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International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963

NUMERICAL SOLUTION OF ONE-DIMENSIONAL
GROUNDWATER RECHARGE THROUGH POROUS MEDIA
WITH PARABOLIC PERMEABILITY USING VARIATIONAL
ITERATION METHOD
Tailor Kruti S.1Rathva G.A.1, Bhathawala P.H.2
1

Department of Mathematics, Government Engineering College, Valsad, Gujarat, India
2
Professor &amp; Former Head, Dept. of Mathematics, VNSGU, Surat, Gujarat, India

ABSTRACT
The present paper deals with the approximate solution of one dimensional ground water recharge problem
through porous media with parabolic permeability. The phenomenon is formulated using Variational Iteration
Method. . This method is based on Lagrange multipliers for identification of optimal values of parameters in a
functional. Using this method creates a sequence which tends to the exact solution of the problem. The
Variational Iteration Method (VIM) has been shown to solve effectively, easily and accurately a large class of
linear problem with approximations converging rapidly to exact solutions. The solution of nonlinear partial
differential equation is in the term of ascending series and it is obtained by using Mathematica.

KEYWORDS:

Variational Iteration method, Lagrange multiplier, Taylor’s series, Partial Differential

equation.

I.

INTRODUCTION

A very large fraction of the water falling as rain on the land surfaces of the sort moves through
unsaturated soil during the subsequent processes of infiltration, drainage, evaporation and absorption
of soil water by plant roots. Hydrologists have tended, nevertheless, to pay relatively little attention to
the phenomenon of water movement in unsaturated soils. Most research on this topic has been done
by soil physics concerned ultimately with agronomic or ecological aspects of hydrology.
The unsteady and unsaturated flow of water through soils is due to content changes as a function of
time and entire pore spaces are not completely filled with flowing liquid respectively. Knowledge
concerning such flows some helps some workers like hydrologist, agriculturalists, many fields of
science and engineering. The water infiltrations system and the underground disposal of seepage and
waste water are encountered by these flows, which are described by nonlinear partial differential
equation.
The mathematical model conforms to the hydrological situation of one dimensional vertical ground
water recharge by Spreading . Such flows are of great importance in water resources science, soil
engineering and agricultural sciences.
Many researchers have discussed this phenomena from different aspects, for example, Klute  and
Hank Bower  employ a finite difference method; Philips  uses a transformation of variable
technique; Mehta  discussed multiple scale method; Verma [1,6] has obtained Laplace
transformation and similarity solution and Sharma  discusses a varistional approach. The
experimental investigations has been discussed by Bruce and Klute  ; Gardner and Mayhugh 
;Nielson and Bigger  ; Rawlins and Garener , Terwilliger  ; Van Vorts  and Rahme
 have described the phenomena of gravity drainage of liquids through porous media and
supported their theoretical investigation by experimental results.
In present research paper one ground water recharge problem with parabolic permeability is solved by
Variational Iteration Method. He (1999, 2000, 2006) developed the variational iteration method for

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International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
solving linear, nonlinear and boundary value problems. The method was first considered by Inokuti,
Sekine and Mura (1978) and fully explored by He. J. H. In this method, the solution is given in an
infinite series usually converging to an accurate solution. Olayiwolaetal (2009) used modified power
series method for the solution of systems of differential equations. It is observed that the method solve
effectively, easily and accurately a class of linear, nonlinear, ordinary differential equations with
approximate solution which converge very rapidly to accurate solution. Recently introduced
variational iteration method by He [16,17-19], which gives rapidly convergent successive
approximations of the exact solution if such a solution exists, has proved successful in deriving
analytical solutions of linear and nonlinear differential equations. This method is preferable over
numerical methods as it is free from rounding off errors and neither requires large computer
power/memory. He [16, 18] has applied this method for obtaining analytical solutions of autonomous
ordinary differential equation, nonlinear partial differential equations with variable coefficients and
integro-differential equations. The variational iteration method was successfully applied to seventh
order Sawada-Kotera equations , to Schruodinger-KdV, generalized KdV and shallow water
equations , to linear Helmholtz partial differential equation . Linear and nonlinear wave
equations, KdV, K(2,2), Burgers, and cubic Boussinesq equations have been solved by Wazwaz
[21,22] using the variational iteration method.
The present problem is solved by using MATHEMATICA and then numerical and graphical
presentations are given.

