CHAPTER II COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS …shodhganga.inflibnet.ac.in/bitstream/10603/33030/10/10_chapter 2.pdf · COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS FOR TIME-DEPENDENT

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CHAPTER II

COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS FOR TIME-

DEPENDENT SOLUTE TRANSPORT ALONG UNSTEADY GROUNDWATER

FLOW IN SEMI-INFINITE AQUIFER

2.1 Introduction

Groundwater constituents are important components of many natural water resource

systems which supply water for domestic, industrial and agricultural purposes. It is

generally a good source of drinking water. It is believed that groundwater is more risk free

in compare to the surface water. But these days groundwater contamination is growing

continuously in the developing countries particularly in India due to the indiscriminate

discharge of waste water from the various industries, especially coal based industries,

which do not have sufficient treatment facilities. These industries discharge their waste

water into the neighboring ponds, streams, rivers etc. The chemical constituents of the

waste material often infiltrate from these ponds and mixed with the groundwater system

causes groundwater contamination (Mohan and Muthukumaran, 2004; Sharma and Reddy,

2004; Rausch et al., 2005; Thangarajan, 2006).

Groundwater modeling is specially used in the hydrological sciences for the

assessment of the resource potential and prediction of future impact under different

conditions. Many experimental and theoretical studies were undertaken to improve the

understanding, management, and prediction of the movement of contaminant behavior in

groundwater system. These investigations are primarily motivated by concerns about

possible contamination of the subsurface environment. Hydrologist, Civil engineers,

Scientists etc. are doing their best to solve this type of serious problem by various means.

The subsurface solute transport is generally described with the advection-diffusion

equation. In the deterministic approach, explicit closed-form solutions for transport

problem can often be derived subjected to the model parameters remains constant with

respect to time and position (Leij et al., 1993). Mathematical modeling is one of the

powerful tools to project the existing problems and its appropriate solutions. Although

many transport problems must be solved numerically, analytical solutions are still pursued

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by many scientists because they can provide better physical insight into problems (Mahato,

2012).

As we all know analytical solution of the problem provide closed form solution

which gives more realistic result rather than numerical solution which provide approximate

solution confining the percentage of error. In 1961, Ogata and Banks introduced a direct

method for solving the differential equation governing the process of dispersion in porous

media. In that, the medium was considered as homogeneous and isotropic and it was

assumed that no mass transfer occur between the solid and liquid phases. Dispersion of

pollutants in semi-infinite porous media with unsteady velocity distribution was discussed

by Kumar (1983). The solution was obtained by Laplace transform technique for both non-

adsorbing and adsorbing porous medium subjected to temporally dependent input

concentration. Lindstrom and Boersma (1989) studied the analytical solutions for

convective-dispersive transport in confined aquifers with different initial and boundary

conditions. Considering time-dependent inactivation rate coefficients, a mathematical

model was developed for virus transport in one-dimensional homogeneous porous media

(Sim and Chrysikopoulos, 1996). The solutions were derived with the help of Laplace

transformations using the binomial theorem. Sometimes, the dispersion coefficient and

seepage velocity may vary with time as well as distance. The solutions obtained in these

studies were obtained with a variety of integral transforms. A generalized analytical

solution was developed using Laplace transform technique for one-dimensional solute

transport in heterogeneous porous media with scale-dependent dispersion (Huang et al.,

1996). The analytical solution for solute transport with depth dependent transformation or

sorption coefficient was presented by Flury et al. (1998) in Laplace space and inverted

numerically. Considering scale and time-dependent dispersivity, Sander and Braddock

(2005) presented a range of analytical solutions to the combined transient water and solute

transport for horizontal flow. The scale and time dispersivity was applied to transient,

unsaturated flow to develop similarity solutions for both constant solute concentration and

solute flux boundary conditions. The Investigation of consolidation-induced solute

transport, effects of consolidation on solute transport parameters were discussed and further

extended in which experimental and numerical results were explored by Lee et al. (2009)

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and Lee and Fox (2009). Li and Cleall (2010) presented the analytical solutions for

contaminant diffusion in double-layered porous media subjected to arbitrary initial and

boundary conditions. The analytical solutions were verified against numerical solutions

from a finite-element method based model. All these analytical solutions are having some

limitations though significant contribution for the scientific community is very well

reported.

In recent years, numerical solution of the complicated problem for which analytical

solution is not available, is being obtained frequently by the various scientists and

researchers in India and abroad. The finite difference method is the well-known numerical

method to solve the partial differential equations. The numerical solutions of one-

dimensional solute transport equation was obtained by finite element technique and finite

difference method and compared with each other (van Genuchten, 1982). Celia et al. (1990)

developed a generalization of characteristic method named as Eulerian-Lagrangian

localized adjoint method to provide a consistent formulation by defining test functions as

specific solutions of the localized homogeneous adjoint equation. Ataie-Ashtiani et al.

(1996) presented the numerical correction for finite-difference solution of the advection-

dispersion equation with reaction in which the numerical and analytical solutions were

compared. Assuming the dispersion coefficient and groundwater velocity as temporally and

spatially dependent, the effect of solute dispersion along unsteady groundwater flow in a

semi-infinite aquifer was presented by Kumar and Kumar (1998) for both homogeneous

and inhomogeneous formations. Here the analytical solution was obtained by Laplace

transform technique and it was compared with two-level explicit finite-difference method.

The truncation errors in finite difference models for one-dimensional solute transport

equation with first-order reaction were discussed by Ataie-Ashtiani et al. (1999). Using

mesh/grid free explicit and implicit numerical schemes Zerroukat et al. (2000) developed

solution of liner advection-diffusion problem. Campbell and Yin (2006) examined the

stability of alternating direction explicit method for one-dimensional advection-diffusion

equations. With the help of meshless method also known as element-free Galerkin method,

Kumar et al. (2007) modeled the numerical solution of contaminant transport through

unsaturated porous media with transient flow condition. The effect of linear first order

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degradation was also taken into consideration. Rouholahnejad and Sadrnejad (2009) studied

the numerical simulation of leachate transport into the groundwater at the landfill sites. To

predict the quality of water in rivers, Ahsan (2012) presented a numerical solution of one-

dimensional advection-diffusion equation with first order decay coefficient using Laplace

transform finite analytical method. The initial concentration was taken as space dependent

function with uniform boundary condition.

The analytical solutions obtained in these studies were obtained with a variety of

integral transforms. However, to find the analytical solution with the help of Fourier

transform technique may help to benchmark against the other analytical methods. The

numerical solution can be used to verify the analytical method applied in the problem.

Keeping these facts, the chapter has been made. This chapter deals with the one-

dimensional advection dispersion problem subject to Dirichlet and Robin type boundary

conditions in both homogeneous and inhomogeneous formations which contain three

problems.

