  We study the asymptotic behaviour of sharp front solutions arising from the
nonlinear diffusion equation \theta_t = (D(\theta)\theta_x)_x, where the
diffusivity is an exponential function D({\theta}) = D_o exp(\beta\theta). This
problem arises for example in the study of unsaturated flow in porous media
where {\theta} represents the liquid saturation. For physical parameters
corresponding to actual porous media, the diffusivity at the residual
saturation is D(0) = D_o << 1 so that the diffusion problem is nearly
degenerate. Such problems are characterised by wetting fronts that sharply
delineate regions of saturated and unsaturated flow, and that propagate with a
well-defined speed. Using matched asymptotic expansions in the limit of large
{\beta}, we derive an analytical description of the solution that is uniformly
valid throughout the wetting front. This is in contrast with most other related
analyses that instead truncate the solution at some specific wetting front
location, which is then calculated as part of the solution, and beyond that
location the solution is undefined. Our asymptotic analysis demonstrates that
the solution has a four-layer structure, and by matching through the adjacent
layers we obtain an estimate of the wetting front location in terms of the
material parameters describing the porous medium. Using numerical simulations
of the original nonlinear diffusion equation, we demonstrate that the first few
terms in our series solution provide approximations of physical quantities such
as wetting front location and speed of propagation that are more accurate (over
a wide range of admissible {\beta} values) than other asymptotic approximations
reported in the literature.
