CHAPTER III COMPARATIVE STUDY OF ONE-DIMENSIONAL …shodhganga.inflibnet.ac.in/bitstream/10603/33030/11... · reservoirs. To date, many analytical models have been developed for simulating

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CHAPTER III

COMPARATIVE STUDY OF ONE-DIMENSIONAL SOLUTE TRANSPORT

MODEL WITH ZERO-ORDER PRODUCTION TERM FOR DIRICHLET AND

ROBIN TYPE BOUNDARY CONDITIONS

3.1 Introduction

Over the years, there has been considerable interest in the migration of solutes in

groundwater. Many of the current pressing problems in groundwater and of environmental

concern involve transport issues including groundwater contamination, seawater intrusion

in coastal aquifers, radioactive waste disposal, geothermal energy development,

groundwater-surface water interaction, and subsurface storage of materials and fluids. An

ever increasing pressure on the waste assimilating capacity of our water resources does

contamination of groundwater by the various types of waste disposal has become a concern

issue in the recent years.

The remediation of groundwater contamination usually requires a quantitative

knowledge about the distribution and fate of the concerned contaminants. This kind of

knowledge may be obtained by means of mathematical modeling which solves the

advection-dispersion equation either analytically or numerically or both. The advection-

dispersion equation is derived on the principle of conservation of mass and Fick’s law of

diffusion. If the medium is porous then it also satisfies Darcy’s law. This equation is widely

used to describe pollutant distribution behavior in aquifers, rivers, lakes, streams and oil

reservoirs. To date, many analytical models have been developed for simulating transport

problems under different initial and boundary conditions.

Banks and Jerasate (1962) studied the solution for linear and exponentially

decreasing time-dependent expressions for seepage velocity through porous media. van

Genuchten and Weirenga (1986) explored about five different techniques to determine the

dispersion coefficient and retardation factor from observed solute concentration distribution

in one-dimensional semi-infinite media for both laboratory and field displacement

experiments. Analytical solutions were developed by Yates (1990) for describing the

transport of dissolved substances in heterogeneous semi-infinite porous media with a

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distance dependent dispersion of exponential nature along the uniform flow. The

hydrodynamic dispersion in the porous medium was assumed to be constant.

Considering the scale dependent dispersivity, Serrano (1992) developed an

analytical solution for dispersion equation under recharge and variable velocity. The aquifer

was subjected to natural recharge from rainfall, exhibit groundwater velocities which vary

with distance and with the recharge intensity. Logan (1996) extended the work of Yates

(1990) by including the adsorption and decay effects. The boundary condition was taken as

periodic type and it was assumed that dispersion varies with distance and increases up to

some constant value. A specific form of the advection-dispersion equation with time and

scale dependent dispersivity was presented for solute transport modeling in saturated

heterogeneous porous media (Su et al., 2005). The solutions were derived for an

instantaneous point-source (Dirac-delta function) input, and for constant concentration and

constant flux boundary conditions in a semi-infinite domain. Considering temporally and

spatially dependent flow through horizontal semi-infinite media and solute dispersion

proportional to the square of the velocities, Jaiswal et al. (2011) presented a one-

dimensional analytical solution for the advection-diffusion equation. The analytical solution

was obtained by Laplace transform technique for uniform and varying pulse type input

concentration. These analytical solutions were obtained only for relatively simple problem

i.e. the geometry of the problem must be regular. However for irregular shape of the

boundaries, the spatial variability of the co-efficient appearing in the equations and in the

boundary conditions, the non-uniformity of the initial conditions and the non analytic form

of the various sources analytical solutions are virtually impossible. In those cases,

numerical approaches can be applicable to obtain the solution.

To find the numerical solution of the partial differential equation various methods

are available in the literature, for example Finite element method, Finite volume method,

Finite difference method etc. Among these, Finite difference method is a well-established

numerical method which has been applied in flow and transport modeling. The finite

difference method is the oldest numerical method to solve the partial differential equations.

