Generalizing Source Geometry of Site Contamination by Simulating and Analyzing Analytical Solution of Three-Dimensional Solute Transport Model

Due to the uneven distribution of pollutions and blur edge of pollutant area, there will exist uncertainty of source term shape in advective-diffusion equation model of contaminant transport. How to generalize those irregular source terms and deal with those uncertainties is very critical but rarely studied in previous research. In this study, the fate and transport of contaminant from rectangular and elliptic source geometry were simulated based on a three-dimensional analytical solute transport model, and the source geometry generalization guideline was developed by comparing the migration of contaminant. The result indicated that the variation of source area size had no effect on pollution plumemigration when the plumemigrated as far as five times of source side length. The migration of pollution plume became slower with the increase of aquifer thickness. The contaminant concentration was decreasing with scale factor rising, and the differences among various scale factors became smaller with the distance to field increasing.


Introduction
Concern about contamination of the subsurface environment has greatly stimulated research of solute transport phenomena in porous media.A number of solute transport studies aimed at solving the advective-diffusion equation (ADE) for nonreactive and reactive solutes, subject to various initial and boundary conditions [1].Several analytical solutions for one-, two-, and three-dimensional ADEs have been developed for predicting the transport of various contaminants in the subsurface [2].For instance, Ogata and Banks [3], Sauty [4], and Van Genuchten [5] formulated several analytical solutions for the one-dimensional ADE subject to the first-type (Dirichlet), second-type (Neumann), and third-type (Cauchy) boundary conditions, respectively.Batu [6,7] compiled analytical solutions to two-dimensional ADE with various source boundary conditions.The analytical solutions for three-dimensional ADE have been derived by Sagar [8], Domenico [9], Leij et al., [10], Batu [11], Sim and Chrysikopoulos [12], and Park and Zhan [13].Analytical solutions for ADE play important roles in giving initial or approximate estimates of contaminant distributions in soil or aquifer systems [14].Although large amounts of two-or three-dimensional analytical models are available at present [15], those models illustrated the surface condition for transport from a regular source.
The regular sources included point source, line source, and area source (rectangular or elliptic).For example, Sim and Chrysikopoulos [12] investigated the effect of aquifer boundary conditions and the source geometry on solute transport in subsurface porous formations.Porous media with either semi-infinite or finite thickness and source geometry with either a point source or an elliptic source were examined.Park and Zhan [13] tested the sensitivity of the line source solutions to source geometry, dispersion coefficients, and distance to the source.The results indicated that the concentration at a near field point was sensitive to the source geometry when the dispersion coefficients are anisotropic, and it was less sensitive to the source geometry when the dispersion coefficients were isotropic.The concentration at a far field was found to be almost independent of the source geometry.Zhao et al. [16] investigated the effects of different domain shapes in general and trapezoidal domain shape in particular on the morphological evolution of NAPL dissolution fronts in two-dimensional fluid-saturated porous media.Domain shapes had a significant effect on the propagating speed, and an increase in the divergent angle of a trapezoidal domain could lead to a decrease in the propagating speed of the NAPL dissolution front.The above-mentioned studies commendably analyzed the fate and transport of contaminant under various conditions from a regular source geometry.However, due to the complex condition in reality, such as uneven distribution of pollutions and blur edge of pollutant area, the shape of pollution plume is not explicit in the initial period of contaminated site investigation.On this background, there will exist uncertainty of source term shape in ADE model of contaminant transport.How to generalize those irregular source terms and deal with those uncertainties is very critical and can influence the accuracy of final calculation result.Nevertheless, few previous studies have been conducted concerning those problems and then provided decision-support for site remediation practice, through numerical simulation methods.
Therefore, the objective of this study is to develop a guideline for source geometry generalization, which differs from previous research in two aspects.First, it can answer the questions of "how to use the regular source geometry to substitute the irregular source geometry" and "what regular source geometry can be used for substitution." Second, it can help technicians examine and predict the contaminant transport in subsurface flow systems in actual site remediation.We start from identifying contaminant from regular (i.e., rectangular and elliptic) source geometry, investigating its application scope and condition from its analytical solutions of ADE as well as the corresponding influence factors.Then, simulating flow and transport of contaminants from two regular source geometries under the same condition compare flow and transport of contaminant based on the simulation results to develop source geometry generalization guideline.The obtained results will provide useful information and technical support for estimating the distribution of contaminant concentration in the initial period of contaminated site investigation.

