In this study, the reactive transport of each organic compound and pseudocompounds was simulated numerically using the finite-difference groundwater flow model, MODFLOW [

18], and the associated reactive transport code, RT3D [

19]. In column tests, the numerical model assumed saturated flow through homogeneous and isotropic simulated aquifer sands, and the specific MODFLOW model considered a rectangular two-dimensional confined flow system with length of 400 mm by width of 70 mm in the Cartesian

x- and

y-directions, respectively. The numerical grid was based on nodal spacings of Δ

x = 10 mm, Δ

y = 10 mm, and Δ

z = 72 mm. Since biological degradation was not considered due to the addition of the biocide (200 mg·L

^{−1} NaN

_{3}) to the permeant liquids, advection, dispersion, and equilibrium sorption to simulated aquifer sands were considered as the processes governing the reactive transport of the organic compounds in mixtures. Thus, the governing equation for one-dimensional transport of the organic compounds in mixtures through simulated aquifer sands can be written as follows:

where

C and

S are the concentration of an organic compound in the aqueous phase (mg·L

^{−1}) and the solid phase (mg·kg

^{−1}), respectively,

x is the longitudinal distance of transport (mm),

t is the elapsed time (d),

D_{h} is the hydrodynamic dispersion coefficient obtained from the results of the salt tracer (KCl) (mm

^{2}·day

^{−1}),

v_{s} is the average linear seepage velocity (mm·day

^{−1}),

ρ_{b} is the bulk (dry) density (g·mL

^{−1}), and

n is the total porosity. According to these considerations and

Figure 1, the column was assumed to be initially free of any organic compound (

C(x,0) = 0), a constant source concentration using injection wells was assumed for the upper boundary condition (

C(0,t) =

C_{0}), and a zero concentration gradient using constant head was assumed for the lower boundary condition (

∂C(L,t)/∂x = 0).

According to the results of a previous study evaluating the equilibrium sorption of the same organic compounds to the identical simulated aquifer sands [

1,

2,

3], the sorption of the organic compounds in mixtures to simulated aquifer sands was assumed to follow the Freundlich sorption isotherm equation as follows:

where

K_{f} ((mg·kg

^{−1})/(mg·L

^{−1})

^{N}) is the Freundlich unit sorption capacity, and

N (dimensionless) is the joint measure of the relative magnitude and diversity of energies [

1].

Since the root-mean-square error (RMSE) is more appropriate to represent model performance than the mean absolute error (MAE) when the error distribution is expected to be Gaussian [

21], the agreement between the experimentally measured concentrations according to the effluent/pore-water samples and the fitted (or predicted) concentrations estimated from MODFLOW-RT3D was evaluated in terms of the root-mean-square error (RMSE) defined as follows:

where

X_{i,mes} and

X_{i,fit} are the measured and fitted (or predicted) values for

C(x,t), respectively, and

p is the sample size.

This error function attempts to minimize the fractional error distribution across the entire concentration ranges from the comparison of measured values with the corresponding fitted (or predicted) values. Once the minimum of the error function was achieved, the values for the Freundlich sorption parameters (K_{f} and N) were obtained. The values for the Freundlich sorption parameters obtained from the column tests were estimated using the overall effluent data, and the resulting predicted concentration profiles within a column were compared to the measured concentration profiles to validate the effectiveness of lumped approach.