Comparing and Contrasting Travelling Wave Behaviour for Groundwater Flow and Foam Drainage

Abstract

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.

Similarity solutions for early-time constant boundary flux imbibition in foams and soils

“…Equation (18) still predicts a Φ value that tends to zero at a finite η max but the approach is not abrupt, since Eq. (18) now gives moisture content varying like the square of distance from the leading edge (again the same behaviour is observed close to zero moisture content in the longtime channel-dominated case [1]).…”

“…The third time that we identify is the time at which moisture content at the top boundary would be predicted via the purely diffusive model to be so large Θ ≈ 1, that the model has certainly broken down. Indeed, if we ever obtain Θ ≈ 1 at the top boundary then since the relative hydraulic conductivity K r approaches unity (by construction) in the Θ → 1 limit, the flux will definitely be dominated by conduction not diffusion (this is the long-time limit identified by [1]). Note that the soil material property functions we have employed K r ≈ m 2 Θ 1/2+2/m and D r ≈ m −1 (1 − m)Θ 1/2+1/m are unreliable in this limit (we used Θ 1 approximations to the material functions of [9] here).…”

“…Data obtained from the shooting method in the channeldominated case (N = 1 2 ) are presented in Fig. 3.…”

“…Infiltration in soils is an important problem that arises in multiple applications in agriculture, hydrology, soil sciences, and environmental engineering, among others [ 3 , 4 ]. In work published previously [ 1 ], we presented solutions for the long-time propagation of moisture into porous media (soils or foams) which yields travelling waves as solutions of a convection–diffusion transport equation. In this present work, we focus instead on (nonlinear) diffusion equations as given for early-time or short-time infiltration (also called imbibition) processes.…”

“…In this present work, we focus instead on (nonlinear) diffusion equations as given for early-time or short-time infiltration (also called imbibition) processes. Early-time imbibition presents a rather different problem from general infiltration problems since the dominant driving force is a diffusive capillary force, instead of a balance between gravity and capillary diffusivity as happens subsequently, up to the point where the system is nearly fully saturated [ 1 , 14 ]. Similarly for horizontal flow, the dominant force affecting flow is capillary diffusivity [ 14 ], and hence solutions obtained for early-time problems can describe this phenomenon as well.…”