II.

VARIATIONAL ITERATION METHOD

According to the variational iteration method, we consider the following differential equation
Lu + Nu = g(x,t)
(2.1)
t
un+1 (x, t) = un (x, t) + ∫0 λ(ξ)(Lun (ξ) + Nǔ(ξ)
− g(x, ξ))dξ
n

(2.2)

Where λ is a general Lagrangian multiplier, which can be identified optimally via the variational
theory, the second term on the right is called the correction and ǔn is considered as a restricted
̌n (ξ) = 0. So, we first determine the Lagrange multiplier λ that will be identified
variation, i.e.,δu
optimally via integration by parts. The successive approximationsun+1 (x, t), n ≥ 0 of the solution will
be readily obtained upon using the obtained Lagrange multiplier and by using any selective function
u0 consequently, the solution,
u(x,t)=lim un (x, t)
(2.3)
n→∞

III.

STATEMENT OF THE PROBLEM

In the investigated mathematical model, we consider that the groundwater recharge takes place over a
large basin of such geological location that sides are limited by rigid boundaries and the bottom by a
thick layer of water table. In this case, take flow is assumed vertically downloads through unsaturated
porous media. It is assumed that diffusivity coefficient is equivalent to its average value over the
whole range of moisture content, and the permeability of the media is continuous linear function of
the problem yields a nonlinear partial differential equation for the moisture content.

IV.

MATHEMATICAL FORMULATION OF THE PROBLEM

Following Klute,
We may write fundamental equation as below. The equation of continuity for unsaturated medium is
given by

(ρs θ)
∂t

= −∇M

(4.1)

Where ρs is the bulk density of the medium, θ is its moisture content on a dry weight basis, and

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International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
M is the mass flux of moisture.
From Darcy’s law for the motion of water in a porous medium we get,
V = −k∇∅

(4.2)

Where ∇∅ represents the gradient of whole moisture potential, V the volume flux of moisture
potential, and k the coefficient of aqueous conductivity. Combining equation (4.1) &amp; (4.2) we obtain,

(ρs θ)
∂t

= −∇(ρk∇∅)

(4.3)

Where ρ is the fluid density. Since in the present case we consider that the flow takes place only in the
vertical direction, equation (4.3) reduces to,
ρx

∂θ
∂t

=

∂ψ

(ρk ) −
∂z
∂z
∂z

(ρkg)

(4.4)

Where 𝜓 the capillary pressure potential, g is is the gravitational constant and ∅ = ψ − gz.
The positive direction of the z-axis is the same as that of the gravity.
Considering θ and ψ to be conducted by a single valued function, we may write (4.4) as,
∂θ
∂t
ρ

=

∂θ
ρ ∂k
(D ) − g
∂z
∂z
ρs ∂z

(4.5)

∂ψ

Where D= ρ k ∂θ and is called diffusivity coefficient.
s

Replacing D by its average value Da and assuming,k = k 0 θ, k 0 = 0.232, we have
∂θ
∂t

= Da

∂2 θ
ρ
∂θ
− k0
∂z2
ρs
∂z

(4.6)

Considering water table to be situated at a depth L, and putting:
z
tDa
ρ
∂θ
= ξ, 2 = T, β0 = k 0
L
L
ρs ∂z
One-dimensional Groundwater recharge through porous media with linear permeability is,
∂θ
∂T

=

∂2 θ

∂ξ2

∂θ

β0 ∂ξ

Where ξ = penetration depth (dimensionless)
T=time (dimensionless)
β0 =flow parameter (cm2)
∂θ

Set of appropriate boundary conditions are,θ(0, T) = θ0 , ∂ξ (1, T) = 0, θ(ξ, 0) = 0,
In above model, if we assumed, the permeability of the moisture content to have a parabolic
distribution, i.e. K=K i + K 0 θ2 (K 0 =0.232) where K i and K 0 are constants. Then equation (4.5)
becomes,
∂θ
∂t

806

∂2 θ

ρ

∂θ

= Dα ∂z2 − ρ K 0 θ ∂z
s

Vol. 6, Issue 2, pp. 804-811

International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
Considering the water table to be situated at a depth L, and put:
z
L

