ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-8-921-2017Fractional governing equations of transient groundwater flow in confined aquifers with multi-fractional dimensions in fractional timeKavvasM. Leventmlkavvas@ucdavis.eduTuTongbiErcanAlihttps://orcid.org/0000-0003-1052-4302PolsinelliJamesHydrologic Research Laboratory, Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USAM. Levent Kavvas (mlkavvas@ucdavis.edu)16October20178492192919May20179June201718August20174September2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/articles/8/921/2017/esd-8-921-2017.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/8/921/2017/esd-8-921-2017.pdf
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.
Introduction
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.
Derivation of the continuity equation for transient groundwater flow in a multi-fractional confined aquifer in fractional time
Let Dakβf(x) be a Caputo fractional derivative
of the function f(x), defined as (Li et al., 2009; Odibat and Shawagfeh,
2007; Podlubny, 1998; Usero, 2008)
Dakβf(x)=1Γ(m-kβ)∫axfm(ξ)(x-ξ)kβ+1-mdξ,m-1<β<m,mϵN,x≥a.
Specializing the integer m= 1 reduces Eq. (1) to
Dakβf(x)=1Γ(1-kβ)∫axf′(ξ)(x-ξ)kβdξ,0<β<1,x≥a,
then to β-order
Daβf(x)=1Γ(1-β)∫axf′(ξ)(x-ξ)βdξ,0<β<1,x≥a.
One can obtain a βxi-order approximation (i= 1, 2, 3;
x1=x, x2=y, x3=z) to a function f (.)
around a as
fxi=f(a)+xi-aβxiΓβxi+1Daβxifxi,0<βxi<1;i=1,2,3x1=x,x2=y,x3=z.
This result may be obtained by taking in the mean value representation of a
function in terms of the fractional Caputo derivative (Odibat and Shawagfeh,
2007; Usero, 2008; Li et al., 2009) the upper limit value
of the Caputo
derivative at xi (i= 1, 2, 3; x1=x, x2=y, x3= z) to
have a distinct value for the βxi-order approximation above
(i= 1, 2, 3; x1=x, x2=y, x3=z) of the function f
around a. Based on this approximation, for the whole modeling domain in
time and space, the governing equations become prognostic equations that
shall be known from the outset of model simulation. The next issue is what
to take for the value of a. If one expresses Eq. (4) with a=xi-Δxi, that is,
fxi=fxi-Δxi+ΔxiβxiΓβxi+1Dxi-Δxiβxifxi;i=1,2,3;x1=x,x2=y,x3=z,
then the question becomes what to take for the value of Δxi in
Eq. (5). In order to obtain fractional governing equations as purely
differential equations, an analytical relationship between Δxi
and (Δxi)β (i= 1, 2, 3; x1=x,
x2=y, x3=z) that will be universally applicable throughout
the modeling domain must be established. Such an analytical relationship
is possible when the lower limit in the Caputo derivative above in Eq. (5)
is taken as zero (that is, Δxi=xi) for
f(xi)=xi. As will be shown below, it will be possible to
develop purely differential forms (with no finite difference operators) for
the fractional governing equations of confined groundwater flow by following
the construct above.
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 qvΔxΔyΔz, can be formulated
within the framework above as
ρqx(x,y,z;t)-ρqx(x-Δx,y,z;t)ΔyΔz+ρqy(x,y,z;t)-ρqy(x,y-Δy,z;t)ΔxΔz+ρqz(x,y,z;t)-ρqz(x,y,z-Δz;t)ΔxΔy+ρqvΔxΔyΔz.
