Analytical Simulation of Two Dimensional Advection Dispersion Equation of Contaminant Transport

: The study was designed to investigate the analytical simulation of two dimensional advection dispersion equation of contaminant transport. The steady state flow condition of the contaminant transport where inorganic contaminants in aqueous waste solutions are disposed of at the land surface where it would migrate through the verdoze zone to underground water is considered. We solved the two dimensional advection dispersion equation analytically which is solute transport model without sorption or degradation using change of variable method. We critically reviewed two dimensional equations depicting the transport of contaminant in groundwater and investigate with the help of graphical representation the effect of Peclet number on the concentration of contaminant and established real life interpretation of contaminant transport. Two cases were considered, when Peclet number is less than one and when Peclet number is greater than one. The result obtained revealed that the contaminant concentration increases along x direction and decreases along y direction for both values of peclet number greater than one and less than one. The study has contributed to knowledge through the method utilized to achieve the model analytical solution and the Physical interpretation of Peclet number. From the analysis, we recommend for further studies on the contaminant transport which also depends on the available data, that the extension of advection –dispersion model to three dimensions and comparison of travel time of contaminant transport solution to Kinetic or multi-component mode .©

The most usual way of "managing" town refuses is achieve by depositing it in sanitary landfills.In moist and semi-humid region, the movement of contaminants from the refuse into the ground water zone lying underneath normally occur due to infiltration through landfills, the zones of contaminated ground water expand many hundreds meters.In a few circumstances, contaminant can lead to critical deterioration of aquifer used for ground water source; therefore, we need to know how contaminants moves from place at which they are created, leading to impacts, which can be far from the contaminant source.On the other hand, some contaminant, such as sewage sludge, can be degrading in the environment if they are sufficiently dilute.For these contaminants, slow transport slow dilution can result in excessively high contaminant concentrations, with resulting increased adverse impacts.It is therefore important to have a simple method of making contaminant transport calculations for contaminant at concentration other than the average (for contaminant fronts) or the peak (for contaminant pulse).The steady state flow condition of the contaminant transport in a point source where inorganic contaminants in aqueous waste solutions are disposed of at the land surface must migrate through the verdoze zone to underground water is considered.
It known here that, if a model is developed and used in application to a problem, the solution will be analyses and interpreted with regard to the problem in question.Aiyesimi, (2004) studied the mathematical analysis of environmental contamination of the Freudlich non-linear contaminant transport formulation.Gideon and Aiyesimi (2005) lucubrated on the non-linear contaminant flow and the influence of retardation factor on it.Mulligan et al, (1999) Groundwater flow below surface layer is pushed by difference in energy water flows from elevated-energy regions to ground energy regions.The energy component of a unit volume of water is ascertained by the addition of gravitational potential energy, kinetic energy, and pressure energy.
The aim of the study is to derive analytical solution of two dimensional equations and simulation of contaminant transport in underground water.
Critically review two -dimensional equation depicting the transport of contaminant in groundwater.Solve the equation analytically using change of variable method and direct integration.
investigate with the help of graphical representation the effects of Peclet number on the concentration of contaminant.vi.Establish real life interpretation of the contaminant transport Mathematical formulation: Patil and Chole (2014), forecast the environmental repercussion of contamination of groundwater, there should be information as to where there will be interference of the contaminant, the time it will arrive and the concentrations potential.The introduction of contaminant into groundwater is by: (i) advection due to groundwater flow; (ii) dispersion due to molecular diffusion and mechanical mixing; and (iii) retardation due to adsorption.
The relation of these processes with one another can be expressed mathematically as follows: ( ) An evaluated two-dimensional form shows the contaminant transport in groundwater in an isotropic, homogeneous medium having a steady-state flow that is unidirectional with seepage velocity v is given as: Where; , and the value for R becomes equal to one and the equation ( ) The initial and boundary conditions are formulated as follows: we have: Together with boundary conditions: Together with boundary conditions: , 1 0, 0 , 0 x y on x y x y as x y Where, P em =

