## 2 Mathematical Model of the Freezing Temperature Field and Related Physical Parameters
### Physical Parameters of the Soil Layer  
(1) Physical parameters of the soil layer  
In this study, the sandy clay is the main soil type, its freezing temperature is around 0 \({}^{\circ}\)C\(\sim\)\(-\)0.5 \({}^{\circ}\)C as obtained by a series of frozen soil physical tests, and the calculation equation of the freezing temperature for the frozen soil [42] is as follows:  
\[T_{d}=T_{s}+\eta p \tag{7}\]  
where \(T_{s}\) is the freezing temperature of wet soil without salt and external load (\({}^{\circ}\)C), \(\eta\) is the average rate of change of the freezing temperature of wet soil without salt with external loading, generally taken as \(-\)0.075 \({}^{\circ}\)C\(\cdot\)MPa\({}^{-1}\), and \(p\) is the pore water pressure of the soil (MPa).  
(2) Thermal Conductivity of Soil and Frozen Soil  
The thermal conductivity of unfrozen soil and frozen soil decreases by 0.2\(-\)0.5% as the temperature decreases by 1 \({}^{\circ}\)C, which is taken as 0.3%. The thermal conductivity coefficient at the corresponding temperature can be calculated according to the \"Thermophysics of Frozen Soil\" in the previous literature [43].  
(3) Specific heat of soil and frozen soil  
The calculation equation for the unfrozen and frozen soil is shown below:  
\[C_{dt}=\frac{C_{su}+WC_{w}}{1+W} \tag{8}\]  
\[C_{df}=\frac{C_{sf}+(W-W_{u})C_{i}+W_{u}C_{w}}{1+W} \tag{9}\]  
where \(C_{dt}\), \(C_{df}\), \(C_{su}\), \(C_{sf}\), \(C_{w}\), and \(C_{i}\) are the specific heat of the soil, frozen soil, soil skeleton, frozen soil skeleton, water, and ice (J\(\cdot\)kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\)), respectively, and \(W\) and \(W_{u}\) are the total water content and unfrozen water content, respectively. Specifically, the values of \(C_{w}\) and \(C_{i}\) can be taken as 4182 and 2090, respectively, while the value of \(W_{\rm u}\) is close to 0. In addition, the specific heat capacity of the ice-water mixture is generally ignored due to its relatively weak effect on the calculation of freezing temperature field.  
Phase transformation latent heat of soil  
The amount of heat emitted or absorbed due to the phase transformation of water in unit volume soil can be calculated by:  
\[Q=L\rho_{d}(W-W_{u}) \tag{10}\]  
\[\rho_{d}=\rho/(1+W) \tag{11}\]  
where \(Q\) is the phase transformation latent heat of the soil (kJ\(\cdot\)m\({}^{-3}\)), \(L\) is the freezing latent heat (334.56 kJ\(\cdot\)kg\({}^{-1}\)), \(\rho_{\rm d}\) is the dry density of the soil (kg\(\cdot\)m\({}^{-3}\)), and \(\rho\) is the wet density of the soil (kg\(\cdot\)m\({}^{-3}\)).  
Enthalpy  
The phase transformation latent heat of soil can be taken as the constant value when its water content stays at a lower stage, while the phase transformation latent heat of soil can be expressed by the change in enthalpy, as this method could solve phase transition problems with complex boundary conditions and multiple nonmonotonic interfaces. The unfrozen water content is relatively low and its effect on the phase transition is considered in the calculation. When the enthalpy of unit volume soil at \(T_{0}\) is zero, the enthalpy of unit volume soil at \(T\) can be expressed by:  
\[\Delta H_{T_{-}}=\int_{T_{0}}^{T_{-}}\rho C_{df}dT=\rho C_{df}(T_{-}-T_{0}) \tag{12}\]  
The enthalpy increment of unit volume soil from \(T_{-}\) to \(T_{+}\) is the phase transformation latent heat \(Q\); thus, the enthalpy increment from \(T_{+}\) to \(T\) is:  
\[\Delta H_{T_{+}}=\int_{T_{+}}^{T}\rho C_{du}dT=\rho C_{du}(T-T_{+}) \tag{13}\]  
where \(T_{-}\) and \(T_{+}\) are the temperatures of the phase transition interval, which can be taken as the freezing points \(T_{d}\) and \(T_{d}\) minus 0.5 \({}^{\circ}\)C, while these two values could be applicable to the sandy clay and determined by an enthalpy change interval requirement in ANSYS software.