In the first problem, the uniform initial concentration has been taken into account

which is invariant with time or distance. The boundary condition is taken as exponential

decreasing Dirichlet type time-dependent function. In case of homogenous formation, the

problem is solved analytically using Fourier transform technique and numerically using

two-level explicit finite difference method and the results are compared with the solution

obtained by Laplace transform technique by Singh et al. (2008). For inhomogeneous

formation the dispersion coefficient is assumed to be function of both space and time and

the concentration pattern is obtained using the same numerical technique. To predict the

nature of the contaminant concentration along unsteady groundwater flow in semi-infinite

aquifer, a comparative study is made by the proposed methods. Time-dependent velocity

expressions are considered to illustrate the obtained result.

Due to leachate and landfills, the initial concentration of the aquifer may depend on

the distance. The second problem represents one-dimensional advection-dispersion

equation with space dependent initial concentration. The input point source concentration

has been taken as Dirichlet type time-dependent in the form of logistic sigmoid function

different from the first problem. The logistic sigmoid function is horizontally asymptotic in

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nature, i.e., it increases continuously for 0t and tends to 1 as t . In the solute

transport modeling context, the input point source concentration can be taken as of this

form assuming that input concentration would initially increase with time and after a

certain time period it would stabilize at an asymptotic value. The analytical and numerical

solutions are obtained for homogeneous and inhomogeneous formations. In case of

homogeneous formation, the analytical solution is obtained by Laplace transform technique

while numerical solution is obtained by two-level explicit finite-difference method and it is

compared with the numerical solution obtained for inhomogeneous formation.

Considering same initial concentration as discussed in second problem, third

problem is solved for Robin type boundary condition with time-dependent logistic sigmoid

function. The analytical and numerical solutions for homogeneous and inhomogeneous

formations follow the same approach as discussed in second problem. The physical model

of the problem is represented in Fig. 2.1.

2. 2. One-dimensional advection-dispersion equation with uniform initial concentration

2.2.1. Mathematical Formulation

Consider a one-dimensional isotropic semi-infinite aquifer. The Dirichlet type time-

dependent source of contaminant concentration is considered at the origin, i.e., at 0x

and at the other end of the aquifer it is supposed to be zero. In order to mathematically

formulate the problem, let 3c ML be the concentration of contaminants in the aquifer,

1u LT be the groundwater velocity, and 2 1D L T is the dispersion coefficient at time

t T . Initially, the groundwater is not supposed to be solute free i.e., at time 0t , the

aquifer is not clean which means that some initial background concentration exists in

aquifer. It is represented by uniform concentration ic . The one-dimensional advection

dispersion equation can be written as

c cD uc

t x x

(2.1)

The initial and boundary conditions can be expressed as

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, ; 0, 0ic x t c x t (2.2)

0, [1 exp ]; > 0, 0 c x t c qt t x (2.3a)

= 0; 0,t x (2.3b)

0; 0c

tx

x (2.4)

where 3

ic ML is the initial concentration describing distribution of the contaminant

concentration at all point i.e., at 0x , 3

0c ML is the solute concentration and1q T is

the contaminant decay rate coefficients.

2.2.2 Dispersion along Homogeneous Aquifer

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

velocity is function of time only.

Therefore, Eq. (2.1) can be written as

2

2

c c cD u

t x x

(2.5)

Let 0u u f t (2.6)

where 1

0u LT is the initial groundwater velocity at distance x L . The two forms of

f t are considered such as 1 sinf t mt and exp , 1f t mt mt , where 1m T

is the flow resistance coefficient.

The groundwater flow in the aquifer is unsteady where the velocity follows either a

sinusoidal form or an exponential decreasing form. The sinusoidal form of velocity

represents the seasonal variation in a year often observed in tropical regions like Indian

sub-continent. In aquifers in tropical regions, groundwater velocity and water level may

exhibit seasonally sinusoidal behavior. In tropical regions like in Indian sub-continent,

groundwater velocity and water level are minimum during the peak of the summer season

(the period of greatest pumping), which falls in the month of June, just before rainy season.

Maximum values are observed during the peak of winter season around December, after the

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rainy season (the period of lowest pumping). In these regions, groundwater infiltration is

from rainfall and rivers. However, exponentially decreasing velocity expression is taken

into consideration, from Banks and Jerasate (1962).

The dispersion coefficient, vary approximately directly to seepage velocity for

various types of porous media (Ebach and White, 1958). Also it was found that such

relationship established for steady flow was also valid for unsteady flow with sinusoidally

varying seepage velocity (Rumer, 1962; Kumar, 1983).

Let, auD where a [L] is the dispersivity. It depends upon pore system geometry

and average pore size diameter of porous medium. However, molecular diffusion is not

included in the present discussion only because the value of molecular diffusion does not

vary significantly for different soil and contaminant combinations and they range from

1×10-9

to 2×10-9

m2/sec (Mitchell, 1976).

Using Eq. (2.6), we get 0D D f t (2.7)

where 00 auD is an initial dispersion coefficient. Using Eqs. (2.6) and (2.7), Eq. (2.5)

can be written as follows:

2

0 02

1 c c cD u

f t t x x

(2.8)

A new time variable is introduced by the transformation (Crank, 1975)

*

0

t

T f t dt (2.9)

Therefore, Eq. (2.8) becomes

2

0 0* 2

c c cD u

T x x

(2.10)

Now the set of dimensionless parameters are defined as follows:

2 *

0 0 0

2

0 0 0 0

, , ,

x u u T qDcC X T Q

c D D u (2.11)

The PDE given in Eq. (2.10) in non-dimensional form can be written as

2

2

C C C

T X X

(2.12)

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0

, ; 0, 0icC X T X T

c (2.13)

, 2 ; > 0, X 0 C X T QT T (2.14a)

= 0; 0, XT (2.14b)

,0; 0,

C X TT X

X

(2.15)

2.2.2a Analytical Solution

To obtain the analytical solution, we reduce the convective term present in Eq.

(2.12) by using the transformation

, , exp2 4

X TC X T K X T

(2.16)

Using Eq. (2.16), Eqs. (2.12) to (2.15) can be written as

2

2

K K

T X

(2.17)

0

, exp ; 0, 02

ic XK X T X T

c

(2.18)

, 2 exp ; > 0, X 0 4

TK X T QT T

(2.19a)

0; 0, XT (2.19b)

,0; 0,

2

K X T KT X

X

(2.20)

The analytical solution is obtained with the help of Fourier transform technique and

it is compared with the solution obtained by Singh et al. (2008) with the help of Laplace

transform technique.

Case I- Solution Using Fourier Transform

In the given problem as ,K X T is specified at 0X , thus Fourier sine transform

is applicable for this problem.

