Wetting Fronts in One-Dimensional Periodically Layered Soils
G.G. Fennemore, Jack X. Xin · SIAM Journal on Applied Mathematics · 1998
We study wetting front (traveling wave) solutions to the Richards equation that describe the vertical infiltration of water through one-dimensional periodically layered unsaturated soils. We prove the existence, uniqueness, and large time asymptotic stability of the traveling wave solutions under prescribed flux boundary conditions and certain constitutive conditions. The traveling waves are connections between two steady state solutions that form near the ground surface and towards the underground water table. We found a closed form expression of the wave speed. The speed of a traveling wave is equal to the ratio of the flux difference and the difference of the spatial averages of the two steady states. We give both analytical and numerical examples showing that the wave speeds in the periodic soils can be larger or smaller than those in the homogeneous soils which have the same mean diffusivity and conductivity. In our examples, if the phases of inhomogeneities in diffusivity and conductivity functions differ by half the period, then the periodic soils speed up the waves; if the phases are the same, then the periodic soils slow down the waves. We also present numerical solutions to the Richards equation using the finite difference method in regimes where our constitutive conditions are no longer valid, and we observe similar stable fronts.