In recent years, other powerful numerical methods have been developed but due to simple

basic theory, less computer memory and less computation time, the finite difference method

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is still preferred. The application of finite difference method on solute transport modeling

was studied by very few researchers. In general, standard finite-difference method is used

for approximation of the time derivative. Such an approach yield a scheme which is at best

second order correct in time. van Genuchten and Grey (1978) developed several higher

order approximations of the time derivative and analyzed using finite difference

approximation and Galerkin-type finite element approximations in conjunction with several

sets of basis function. Considering dispersion coefficient as directly proportional to the

seepage velocity, Perrochet and Berod (1993) had done the stability analysis of the classical

Crank-Nicolson-Galerkin scheme applied to the one-dimensional solute transport equation

on the basis of two fairly different approaches such as space time Eigen value analysis and

error amplification of the Crank-Nicolson-Galerkin scheme. Navarro et al. (2000)

developed a model of solute transport in overland flow in which water flow is computed

using the explicit two-step McCormack method. Based on the obtained velocity field,

solute transport was explicitly determined from the advection-diffusion equation using the

operator split technique. Kumar et al. (2005) presented a mathematical model for solute

concentration distribution along unsteady groundwater flow in inhomogeneous aquifer of

finite and infinite length. The input concentration was taken as uniform pulse type. The

numerical solution was obtained by finite difference method and compared with the

analytical solution obtained by Laplace transform technique. Thongmoon and Mckibbin

(2006) made a comparison among some of the numerical methods such as cubic-spline

method, forward time centered space and Crank-Nicolson method in finite domain for the

advection-diffusion equation with space dependent initial concentration and a combination

of boundary conditions. Najafi and Hajinezhad (2008) solved the one-dimensional

advection-dispersion equation with reaction using some finite-difference methods namely-

forward time centered space method, the upstream method, DuFort-Frankel method and

Crank-Nicolson method. It was observed that Crank-Nicolson method works more

accurately than the other methods. On the basis of numerical solution of transport of a

conservative and non reactive tracer, Zhang et al. (2012) presented a semi-analytical

solution. It was observed that the concentrations derived from the semi-analytical solution

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were identical to those derived from the individual numerical fate and transport model

simulations.

In the present chapter, a comparative study has been made for one-dimensional

advection-dispersion equation with temporally and spatially dependent dispersion

coefficients. The initial concentration has been taken as linear function of space-dependent

term. The effect of zero-order production term has also been taken into consideration. The

solution is obtained for homogeneous aquifer where the dispersion coefficient and seepage

velocity are temporally dependent and inhomogeneous aquifer where the dispersion

coefficient and seepage velocity are temporally and spatially dependent. The chapter is

divided into two parts: In the first part, the boundary condition has been taken as Dirichlet

type time-dependent logistic sigmoid function at the origin, while in the second part the

boundary condition has been taken as Robin type time-dependent logistic sigmoid function.

The analytical solution is obtained with the help of Laplace transform technique for

homogenous aquifer and it is compared with the numerical solution obtained by two-level

explicit finite-difference method for both homogeneous and inhomogeneous aquifer. The

physical model of the problem is shown in Fig. 3.1.

3.2. One-dimensional advection-dispersion equation with zero-order production term and

Dirichlet type time-dependent logistic sigmoid input concentration

3.2.1 Mathematical Formulation

Consider a one-dimensional advection-dispersion equation in an isotropic semi-

infinite aquifer with zero-order production term. Initially the aquifer is not solute free, it is

a function of space dependent term with zero-order production term. The input

concentration has been taken as Dirichlet type time-dependent logistic sigmoid function at

the origin while at the other end, the concentration gradient is assumed to be zero. The

unsteady velocity expressions are taken as of two types: 1) sinusoidal varying function 2)

and exponentially decreasing function for numerical discussions.

The partial differential equation describing the solute concentration distribution due

to advection and dispersion in one-dimension with a zero order liquid phase source can be

written as

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p

c cD uc

t x x (3.1)

where, -3[ ]c ML is the solute concentration of contaminants in the aquifer, 1[ ]u LT is the

groundwater velocity of position x at time t , 2 1[ ]D L T is the dispersion coefficient at time

[ ]t T , 3 1

[ ]p ML T is a zero order production term.

