## 2 Materials and Methods
### Groundwater Flow Simulation  
Groundwater models, also known as groundwater flow simulation models, are used to forecast the consequences of hydrological changes on the behavior of the aquifer [4]. Aspects of groundwater quality are included in specific groundwater models in other studies [35]. Groundwater models are employed in a variety of urban water management programs. Some groundwater models attempt to forecast the chemical's fate and movement in natural, urban, and speculative scenarios [36]. Three-dimensional groundwater flow is based on the governing equation of groundwater based on [PERSON]'s law represented by the partial differential equation written in Sepulveda 2015 [37] as Equation (1).  
\[\frac{\partial}{\partial x}\bigg{(}K_{xx}\frac{\partial h}{\partial x}\bigg{)} +\frac{\partial}{\partial y}\bigg{(}K_{yy}\frac{\partial h}{\partial y}\bigg{)} +\frac{\partial}{\partial y}\bigg{(}K_{zz}\frac{\partial h}{\partial z}\bigg{)} -W=(S_{c})\frac{\partial h}{\partial t}, \tag{1}\]  
Figure 3: Discretization using a regular grid.  
where \(x\), \(y\), and \(z\) are the permeability direction on horizontal dimension, \(t\) is the time, \(h\) is the hydraulic head, \(K\) represents the hydraulic conductivity, \(W\) is the sink term, and \(S_{c}\) is the storage.  
Hydraulic heads for one stress period were simulated by the interface of the Layer-Property Flow package (LPF) on a steady-state condition. The unsteady state simulation based on MODFLOW USGS has been investigated [38]. The model hydraulic properties were assigned using the zonation, in our case, the downgradient, and the upgradient by the graphic user interface (GUI). Manual calibration was carried out to approach the model output results. Accordingly, manual calibration was conducted to obtain the same trend of errors; it could considerably reduce the problem of local minima and approach the sought calibration parameters values. Afterwards, the mechanic calibration using the pilot point methodology could be processed. The non-linearity of the hydraulic conductivity during the computation iteration is the main challenge in calibrating the groundwater flow model in a heterogeneous medium. Whereas the standard simulation with manual calibration, the main objective is to achieve less or equal to 50% accuracy in minimum residual error. The statistical criteria are used to evaluate the model accuracy, such as root-mean squared (RMS) and R-squared. The RMS is traditionally the primary calibration measure for the groundwater flow model. The square root of the average square of the residual error represents the difference between the onsite observed value and the numerically simulated hydraulic head (Equation (2)) [39].  
\[\mathrm{r}=\mathrm{h}_{0}-\mathrm{h}_{\mathrm{s}}, \tag{2}\]  
where r is the residual, \(\mathrm{h}_{0}\) is the observed hydraulic head, and \(\mathrm{h}_{\mathrm{s}}\) is the numerically simulated.  
\[\mathrm{RMS}=\sqrt{\frac{1}{2}\sum_{t=1}^{n}\left(\mathrm{h}_{0}-\mathrm{h}_ {\mathrm{s}}\right)_{t}^{2}}, \tag{3}\]  
The groundwater flow and the model comparison are measured from their R-squared described by [PERSON] (2013) [40] (Equation (4)).  
\[R^{2}=\frac{\sum_{i=1}^{n}\left(\,y_{i}-\hat{y}_{i}\right)^{2}}{\sum_{i=1}^{n }\left(\,y_{i}-\overline{y}_{i}\right)^{2}}, \tag{4}\]  
where \(R^{2}\) is the R-squared, \(\hat{y}\) is the predicted value of \(y\), and \(\overline{y}\) is the mean value of \(y\).