Research Article Analytical Solutions of a Space-Time...

Research ArticleAnalytical Solutions of a Space-Time FractionalDerivative of Groundwater Flow Equation

Abdon Atangana and P. D. Vermeulen

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,P.O. Box 399, Bloemfontein, South Africa

Correspondence should be addressed to Abdon Atangana; [email protected]

Received 5 September 2013; Accepted 12 September 2013; Published 21 January 2014

Academic Editor: Hossein Jafari

Copyright © 2014 A. Atangana and P. D. Vermeulen. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometrichead. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution tothe generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole.This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwaterstudy on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of theborehole at the constant discharge rate 𝑄 and monitoring the piezometric head for 350 minutes. Numerical solutions obtainedvia Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow isassumed. Proposition for uncertainties in groundwater studies was given.

1. Introduction

Thereal problem encounter in groundwater studies up to nowis the real shape of the geological formation in which waterflows in the aquifer under investigation. However, there aremany fractured rock aquifers where the flow of groundwaterdoes not fit conventional geometries [1], for example, in SouthAfrica, the Karoo aquifers, characterized by the presence ofa very few bedding parallel fractures that serve as the mainconduits of water in the aquifers [2]. With a challenge to fitthe solution of the groundwater flow equation with experi-mental data fromfield observation in particular, the observeddrawdown see [3], for all time yields a fit that undervaluesthe observed drawdown at early times and overvalues it atlater times. The variation of observations from theoreticallypredictable values is usually an indication of uncertainties inthe predictable. To investigate the first possibility Botha etal. [2] developed a three-dimensional model for the Karooaquifer on the campus of the University of the Free State.

This model is based on the conventional, saturated ground-water flow equation for density-independent flow:

𝑆0(𝑥, 𝑡) 𝜕

𝑡Φ(𝑥, 𝑡) = ∇ ⋅ [𝐾 (𝑥, 𝑡) ∇Φ (𝑥, 𝑡)] + 𝑓 (𝑥, 𝑡) , (1)

where 𝑆0is the specific storativity,𝐾 the hydraulic conductiv-

ity tensor of the aquifer,Φ(𝑥, 𝑡) the piezometric head, 𝑓(𝑥, 𝑡)the strength of any sources or sinks, with 𝑥 and 𝑡 the usualspatial and time coordinates; ∇ the gradient operator, and 𝜕

𝑡

the time derivative.This model showed that the dominant flow field in these

aquifers is vertical and linear and not horizontal and radialas commonly assumed. However, more recent investigations[3] suggest that the flow is also influenced by the geometryof the bedding parallel fractures, a feature that (1) cannotaccount for. It is therefore possible that the equation may notbe applicable to flow in these fractured aquifers.

In an attempt to circumvent this problem, Barker [4]introduced a model in which the geometry of the aquifer

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 381753, 11 pageshttp://dx.doi.org/10.1155/2014/381753

2 Abstract and Applied Analysis

is regarded as a fractal. Although this model has been appliedwith reasonable success in the analysis of hydraulic tests fromboreholes in Karoo aquifers, it introduces parameters forwhich no sound definition exists in the case of nonintegerflow dimensions.

As a review of the derivation of (1) will show, [5] Darcylaw

𝑞 (𝑥, 𝑡) = −𝐾∇Φ (𝑥, 𝑡) (2)

is used as a keystone in the derivation of (1). This law,proposed by Darcy early in the 19th century, is relyingon experimental results obtained from the flow of waterthrough a one-dimensional sand column, the geometry ofwhich differs completely from that of a fracture [6]. There istherefore a possibility that the Darcy law may not be validfor flow in fractured rock formations but is only a verycrude idealization of the reality [6]. Nevertheless, the relativesuccess achieved by Botha et al. [2], to describe many ofthe properties of Karoo aquifers, suggests also that the basicprinciple underlying this law may be correct: the observeddrawdown is to be related to either a variation in the hydraulicconductivity of the aquifer or a change in the piezometrichead. Any new form of the law should therefore be reducedto the classical form under the more common conditions.Because 𝐾 is essentially determined by the permeability ofthe rocks, and not the flow pattern, the gradient term in (2) isthe most likely cause for the deviation between the observedand theoretical drawdown observed in the Karoo formations[6]. In the same direction, Cloot and Botha introduced theconcept of non-integer fractional derivative to investigate aradially symmetric form of (1), by replacing the classical firstorder derivative of the piezometric head by a complementaryfractional derivative [6]. However the generalized modelfor groundwater flow equation was solved numerically. Inthis work, more general form of groundwater flow equationwill be introduced; the Frobenius and Adomian decompo-sition will be used to give an asymptotic solution of thegeneralized model for groundwater flow equation. Becausethe concepts of fractional (or non-integer) order derivativesand complementary fractional order derivatives may notbe widely known, both concepts are first briefly discussedbelow.

2. Fractional Order Derivatives

On one hand, the concept of fractional calculus is popularlybelieved to have stemmed from a question raised in theyear 1695 by Marquis de L Hospital (1661–1704) to GottfriedWilhelm Leibniz (1646–1716), which sought the meaning ofLeibniz’s currently popular notation 𝑑

𝑛𝑦/𝑑𝑥𝑛 for derivative

of order 𝑛 ∈ N0

:= {0, 1, 2, . . .} when 𝑛 = 1/2 (what if𝑛 = 1/2). In his reply, dated September 30, 1695, Leibnizwrote to L’Hospital as follows: “This is an apparent paradoxfrom which, one day, useful consequences will drawn. . ..” Onthe other hand, the concept of fractional order derivatives for

a function, say 𝑓(𝑥), is based on a generalization of the Abelintegral [7]:

𝐷−𝑛𝑓 (𝑥) = ∭𝑓(𝑥) 𝑑𝑥

𝑛=

1

Γ (𝑛)∫

𝑥

0

(𝑥 − 𝑡)𝑛−1

𝑓 (𝑡) 𝑑𝑡,

(3)

where 𝑛 is a nonzero positive integer [8–12] and Γ(𝑛) theGamma function [13]. This represents an integral of order𝑛 for the continuous function 𝑓(𝑥), whenever 𝑓 and all itsderivatives vanish at the origin, 𝑥 = 0. This result can beextended to the concept of an integral of arbitrary order 𝑐,defined as:

𝐷−𝑐𝑓 (𝑥) = 𝐷

−𝑗−𝑠𝑓 (𝑥) =

1

Γ (𝑐)∫

𝑥

0

(𝑥 − 𝑡)𝑐−1

𝑓 (𝑡) 𝑑𝑡, (4)

where 𝑐 is a positive real number and 𝑗 an integer such that0 < 𝑠 ≤ 1.

