Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The governing equation of transient saturated groundwater flow in a multi-fractional, multidimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations. To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed.

Previous laboratory and field studies (Levy and Berkowitz, 2003; Silliman and Simpson, 1987; Peaudecerf and Sauty, 1978; Sidle et al., 1998; Sudicky et al., 1983) demonstrated substantial deviations from Fickian behavior in transport in subsurface porous media. Various authors (Meerschaert et al., 1999, 2002, 2006; Benson et al., 2000a, b; Schumer et al., 2001, 2009; Baeumer et al., 2005; Baeumer and Meerschaert, 2007; 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 as one approach to the modeling of the generally non-Fickian behavior of transport. As was demonstrated by the studies above, the heavy-tailed non-Fickian dispersion in subsurface media can be modeled well by a fractional spatial derivative, and the long particle waiting times in transport can be modeled well by means of a fractional time derivative within fADE. However, the abovementioned studies focused on the fractional differential equation modeling of solute transport in fractional time and space, and not on the modeling of the underlying subsurface flows that transport the solutes. Also, as shown by Kim et al. (2014), non-Fickian behavior in transport can also be obtained if the underlying flow field has a long memory in time, which can be described by a time-fractional governing equation of the specific flow field (Ercan and Kavvas, 2014, 2016). Kang et al. (2015) also showed that velocity correlation and distribution in fractured media may lead to non-Fickian transport and proposed a continuous-time random walk model (see Metzler and Klafter, 2000, for details of such models) that can account for velocity correlation and distribution.

Cloot and Botha (2006) argued that there are many fractured rock aquifers in which the groundwater flow does not fit conventional geometries (Black et al., 1986), and in such aquifers the conventional radial groundwater flow model underestimates the observed drawdown in early times and overestimates it at later times (Van Tonder et al., 2001). Based on this argument, which they supported with some radial flow field data, Cloot and Botha (2006) then formulated a fractional governing equation for radial groundwater flow in integer time but fractional space and provided some numerical applications of this model. In that formulation they also provided a formulation of the Darcy flux in radial fractional space. However, in addition to taking the time as an integer, they also considered a uniform homogeneous aquifer with a constant hydraulic conductivity. In the formulation of their radial groundwater flow model, they did not provide a derivation of the mass conservation equation for groundwater flow in fractional time and space. Also, they utilized the Riemann–Liouville form of the fractional derivative. Later, Atangana and his co-workers (Atangana, 2014; Atangana and Bildik, 2013; Atangana and Vermeulen, 2014) developed the fractional radial groundwater flow formulation of Cloot and Botha (2006) in terms of the Caputo derivative and claimed it yielded superior performance when compared to the Riemann–Liouville derivative formulation. The fundamental advantage of the Caputo derivative over the Riemann–Liouville derivative is that it can accommodate the real-life initial and boundary conditions, while the Riemann–Liouville derivative cannot (Podlubny, 1998). That is, the fractional differential equations with Caputo derivatives contain the physically interpretable integer-order derivatives at the initial times and at the upstream spatial boundaries, whereas the Riemann–Liouville derivatives do not (Podlubny, 1998). More recently, Atangana and Baleanu (2014) utilized a new definition of the fractional derivative, called the “conformable derivative” (Khalil et al., 2014), for the modeling of radial groundwater flow in fractional time but integer space. In all the studies above, the authors formulated their fractional governing equations instead of providing derivations of their groundwater flow equations from the basic conservation principles.

Wheatcraft and Meerschaert (2008) were the first to provide a comprehensive derivation of the continuity equation for groundwater flow. These authors have shown that since a first-order Taylor series approximation is used to represent the change in the mass flux through a control volume, the traditional continuity equation in an infinitesimal control volume is exact only when the change in flux in the control volume is linear. They also showed that, analogous to a first-order Taylor series, a fractional Taylor series is able to represent the nonlinear flux in a control volume by exactly only two terms. By replacing the integer-order Taylor series approximation for flux with the fractional-order Taylor series approximation, they derived a fractional form of the continuity equation for groundwater flow, removing the linearity or piecewise linearity restriction for the flux and the restriction that the control volume must be infinitesimal. In their development of the continuity equation, Wheatcraft and Meerschaert (2008) considered the porous medium in fractional space but the flow process in integer time. They also considered the fractional porous media space to have the same fractional power in all directions. Furthermore, their derivation is confined to only the mass conservation. It does not address the fractional water flux (motion) equation, nor the complete governing equation of groundwater flow.

Groundwater level fluctuations through time at certain locations exhibit long-range time correlation, which implies the need for the incorporation of time-fractional operation in the standard groundwater flow governing equations in order to accommodate the long-range time dependence (Li and Zhang, 2007; Rakhshandehroo and Amiri, 2012; Tu et al., 2017; Yu et al., 2016). Hence, in order to provide a general modeling structure, it is necessary to develop the governing equations of confined groundwater flow in fractional time as well as in fractional space. Also, different fractional powers should be considered in different spatial directions in order to accommodate the anisotropy of a confined aquifer medium.

In parallel to the conventional governing equations of groundwater flow processes (Bear, 1979; Freeze and Cherry, 1979), the corresponding time–space fractional governing equations of the confined groundwater flow must have certain characteristics (Kavvas et al., 2017): (a) from the outset, the form of the governing equation must be known completely. As such, it must be a prognostic equation. That is, in order to describe the evolution of the flow field in time and space it is solved from the initial conditions and boundary conditions. The governing equation is fixed throughout the simulation time and space for the simulation of the groundwater flow in question once its physical parameters, such as porosity, saturated hydraulic conductivity, etc., are estimated. (b) The fractional governing equations must be purely differential equations, containing only differential operators and no difference operators. (c) These equations must be dimensionally consistent. (d) As the orders of the fractional derivatives in the equations approach the corresponding integer powers, the fractional governing equations of confined groundwater flow with fractional powers must converge to the corresponding conventional governing equations with integer powers. The following development of the fractional governing equations of confined groundwater flow will be performed within the framework above.

