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One dimensional advection dispersion equation is analytically solved initially in solute free domain by considering uniform exponential decay input condition at origin. Heterogeneous medium of semi infinite extent is considered. Due to heterogeneity velocity and dispersivity coefficient of the advection dispersion equation are considered functions of space variable and time variable. Analytical solution is obtained using Laplace transform technique when dispersivity depended on velocity. The effects of first order decay term and adsorption are studied. The graphical representations are made using MATLAB

Managing the groundwater resources and rehabilitation of polluted aquifers, mathematical modeling is a powerful tool. The contaminant concentration distribution behaviour along/against unsteady groundwater flow in aquifer is studied through mathematical modeling as it is an important approach to formulate the geo-environmental problems and provides the best possible solution for reducing its impact on the environment. The pollutant’s solute transport from a source through a medium of air or water is described by a partial differential equation of parabolic type derived on the principle of conservation of mass, and is known as advection–diffusion equation (ADE). In one-dimension it contains two coefficients, one represents the diffusion parameter and the second represents the velocity of the advection of the medium like air or water. In case of porous medium, like aquifer, velocity satisfies the Darcy law and in non-porous medium, like air it satisfies the laminar conditions. The dispersive property differs from pollutant to pollutant.

The literature contains analytical solutions for solute transport in homogenous and heterogeneous porous media. Analytical solutions in one-, two-, and three-dimensional advection-dispersion transport equations with constant coefficients in homogeneous medium which have been collected in various compendiums [1-4]. Some more works in homogeneous medium has been compiled [5-10]. Using the theory [

In the present work one-dimensional advection diffusion equation is solved for dispersivity depended on square of velocity. The medium is of inhomogeneous nature and is of semi infinite extent. Due to inhomogeneous medium both the parameters dispersion and fluid velocity depends on space and time. Initially aquifer is considered to be solute free. The input point source is of exponentially decreasing nature at the origin and at the other end its concentration gradient is considered to be zero. The effect of first order decay of temporally dependent and adsorption is also considered in this work to get the physical insite of the problem. Laplace transform technique is used to obtain the analytical solution.

The linear Advection-Diffusion partial differential equation in one dimension in general form with absorption and decay term may be written as

where is the solute concentration at a position at time, represents the solute dispersion and is velocity of the medium transporting the solute particles, is first order decay or production term [T^{–}^{1}], is source/sink of dimension [ML^{–3}T^{–1}], is empirical constant and is the porosity. Initially the medium is solute free. An exponential decay type input point source concentration is assumed at the origin of the medium of uniform nature where is the contaminant decay rate of dimension inverse of time [T^{–1}]. It means that the input concentration decreases with time at the source. The second boundary condition is assumed to be of second type (flux type) of homogeneous nature. Thus the initial and two boundary conditions are as follows:

and

In [

where the coefficient is the heterogeneity parameter of dimension inverse of that of space variable, and is an unsteadiness parameter of dimension inverse of that of time variable, , and in above expressions referred as initial dispersion coefficient of dimension [L^{2 }T^{–1}], initial velocity of dimension [L^{2} T^{–1}] and initial firs order time decay rate of dimension of inverse of time [T^{–1}].

Using the expressions (5), the advection-diffusion Equation (1) can now be written as

or

Let us introduce a new time variable defined by [

The dimension of is same as dimension of, so it is referred to as a new time variable. An expression for chosen such that for, we get the value of, so that the initial condition not affected in new time domain. Also a space variable transformation is introduced [23,46] as

The initial value problem together with their initial and boundary conditions in new time and space variable becomes

and

where is another time dependent expression in non-dimensional variable and is non dimensional coefficient.

To eliminate the first order decay term form the Equation (10), introducing the transformation as:

With the use of Equation (14), Equation (10) becomes

Further using a space variable and time variable through the transformations as:

and

The one-dimensional advection-diffusion Equation (15) with their initial condition (11) and boundary conditions (12)-(13) may now be written as

The time variable has to be expressed explicitly in terms of. An expression of exponentially decreasing nature is chosen as

So from Equation (8), we get

or

Also using the transformation in Equation (17) we get

or

or

In, is much smaller than one, so its second and higher degree terms in the logarithmic and binomial expansions in above equations are omitted. So we get

, where (23)

Thus the initial value problem (18) and their conditions (19)-(21), becomes

where.

Now to find the analytical solution for Equation (24), Laplace transform technique is used, but to apply it more conveniently the convective term from the Equation (24) is to be removed by the use of the transformation as

The initial and boundary value problem from (24-27) in terms of new dependent variable may now be written as

where and

Applying Laplace transformation on the above boundary value problem, the problems become in second order ordinary differential equation in the Laplacian domain as :

and

After using the boundary conditions (34) and (35), its particular solution may be obtained as

where

Now taking the inverse Laplace transform of Equation (36), the solution in may be obtained. Using the transformation (28) and (14) the desired solution may be obtained as

where

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The solution defined by Equation (37) describes the solute transport for exponential decay type input condition at origin in heterogeneous semi infinite domain.

The analytical solution of the present hydrodynamics dispersion is obtained as given in Equation (37). The concentration values are evaluated from the solution for the input values: reference concentration = 1, initial velocity = 0.61 (km/year), initial dispersivity = 0.71 (km^{2}/year), heterogeneity parameter () = 0.1 (km^{–1}), unsteady parameter () = 0.1 (km^{-1}), contaminant decay rate () = 0.1 (km^{–1}), initial first order decay () = 0.5 (year^{–1}) , and the initial source/sink () = 0.2. Concentration attenuation with position and time is studied in the domain, at = 0.4, 0.7 and 1.0 (year). It is illustrated in

The effect of first order decay and zero order production are studied through

and. It may be observed that solute transport is fastest in the absence of both the parameters. It is slowest in the presence of first order decay but in the absence of the production term.

One-dimensional analytical solution of advection—diffusion equation with variable coefficients is obtained using Laplace transformation technique. The source con-

centration is a point uniform source of exponentially decay nature. The expressions for both the coefficients are considered in both the independent variables but in degenerate forms given by Equation (5). With the help of certain transformations the variable coefficients are reduced into constant coefficients. Such forms of the two coefficients are conceived which correspond to the different dispersion theory (Scheidegger, 1957). The change in velocity due to heterogeneity and unsteadiness may be varied by assigning appropriate values to the separate parameters of the both. It may be concluded from the present study that the concentration level in case of accelerating diffusive source along decelerating flow domain are the least. From engineering point of view this observation may be important to keep the emission of polluting solute particles from a source of accelerating nature. The effects of first order decay and adsorption are considered and their impact illustrated by graph.

The author gratefully acknowledges the financial support in the form of Dr D. S. Kothari post doctoral fellowship by the University Grants Commission, Government of India.