= ξ,

tDα
L2

ρ K0
s Dα

= T, β0 = ρ

We may write the boundary value problem as:
∂θ
∂T

=

∂2 θ

∂ξ2

∂θ

β0 θ ∂ξ

(4.7)

It may be mentioned for definiteness that a set of appropriate boundary conditions are
∂θ
(1, T)
∂ξ

θ(0, T) = θ0 ,

= 0, θ(ξ, 0) = 0

Where the moisture content throughout the region is zero initially, at the layer z=0 it isθ0 , and at the
water table (Z=L) it is assumed to remain 100% throughout the process of investigation. It may be
remarked that the effect of capillary action at the stationary groundwater level, being small is
neglected.
From (4.7) θT = θξξ − β0 θθξ

V.



SOLUTION OF THE PROBLEM USING VARIATIONAL ITERATION METHOD

Applying Variational iteration method to (4.7) we get,
t

θn+1 (ξ, T) = θ(ξ, T) + ∫ λ (
0

∂θn
∂θn ∂2 θn
+ β0 θ
− 2 ) dT
∂T
∂ξ
∂ξ

After simplification we get 𝜆 = −1
Take the initial condition as θ(ξ, 0) = θ0 = f(ξ)
Let θ0 =

eξ −1
e−1

Using Mathematica Solutions of (4.7) are,
θ1 =

eξ (−1+β0 +e−β0 eξ )T
eξ −1
+
(e−1)2
e−1

θ2 =

eξ (−1+β0 +e−β0 eξ )T
eξ −1
1
+
− 6(𝑒−1)4 (𝑒 𝜉 𝑡 2 (−3(1 +
(e−1)2
e−1

(5.1)
𝑒)3 + 2β0 3 𝑒 𝜉 (1 − 3𝑒 𝜉 + 2𝑒 2𝜉 )𝑇 +

2β0 (𝑒 − 1)2 (−3 + 𝑒 𝜉 (9 + 𝑇)) − β0 2 (−1 + 𝑒) (3 − 4𝑒 𝜉 (3 + 𝑇) + 𝑒 2𝜉 (9 + 6𝑇))))

θ3 =

eξ (−1+β0 +e−β0 eξ )T
eξ −1
1
+
− 6(𝑒−1)4 (𝑒 𝜉 𝑡 2 (−3(1 +
(e−1)2
e−1

(5.2)

𝑒)3 + 2β0 3 𝑒 𝜉 (1 − 3𝑒 𝜉 + 2𝑒 2𝜉 )𝑇 +

2β0 (𝑒 − 1)2 (−3 + 𝑒 𝜉 (9 + 𝑇)) − β0 2 (−1 + 𝑒) (3 − 4𝑒 𝜉 (3 + 𝑇) + 𝑒 2𝜉 (9 + 6𝑇)))) −
1
(𝑒 𝜉 𝑇 3 (−420(𝑒
2520(𝑒−1)8

− 1)7 + 40β0 7 𝑒 3𝜉 (2 − 15𝑒 𝜉 + 39𝑒 2𝜉 − 42𝑒 3𝜉 + 16𝑒 4𝜉 )𝑇 4 −

10β0 6 (𝑒 − 1)𝑒 2𝜉 𝑇 3 (21 + 42𝑒 4𝜉 (7 + 4𝑇) − 4𝑒 𝜉 (49 + 8𝑇) + 5𝑒 2𝜉 (119 + 36𝑇) − 6𝑒 3𝜉 (119 +
52𝑇)) + 42β0 (𝑒 − 1)6 (−30 + 𝑒 𝜉 (260 + 35𝑇 + 3𝑇 2 )) + 2β0 5 (𝑒 − 1)2 𝑒 𝜉 𝑇 2 (63 − 84𝑒 𝜉 (12 +

