Abstract
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
Highlights
To summarise, at short times, diffusivity must dominate flow in soils since gravitydriven conduction of moisture is weak in dry systems
The early-time diffusion equations obtained for moisture propagation in foams and soils may be either linear or nonlinear equations based on whether we have constant or variable diffusivity functions describing the
We have given the analytical solution for the nodedominated foam drainage, and the numerical solution for drainage in channel-dominated foams and imbibition in soils, the latter based on the diffusivity function of [9]
Summary
At short times, diffusivity (due to capillary action) must dominate flow in soils since gravitydriven conduction of moisture is weak in dry systems. Various self-similar solutions have been obtained for this nonlinear problem for soils [8,14,21] and analogous systems [6] All these various solutions recognise that the initial stage of moisture evolution into porous media describes entry from a source for very short times. The SWRC, RHC and RD’ all reduce to power laws Even with these simple material properties, the governing equation for evolution of moisture content remains nonlinear, and solutions of it (even self-similar ones) can usually only be obtained numerically. Unlike other situations that we consider here, for foam drainage in the so-called node-dominated (ND) case [11,12], relative diffusivity is unity at all liquid contents This presents a special case for the earlytime self-similar solution which can be solved analytically.
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