Distributed-order infiltration, absorption and water exchange in mobile and immobile zones of swelling soils

Abstract

We present new equations for infiltration and absorption based on the distributed-order fractional Fokker–Planck equation of flow in swelling porous media in a material coordinate. We show that the cumulative infiltration into dual porous swelling soils is I(t)=At+St^β2/2-β1/4, where β1 and β2 are the orders of fractional derivatives for immobile and mobile zones respectively, A the final infiltration rate, and S the sorptivity. When the single porosity model is assumed by neglecting β1, the infiltration equation degenerates to the anomalous infiltration equation presented earlier, i.e., I(t)=At+St^β2/2, where β2 is the order of the fractional derivative for single porous media. Furthermore, when β2 = 1 and β1 is neglected, the infiltration equation becomes Philip's two-term infiltration equation. We have also derived the cumulative absorption or horizontal infiltration as I(t)=St^β2/2-β1/4, and established a fractional relationship for water exchange between the mobile and immobile zones, and present solutions subject to two types of conditions: one for the known initial water content and the other with the known rate of exchange of water between the immobile and mobile zones. Using the published data collected under the field and laboratory conditions for infiltration and absorption, we have found that the values of β1 and β2 change (0 < β2 < 2 and β1 < 1) with different classes of diffusion mechanisms operating in the different soils.