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International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
5𝑇) − 20𝑒 3𝜉 (294 + 350𝑇 + 45𝑇 2 ) + 4𝑒 2𝜉 (1029 + 805𝑇 + 60𝑇 2 ) + 3𝑒 4𝜉 (903 + 1470𝑇 +
260𝑇 2 )) − 21β0 2 (𝑒 − 1)5 (60 − 2𝑒 𝜉 (320 + 95𝑇 + 12𝑇 2 ) + 𝑒 2𝜉 (900 + 615𝑇 + 132𝑇 2 +
10𝑇 3 )) − β0 4 (𝑒 − 1)3 𝑒 𝜉 𝑇 (−42(25 + 12𝑇) + 63𝑒 𝜉 (115 + 108𝑇 + 20𝑇 2 ) − 4𝑒 2𝜉 (3255 +
5124𝑇 + 1750𝑇 2 + 80𝑇 3 ) + 5𝑒 3𝜉 (1365 + 3108𝑇 + 1610𝑇 2 + 120𝑇 3 )) + 2β0 3 (𝑒 −
1)4 (−210 + 21𝑒 𝜉 (100 + 85𝑇 + 18𝑇 2 ) − 210𝑒 2𝜉 (21 + 48𝑇 + 18𝑇 2 + 2𝑇 3 ) + 2𝑒 3𝜉 (1260 +
5145𝑇 + 3318𝑇 2 + 630𝑇 3 + 20𝑇 4 ))))

(5.3)

……………….

VI.

RESULTS

Following table shows the approximate solution for equation (5.3) for different values of ξ at time
T=0, T=0.01, T=0.02, T=0.03, T=0.04, T=0.05 seconds and for β0 = 0.05 using variational Iteration
method.
Table1: moisture content at different time
ξ

T=0

T=0.01

T=0.02

T=0.03

T=0.04

T=0.05

0

0

0

0

0

0

0

0.1

0.006121

0.006691

0.007253

0.007807

0.008353

0.008891

0.2

0.02577

0.026889

0.027991

0.029079

0.030153

0.031213

0.3

0.061083

0.062706

0.064308

0.065891

0.067455

0.069001

0.4

0.114492

0.11655

0.118584

0.120595

0.122584

0.12455

0.5

0.18877

0.191156

0.193516

0.195851

0.198162

0.200447

0.6

0.287072

0.28963

0.292159

0.294664

0.297145

0.299599

0.7

0.412986

0.415492

0.417971

0.420427

0.42286

0.425266

0.8

0.570589

0.572733

0.574854

0.576955

0.579037

0.581095

0.9

0.76451

0.765869

0.76721

0.76854

0.76986

0.77116

1

1

1

1

1

1

1

moisture content at different time
1.2
1
T=0

θ

0.8

T=0.1

0.6

T=0.2

0.4

T=0.3

0.2

T=0.4

0

T=0.5

0

0.2

0.4

0.6

0.8

1

1.2

ξ
Figure1: moisture content at different time solved by Variational Iteration Method

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Vol. 6, Issue 2, pp. 804-811

International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
moisture content at different time
1.2
1

θ

0.8

T=0.1
T=0.2

0.6

T=0.3
T=0.4

0.4

T=0.5
0.2

T=0.6

0

0

0.2

0.4

0.6

0.8

1

1.2

ξ
Figure 2: moisture content at different time solved by Adomian Decomposition Method

VII.

CONCLUDING REMARKS AND UTILITIES

A specific problem of one-dimensional flow in unsaturated porous media under certain assumptions is
discussed and its solution is obtained by Variational Iteration Method. The equations (5.3) represent
an approximate solution of moisture content in terms of 𝜉 and T. For the phenomenon of definiteness
in the analysis, we have assumed certain specific relationships, which are consistent with the physical
problem. The numerical has been obtained by using Mathematica of equation (5.3). The graphical
behaviour of solutions obtained by Variational Iteration Methods is compared with the behaviour of
graphical presentation which is obtained by Adomian Decomposition method and as a result the
behaviour is approximately same in both the methods.Figure1 and 2 presents the graphical behaviour
of the solutions obtained by variational Iteration Method and Adomian decomposition Method
respectively.
It is interpreted from the graph that as time increases, the moisture content also increases at each point
in the basin and after sometime, it become constant. Also, at particular time, optimum moisture
content rises with increase in length.
Notwithstanding the specific reference to the physical problem of vertical groundwater recharge the
present discussions hold equally true to other flow system in porous media, which confirm to the basic
specifications of the investigated case.

