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

- Mass-time and space-time fractional partial different equations (fPDEs).
- New equations for infiltration based on the solutions of fPDEs.
- The new equations in different forms for different types of soils.
- Either swelling or non-swelling soils with or without mobile-immobile zones.
- The orders of mass-time fractional derivative are derived as examples.

SummaryThis 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+Stβ2+β1S1tβ2+S2tβ11/(2λ-1), where A is the final infiltration rate, S is the sorptivity which differs between swelling and non-swelling soils, β2 and β1 are orders of temporal fractional derivatives for mobile and immobile zones, respectively, λ is the order of spatial fractional derivatives, and S2 and S1 are parameters incorporating relative porosities and β2 and β1, respectively. The equation of cumulative infiltration without an immobile zone is I(t)=At+Stβ/(2λ-1), where β 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 μ=2(β2+β1)/λ, and μ=2β/λ for soils without an immobile zone. The transport exponent is the criteria for defining flow patterns: for μ < 1, the flow process is sub-diffusion as compared to μ = 1 for classic diffusion and μ > 1 for super-diffusion.