CHAPTER II COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS...
Transcript of CHAPTER II COMPARATIVE STUDY OF ANALYTICAL SOLUTIONS...
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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
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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)
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,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).
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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
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. 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.
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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
2.3.2 Dispersion along Homogeneous Aquifer
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)
2.3.2b Numerical Solution
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.
2.4.2 Dispersion along Homogeneous Aquifer
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)
2.4.2b Numerical Solution
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
Curve time in days Duration
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
Increasing function of F(X)
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 10.5004
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
Curve time in days Duration
848 3rd Year June
1030 3rd Year Dec
1212 4th year June
78
Fig. 2.8 Contaminant concentration pattern for both sinusoidal and exponential form of
velocity expression in homogeneous aquifer using Finite difference method
Fig. 2.9 Contaminant concentration pattern for both sinusoidal and exponential form of
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
Curve time in days Duration
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
Increasing function of F(X)
Decreasing function of F(X)
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
0 00.036 / , 0.36 /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) 1.8187 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
0 00.039 / , 0.39 /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) 1.7872 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
0 00.042 / , 0.42 /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) 1.7532 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