Let 𝜌 now be the least positive integer larger than 𝛼 suchthat 𝛼 = 𝑚 − 𝜌; 0 < 𝜌 ≤ 1. Equation (4) can then be used todefine the derivative of (positive) fractional order, say 𝛼, of afunction 𝑓(𝑥) as

𝐷𝑐𝑓 (𝑥) = 𝐷

𝑝−𝜌𝑓 (𝑥) =

1

Γ (𝜌)∫

𝑥

0

(𝑥 − 𝑡)𝜌−1 𝑑𝑝𝑓 (𝑡)

𝑑𝑡𝑝𝑑𝑡. (5)

Note that these results, like Abel’s integral, are only validsubject to the condition that 𝑓(𝑘)(𝑥) | 𝑥 = 0 = 0 for𝑘 = 0, 1, 2, . . . , 𝑝 [8–11].

2.1. Properties. Properties of the operator can be found in [14,15], wemention only the following: for𝑓 ∈ 𝐶

𝜇, 𝜇 ≥ −1, 𝛼, 𝛽 ≥

0 and 𝛾 > −1:

𝐷−𝛼𝐷−𝛽𝑓 (𝑥) = 𝐷

−𝛼−𝛽𝑓 (𝑥) ,

𝐷−𝛼𝐷−𝛽𝑓 (𝑥) = 𝐷

−𝛽𝐷−𝛼𝑓 (𝑥) ,

𝐷−𝛼𝑥𝛾

=Γ (𝛾 + 1)

Γ (𝛼 + 𝛾 + 1)𝑥𝛼+𝛾

.

(6)

3. A Generalized Mathematical GroundwaterFlow Model

For the sake of clarity the generalization of the classicalmodelfor groundwater flow in the case of density-independent flowin the uniform homogeneous aquifer is considered in thispaper [4]. Consider the following groundwater flow equation

𝑆0𝜕𝑡Φ (𝑟, 𝑡) =

𝐾

𝑟𝑛−1𝜕𝑟[𝑟𝑛−1

𝜕𝑟Φ (𝑟, 𝑡)] , (7)

where both the specific storability, 𝑆0, and hydraulic conduc-

tivity, 𝐾, are scalar and constant quantities and 𝑛 = 1, 2 or 3is the radial dimension. To be complete, the following set ofinitial and boundary conditions is added:

Φ (𝑟, 0) = 𝑎𝑟, lim𝑟→∞

Φ (𝑟, 𝑡) = 𝑎𝑟,

𝑄 =2𝜋𝑛/2

Γ (𝑛/2)𝑟𝑛−1

𝑏𝐾𝑑3−𝑛

𝜕𝑟Φ(𝑟𝑏, 𝑡) .

(8)

Abstract and Applied Analysis 3

Here Φ(𝑟, 0) = 𝑎𝑟 means that before a pumping test beginsthe level of water or the initial hydraulic head in the aquifer islinear function of space with a positive gradient 𝑎 to be found,lim𝑟→∞

Φ(𝑟, 𝑡) = 𝑎𝑟means that during the pumping test thelevel of water is not affected for a very long distance from theborehole, and 𝑄 = (2𝜋

𝑛/2/Γ(𝑛/2))𝑟

𝑛−1

𝑏𝐾𝑑3−𝑛

𝜕𝑟Φ(𝑟𝑏, 𝑡)means

that the rate of pumping is proportional to the hydraulicconductivity.

Hence𝑄 is the discharge rate of the borehole, with radius𝑟𝑏and 𝑑 the thickness of the aquifer from which the borehole

taps. In order to include explicitly the possible effect of thegeometry into mathematical model the radial component ofthe gradient of the piezometric head, 𝜕

𝑟Φ(𝑟, 𝑡) is replaced

by the Weyl-fractional derivatives of order 𝛼 = 𝑚 − 𝜌; thefractional derivative in this paper is Caputo derivative and isdefined as follows:

𝜕𝛼

𝑟Φ (𝑟, 𝑡) =

(−1)𝑚+1

Γ (𝜌)∫

∞

𝑟

(𝑠 − 𝑟)𝜌−1

𝜕𝑚

𝑠Φ (𝑠, 𝑡) 𝑑𝑠. (9)

This provides a generalized form of the classical equationgoverning the flow equation (1):

𝑆0𝜕1

𝑡Φ (𝑟, 𝑡) =

𝐾

𝑟𝑛−1𝜕1

𝑟[𝑟𝑛−1

𝜕𝛼

𝑟Φ (𝑟, 𝑡)] . (10)

The integrodifferential equation does contain the additionalparameter𝛼 = 𝑚−𝜌, 0 < 𝜌 ≤ 1, which can be viewed as a newphysical parameter that characterizes the flow through thegeological formations [6].The same transformation generatesalso a more general form for the boundary condition at theborehole [6]:

𝑄 = (−1)𝑚+1 2𝜋

𝑛/2

Γ (𝑛/2) Γ (𝜌)𝑟𝑛−1

𝑏𝐾𝑑3−𝑛+𝛼

𝜕𝛼

𝑟Φ(𝑟𝑏, 𝑡) . (11)

The relations (10) and (11), together with the initial conditiondescribed in (10), represent a complete set of equations forwhich a solution exists. The integro-differential character ofthe relations makes the search for analytical solution verydifficult however. Nevertheless, in this paper we make use ofFrobenius and Adomian decomposition methods to give anasymptotic solution.