Then combining Eq. (5) with Eq. (6) with Δxi=xi(i= 1, 2, 3;
x1=x, x2=y, x3=z)
and expressing the resulting Caputo derivative D0βxif(xi)
(taking Δxi=xi causes the lower limit in the Caputo derivative of Eq. 5 to become 0)
by ∂βxif(xi)(∂xi)βxi,
(i= 1, 2, 3; x1=x, x2=y, x3=z)
for convenience, yields the net mass flux through the control volume in
Fig. 1 to the orders of (Δx)βx, (Δy)βy,
and (Δz)βz as
1Γβx+1∂∂xβxρqx(x,y,z;t)(Δx)βxΔyΔz+1Γβy+1∂∂yβyρqy(x,y,z;t)Δx(Δy)βyΔz+1Γβz+1∂∂zβzρqz(x,y,z;t)ΔxΔy(Δz)βz+ρqvΔxΔyΔz,
where, due to the anisotropy in the hydraulic conductivities and in the
subsequent flows in the porous media, different powers for fractional
derivatives are considered in the three Cartesian directions in space.
From Eq. (5) it also follows with f(xi)=xi that to the order of
(Δxi)βxi, i= 1, 2, 3,
Δxi=ΔxiβxiΓβxi+1∂βxixi∂xiβxi,i=1,2,3;x1=x,x2=y,x3=z.
Also for the Caputo derivative,
∂βxixi∂xiβxi=xi1-βxiΓ2-βxi,i=1,2,3;x1=x,x2=y,x3=z.
Hence, introducing Eq. (9) into Eq. (8) results in βxi-order fractional
increments in space in the ith direction, i= 1, 2, 3,
Δxiβxi=Γβxi+1Γ2-βxixi1-βxiΔxi,x1=x,x2=y,x3=z;βx1=βx,βx2=βy,βx3=βz.
For the net mass outflow through the
control volume in Fig. 1 (to the order of (Δxi)βxi,
i= 1, 2, 3; x1=x, x2=y, x3=z), combining Eqs. (10) and (7) yields
Γ2-βxx1-βx∂∂xβxρqx(x‾;t)ΔxΔyΔz+Γ2-βyy1-βy∂∂yβyρqy(x‾;t)ΔyΔxΔz+Γ2-βzz1-βz∂∂zβzρqz(x‾;t)ΔzΔxΔy+ρqvΔxΔyΔz,x‾=(x,y,z).
Denoting the porosity, which is the water volume per volume of the control
volume in Fig. 1 under saturated conditions, using n, the change in mass
within the control volume in Fig. 1 per time increment Δt may be
expressed as (Freeze and Cherry, 1979)
ρn|t-ρn|t-Δt/Δt.
Meanwhile, the specific storage Ss of a saturated aquifer may be defined
as the volume of water that is released from a unit volume of the aquifer
under a unit decline in the hydraulic head h (Freeze and Cherry, 1979).
Under this definition the change in mass in the control volume of Fig. 1
per time increment Δt may be expressed as (Freeze and Cherry, 1979)
ρn|t-ρn|t-ΔtΔt=ρSsh|t-h|t-ΔtΔtΔxΔyΔz=ρSsΔhΔtΔxΔyΔz.
Expressing the relationship (Eq. 10) to α-order fractional increments in
time,
(Δt)α=Γ(α+1)Γ(2-α)t1-αΔt.
Meanwhile, using the approximation (Eq. 5) in the time dimension to the order
of (Δt)α, for any function g of time,
g(t)-g(t-Δt)=(Δt)αΓ(α+1)∂∂tαg(t).
Introducing Eq. (15) into the right-hand side (RHS) of Eq. (13) yields to the order of (Δt)α,
ρSs1Δt(Δt)αΓ(α+1)∂∂tα(h)ΔxΔyΔz.
Then introducing Eq. (14) into the expression (Eq. 16) yields
ρSsΓ(2-α)t1-α∂∂tα(h)ΔxΔyΔz
as the time rate of change of mass in the control volume of size ΔxΔyΔz.
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
ρSsΓ(2-α)t1-α∂∂tα(h)=-Γ2-βxx1-βx∂∂xβxρ(x‾;t)qx(x‾;t)+Γ2-βyy1-βy∂∂yβyρ(x‾;t)qy(x‾;t)+Γ2-βzz1-βz∂∂zβzρ(x‾;t)qz(x‾;t)+ρqv.
