Analytical Solutions for One-dimensional Advection-dispersion Equation with Uniform and Varying poin by ijmrem journal

International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 2 ||Issue|| 4 ||Pages|| 39-50 || April 2019 || ISSN: 2581-4540

Analytical Solutions for One-dimensional Advection-dispersion Equation with Uniform and Varying point source in a Heterogeneous Porous medium 1, 1, 2,

R. R. Yadav, 2,Vijayshree Yadav

Department of Mathematics and Astronomy, Lucknow University, Lucknow-226007, India Department of Mathematics and Astronomy, Lucknow University, Lucknow-226007, India

-----------------------------------------------------ABSTRACT----------------------------------------------------In this study, analytical solutions for one-dimensional advection-dispersion equation in a semi-infinite porous domain with temporally dependent dispersion coefficients are considered. Seepage velocity is taken inversely proportional to spatially and temporally dependent function. It assumed that retardation factor is inversely proportional to square of the spatially dependent function. In this study, two cases for the boundary conditions are taken into consideration. The input condition is considered continuous of uniform and varying input of increasing nature both. The solution is obtained for the proposed mathematical model in a semi-infinite initially not solute free domain. The solution has been obtained by using Laplace Transform Technique. A comprehensive study on the effect of different geological parameters and heterogeneity of the medium on the solute movement during movement processes is carried out for heterogeneous porous medium. The developed analytical solution is illustrated using a hypothetical example and may help benchmark numerical codes and solutions. The effects of parameters on the solute concentrations with respect to position and time are explained with help of different graphs. In real scenario, such mathematical model is helpful for the assessment and effective remediation process. A new time variable is introduced through a known transformation.

KEY WORDS: Advection, dispersion, heterogeneous, point source, retardation factor. --------------------------------------------------------------------------------------------------------------------------------------Date of Submission: Date, 11 April 2019 Date of Publication: 18 April 2019 ---------------------------------------------------------------------------------------------------------------------------------------

INTRODUCTION

Solute transport in groundwater is affected by a various physical, chemical, biological and environmental processes and properties of the medium. Once the groundwater polluted by any means it becomes very difficult to improve its quality. So its management and redress becomes very challenging task for the engineers and hydrologist. Mathematical models of solute transport in porous media are the basic tools to understand the mechanism of transport phenomena in the geological formation. They can be used to forecast the movement of contaminant plumes and remedial mechanism in the field. A number of analytical solutions for conservative point, non-source are available in the literature. [1] observed that longitudinal dispersion coefficient is directly proportional to the mean velocity and depends with position. [2] observed that the longitudinal dispersion varies with time. [3] derived analytical solutions for one-dimensional solute transport with first-order decay and zeroorder production in a semi-infinite porous domain. [4] developed a finite element model adopting the concept that dispersivity varies temporally dependent function. [5] suggested that ground water velocity varies linearly with space and dispersion coefficient varies proportion to square of space. [6], [7] developed a method that describing linear time-dependent dispersivity and scale dependent dispersivity. [8] used Laplace Transformation technique to solved advection-dispersion equation in cylindrical coordinate in radially convergent flow. [9] developed a method for solving temporally dependent advection dispersion equation without discretize the derivative terms. The spatial dependence of groundwater velocity is causes due to the heterogeneous nature of the medium through which the solute transport takes place. The temporal dependence is due to unsteadiness of the groundwater flow. [10] worked on one-dimensional scale-dependent fractional advection-dispersion. [11] investigated the consequences of immobile water content on contaminant advection and dispersion in unsaturated porous formation. [12, 13,14] developed a numerical mathematical model by fractional advection diffusion equation in a heterogeneous porous formation. [15] developed an analytical solution with hyperbolic distance-dependent dispersivity on solute transport through porous media. [16] obtained analytical solutions for temporally and spatially dependent solute dispersion with pulse type input concentration in one-dimensional homogenous porous media. [17] developed analytical solutions with variable coefficients in one-dimensional porous media. [18] developed one-dimensional advection-dispersion equation in finite domain with time dependent inlet condition. [19] developed analytical solution to predict concentration distribution along unsteady groundwater flow in aquifer. [20] developed an analytical solution of advection diffusion equation in

