In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Nearly 70 years ago in his hydrologic studies of the Aswan High Dam, Hurst (1951) discovered that the flow time series of the Nile River
demonstrated fluctuations whose rescaled range may not be proportional to
the square root of the observation duration but may be proportional to the
duration raised to a power

Reporting that conventional geometries cannot characterize groundwater flow in many fractured rock aquifers (Black et al., 1986) and the observed drawdown tends to be underestimated in early times and overestimated at later times by the conventional radial groundwater flow model (Van Tonder et al., 2001), Cloot and Botha (2006) developed a fractional governing equation for radial groundwater flow in integer time and fractional space in a uniform homogeneous aquifer. They used the Riemann–Liouville (RL) fractional derivative form (please see Podlubny, 1998, pp. 62–77, for a comprehensive explanation of the RL fractional derivative) in their model formulation. Atangana and Bildik (2013), Atangana (2014), and Atangana and Vermeulen (2014) then reformulated the fractional radial groundwater flow model of Cloot and Botha (2006) by using the Caputo differentiation framework (to be detailed in the next section) and reported better performance. Compared to the Riemann–Liouville derivative approach, the Caputo framework has a fundamental advantage of being able to accommodate physically interpretable real-life initial and boundary conditions (Podlubny, 1998). In simple terms, a differential equation that is based on the RL fractional derivative requires the limit values of the RL fractional derivative for its initial and boundary values, which have no known physical interpretation (Podlubny, 1998, p. 78). Meanwhile, “Caputo derivatives take on the same form as for integer-order differential equations, i.e., contain the limit values of integer-order derivatives” (Podlubny, 1998, p. 79), incorporating the real-world initial and boundary conditions into the solution to a fractional governing equation. Atangana and Baleanu (2014) presented a new radial groundwater flow model in fractional time based on a new fractional derivative definition, “conformable derivative” (Khalil et al., 2014). Most recently, Su (2017) proposed a time–space fractional Boussinesq equation, and he claimed this fractional equation is a general groundwater flow equation and can be applied to groundwater flow in both confined and unconfined aquifers. However, all of the aforementioned studies only presented the formulated fractional governing groundwater flow equations, and no detailed derivations of these governing equations from the fundamental conservation principles were provided.

Wheatcraft and Meerschaert (2008) derived the groundwater flow continuity equation in the fractional form by using the fractional Taylor series approximation. They further removed the linearity, or piecewise linearity, restriction for the flux and the infinitesimal control volume restriction. When developing the fractional continuity equation, the groundwater flow process was considered in fractional space but in integer time by Wheatcraft and Meerschaert (2008). They further assumed the same fractional power in every direction of the fractional porous media space. Furthermore, only the mass conservation was considered in their derivation, not the fractional water flux equation. Mehdinejadiani et al. (2013) expanded the approach of Wheatcraft and Meerschaert (2008) to the derivation of a governing equation of groundwater flow in an unconfined aquifer in fractional space but in integer time. In their derivation, they used the conventional Darcy formulation for the water flux with an integer spatial derivative, while utilizing fractional spatial derivatives in their continuity equation.

Olsen et al. (2016) pointed out that the derivations in Wheatcraft and Meerschaert (2008) and Mehdinejadiani et al. (2013) utilized the fractional Taylor series, as formulated by Odibat and Shawagfeh (2007), which utilized local Caputo derivatives. In order to expand the local Caputo derivatives in the abovementioned studies, Olsen et al. (2016) utilized the fractional mean value theorem from Diethelm (2012) to develop a continuity equation of groundwater flow with left and right fractional nonlocal Caputo derivatives in fractional space but in integer time. Olsen et al. (2016) did not address the water flux formulation in fractional space and, hence, did not develop a complete governing equation of groundwater flow. They also did not address the multi-fractional spatial derivatives in order to address anisotropy within an aquifer. Around that time, Kavvas et al. (2017a) utilized the mean value formulation from Usero (2008), Odibat and Shawagfeh (2007), and Li et al. (2009) to derive a complete governing equation of transient groundwater flow in an anisotropic confined aquifer with fractional time and multi-fractional space derivatives which addressed not only the continuity but also the water flux (motion) in fractional time–space and the effect of a sink/source term. By employing the abovementioned fractional mean value formulations, Kavvas et al. (2017a) developed the governing equation of confined groundwater flow in fractional time–space in nonlocal form.