Let

The control volume for the three-dimensional groundwater flow in confined aquifers.

The net mass flux through the control volume in Fig. 1, which also has a
sink–source mass flux

From Eq. (5) it also follows with

Since the net flux through the control volume is inversely related to the
time rate of change of mass within the control volume of Fig. 1, one may
combine Eqs. (11) and (17) to obtain

Performing a dimensional analysis of Eq. (21), one obtains

It was shown by Podlubny (1998) that for

A governing equation for water flux (specific discharge)

A dimensional analysis on Eq. (24) yields

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

One can combine the specific discharge Eq. (24) for groundwater flow
(the motion equation) in a fractional confined aquifer with the time–space
fractional continuity Eq. (21) of groundwater flow in fractional
time and space in confined aquifers to obtain

Performing a dimensional analysis on the governing fractional Eq. (26)
for confined groundwater flow results in

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

The reservoir example modified based on Wang and Anderson (1995).

Let us consider the Caputo fractional time derivative of the function

Nondimensional groundwater hydraulic heads through time at

To illustrate the capability of the proposed governing equation of
groundwater flow in a confined aquifer, a numerical application of the
fractional governing equation to the physical setting of an example from
Wang and Anderson (1995) is provided as shown in Fig. 2. In this
example, groundwater flow in a confined aquifer is simplified to be
one-dimensional. The length of the confined aquifer is 100 m. The hydraulic
transmissivity (

Nondimensional groundwater hydraulic heads (

The conventional governing equations of porous media flows in geosciences in
various environments are all local-scale equations in which only the
interactions among nearest neighbors in time and space are described. All
of these governing equations are differential equations where the powers of
the derivative terms that appear in these equations take integer values. In
the case that a porous media flow field shows interactions among time–space
locations that are separated by substantial distances in time or space, the
local-scale conventional governing flow equations for such media, because
they are based on local interactions, may not be able to describe such
long-distance interactions adequately. A more efficient approach for
modeling such long-distance interactions in time and space may be the use of
fractional governing equations of porous media flows. Such fractional
governing equations, as those developed in this study, utilize time–space
derivatives with fractional powers. As already shown in Sect. 5 above, the
fractional Caputo time derivative is nonlocal, and, as such, can accommodate
the effect of the initial conditions on the groundwater flow process for
times that are substantially later than the initial time. Similarly, the
fractional Caputo space derivatives in the governing Eqs. (21), (24),
and (26) of this study are also nonlocal derivatives. To observe this, consider
the Caputo fractional space derivative

As shown in the previous sections, the fractional governing equations converge to their conventional integer counterparts as the fractional derivative powers take integer values. Consequently, the conventional governing equations of porous media flows may be considered as special cases of the corresponding fractional governing equations, corresponding to the integer values of the derivative powers. While the fractional powers of the derivatives in the governing Eq. (26) may take any fractional value within the interval (0, 1), the integer powers of the derivatives in the conventional governing Eq. (28) are restricted to the value of unity. Within this context, the fractional governing equations of porous media flows may be thought of as the generalizations of the conventional governing equations of porous media flows with integer powers.

From the information above, it follows that the fractional governing equations developed in this study are nonlocal. Accordingly, they can account for the influence of the initial and boundary conditions on the flow process more effectively than the corresponding local-scale integer-order conventional governing equations since the conventional governing equations consider the effect of initial and boundary conditions on the flow processes within shorter time–space ranges.

From Eq. (28) it may be noted that the saturated hydraulic conductivity
plays the role of a diffusion coefficient in the conventional governing
equation of transient groundwater flow in an anisotropic confined aquifer in
integer time and space. For discussion purposes, let us rewrite Eq. (26)
for the governing equation of transient saturated groundwater flow in an anisotropic
confined aquifer in fractional time and space:

Kavvas et al. (2014) argued, and Kim et al. (2014) have shown by numerical simulations, that non-Fickian behavior in solute transport can also be obtained if the underlying flow field has a long memory, which can be described by a fractional governing equation of the specific flow field. Ercan and Kavvas (2014, 2016) have shown by numerical simulations that it is possible to obtain long waves in time and in space by means of the fractional governing equations of unsteady open channel flow.

In this study, a dimensionally consistent continuity equation for transient saturated groundwater flow in multi-fractional, multidimensional confined aquifers in fractional time was developed. It was then shown that as the fractional powers of time and space derivatives approach unity, the time–space fractional continuity equation approaches the conventional continuity equation for transient groundwater flow in a confined aquifer. For the motion equation of confined saturated groundwater flow, or the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation was also developed. It was shown that as the fractional powers of the spatial derivatives approach unity, the fractional water flux equation approaches the conventional Darcy equation for groundwater specific discharge.

The governing equation of transient saturated groundwater flow in multi-fractional, multidimensional confined aquifers and in fractional time was then obtained by combining the fractional continuity and water flux equations. It was then shown that as the fractional powers of time and space derivatives approach unity, the time–space fractional governing equation of transient saturated confined groundwater flow approaches the conventional governing equation with integer derivatives for transient saturated groundwater flow in an anisotropic confined aquifer.

To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed. The modeling results indicate that the proposed governing equations may help explain the nonlocal effects in groundwater flow and may further help illustrate the associated non-Fickian transport in groundwater flow.

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

The authors declare that they have no conflict of interest.

This article is part of the special issue “Hydro-climate dynamics, analytics and predictability”. It is not associated with a conference.