Theory of infiltration: infiltration into swelling soils in a material coordinate

Abstract

The majority of infiltration equations in use today were developed for infiltration into rigid porous media based either on the classic diffusion equation or empirical formulation. In this paper, we present new equations of infiltration into swelling soils derived from the fractional Fokker–Planck equation (fFPE) of flow in swelling porous media formulated in a material coordinate. The cumulative infiltration we derived is I(t) = At + Stβ/2, where S is the fractional sorptivity, A the final infiltration rate, and β the order of fractional derivative in the fFPE. Using the data published in the literature on cumulative infiltration against time, β, S and A are determined. The determined value of β = 0.2385 for this reported soil is much less than 1.0 which implies that infiltration into this specific swelling soil belongs to the category of sub-diffusion. We show that the new cumulative infiltration equation not only fits the reported data exceptionally better than Philip’s two-term infiltration equation, more importantly, it provides fresh explanation and understanding of infiltration into swelling media. The new infiltration equation deviates from the majority of infiltration equations which were developed for rigid soils and ignore the realistic swelling and shrinking properties of the soils.