The range of β in the distributed-order infiltration into swelling soils and ponding time texted in the field experiments

Theoretical research and experimental analysis of the process that how water into the soil surface, that is infiltration, has profound implications for water-efficient irrigation both in hydrology science and soil physics due to the global water shortage and the rising demand for food. However, the majorities of previous work on infiltration equations are generally based on the assumption that soils are rigid objects and then overlook the realistic properties of swelling-shrinking properties and anomalous diffusion of natural soils, resulting in inaccurate prediction of infiltration. After the new general equation of flow in swelling soils based on the fractional calculus in a material coordinate was presented in 2010, heated research are conducted in the following years among which the parameter β is the hottest topic that has no doubt responses the physical properties of the soil and need further discussion. In such cases, we measured, for the first time, the range of parameter β with data collected from field experiments and the results show that the range of β is 0 to 2. In addition, we simulate the infiltration and analyze the spatial and temporal variability of parameters. Afterwards, we presented a new formula for the time to ponding with parameters from fractional Fokker-Planck equation (fFPE) and simulated ponding time under different rainfall intensity in Loess Plateau.