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Analytical Solutions for One-dimensional Advection-dispersion… heterogeneous porous medium using Green’s function method. There are many mathematical studies executed earlier to estimate contamination levels at various position and time in aquifer. Most of the groundwater contaminants are non-conservative in nature and they normally migrate in the direction of groundwater flow. The transport of solute in surface and subsurface has long been a major attraction of experimental and theoretical study in subsurface hydrology. Attenuation of pollutant by dilution is the only mechanism which promptly minimizes the pollutant concentration in ground water and best remedy also. The objective of the present study is to develop the mathematical model to establish how solute transport, flow of pollutants and other parameters can be applied to determine the conduct of pollutants in the heterogeneous porous formation. The dispersion coefficients, seepage velocity of the advection-dispersion equation are considered as function of space and time variable while retardation factor is considered only space dependent. New space and time variables are introduced to reduce the variable coefficients into constant coefficients. The Laplace transform technique is applied to derive the analytical solutions. The retardation factor is also considered. Two cases for the boundary conditions are taken into consideration. The first case is of uniform continuous input condition and second one is of varying input of increasing nature. The aquifer being initially kept uniform concentration and exit boundary being kept concentration gradient zero at infinity. The effects of the parameters on the solute transport are studied separately with the help of graphs.

II. MATHEMATICAL FORMULATION AND ANALYTICAL SOLUTION The aquifer is assumed horizontal and semi-infinitely long in the x -direction. There is no solute flux at end of exist boundary. The problem is formulated mathematically as the point source in one-dimension. The domain is considered heterogeneous semi-infinite. The aquifer is supposed initially not solute free. The advectiondispersion equation in one-dimension may be written as follows:

R( x, t )

c( x, t )   c( x, t )    D( x, t )  u( x, t )c(x, t )  t x  x 

(1)

where c [ML3 ] be the concentration of solutes in the aquifer at position x L and time t T  . D [L2 T 1 ] and u [LT 1 ] are the dispersion parameter and seepage velocity along the x -axis respectively. Retardation factor R is also assumed space dependent which is a dimensionless quantity. First term on the left hand side of the (1) is represents change in concentration with time in liquid phase. The effect of molecular diffusion is not taken into account due to dominance of the mechanical dispersion on the hydrodynamic dispersion during solute transport. The medium is supposed to have a uniform solute concentration

ci before an injection of pollutant in the

domain. The constant contamination c0 is being entered through the left boundary condition. Two types input conditions are considered. First one is continuous uniform and second is varying type. [21] prompted that the Cauchy boundary is more realistic than the Dirichlet boundary. The uniform and varying type input are discussed in two separate cases. A solute of mass

is released into the flow field at time t  0 .

Uniform input point source condition: Mathematically initial and boundary conditions may be written as:

c( x, t )  ci ; x  0 , t  0 c( x, t )  c0 ; x  0 , t  0

(2) (3)

c ( x , t )  0; x   , t  0 x where

(4)

ci is solute concentration at origin. The constant contamination c0 is considered through the point source

at t  0, as the boundary condition. The contaminant flux is zero when x   . The dispersion, retardation and seepage velocity are temporally and spatially dependent along x -axis, are considered as follows:

u

u0 k1 1  ax k1  mt

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Analytical Solutions for One-dimensional Advection-dispersion…

 k1 D  D0   k1  mt

R

  

(6)

R0 1  ax 2 1

1

(7) ] is heterogeneity parameter and k 1 is

where m [ T ] may be termed as an unsteady parameter, a[ km arbitrary constant. To get free of the old-time dependent coefficient t following transformation [22] is used: t

k1 dt  mt 0 1 The dimension of T will be same as that of t . Let us introduce a new independent variable X using a transformation as: T 

k

dx 1 1 Or X  Or dX   (1  ax ) 2 a (1  ax ) dx (1  ax ) 2 Now using transformations (8,9), (1-4) reduce into following form: c ( X , T )  2c( X , T ) c ( X , T ) R0  D0a 2 X 2  2 D0 a 2  u 0 a X  au 0 c ( X , T ) T X 2 X c ( X , T )  c i ; X  0, T  0 X  









