Abstract
Abstract. In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks–Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The governing fractional equation is then specialized to the case of only vertical soil water flow and of only horizontal soil water flow in fractional time–space. It is shown that the developed governing equations, in their fractional time but integer space forms, show behavior consistent with the previous experimental observations concerning the diffusive behavior of soil water flow.
Highlights
Various laboratory (Silliman and Simpson, 1987; Levy and Berkowitz, 2003) and field studies (Peaudecerf and Sauty, 1978; Sudicky et al, 1983; Sidle et al, 1998) of transport in subsurface porous media have shown significant deviations from Fickian behavior
Kavvas et al.: Governing equations of transient soil water flow and soil water flux while, they have shown that fractional advection–dispersion equation (fADE), with a fractional time derivative, can model well the long particle waiting times in transport in both surface and subsurface environments
The governing equations that were developed in this study are for the fractional time dimension and for multidimensional fractional space that represents the fractal spatial structure of a soil field
Summary
Various laboratory (Silliman and Simpson, 1987; Levy and Berkowitz, 2003) and field studies (Peaudecerf and Sauty, 1978; Sudicky et al, 1983; Sidle et al, 1998) of transport in subsurface porous media have shown significant deviations from Fickian behavior. (41) and (43) into Eq (35) results in an explicit form of the governing equation of transient soil water flow in anisotropic multi-dimensional fractional soil space and fractional time in terms of the volumetric water content θ as
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