Taking the Fourier sine transform of Eq. (2.17) and using the notation

50

0

2, , sinsK s T K X T sXdX

(2.21)

and using the conditions , 0K X T , ,

0K X T

X

as X ; Eq. (2.17) can be

written as

2, 2

0, ,s

s

dK s TsK T s K s T

dT

Or,

2, 22 exp ,

4

s

s

dK s T Ts QT s K s T

dT

[Using Eq.(2.19a)]

Or, 2, 2

, 2 exp4

s

s

dK s T Ts K s T s QT

dT (2.22)

The auxiliary equation of Eq. (2.22) can be written as

2 2

1 10m s m s

Therefore, it’s Complementary Function (C.F) is given by

C.F 2

1 expc s T (2.23a)

and

Particular Integral (P.I) ' 2

1 22 exp

4

Ts QT

D s

2

2 2

2 2exp

1 41

4 4

QT Q Ts

s s

(2.23b)

Thus, one can get the general solution as follows:

2

122 2

2 2, exp exp

1 41

4 4

s

QT Q TK s T s c s T

s s

(2.24)

Using initial condition given in Eq. (2.18), in Eq. (2.24), it gives

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1 220 2

2 12

1 1

4 4

ic Qc s

cs s

(2.25)

Using, Eq. (2.25), Eq. (2.24) become

0 2

2 22 22 2

22 2

, exp exp1 141 1

4 44 4

i

s

c

cQT Q T QK s T s s T

s ss s

2 2

2 22 202 2

exp expexp exp2 4 4

( , ) (2 ) 21 11 1

4 44 4

is

T Ts s

s s T s s TcK s T QT Q Q

cs ss s

(2.26)

Taking the inverse Fourier transform on both the sides of Eq. (2.26) and using the

transformation given in Eq. (2.16), one can get

0

1, 2 2 exp

2 2 22 2

exp2 2 22 2

2exp exp2 2 2

ic T X T XC X T QT erfc X erfc

c T T

Q T X T XQX T erfc X erfc

T T

Q T X T TXerfc X

T

2

2

2 2

exp 2exp exp2 2 22 2

X

T

Q T X T T XX X erfc X

T T

(2.27)

or

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1

0

1 2 3

1, 2 2 ,

2

, , exp ,2

icC X T QT QX F X T

c

QTF X T F X T X F X T

(2.28)

where 1 , exp2 22 2

T X T XF X T erfc X erfc

T T

2

2

2, exp

2 22 2

T X T T XF X T Xerfc

T T

2

3

2, exp

2 22 2

T X T T XF X T Xerfc

T T

Case II- Solution Using Laplace Transform

The solution of above problem was obtained with same initial and boundary

conditions by Singh et al. (2008) given below:

0

0

1, 2 exp

2 2 22 2

2 2 22 2

i

i

c X T X TC X T erfc X erfc

c T T

c Q X T X TT X erfc T X erfc

c T T

(2.29)

2.2.2b Numerical Solution

Eqs. (2.12) - (2.15), is a one-dimensional solute transport problem of semi-infinite

domain. To obtain the numerical solution, the semi-infinite domain is converted into finite

domain. In order to convert the problem of semi-infinite domain, 0,X into a finite

domain ' 0,1X ; the following transformation is used.

'1 expX X (2.30)

Here, 'X has the same variation '0,1X as of X in the domain 0,X . Applying Eq.

(2.30) in Eq. (2.12), it reduces to

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22

' '

'2 '1 2 1

C C CX X

T X X

(2.31)

On converting the initial and boundary conditions Eqs. (2.13) to (2.15) in the '

X domain, it

becomes as follows:

' '

0

, , 0 , 0,icC X T X T

c

(2.32)

' ', 2 ; > 0, 0 C X T QT T X (2.33a)

'0; 0, 1T X (2.33b)

'

'0; 1, 0

CX T

X

(2.34)

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

is not possible. Therefore, the values are evaluated 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

follows:

' ' ' ' '

1 0, 1,2,..., , 0, 0.05

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. (2.31) is approximated as central

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

difference approximation respectively. Using two-level explicit finite difference method,

Eqs. (2.31) to (2.34) can be written as

2' '

, 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

X X

(2.35)

,0

0

, 0ii

cC i

c

(2.36)

0 , 2 0j jC Q T j (2.37a)

54

,0 0

M jC j

(2.37b)

, 1, 0M j M jC C j

(2.38)

To find the numerical solution, two boundary conditions Eqs. (2.37a) and (2.38) have been

used.

2.2.3 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

follows:

and D D t F x u u t F x (2.39)

Using Eq. (2.39), the advection-dispersion equation given in Eq. (2.1) in inhomogeneous

form can be written as

2

2

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

t x x dx x

(2.40)

Using new time variable defined in Eq. (2.9) and non-dimensional variables defined in Eq.

(2.11), Eq. (2.40) becomes

2

2

C C C d CF X F X C

T X X dX X

(2.41)

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

(2.42)

0.05exp0.8

1.25 exp

XF X

X

(2.43)

In such a form, 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

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

same transformation defined in Eq. (2.30) is used and therefore, Eq. (2.41) reduces to

2' ' ' ' '

'2 ' ' '1 1 2 1

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

T X X dX X

(2.44)

where, ''

'

0.5 11

2.5

XF X

X

(2.45)

and ''

'

0.05 10.8

0.25

XF X

X

(2.46)

The function 'F X has the same variation '0,1X 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, Eqs. (2.44) becomes

' ' '

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

' '

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

dX X

(2.47)

Eq. (2.47) subjected to initial and boundary conditions given in Eqs. (2.36) - (2.38), is

solved by using the 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 same, 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

T

D u

XX

(2.48)

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

56

2.2.4. Illustration and Discussion

We consider the sinusoidally varying and exponentially decreasing forms of

velocities which are valid for transient groundwater flow too (Banks and Jerasate, 1962;

Kumar, 1983). Now from Eq. (2.6) the velocity expressions are as follows:

0 1 sinu t u mt (2.49a)

0 expu t u mt , mt < 1 (2.49b)

For both the expressions, the non-dimensional time variable T can be written as follows:

2

0

0

1 cosu

T mt mtmD

(2.50a)

2

0

0

1 expu

T mtmD

(2.50b)

where 13 2mt k , 1k is a whole number. Considering 0.0165 (/ )m day gives

1 182 121 t k (days), approximately. For these values of mt , the velocity u , is

alternatively minimum and maximum. It represents that the groundwater level and velocity

minimum during the month of June and maximum during December just after six months

(Approximately 182 days) in one year. The next data of t , represents minimum and

maximum records during June and December respectively in the subsequent years. The

sinusoidally varying and exponentially decreasing form of velocity representations are

made graphically with respect to time at different values of seepage velocity and dispersion

parameters and shown in the Fig.2.2 (a, b). As we increase the seepage velocity parameter,

the peak of sinusoidal form of velocity increases which reveals in Fig. 2.2a. This

representation can often be observed in tropical region of India. Eqs. (2.28) and (2.29) are

the analytical solutions with Fourier transform and Laplace transform techniques are

computed respectively for the input values 0.1,ic 0 1.0, c 0 0.033 0.045 / ,u km day

2

00.33 0.45 / ,D km day 0.0009 /q day and 100 x km . The time-dependent

concentration values are depicted from the table 2.1(a-d) for sinusoidal form of velocity

expression given in Eqn. 2.49(a) at the seepage velocity 0u ranging from 0.033 /km day to

0.042 /km day and dispersions parameter 0D ranging from 20.33 /km day to 20.42 /km day

57

. The concentration values at different positions are obtained for both the methods Laplace

transform technique and Fourier transform technique in row (i) and row (ii) respectively

shown in Table 2.1(a, b). It is observed that concentration values decreases rapidly in row

(i) in comparison to row (ii). However, in Table 2.1 (c, d) the concentration values also

decreases rapidly in row (i) and slowly and gradually converges at a common point near by

the source and after that it further decreases and reached towards minimum or harmless

concentration. But in row (ii), the concentration values decreases and goes on decreasing

towards minimum or harmless concentration. The contaminant concentration value

represented by tabular form gives more clarity to understand the distribution pattern of

solute concentration in groundwater reservoir i.e., aquifer.