1 a c ( X , T )  0; X  0, T  0

c( X , T )  c0 ; X  , T  0

(8)

(9)

(10) (11) (12) (13)

X Now using another transformation as:

(14)

Z   log aX (10-13) reduces into new independent variable Z , T  as:

 c Z , T   2 c Z , T   c Z , T   D0a 2  w0  au 0 c Z , T  T Z 2 Z where, w 0  2 a 2 D 0  au 0





cZ , T   ci ; Z  0, T  0

(15)

(16) (17)

cZ , T   c0 ; Z  0, T  0 c Z , T   0; Z   , T  0

(18)

Z

To eliminate the convective term from (15), we introduce following transformation: 2  w   1  w 0 c ( Z , T )  K ( Z , T ) exp  2 0 Z   au 0 T  2   R 0  4 a D 0  2 a D 0   The (15-18) reduce as in new dependent variables K (Z , T ) :

 K Z , T   2 K Z , T   a 2 D0 T Z 2

(20)

w Z  K Z , T   ci exp 2 0 ; Z  0 , T  0  2a D0   1  w 2   K Z , T   c0 exp  20  au0 T ; Z  0, T  0  R0  4a D0  

w K Z , T   20 K Z , T   0; Z   , T  0 Z 2 a D0 www.ijmrem.com

(19)

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(22)

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Analytical Solutions for One-dimensional Advection-dispersion… Applying Laplace transformation technique (20-23) reduces into ordinary differential equation as:

w Z  a 2 D 0 d 2 K Z , p   p K Z , p   c i exp  2 0  2 R0 dZ  2 a D0  c0 ;Z  0 K Z , p   2   1  w0 p  au 0  R 0  4 a 2 D0  w0 d K Z , p   K Z , p   0 ; Z   dZ 2a 2 D0

(24) (25)

(26)

Solving (24-26), first we obtain K ( Z , p ) and then taking inverse Laplace transformation of K ( Z , p ) and using (19) we get c Z , T  as a solution:

  R p  R0 p  w Z  c0 exp  20 Z  Z ci exp  2 0  ci exp  2 a D a D 2 a D 0 0     0   K Z , p    2 2 2  w0 w0 1  w0 p 2 p 2 p   2  au 0  4a D0 R0 4a D0 R0 R0  4a D0 

(27)

            R0   Z     T    exp Z a2 D erfc 2 0  2 a D0T        c   R0    0  expT    2            Z   expZ R0 erfc   T    2 2  a D   0    2 a D0T         2 R    0     exp w0 Z  1  w0  au T  cZ ,T    0 2 2      2a D0 R0  4a D0            R0   Z     T    exp Z a2 D erfc 2  0  2 a D0T         R0   c  w Z     i expT  ci exp 2 0  T     2    2a D0         Z   expZ R0 erfc  T    2 2    a D0   a D0T      2     R 0     

(28) 2

where,  2 

w0 1 and  2  2 4a D0 R0 R0

 w0     4 a 2 D  au 0  0  

Varying input point source condition: Uniform nature input condition on the earth surface exist at all time is not possible in real situations. In fact, due to human or other natural activities on the surface, the pollutant at the source may increase with time. Such situation may mathematically defined by mixed type condition:

 D ( x, t )

c ( x , t )  u ( x , t ) c ( x , t )  u 0 c 0 ; x  0, t  0 x

Using same transformation (8,9), (30) reduces into following form: 1  mT  ; c ( X , T ) aD 0 X  u 0 c ( X , T )  u 0 c 0 exp   X  ,T 0 a X  k1  With the help of (14), (30) may be written in another variable c ( Z , T )

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Analytical Solutions for One-dimensional Advection-dispersion…

 aD0

 mT  c ( Z , T )  u 0 c ( Z , T )  u 0 c0 exp   ; Z  0, T  0 Z  k1 

(31)