4. Solution of the Generalized Flow Equation

4.1. FrobeniusMethod. In this work to perform the Frobeniusmethod, we consider the groundwater flow governed by thefollowing fractional Caputo-Weyl derivation partial differen-tial of order 2𝛼, where 𝛼 is real number, 0 < 𝛼 ≤ 1. Also,we consider the dimension of the flow to be 2. Therefore,for 𝑛 = 2 (10) can then be transformed into the followingequation:

𝜕2𝛼

𝑟Φ (𝑟, 𝑡) +

1

𝑟𝜕𝛼

𝑟Φ (𝑟, 𝑡) −

𝑆0

𝐾𝜕𝑡Φ (𝑟, 𝑡) = 0. (12)

Applying the Laplace operator on both sides of the aboveequation, we have the following ordinary differential equa-tion

𝜕2𝛼

𝑟Φ (𝑟, 𝑠) +

1

𝑟𝜕𝛼

𝑟Φ (𝑟, 𝑠) −

𝑆0

𝐾𝑠Φ (𝑟, 𝑡) +

𝑆0

𝐾Φ (𝑟, 0) = 0,

(13)

where 𝑠 is the variable of Laplace. In this matter we chooseΦ(𝑟, 0) = 0 or we choose 𝑎 = 0, meaning that the level ofwater is the same everywhere in the aquifer if the water is nottaken out from the aquifer. If we let

Φ (𝑟, 𝑠) = Ψ (𝑟) (14)

then (13) becomes

𝜕2𝛼

𝑟Ψ (𝑟) +

1

𝑟𝜕𝛼

𝑟Ψ (𝑟) −

𝑆0

𝐾𝑠Ψ (𝑟) = 0. (15)

We put 𝑝(𝑟) = 1/𝑟 and 𝑞(𝑟) = −(𝑆0/𝐾)𝑠. To meet the

condition under which Frobenius method can be used, wehave to prove that 𝑝(𝑟) and 𝑞(𝑟) are analytical around 𝑟

𝑏that

means we have to prove that 𝑝(𝑟) and 𝑞(𝑟) can be writtenas series. It is very obvious to see that 𝑝(𝑟) and 𝑞(𝑟) can beexpressed as follows:

𝑝 (𝑟) =

∞

∑𝑛=0

𝑝𝑛(𝑟 − 𝑟𝑏)𝑛𝛼, 𝑞 (𝑟) =

∞

∑𝑛=0

𝑞𝑛(𝑟 − 𝑟𝑏)𝑛𝛼. (16)

We start here with the coefficients of 𝑝(𝑟):

𝑝 (𝑟) =

∞

∑𝑛=0

𝑝𝑛(𝑟 − 𝑟𝑏)𝑛𝛼 (17)

implying that 1 = ∑∞

𝑛=0𝑝𝑛(𝑟 − 𝑟𝑏)𝑛𝛼𝑟.

Putting 𝑅 = 𝑟 − 𝑟𝑏we have that 1 = ∑

∞

𝑛=0𝑝𝑛𝑅𝑛𝛼(𝑅 + 𝑟

𝑏)

and equating the coefficients of same power we obtain thefollowing set of equations:

𝑝0𝑟𝑏= 1,

𝑝1/𝛼

𝑟𝑏+ 𝑝0= 0,

𝑝(𝑛+1)1/𝛼

𝑟𝑏+ 𝑝𝑛/𝛼

= 0.

(18)

Therefore, the coefficients can be given with the generalfollowing recursive formula:

𝑝𝑛={

{

{

(−1)𝑘𝑟1/(𝑘+1)

𝑏for 𝑛 = 𝑘

1

𝛼0 otherwise.

(19)

And obviously the coefficients 𝑞𝑛are given below as

𝑞𝑛={

{

{

−𝑆0

𝐾𝑠 for 𝑛 = 0

0 for 𝑛 ≥ 1.

(20)

From the above expression we can see that 𝑝(𝑟) and 𝑞(𝑟) areanalytical around 𝑟

𝑏, which follows from Frobenius method

that the solution of (15) can be in the form

Ψ (𝑟) =

∞

∑𝑛=0

Ψ𝑛(𝑟 − 𝑟𝑏)𝑛𝛼+𝛼−1

. (21)


This solution is not convergent for a large 𝑟; that is, thesolutionwill diverge if we observe the drawdown at a positionvery far from the borehole from which the water is takenout. Therefore, we restrict our solution in the vicinity of theborehole; more precisely we investigate the solution when|𝑟 − 𝑟𝑏| < 1. That means we pump water from the borehole

and we observe the drawdown in the vicinity of the borehole.Thus substituting (21), 𝑝(𝑟), and 𝑞(𝑟) into (15) and

equating the coefficients of the same power, we obtain thefollowing recursive formula for which the coefficientsΨ

𝑛(𝑛 ≥

0), coefficients of our series:

Ψ2𝑛+2 (𝑟, 𝑠) =

(−1)𝑛(−𝑆0𝑠/𝐾)𝑛Γ (𝛼)Ψ

1/𝛼

Γ [(2𝑛 + 3) 𝛼]. (22)

Here we have the following:

𝜕𝛼

𝑟Ψ (𝑟) =

∞

∑𝑛=0

Ψ𝑛

Γ (𝑛𝛼 + 𝛼)

Γ (𝑛𝛼)(𝑟 − 𝑟𝑏)𝑛𝛼−1

,

lim𝑟→𝑟𝑏

𝜕𝛼

𝑟Ψ (𝑟) = Γ (𝛼 + 1)Ψ

1/𝛼.