However, the concentration values are also depicted graphically in the presence of

time-dependent source of contaminant concentration at 13 2mt k , 18 13k which

represents minimum and maximum records of groundwater level and velocity during June

and December in 5th

, 6th

and 7th

years respectively. The contaminant concentration

distribution behavior along transient groundwater flow of sinusoidally varying velocity is

shown in the Fig. 2.3a at the seepage velocity 0 0.045 /u km day

and dispersions parameter

2

00.45 /D km day

. It is observed that the contaminant concentration decreases at the

source and emerges at a point nearby origin. After emergence tendency of the contaminant

concentration is same reaching towards the minimum or harmless concentration. But the

values of the contaminant concentration decreases and increases with time just before and

after the emergence respectively. For example, before emergence 5th

year Dec.

concentration is less than 5th

year June concentration while after emergence the trend is just

reverse. For the same set of inputs except 0.0002 / m day as 1mt , Eqs. (2.28) and

(2.29) are also computed for exponentially decreasing form of velocity and shown in the

Fig. 2.3b. It is also observed that the trend of contaminant concentration is almost same as

discussed in sinusoidally varying velocity but the decreasing rate is little slower at the

source and nearby the origin. The decreasing tendency of concentration values depicted

through the Table 2.1(a, d) and Fig. 2.3 (a, b) reveals that Fourier transform technique is

more effective in case of increasing the seepage velocity and dispersion parameters.

58

However, Laplace transform technique is preferable in the case of decreasing seepage

velocity and dispersion parameters.

The numerical solutions for homogeneous and inhomogeneous aquifer have also

been obtained for the same set of values at 13 2mt k , 18 10k . The velocity and

dispersion coefficient have been taken as 0 0.025 /u km day

and 2

00.25 /D km day

for

sinusoidal and exponential form of velocity expressions. Here, from Eq. (2.48) the stability

condition for which 0.001T is less than 0.005 satisfy the stability criteria. The

contaminant concentration patterns can easily be observed from Fig. 2.4a and Fig. 2.4b for

sinusoidal and exponential form of velocity expressions respectively. The length of the

aquifer is assumed as 10 x km . It is observed that in both the homogeneous and

inhomogeneous cases the contaminant concentration decreases with distance throughout the

aquifer. At the origin initially the concentration decreases with time and then it emerge to a

point and shows reverse pattern i.e. after emergence the contaminant concentration

increases with time. Due to occurrence of inhomogenity, the contaminant concentration

decreases more rapidly with distance as compared to the case of homogeneous formation.

However, the case, in which the decreasing form of F X has been taken, the

concentration decreases faster with distance as compared to increasing form of F X . On

comparing the two-types of velocity expressions, it is observed that contaminant

concentration decreases more rapidly in case of exponentially decreasing form of velocity

expression as compared to sinusoidal form of velocity expression.

2.2.5 Conclusion

A comparative study is made to obtain the analytical and numerical solutions of

solute transport modeling in homogeneous and inhomogeneous groundwater system using,

Laplace transform technique, Fourier transform technique and Finite difference method. A

solute transport model is formulated with time-dependent source concentration in one-

dimensional isotropic semi-infinite aquifer with suitable initial and boundary conditions. To

predict contaminants concentration along transient groundwater flow in homogeneous,

semi-infinite aquifer Fourier transform technique is more preferable than Laplace transform

59

technique with respect to sensitivity of seepage velocity and dispersion parameters. The

numerical method can also be used to verify the analytical methods applied to solve the

problem. If the aquifer is inhomogeneous, then the analytical methods become more

complex, on that case two-level explicit finite difference method can be used as an

alternative to find the solution. The dispersion coefficient is directly proportional to

seepage velocity concept is used.

60

2.3. One-dimensional advection-dispersion equation with space dependent initial

concentration and Dirichlet type time-dependent logistic sigmoid input concentration

2.3.1 Mathematical Formulation

Consider an isotropic one-dimensional semi-infinite aquifer, which is initially not

solute free i.e. a space dependent exponentially decreasing type background concentration

exists in the aquifer. Let the aquifer be subjected to a Dirichlet-type time-dependent point

source contamination at the uppermost groundwater level. The direction of groundwater

flow is along x direction. The problem is solved for both homogeneous as well as

inhomogeneous formations. In case of homogeneous media, the analytical solution is

obtained by Laplace transform technique and it is compared with the numerical solution

obtained by two-level explicit finite-difference method. The solutions obtained for

homogeneous case is also compared with that of inhomogeneous case. In this problem, the

one-dimensional advection-dispersion equation given in Eq. (2.1) with the initial and

boundary conditions considered below. This problem is discussed with different initial and

boundary conditions.

Initially, groundwater is not solute free i.e. some initial background concentration

exists in the aquifer, therefore an appropriate initial condition in the form of exponentially

decreasing function of space variable, is considered as

*, exp , 0 , 0ic x t c x x t (2.51)

The source of input concentration at the origin, where the pollutants reach the groundwater

level, is taken of temporally dependent form and the concentration gradient at the infinite

extent is supposed to be zero. Therefore, the boundary conditions are considered as follows:

0, , 0, 0[1 exp ]

cc x t x t

qt (2.52)

0 , , 0c

x tx

(2.53)

where, * 1L is the decay parameter and the symbols have their usual meaning as defined

in the first problem.

61


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

vary with time. Using Eqs. (2.6) and (2.7), introducing the new time variable *

T defined in

Eq. (2.9), and non-dimensional parameters defined in Eq. (2.11) in addition to non-

dimensional variable*

0

0

D

u

, Eq. (2.1) with initial and boundary conditions Eqs. (2.51)-

(2.53) becomes

2

2

C C C

T X X

(2.54)

0

, exp , 0 , 0icC X T X X T

c (2.55)

1, 0, 0

2C X T X T

QT

or, 1, 1 , 0, 0,

2 2

QTC X T X T

(2.56)

and, 0 , , 0.C

X TX

(2.57)

2. 3.2a Analytical Solution

Using the transformation defined in Eq. (2.16), Eqs. (2.54) - (2.57) become

2

2

K K

T X

(2.58)

0

1, exp , 0, 0

2

icK X T X X T

c (2.59)

1

, 1 exp , 0, 0,2 2 4

QT TK X T X T

(2.60)

and, , , 02

K KX T

X

(2.61)