Mathematical Model
This study considers the problem of contaminant transport in saturated, homogeneous porous media, accounting for threedimensional hydrodynamic dispersion in a uniform flow field, and first-order decay of liquid phase and sorbed contaminants with different decay rates.Before formulating the model, several assumptions must be made.First, it is assumed that the upper and lower boundaries of homogeneous porous media are impermeable (no-flow boundary); if a water table boundary exists, the slope of the water table is small and the aquifer is horizontal without curvature and parallel to the lower boundary.Second, there exists one-dimensional steady-state ground water flow along the -axis, and the averaged actual flow velocity is .Third, cardinal direction of dispersion coefficient is consistent with the coordinate axis.Fourth, there is no pollution in the study area at initial moments, and the continuous injection strength is   .
Based on the above assumptions, the advective-diffusion equation can be formulated by the following partial differential equation: The corresponding initial and boundary conditions are as follows: where  denotes the solute concentration;   ,   , and   are the longitudinal, lateral, and vertical hydrodynamic dispersion coefficients, respectively;  represents the averaged actual steady-state pore water velocity;  is time; , , and  stands for the spatial coordinates in the longitudinal, lateral, and vertical directions, respectively;  is the decay rate of solutes;  is the porosity of the porous medium;   is the continuous injection strength;  denotes aquifer thickness.

Analytical Solution of Rectangular Source.
The general geometry of the rectangular source is show in Figure 1.The origin of the coordinate system is at the upper boundary.The positive -axis is downward.The aquifer is assumed infinite in the and -directions but finite in the -direction with a thickness of .The rectangular source of contamination is located on top of an aquifer, without considering its thickness influence, with The general geometry of the elliptic source.
and  ∈ [0, ], which is described mathematically by the following expression: Through using Green's function methods, three-dimensional analytical solution for rectangular source can be obtained [13] as follows: where  0 is a constant and () is a function of time; erfc[] denotes residual error function, equal to 1−(2/√) ∫  0  − 2 .

Analytical Solution of Elliptic Source.
Consider solute movement from an elliptic source as sketched in Figure 2. The set of coordinate system is similar to the rectangular source, the source contaminant is located on top of an aquifer, and the solute may move from the source, which has a negligible thickness, by diffusion or advection, with  ∈ [0,  1 ],  ∈ [− 2 ,  2 ], and  ∈ [0,]; the elliptic source geometry is defined mathematically by the following expression: The governing solute transport equations are solved analytically by employing Laplace, Fourier, and finite Fourier cosine transform techniques: where  1 and  2 are the forward and reverse rate coefficients;  is decay rate of liquid phase solute;  5 is decay rate of sorbed contaminant.

Model Simulations and Discussion
Model simulations are performed for two different source configurations.The integrals present in the analytical solutions are evaluated numerically by using globally adaptive quadrature algorithms.The number of terms is selected so that additional terms do not alter the summation more than  1.0 − 7.In order to compare the fate and transport of contaminant from rectangular and elliptic source geometry, it is assumed that aspect ratio (length-width ratio for rectangular source and major axis-minor axis ratio for elliptic source) of rectangular source geometry is the same as that of elliptic source geometry; besides that, the area of two disparate sources is equal, which can guarantee the uniform total pollution load under the same continuous injection strength condition.