RESULT AND DISCUSSION
Two dimensional advection dispersion equations describing the contaminant transport are solve analytically using the change of variable method.Analytical solution of equation ( 9) is computed.
The following figure explains contaminant concentration against the different parameters.
Figure 1 display the graph of contaminant concentration against distance x when peclet number is less than one.It shows that Contaminant concentration increases along x direction, but decreases as Peclet numbers increases.The real life interpretation here is that the dispersion is noticeably larger than advection, thus dispersion dominates and advection is negligible and this implies that contaminant concentration moves slowly to receptor.The real life interpretation here is that the dispersion is noticeably larger than advection along y direction, thus dispersion dominates and advection is negligible along y direction and this implies that contaminant concentration moves slowly to receptor.The real life interpretation here is that the advection is noticeably larger than dispersion along x direction, thus advection dominates and dispersion is negligible and this implies that contaminant concentration moves faster along x direction to receptor.Figure 6 display the graph of contaminant concentration against distance x and y.It shows that Contaminant concentration increases along x direction and decreases along y direction but decreases as peclet numbers increases.The real life interpretation here is that the advection is noticeably larger than dispersion along x and y direction, thus advection dominates and dispersion is negligible in both direction and this implies that contaminant concentration moves faster along x and y direction to receptor.Conclusions: We concluded that the contaminant concentration in two directions moves either fast or slow to receptor as affected by the values of Peclet number.The relative error committed in the solution by so doing is expected to be on the order of the Peclet number, and the smaller Peclet number, the smaller the error when peclet number is less than one,on the other hand, The relative error perpetrated in the solution by so doing is expected to be on the Concentration in the source or sink fluid " w = Volume Flow rate per unit volume of the source or sink n ij D = Dispersion Coefficient ij k = Hydraulic Conductivity c = Contaminant Concentration v = Seepage or Average pore water velocity volume L D = Coefficient of Longitudinal Dispersion T D = transversal dispersion coefficient R= retardation Since sorption onto the sediment reduces the apparent advective -dispersive flux by a factor equal to R .The sorption considered negligible, in which case 0 = d k

Fig 1 :
Fig 1: Graphical Illustration of Contaminant Concentration against distance travel for various values of Peclet number less than one Figure 2 display the graph of contaminant concentration against distance y.It shows that Contaminant concentration decreases along y direction, but decreases as Peclet numbers increases.The real life interpretation here is that the dispersion is noticeably larger than advection along y direction, thus dispersion dominates and advection is negligible along y direction and this implies that contaminant concentration moves slowly to receptor.

Fig. 2 :
Fig.2: Graphical Illustration of Contaminant Concentration against distance travel for various values of Peclet number less than one.

Figure 3
Figure3display the graph of contaminant concentration against distance x and y.It shows that Contaminant concentration increases along x direction and decreases along y direction but decreases as peclet numbers increases.The real life interpretation here is that the dispersion is noticeably larger than advection along x and y direction, thus dispersion dominates and advection is negligible alongx and y direction and this implies that contaminant concentration moves slowly to receptor in both directions.

Figure 4
Figure 4 display the graph of contaminant concentration against distance x.It shows that Contaminant concentration increases along x direction, but decreases as Peclet numbers increases.The real life interpretation here is that the advection is noticeably larger than dispersion along x direction, thus advection dominates and dispersion is negligible and this implies that contaminant concentration moves faster along x direction to receptor.

Fig3:
Fig3: Graphical Illustration of Contaminant Concentration against distances travel for various values of Peclet number less than one.

Fig 4 :
Fig 4: Graphical Illustration of Contaminant Concentration against distance travel for various values of Peclet number greater than one Figure 5 display the graph of contaminant concentration against distance y.It shows that Contaminant concentration decreases along y direction, but decreases as Peclet numbers increases.The real life interpretation here is that the advection is noticeably larger than dispersion along y direction, thus advection dominates and dispersion is negligible and this implies that contaminant concentration moves faster along y direction to receptor.

Fig 5 :
Fig 5: Graphical Illustration of Contaminant Concentration against distance travel for various values of Peclet number greater than one.

Fig 6 :
Fig 6: Graphical Illustration of Contaminant Concentration against distance travel for various values of Peclet number greater than one