In the conventional case with the integer derivatives (Freeze and Cherry, 1979),
ρ∂qxi∂xi≫qxi∂ρ∂xi,i=1,2,3;x1=x,x2=y,x3=z.
Hence, it is also expected that
ρ∂βiqxi∂xiβi≫qxi∂βiρ∂xiβi,i=1,2,3;x1=x,x2=y,x3=z;β1=βx,β2=βy,β3=βz.
Combining the inequality (Eq. 20) with Eq. (18) yields
Ss∂αh(∂t)α=-Γ2-βxΓ(2-α)t1-αx1-βx∂∂xβxqx(x‾;t)-Γ2-βyΓ(2-α)t1-αy1-βy∂∂yβyqy(x‾;t)-Γ2-βzΓ(2-α)t1-αz1-βz∂∂zβzqz(x‾;t)-qvt1-αΓ(2-α)0<α,βx,βy,βz<1,x‾=x1,x2,x3
as the time–space fractional continuity equation of transient saturated groundwater
flow in an anisotropic confined aquifer with fractional dimensions and in
fractional time.
Performing a dimensional analysis of Eq. (21), one obtains
1Tα=1L⋅LTα=T1-αL1-βx1LβxLT=T1-αL1-βy1LβyLT=T1-αL1-βz1LβzLT=1Tα,
where L denotes length and T denotes time. Hence, the left-hand side (LHS) and RHS of the continuity Eq. (21) for transient groundwater flow in
multi-fractional space and fractional time are shown to be consistent by
means of Eq. (22).
It was shown by Podlubny (1998) that for
n- 1 <α, βi<n,
where n is any positive integer, as α and βi→n, the
Caputo fractional derivative of a function f(y) to order α or
βi(i= 1, 2, 3; β1=βx, β2=βy,
β3=βz) becomes the conventional nth derivative of
the function f(y). Specializing the Podlubny (1998) result to n= 1, for
α and βi→ 1 (i= 1, 2, 3; β1=βx,
β2=βy, β3=βz), reduces the
continuity Eq. (21) to the conventional continuity equation for
transient groundwater flow in a confined aquifer:
Ss∂h∂t=-∂∂xqx(x‾;t)-∂∂yqy(x‾;t)-∂∂zqz(x‾;t)-qv.
An equation for specific discharge (motion equation) in fractional multidimensional confined aquifers
A governing equation for water flux (specific discharge) qxi, (i= 1,
2, 3; x1=x, x2=y, x3=z) in a saturated or unsaturated
porous medium with fractional dimensions was recently developed
(Kavvas et al., 2017). For the case of transient saturated
groundwater flow in an anisotropic confined aquifer with multi-fractional
dimensions, that equation for the specific discharge takes the form
qi(x‾,t)=-Ks,xi(x‾)Γ2-βixi1-βi∂βih∂xiβi,i=1,2,3;x1=x,x2=y,x3=z,
where Ks,xi(x‾) denotes the saturated hydraulic
conductivity in the ith spatial direction (i= 1, 2, 3; x1=x,
x2=y, x3=z). Due to the groundwater flow being in the direction of
decreasing hydraulic head, the RHS of Eq. (24) takes a negative sign.
A dimensional analysis on Eq. (24) yields L/T for the units of both the
LHS and the RHS of the equation, establishing its dimensional consistency.
Applying the abovementioned result of Podlubny (1998) for the convergence
of a fractional derivative to a corresponding integer derivative, for
βi→ 1 (i= 1, 2, 3; β1=βx,
β2=βy, β3=βz), reduces the fractional specific discharge
(Eq. 24) for groundwater flow to the conventional Darcy equation for
groundwater specific discharge:
qi(x‾,t)=-Ks,xi(x‾)∂h(x‾,t)∂xi,i=1,2,3;x1=x,x2=y,x3=z
for the case of integer spatial dimensions. As such, the fractional specific
discharge (Eq. 24) for confined groundwater flow in fractional spatial
dimensions is consistent with the conventional Darcy equation for the
integer spatial dimensions.