Rectangular Source Type.
The groundwater table and the bottom of the finite thickness aquifer are assumed to be located at  =  table = 0cm and  =  = 200cm, respectively.Other parameters are shown in Table 1; with them, concentrations can be calculated at any given time for continuous source with consideration of the first-order decay using FORTRAN language.
Figure 3 displays the fate and transport of contaminant from rectangular source on the -plane ( = 0) at  = 10,  = 100, and  = 500 days under conditions by continuous injection, where the black line stands for  = 10 days, the red line represents  = 100 days, and the blue line denotes  = 500 days.As expected, the pollution plume spreads out with time, and the plume has already reached the lower boundary of aquifers at  = 100 days.The degree of spreading along -axis is more than that along -axis, and the migration speed of contaminant at the near field is lower than that of contaminant at the far field.The reason is due to the fact  that there exists one-dimensional steady-state ground water flow along the -axis, and the dispersion effect along -axis (  = 2000) is larger than that along -axis (  = 500).At initial time, due to the higher concentration gradient and gravity, the plume spread quickly both along -axis and axis; however, this phenomenon is insignificant over time by continuous injection.These results agree with the findings by Cianci et al. [17] investigating contaminant transport through a saturated porous medium in a semi-infinite domain.In comparison with the work by Leij et al. [18], which illustrate that the relatively high maximum concentration occurs at the surface.This conclusion is obtained based on the surface condition transporting from a rectangular source, which is also indicated in this research.
Figure 4 illustrates contaminant concentration contours on the -plane ( = 0) under larger source area (red line), unchanged source area (black line), and smaller source area conditions (blue line) at  = 100 days.The result of larger area is obtained under condition that rectangular source area is double, while the continuous injection strength reduces to half compared to the original source area.Similarly, smaller area means that rectangular source area reduces to half, while the continuous injection strength is double.From the figure, we can draw a conclusion that the influence of source area size decreases with the increase of distance to field; when the pollution plume migrates to a certain distance, the variation of source area size has no effect on plume migration.In order to make this problem clear, we conduct the following study.
Figure 5 presents contaminant concentration contours on the -plane ( = 0) under  = 150 cm (red line),  = 200 cm (black line), and  = 250 cm (blue line) conditions at  = 100 days, which shows the influence of aquifer thickness on pollutant migration.The pollution plume migrates slower with aquifer thickness increasing.In order to analyze Table 2: The related calculation parameters for Figure 6. the comprehensive influence of both source area and aquifer thickness, we define a scale factor as follows: where SF denotes scale factor;  is aquifer thickness;   stands for width of rectangular source (or major axis of elliptic source).Figure 6 shows contaminant concentration at half aquifer thickness under various scale factors at  = 100 days.The calculation parameters are listed in Table 2.The results in Figure 6(a) are obtained with a fixed rectangular source area under various aquifer thicknesses conditions (as shown in Table 2(a)).From the vertical perspective, the contaminant concentration at half aquifer thickness is decreasing with scale factor rising; besides differences of contaminant concentration decrease among various scale factors.From the horizontal perspective, differences of contaminant concentration among various scale factors become smaller with the distance to field increasing.conditions (as shown in Table 2(b)).The contaminant concentration at half aquifer thickness is almost unchangeable with scale factor increasing when the distance to field is greater than 250 cm.In other words, the influence of source area size can be neglected when the distance to field is greater than 250 cm. 1 lists the related parameters for model simulation of elliptic source.Figure 7 illustrates contaminant concentration contours on the -plane ( = 0) under  = 10 days (black line),  = 100 days (red line), and  = 500 days (blue line) conditions by continuous injection.The pollution plume spreads similarly to that of rectangular source geometry, but the speed of contaminant migration is slower than that of rectangular source type in both transverse and longitudinal directions.These results agree with the findings by Sim and Chrysikopoulos [12] based on a continuous source loading from elliptic source geometry in saturated, homogeneous porous media.

Elliptic Source Type. Table
Figure 8 displays contaminant concentration contours on the -plane ( = 0) under various elliptic source areas at  = 100 days (larger source area (red line), unchanged source area (black line), and smaller source area conditions (blue line)).The influence of source area size on pollution plume migration is insignificant.It is consistent with that of rectangular source type.at  = 100 days.The influence of aquifer thickness for elliptic source is the same as that of rectangular source; the pollution plume migrates fast with aquifer thickness decreasing.The influence of scale factor on pollution plume migration from elliptic source geometry is the same as that of scale factor from rectangular source geometry under the two parameter conditions listed in Table 2.

