ANALYTICAL APPROACH OF ONE-DIMENSIONAL SOLUTE TRANSPORT THROUGH INHOMOGENEOUS SEMI-INFINITE POROUS DOMAIN FOR UNSTEADY FLOW : DISPERSION BEING PROPORTIONAL TO SQUARE OF VELOCITY

In this study, we present an analytical solution for solute transport in a semi-infinite inhomogeneous porous domain and a time-varying boundary condition. Dispersion is considered directly proportional to the square of velocity whereas the velocity is time and spatially dependent function. It is expressed in degenerate form. Initially the domain is solute free. The input condition is considered pulse type at the origin and flux type at the other end of the domain. Certain new independent variables are introduced through separate transformation to eliminate the variable coefficients of Advection Diffusion Equation (ADE) into constant coefficient. Laplace transform technique (LTT) is used to get the analytical solution of ADE concentration values are illustrated graphically.

1. Introduction.Since last five-six decades analytical solutions for a solute transport problem have been discussed under various conditions.Solute transport through a porous media is normally mathematically modeled by means of the advectiondispersion equation which is a partial differential equation of parabolic type.The movement of solute in subsurface involves mainly two processes, firstly convection and secondly hydrodynamic dispersion.Developing a mathematical model that predicts solute transport in groundwater is important in establishing optimal situations for solute transport and in the assessment of its effect to human health and environment.If groundwater becomes polluted it is very difficult to rehabilitate.It is due to slow rate of groundwater flow and low microbiological activities limit any self-purification processes which take place in days or weeks in surface water systems can take decades in groundwater.Mathematical modeling of contaminant behavior in porous media is considered to be a powerful tool for a wide range of pollution problems related with groundwater quality rehabilitation.
Analytical solutions describing solute transport in one-dimensional porous media, considering adsorption, first-order decay and zero-order production, ( [10] and [20]).[23] and [21] considered the uniform flow through a homogeneous and isotropic porous domain.[6] considered the longitudinal dispersion coefficient is linearly proportional to the product of the fluid velocity and the particle diameter.[24], [25] and [9] obtained analytical solutions of one-dimensional transport with firsttype, second-type and third-type boundary conditions respectively.
The solute transport of dissolved solids and the analysis of tracer tests, discussed by recently [15] and [16].[30] obtained a solution considering time and scale-dependent dispersivity for solute transport in saturated heterogeneous porous media.[31] presented an analytical solution for advection-diffusion equation to simulate the pollutant dispersion in the planetary boundary layer.[7] considered a point source pollutant and obtained analytical solution for wind speed and eddy diffusivity.[29] presented a model for solute transport in rivers including transient storage in hyporheic zones.[12] obtained analytical solution with distance dependent dispersion.[1] and [2] solved solute transport equation analytically in a convergent and radially convergent flow field with scale-dependent dispersion.[19] investigated transport and fate of reactive components in the unsaturated soil (vadose zone) using numerical simulation of steadystate and transient flow scenario.[22] presented a review of the GILTT solutions focusing the applications to pollutant dispersion in atmosphere.[11] presented an analytical in finite domain for both transient and steady-state regimes.[14] studied the effect of immobile water content on contaminant advection and dispersion in unsaturated porous media.
Recently ( [13]; [17]; [18] and [3]) obtained analytical for specially and time dependent dispersion in one-dimensional porous domain.[27] solved a solute transport equation analytically for along and against transient groundwater flow.The horizontal dispersion along and against transient groundwater flow in homogeneous and finite aquifer and velocity is considered the time-dependent forms which one form, sinusoidal form, represented the seasonal variation in a year in tropical regions.[8] presented an analytical solution to two-dimensional advection-dispersion equation in semi-infinite and laterally boundary domain.[28] solved one-dimensional advection-dispersion equation is analytically heterogeneous semi-infinite medium and heterogeneity velocity and dispersivity coefficient of are considered functions of space variable and time variable.[4] presented a generalized analytical solution for one-dimensional solute transport in finite spatial domain subject to arbitrary time-dependent inlet boundary condition with linear equilibrium sorption and first order decay processes and also developed generalized solution offers a convenient tool for further development of analytical solution of specified time-dependent inlet boundary conditions or numerical evaluation of the concentration field for arbitrary time-dependent inlet boundary problem.