Initially, groundwater is not solute free due to some internal cause or effect in the

aquifer. Therefore, an appropriate initial condition may be chosen as

( , ) , 0 , 0p

i

xc x t c x t

u

(3.2)

The input concentration at the origin, where pollutants reach the groundwater level is taken

as time-dependent function and it can be considered as

0( , ) , 0 , 0[1 exp ]

cc x t t x

qt (3.3)

The other boundary condition for a semi-infinite aquifer can be written as

0 , , 0c

x tx

(3.4)

where, 3[ ]i

c ML is the initial concentration and 3

0[ ]c ML is the solute concentration. The

solution is obtained for the dispersion along homogeneous as well as inhomogeneous

aquifers.

3.2.2 Dispersion along Homogeneous Aquifer

In case of homogeneous formation, the dispersion coefficient D and seepage

velocity u are independent of x . Therefore, the partial differential equation Eq. (3.1)

describing solute transport in homogeneous aquifer can be written as

2

2 p

c c cD u

t x x (3.5)

Here, 0u u f t , (3.6)

and, 0 0

0

, , ,

p

D u D au D au f t D D f t

and f t

(3.7)

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where, a is the dispersivity depends upon pore system geometry and on the average pore

size diameter of an aquifer, 0D is the initial dispersion coefficient and 0 is the initial zero

order production term.

Using Eqs. (3.6) and (3.7), Eq. (3.5) can be written as

2

0 0 02

1 c c cD u

f t t x x (3.8)

A new time variable *

T has been introduced by the following transformation (Crank, 1975)

*

0

t

T f t dt (3.9)

Using Eq. (3.9), Eq. (3.8) becomes

2

0 0 0* 2

c c cD u

T x x (3.10)

and, the initial and boundary conditions Eqs. (3.2) - (3.4) can be written as

* *0

0

, , 0 , 0i

xc x T c x T

u

(3.11)

* *0

*, , 0, 0

[2 ]

cc x T T x

qT (3.12)

*0 , , 0c

x Tx

(3.13)

Introducing a set of non-dimensional variables as follows:

2*0 0 0 0 0

1 2 2

0 0 0 0 0 0

, , , ,xu u D qDc

C X T T Qc D D u c u

(3.14)

Using Eq. (3.14), Eq. (3.10) together with initial and boundary conditions Eqs. (3.11) -

(3.13) reduces to

2

12

C C C

T X X (3.15)

1

0

, , 0, 0icC X T X T X

c (3.16)

1, 0, 0

2C X T X T

QT

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or, 1, 1 , 0, 0

2 2

QTC X T X T

(3.17)

0 , , 0.C

X TX

(3.18)

3.2.2a Analytical Solution

The following transformation has been used to reduce the advective term from Eq. (3.15)

, , exp2 4

X TC X T K X T

(3.19)

With the help of Eq. (3.19), Eqs. (3.15) - (3.18) become

2

12exp

4 2

K K T X

T X (3.20)

1

0

, exp , 0, 0,2

ic XK X T X T X

c (3.21)

1, 1 exp , 0, 0,

2 2 4

QT TK X T X T

(3.22)

and, , , 02

K KX T

X

(3.23)

Taking Laplace transformation of Eq. (3.20), we get

2

12

0 0

exp4 2

pT pTK K T Xe dT e dT

T X (3.24)

12

42120

0 0

X p TpT pT K

Ke dT p Ke dT e e dTX

2

21 12

0

1exp

12

4

X

ic X KX pK e

c Xp

[Using Eq. (3.21)]

2

112

0

exp1 2

4

icK XpK X

X cp

(3.25)

To solve the Eq. (3.25), its auxiliary equation can be written as

90

2

1 10m p m p

Therefore,

C.F. 1 2

pX pXc e c e

(3.26a)

and, P.I. 11'2

0

1exp

1 2

4

ic XX

D p cp

2

1 1 12 2 22 2

0

1

1 11 1

4 44 4

XX X X

ic Xee e e

cp pp p

2

1

01

4

X

ic eX

cp

(3.26b)