Taking Laplace transform on both sides of Eq. (2.58), it gives

62

2

2

0

pT K Ke dT

T X

2

200

pT pT KKe pKe dT

X

2

2,0

KK X pK

X

2

2,0

KpK K X

X

2

2

0

1exp

2

icKpK X

X c [Using Eq. (2.59)] (2.62)

where, 0

, , pTK X p L K X T Ke dT

The auxiliary equation of Eq. (2.62) can be written as

2

1 10m p m p

Therefore, the C. F. will be

C.F = 1 2

pX pXc e c e

(2.63a)

and P.I is given as

P.I '2

0

1 1exp

2

icX

c D p

2

0

1 1exp

21

2

icX

cp

(2.63b)

Hence, the general solution can be written as

1 2 2

0

1 1, exp

21

2

p X p X icK X p c e c e X

cp

(2.64)

Using Eq. (2.61) in Eq. (2.64), it gives 1 0c . Therefore, Eq. (2.64) reduces to

63

2 2

0

1 1, exp

21

2

pX icK X p c e X

cp

(2.65)

Now, on applying Laplace transform in Eq. (2.60),

2

1 1 10, ,

12 4 1

4 4

QK p

p p

(2.66)

Using Eq. (2.66) in Eq. (2.65), one can get

2 2 2

0

1 1 1 1

12 4 1 1

4 4 2

icQc

cp p p

(2.67)

Using Eq. (2.67) in Eq. (2.65), it reduces to

2 2 2

0 0

1 1 1 1 1 1, exp

12 4 21 1 1

4 4 2 2

pXi ic cQK X p e X

c cp p p p

(2.68)

Taking inverse Laplace transform of Eq. (2.68), the value of ,K X T can be written as

2

0

1, exp exp

4 4 2 2 4 2 22 2

exp exp4 2 2 4 22

8

22

exp2 2

i

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

T T

T X X T T XT X erfc T X

TQ

X Terfc

T

c XT X erfc T

c T

2

2

0

exp2

expi

XT X erfc T

T

cT X

c

(2.69)

where, 1

2 .

Now applying the transformation given in Eq. (2.16), we get the solution as

64

2

0 2

1, exp

4 2 22 2

exp8 2 22 2

exp22

2exp

22

i

X T X TC X T erfc X erfc

T T

Q X T X TT X erfc T X X erfc

T T

X TT X erfc

TC

C X TT X X erfc

T

2

0

expiCT X

C

(2.70)


On converting the problem defined in Eqs. (2.54)- (2.57) of semi-infinite domain,

0,X into a finite domain '0,1X by using the same transformation given in Eq.

(2.30), the non-dimensional advection-dispersion equation Eq. (2.54), together with initial

and boundary conditions Eqs. (2.55) - (2.57) become

22

' '

'2 '1 2 1

C C CX X

T X X

(2.71)

' '

'

0

1, exp log , 0 , 0,

1

icC X T X T

c X (2.72)

' '1, 1 , 0 , 0,

2 2

QTC X T X T

(2.73)

'

'0 , 1 , 0

CX T

X

(2.74)

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, 0.0001

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 .

Using two-level explicit finite difference methods, Eqs. (2.71) to (2.74) can be written as

65

2' '

, 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

X X

(2.75)

,0 '

0

1exp log , 0

1

ii

i

cC i

c X

(2.76)

0,

1 10

2 2 2

j

j

QTC j

(2.77)

, 1,0

M j M jC C j

(2.78)

The numerical solution of Eq. (2.75) can now be computed with initial and boundary

conditions, given in Eqs. (2.76) - (2.78).

2.3.3 Dispersion along Inhomogeneous Aquifer

To find the numerical solution for inhomogeneous formation, the same process has

been followed as discussed in 2.2.3. The inhomogeneous advection-dispersion equation in

grid form can be expressed in a similar manner and given in Eq. (2.47). The same initial

and boundary conditions are considered as that of the homogeneous case given in Eqs.

(2.76) to (2.78). The stability criterion follows the same for the advection-dispersion

equation and given in Eq. (2.48).

2.3.4. Illustration and Discussion

Considering the similar form of time-dependent unsteady velocity expressions i.e.

sinusoidal form and exponential decreasing form the analytical and numerical solutions for

homogeneous aquifer are computed for the input values of 0 0.01 /u km day ,

2

00.1 /D km day

, *

0.001 / km

, 0 1.0c , 0.01ic , 0.0001 /q day and it is

compared with that of inhomogeneous case for both the increasing and decreasing function

of F X as given in Eqs.(2.42) and (2.43) respectively. The flow resistance coefficient m

is chosen as 0.0165 / day for sinusoidal form of velocity. In case of the exponential form

of velocity, the same sets of values are considered except 0.0002 /m day . The value of

mt are chosen in the form of 13 2k , where 1k is whole number and in particular,

66

18 10k is taken into consideration for the present discussion. Here u t given in Eq.

(2.49a) is minimum and maximum alternatively for these values of mt which means that

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

number at regular interval of 182days. The groundwater level and the velocity are

minimum during the period in the month of June. This period is the peak of summer season

just before rainy season. However, the next of t corresponds to approximately in the month

of December, the peak of winter season, just after the rainy season, during which

groundwater level and velocity are maximum. From Eq. (2.48) the stability condition is

T should be less than 0.002. Here 0.0001T is considered which satisfy the stability

criterion. The concentration distribution behavior for sinusoidal form of velocity expression

in homogeneous formation is shown in Fig. 2.5a. The solid line and dashed line in Fig. 2.5a

represents the analytical and numerical solution for homogeneous aquifer respectively.

Here, it is observed that the contaminant concentration decreases with distance and

increases with time. Due to presence of some numerical error, the numerical solution

deviates from that of analytical one. The contaminant concentration has obtained large

value for numerical solution as compared to analytical solution. Fig. 2.5b represents the

concentration distribution behavior for exponential form of velocity expression in

homogeneous formation both analytically and numerically. Here, the same concentration

pattern has been observed as in sinusoidal form of velocity expression. However, the

concentration values at the origin for exponential form of velocity expression are small as

compared to the sinusoidal form of velocity expression. Fig. 2.6a and Fig. 2.6b represent

the numerical solution for inhomogeneous aquifer for sinusoidal and exponential form of

velocity expression respectively. The effect of inhomogenity has been taken into the

problem by considering increasing and decreasing nature of F X . From the Figs. 2.6a and

2.6b, it reveals that the contaminant concentration decreases with distance and increases

with time throughout the aquifer. Also, the contaminant concentration decreases more

rapidly in case of inhomogeneous aquifer as compared to that of homogeneous aquifer. It

shows that the effect of inhomogenity in the aquifer supports the concentration decreasing

behavior of groundwater flow. The contaminant concentration distribution pattern with

67

respect to distance and time in sinusoidal and exponential form of velocity expressions

obtained by numerical method for different form of F X is also depicted. On comparing

the two different types of velocity expressions, it is observed that the concentration

decreases more rapidly in case of exponential form of velocity expression as compared to

the sinusoidal form of velocity expression. The contaminant concentration values at the

boundary for analytical and numerical results are deviated due to some numerical error.