As mentioned above, unconfined groundwater flow is the fundamental component of the watershed runoff baseflow since it is the fundamental contributor to the network streamflow within a watershed during dry periods. As such, the behavior of unconfined groundwater flow is key to the physically based understanding of the long memory in watershed runoff. Meanwhile, as will be seen in the following derivation of its governing equation, unconfined aquifer groundwater flow is uniquely different from the confined aquifer groundwater flow. The fundamental difference between the two aquifer flows is that while the flow in a confined aquifer is linear and compressible, the flow in an unconfined aquifer is nonlinear and incompressible due to the unconfined aquifer being phreatic, its top surface boundary being open to the atmosphere. Accordingly, hydrologists have developed unique governing equations of unconfined aquifer groundwater flow (Bear, 1979; Freeze and Cherry, 1979). Starting with the next section, first the continuity equation of transient unconfined groundwater flow within an anisotropic heterogeneous aquifer under a time–space varying sink/source will be developed in fractional time and fractional space. Then, this fractional continuity equation will be combined with a fractional groundwater motion equation to obtain a transient groundwater flow equation in fractional time and multi-fractional space within an anisotropic heterogeneous unconfined aquifer.

Analogous to the traditional governing groundwater flow equations, as
outlined by Freeze and Cherry (1979) and Bear (1979), the fractional
unconfined groundwater flow equations must have specific features (Kavvas et
al., 2017a):

In order for the governing equation to be prognostic, the form of the equation must be known completely from the outset.

The fractional governing equations must be dimensionally consistent and be purely differential equations, containing only differential operators without difference operators.

As the fractional derivative powers go to integer values, the fractional unconfined groundwater flow equations must converge to the corresponding conventional integer-order governing equations.

The

It was shown in Kavvas et al. (2017b) that one can obtain a

The mass flux through the control volume of an unconfined aquifer.

Under the Dupuit approximation of horizontal flow streamlines (for a very small
water table gradient) (Bear, 1979), the net mass flux through the control
volume of an unconfined aquifer with a flat bottom confining layer, as
depicted in Fig. 1, which also has a sink/source mass flux,

Kavvas et al. (2017b) have shown that, to

Combining Eqs. (5) and (4) yields, for the net mass outflow through the
control volume in Fig. 1 (to the orders of

As the time rate of change of mass within the control volume, as shown in
Fig. 1, must be inversely proportional to the net mass flux passing
through the control volume, one may combine Eqs. (6) and (11) to obtain

Within the framework of fluid incompressibility in the unconfined aquifer,
Eq. (13) reduces further to

Performing a dimensional analysis of Eq. (14) yields

For

Recently, Kavvas et al. (2017a, b) derived a governing equation for
water flux (specific discharge),

A dimensional analysis of Eq. (19) yields

Applying the abovementioned result of Podlubny (1998) to the convergence
of a fractional derivative to a corresponding integer derivative for

Combining the fractional motion Eq. (19) of groundwater flow in an
unconfined aquifer with the fractional continuity Eq. (14) of
unconfined groundwater flow results in the equation

Performing a dimensional analysis of Eq. (21) yields

Specializing the above-discussed result of Podlubny (1998) to

The sketch of numerical application 1; water seepage through the body of a dam as an unconfined groundwater flow.

