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.

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. As one approach to the modeling of the generally non-Fickian behavior of transport, Meerschaert, Benson, Baumer, Schumer, Zhang and their co-workers (Meerschaert et al., 1999, 2002, 2006; Benson et al., 2000a, b; Baumer et al., 2005, 2007; Schumer et al., 2001, 2009; Zhang et al., 2007, 2009; Zhang and Benson, 2008) have introduced the fractional advection–dispersion equation (fADE) as a model for transport in heterogeneous subsurface media. By theoretical and numerical studies the above authors have shown that fADE has a nonlocal structure that can model well the heavy-tailed non-Fickian dispersion in subsurface media, mainly by means of a fractional spatial derivative in the dispersion term of the equation. Meanwhile, they have also shown that fADE, with a fractional time derivative, can also model well the long particle waiting times in transport in both surface and subsurface environments. However, while the above-mentioned studies provided extensive treatment of the fractional differential equation modeling of transport in fractional time–space by subsurface flows, few studies have addressed the detailed modeling of the actual subsurface flows in porous media in fractional time–space.

He (1998) seems to be the first scholar who proposed a fractional form of Darcy's equation for water flux in porous media. Based on this fractional water flux equation, in his pioneering work He (1998) then proposed a fractional governing equation of flow through saturated porous media. The left-hand side (LHS) and the right-hand side (RHS) of He's fractional Darcy flux formulation have different units. As saturated flow equations, He's proposed governing equations address the groundwater flow instead of the unsaturated soil water flow. Since the focus of our study is soil water flow in fractional time–space, below we shall discuss the literature that specifically addresses the fractional soil water flow equations.

As early as in 1960s Gardner and his co-workers (Ferguson and Gardner,
1963; Rawlins and Gardner, 1963) questioned the classical diffusivity
expression in the diffusion form of the conventional Richards equation for
soil water flow being only dependent on the soil water content. Based on
their experimental observations, they reported that diffusivity was also
dependent explicitly on time besides being dependent on the soil water
content. Following on these experimental observations, Guerrini and
Swartzendruber (1992) hypothesized a new form for Richards equation for
horizontal unsaturated soil water flow in semi-rigid soils. Unlike the
assumption that the soil hydraulic conductivity

The study by Pachepsky et al. (2003) appears to be the first to propose a
fractional model of horizontal, unsaturated soil water flow in field soils.
Motivated by the observations of Nielsen et al. (1962) on the jerky
movements of the infiltration front in field soils, which can be explained by
long recurrence time intervals in between motions, Pachepsky et al. (2003)
proposed a time-fractional model of horizontal soil water flow in field
soils. While the space component of their model has integer derivatives,
they proposed a fractional form for the diffusivity, and expressed the Darcy
water flux formulation in diffusive form with their proposed fractional
diffusivity. Pachepsky et al. (2003) showed that the cause for fractional
diffusivity is the scaling of time in the Boltzmann relationship not with
the power of 0.5 (which corresponds to Brownian motion) but with a power
less than 0.5, an experimental observation that was already made by Guerrini
and Swartzendruber (1992). Pachepsky et al. (2003) supported their claim by
various previous experimental studies' results, and showed that their
proposed time-fractional form of the Richards equation with fractional
diffusivity can explain experimental data. Meanwhile, Gerolymatou
et al. (2006) proposed a fractional integral form for the Richards equation in
fractional time but in integer horizontal space for unsaturated soil water
flow in one horizontal dimension. Comparing their model simulations against
the field experimental data of El-Abd and Milczarek (2004), they showed that
their fractional Richards equation describes the evolution of soil water
content in time and space better than the corresponding integer Richards
equation. Again considering horizontal unsaturated soil water flow in
fractional time but integer space, Sun et al. (2013) utilized the concept of
fractal ruler in time, due to Cushman et al. (2009), to define a fractional
derivative in time which they used to modify the integer time derivative in
the conventional Richards equation. By means of this fractional derivative
definition they were able to model the anomalous Boltzmann scaling in the
wetting front movement and were able to obtain good fits to water content
experimental data. Sun et al. (2013) conjectured that the time-dependent
diffusivity

The above-cited studies on the governing equations of soil water flow only treat time with fractional dimension, while keeping space with integer dimension. Furthermore, these studies address only one spatial dimension. Accordingly, our study in the following will attempt to develop a fractional continuity equation and a fractional water flux (motion) equation for unsaturated soil water flow in both fractional time and in multi-dimensional fractional space, starting from the conventional mass conservation and Darcy's law. 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 will be considered different, resulting in multi-fractional 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-fractional, multi-dimensional space and fractional time will be developed. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-fractional multi-dimensional space and fractional time, a dimensionally consistent equation will also be developed. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-fractional, multi-dimensional space and fractional time will be obtained. It will be 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 (Brooks and Corey, 1964) into the fractional transient soil water flow equation, an explicit form of the equation will be obtained in multi-dimensional, multi-fractional 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.

Let

Specializing the integer

Then to

One can obtain a

Within the above framework one can express the net mass outflow rate from
the control volume in Fig. 1 as

The control volume for the three-dimensional soil water flow.

With respect to the Caputo derivative

Introducing Eq. (10) into Eq. (7) yields for the net mass outflow
rate

Denoting the volumetric water content by

If one further assumes an incompressible soil medium with constant density,
then the fractional soil water flow continuity Eq. (18) simplifies
further to

In the following, Eq. (19) will be used as the fractional continuity equation for soil water flow for further study.

