分野別セミナー

Classically, one phase infiltration of water into soil is governed by a strongly nonlinear convection-diffusion equation called Richards' equation that governs the behaviour of the soil volumetric moisture content. Soil properties can be expressed through specifying nonlinear diffusivity and conductivity functions. In this seminar I will focus on a particular model soil, introduced about 30 years go, that produces an integrable form of Richards' equation. This soil model matches the properties of many real soils well, and is of immediate use to hydrological scientists.


The utility of this integrable form of Richards' equation depends on the system boundary conditions, which must remain tractable after transformation. Constant concentration boundary conditions are an approximation to shallow flood irrigation of agricultural land, and analytical solutions have only recently been obtained for such cases. The solutions take the form of a small-time series expansion. Each order of the expansion requires specification of two constants, which can be shown to be related to all lower-order constants by a complicated iterative relation involving sums over partitions of the natural numbers. Efficient iterative algorithms have been developed for the symbolic or numerical calculation of successive constants.


The free parameters of this family of integrable soils admit a `nonlinear limit', resulting in a diffusivity that approaches a delta-function and extremely steep soil-moisture profiles. Soil moisture profiles of this type occur in the simple but very popular Green--Ampt infiltration model. Considering the delta-function diffusivity limit analytically using the integrable model has identified non-intuitive behaviour that was not understood using simpler analyses. In particular the Green--Ampt model for one-dimensional flow governed by Richards' equation has been identified as a fundamentally unphysical member of a larger class of delta-function diffusivity soils.