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

Solute dispersivity defined from the classical advective-dispersive equatio n (ADE) was found to increase as the length of a soil column or the soil de pth increased, The heterogeneity of soil is a physical reason for this scal e dependence. Such transport can be described assuming that the random move ment of solute particles belongs to the family of so-called Levy motions. R ecently a differential solute transport equation was derived for Levy motio ns using fractional derivatives to describe advective dispersion. Our objec tive was to test applicability of the fractional ADE, or FADE, to solute tr ansport in soils and to compare results of FADE and ADE applications. The o ne-dimensional FADE with symmetrical dispersion included two parameters: th e fractional dispersion coefficient and the order of fractional differentia tion 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 struc tured 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 o n 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 t he order of the fractional derivative, and the fractional dispersion coeffi cient needs to be found at only one scale.