 Open Access
 Authors : Dr. Ramesh T , Rekha J , Dr . Suma S P , Dr. S R Sudheendra
 Paper ID : IJERTV10IS090196
 Volume & Issue : Volume 10, Issue 09 (September 2021)
 Published (First Online): 29092021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mathematical Modelling of Reactive Transport of Contaminants in Heterogeneous Flow of Saturated and Unsaturated Media
Dr. Ramesh. T1
Department of Mathematics .
Cambridge institute of Technology, Bengaluru ,India
Dr. Suma S P 3
Department of Mathematics .
Cambridge institute of Technology,Bangaluru India
Rekha J2
Department of Mathematics
Cambridge institute of Technology, Bengaluru India
Dr. S R Sudheendra4
Department of Mathematics .
Presidency University Bengaluru India
Abstract In this paper, an analytical first order solution to the onedimensional advectiondispersion equation
0
0
with adsorption term C 1 et to study the transport of
II MATHEMATICAL MODEL
The Advection Dispersion Equation along with Initial conditions and Boundary conditions can be written as
pollutant vary exponentially with time using a generalized
C 2C C 1 n
integral transform method. In this chapter, we have investigated the transport of pollutant for adsorbing medium and non adsorbing for reactive and nonreactive flow of saturated and unsaturated porous media. The solution is derived under
t D z2
w z
Kd C
n
n
(1)
conditions of steadystate flow and arbitrary initial and inlet boundary conditions using Duhamels theorem and integral transform methods.
Keywords Integral Transforms Method, Mathematical Modeling, Advection Dispersion Equation,Saturated porous media, unsaturated porous media, heterogeneous flow.

INTRODUCTION
In this study, the Advection Dispersion Equation has been resolved by using analytical methods to measure the solute transport of contaminants by considering the porosity and dissipation coefficient using input concentrations.
The Advection Dispersion Equation has been extensively used in simulation and modelling of reactive transference in groundwater. The convectiondispersion [Bear (1979), Gelhar (1973), Domenico (1998), Fetter (1999), Sudheendra (2010, 2011)] included hydrodynamic dispersion, advection, adsorption, firstorder decay reaction, and possibly zeroth order production.
Analytical solutions are generally derived from the basic physical principles and different from numerical dispersion. The other truncation errors usually occur in numerical simulations [Zheng and Bennett (1995)].
Consider a semiinfinite porous medium in a unidirectional flow field in which the input tracer concentration is C 1 et , where C0 is a reference concentration and
0
0
is a constant as show in the following figure (1).
Figure 1: Schematic representation of semiinfinite porous medium in unidirectional flow field. Source concentrations is
Hence, the suitable Boundary conditions for the given model
The solutions of onedimensional Advection Dispersion Equation have been examined in many previously and are still
actively studied. For example, Sauty (1980) and van
C(z, 0) 0
t
t
C(0, t) C0 1 e
z 0
t 0 .
Genuchten (1981), Sudheendra (2014) have been providing analytical solutions of the solute transference equation with
C(, t) 0
t 0
(2)
the first, second and third type Boundary conditions. Yeh (1981) has given by analytical method and computer code for assessing the waste transportation in groundwater aquifers.
Then the problem is describe the concentration as a function of z and t, where the input condition is supposed at the source and a 2nd type condition is supposed. C0 is initial
concentration. To reduce equation (3) in a more conversant form, we take
1 erf (
2
z ) 2
Dt
e 2 d
wz
w2t K 1 n t z
C z, t z, t
Exp d
2 Dt
2D 4D
n
(3)
By Duhamels theorem, the result of the problem with C0 zero and the time dependent surface condition at z = 0 is
D
By substituting equation (3) in (1), we get
2
(4)
Since e 2 is a constant function, , it becomes
t z 2
The Initial Conditions and Boundary conditions transform
equation (2) to
2
e 2 d z
Exp
z 2
t
2 D t 32
4D t
w2t
K 1 n t z
d
d
0, t
C0
Exp
4D n
t
t 0
2 Dt
The solution to the problem is
z, 0 0
, t 0
z 0
t 0
z t
z 2 d
2 D
Exp 4D t t 32
(5)
0
(7)
Equation (4) can be resolved for a time dependent arrival of the fluid at z = 0. The solution of equation (4) can be attained quickly by use of Duhamels theorem, the same method already been used in the previous chapters.
Putting
2
becomes
z
D t
then the Equation (7)
The boundary conditions are
2
z 2
2
0, t 0
t 0
t 4D 2 e d .
z
z, 0 1
, t 0
z 0
t 0
2 Dt
(8)
The Laplace Transform of equation (4) is
w2t
K 1 nt
t C0 Exp d t
the particular
2
4D n
L t D z 2
solution of problem be written as
Later, it is reduced to an Ordinary Differential Equation
2 p
2C w2t
K 1 nt
2
2
z, t 0 Exp
d t Exp 2
d
Exp 2
d
4D n
2
2
z2 D
(6)
0
0
(9)
The resulting equation is
p
Aeqz Beqz
where,
Where, z
2 D t
and
q .
D
w2 K 1 n z
d
The Boundary conditions as z requires that B = 0 and
4D n
2 D
B C at z = 0 requires that
A 1
p
thus the solution of
III EVALUATION OF THE INTEGRAL SOLUTION
By integrating the 1st term of equation (9) we get
the problem using Laplace transform method is
2
2
2
Exp 2 d
e2 .
1 eq z
p
0 2
(10)
The inverse Laplace transform of the given function using the table. The result is
For convenience the 2nd integral can state in terms of the error function, thus the function was perfectly tabulated.
Observing that
2 2
2
2
2
2
2 .
2 2
2
2C w2t
K 1 n t
2 2
2 .
z, t 0 Exp d t
2
4D n
The 2nd integral of equation (9) may be written as
2
2
2
p/>
1 2
2
2
2
1 2
2
I Exp
2 d 2 e
Exp d e
Exp d
e2
e2
e
d e2
e
d
0
0
0
2
(11)
2
Only by considering the 1st term of Equation. (11). Let
(15)
a and the integral can be stated as
But, by definition
2
I1 e2 Exp
d
2
0
e2
e
d
e2
2
erfc
2
2
e2 1 Exp
2
a da e2 Exp

