### Abstract

We study the asymptotic behavior of sharp front solutions arising from the nonlinear diffusion equation θt=(D(θ)θx)x, where the diffusivity is an exponential function D(θ)=Doexp(βθ). This problem arises, for example, in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is D(0)=Do≪1 so that the diffusion problem is nearly degenerate. Such problems are characterized 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 β, 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 β values) than other asymptotic approximations reported in the literature.

Original language | English |
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Pages (from-to) | 424-458 |

Journal | Studies in Applied Mathematics |

Volume | 136 |

Issue number | 4 |

Early online date | 30 Dec 2015 |

DOIs | |

Publication status | Published - May 2016 |

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### Cite this

*Studies in Applied Mathematics*,

*136*(4), 424-458. https://doi.org/10.1111/sapm.12111