## 2 Materials and Methods
### Flow and Transport Simulation  
#### 2.2.1 Governing Equation  
In this study, the aquifer was assumed to be a single layer with a thickness of 5 m because it was difficult to obtain a detailed geological cross section of the aquifer. The flow in the aquifer was modeled as a 2D two-phase flow of water and non-aqueous phase liquid (NAPL) in the x-y coordinates, where NAPL is 1,4-dioxane [27]. However, the flow potential of the model takes into account the groundwater level (i.e., difference between land surface elevation and depth to water). As a result, the model is a quasi 3D model. The governing equations for the flows for NAPL and groundwater with/without dissolved NAPL and the advection-diffusion equation for NAPL in water are respectively represented by Equations (1)-(3), as follows. \[\frac{\partial}{\partial x}\left(\frac{Kk_{ro}}{\mu_{o}}\rho_{o}\frac{ \partial}{\partial x}\Phi_{o}\right)+\frac{\partial}{\partial y}\left(\frac{ Kk_{ro}}{\mu_{o}}\rho_{o}\frac{\partial}{\partial y}\[PERSON]_{o}\right)=\frac{ \partial(\varpi\rho_{o}S_{o})}{\partial t} \tag{1}\]  
\[\frac{\partial}{\partial x}\left(\frac{Kk_{rw}}{\mu_{w}}\rho_{w}\frac{ \partial}{\partial x}\[PERSON]_{w}\right)+\frac{\partial}{\partial y}\left(\frac{ Kk_{rw}}{\mu_{w}}\rho_{w}\frac{\partial}{\partial y}\Phi_{w}\right)=\frac{ \partial(\varpi\rho_{w}S_{w})}{\partial t} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline Year & Events \\ \hline
1990 & Started illegally dumping industrial waste \\
2000 & Started removing waste \\
2005–2007 & Installed impermeable wall \\
2009 & Implemented a groundwater treatment plant \\
2010 & Installed pumping well \\
2013 & Started monitoring of 1,4-dioxane \\
2014 & (Under survey (2021)) \\ \hline \end{tabular}
\end{table} 
Table 1: History of site modifications (well installation and monitoring period). Figure 1: Map of an illegal waste dumping site. The blue grid means the calculated area. (**a**) Site map and simulation area. (**b**) Distribution of elevation level (EL) from a GIS. (**c**) Distribution of the groundwater level (GL). The EL and GL is the distance from the sea surface. \[ \begin{array}{ll}\frac{\partial}{\partial x}\Big{(}x_{ocw,k}\frac{kk_{kv}}{\mu_{w}} \rho_{w}\frac{\partial}{\partial x}\Phi_{w}\Big{)}&+\frac{\partial}{\partial y} \Big{(}x_{ocw,k}\frac{kk_{kv}}{\mu_{w}}\rho_{w}\frac{\partial}{\partial y}\Phi_ {w}\Big{)}\\ &+\frac{\partial}{\partial x}\Big{[}D_{ocw,k}\frac{\partial}{\partial y}(x_{ocw,k }\varphi\rho_{w}S_{w})\Big{]}\\ &+\frac{\partial}{\partial y}\Big{[}D_{ocw,k}\frac{\partial}{\partial y}(x_{ocw,k }\varphi\rho_{w}S_{w})\Big{]}=\frac{\partial(x_{ocw,k}\varphi\rho_{w}S_{w})}{ \partial t}\end{array}  \tag{3}\]  
where \(K\) is the absolute permeability (m\({}^{2}\)); \(k_{vo}\) is the relative permeability for NAPL (fraction) [28; 29; 30]; \(k_{rw}\) is the relative permeability for water (fraction) [28; 29; 30]; \(\mu_{o}\) is the NAPL viscosity (Pa-s); \(\mu_{w}\) is the water viscosity (Pa-s); \(\rho_{o}\) is the molar density of NAPL (mol/m\({}^{3}\)); \(\rho_{w}\) is the molar density of water (mol/m\({}^{3}\)); \(\Phi_{o}\) is the flow potential of NAPL (Pa); \(\Phi_{w}\) is the flow potential of water (Pa); \(S_{o}\) is the NAPL saturation (fraction); \(S_{w}\) is the water saturation (fraction); \(\varphi\) is the porosity (fraction); \(D_{ocw,k}\) is the diffusion coefficient (m\({}^{2}\)/s); \(x_{ocw,k}\) is the concentration of NAPL dissolved in water (fraction); and \(t\) is time (s). The parameter values used in this study are listed in Table 2. The parameter values for the groundwater layer were set by assuming a clay soil, and the parameter values for 1,4-dioxane were acquired from a database [31; 32; 33]. Note that the biodegradation and adsorption to soil for 1,4-dioxane were neglected based on the chemical properties of 1,4-dioxane [34].