## 3 Coupling Mathematical Model for Tunnel Freezing-Frost Heave  
### Mathematical Model for the Freezing Temperature Field  
The transient heat conduction issue with phase transition includes the fluctuation of the freezing temperature field in the horizontal freezing tunnel [28; 29]. Based on heat transfer and permafrost theories, and assuming that the temperature gradient of the low-temperature refrigerant in the freezing tube along the longitudinal direction (length direction) of the tunnel is approximately zero during the freezing process [30; 31], the two  
Figure 3: Schematic presentation of FSPR applied in the Gongbei Tunnel. -dimensional heat balance control differential equation of the freezing temperature field can be expressed as follows:  
\[\left\{ \begin{array}{l}C_{f}\frac{\partial T_{f}}{\partial t}=\frac{\partial}{ \partial x}\left(\lambda/\frac{\partial T_{f}}{\partial x}\right)+\frac{ \partial}{\partial y}\left(\lambda_{f}\frac{\partial T_{f}}{\partial y}\right) \text{, in }\Omega_{f}\\ C_{u}\frac{\partial T_{u}}{\partial t}=\frac{\partial}{\partial x}\left( \lambda_{u}\frac{\partial T_{u}}{\partial x}\right)+\frac{\partial}{\partial y }\left(\lambda_{u}\frac{\partial T_{u}}{\partial y}\right)\text{, in }\Omega_{u}\\ C_{L}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(\lambda_{L} \frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left( \lambda_{L}\frac{\partial T}{\partial y}\right)+L_{w}\rho_{i}\frac{\partial \partial t}{\partial t}\text{, in }\Omega_{L}\end{array} \right. \tag{1}\]  
where  
\(\Omega_{f},\Omega_{u},\Omega_{L}\)--Frozenzone, Unfrozen zone, Phase transition zone  
\(C_{f},C_{u},C_{L}\)--Specific heat of soil volume in \(\Omega_{f},\Omega_{u},\Omega_{L},\text{kJ}/(\text{m}^{3}\cdot\text{{}^{ \circ}C})\)  
\(T_{f},T_{u}\)--Soil temperature in \(\Omega_{f},\Omega_{u},\text{{}^{\circ}C}\)  
\(\lambda_{f},\lambda_{u},\lambda_{L}\)--Soil thermal conductivity in \(\Omega_{f},\Omega_{u},\Omega_{L},\text{W}/(\text{m}^{3}\cdot\text{{}^{\circ}C})\)  
\(L_{w}\)--Latent heat of phase transition of water, 333 kJ/kg  
\(\rho_{i}\)--The density of ice, kg/m\({}^{3}\)  
\(\theta_{i}\)--Volume ice content of soil, %  
\(t\)--Freezing time, s  
Assuming that the soil's freezing and melting temperatures are \(T_{d}\) and \(T_{r}\), respectively, and that no phase transition occurs when the soil temperature \(T<T_{d}\) or T \(>T_{r}\), the phase transition latent heat of water in \(\Omega_{L}=[T_{d},\,T_{r}]\) in Equation (1) can be derived and transformed into the phase transition latent heat of water-bearing soil, namely:  
\[L_{w}\rho_{i}\frac{\partial\theta_{i}}{\partial t}=-L_{w}\rho_{w}\frac{ \partial\theta_{w}}{\partial T}\frac{\partial T}{\partial t}=-L_{w}\rho_{w} \frac{\rho_{d}(w_{0}-w_{u})}{\rho_{w}(T_{r}-T_{d})}\frac{\partial T}{\partial t }=-\frac{L}{T_{r}-T_{d}}\frac{\partial T}{\partial t} \tag{2}\]  
where  
\(\rho_{w},\rho_{d}\)--Density of water, Dry density of soil, kg/m\({}^{3}\)  
\(\theta_{w}\)--Volume moisture content of soil, %  
\(w_{0},w_{u}\)--Initial moisture content of soil, Unfrozen water content in frozen soil, %  
\(L\)--Latent heat of phase transition in soil, kJ/m\({}^{3}\)  
The differential equation for managing the heat balance of permafrost in the phase transition zone \(\Omega_{L}=[T_{d},\,T_{r}]\) can be found by substituting the derivation of Equation (2) into Equation (1):  
\[\left(C_{L}+\frac{L}{T_{r}-T_{d}}\right)\frac{\partial T}{\partial t}=\frac{ \partial}{\partial x}\left(\lambda_{L}\frac{\partial T}{\partial x}\right)+ \frac{\partial}{\partial y}\left(\lambda_{L}\frac{\partial T}{\partial y}\right) \tag{3}\]  
By linear interpolating according to the temperature interval, the mass volume-specific heat \(C_{L}\) and thermal conductivity \(\lambda_{L}\) in Equation (3) can be obtained:  
\[C^{*}=\left\{ \begin{array}{ccc}C_{f}&(T<T_{d})\\ \frac{C_{f}+C_{u}}{2}+\frac{L}{T_{r}-T_{d}}&(T_{d}\leq T\leq T_{r})\\ C_{u}&(T>T_{d})\end{array} \right. \tag{4}\]  
\[\lambda^{*}=\left\{