(23)

However, making use of the boundary condition in (11), wehave the following:

Ψ1/𝛼

=𝑄

2𝜋𝑟𝑏𝐾𝑆0𝑠𝑑Γ (𝛼 + 1)

. (24)

Then the general solution of the fractional Caputo-Weylderivation of groundwater flow of order 2𝛼 in Laplace spaceis given below as:

Φ (𝑟, 𝑠) =

∞

∑𝑛=0

(−1)𝑛+1

(𝑆0𝑠)𝑛−1

𝑄

2𝜋𝑟𝑏𝑑𝐾𝑛Γ (𝛼 + 1)

(𝑟 − 𝑟𝑏)𝑛𝛼+𝛼−1

. (25)

To observe the behavior of the solution at the borehole, whichcorresponds to 𝑟 = 𝑟

𝑏, the series solution is reduced here to

the coefficient with order zero which is obtained when 𝑛 =

1/𝛼 − 1 and it’s given below as

Φ (𝑟, 𝑠) =(−1)1/𝛼

(𝑆0𝑠)1/𝛼−2

𝑄

2𝜋𝑟𝑏𝑑𝐾1/𝛼−2Γ (𝛼 + 1)

. (26)

In the following section the analytical asymptotic solutionobtained in Laplace space via Frobenius method will becompared with the experimental data.

4.2. Numerical Results. In order to examine the validationof this solution, the above asymptotic solution is comparedwith the experimental data from the pumping test performedby the Institute for Groundwater Studies on one of theirborehole settled on the campus test site of the Universityof the Free State. The test consisted of the pumping of theborehole at the constant discharge rate 𝑄 and monitoringthe piezometric head for 350 minutes. The first step in thesection is to discretize the range of Laplace transform since

the exact Laplace transform cannot be obtained in practice.This is done as follows:

L (experimental)

= ∫

𝑡𝑛

0

exp (−𝑠𝑡) Φmeasured data (𝑟, 𝑡) 𝑑𝑡

=

𝑛−1

∑

𝑖=0

∫

𝑡𝑖+1

𝑡𝑖

exp (−𝑠𝑡)Φ (𝑟, 𝑡) 𝑑𝑡.

(27)

For 𝑡𝑖≤ 𝑡 ≤ 𝑡

𝑖+1we approximate Φ(𝑟, 𝑡) = (Φ(𝑟, 𝑡

𝑖) +

Φ(𝑡𝑖+1

))/2 where Φ(𝑟, 𝑡𝑖) for 0 ≤ 𝑖 ≤ 𝑛; then the results

obtained in the fields and Laplace transform become

L (experimental)

=

𝑛−1

∑

𝑖=0

[(Φ (𝑟, 𝑡

𝑖) + Φ (𝑡

𝑖+1)) (exp (−𝑠𝑡

𝑖) − exp (−𝑠𝑡

𝑖+1))

2] .

(28)

Using this numerical scheme, the physical data was trans-formed into Laplace space. A comparison between thesevalues and asymptotic computed data can only be providedin the in real space not in Laplace space. Since it is not worthconcluding the validity of this solution in Laplace space,the inverse Laplace transform is applied in (26). To test thevalidity of this solution in real space and applying the inverseLaplace transform on (26), (29) is obtained

Φ (𝑟, 𝑡) =(−1)1/𝛼

(𝑆0)1/𝛼−2

𝑄𝑡1−1/𝛼

2𝜋𝑟𝑏𝑑𝐾1/𝛼−2Γ (𝛼 + 1) Γ (2 − 1/𝛼)

. (29)

The above solution is compared graphically to the experi-mental data from the pumping test performed by the institutefor groundwater study on one of their borehole settled on thetest site of theUniversity of the Free State.The small differenceobserved in the above graph (Figure 1) is due to uncertaintiesin measurement and this will be discussed in Section 6 ofthis work. The aquifer parameters used in this models arerecorded in Table 1, the observed data from field observationwill be attached to this paper. Although this solution is inagreement with the experimental data, there will be the needto investigate the case where the observation can be done fora long distance. In the next section another approach willbe introduced to solve the space-time-fractional derivative ofgroundwater flow equation, and this method is the Adomiandecomposition method.

5. Adomian Decomposition Method

5.1. Example 1. This section is concerned with the ground-water equation with time- and space-fractional derivatives ofthe form

𝑆0𝜕𝑡Φ (𝑟, 𝑡) =

𝐾

𝑟𝜕𝛼

𝑟Φ (𝑟, 𝑡) + 𝐾𝜕

𝜇

𝑟Φ (𝑟, 𝑡) ,

0 < 𝛼 ≤ 1 < 𝜇 ≤ 2.

(30)


Table 1: Aquifer parameters used in the model.

Parameters Values Units𝑄 83.10−3 (m3 s−1)𝑟𝑏

0.025 (m)𝐾 13.10−4 (m s−1)𝑑 1 (m)𝑆0

13.10−4 (m−1)Φ0

0 (m)2𝛼 = 1.46

Drawdo

wn

2.5

2.0

1.5

1.0

0.5

Time350300250200150100500

Figure 1: Comparison of experimental data from real world obser-vation with solution of fractional groundwater flow equation.

Subject to the initial and boundary conditions described in(8), the level of water is assumed to be the same throughoutthe aquifer before the pumping so that the gradient 𝑎

described in (8) is zero. Furthermore, it is assumed that afractional change in drawdown is constant for 𝑡 = 0meaning𝜕𝛼

𝑟Φ(𝑟, 0) = constant.The method used here is based on applying the operator

∫𝑡

0𝑑𝜏 on both sides of (30) to obtain

Φ (𝑟, 𝑡) = Φ (𝑟, 0) + ∫

𝑡

0

(𝜕𝜇

𝑟Φ (𝑟, 𝜏) +

1

𝑟𝜕𝛼

𝑟Φ (𝑟, 𝜏))

𝐾

𝑆0

𝑑𝜏.