Using (19) in (31), we get:

 aD0

 m 1  w 2  w  K (Z , T )    u0  0 K (Z , T )  u0c0 exp   20  au0 T ; Z  0, T  0 Z 2a    K1 R0  4a D0 

(32)

Taking the Laplace transformation of (32), it reduces into following form:

 u c w  d K Z , p     u 0  0  K Z , p    0 0 2 dZ 2 a    p ' 2  m 1  w0 where,  ' 2    au 0  2  K 1 R0  4 a D 0   aD0

  ; Z  0 

(33)

Now, we first solve boundary value problem represented by (24, 33, 26) for value of K ( Z , p ) .  w Z  ci exp  2 0   2 a D0   K (Z , p)  2   w0 p  2    4 a D R 0 0   p  



  u 0ci   R D 0 0 

    exp   R 0 p Z  2  a D 0     w0    u0  2  w0 2a    p  4 a 2 D 0 R 0   R0 D 0  

     

(34)

 u 0c0     exp    R D  0 0       p   m  1   K 1 R0     

R0 p  Z a 2 D0   w    u0  0 2     w0 2 a  au 0   p    4 a 2 D 0 R0 R D    0 0 

     

Applying inverse Laplace transform on (34) we get K (Z , T ) and by (19), we get solution in variable c ( Z , T ).

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Analytical Solutions for One-dimensional Advection-dispersion…             2 2 2    w T      1 w0 R0 w0 T    0     exp  Z erfc Z       2  4 w0    4a 2 D0 R0  4a 2 D0T 4a 2 D0 R0   4 a D      0     u   2 0        w0 2a     2   2 4 a D R R D      0 0 0 0            2 2 2 2             w Z w T  u0ci  w0 R0 w0 T  1   w0 T    ci exp 2 0  2 0   Z erfc Z    exp 2      2 2 2 4 w0   4a D0 R0 4 a D T 4 a D R  2a D0 4a D0 R0   R0 D0   4 a D    0 0 0   0  u0     2   w0     2a 2        2 4 a D R R D  0 0    0 0             2      w0  w0     w0   u  u          u0  0 0  R0 w0  T  2a  2a        2a     T Z erfc Z   u0   exp  2   u0  w0 aD0 4 a D T 2 a D R     R0 D0      0 0 0     u     0    R0 D0  a              cZ , T               2     w0 2  R0    w0 2    R0 1 m 1 m 1    m 1  w0  T      T  exp   au   au Z erfc Z    au      0 0 0  2 2 2 2 2     a D  w  k1 R0  4a D0 k1 R0  4a D0 4a D0T k R0  4a D0 0          u0  0    1   2    w m 1   2  2a    0  au0      k R  4a 2 D    R0 D0  0  0    1          2 2 2             u c 1 m 1 w m 1 w R R m 1 w         0 0 0 0  au0 T    20  au0  2 0 Z erfc Z   20  au0  T  exp   2  R D     4a D  a D  4a D  w0    k1 R0  4a 2 D0 k R 4 a D T k R 1 0 0 0 0 1 0 0 0 0              2  u0  2a      m 1  w0    2   au   0     k1 R0  4a 2 D0 R0 D0              2     w0  w0      w0     u0     u0    u0     R0 w0  T  2a  2a        2a    exp T  Z erfc Z  u         0 2  2 aD0 2a  D0 R0    w0    R0 D0      4a D0T u      2 0      2a      R D  m  1  w0  au       0 0 0  2    R0 D0   k1 R0  4a D0     2  w0   1  w0   exp 2 Z   au0 T    R0  4a 2 D0  2a D0  

(35)

III. RESULTS AND DISCUSSION Fig.(1-5) and Fig.(6-10) illustrate the analytical solutions (28) for uniform input point source and (35) for varying input point source in a longitudinal domain 0  x  6 respectively. The common input values are chosen same as published in literature. These are c 0  1000 mg / l ,

c i  10 mg / l , m( year 1 )  0.1 , u 0 ( km / year )  0 .15 , D 0 ( km 2 / year )  0 . 55 , R 0  1 . 21 , k 1  0 . 1 and value of heterogeneity parameter a(km) 1  0.1 which lies within the range between 0.05 and 0.1 for a heterogeneous porous formation [4]. Uniform input source graphs:

