分野別セミナー

Flow in an unsaturated porous medium is typically modelled using the
Richards equation, which is unquestioningly believed to be an accurate
enough approximation when the viscosity of the fluid being displaced,
e.g. air, is much smaller than that of the infiltrating fluid, e.g. oil
or water. Here, we apply asymptotic and numerical methods to a
one-dimensional problem when this is not the case. With the viscosity
ratio as a small parameter, we find that the Richards equation gives a
leading-order solution that is not uniformly valid over the whole domain
of interest. Instead, whilst the Richards equation holds for the bulk
flow, the problem has a derivative (or one-sided corner) layer for the
saturation function at the infiltration boundary, i.e. there is a
boundary layer in the spatial derivative of the saturation, but not in
saturation itself. Although seemingly insignificant, this has a dramatic
effect on the time taken to fill the porous medium: instead of filling
exponentially quickly, it fills algebraically slowly. As a consequence,
using the Richards equation will dramatically underestimate the time
taken to fill a porous medium. Numerical computations are provided to
underscore these asymptotic predictions.