Approximate analytical solutions for solute transport in two-layer porous media

Abstract

Abstract. Mathematical models for transport in layered media are important for investigating how restricting layers affect rates of solute migration in soil profiles; they may also improve the analysis of solute displacement experiments. This study reports an (approximate) analytical solution for solute transport during steady-state flow in a two-layer medium requiring continuity of solute fluxes and resident concentrations at the interface. The solutions were derived with Laplace transformations making use of the binomial theorem. Results based on this solution were found to be in relatively good agreement with those obtained using numerical inversion of the Laplace transform. An expression for the flux-averaged concentration in the second layer was also obtained. Zero-and first-order approximations for the solute distribution in the second layer were derived for a thin first layer representing a water film or crust on top of the medium. These thin-layer approximations did not perform as well as the 'binomial' solution, except for the first-order approximation when the Peclet number, P, of the first layer, was low (P < 5). Results of this study indicate that the ordering of two layers will affect the predicted breakthrough curves at the outlet of the medium. The two-layer solution was used to illustrate the effects of dispersion in the inlet or outlet reservoirs using previously published data on apparatus-induced dispersion.

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

“…But the conclusion is clearly different here: the concentration can be correctly calculated only if the right transfer matrix is considered. We share the same conclusion as Leij and Van Genuchten [23]: transport phenomena in the upstream influent or downstream effluent reservoirs, as well as apparatus induced-dispersion affect breakthrough curves from a one-layer experimental column. These effects can be taken into account and analysed with the two layers solution.…”

“…Attention has been paid by Kreft and Zuber [21] and Parker and Van Genuchten [30] to the choice of the boundary conditions and its consequence for the resulting solution, especially when the solution is used for parameter estimation. It is reported by Leij and Van Genuchten [23] that the predicted temporal concentration curve will be too steep if dispersion in the effluent reservoir is present but ignored. The different cases corresponding on whether the upstream and downstream limits of the medium are open or closed to diffusion are investigated by Villermaux [39], Barry and Sposito [5], Parlange et al [31] and Maillet et al [26].…”

“…porosity of layer i e eq porosity of the equivalent homogeneous porous medium composed of two layers, one being semi-infinite has been proposed first by Barry and Parker [4] and by Leij and Van Genuchten [23]. The authors derived the solution using the Laplace transformation.…”

Solute and sediment transport at laboratory and field scale: Contributions of J.-Y. Parlange

“…Because the soil structure near the column exit is usually not known a priori, Parlange et al [1992] considered that the semi-infinite column or (5) defined a range of possible exit boundary conditions, with the lowest value of c r at the boundary given by the semiinfinite case, and the maximum when (5) is used. Another approach is to model a soil column as a two-layer medium (i.e., the exit apparatus is modeled as a layer with different transport properties), as investigated in various studies including Shamir and Harleman [1967], Barry and Parker [1987], Barry et al [1987a], Leij and Van Genuchten [1995], and Schwartz et al [1999].…”

On different numerical inverse Laplace methods for solute transport problems

“…Moench [18] pointed out that the Talbot [19] algorithm worked very well for Pe smaller than 100, but might become unstable when the function being inverted had a steep front. For the uniform flow field where the Pe is spatially constant, Bullivant and O'Sullivan [20], and Zhan et al [21][22] employed the Stehfest method to carry out the numerical inverse Laplace transform; Cornaton and Perrochet [23] and Leij et al [24] used the Crump [15] technique for the inverse Laplace transform; Schwartz et al [25] used the Weeks method; Leij et al [24], Leij and van Genuchten [26] and Gao et al [27] applied the de Hoog algorithm for the inverse Laplace transform.…”