Hence, the general solution of ,K X p can be written as

2 21 2 1

0

1 1( , )

1 1

4 4

X X

pX pX icK X p c e c e e Xe

cp p

(3.27)

Now applying the Laplace transform technique on boundary condition Eq. (3.22), it gives

2

1 1 10,

12 4 14 4

QK p

p p

(3.28)

Using Eq. (3.27) and applying Laplace transform technique on Eq. (3.23), we get

1 0c (3.29a)

And, applying Eq. (3.28) in Eq. (3.27), it becomes

22

0

1 1 1 1

1 12 4 14 44

icQc

cp pp

2 2

0

1 1 1 1

1 12 4 14 44

icQc

cp pp

(3.29b)

Computing the values of 1 2and c c

from Eq. (3.29a) and (3.29b) we get

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22 1

2

0 0

1 1 1,

1 1 1 112 44 4 4 44

XX

pXi ic c XeQK X p e e

c cp p p pp

(3.30)

Taking inverse Laplace transform of Eq. (3.30), it becomes

0

1

0

1 1, exp exp

2 2 4 2 2 4 2 22 2

exp exp4 2 2 4 22

8

22

ex

i

i

c T X X T T X X TK X T erfc erfc

c T T

T X X T T XT X erfc T X

TQ

X Terfc

T

cX

c

p4 2

T X

(3.31)

Now applying the transformation of Eq. (3.19) we get the solution as follows:

0

1

0

1 1, exp

2 2 2 22 2

exp8 2 22 2

i

i

c X T X TC X T erfc X erfc

c T T

Q X T X TT X erfc T X X erfc

T T

cX

c

(3.32)

3.2.2b Numerical Solution

The non-dimensional problem defined in Eq. (3.15) subjected to initial and boundary

conditions given in Eqs. (3.16) to (3.18) is of semi-infinite domain 0,X . To find the

numerical solution, the problem has been converted into finite domain '0,1X using the

following transformation

'1 expX X (3.33)

Using Eq. (3.33) in Eqs. (3.15) to (3.18), the problem can be written in finite domain as

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22

' '

1'2 '1 2 1

C C CX X

T X X (3.34)

' '

1 '

0

1, log , 0 , 0,

1

icC X T X T

c X

(3.35)

' '1, 1 , 0 , 0,

2 2

QTC X T X T

(3.36)

'

'0 , 1 , 0

CX T

X

(3.37)

Considering, '

1X results as X i.e., x , but to get concentration values at

infinity may not be possible. Therefore, the concentration values are obtained up to some

finite length along the longitudinal direction. Let the values be computed up to x l , which

corresponds to '

0 01 exp /X u l D in the domain 0,1 .

The '

X and T domain are divided into equal number of subintervals and represented as

' ' ' ' '

1 0, 1, 2,..., , 0, 0.02

i i iX X X i M X X

1 0, 1, 2,..., , 0, 0.001

j jT T T j I T T

The contaminant concentration at a point '

iX at thj sub-interval of time T is denoted as

,i jC

. The first and second order space derivative in Eq. (3.34) is approximated as central

difference approximation and first order time-derivative is approximated as forward

difference approximation respectively. Using finite difference approximation, Eqs. (3.34)

to (3.37) can be written as follows:

2' '

, 1 , 1, , 1, 1, 1, 1'2 '1 2 2 1

2i j i j i i j i j i j i i j i j

T TC C X C C C X C C T

X X

(3.38)

,0 1 '

0

1log , 0

1

ii

i

cC i

c X

(3.39)

0,

11 0

2 2

j

j

QTC j

(3.40)

, 1,0

M j M jC C j

(3.41)

The numerical solution has been obtained with the help of two-level explicit finite-

difference method.