VIII.

UTILITIES

Because of the ground water recharge the salinity of the soil can be reduced because of the increase of
moisture content. due to the increase in moisture content the fertility of soil increases which helps the
farmer in growing up a qualitative crop and in this case production of the crop will also increase and
quality of the ground water also increase.

REFERENCES
. Verma, A. P., (1969) : The Laplace transform solution of a one dimensional groundwater recharge by
spreading Annali Di Geofision, 22, 1, p.25.
. Klute, A., (1952): A numerical method for solving the flow equation of water in unsaturated materials,
Soil sciences, 73, 2, p.105.

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International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
. Hanks, R.J., and Bowers, S.A. (1962) : Numerical solution of the moisture flow equation for
infiltration into the moisture flow equation for infiltration into layered soil, Soil Sci. Sec., Am.Prac.26,
pp.530-35.
. Phillips, R., (1970): Advances in hydrosciences, Ed.Ven Fe Chow, Acad. P. Vol 6, New York.
. Mehta, M.N., (1976): A multiple scale solution of 1-D flow in unsaturated porous media, Indian J.
Engg. Maths
. Verma, A.O., and Mishra, S.K., (1973): A similarity solution of unidimensional vertical groundwater
recharge through porous media, Rev. Roum. Scien. Techn. Ser. Mec. Appl.18, 2, p.345.
. Sharma, S.V.K., (1965) : Problem of partially saturated unsteady state of flow through porous media
with special reference to groundwater recharge by spreading., J. Sci., Engg., Research, 9, 69.
. Bruce, R.R. and Klute, A., (1956): The measurement of soil moisture diffusivity, Soil Sci., Sec., Amer.
Prac. 20, pp. 458-62.
. Gardner, W.H., and Mayhugh, M.S., (1958): Solution and tests o the diffusion equation for the
movement of water in soil, Soil Sci., Sec. Amer. Prac. 22, pp.197-201.
. Rawlins, S.L., and Gardner, W.H., (1963): The test of the validity of the diffusion equation for
unsaturated flow of soil water, Soil, Sci., Amer. Prac. 26, pp.107-112.
. Rawlins, S.L., and Gardner, W.H., (1963): The test of the validity of the diffusion equation for
unsaturated flow of soil water, Soil, Sci., Amer. Prac. 25, pp.507-10.
. Terwilliger, R. D., (1952): An experimental and theoretical investigation of gravity drainage
performance, Trans. AIME, p.195.
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V.C. les Angelos.
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(2006) 1141-1199.
. H. Jafari, et al., Application of He’s Variational Iteration Method for Solving Seventh Order SawadaKotera Equations, Applied Mathematical Sciences, 2 (10), 2008, 471 - 477.
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cubic Boussinesq equations, J. Comput. Appl. Math., 207 (1)(2007) 18-23.
. Tailor Kruti S. Patel K.B., Bhatahwala P.H., Numerical Solution of One dimensional ground water
recharge problem using with parabolic permeability fractional calculus, IJAET/Vol.II/ Issue
IV/October-December, 2011/157-160.

AUTHORS
TailorKruti S. is working as an Assistant Professor at Government Engineering College,
Valsad, Gujarat, since 2years. She completed her Ph. D. under the guidance of Dr. P. H.
Bhathawala and co guidance of Dr. Kaushal B. Patel. Her subjects of interest are Biomedical
Mathematics, Fluid Dynamics &amp; Numerical Analysis.

Rathva G.A. is working as an Assistant Professor at Government Engineering College,
Valsad, Gujarat, since 4 years. He is pursuing Ph. D. under the guidance of Dr. P. H.
Bhathawala. His subject of interest is Fluid Dynamics.

810

Vol. 6, Issue 2, pp. 804-811

International Journal of Advances in Engineering &amp; Technology, May 2013.
ISSN: 2231-1963
Pravin H. Bhathawala is a Retired Professor and Head of the Department of Mathematics,
Veer Narmad South Gujarat University, Surat, Gujarat, India. He has served the department for
32 years. He has qualified about 23 Ph. D. and 18 M. Phil. students under his guidance. His
area of interest is Biomedical Mathematics and Fluid dynamics of porous media.

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