Comparative Analysis.
Figure 10 displays the variation of CR/CE, defined as dimensionless coefficient indicating contaminant concentration of rectangular source type divided by contaminant concentration of elliptic source type, under various scale factors along transverse direction.The CR/CE in the near field is smaller than that in the far field, and especially for  = 100 cm, the value is almost equal to 1.In other words, the contaminant concentration from rectangular source is nearly the same as that from elliptic source at  = 100 cm on water table, and the CR/CE is slightly decreasing with scale factor increasing.The research studied by Chrysikopoulos [19] also indicated that predictions of contaminant concentrations were sensitive to the source geometry for short downstream distance.However, when scale factor increases to a certain degree (i.e., SF = 7.0), the CR/CE keeps constant.In Figure 10(b), the situation is slightly different that the CR/CE increases with scale factor rising.However, the CR/CE keeps constant when scale factor increased to SF = 9.0.The main reason is attributed to the fact that the same and continuous injection patterns make the differences of contaminant concentration between two source geometries in the near field insignificant on water table.Since contaminant migration from rectangular source geometry is slightly faster than that from elliptic source geometry as previously studied, so the difference between the two source geometries in the far field is significant, while it keeps constant if scale factor reaches a certain value.Assuming that if the lower boundary is infinite (i.e., the scale factor is big enough), both source geometries could be regarded as point source geometry.In this context, there would be no difference on contaminant migration between the two source geometries.
From the above discussion, a conclusion is obtained that the CR/CE keeps constant under condition that the scale factor is equal to or greater than 9. Meaning that the rectangular source geometry can substitute for the elliptic source geometry under condition that scale factor is equal to or greater than 9.In other words, we can use the data of contaminant migration from rectangular source geometry and the corresponding value of CR/CE to analyze and estimate the contaminant migration from elliptic source geometry.

Summary
Three-dimensional analytical solutions for contaminant transport in subsurface porous media from rectangular source geometry and elliptic source geometry were investigated, accounting for three-dimensional hydrodynamic advection-dispersion in a uniform flow field, first-order decay rates.The fate and transport of contaminant were simulated based on a continuous source loading.Several interesting solutions were obtained as follows.
(1) The migration of pollution plume from the two source geometries shows the same trend under the same aspect ratio, the equal source area size, and the uniform total pollution load condition, except that the migration of pollution plume from rectangular source geometry is faster than that from elliptic source geometry.
(2) The influence of source area size on pollution plume migration decreases with the distance to field increasing.In particular when pollution plume migrates to a certain distance, as far as five times of source side length (  ), the variation of source area size has no effect on pollution plume migration.
(3) The migration of pollution plume becomes slower with the increase of aquifer thickness for both source geometries.Compared with elliptic source geometry, the phenomenon of rectangular source geometry is more significant.
(4) The contaminant concentration is decreasing with scale factor rising, and the differences among various scale factors get smaller with the distance to field increasing.
(5) When scale factor is equal to or greater than 9, the CR/CE would keep constant; meaning that we can use the data of contaminant migration from rectangular source geometry and the corresponding value of CR/CE to analyze and estimate the contaminant migration from elliptic source geometry.Since calculations of three-dimensional analytical solutions for contaminant transport from rectangular source geometry are simpler than those from elliptic source geometry (i.e., the rectangular source geometry requires little information on contaminated site and little calculation), we can use the data of contaminant migration from rectangular source geometry to obtain the contaminant transport from elliptic source geometry to simplify the actual engineering problem.The results would provide useful information and technical support for estimating the distribution of contaminant in the initial period of contaminated site investigation.

Figure 1 :
Figure 1: The general geometry of the rectangular source.

Figure 4 :
Figure 4: Contaminant concentration contours on the -plane ( = 0) under larger source area, unchanged source area, and smaller source area conditions at  = 100 days for rectangular source geometry.

Figure 6 :
Figure 6: Contaminant concentration at half aquifer thickness under various scale factors at  = 100 days for rectangular source geometry.

Figure 9 Figure 8 :
Figure 8: Contaminant concentration contours on the -plane ( = 0) under larger source area, unchanged source area, and smaller source area conditions at  = 100 days for elliptic source geometry.

Figure 10 :
Figure 10: The variation of CR/CE with the increase of scale factor at various locations along transverse direction.

Table 1 :
Model simulation parameters for two source geometries.