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3.2.3 Dispersion along Inhomogeneous Aquifer

In an inhomogeneous aquifer, the solute dispersion coefficient and the groundwater

velocity are both temporally and spatially dependent i.e., both the coefficients are functions

of x and t . The dispersion coefficient D and seepage velocity u may be defined as

D D t F x and u u t F x (3.42)

Using Eq. (3.42), the advection-dispersion equation given in Eq. (3.1) in inhomogeneous

form can be written as

2

2( ) ( ) p

c c c d cF x D u F x D uc

t x x dx x (3.43)

Using new time variable defined in Eq. (3.9) and non-dimensional variables defined in Eq.

(3.14), Eq. (3.43) becomes

2

12

C C C d CF X F X C

T X X dX X (3.44)

Considering hyperbolic space dependent dispersivity, Chen et al. (2008) obtained a power

series solution for one-dimensional finite aquifer. Here, two expressions of F X , similar

to the expressions given by Lin (1977 a, b) are considered.

0.5exp1

1.5 exp

XF X

X

(3.45)

0.05exp0.8

1.25 exp

XF X

X

(3.46)

In such a form that the first expression is of increasing nature from 0.8 at 0X to 1.0 as

X and the second expression is of decreasing nature having reverse tendency. To

convert the problem of semi-infinite domain, 0,X into a finite domain ' 0,1X the

same transformation defined in Eq. (3.33) is used and therefore, Eq. (3.44) reduces to

2' ' ' ' '

1'2 ' ' '1 1 2 ( ) 1

C C C d CX F X X F X X C

T X X dX X (3.47)

where, ''

'

0.5 11

2.5

XF X

X

(3.48)

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and ''

'

0.05 10.8

0.25

XF X

X

(3.49)

The function 'F X has the same variation ' (0,1)X as of ( )F X in the domain 0,X . As the initial and boundary conditions are independent of dispersion coefficient and

seepage velocity, therefore they are same as given for homogeneous formation.

Using two level explicit finite difference scheme, Eq. (3.47) becomes

' ' '

, 1 , 1, , 1, 1, 1,'2 '

' '

1, 1, , 1' '

1 ( ) 1 2 22

12

i j i j i i i i j i j i j i j i j

i i i j i j i j

i

T TC C X F X X C C C C C

X X

d TF X X C C C T T

dX X

(3.50)

Eq. (3.50) subjected to initial and boundary conditions given in Eqs. (3.39) - (3.41) is

solved by two-level explicit finite difference method. The limitation of an explicit scheme

is that there is a certain stability criterion associated with it, so that the size of time step

cannot exceed a certain value. For the present problem, the stability analysis has been done

to improve the accuracy of the numerical solution (Bear and Verrujit, 1987) and the

stability condition for the size of time step is obtained as

0 0

2

10

22

TD u

XX

(3.51)

which satisfy the results and conditions obtained by Ataie-Ashtiani et al. (1999).

3.2.4 Illustration and Discussion

For the given problem, two forms of unsteady groundwater velocity are considered

(Banks and Jerasate 1962; Kumar 1983).

0 1 sinu t u mt (3.52)

0 exp , 1u t u mt mt (3.53)

The analytical and numerical solutions for homogeneous and inhomogeneous aquifers (for

both the increasing and decreasing function of F X given in Eqs. (3.45) and (3.46)

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respectively) are computed for the input values of 0 0.01 /u km day , 2

00.1 /D km day

,

5 -3

00.5 10 /km day , 0 1.0c , 0.01ic , 0.00001 / .q day The length of the aquifer is

assumed as 10x km

. The flow resistance coefficient m is chosen as 0.0165 / day . The

value of mt are chosen as13 2k , where 1k is whole number and in particular,

10 2k is

taken in the present discussion. Here u t having the expression given in Eq. (3.52) is

minimum and maximum value alternatively for these values of mt that means the velocity

has this kind of tendency at 1182 121t k days, where 1k is the whole number at regular

interval of 182days. Let 121t days correspond to some day in the month of June during

which the groundwater level and the velocity is minimum; this period is the peak of

summer season just before rainy season. Then next value 303 t days corresponds to

approximately the same day in the month of December, the peak of winter season, just after

the rainy season, during which groundwater level and velocity are maximum. Further, the

next value 485 t days corresponds to almost the same date in the month of June in the

next year and so on. The numerical solution has been obtained for ' 0.02, 0.001X T .

