## 2 Migration Modelling and Numerical Simulation of Contaminated Fields
### Numerical Simulation of the Migration Characteristics of the Contaminated Site  
Based on the basic law of groundwater flow and the mechanism of pollutant migration, based on the finite unit method, the global flow coupling of [PERSON]'s law module and the porous medium dilute matter transfer module was carried out by using the COMSOL software. A 250 m \(\times\) 10 m two-dimensional chromium heavy metal contamination field model was constructed (see Figure 1), with poorly permeable pulverized clay in material 1, fine sand with better permeability in material 2, and medium sand with good permeability in material 3. Only the parameters such as porosity, density, mass water content, and permeability coefficient of the material are set individually in the simulation. The resistivity of the whole model area is uniformly set to 390 \(\Omega\cdot\)m. Detailed model parameter settings are shown in Table 1. The raised part at the surface is a static storage of chromium pollution source, and the pollutant is a solution of Cr ions with relative concentration of 3 mol/m\({}^{3}\), and its projected orientation on the ground is within 50 m to 100 m. The upper boundary (y = 0 m) is set to have a rainfall of 500 mm. The pressure head is set to be at x = 250 m, 5.4 m from the surface. The groundwater demarcation line is set to be at x = 0, and no fluid passes through the two sides of the demarcation line. The bottom of aquifer (y = \(-\)10 m) is set without flow. The rest of the parameter settings are shown in Table 1. In order to improve the solution accuracy and computational efficiency, the quadrilateral mesh adaptive algorithm is used to mesh the constructed model area [21]. Then, the mesh is automatically encrypted in the pollutant area and near the electrode arrangement line, and the mesh is automatically sparsed in the non-data collection area. The boundary conditions and parameters are set with reference to the numerical forward simulation in [17; 22]. Figure 1: Initial model of contaminated field (not migrated). Studies have shown that porosity \(\varphi\), mass water content \(\theta_{w}\), and Cr ion concentration \(\lambda(mg/kg)\) are the main factors influencing soil resistivity [23; 24], and the relationship between these and soil resistivity can be described by the Alchian equation [25]:  
\[\rho_{0}=k\varphi^{-m}S_{r}{}^{-n}\rho_{w} \tag{2}\]  
where \(\rho_{0}\) is the rock resistivity, \(\Omega\cdot\)m; \(\rho_{w}\) is the pore water resistivity in the soil, \(\Omega\cdot\)m; \(k\), \(m\), and \(n\) are the soil property correlation coefficients. In Equation (2): \(\rho_{w}=\frac{1}{\lambda n^{n}}\); \(S_{r}=\frac{\theta_{w}}{\varphi}\); \(n^{0}=\frac{\lambda}{52\times\theta_{w}}\); \(\theta_{w}=\frac{\rho\theta_{w}}{1+\theta_{w}}\), where \(\Lambda\) is the molar conductivity of the soil solution, S\(\cdot\)m\({}^{2}\)/mol; \(n^{0}\) is the mass concentration of Cr ions, mol/L; \(\theta_{v}\) is the volumetric water content; \(\lambda\) is the concentration of Cr contaminants, mg/kg; \(\rho\) is the density of the soil medium, g/cm\({}^{3}\). The empirical equation for soil resistivity derived from the literature shows that [26]:  
\[\rho_{0}=1.8\times 10^{7}\varphi^{1.762}\times\theta_{w}\times\lambda^{-1} \times(\frac{\rho\theta_{w}}{1+\theta_{w}})^{-0.474} \tag{3}\]  
where \(\varphi\) is the porosity, \(\theta_{w}\) is the mass water content, and \(\lambda\) is the concentration of Cr ions in the soil (mg/kg), which are the main factors affecting the resistivity of the soil. According to Equation (3), the resistivity of the contaminated site can be calculated based on the porosity and mass water content of the contaminated site in a soil with a certain density. The migration process of the contaminated field over time was simulated using COMSOL software.