Performing a dimensional analysis of Eq. (19), one obtains

Podlubny (1999) has shown that for

The experiments of Darcy (1856) showed that the specific discharge

In Eq. (22), using the

Performing a dimensional analysis on Eq. (25), one obtains

As noted above, Podlubny (1999) has shown that for

Combining the fractional continuity Eq. (19) with the fractional soil
water flux Eq. (28) yields

Meanwhile, the soil hydraulic head h is related to the elevation head

Substituting Eq. (32) into Eq. (31) results in

As noted above, Podlubny (1999) has shown that for

With respect to dimensional consistency, one may note that both sides of the
fractional governing Eqs. (33) or (35) for transient soil water
flow have the unit

In the case of vertical transient unsaturated flow for the infiltration
process in soils in fractional time–space, Eq. (35) reduces further to

One can utilize the Brooks and Corey (1964) formula for the soil characteristic
relationship between the capillary soil water pressure head

Substituting Eqs. (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

Upon dimensional analysis of Eq. (44) one can see that it is
dimensionally consistent since both of its sides have the unit of

Specializing Eq. (45) to only the vertical direction, the governing
equation of transient soil water flow in the vertical direction in
fractional space–time may be expressed as

Upon dimensional analysis of Eq. (46) one can find that both sides of
this equation have the unit of

Finally, specializing Eq. (45) to only the horizontal directions, the
governing equation of transient soil water flow in the horizontal directions
in fractional space–time may be expressed as

Upon dimensional analysis of Eq. (47) one can find that both sides of
this equation have the unit of

In parallel to the conventional governing equations of soil water flow processes (Freeze and Cherry, 1979; Bear, 1979), the corresponding governing equations of the soil water flow processes in fractional time–space must have certain properties. (i) The fractional governing equations must be purely differential equations, containing only differential operators, and no difference operators. (ii) They must be prognostic equations. That is, they are solved from the initial conditions and boundary conditions in order to describe the evolution of the flow field in time and space. As such, from the outset the form of the governing equation must be known in its entirety. Once its physical parameters (such as the saturated hydraulic conductivity, etc.) are estimated, the governing equation is fixed throughout the simulation time and the simulation space for the simulation of the soil water flow in question. (iii) These equations must be dimensionally consistent. (iv) The fractional governing equations of soil water flow with fractional powers must converge to the corresponding conventional governing equations with integer powers as the fractional powers approach the corresponding integer powers.

However, a distinct difference of the fractional governing equations of soil
water flow from the corresponding conventional equations is that they are
based on fractional derivatives which are nonlocal. Being nonlocal, the
fractional governing equations of soil water flow have the potential to
account for the effect of the initial conditions on the soil water flow for
long times, and for the effect of the upstream boundary conditions on the
flow for long distances from the upstream boundary. The physical meaning of
the fractional governing equation may be explained most easily in the case
of vertical soil water flow. In the context of upstream-to-downstream
vertical soil water flow from the soil surface downward, in the integer form
of the soil water flow mass conservation equation (the conventional
equation) the time rate of change of mass within a control volume grid
(

As such, each local integer derivative

Referring to Eq. (4) above, it is necessary to take the upper limit
value of the Caputo derivative at “

The governing equations that were developed in this study are for the
fractional time dimension and for multi-dimensional fractional space that
represents the fractal spatial structure of a soil field. If one were to
simplify the developed theory above to only fractional time but
integer-space soil fields, then the developed equations would simplify
substantially. The governing Eq. (36) of transient soil water flow in
anisotropic multi-dimensional fractional soil media in fractional time would
simplify to (with

As mentioned before, Guerrini and Swartzendruber (1992) and El-Abd and
Milczarek (2004), in their explanation of the anomalous behavior of the
diffusivity coefficient in their experiments, have proposed that the
diffusivity coefficient in the diffusion-based formulation of the Richards
equation of soil water flow must depend not only on the water content but
also on time. Hence, they formulated this diffusivity coefficient

Sun et al. (2013) conjectured that the time-dependent diffusivity

Equation (53) shows that the fractional soil water flow Eq. (51) which accounts for the time-dependent diffusivity expression of Guerrini and Swartzendruber (1992) and El-Abd and Milczarek (2004) does have an equivalent form where the integer time derivative of the soil water content in the conventional Richards equation is replaced by a product of the fractional time derivative of the soil water content and a fractional power of time, thereby supporting Sun et al.'s (2013) conjecture, although in this study the fractional derivative is defined in the Caputo sense while in Sun et al. (2013) the fractional derivative is defined with respect to a fractal ruler in time.

In conclusion, in this study first a dimensionally consistent continuity equation for soil water flow in multi-fractional, multi-dimensional space and fractional time was developed. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-fractional multi-dimensional space and fractional time, a dimensionally consistent equation was also developed. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-fractional, multi-dimensional space and fractional time was then 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 (Brooks and Corey, 1964) into the fractional transient soil water flow equation, an explicit form of the equation was obtained in multi-dimensional, multi-fractional space and fractional time. Finally, the governing fractional equation was specialized to the cases of vertical soil water flow and 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.

The authors declare that they have no conflict of interest.