# Configuration of Non-Pumping Reactive Wells Considering Minimum Well Spacing
## 2 Materials and Methods
### Modeling Groundwater Flow and Contaminant Transport  
A numerical simulation on a hypothetical contaminated site was conducted to evaluate the remediation of contaminated groundwater and prevention of contaminant spreading. The conceptual model of the contaminated aquifer assumed that groundwater, which was in an unconfined aquifer, was in steady state and was contaminated by heavy metals. The aquifer had 10 m of saturated thickness and 0.01 hydraulic gradient. The geologic media was modeled with three layers: an unsaturated zone, unconfined aquifer, and impermeable layer. In the simulation, the NPRW was inserted by boring wells in various configurations from the unsaturated zone to the unconfined aquifer for contaminant remediation. A three-dimensional modeling domain was constructed to numerically simulate the groundwater flow and contaminant transport. The model domain (20 \(\times\) 5 \(\times\) 10 m) was divided into 40,000 square grid elements. The specified head boundary at the west and east edges of the model produced a regional hydraulic gradient of 0.01 eastward. The north, south, and bottom edges of the model were defined as no-flow boundaries. Hydraulic conductivities of 3 \(\times\) 10\({}^{-4}\) cm/s and 0.1 cm/s were assigned to the aquifer and wells, respectively. The hydraulic conductivity of aquifer was selected within the range of values (0.00864-0.864 m/d) generally observed in the porous medium mainly composed of sand [38]. Wells containing the reactive media were arranged in the middle of the domain following a saw-tooth array (Figure 1e,f). The effective porosity was 0.30 and 0.35 for the aquifer and the reactive material, respectively. The contaminant source was located on the west edge of the domain and continuously supplied water with 10 mg/L of the contaminant to all the layers in the domain. The plume width (S\({}_{\text{w}}\)) was set equal to the distance between the centers of the wells on the edges of the upgradient transect. Longitudinal and transverse horizontal dispersivity were given as 0.01 m and 0.001 m, respectively. Wells were simulated as contaminant sinks with a concentration of 0 mg/L; the model assumed that reactive media completely removed any contaminants entering the wells. This study employed three numerical models, MODFLOW [39], MODPATH [40], and MT3 DMS [41] to simulate groundwater flow, particle tracking, and solute transport, respectively, in a hypothetical unconfined aquifer. MT3 DMS [42] can be used to simulate changes in concentrations of miscible contaminants in groundwater considering advection, dispersion, diffusion, and some basic chemical reactions with various types of boundary conditions and external sources or sinks [42]. It uses the following equation:  
\[\frac{\partial\text{C}}{\partial\text{t}}=\frac{\partial}{\partial\text{x}_{ 1}}\left(\text{D}_{\text{ij}}\frac{\partial\text{C}}{\partial\text{x}_{ \text{j}}}\right)-\frac{\partial}{\partial\text{x}_{\text{i}}}\left(\text{v} _{\text{i}}\text{C}\right)+\frac{\text{q}_{\text{s}}}{\text{\theta}}\text{C} _{\text{s}}+\sum\limits_{\text{k}=1}^{\text{N}}\text{R}_{\text{k}} \tag{1}\]  
where C is the concentration of contaminants dissolved in groundwater, t is the time, x\({}_{\text{i}}\) is the distance along the respective Cartesian coordinate axis, D\({}_{\text{ij}}\) is the hydrodynamic dispersion coefficient, v\({}_{\text{i}}\) is the seepage or linear pore water velocity, q\({}_{\text{s}}\) is the volumetric flux of water per unit volume of aquifer representing sources (positive) and sinks (negative), C\({}_{\text{s}}\) is the concentration of the sources or sinks, \(\theta\) is the porosity of the porous medium, and \(\sum\limits_{\text{K}=1}^{\text{N}}\text{R}_{\text{k}}\) is the chemical reaction term. The transport equation is related to the flow equation as follows:  
\[\text{v}_{\text{i}}=-\frac{\text{K}_{\text{ii}}}{\theta}\frac{\partial\text{h }}{\partial\text{x}_{\text{i}}} \tag{2}\]where \(\mathrm{K_{ii}}\) is the hydraulic conductivity value along the i coordinate axes (i = x, y, z).