(31)

TheAdomian decompositionmethod [16, 17] assumes a seriessolution for (31) to be

Φ (𝑟, 𝑡) =

∞

∑𝑛=0

Φ𝑛 (𝑟, 𝑡) , (32)

where the components Φ𝑛(𝑟, 𝑡) are determined recursively.

Substituting (32) into both sides of (30) gives∞

∑𝑛=0

Φ𝑛(𝑟, 𝑡)

= Φ (𝑟, 0)

+ ∫

𝑡

0

(𝜕𝜇

𝑟(

∞

∑𝑛=0

Φ𝑛 (𝑟, 𝜏)) +

1

𝑟𝜕𝛼

𝑟(

∞

∑𝑛=0

Φ𝑛 (𝑟, 𝜏)))

𝐾

𝑆0

𝑑𝜏.

(33)

Following the decomposition method, the recursive relationsare introduced as

Φ0(𝑟, 𝑡) = Φ (𝑟, 0) , (34)

𝑆0Φ𝑛+1

(𝑟, 𝑡) = ∫

𝑡

0

(𝐾𝜕𝜇

𝑟(Φ𝑛(𝑟, 𝜏)) +

𝐾

𝑟𝜕𝛼

𝑟(Φ𝑛(𝑟, 𝜏))) 𝑑𝜏.

(35)

It is worth noting that if the component Φ0(𝑟, 𝑡) is defined,

then the remaining components 𝑛 ≥ 1 can be completelydetermined such that each term is determined by using theprevious terms, and the series solutions are thus entirelydetermined. Finally, the solution Φ(𝑟, 𝑡) is approximated bythe truncated series

Φ𝑁 (𝑟, 𝑡) =

𝑁−1

∑𝑛=0

Φ𝑛 (𝑟, 𝑡) , lim

𝑁→∞

Φ𝑁 (𝑟, 𝑡) = Φ (𝑟, 𝑡) .

(36)

However, the inclusion of boundary conditions in fractionaldifferential equations introduces additional difficulties. TheAdomian decomposition method can handle these difficul-ties by using the space-fractional operator 𝜕𝛼

𝑟and the initial

conditions only. The method provides the solution in theform of a rapidly convergent series that may lead to the exactsolution in the case of integer derivatives (𝛼 = 1, 𝜇 = 2)

and to an efficient numerical solution with high accuracy for0 ≤ 𝛼 < 1. The convergence of the decomposition series hasbeen investigated in [18, 19].

Following the recursive formula equation (35) and usingthe fact that 𝜕𝛼

𝑟Φ(𝑟, 0) = constant = 𝑎, the equations below

are obtained:

Φ1(𝑟, 𝑡) =

1

𝑆0

∫

𝑡

0

(𝐾𝜕𝜇

𝑟(Φ0(𝑟, 𝜏)) +

𝐾

𝑟𝜕𝛼

𝑟(Φ0(𝑟, 𝜏))) 𝑑𝜏

=𝐾𝑎𝑡

𝑆0𝑟,

Φ2(𝑟, 𝑡) =

𝐾2𝑎𝑡2Γ (1 + 𝜇)

2𝑆20𝑟1+𝜇

+𝐾2𝑎𝑡2Γ (1 + 𝛼)

2𝑆20𝑟2+𝛼

,

Φ3 (𝑟, 𝑡) =

𝐾3𝑎𝑡3

6𝑆30

[Γ (1 + 2𝜇)

𝑟1+2𝜇+Γ (2𝛼 + 1)

𝛼𝑟2𝛼+2] .

(37)

The component Φ4(𝑟, 𝑡) was also determined and will be

used, but for brevity it is not listed. In this matter five com-ponents of the decomposition series (30) were obtained forwhichΦ(𝑟, 𝑡) was evaluated to have the following expansion:

Φ (𝑟, 𝑡) = Φ0 (𝑟, 𝑡) +

𝐾𝑎𝑡

𝑆0𝑟+𝐾2𝑎𝑡2Γ (1 + 𝜇)

2𝑆20𝑟1+𝜇

+𝐾2𝑎𝑡2Γ (1 + 𝛼)

2𝑆20𝑟2+𝛼

+𝐾3𝑎𝑡3

6𝑆30

[Γ (1 + 2𝜇)

𝑟1+2𝜇+Γ (2𝛼 + 1)

𝛼𝑟2𝛼+2] .

(38)


Applying the boundary condition yields

𝑎 =𝑄

2𝜋𝑟𝑏𝐾𝑑𝛼𝑓 (𝑟

𝑏, 𝑡)

, (39)

where

𝑓 (𝑟𝑏, 𝑡)

= 1 +𝐾𝑡Γ (1 + 𝛼)

𝑆0𝑟1+𝛼𝑏

+(𝐾𝑡)2Γ (1 + 𝜇 + 𝛼)

2𝑆20𝑟1+𝛼+𝜇

𝑏

+(𝐾𝑡)2Γ (2 + 2𝛼)

2𝑆20𝑟2+2𝛼𝑏

(1 + 𝛼)+(𝐾𝑡)3

6𝑆30

× [Γ (1 + 2𝜇 + 𝛼)

𝑟1+2𝜇+𝛼

0

+Γ (2 + 3𝛼)

𝑟2+3𝛼0

(𝛼 + 1) (2𝛼 + 1)] .

