Ninghu Su
School of Earth and Environmental Sciences, James Cook University, Australia

Infiltration is the process by which water enters the soil surface. It interfaces the surface and subsurface waters, and partitions water into surface runoff and soil water which further modulates recharge to aquifers. For its importance as an interfacing process, infiltration is one of the most intensively investigated topics in hydrology and soil physics.
Since the first infiltration model pioneered by Green and Ampt (1911), more physically-based and empirical models were proposed, which include the two-term infiltration equation by Philip (1957) based on the Richards equation (1931) that explains the physics of infiltration into the soil in addition to other empirical models by Kostiakov (1932) and Horton (1933) etc.

Research on infiltration treated soils as rigid aggregates until the late 1960s when more realistic swelling-shrinking properties were considered (Raats, 1965; Raats & Klute, 1968; Philip, 1969; Smiles, 1974). In these models, however, the underlying deterministic principles of the Richards equation remain unchanged.

Recently, new forms of infiltration and absorption equations have been presented (Su, 2012), which are based on the distributed-order fractional Fokker-Planck equation of flow in swelling porous media formulated in a material coordinate. The new equation of cumulative infiltration takes the form of , and the cumulative absorption equation is , where  and  are the orders of fractional derivatives for immobile and mobile zones respectively,  the final infiltration rate, and  the sorptivity. When the single porosity model is considered by neglecting , the infiltration equation degenerates to the anomalous infiltration equation (Su, 2010), i.e., , where  is the order of the fractional derivative for single porous media. Furthermore, when  and  is neglected, the infiltration equation becomes Philip’s infiltration equation. The new equations were verified using data collected under field and laboratory conditions. In the new infiltration and absorption equations, the fractional distributed orders account for stochastic processes during infiltration and absorption into mobile and immobile zones, and the material coordinate represents the swelling-shrinking processes. These models provide new insights into the mechanics of infiltration, absorption, and related processes such as runoff generation.