2.3.5 Conclusion

The solution of one-dimensional advection-dispersion equation is obtained by both

analytical and numerical approach for homogeneous porous formation and it is compared

with numerical approach of inhomogeneous porous formations. Initially, the domain is not

solute free i.e., some initial concentration exists in the aquifer. The initial concentration of

the aquifer is considered as exponentially decreasing function of space parameter x . The

temporally dependent groundwater velocity is chosen of two types: 1) the sinusoidal form

of velocity which describes the seasonal pattern of groundwater in a tropical region over a

year at a uniform time interval, 2) the exponential form of velocity which is decreasing in

nature. The analytical solution of the problem is obtained by Laplace transform technique

whereas for the numerical solution, explicit finite difference method is used. On comparing

the analytical and numerical solution, it is observed that nature of contaminant distribution

pattern is almost identical in both the cases. But the concentration value varies with respect

to position and time. The contaminant concentration decreases more rapidly for the solution

obtained for inhomogeneous aquifer as compared to the homogeneous aquifer. In most of

the cases, the contaminant concentration decreases with distance but increases with time

throughout the aquifer.

68

2.4 One-dimensional advection-dispersion equation with space dependent initial

concentration and Robin type time-dependent logistic sigmoid input concentration

2.4.1 Mathematical Formulation

Considering exponentially decreasing type space dependent concentration as

discussed in the second problem and time-dependent dispersion coefficient and seepage

velocity, one-dimensional isotropic semi-infinite aquifer is considered. The input

concentration is taken as Robin type time-dependent logistic sigmoid concentration at the

origin and the concentration gradient at an infinite extent is supposed to be zero. Hence,

one can write as

0 , 0, 01 exp

uccD uc x t

x qt

(2.79)

The analytical and numerical solution of advection-dispersion equation given in Eq. (2.1)

subjected to initial and boundary conditions given in Eqs. (2.51), (2.53) and (2.79); have

been obtained by using the Laplace transform technique and Finite-difference method

respectively.


Introducing the effect of homogeneity in similar manner as discussed earlier, from

Eqs. (2.6) and (2.7), the new time variable transformation defined in Eq. (2.9) and set of

non-dimensional variables given in Eq. (2.11), Eq. (2.79) becomes

11 , 0, 0,

2 2

C QTC X T

X

(2.80)

Eq. (2.54) subjected to initial and boundary conditions given in Eqs. (2.55), (2.57) and

(2.80) represent the problem in non-dimensional form. The analytical solution and

numerical solution of the problem can be obtained with the same methodology for

homogeneous and inhomogeneous aquifers. The significance of Robin type boundary

conditions over Dirichlet type boundary conditions may be observed in the numerical result

and discussions.

69

2.4.2a Analytical Solution

The analytical solution has been obtained with the help of Laplace transform

technique and it can be written as follows:

2

2

1exp

2 2 22 21( , )

21

1 exp2 22

11 exp 1

2 2 2 2 22 2

41

12

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

2

2

0 0

exp2 22

1exp

2 22

1exp e

2 22

1exp

2 22

i i

X T X TX erfc

T

X TT T X erfc T

T

c cX TT T X X erfc T

c cT

X TX erfc

T

2xp T T X

(2.81)


The numerical solution for homogeneous aquifer has been obtained by explicit

finite-difference method as discussed in earlier sections. Using two-level explicit finite

difference method, the advection-dispersion equation in grid form for homogeneous case

can be expressed as Eq. (2.75), together with initial condition Eq. (2.76) and boundary

condition at the other end of the aquifer given in Eq. (2.78). The Robin type temporally

dependent boundary condition given in Eq. (2.80) in non-dimensional form in finite region

0,1 can be written as

' '

'

11 1 , 0, 0,

2 2

C QTX C X T

X

(2.82)

Using finite difference scheme Eq. (2.82) becomes

70

'

1,

0, ' '1 , 0, 0,

21 2 1

j j

j

i i

C QTXC i j

X X

(2.83)

The numerical solution has been obtained for homogeneous aquifer by solving Eq. (2.75)

subjected to initial and boundary conditions given in Eqs. (2.76), (2.78) and (2.83).

2.4.3. Dispersion along Inhomogeneous Aquifer

In case of inhomogeneous formation the problem is described by Eq. (2.47) with initial and

boundary conditions given in Eqs. (2.76), (2.78) and (2.83). The numerical solution has

been obtained in a similar manner with Robin type boundary condition. The methodology

to solve and compute the problems remains same as discussed in 2.2.3.

2.4.4 Illustration and Discussion

For the illustration and discussion, two forms of unsteady groundwater velocities are

considered as given in Eqs. (2.49a) and (2.49b). The analytical and numerical solutions for

both homogenous and inhomogeneous formation (for both the increasing and decreasing

function of F X as given in Eq. (2.42) and (2.43) respectively) are computed for the input

values of 0 0.01 /u km day , 2

00.1 /D km day

, *

0.00001 / km

, 0 1.0c , 0.5ic ,

0.00001 / .q day The flow resistance coefficient m is chosen as 0.0165 / day for

sinusoidal form of velocity. In case of the exponential form of velocity the same sets of

values are considered except 0.0002 /m day . The value of mt are chosen in the form of

13 2k , where 1k is whole number and in particular, 14 6k is taken in the present

discussion. Here u t given in Eq. (2.49a) is minimum and maximum alternatively. For

these values of mt at 1182 121t k days, where 1k is the whole number at regular

interval of 182 days, the velocity profile has this kind of tendency. The groundwater level

and the velocity are minimum during the period in the month of June, the peak of summer

season just before rainy season. However, the next value of t corresponds to approximately

in the month of December, the peak of winter season, just after the rainy season, during

which groundwater level and velocity are maximum. The numerical solution has been

71

obtained for ' 0.02, 0.0001X T . From Eq. (2.48) the stability condition is T should

be less than 0.002 hence it can be observed that the stability criterion has been satisfied. Fig.

2.7 shows that the analytical solution for both sinusoidal and exponential form of velocity

expression in homogeneous aquifer. It is observed that the contaminant concentration

decreases with distance and increases with time. Near the other end of the aquifer the

concentration emerges at a point then shows the reverse pattern with respect to time, i. e.

after emergences the concentration decreases with time. It is also observed that the

concentration values in exponential form of velocity at each of the position are lower than

that of the sinusoidal form of velocity. Fig. 2.8 represents the numerical solution for

homogeneous formation with both sinusoidal and exponential types of velocity expression.

Here, it is observed that the contaminant concentration decreases with distance, but increases

with time throughout the aquifer. The contaminant concentration patterns for

inhomogeneous formation for both increasing and decreasing types of function F X have

been depicted in Fig.2.9. It is observed that the contaminant concentration decreases with

both distance and time throughout the aquifer. On comparing the two different types of

inhomogeneous function it shows that the contaminant concentration decreases more rapidly

for increasing function of F X as compared to the decreasing function of F X . Also, the

nature of contaminant concentration for both type of seepage velocities (sinusoidal and

exponential form) are almost identical to each other in all the three cases.