(40)

5.2. Example 2. Consider the groundwater flow equationwith time- and space-fractional derivatives of the form

𝐾𝜕𝛽

𝑟Φ (𝑟, 𝑡) +

𝐾

𝑟𝜕𝜇

𝑟Φ (𝑟, 𝑡) − 𝑆

0𝜕𝛼

𝑡Φ (𝑟, 𝑡) = 0, (41)

subject to the initial condition described in (11). Furthermorewe suppose that the gradient 𝑎 = 0, meaning that the waterlevel is everywhere in the aquifer at 𝜕𝜇

𝑟Φ(𝑟, 0) = constant

and the fractional change in drawdown is a constant andboundary condition:

𝑄 = 2𝜋𝐾𝑑𝜇𝑟𝑏𝜕𝜇

𝑟Φ(𝑟𝑏, 𝑡) . (42)

Following the discussion earlier, we have the below recursiveformula:

Φ0(𝑟, 𝑡) = Φ (𝑟, 0) ,

𝑆0Φ𝑛+1

(𝑟, 𝑡) = 𝐷−𝛼

𝑡[𝐾𝜕𝛽

𝑟Φ𝑛 (𝑟, 𝑡) +

𝐾

𝑟𝜕𝜇

𝑟Φ𝑛 (𝑟, 𝑡)] .

(43)

It follows from the recursive formula that

Φ1(𝑟, 𝑡) =

𝑎𝐾𝑡𝛼

Γ (𝛼 + 1) 𝑆0𝑟,

Φ2(𝑟, 𝑡) =

𝐾2𝑎𝑡2𝛼

𝑆20Γ (2𝛼 + 1)

[Γ (1 + 𝜇)

𝑟1+𝜇+Γ (1 + 𝛽)

𝑟1+𝛽] ,

Φ3(𝑟, 𝑡)

=𝐾3𝑎𝑡3𝛼Γ (2 + 𝜇 + 𝛽)

𝑆30Γ (3𝛼 + 1) (1 + 𝜇) 𝑟2+𝜇+𝛽

+𝐾3𝑎𝑡3𝛼Γ (2𝛽 + 1)

𝑆30Γ (3𝛼 + 1) 𝑟2𝛽+1

+𝐾3𝑎𝑡3𝛼Γ (2𝜇 + 2)

𝑆30𝑟2𝜇+3Γ (3𝛼 + 1) (1 + 𝜇)

+𝑡3𝛼𝐾3𝑎Γ (1 + 𝜇 + 𝛽)

𝑆20Γ (3𝛼 + 1) 𝑟2+𝜇+𝛽

.

(44)


used, but for brevity it is not listed. In this matter five compo-nents of the decomposition series (41) were obtained of whichΦ(𝑟, 𝑡) was evaluated to have the following expansion:

Φ (𝑟, 𝑡)

= Φ (𝑟, 0) +𝑎𝐾𝑡𝛼

Γ (𝛼 + 1) 𝑆0𝑟+

𝐾2𝑎𝑡2𝛼

𝑆20Γ (2𝛼 + 1)

× [Γ (1 + 𝜇)

𝑟1+𝜇+Γ (1 + 𝛽)

𝑟1+𝛽]

+𝐾3𝑎𝑡3𝛼Γ (2 + 𝜇 + 𝛽)

𝑆30Γ (3𝛼 + 1) (1 + 𝜇) 𝑟2+𝜇+𝛽

+𝐾3𝑎𝑡3𝛼Γ (2𝛽 + 1)

𝑆30Γ (3𝛼 + 1) 𝑟2𝛽+1

+𝐾3𝑎𝑡3𝛼Γ (2𝜇 + 2)

𝑆30𝑟2𝜇+3Γ (3𝛼 + 1) (1 + 𝜇)

+𝑡3𝛼𝐾3𝑎Γ (1 + 𝜇 + 𝛽)

𝑆30Γ (3𝛼 + 1) 𝑟2+𝜇+𝛽

.

(45)

Applying the boundary condition yields

𝑎 =𝑄

2𝜋𝑟𝑏𝐾𝑑𝛼𝑓 (𝑟

𝑏, 𝑡)

, (46)

where𝑓 (𝑟𝑏, 𝑡)

= 1 +𝐾Γ (1 + 𝜇) 𝑡

𝛼

Γ (1 + 𝛼) 𝑆0𝑟1+𝜇

𝑏

+𝐾2𝑡2𝛼

𝑆20Γ (2𝛼 + 1)

[𝜇Γ (1 + 2𝜇)

𝑟1+2𝜇

𝑏

+𝛽Γ (1 + 𝛽2)

𝑟1+2𝛽

𝑏

] + ⋅ ⋅ ⋅ .

(47)

5.3. Example 3. Consider time fractional derivative of thegroundwater flow equation with time-fractional derivativesof the form

𝐾𝜕2

𝑟Φ (𝑟, 𝑡) +

𝐾

𝑟𝜕𝑟Φ (𝑟, 𝑡) − 𝑆

0𝜕𝛼

𝑡Φ (𝑟, 𝑡) = 0. (48)

Subject to the initial and boundary conditions describedin (8) it is assumed that 𝑎 ̸= 0. Following the discussionpresented earlier, we obtained the below recursive formula:

𝑆0Φ𝑛+1

(𝑟, 𝑡) = 𝐷−𝛼

𝑡[𝐾

𝑟𝜕𝑟Φ𝑛(𝑟, 𝑡) + 𝐾𝜕

2

𝑟Φ𝑛(𝑟, 𝑡)] ,

Φ0 (𝑟, 𝑡) = Φ (𝑟, 0) ,

Φ1(𝑟, 𝑡) =

𝐾𝑎𝑡𝛼

𝑟𝑆0Γ (𝛼 + 1)

,

Φ2 (𝑟, 𝑡) =

𝐾2𝑎𝑡2𝛼

𝑆20Γ (2𝛼 + 1) 𝑟3

,

Φ3(𝑟, 𝑡) =

9𝑎𝐾3𝑡3𝛼

𝑟5𝑆3

0Γ (3𝛼 + 1)

.

(49)


Table 2: Theoretical values.

