Contaminant Diffusion along uniform flow velocity with pulse type input sources in finite porous medium

Solute transport inside pore system occurs due to advection and diffusion which are the important mechanisms of contaminant transport in porous medium. Analytical solutions of one-dimensional advection-diffusion equation (the coefficient of second order space derivative being temporally dependent) are obtained in a finite domain for two sets of pulse type input boundary conditions. Initially the domain is not solute free. It is supposed uniformly distributed at the initial stage. The Laplace transform technique is used with the help of new space and time variables. The solutions are graphically illustrated and compared solute distribution for finite and semi-infinite domain.


Introduction
Many examples of porous material are seen in everyday of life and environment. Soil, paper towels, Textiles, leather and tissue paper are highly porous. There are many examples where porous media play an important role in science and technology. The most important area of science and technology that to great extent depend on the properties of porous media is hydrology, which relates to water movement in earth and sand structures such as dame, flow to wells from water bearing formations, intrusion of sea water in coastal area and nutrient transport in soil. Contaminant (e.g. pesticides, chemicals, fertilizers, hygienic substance etc.) transport in subsurface, in soil and its analysis is complicated due to the complex behaviour of porous medium. Knowledge of various physical, chemical, and biological processes, which affect the movement of subsurface contaminants, is necessary for soil, blood, aquifer and groundwater remediation research and practice. From several decades the uncontrolled use of pesticides in agriculture, human and factory activities cause serious damages the environment and affected flora-fauna (forest, animals, human body etc.), soil and groundwater. Contaminant behaviours in the soil/aquifer system are subject to many processes. The study of contaminant transport requires the fundamental knowledge of many of the basic principle of physics and mathematics. A large number of theoretical and mathematical models have been developed and deployed to study the hydrodynamic processes involved in groundwater and surface water. In previous literatures, the longitudinal dispersion coefficient have been considered either linearly or squarely proportional to the fluid velocity. Banks and Jerasate [1] observed well agreement between concentration distributions and theoretical values except at very low concentrations. Ogata and Banks, Lin, Al-Niami and Rushton [2][3][4] obtained analytical solutions for dispersion in a porous media. Harleman and Rumer [5] obtained solution for longitudinal and lateral dispersion in an isotropic porous medium i.e. the permeability does not change with direction. Bruch [6] derived a series of two-dimensional dispersion problems in one and two layered porous medium. The experimental results were compared with theoretical and numerical solutions both of which describe the two-dimensional dispersion of a miscible. Most of such works have been compiled by van Genutchen and Alves [7]. Chen and Liu [8] dealt with solute transport form an injection well into an aquifer. A macroscopic boundary condition of the Cauchy type (the third type) can be formulated at the well-aquifer interface if the mass balance principle is invoked. Tracy [9] first gave some simple one-dimensional solutions. Next, by use of a transformation, the nonlinear partial differential equation is converted to a linear one for a specific form of the moisture content vs. pressure head and relative hydraulic conductivity vs. pressure head curves. This allows both two and three dimensional solutions to be derived. Aral and Liao [10] examined solutions of two dimensional advection-dispersion equation with time dependent dispersion coefficients and demonstrated the time and space dependent nature of the dispersion coefficient in subsurface contaminant transport problems. They developed instantaneous and continuous point source solutions for constant, linear, asymptotic, and exponentially varying dispersion coefficients. Ataie-Ashtiani et al. [11] studied the influence of tidal fluctuation effects on groundwater dynamics and contaminant transport in unconfined coastal aquifers. Sander and Braddock [12] obtained analytical solutions of advection-dispersion equation in one-dimension with scale and time dependent dispersivities. In order to perform a general analysis of groundwater contaminant transport, Sirin [13] considered (i) non-divergence-free pore flow velocity since nondivergence-free pore flow velocity occurs during density department flows, (ii) unsteady pore flow velocity. Chen and Liu [14], studied an analytical solution for one-dimensional advective-dispersive transport in finite spatial domain with timedependent inlet conditions including constant, exponentially decaying and sinusoidally periodic input functions and demonstrate the applicability of solution. Yadav et al. and Jaiswal et al. [15][16] obtained analytical solutions for solute dispersion in finite porous media. Sharma [17] obtained solution of an advective dispersive transport equation, including equilibrium sorption and first-order degradation coefficient in the fracture and simultaneously a diffusive transport equation for porous media using numerical implicit finite difference method and discussed numerical results of various temporal moments have been predicted to investigate the behaviour of reactive solute in the fracture. In the present study, advection-diffusion equation is considered one dimensional. The solute dispersion parameter is considered temporally dependent along uniform flow in longitudinal finite domain of length, . The input source condition is assumed to be of pulse type, introduced at the origin of the domain. The second condition is considered at the other end of the domain which is of second type (flux type) of homogeneous nature. The domain is assumed initially not solute free i.e. the domain is supposed to have uniformly distributed solutes at the initial stage. Laplace transformation technique is used to getting the analytical solutions.

