Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation

Solute dispersivity defined from the classical advective-dispersive equation (ADE) was found to increase as the length of a soil column or the soil depth increased, The heterogeneity of soil is a physical reason for this scale dependence. Such transport can be described assuming that the random movement of solute particles belongs to the family of so-called Levy motions. Recently a differential solute transport equation was derived for Levy motions using fractional derivatives to describe advective dispersion. Our objective was to test applicability of the fractional ADE, or FADE, to solute transport in soils and to compare results of FADE and ADE applications. The one-dimensional FADE with symmetrical dispersion included two parameters: the fractional dispersion coefficient and the order of fractional differentiation alpha, 0 < alpha less than or equal to 2. The FADE reduces to the ADE when the parameter alpha = 2. Analytical solutions of the FADE and the ADE were fitted to the data from experiments on Cl- transport in sand, in structured clay soil, and in columns made of soil aggregates, The FADE simulated scale effects and tails on the breakthrough curves (BTCs) better than, or as well as, the ADE. The fractional dispersion coefficient did not depend on the distance. In the clay soil column, the parameter alpha did not change significantly when the flow rate changed provided the degree of saturation changed only slightly. With the FADE, the scale effects are reflected by the order of the fractional derivative, and the fractional dispersion coefficient needs to be found at only one scale.