Results for numerical application 1:

To demonstrate the skills of the proposed fractional governing equation of
unconfined aquifer groundwater flow, two numerical applications are
performed using the proposed fractional governing equation. The first
application (numerical application 1) follows the physical setting of an example from Wang and
Anderson (1995), as depicted in Fig. 2. The numerical problem of seepage
through a dam under a sudden change in the water surface elevation at the
downstream section of the dam is modified based on seepage through a dam,
p. 53 and Problem 4.4(a), p. 89 in Wang and Anderson (1995), as
shown in Fig. 2. The water seepage through the body of the dam may be
interpreted as one-dimensional groundwater flow through an unconfined
aquifer. The unconfined flow system locates the top boundary of the
saturated zone in an earthen dam and the bottom of the dam rests on
impermeable rock. For this example, the unconfined aquifer length

In Fig. 3a and b, the normalized groundwater head and normalized
groundwater discharge per unit width at location

The sketch numerical application 2; the downstream groundwater head is fixed at 11 m and the initial upstream groundwater head is 16 m.

Results for numerical application 2:

The second application (numerical application 2) deals with a transient unconfined groundwater flow
from a hillslope toward a stream (Fig. 4). The upstream boundary plane
separates the region of flow from the adjacent hillslope that feeds the
adjacent tributary system; therefore

From the standard governing Eq. (23) of unconfined groundwater flow in
integer time–space the saturated hydraulic conductivity may be interpreted
as a diffusion coefficient for the nonlinear diffusion of groundwater in an
unconfined aquifer. The basic difference between confined and unconfined
groundwater flow is that the former can be interpreted as a linear diffusion
of groundwater while the latter is a nonlinear diffusion of groundwater
within an aquifer. Similar to saturated hydraulic conductivities in Eq. (26) in Kavvas et al. (2017a) for the fractional confined aquifer
groundwater flow, the saturated hydraulic conductivities in Eq. (21), which governs fractional unconfined aquifer groundwater flow, are
modulated by the ratios of fractional time to fractional space,

Numerical applications demonstrated that as the powers of the space and time fractional derivatives decrease from 1, the recession rate of the nondimensional groundwater hydraulic head slows down compared to the case by the conventional governing equation (i.e., with integer-order derivatives). This behavior also indicates the modulation of the nonlinear diffusion of the groundwater by the fractional powers of time and space.

As mentioned in the Introduction section, unconfined groundwater flow is the
fundamental component of the watershed runoff baseflow since it is the
fundamental contributor to the streamflow network within a watershed during
dry periods. As such, the behavior of unconfined groundwater flow is key to
the physically based understanding of the long memory in watershed runoff.
As seen from the numerical applications in Figs. 3 and 5, the powers of
the fractional derivatives in time and space can modulate the speed of the
groundwater discharge evolution. Hence, they can modulate the memory of the
unconfined aquifer flow, which, in turn, can modulate the memory of the
watershed baseflow. Meanwhile, the Caputo derivative, as defined in its
special form

A dimensionally consistent fractional governing equation of transient unconfined aquifer groundwater flow was derived within a fractional differentiation framework. After developing a fractional continuity equation, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media was combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. By combining the fractional continuity and motion equations, the governing equation of transient unconfined aquifer groundwater flow in a multi-fractional medium in fractional time was obtained. To demonstrate the skills of the proposed fractional governing equation of unconfined aquifer groundwater flow, two numerical applications were performed. As demonstrated in the numerical application results, the orders of the fractional space and time derivatives modulate the speed of groundwater discharge and groundwater flow evolution, slowing the process with the decrease in the powers of the fractional derivatives from 1. It is also shown that the proposed dimensionally consistent fractional governing equations approach the corresponding conventional equations as the fractional orders of the derivatives go to 1.

The data used in this article can be accessed by contacting the corresponding author.

One-dimensional time–space fractional groundwater flow in the unconfined
aquifer with no recharge or leakage can be written as

The Caputo fractional time derivative

for

for

MLK wrote the main body of the paper. TT and AE wrote the numerical applications section. MLK, TT, AE, and JP discussed the results and responded to the reviewers.

The authors declare that they have no conflict of interest.

Authors thank the editor Rui A. P. Perdigão and two anonymous referees for their valuable comments and suggestions.

This paper was edited by Rui A. P. Perdigão and reviewed by two anonymous referees.