuum-scale Richards equation (RE) with standard monotonic constitutive relations and hysteretic equations of state to model gravity-driven fingers in unsaturated porous media and, in particular, the underlying nonmonotonicity observed along the vertical extent of fingers. As published in a number of recent papers, finger-like solutions with nonmonotonic profiles can be simulated numerically using a downwind averaging method. However, we find these fingers to be artifacts, generated by the combined effects of a truncation error induced oscillation or oversaturation at the wetting front, and capillary hysteresis. As the oscillation is removed, either through grid refinement or the use of inherently monotone schemes, the numerically generated fingers disappear, and the solution becomes monotonic. Thus the RE along with standard monotonic hydraulic properties does not contain the critical physics required to model gravity-driven fingers and must be considered inadequate for unsaturated flow in initially dry, highly nonlinear, and hysteretic media where these fingers occur.