From the stability condition given in Eq. (3.51) the value of T should be less than 0.002

which satisfy the stability condition. Fig. 3.2a shows the analytical and numerical solution

of the contaminant concentration for the sinusoidal form of velocity expression in

homogeneous formation. It is observed that the source concentration decreases with

distance and increases with time. The decreasing tendency of the contaminant concentration

is faster in time 121t days

and same will continue for the further time of intervals. The

numerical solution of the problem obtained is compared with the analytical solution of the

problem obtained for homogeneous aquifer, in which it is shown that the concentration

distribution behaviors for both the solutions are approximately identical. Fig. 3.2b depicts

the concentration behavior for exponentially decreasing form of velocities expression in

homogeneous formation. It also shows the decreasing tendency of contaminant

concentration which reaches the minimum or harmless concentration for 0.0002 /m day

, 1mt are similar to that of sinusoidal velocity for 0.0165 /m day . The analytical and

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numerical solutions of the problem have also been compared for exponential form of

velocity. Due to presence of some numerical error, the numerical solution slightly deviates

from analytical solution. Fig. 3.3a represents contaminant concentration pattern for

sinusoidal varying unsteady velocity in inhomogeneous formation for both the increasing

and decreasing function of F X given in Eqs. (3.45) and (3.46) respectively. Here, it is

observed that the contaminant concentration decreases with distance and increases with

time throughout the aquifer. The contaminant concentration for increasing function of

F X has large value as compared to the decreasing function of F X . The contaminant

concentration pattern for exponential decreasing form of velocity expression in

inhomogeneous aquifer is shown in Fig. 3.3b. Here, the same pattern has been observed as

that in sinusoidal case. Hence, we can say that on changing the groundwater velocity

expression does not change the contaminant concentration pattern. But it certainly changes

the concentration values at each of the positions.

3.2.5 Conclusion

The contaminant concentration pattern of one-dimensional advection-dispersion

equation with zero-order production term has been studied for homogeneous and

inhomogeneous formation by analytical and numerical methods. The initial concentration

has been taken as space-dependent with zero-order production term and Dirichlet type

logistic sigmoid function time-dependent boundary condition has been taken at the origin.

In case of homogeneous formation, the contaminant concentration pattern has been

observed for sinusoidal and exponentially decreasing form of velocity expressions. The

contaminant concentration patterns for increasing and decreasing function of F X have

been depicted for inhomogeneous cases. It is observed that the contaminant concentration

decreases more rapidly for decreasing function of F X as compared to the increasing

function of F X . It is also observed that the contaminant concentration follows the same

pattern for both homogeneous and inhomogeneous cases with sinusoidal and exponential

form of velocity expressions.

97

3.3. One-dimensional advection-dispersion equation with zero-order production term and

Robin type time-dependent logistic sigmoid input concentration

3.3.1 Mathematical Formulation

The mathematical formulation of second problem consists of same advection-

dispersion equation defined in Eq. (3.1) along with the initial condition Eq. (3.2) and

boundary condition at the infinite extent defined in Eq. (3.4). The boundary condition at the

origin is taken as Robin type time-dependent logistic sigmoid input concentration and it can

be written as

0 , 0, 01 exp

uccD uc x t

x qt

(3.54)

The problem is solved by both analytical and numerical techniques for homogeneous and

inhomogeneous medium with same procedure as discussed earlier in 3.2.

3.3.2. Dispersion along Homogeneous Aquifer

In case of homogeneous formation, the dispersion coefficient and seepage velocity

is assumed as temporally dependent given in Eqs. (3.6) and (3.7). Using the new time

variable *

T defined in Eq. (3.9), and set of non-dimensional variables defined in Eq. (3.14),

Eq. (3.54) reduces to

11 , 0, 0,

2 2

C QTC X T

X

(3.55)

The analytical and numerical solution is obtained by Laplace transform technique and finite

difference method respectively.