a da
/ a
a
/
a
2
2
(12)
also, e
e
d
e 2 1 erf
2
Added, let, a in the a
1st term of the
Consider equation.(15) in terms of error functions, we have
a a
C w2t K 1 n t
above Equation, then
z, t 0 Exp d t
2
2 4D n
I1 e2
e 2 d e2 Exp a da .
2 2
a
e erfc e
erfc ………..(16)
(13)
Similar evaluation of the 2nd integral of Equation (11) gives
By substituting this in equation (3) which gives
2
2
2
I e2
Exp
a da e2
Exp

a da
e erfc
2 a
a
C 1 wz
/
/
Exp
t
……..(17)
. C0 2
2D
e2 erfc
Again substituting
a a
into the 1st term, the
I 2 e2
solution is
e 2 d e2
Exp a
2
da .
Resubstituting for and gives
/
a
IV ADSORBING MEDIUM:
Observing that
C0 z w
z t
2 2
Cz, t
2 exp 2D erfc 2 Dt
Exp a 2 da Exp a 2 da
/
a
/
a

exp z w erfc z t exp t
2D
Dt
Substitute in equation (11) gives
z w
z t
z w
z t
1
I e
2
e 2
d e2
e 2
d
exp
2D
erfc 2
exp
Dt 2D
erfc 2
Dt
2
..(14)
in which
1
.(18)
Hence, equation (9) can be expressed as
w2 n 4D K (1 n) 2
d
(19)
n
And
d
d
1
1
w2 n 4D K (1 n) n 2
realistically in figures 5 to 9. Figures 5 to 9 represent the Break through Curves for C/Co vs time for different depth z. It is seen that the concentration field increases in the initial and
C0
n
z wt
wz
(20)
z wt
reaches a constant state value for a static z but decreases with an increase in the layer width. Similar pattern is observed in figures 5 to 9 for distinct values of w and D.
Figures 5 to 9 represent the BreakThroughCurves for C/C0, and is maximum at the surface z=0 and decreases to reaches zero at the depth of 100 meters. With an increase in most of
Cz, t
erfc
2 2
exp erfc
Dt D 2 Dt
the contaminants get absorbed by the solid surface and thus
suspending the movements of the contaminants as evident from the graphs. The majority of the pollutants are reduced in
exp t xp z w erfc z t exp z w erfc z t
unsaturated area itself, hence the risk of ground water being
Dt
Dt
e
2D
2 Dt
2D
2
polluted is diminished.
(21)
in which in which
w2 4D 12 . (22)
where boundaries are symmetrical then the solution of problem is specified by the 1st term in Equation (18) and (21).and the 2nd term in equation (18) and (22) is thus due to the asymmetric boundary imposed in the more general problem. Though, it should also be noted that if a point an immense distance away from the basis is measured, then it is probable to estimate the boundary condition
by C , t C0 , which tends to a symmetrical solution.
VI RESULTS AND DISCUSSIONS
This water finally enters the groundwater storage basin – a basis for drinking water. Throughout the passageway of water through the soil, the pollution is combined, and then mixing takes place in the soil medium by two processes, viz., molecular diffusion and dispersion. Molecular diffusion is a physical process, which depends upon the kinetic properties of the fluid particles and causes mixing at the contact front between the two fluids.
Dispersion, however, is defined as a mechanical mixing process caused by the twisting path followed by the fluid owing in the geometrically complex interconnections of the flow channels and by the variations in equations relating to solute transport are resolved analytically and numerically. Analytical solutions for the onedimensional model are obtained using Laplace transformation techniques.
To estimate the magnitude of the danger posed by some of these chemicals, it is important to investigate the processes that control their movement from the soil surface through the root zone down to the ground water table. At present, foremost drive on the transportation of pollutants and research is directed towards the definition and quantification of the process governing the behaviour of pollutants in subsurface environment, combined with the development of mathematical models that integrate process descriptions with the pollutant properties and site characteristics.
From the equation (18) and (22), C/C0 was numerically computed using `Mathematica' and the outcomes are presented
Table 1:Esimated parameters and statistical criteria for advection dispersion equation and spatial fractional advection dispersion equation at various distances in hetrogeneous soil
Figure 2:Comparision of the observed breakthrough curves with those fitted by advection dispersion equation and spatial fractional advection dispersion equation for heterogeneous soil at different distances
Table 2: Values of statistical criteria as the indicators of performance for advection dispersion equation and spatial fractional advection dispersion equation at subsequent distancesusing the best estimated parameters at 30 cm
Figure 3: Predicted breakthrough curves in the homogeneous soil at different distances by advection dispersin equation and spatial fractional advection dispersion equation using parameters determined at 30 cm
Figure 4: Predicted breakthrough curves in the heterogeneous soil at different distances by advection dispersion equation and spatial fractional advection dispersion equation using parameters determined at 30 cm
Figure 5: BTC for
,
Figure 6: BTC for
,
Figure 7: BTC for
,
Figure 8: BTC for
,
Figure 9: BTC for
5,
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