In the present study advection-diffusion equation is solved analytical for onedimensional solute transport for uniform pulse type point source along unsteady flow through inhomogeneous semi-infinite porous medium.Dispersion parameter is considered directly proportional to the square of velocity whereas the velocity is time and spatially dependent function.It is expressed in degenerate form.It is expressed in degenerate form.The pulse type conservative solute is introduced at the origin of the domain and other end considered flux type.Initially the domain is considered solute free.It means before solute entering in the domain medium is solute free.New independent variables are introduced to eliminate the variable coefficients in Advection Diffusion Equation into constant coefficient.Laplace transform technique (LTT) is used to get the analytical solution of ADE.Concentration behaviors are illustrated graphically.
2. Transport of solute particle along the unsteady flow through inhomogeneous medium.We consider advective-dispersive transport of a conservative tracer with concentration c introduced by a continuous point source in a onedimensional porous medium.The transport of this tracer can be described by the well-known advection-dispersion equation (ADE) where c is the solute concentration at a position x at time t, D(x, t) and u(x, t) are the solute dispersion parameter and velocity of the medium transporting the solute particles along the longitudinal direction.This equation is solved for a dispersion problem in which both the coefficients remain functions of independent variables.The medium is of semi-infinite and inhomogeneous nature.Due to inhomogeneous medium, the velocity of the flow field transporting the solute particles along its downstream is considered spatially dependent.Its expression of increasing nature is linearly interpolated in position variable in a finite longitudinal region, in which concentration values are evaluated ( [18]).Further the velocity is also considered temporally dependent.The expression for velocity is written in degenerate form as where a is the inhomogeneity parameter along the longitudinal direction, and is of the dimension inverse of space variable and m is an unsteady flow parameter of dimension inverse of the dimension of t .While choosing an expression for f (mt) it is insured that f (mt) = 1 for m = 0 and t = 0 .The solute dispersion parameter is considered proportional to square of the velocity ( [26]), that is we consider where u 0 and D 0 in Eqs ( 2) and (3) may be referred to as uniform velocity of dimension (LT −1 ) and the initial dispersion coefficient of dimension (L 2 T −1 ), respectively.
2.1.Initial and boundary value problem.The point source of the pollutants is considered and entering in the domain from the origin of the medium.It is of pulse type of uniform nature.In other words, the input concentration at the origin remains uniform up to certain time period till the source is present, and just after the source is eliminated it becomes zero.For example, smoke coming out of a chimney of a factory or waste meeting a river bed at a uniform rate vanishes as soon as the factory or the drainage system is closed.Initially the medium is considered solute free.Under these assumptions mathematical formulation of the proposed problem may be written as To solve the Advection-diffusion equation (4), one initial and two boundary conditions are required.The source of the solute mass is a uniform pulse type point source at the origin.It serves as the first boundary condition known as the input condition.A flux type homogeneous condition is assumed at the end of the medium.Thus initial and boundary conditions are Now, using the transformation ( [5]) Then the advection-diffusion equation ( 4) and initial and boundary conditions ( 5) to (7) becomes For an expression of f (mt), the dimension of T * will be that of t hence it is referred to as new time variable.Also an expression of f (mt) is chosen such that for t = 0 , we get T * = 0 so that the nature of initial condition does not change.Further another space variable is introduced through a transformation Eq. ( 9) becomes where ) is defined another time dependent expression in non-dimensional variable (term), mt , and λ = (aD 0 /u 0 ), is a non-dimensional parameter.Now, the first order decay term in Eq. ( 14) is eliminated by using a transformation c = C exp {−au 0 T * } (16) Lastly with the help of another independent variables introduced through the transformations and respectively.