3.3.2a Analytical Solution

The analytical solution of this problem in non-dimensional form along with initial and

boundary conditions given in Eqs. (3.15), (3.16), (3.18) and (3.55) has been obtained with

the help of Laplace transform technique using same procedure as discussed earlier in 3.2.2a

and it can be written as follows:

98

2

1

2

1exp

2 2 22 21 2,

21

1 exp2 22

11 exp 1

2 2 2 2 22 2

41

2

T X T X Terfc

T TC X T

X TX T X erfc

T

T X T X T X TT X erfc

T TQ

T

2

1

0

1 exp2 22

i

X T X TX erfc

T

cX

c

(3.56)

3.3.2b Numerical Solution

The numerical solution of the non-dimensional advection dispersion equation

defined in Eq. (3.15) together with initial and boundary conditions given in Eq. (3.16),

(3.18) and (3.55) has also been obtained by two-level explicit finite-difference method.

Using the transformation defined in Eq. (3.33), Eq. (3.55) becomes

' '

'

11 1 , 0, 0,

2 2

C QTX C X T

X

(3.57)

By using the finite difference approximation Eq. (3.57) in grid form can be written as

'

1,

0, ' '1 , 0, 0,

21 2 1

j j

j

i i

C QTXC i j

X X

(3.58)

The numerical solution is derived by solving Eq. (3.38) subjected to initial and boundary

conditions given in Eqs. (3.39), (3.41) and (3.58).

3.3.3 Dispersion along Inhomogeneous Aquifer

In case of inhomogeneous aquifer, the dispersion coefficient and seepage velocity

will be both temporally and spatially dependent. In this case dispersion coefficient D and

seepage velocity u is defined as in Eq. (3.42). The numerical solution is obtained by

99

solving Eq. (3.50), with initial and boundary conditions given in Eqs. (3.39), (3.41) and

(3.58) using two-level explicit finite difference method.

As in both the problems, the fundamental advection-dispersion equation remains

same i.e., one-dimensional advection-dispersion equation with zero-order production term,

therefore, the stability criterion for the error approximation also remains same in both the

cases.

3.3.4 Illustration and Discussion

The unsteady groundwater velocity expression has been taken as the same type i.e.

sinusoidal varying and exponentially decreasing form given in Eqs. (3.52) and (3.53). The

analytical and numerical solutions are computed for the input values of 0 0.003km/dayu ,

2

00.03 km /dayD

, 5 -3

00.1 10 km /day , 0 1.0c , 5

0.1 10 / day .q The length of the

aquifer is assumed as 10x km

. The flow resistance coefficient m is taken as

0.0165 / day and 0.0002 / day for sinusoidal and exponentially decreasing velocity

respectively. The value of mt is chosen as13 2k , where

1k is whole number. Here,

18 10k is taken in the present discussion. The numerical solution has been obtained for

'0.001and 0.1T X in case of homogeneous formation while for inhomogeneous

case '0.05X is taken. From the stability condition given in Eq. (3.51) the value of T

should be less than 0.1658 and 0.0416 for homogeneous and inhomogeneous medium

respectively. Therefore, it can be observed that the present problem satisfy the stability

criterion. The concentration distribution pattern for homogeneous aquifer by analytical

method has been represented in Fig. 3.4 considering the initial concentration value 0.2ic .

It is observed that the contaminant concentration decreases with distance but increases with

time throughout the aquifer. On comparing the two different types of velocity expression it

has been observed that the contaminant concentration value is less in case of exponential

form of velocity as compared to the sinusoidal one. For graphical representation of the

contaminant concentration by numerical technique in homogeneous and inhomogeneous

aquifers both, the initial concentration has been taken as 0.4ic . Fig. 3.5 represents the

100

concentration pattern for homogeneous formation by numerical method. Here, the same

pattern has been observed as that of analytical case. However, the contaminant

concentration decreases more rapidly in case of analytical method. This is due to error

possibly up to 15-20% present in the solution obtained by numerical technique. The

numerical solution of contaminant concentration in inhomogeneous formation is presented

in Fig. 3.6 for increasing and decreasing function of F X given in Eqs. (3.45) and (3.46).