Parameters Values Units Parameters Values Units𝑄 10.8 (m3 s−1) 𝛼 0.005 None𝑟𝑏

0.15 (m) 𝛽 1.05 None𝐾 0.65 (m s−1) 𝜇 1.005 None𝑑 40 (m) 𝜆 0.012 None𝑆0

0.75 (m−1) 𝛼1

0.5 NoneΦ0

0 (m) 𝛼2

0.005 None𝜇1

0.025 None


used, but for brevity not listed. In thismatter five componentsof the decomposition series (31) were obtained of whichΦ(𝑟, 𝑡) was evaluated to have the following expansion

Φ (𝑟, 𝑡) = Φ (𝑟, 0) +𝐾𝑎𝑡𝛼

𝑟𝑆0Γ (𝛼 + 1)

+𝐾2𝑎𝑡2𝛼

𝑆20Γ (2𝛼 + 1) 𝑟3

+9𝑎𝐾3𝑡3𝛼

𝑟5𝑆3

0Γ (3𝛼 + 1)

.

(50)

Applying the boundary conditions yield to

𝑎 =𝑄

2𝜋𝑟𝑏𝐾𝑑𝑓 (𝑟

𝑏, 𝑡)

, (51)

where

𝑓 (𝑟𝑏, 𝑡) = 1 −

𝐾𝑡𝛼

𝑟20𝑆0Γ (1 + 𝛼)

+3𝐾𝑡2𝛼

𝑆20Γ (1 + 2𝛼) 𝑟4

𝑏

+ ⋅ ⋅ ⋅ . (52)

The normalized solutions with Φ(𝑟, 0) of (50), (51) and (52)are illustrated graphically in Figures 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12, and 13 for different values of various parameters;these solutions are compared to the solution proposed byBarker [4].These graphs show the behaviour of the drawdownduring the pumping test, first as a function of space and timefrom the borehole to the point of observation, secondly asfunction of time for a fixed distance from the borehole, andfinally as a function of space for a fixed time. Based on theassumption that groundwater in an aquifer flows through anequipotential surface that are projections of 𝑙-dimensionalspheres onto two-dimensional space, Barker [4] obtained thefollowing analytical solution for an infinite aquifer with a linesource:

Φ (𝑟, 𝑡) =𝑄 𝑟2(1−𝜆/2)

4 𝜋𝜆/2𝐾𝑑1−𝜆Γ [

𝜆

2− 1,

𝑟2𝑆0

4𝐾𝑡] , (53)

where Γ is the incomplete gamma function and 𝜆 thedimension of the flow which equals the special dimension, 𝑛being an integer, and the other quantities all have the samemeaning as before. Although this model has been appliedwith reasonable success in the analysis of hydraulic tests fromboreholes in Karoo aquifers, it introduces parameters forwhich no sound definition exists in the case of non-integerflow dimensions. The main raison of comparison of theseresults obtained via the Adomian decomposition methods

0.06

0.04

0.02

0.00

0

10

20

30

400

10

20

30

40

Figure 2: Solution of (30) (piezometric head).

with those of Barker’s fractal radial flow model is to establisha possible relationship between the fractional order of thederivative and the parameter fractal introduced earlier byBarker.

Table 2 shows the theoretical values of the discharge rateand aquifer parameters used in the numerical simulations.

6. Discussion and Propositions

Although the analytical solution obtained via Frobeniusmethod fit the experimental data or have described success-fully the events taking place in the vicinity of the borehole, onone hand, and the analytical solutions obtained via Adomiandecomposition method was successfully compared to thesolution proposed by Barker, on the other hand, the problemof choosing an appropriate geometry for the geologicalsystem in which the flow occurs still remains a challenge ingroundwater studies. We personally believe that we do notpropose solution to a problem because it going to be usefulfor the generation in which we are, but we propose solutionbecause we hope that theywill one day be useful formankind;therefore the following propositionmay not be useful for thisgeneration because wemay not have the adequate technologyto perform the steps involved but it will be in the future.


0.06

0.04

0.02

0.00

0

10

20

30

400

10

20

30

40

4

2

00

0

10

20

30

0

10

20

30

40


10

20

30

400

10

20

30

40

0.000

0.002

0.004

10

20

30

0

10

20

30

4

0

2

Figure 4: Barker’s solution.

We think that describing the groundwater flow with oneequation for the whole aquifer is unrealistic, because fromone point of the aquifer to another properties includinggeology and geometry change, therefore the flow. Assump-tion such as homogeneity, isotropy, uniform thickness overthe area under investigation and so on render the study ofgroundwater uncertain. In order to study the geology orthe geometry of an aquifer, we have to divide the aquiferin small portions, from north to south and from east towest. The geology and the geometry of each portion can

0

10

20

30

40 0

10

20

30

40

0.15

0.10

0.05

0.000

10

20

30

10

20

30

4

5

0


0

10

20

30

40

10

20

30

40

0.5

0.4

0.3

0

10

20

30

4

10

20

30

3

Figure 6: Solution of (30) (drawdown).

then be studied. If the results of the study reveal that theportion under investigation is for instance a hyperboloid,since there is no exact solution to groundwater flowmodel byhyperboloid flow and there are solutions to circular flow, thena suitable transformation can be done including transfor-mation of hyperboloid coordinates to Cartesian coordinates,then to Cartesian coordinates to cylindrical coordinate suchthat this new coordinate can now be put in the equation


10

20

30

40

0

10

20

30

40

0.5

0.4

0.3

10

20

30

0

10

20

30

4

3


0.06

0.04

0.02

0.00

0

10

20

30

40

0

10

20

30

40

2

0

0

10

20

30

4

0

10

20

30


describing groundwater flow, and the solution to the newequation can then be investigated. Henceforth knowing thereal geology and geometry of this portion, the real pathsflow will be known. Then having a good knowledge of eachsmall portion including its geometry and geology, the realgeometry and geology of the aquifer can be not exactly butmore accurately known.