Then the variable coefficient of the advection-diffusion equation ( 14) is reduced into constant coefficients.
Thus the initial and boundary conditions defined by Eqs. ( 10) to ( 12) is reduced in (Z, T ) domain where D 0 and u 0 are dispersion coefficient and uniform velocity and γ = (1 − λ) −2 in Eq. ( 21) is another non-dimensional parameter.Applying the transformations defined by Eqs. ( 13), ( 16) and ( 17), input condition (11) becomes where new time variable T * is defined by Eq. ( 8).To write it in terms of another time variable T defined by Eq. ( 18), we chose an expression of f (mt) as For it the both new time variables (T * and T ) becomes and Eliminating old time variable t , from Eqs. ( 25) and ( 26) we get To get unsteady flow, the parameter m is considered much smaller than one, in f (mt) , i.e., m << 1 , so its second and higher degree terms in any exponential or logarithmic or binomial expansions can be neglected.Also while choosing an expression of f (mt) , it is also ascertained that through the transformation (18), we get T = 0 for t = 0 , so that the nature of the initial condition does not change in this time domain.
Applying the transformation (28) in Equations (19)(20)(21)(22) may be write new dependent variable K(Z, T ) as Thus the initial and boundary value problem defined by equations (19 -22) and Now applying the Laplace transformation technique the desired solution may be obtained as where and by using Eq. ( 8), T * may be expressed in terms of t for an expression of f (mt) .It may be verified that Eq. ( 27), obtained for f (mt) = exp (−mt), may also valid for f (mt) = exp (mt) .So this solution is applicable for both the expressions of f (mt) .Also for m = 0 , solution (33 and 34) reduces to solution given by Eq. ( 18) of our previous work ([18]).
3. Results and discussions.The concentration values are evaluated from analytical solutions (33) and (34) for uniform pulse type input condition at the presence of the source of pollution t ≤ t 0 at t (year) = 0.1, 0.4, 0.7 and 1.0, and absence of the source of pollution t > t 0 at t (year) = 1.3, 1.6, 1.9 and 2.2 respectively, in finite domain 0 ≤ x ≤ 1.The set of input data are C 0 = 1.0, initial velocity u 0 = 0.60 (km/year) and initial dispersion coefficient D 0 = 0.71 km 2 /year , inhomogeneity parameter a = 0.1 (km) −1 and unsteady flow parameter m = 0.1 (year) −1 .The concentration values (c/C 0 ) are evaluated in a finite domain 0 ≤ x ≤ 1, of the semiinfinite medium.For f (mt) = exp (−mt), these values are depicted in Fig. 1 at at t (year) = 0.1, 0.4, 0.7 and 1.0 for the t ≤ t 0 and in Fig. 2 at t (year) = 1.3, 1.6, 1.9 and 2.2 for the t > t 0 and compared with the exponentially increasing function f (mt) = exp (mt) for the t (year) = 0.1 (t ≤ t 0 : presence of the source) and 1.3 (t > t 0 : absence of the source), respectively.It observed that the concentration values decrease with position and increase with time.Fig. 3 and Fig. 4 shows the comparison of concentration values evaluated from solution (33 and 34) at t (year) = 0.4 (t ≤ t 0 : presence of the source) and 1.6 ( t > t 0 : absence of the source) for the three expressions (i) f (mt) = 1, (ii) f (mt) = exp (−mt) and (iii) f (mt) = exp (mt).
Fig 4  shows that concentration levels near the source boundary (up to x = 0.4 (year)) are almost same but after x = 0.4 (year) the concentrating levels deceases for decreasing function of f (mt) = exp (−mt) and increases for increasing function of f (mt) = exp (mt).More expressions for f (mt) may be chosen but for one such expression an explicitly relation between new time variables T * and T like Eq. (27), should occur.

4 .
Conclusions.Partial differential equations are the basis of many mathematical models of physical, chemical and biological phenomena, and their use has also spread into economics, financial forecasting and other fields.The advection-diffusion equation is solved analytically with variable coefficients using Laplace Transformation Technique by introducing new independent variables at the different stages.The effect of temporal dependence of the flow is illustrated by evaluating the concentration values for exponentially decreasing, exponentially increasing unsteady and steady flow fields.Concentration values are illustrated graphically with appropriate numerical field data.The derived result may helpful to predict the concentration levels at space and time which may help to reduce/eliminate the concentration levels.

Figure 2 .
Figure 2. Solute concentration distribution behavior for Eq.(34) at absence of the source of pollution.

Figure 3 .
Figure 3.Comparison of solute concentration distribution behavior for Eq.(33) at presence of the source of pollution.

Figure 4 .
Figure 4. Comparison of solute concentration distribution behavior for Eq.(34) at presence of the source of pollution.