This graphical representation shows that the contaminant concentration decreases with

distance in first half length of the aquifer in both the cases. In case of increasing function of

F X , the concentration pattern shows the stagnant nature in second half length of the

aquifer. However, in case of decreasing function of F X the concentration start increases

slightly and then stagnant in the second half length of the aquifer. It is observed from the

figure that due to effect of inhomogenity the concentration pattern shows reverse nature

with respect to time, i.e. contaminant concentration decreases with time also. In case of

inhomogeneous formation the sinusoidal form of velocity expression has less concentration

value as compared to the exponential one i.e., just reverse as that of homogeneous

formation.

3.3.5 Conclusion

The contaminant concentration pattern for one-dimensional semi-infinite aquifer in

homogeneous and inhomogeneous formation with zero-order production term has been

studied and compared. The boundary condition has been taken as Robin type temporally

dependent in the form of logistic sigmoid function. It is observed that in case of

homogeneous aquifer the contaminant concentration decreases with distance but increase

with time. Also, the concentration pattern for exponential form of velocity decreases more

rapidly as compared to the sinusoidal form. On introducing the inhomogenity in the aquifer

the reverse pattern has been observed. In that case, the contaminant concentration decreases

with both distance and time. Also, the contaminant concentration decreases more rapidly in

case of sinusoidal form as compared to the exponential case.

101

Fig. 3.1 Physical model of the problem

Fig. 3.2a Contaminant concentration pattern for sinusoidal varying form of velocity

expression in homogeneous aquifer

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance Variable X

Conce

ntr

atio

n C

Curve mt Duration

2 1st Year June

5 1st Year Dec

8 2nd Year June

Analytical Solution

Numerical Solution

102

Fig. 3.2b Contaminant concentration pattern for exponential decreasing form of velocity

expression in homogeneous aquifer

Fig. 3.3a Contaminant concentration pattern for sinusoidal varying form of velocity

expression in inhomogeneous aquifer

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance Variable X

Conce

ntr

atio

n C

Analytical Solution Numerical Solution

Curve time in days Duration 121 1st Year June 303 1st Year Dec

485 2nd Year June

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

Distance Variable X

Conce

ntr

atio

n C

Increasing function of F(X) Decreasing function of F(X)

Curve mt Duration

2 1st Year June 5 1st Year Dec 8 2nd Year June

103

Fig. 3.3b Contaminant concentration pattern for exponentially decreasing form of velocity

expression in inhomogeneous aquifer

Fig. 3.4 Contaminant concentration pattern along homogeneous aquifer by analytical

method for both sinusoidal and exponential form of velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

Distance Variable X

Conce

ntr

atio

n C

Increasing function of F(X)

Decreasing function of F(X)

Curve time in days Duration

121 1st Year June

303 1st Year Dec

485 2nd Year June

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Distance Variable X

Conce

ntr

atio

n C

Curve time in days Duration

1576 5th Year June

1758 5th Year Dec

1940 6th Year June

Sinusoidal Velocity

Exponential Velocity

104

Fig. 3.5 Contaminant concentration pattern along homogeneous aquifer by numerical

method for both sinusoidal and exponential form of velocity

Fig. 3.6 Contaminant concentration pattern along inhomogeneous aquifer by numerical

method for both sinusoidal and exponential form of velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4055

0.406

0.4065

0.407

0.4075

0.408

0.4085

0.409

0.4095

0.41

Distance Variable X

Concentration C

Curve time in days Duration

1576 5th Year June

1758 5th Year Dec

1940 6th Year June

Sinusoidal Velocity

Exponential Velocity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.392

0.394

0.396

0.398

0.4

0.402

0.404

0.406

Distance Variable X

Conce

ntr

atio

n C

Sinusoidal Velocity

Exponential Velocity

Curve time in days Duration

1576 5th Year June

1758 5th Year Dec

1940 6th Year June

Increasing function of F(X)

Decreasing function of F(X)