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Analytical Solutions for One-dimensional Advection-dispersion… 1.0

0.8

cc0 

D0 =0.55 km2 year1 0.6

D0 =0.75 km2 year1 0.4

D0 =0.95 km2 year1

0.2

0.0

Distance x km 

Figure1 Contaminant concentration for uniform input described by (28) for different values of dispersion coefficient with fixed retardation R 0  1 . 21 and at time t  3 years Fig. 1 illustrate the dimensionless concentration profiles computed for different values of dispersion parameter D 0 ( km 2 / year )  0 .55 , 0 . 75 , 0 .95 at fixed retardation R 0  1 . 21 and at time t  3 years . It reveals the concentration distribution pattern originating from a continuous source. The concentration level is recorded higher for higher value of dispersion inside the study domain.

1.0

0.8

cc0

t=3 years 0.6

t=4 years 0.4

t=5 years 0.2

0.0

Distance x km 

Figure 2 Contaminant concentration for uniform input described by (28) for a varied value of time t ( year )  3, 4,5 with fixed retardation R 0  1 . 21 and D 0  0 .55 km 2 / year Fig.2 shows the concentration profiles at different times t ( year )  3, 4,5 is described. It is clear that as x increases (longitudinal distance) the value of the concentration decreases for any time. The solute concentration is recorded higher for higher value of time and lower for lower value of time. Concentration patterns for all the three values of time are same.

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Analytical Solutions for One-dimensional Advection-dispersion… 1.0

cc0 

0.8

R0 =1.21

0.6

R0 =1.51 0.4

R0 =1.81

0.2

0.0

Distance x km 

Figure 3 Contaminant concentration for uniform input described by (28) for different values of retardation with fixed D 0  0 .55 km 2 / year and at time t  3 years Fig.3, shows the variation of concentration profile at different retardation factor. It is observed that concentration increases as the retardation factor decreases. Concentration pattern remains same for all values of retardation. For any fixed value of retardation as x increases, we notice that concentration decreases and reaches its minimum value nearly at x  4.2 km.

1.0

cc0 

0.8

m  0.1 year1

0.6

m  0.2 year1

0.4

m  0.3 year1 0.2

0.0

Distance x km 

Figure 4 Contaminant concentration for uniform input described by (28) for different values of unsteady parameter time t  3 years and D 0  0 .55 km 2 / year .

Fig. 4 shows the effect of the unsteady parameter on the solute transport. From the figure it is clear that as the value of parameter increases, the concentration level decreases. No significant change in pattern was noticed for different values of unsteady parameter. It is observed that the solute concentration patterns follow almost the same trends at different unsteady parameter in the domain. In all the three values of unsteady parameter concentration rehabilitates approximately upto 4 km .

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Analytical Solutions for One-dimensional Advection-dispersion… 1.0

a =0.1 km 1 c c 0 

0.8

a =0.3 km1

0.6

a =0.5 km1

0.4

0.2

0.0

Distance x km 

Figure 5 Contaminant concentration for uniform input described by (28) for different values of heterogeneity parameter a at time t  3 years and D 0  0 . 55 km 2 / year . Fig. 5 demonstrates effects of heterogeneity parameter a over solute distribution in study domain. A significant variation in concentration pattern may be observed for small change in parameter a . Concentration remains higher for higher value of parameter and lower for lower. In all the three values of heterogeneity parameter a concentration rehabilitates trends are similar. Varying input source graphs: 0.8

D0 =0.55 km2 year1

0.6

c c 0 

D0 =0.75 km2 year1 D0 =0.95 km2 year1

0.4

0.2

0.0

Distance x km 

Figure 6 Contaminant concentration for varying input described by (35) for different values of dispersion coefficient with fixed retardation R 0  1 .21 and at time t  3 years Fig. 6 depicts dimensionless concentration profiles evaluated for different values of dispersion parameters

D 0 ( km 2 / year )  0 . 55 , 0 . 75 , 0 . 95 . Dimensionless concentration c / c 0 is recorded lower for higher value of dispersion parameter near the source but after some distance the pattern reverses. It is also observed concentration attenuates up to x  4 km for all three values of dispersion parameter.