2.4.5 Conclusion

Considering Robin type of boundary condition, a comparative study has been made

between analytical and numerical solutions for one-dimensional isotropic semi-infinite

aquifer for both homogeneous and inhomogeneous formations. The aquifer is assumed with

some space dependent initial concentration and Robin type time-dependent sigmoid

function as a boundary condition. The analytical solution has been obtained by Laplace

transform technique and numerical solution is obtained by two-level explicit finite-

difference method. The contaminant concentration distribution pattern is depicted with time

and distance for homogeneous and inhomogeneous aquifer.

72

Fig. 2.1 Physical model of the problem

Fig. 2.2a Velocity pattern for sinusoidal type of unsteady velocity expression with different

values of 0u

1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 25000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Time(days)

Sin

uso

idal

Vel

oci

ty(K

m/d

ay)

u0=0.033

u0=0.036

u0=0.039

u0=0.042

73

Fig. 2.2b Velocity pattern for exponentially decreasing type of unsteady velocity expression

with different values of 0u

Fig. 2.3a Time-dependent contaminant concentration pattern subjected to sinusoidally

varying velocity expression using Laplace transform technique (solid line) and Fourier

transform technique (dotted line) for homogeneous aquifer.

1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 25000.02

0.022

0.024

0.026

0.028

0.03

0.032

Time(days)

Exponen

tial

Vel

oci

ty(K

m/d

ay)

u0=0.033

u0=0.036

u0=0.039

u0=0.042

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Distance Variable X

Conce

ntr

taio

n C

Curve mt Duration 26 5th Year June

29 5th Year Dec 32 6th year June 35 6th Year Dec 38 7th Year June

41 7th Year Dec

74

Fig. 2.3b Time-dependent contaminant concentration pattern subjected to exponentially

decreasing velocity expression using Laplace transform technique (solid line) and Fourier

transform technique (dotted line) for homogeneous aquifer.

Fig. 2.4a Time-dependent contaminant concentration pattern subjected to sinusoidal type

velocity expression using Finite difference method for both homogeneous and

inhomogeneous aquifers.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Distance Variable X

Conce

ntr

atio

n C

Curve time Duration 1576 5th Year June 1758 5th Year Dec 1940 6th year June

2122 6th Year Dec 2304 7th Year June 2486 7th Year Dec

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance Variable X

Conce

ntr

atio

n C

Numerical Solution for homogenous aquifer

Numerical Solution for inhomogeneous aquifer

for increasing function of F(X)

Numerical Solution for inhomogenous aquifer

for decreasing function of F(X)

Curve mt Duration

26 5th Year June

29 5th Year Dec

32 6th Year June

75

Fig. 2.4b Time-dependent contaminant concentration pattern subjected to exponentially

decreasing velocity expression using Finite difference method for both homogeneous and

inhomogeneous aquifers.

Fig. 2.5a Contaminant concentration pattern for sinusoidal form of velocity expression

using analytical (Laplace transform technique) and numerical (Finite difference method)

methods in homogeneous aquifer.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance Variable X

Conce

ntr

atio

n C

Numerical Solution for homogenous aquifer

Numerical Solution for inhomogeneous aquifer

for increasing function of F(X)

Numerical Solution for inhomogenous aquifer

for decreasing function of F(X)

Curve time in days Duration

1576 5th Year June

1758 5th Year Dec 1940 6th Year June

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

Distance Variable X

Concen

trat

ion C

Analytical Solution

Numerical Solution

Curve mt Duration

26 5th Year June

29 5th Year Dec

32 6th year June

76

Fig. 2.5b Contaminant concentration pattern for exponential form of velocity expression

using analytical (Laplace transform technique) and numerical (Finite difference method)

methods in homogeneous aquifer.

Fig. 2.6a Contaminant concentration pattern for sinusoidal form of velocity expression in

inhomogeneous aquifer subjected to increasing and decreasing function of F X .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Distance Variable X

Conce

ntr

atio

n C


1576 5th Year June

1758 5th Year Dec

1940 6th year June

Analytical Solution Numerical Solution

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

Con

cen

trat

ion

C

Curve mt Duration

26 5th Year June

29 5th Year Dec

32 6th year June

Increasing function of F(X)

Decreasing function of F(X)

77

Fig. 2.6b Contaminant concentration pattern for exponential form of velocity expression in

inhomogeneous aquifer subjected to increasing and decreasing function of F X .

Fig. 2.7 Contaminant concentration pattern for both sinusoidal and exponential form of

velocity expression in homogeneous aquifer using Laplace transform technique.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.35

0.4

0.45

0.5

0.55

Distance Variable X

Conce

ntr

atio

n C




1576 5th Year June

1758 5th Year Dec

1940 6th year June

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5004

0.5006

0.5008

0.501

0.5012

0.5014

0.5016

0.5018

Distance Variable X

Conce

ntr

atio

n C

Sinusoidal Velocity

Exponential Velocity


848 3rd Year June

1030 3rd Year Dec

1212 4th year June

78


velocity expression in homogeneous aquifer using Finite difference method


velocity expression in inhomogeneous aquifer using Finite difference method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5001

0.5001

Distance Variable X

Conce

ntr

atio

n C

Sinusoidal Velocity

Exponential Velocity


848 3rd Year June

1030 3rd Year Dec

1212 4th year June

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

Distance Variable X

Conce

ntr

atio

n C

Curve time in days Duration 848 3rd Year June 1030 3rd Year Dec

1212 4th year June



79

Table 2.1a Contaminant concentration values in sinusoidal form of velocity with

2

0 00.033 / , 0.33 /u km day D km day using

(i) Laplace transform technique and (ii) Fourier transform technique

X

(km)

0 1 2 3 4 5 6 7 8 9 10

26, 1576 mt t days

(i) 1.8477 1.4911 1.1343 0.8585 0.5759 0.3340 0.1887 0.1259 0.1058 0.1010 0.1001

(ii) 1.8477 1.7564 1.4278 0.9765 0.5663 0.2966 0.1648 0.1165 0.1032 0.1005 0.1001

29, 1758 mt t days

(i) 1.8381 1.4816 1.0805 0.7616 0.4798 0.2692 0.1572 0.1146 0.1028 0.1004 0.1000

(ii) 1.8381 1.7144 1.3475 0.8798 0.4859 0.2487 0.1441 0.1099 0.1017 0.1002 0.1000

32, 1940 mt t days

(i) 1.8109 1.4586 0.9673 0.5667 0.3056 0.1688 0.1167 0.1029 0.1004 0.1000 0.1000

(ii) 1.8109 1.6049 1.1514 0.6648 0.3281 0.1681 0.1149 0.1024 0.1003 0.1000 0.1000

35, 2122 mt t days

(i) 1.8034 1.4531 0.9447 0.5296 0.2756 0.1541 0.1120 0.1019 0.1002 0.1000 0.1000

(ii) 1.8034 1.5771 1.1048 0.6183 0.2981 0.1552 0.1111 0.1016 0.1002 0.1000 0.1000

