Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time

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.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

“…In Equation 2, an analytical relationship between ∆ and (∆ ) (i=1,2) that will be universally applicable throughout the modelling domain is possible when the lower limit in the above Caputo derivative in equation 2is taken as zero (that is, ∆ = ) for f( ) = (Kavvas et al 2017b).…”

“…where different powers for fractional space derivatives are utilized in different directions due to the anisotropy in the flow medium. Kavvas et al (2017b) have shown that to -order fractional increments in space in the i-th direction, i=1,2, https://doi.org/10.5194/esd-2019-37 Preprint. Discussion started: 29 July 2019 c Author(s) 2019.…”

“…Recently, Kavvas et al, (2017aKavvas et al, ( , 2017b) derived a governing equation for water flux (specific discharge), , (i = 1, 2, 3) in a saturated or unsaturated porous medium with fractional dimensions in the form,…”

Modelling solute dispersion in periodic heterogeneous porous media: Model benchmarking against intermediate scale experiments

“…FAD occurs when the motion of the solute molecules is non-Brownian. Dierent behaviours may be obtained depending on the assumptions made on the characteristic times and lengths of molecule jumps [43,44,41,42]. In the presence of trapping eects, an inverse power law asymptotic behaviour may be observed for the probability density function of solute residence time in the porous media.…”

Time–space fractional governing equations of one‐dimensional unsteady open channel flow process: Numerical solution and exploration

“…A unique characteristic of the fractional governing equations is that they are based on fractional derivatives, which are nonlocal. Being nonlocal, the fractional governing equations of open channel flow process have the capability to account for the effect of the initial conditions on the flow process for long times, and the effect of the upstream boundary conditions for long distances from the boundary (see also the discussion on the physical framework of fractional governing equations of soil water flow in Kavvas, Ercan, & Polsinelli, ). The advantage of the fractional derivatives in Caputo framework is that the traditional initial and boundary conditions, which are physically interpretable, can be utilized (Podlubny, ).…”