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Analytical Solutions for One-dimensional Advection-dispersion… 1.0

0.8

c c 0 

t=3 years 0.6

t=4 years 0.4

t=5 years 0.2

0.0

Distance x km 

Figure 7 Contaminant concentration for varying input described by (35) for different values of time with fixed retardation R 0  1 . 21 and D 0  0 . 55 km 2 / year Changes in concentration pattern with time is represented in Fig. 7. Concentration pattern for all the three values of time are same. The solute concentration is noticed higher for higher value of time and lower for lower value of time. Fig.7 agrees with pattern of varying nature input. 0.8

c  c 0 

0.6

R0 =1.21

0.4

R0 =1.51 R0 =1.81

0.2

0.0

Distance x km 

Figure 8 Contaminant concentration for varying input described by (35) for different values of retardation with fixed dispersion coefficient D 0  0 .55 km 2 / year and at time t  3 years Fig. 8 analyzed concentration variations for different values of retardation factor. It may be observed that concentration increases as the retardation factor decreases. Concentration pattern remains nearly same for all values of retardation factor R 0  1 .21 , 1 . 51 , 1 .81 . Concentration rehabilitates to its lowest level up to distances x  3 . 6 km for all the three values of retardation.

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Analytical Solutions for One-dimensional Advection-dispersion… 1.0

cc0

0.8

0.6

0.4

m  0.1 year1 m  0.2 year1 m  0.3 year1

0.2

0.0

Distance x km

Figure 9 Concentration profile for varying input described by (35) for different values of unsteady parameter coefficient m with fixed retardation R 0  1 . 21 and at time t  3 years Fig. 9 demonstrates the effect of the unsteady parameter m on concentration distribution. Dimensionless concentration increases as the unsteady parameter increases near the origin but this pattern reverses after distance x  0 .5 km . Concentration level deceases to its lowest level up to distance x  4 km for all three values of unsteady parameter. 1.0

0.8

a =0.1 km1

cc0

0.6

a =0.3 km1

0.4

a  0.5 km1 0.2

0.0

Distance x km 

Figure 10 Concentration profile for varying input described by (35) for different values of heterogeneity parameter a with fixed dispersion coefficient D 0  0.55 km 2 / year and at time t  3 years Fig. 10 elaborates the effect of heterogeneity parameter a over solute concentration distribution in the study domain for fixed dispersion coefficient D 0  0 .55 km 2 / year at time t  3 years . Throughout the domain, concentrations level increases as the heterogeneity parameter increases. Concentrations pattern are almost same for all three values of heterogeneity parameter.

IV. CONCLUSION This study derives the analytical solutions for one-dimensional advection dispersion equation ADE with spatiotemporal coefficients in a semi-infinite porous domain for uniform and increasing nature of sources. Simple transformation equation introducing a new time variable was needed to transform the ADE with variable coefficients into constant coefficients. Laplace transformation technique is used to get the analytical solutions. The input source concentration is considered as continuous and varying nature. Aquifer domain is assumed to be heterogeneous and semi-infinitely long. Initially the aquifer is not supposed to solute free. It assumed that retardation factor is inversely proportional to square of the spatially dependent function. The obtained solution of advection-dispersion equation can be imposed to predict the field problems where hydrological properties of the medium, initial and boundary conditions are same as or can be approximated by ones considered in this

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Analytical Solutions for One-dimensional Advection-dispersion… study.These solutions may be applicable as predictive tools in groundwater resource management to predict the field problems where hydrological properties of the medium, initial and boundary conditions are same as or can be approximated by ones considered in this study. The derived result could be possible to show the possible effects of spatial and temporal variations of velocity, dispersion coefficient and other parameters on the solute transport due to continuous and varying point sources.

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[19] [20] [21] [22]

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