Mathematical Model of the Problem
For the analyses presented here, the governing equation for a solute transport model represent a mathematical description of the assumed transport mechanisms and processes in ideal case which include the effect of adsorption, in one dimension may be written as, 1 ( , ) ( , ) where C is the solute concentration in the liquid phase and F is the concentration in the solid phase. As is generally known, the mass transport equation uses hydrodynamic dispersion, which is the combination of mechanical dispersion and diffusion, however molecular diffusion is negligible due to very low seepage velocity.
The advection-diffusion Eq.
(1) has served as the main theoretical framework for modelling and transport of solute in porous media and for addressing critical environmental issues or waste disposal operations during the last few decades Jury and Fluhler [18] respectively, equilibrium and non-equilibrium isotherm between the concentrations in the two phases, where k 1 and k 2 are empirical constants of the medium. The isotherm is linear if n=1, and is non-linear if n>1. For simplicity, the former relationship is adopted in the present analysis. This assumption is generally valid when the adsorption process is fast in relation to the ground-water velocity Cherry et al. [20]. Using Eq. (2) in Eq. (1) for n=1we may get linear advection-diffusion equation, , , the linear advection-diffusion partial differential equation in one dimension in general form is, which in case of temporally dependent dispersion along a uniform flow, i.e., for where m is a unsteady coefficient whose dimension is inverse of Now the time dependent coefficient on left hand side may be got rid of by introducing another new time variable, The partial differential equation (8) reduces into that with constant coefficients as Further using transformations and     2 00 00 , , exp 24 The advection-diffusion equation (11)

Analytical Solutions
To proceed further, let us consider initial and boundary conditions for (5) in a finite longitudinal domain of length L .The analytical solutions are obtained for two cases. The input source is introduced at the origin of the domain. In both cases second boundary condition of flux type and homogeneous nature is imposed at the extreme end xL  of the domain. In case of pulse type input source, the domain is assumed to be not solute free, instead it is assumed to uniformly polluted by solute particles.

Case-I Pulse type input concentration of uniform nature
If the source of the input concentration remains uniform up to certain time period and after its elimination forever the input becomes zero. This type of condition is defined by first type or Dirichlet boundary condition. For uniform pulse type input concentration the initial and boundary conditions are, respectively. Applying Laplace transform on the diffusion equation (14) and using initial condition (18), we may get The boundary conditions (19) and (20) become Thus the general solution of equation (21) may be written as Using conditions (22) and (23) Thus the solution of Eq. (24) in the Laplace parameter may be written as Applying inverse Laplace transform on it, we get ( , ) n K Z T and using back transformation (13), we may get the desired analytical solution in ( , ) n C Z T of the initial and boundary value problem (5), (15), (16) and (17), as follows and n T may be obtained from transformation (10).

Case-II Pulse type input concentration of varying nature
It may happen that if the input concentration continuously uniform up to certain time period and after its elimination forever the input becomes zero. But due to human, industries and some other responsible activities, the source of input concentration increases till certain time period, beyond that it starts decreasing, when source of concentration is eliminated forever. The type of condition defined by third type of boundary condition and is Thus the solution of Eq. (24) in the Laplace parameter may be written as Applying inverse Laplace transform on it, we get ( , ) n K Z T and using back transformation (13), we may get the desired analytical solution in ( , ) n C Z T of the initial and boundary value problem (5), (15), (16) and (17), as follows and n T may be obtained from transformation (10).

Numerical Results and Discussion
Two analytical solutions (28)   and n T may be written in terms of t using the transformation (10) for an expression () f mt .

Conclusion
Analytical solutions of one-dimensional advection-diffusion equation are obtained in a finite domain for two sets of boundary conditions. In the both set, initial condition is non-homogeneous. The input condition is pulse type. The second boundary condition in each set is flux type of homogeneous nature . Laplace transformation technique is utilized in order to attain the analytical solutions. From figures (1,2), distribution of solute constration shows the respective boundary conditions. The pulse type input boundary condition helps predicting the rehabilitation process of a degraded system once the source of the solute contamination is eliminated for ever. Such analytical solutions may serve as tools in validating numerical solutions in more realistic dispersion problems. These solutions are facilitating to assess the transport of pollutants solute concentration away from its source along a flow through soil medium, aquifers, and oil reservoirs etc. which has always been difficult because of the inherent complexities.