For groundwater remediation it will be possible to knowwhere the maximum, minimum, and average chemical con-centrations are found in the aquifer. We have no proof forthis but we believe that this proposition will be useful inreducing uncertainties in groundwater study. It is believedthat the field test gives the characteristic of an aquifer, but

0

10

20

30

40

10

20

30

40

0.5

0.4

0.3

0

10

20

30

4

10

20

30

3


−0.15

−0.10

−0.05

r

10 20 30 40

Drawdo

wn

Figure 10: Solution of (30) as function of space (piezometric head).

we believe that the field test gives both uncertainties andcharacteristics of the aquifer; therefore quantify uncertaintiesin this measurement lead us to the real picture of aquifercharacteristics, henceforth we propose that the studies ingroundwater should focus on both uncertainties and fieldsobservations, because what is known is bounded by what isnot known; knowing what is not known give a real pictureto what was known, and it follows that the knowledge ofuncertainties in groundwater study will give a clear pictureof what we already know in groundwater.

7. Conclusion

The classical Darcy Law has been generalized using theconcept of complementary fractional order derivatives ofWeyl fractional derivative. This leads to the formulation of


−0.15

−0.10

−0.05

r

10 20 30 40

Drawdo

wn


−0.15

−0.10

−0.05

r

10 20 30 40

Drawdo

wn


a new generalized form of the groundwater flow equation [6].The applications of Adomian decomposition and Frobeniusmethods were extended to obtain explicit and numericalsolutions of the space-time fractional groundwater flow.The two methods were very clearly efficient and powerfultechniques in finding the solutions of the proposed equations.The solution obtained via Frobenius takes into account theevents taking place in the vicinity of the borehole duringthe pumping test whereas the solution obtained via Adomiandecomposition methods takes into account the events thattake place far from the point where water is pumped out,that is, in the borehole. The solution obtained via Frobeniusmethod was in perfect agreement with the observed dataobtained from the pumping test performed by the Institutefor Groundwater Studies on one of their borehole settled onthe campus test site of the University of the Free State. TheAdomian decompositionmethod requires less computational

−0.15

−0.10

−0.05

r

10 20 30 40

Drawdo

wn


work than existing approaches while supplying quantitativelyreliable results.The obtained results demonstrate the reliabil-ity of the algorithms and their wider applicability to fractionalevolution equations. A comparison of these results obtainedvia the Adomian decomposition methods with those ofBarker’s fractal radial flow model suggests that there exists arelation between the fractional order of the derivative and thenon-integral dimension of the flow.

Authors’ Contribution

Abdon Atangana wrote the first draft and P. D. Vermeulenread and revised it; the revised versionwas read and correctedby both authors.

Conflict of Interests

The authors declare that there is no conflict of interests forthis paper.

References

[1] J. H. Black, J. A. Barber, andD. J. Noy, “Crosshole investigations:themethod, theory and analysis of crosshole sinusoidal pressuretests in fissured rocks,” Stripa Projects Internal Reports 86-03,SKB, Stockholm, Sweden.

[2] J. F. Botha, I. Verwey van Voort, J. J. P. Viviers, W. P. Collinston,and J. C. Loock, “Karoo aquifers. Their geology, geometry andphysical behaviour,” WRC Report 487/1/98, Water ResearchCommission, Pretoria, 1998.

[3] G. J. van Tonder, J. F. Botha, W.-H. Chiang, H. Kunstmann,and Y. Xu, “Estimation of the sustainable yields of boreholes infractured rock formations,” Journal of Hydrology, vol. 241, no.1-2, pp. 70–90, 2001.

[4] J. A. Barker, “A generalized radial flowmodel for hydraulic testsin fractured rock,”Water Resources Research, vol. 24, no. 10, pp.1796–1804, 1988.


[5] J. Bear, Dynamics of Fluids in Porous Media, American ElsevierEnvironmental Science Series, Elsevier, New York, NY, USA,1972.

[6] A. Cloot and J. F. Botha, “A generalised groundwater flowequation using the concept of non-integer order derivatives,”Water SA, vol. 32, no. 1, pp. 1–7, 2006.

[7] R. Courant and F. John, Introduction to Calculus and Analysis,vol. 2, John Wiley & Sons, New York, NY, USA, 1974.

[8] Y. Cherruault and G. Adomian, “Decomposition methods:a new proof of convergence,” Mathematical and ComputerModelling, vol. 18, no. 12, pp. 103–106, 1993.

[9] I. Podlubny, Fractional Differential Equations, Academic Press,San Diego, Calif, USA, 1999.

[10] I. Podlubny, “Geometric and physical interpretation of frac-tional integration and fractional differentiation,” FractionalCalculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.

[11] K.Adolfsson, “Nonlinear fractional order viscoelasticity at largestrains,”NonlinearDynamics, vol. 38, no. 1–4, pp. 233–246, 2004.

[12] O. P. Agrawal, “Application of fractional derivatives in thermalanalysis of disk brakes,” Nonlinear Dynamics, vol. 38, pp. 191–206, 2004.

[13] G. Afken,Mathematical Methods for Physicists, Academic Press,London, UK, 1985.

[14] K. S. Miller and B. Ross, An Introduction to the FractionalCalculus and Fractional Differential Equations, John Wiley &Sons, New York, NY, USA, 1993.

[15] K. B. Oldham and J. Spanier,The Fractional Calculus, AcademicPress, New York, NY, USA, 1974.

[16] G. Adomian, “A review of the decomposition method inapplied mathematics,” Journal of Mathematical Analysis andApplications, vol. 135, no. 2, pp. 501–544, 1988.

[17] A. Atangana, “New class of boundary value problems,” Informa-tion Science Letters, vol. 1, no. 2, pp. 67–76, 2012.

[18] G. Adomian, Solving Frontier Problems of Physics:TheDecompo-sition Method, Kluwer Academic, Dodrecht, The Netherlands,1994.

[19] Y. Cherruault, “Convergence of Adomian’smethod,”Kybernetes,vol. 18, no. 2, pp. 31–38, 1989.

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