38, 2304 mt t days

(i) 1.7745 1.4349 0.8833 0.4331 0.2031 0.1229 0.1036 0.1004 0.1000 0.1000 0.1000

(ii) 1.7745 1.4783 0.9489 0.4762 0.2169 0.1251 0.1037 0.1004 0.1000 0.1000 0.1000

41, 2486 mt t days

(i) 1.7683 1.4315 0.8745 0.4199 0.1938 0.1194 0.1028 0.1003 0.1000 0.1000 0.1000

(ii) 1.7683 1.4584 0.9194 0.4516 0.2046 0.1212 0.1029 0.1003 0.1000 0.1000 0.1000

80

Table 2.1b Contaminant concentration values in sinusoidal form of velocity with

2



X

(km)

0 1 2 3 4 5 6 7 8 9 10

26, 1576 mt t days

(i) 1.8187 1.4536 0.9357 0.5190 0.2688 0.1514 0.1113 0.1018 0.1002 0.1000 0.1000

(ii) 1.8187 1.5784 1.0993 0.6122 0.2944 0.1538 0.1107 0.1015 0.1002 0.1000 0.1000

29, 1758 mt t days

(i) 1.8074 1.4457 0.9057 0.4712 0.2321 0.1350 0.1066 0.1009 0.1001 0.1000 0.1000

(ii) 1.8074 1.5354 1.0288 0.5455 0.2545 0.1381 0.1066 0.1008 0.1001 0.1000 0.1000

32, 1940 mt t days

(i) 1.7750 1.4274 0.8548 0.3943 0.1777 0.1138 0.1017 0.1001 0.1000 0.1000 0.1000

(ii) 1.7750 1.4245 0.8615 0.4045 0.1824 0.1148 0.1018 0.1001 0.1000 0.1000 0.1000

35, 2122 mt t days

(i) 1.7660 1.4234 0.8477 0.3845 0.1713 0.1117 0.1012 0.1001 0.1000 0.1000 0.1000

(ii) 1.7660 1.3967 0.8227 0.3755 0.1698 0.1115 0.1012 0.1001 0.1000 0.1000 0.1000

38, 2304 mt t days

(i) 1.7317 1.4107 0.8400 0.3764 0.1665 0.1102 0.1010 0.1001 0.1000 0.1000 0.1000

(ii) 1.7317 1.2989 0.6961 0.2903 0.1378 0.1046 0.1003 0.1000 0.1000 0.1000 0.1000

41, 2486 mt t days

(i) 1.7242 1.4085 0.8416 0.3793 0.1684 0.1109 0.1011 0.1001 0.1000 0.1000 0.1000

(ii) 1.7242 1.2793 0.6726 0.2762 0.1333 0.1038 0.1003 0.1000 0.1000 0.1000 0.1000

81

Table 2.1c Contaminant concentration values in sinusoidal form of velocity with

2



X

(km)

0 1 2 3 4 5 6 7 8 9 10

26, 1576 mt t days

(i) 1.7872 1.4274 0.8461 0.3846 0.1723 0.1121 0.1013 0.1001 0.1000 0.1000 0.1000

(ii) 1.7872 1.4145 0.8395 0.3867 0.1744 0.1127 0.1014 0.1001 0.1000 0.1000 0.1000

29, 1758 mt t days

(i) 1.7739 1.4217 0.8384 0.3744 0.1658 0.1100 0.1010 0.1001 0.1000 0.1000 0.1000

(ii) 1.7739 1.3719 0.7811 0.3446 0.1572 0.1085 0.1008 0.1000 0.1000 0.1000 0.1000

32, 1940 mt t days

(i) 1.7359 1.4098 0.8423 0.3830 0.1718 0.1122 0.1014 0.1001 0.1000 0.1000 0.1000

(ii) 1.7359 1.2634 0.6459 0.2596 0.1281 0.1029 0.1002 0.1000 0.1000 0.1000 0.1000

35, 2122 mt t days

(i) 1.7254 1.4075 0.8482 0.3921 0.1777 0.1143 0.1018 0.1002 0.1000 0.1000 0.1000

(ii) 1.7254 1.2365 0.6152 0.2427 0.1233 0.1022 0.1001 0.1000 0.1000 0.1000 0.1000

38, 2304 m t t days

(i) 1.6851 1.4014 0.8830 0.4424 0.2110 0.1269 0.1048 0.1006 0.1001 0.1000 0.1000

(ii) 1.6851 1.1429 0.5172 0.1949 0.1118 0.1008 0.1000 0.1000 0.1000 0.1000 0.1000

41, 2486 m t t days

(i) 1.6763 1.4006 0.8925 0.4555 0.2199 0.1305 0.1058 0.1008 0.1001 0.1000 0.1000

(ii) 1.6763 1.1243 0.4993 0.1872 0.1102 0.1006 0.1000 0.1000 0.1000 0.1000 0.1000

82

Table 2.1d Contaminant concentration values in sinusoidal form of velocity with

2



X

(km)

0 1 2 3 4 5 6 7 8 9 10

26, 1576 mt t days

(i) 1.7532 1.4130 0.8405 0.3824 0.1721 0.1124 0.1014 0.1001 0.1000 0.1000 0.1000

(ii) 1.7532 1.2680 0.6441 0.2576 0.1274 0.1028 0.1002 0.1000 0.1000 0.1000 0.1000

29, 1758 mt t days

(i) 1.7378 1.4100 0.8513 0.3985 0.1825 0.1161 0.1022 0.1002 0.1000 0.1000 0.1000

(ii) 1.7378 1.2270 0.5976 0.2326 0.1206 0.1018 0.1001 0.1000 0.1000 0.1000 0.1000

32, 1940 mt t days

(i) 1.6937 1.4055 0.8978 0.4640 0.2263 0.1333 0.1065 0.1009 0.1001 0.1000 0.1000

(ii) 1.6937 1.1239 0.4923 0.1834 0.1095 0.1006 0.1000 0.1000 0.1000 0.1000 0.1000

35, 2122 mt t days

(i) 1.6816 1.4050 0.9133 0.4851 0.2410 0.1397 0.1084 0.1013 0.1001 0.1000 0.1000

(ii) 1.6816 1.0986 0.4688 0.1739 0.1077 0.1004 0.1000 0.1000 0.1000 0.1000 0.1000

38, 2304 mt t days

(i) 1.6348 1.4055 0.9776 0.5718 0.3046 0.1706 0.1190 0.1039 0.1006 0.1001 0.1000

(ii) 1.6348 1.0112 0.3945 0.1474 0.1036 0.1001 0.1000 0.1000 0.1000 0.1000 0.1000

41, 2486 mt t days

(i) 1.6246 1.4060 0.9921 0.5912 0.3196 0.1786 0.1221 0.1048 0.1008 0.1001 0.1000

(ii) 1.6246 0.9940 0.3811 0.1431 0.1030 0.1001 0.1000 0.1000 0.1000 0.1000 0.1000

CHAPTER II COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS …shodhganga.inflibnet.ac.in/bitstream/10603/33030/10/10_chapter 2.pdf · COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS FOR TIME-DEPENDENT

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