The complete equation for transient confined groundwater flow in multi-fractional space and fractional time
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
Ss∂αh(∂t)α=Γ2-βxx1-βx∂∂xβxKs,x(x‾)t1-αx1-βxΓ2-βxΓ(2-α)∂βxh(∂x)βx+Γ2-βyy1-βy∂∂yβyKs,y(x‾)t1-αy1-βyΓ2-βyΓ(2-α)∂βyh(∂y)βy+Γ2-βzz1-βz∂∂zβzKs,z(x‾)t1-αz1-βzΓ2-βzΓ(2-α)∂βzh(∂z)βz-qvt1-αΓ(2-α);0<α,βx,βy,βz<1;x‾=x1,x2,x3
as the time–space fractional governing equation of transient saturated groundwater
flow in a confined anisotropic aquifer with multi-fractional dimensions and
in fractional time. In Eq. (26), qv may be taken as the pumping rate or recharge rate.
Performing a dimensional analysis on the governing fractional Eq. (26)
for confined groundwater flow results in
1Tα=1L1-βx1LβxLTT1-αL1-βxLLβx=1L1-βy1LβyLTT1-αL1-βyLLβy=1L1-βz1LβzLTT1-αL1-βzLLβz=1Tα,
which shows that both the RHS and the LHS of the equation have the
unit 1Tα,
which verifies its dimensional consistency.
Applying the abovementioned result of Podlubny (1998) for the convergence
of a fractional derivative to a corresponding integer derivative, for α
and βi→ 1 (i= 1, 2, 3; β1=βx,
β2=βy, β3=βz), the governing
Eq. (26) for confined groundwater flow in fractional time and space takes the form
Ss∂h(x‾;t)∂t=∂∂xKs,x(x‾)∂h(x‾;t)∂x+∂∂yKs,y(x‾)∂h(x‾;t)∂y+∂∂zKs,z(x‾)∂h(x‾;t)∂z-qv,x‾=x1,x2,x3,
which is the conventional governing equation for transient saturated groundwater flow
in an anisotropic confined aquifer (Freeze and Cherry, 1979). As such,
the time–space fractional governing Eq. (26) of transient groundwater
flow in a confined anisotropic aquifer with multi-fractional dimensions in
fractional time is consistent with the conventional governing equation for
transient groundwater flow in an anisotropic confined aquifer with integer derivatives.
The reservoir example modified based on Wang and Anderson (1995).
Physical meaning of fractional time derivative in the fractional governing equations of confined transient groundwater flow
Let us consider the Caputo fractional time derivative of the function f(t),
∂αf(∂t)α=D0αf(t),
defined by
D0αf(t)=1Γ(1-α)∫0tf′(s)(t-s)αds,0<α<1,t≥0.
As such, each local integer derivative f′(s) at each time position s
(0 ≤s≤t) in the time interval (0, t) contributes with the weight
(t-s)-α to the Caputo fractional derivative of f(t) during the
time interval (0, t). Hence, the Caputo derivative is a nonlocal
quantity, pertaining to a time interval, vs. the conventional derivative
of f(t), f′(t), which is defined for the particular time location t.
Within this framework, the effect of the initial condition at the initial
time location 0 is still accounted for at any time t (0 ≤t≤T)
during the whole simulation period (0, T) by means of the fractional
time derivative that appears in the governing Eq. (26) above of
confined transient groundwater flow in fractional time. It also follows from
Eq. (30) that this memory effect is modulated by the value of the
fractional power α. As shown by Podlubny (1998), as α→ 1,
the Caputo fractional time derivative of f(t), as given by
Eq. (30), converges to the local time derivative f′(t) at t.
Nondimensional groundwater hydraulic heads through time at
x=L/2, when fractional space and time derivatives are βx=α= 0.8,
0.9, 1.0, where L is the length of the aquifer and βx and α are
the fractional orders in space and time, respectively.
