The 5th IFAC Symposium on Fractional Differentiation Its Applications, 14-17 May 2012, Nanjing, China N-dimensional fractional water infiltration into porous media affected by the concurrent flow of air

摘要

Infiltration in nature is the process by which water enters the surface of the soil. In this paper we investigate the n-dimensional two-phase infiltration into porous media, i.e.,water infiltration affected by the concurrent flow of air. We investigate the n-dimensional fractional Fokker-Planck equation (fFPE),where (9) is the normalised liquid saturation, qt=qw+qa with qw and qa the volumetric flow rates (equivalent to velocity) of liquid and gas, respectively, and fw the fractional flow function. The fFPE of order β is solved for one-, two-and three-dimensional flow patterns, and the generic equation of cumulative infiltration derived here is of the form I(t)=At+Sntβ/2, where A is the final infiltration rate, Sn the fractional sorptivity which differs for different dimensions, and n the number of dimensions. For β=1, these equations reduce to the two-term equations of cumulative infiltration widely used in hydrology and soil physics. The one-dimensional infiltration equation is fitted to the laboratory data of McWhorter (1971) to give β=0.9557 for Poudre sand and β=0.9579 for Berea sandstone, respectively.