## 3 Software and Methodology
### Groundwater Contaminant Transport Mathematical Model  
Contaminants in the subsurface are present on the surfaces of geological particles and within the pore spaces of geological formations, and they move with the flow of groundwater. Contaminants migrate with the natural water cycle in the subsurface and are influenced by human activities such as pumping. Additionally, dissolved contaminants in groundwater undergo diffusion due to concentration differences between different regions. The sorption and desorption of contaminants also occur between groundwater and soil due to concentration disparities. We have simulated the dry winter season, accounting for the effects of moisture evaporation. In layers 1--silty clay and 2--rounded gravel, which are in contact with the air, we employ the Richards equation to simulate the flow of unsaturated groundwater. In layers 3--clay and 4--rounded gravel, as well as the aquifer with relative confining properties, Darcy's Law is commonly used in COMSOL software to describe this flow process. Darcy's law is a fundamental equation for describing fluid flow in porous media, derived from experiments involving water flow through sand layers, and holds significant importance in hydrogeology [22]. It outlines the principles of fluid flow in porous media as follows:  
\[\mathrm{q}=-\frac{k}{\mu L}\Delta p, \tag{1}\]  
This equation defines the absolute permeability for describing single-phase flow. The symbols in the equation include q (flow rate, units: m/s), \(k\) (permeability, units: m\({}^{2}\)), \(\mu\) (fluid viscosity, units: Pa\(\cdot\)s), \(L\) (length of the rock sample, units: m), and \(\Delta p\) (pressure difference between the left and right ends, units: Pa). It exhibits high reliability when dealing with conventional saturated flow. Darcy's Law is an empirical law that mathematically combines various factors influencing fluid flow in porous media. It is primarily applicable to laminar flow conditions, where groundwater flow in fine-grained or coarse-grained sediments is predominantly laminar. Darcy's Law finds extensive application in groundwater flow and petroleum engineering, particularly when addressing saturated flow problems. For formations above the water Table, the formation is unsaturated with both water and gas in the formation. This is usually described in COMSOL using the Richards equation. The Richards equation is used to describe the movement of water in unsaturated soils and was proposed by [PERSON] in 1931 [23]. This equation builds upon Darcy's Law but is suitable for unsaturated flow situations. It involves parameters such as hydraulic conductivity (\(K\)), pressure head (\(h\)), elevation (\(z\)), volumetric water content (\(\theta\)), time (\(t\)), and others. The equation describes the variation in volumetric water content with time and expresses the rate of change of saturation by considering variations in the inflow and outflow fluxes. \[\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}\bigg{(}K(\theta )\bigg{(}\frac{\partial h}{\partial z}\bigg{)}+1\bigg{)}, \tag{2}\]  
In porous media, including both saturated and unsaturated conditions, it is essential to consider the processes of pollutant advection, diffusion, and adsorption simultaneously to perform accurate numerical simulations of pollutant transport. The advection-diffusion equation for pollutants describes the dissolution of pollutants in groundwater and their movement towards regions of lower concentrations in the presence of concentration gradients. The general form of this equation is expressed as follows [24]:  
\[\frac{\partial c}{\partial t}=\
abla(\mathrm{D}\
abla\mathrm{c})-\
abla(v \mathrm{c})+\mathrm{R}, \tag{3}\]  
In this context, '\(c\)' represents the variable under investigation, which could be the concentration of a solute in the case of mass transfer or the temperature in the context of heat transfer.