A numerical application of the developed fractional governing equation of confined groundwater flow
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 (T) of the aquifer is 0.02 m2 min-1 and the specific
storage (S) of the aquifer is 0.002. The groundwater hydraulic head is
initially uniform at 20 m. The water level downstream suddenly drops to
10 m and stays at 10 m. The groundwater level upstream is set to be 20 m
throughout the simulation duration. The total simulation time is 600 min.
Nondimensional groundwater hydraulic heads (H/H0, where H0 is the
initial groundwater hydraulic head) at x= 50 m through time in the aquifer
are shown in Fig. 3, in which fractional derivatives in space and time are
taken as βx=α= 0.8, 0.9, 1.0. As one can see from Fig. 3,
compared to the curve of hydraulic head recession in time that corresponds
to βx=α= 1.0 (the conventional integer derivative case), the
hydraulic head recession in time gets slower with the decrease in
βx=α from 1. The groundwater hydraulic heads in Fig. 3 clearly
show heavier tails as fractional derivative orders in space and time
decrease from 1. Additionally, the smaller the fractional orders are, the
heavier the tails become with the increase in time. The modeling results
may indicate nonlocal effects in groundwater flow and help explain the
long-range dependence characteristics in some groundwater level fluctuation
datasets (Tu et al., 2017). The results may also shed light on the
non-Fickian transport phenomena in groundwater flow.
Discussion on the developed fractional governing equations in the context of broader geosciences
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 D0βf(xi):
D0βfxi=1Γ(1-β)∫0xif′(ξ)xi-ξβdξ.
Hence, each local integer derivative f′(ξ) at each spatial
location ξ in the spatial interval (0, xi) will contribute to
the Caputo fractional derivative of the interval (0, xi) with the weight
(xi-ξ)-β. As such, for groundwater flow in any
i direction, the effect of a boundary condition that is placed at boundary
location 0 in the i direction will be accounted for at any distance xi
from the boundary location 0 by means of the fractional space
derivative that appears in the fractional governing equations above for the
ith direction. It follows from Eq. (31) that this effect will be
modulated by the value of the fractional derivative power β due to
the weight (xi-ξ)-β.
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:
Ss∂αh(∂t)α=Γ2-βxx1-βx∂∂xβxKs,x(x‾)t1-αx1-βxΓ2-βxΓ(2-α)∂βxh(∂x)βx+Γ2-βyy1-βy∂∂yβyKs,y(x‾)t1-αy1-βyΓ2-βyΓ(2-α)∂βyh(∂y)βy+Γ2-βzz1-βz∂∂zβzKs,z(x‾)t1-αz1-βzΓ2-βzΓ(2-α)∂βzh(∂z)βz-qvt1-αΓ(2-α);0<α,βx,βy,βz<1;x‾=x1,x2,x3.
In this governing equation of transient confined groundwater flow in
fractional time and space, the saturated hydraulic conductivities are augmented
by fractional powers of time, t1-α, and of space,
xi1-βxi, i= 1, 2, 3, in terms of the ratios of fractional
time to fractional space, t1-αxi1-βxi,
i= 1, 2, 3, in multiple dimensions. As such the confined groundwater
diffusion in fractional time and space is modulated by the ratios of
fractional time to fractional space above. Accordingly, since the diffusion
coefficient scales with a fractional power of time and a fractional power of
space, the process represented by Eq. (32) may be thought to be
non-Fickian. One can also see from Fig. 3 on the numerical application
of the fractional confined groundwater flow equation to a simple
one-dimensional case, as the fractional powers of the derivatives in space
and time in the governing equation decrease from unity, the recession rate
of the nondimensional hydraulic heads from the initial condition also becomes
slower with respect to the case of the conventional governing equation with
integer derivative powers. Therefore, the speed of the response of the
groundwater system to the external forcings to the system (pumping rates,
recharge rates, etc.) can be modulated in the fractional governing Eq. (26)
of confined aquifer groundwater flow by means of the values that the
fractional derivative power α takes, slowing down with the decrease
in the values of α.
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.
Conclusion
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.
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