Mass-time and space-time fractional partial differential equations of water movement in soils: Theoretical framework and application to infiltration

作者：Su Ninghu^*

来源：Journal of Hydrology, 2014, 519: 1792-1803.

DOI：10.1016/j.jhydrol.2014.09.021

• Water movement in soils
• Mass-time and space-time fractional partial differential equations
• Swelling and non-swelling soils
• Infiltration
• Mobile and immobile zones
• Transport exponent

This paper presents mass-time fractional partial differential equations (fPDEs) formulated in a material coordinate for swelling-shrinking soils, and space-time fPDEs formulated in Cartesian coordinates for non-swelling soils. The fPDEs are capable of incorporating mobile and immobile zones or without immobile zones. As an example of the applications, the solutions of the fPDEs are derived and used to construct equations of infiltration. The new equation of cumulative infiltration into soils with mobile and immobile zones is I(t) = At + S(t(beta 2+beta 1)/S-1(t beta 2)+S-2(beta 1))(1/(2 lambda-1)) where A is the final infiltration rate, S is the sorptivity which differs between swelling and non-swelling soils, beta(2) and beta(1) are orders of temporal fractional derivatives for mobile and immobile zones, respectively, lambda is the order of spatial fractional derivatives, and S-2 and S-1 are parameters incorporating relative porosities and beta(2) and beta(1), respectively. The equation of cumulative infiltration without an immobile zone is I(t) = At + St(beta/(2 lambda-1)), where beta is the order of temporal fractional derivatives. Published data are used to demonstrate the use of the new equations and derive the parameters. The transport exponent for soils with mobile and immobile zones is mu = 2(beta(2) + beta(1))/lambda, and mu = 2 beta/lambda for soils without an immobile zone. The transport exponent is the criteria for defining flow patterns: for mu %26lt; 1, the flow process is sub-diffusion as compared to mu = 1 for classic diffusion and it mu %26gt; 1 for super-diffusion.

• 出版日期2014-11-27
• 单位沈阳师范大学