Liquid drainage within foam is generally described by the foam drainage equation which admits travelling wave solutions. Meanwhile, Richards’ equation has been used to describe liquid flow in unsaturated soil. Travelling wave solutions for Richards equation  are also available using soil material property functions which have been developed by Van Genuchten. In order to compare and contrast these solutions, the travelling waves are expressed as dimensionless height, $$ {\hat{\xi }} $$ ξ ^ , versus moisture content, $$ \varTheta $$ Θ . For low moisture content, $$ {{\hat{\xi }}} $$ ξ ^ exhibits an abrupt change away from the dry state in Richards equation compared to a much more gradual change in foam drainage. When moisture content nears saturation, $$ {\hat{\xi }} $$ ξ ^ reaches large values (i.e. $$ {\hat{\xi }} \gg 1 $$ ξ ^ ≫ 1 ) for both Richards equation and foam drainage, implying a gradual approach of $$ \varTheta $$ Θ towards the saturated state. The $$ {\hat{\xi }} $$ ξ ^ values in Richards equation tend, however, to be larger than those in the foam drainage equation, implying an even more gradual approach towards saturation. The reasons for the difference between foam drainage and Richards equation solutions are traced back to soil material properties and in particular a soil specific parameter “m” which is determined from the soil-water retention curve.

Richards equation describes water transport in soils, but requires as input, soil material property functions specifically relative hydraulic conductivity and relative diffusivity typically obtained from the soil-water retention curve (SWRC) function (involving capillary suction head). These properties are often expressed via particular functional forms, with different soil types from sandstones to loams represented within those functional forms by a free fitting parameter. Travelling wave solutions (profile of height ξ against moisture content Θ ) of Richards equation using van Genuchten's form of the soil material property functions diverge to arbitrarily large height close to full saturation. The value of relative diffusivity itself diverges at full saturation owing to a weak singularity in the SWRC. If however soil material property data are sparse near full saturation, evidence for the nature of that divergence may be limited. Here we rescale the relative diffusivity to approach unity at full saturation, removing a singularity from the original van Genuchten SWRC function by constructing a convex hull around it. A piecewise SWRC function results with capillary suction head approaching zero smoothly at full saturation. We use this SWRC with the Brooks-Corey relative hydraulic conductivity to develop a new relative diffusivity function and proceed to solve Richards equation. We obtain logarithmic relationships between height ξ and moisture content Θ close to saturation. Predicted ξ values are smaller than heights obtained when solving using the original van Genuchten's soil material property functions. Those heights instead exhibit power law behaviour.

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The foam drainage equation and Richards equation are transport equations for foams and soils, respectively. Each reduces to a nonlinear diffusion equation in the early stage of infiltration during which time, flow is predominantly capillary driven, hence is effectively capillary imbibition. Indeed such equations arise quite generally during imbibition processes in porous media. New early-time solutions based on the van Genuchten relative diffusivity function for soils are found and compared with the same for drainage in foams. The moisture profiles which develop when delivering a known flux into these various porous materials are sought. Solutions are found using the principle of self-similarity. Singular profiles that terminate abruptly are obtained for soils, a contrast with solutions obtained for node-dominated foam drainage which are known from the literature (the governing equation being now linear is analogous to the linear equation for heat transfer). As time evolves, the moisture that develops at the top boundary when a known flux is delivered is greater in soils than in foams and is greater still in loamy soils than in sandstones. Similarities and differences between the various solutions for nonlinear and linear diffusion are highlighted. Graphic abstract