Coffee percolation is one of many examples of two-phase flow in an unsaturated porous medium, with the original one perhaps being groundwater flow. This 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. water, in the case of coffee. Here, we apply asymptotic and numerical methods to a one-dimensional problem, akin to that in coffee percolation, when this turns out not to be 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 corner (or derivative) layer for the saturation function at the infiltration boundary, i.e. there is a boundary layer in the spatial derivative of the function, but not in the function 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. Furthermore, these results ought to prove instructive, were a scaled physics-informed neural networks (PINNs) approach to be applied to the coffee percolation problem in the future.