## 3 Uncertain Analysis Method of Thermodynamic Properties
### Coupled Governing Equations
\(H_{\mathrm{f}}\), \(S_{\mathrm{f}}\), \(K_{x}^{\mathrm{f}}\) and \(K_{y}^{\mathrm{f}}\) are the water head, specific yield, and permeability coefficients in the frozen area \(\Omega_{\mathrm{f}}\), respectively. \(Q\) is the source or collection of the seepage field. Variables \(V_{x}\) and \(V_{y}\) are the seepage rates in the \(x\) and \(y\) directions, respectively. \(C_{w}\) and \(\rho_{w}\) are the volumetric heat capacity and bulk density of water. Parameters with subscript u are the corresponding physical components in the unfrozen area \(\Omega_{\mathrm{u}}\). \(t\) is time, and \(x\), \(y\) are distances. At the freezing-front position \(s\) (\(x\);\(t\)), in which \(x=\{x,y\}\), The following equation needs to be satisfied. \[T_{\mathrm{f}}(s(\mathbf{x},t),t)=T_{\mathrm{u}}(s(\mathbf{x},t),t)=T_{\mathrm{ m}} \tag{7}\]  
\[\lambda_{\mathrm{f}}\frac{\partial T_{\mathrm{f}}}{\partial n}-\lambda_{\mathrm{ u}}\frac{\partial T_{\mathrm{u}}}{\partial n}=L\frac{\mathrm{d}s(\mathbf{x},t)}{ \mathrm{d}t} \tag{8}\]  
where \(L\) is the latent heat per unit volume, \(n\) is the direction vector of the moving boundary \(s\) (\(x\);\(t\)), and \(T_{\mathrm{m}}\) is the phase change temperature. The hydrothermal parameters can be expressed as  
\[C^{*}=\left\{ \begin{array}{ll}C_{\mathrm{f}}&T<T_{\mathrm{m}}-\Delta T\\ \frac{C_{\mathrm{u}}+C_{\mathrm{f}}}{2}+\frac{L}{2\Delta T}&T_{\mathrm{m}}- \Delta T\leq T\leq T_{\mathrm{m}}+\Delta T_{\mathrm{u}}\\ C_{\mathrm{u}}&T>T_{\mathrm{m}}+\Delta T_{\mathrm{u}}\end{array} \right. \tag{9}\]  
\[\lambda^{*}=\left\{ \begin{array}{ll}\lambda_{\mathrm{f}}&T<T_{\mathrm{m}}- \Delta T\\ \lambda_{\mathrm{f}}+\frac{\lambda_{\mathrm{u}}-\lambda t}{2\Delta T}[T-(T_{ \mathrm{m}}-\Delta T)]&T_{\mathrm{m}}-\Delta T\leq T\leq T_{\mathrm{m}}+ \Delta T\\ \lambda_{\mathrm{u}}&T>T_{\mathrm{m}}+\Delta T\end{array} \right. \tag{10}\]  
The hydraulic conductivity can be written as follows:  
\[K_{x}^{*}=\left\{ \begin{array}{ll}\bigg{(}\frac{W_{\mathrm{u}}}{W}\bigg{)} ^{\beta}.K_{x}^{\mathrm{u}}&T\leq T_{\mathrm{m}}\\ K_{x}^{\mathrm{u}}&T>T_{\mathrm{m}}\end{array} \right.,\ K_{y}^{*}=\left\{ \begin{array}{ll}\Big{(}\frac{W_{\mathrm{u}}}{W}\Big{)}^{\beta}.K_{y}^{ \mathrm{u}}&T\leq T_{\mathrm{m}}\\ K_{y}^{\mathrm{u}}&T>T_{\mathrm{m}}\end{array} \right. \tag{11}\]  
where \(W\) and \(W_{\mathrm{u}}\) denote the water content and unfrozen water content of soil, respectively. The constant \(\beta\) is determined experimentally. Using Equations (9)-(11), Equations. (1)-(8) can be written simply as  
\[C^{*}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(}\lambda^{*} \frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(} \lambda^{*}\frac{\partial T}{\partial y}\bigg{)}-C_{w}\rho_{w}\bigg{(}V_{x}^{* }\frac{\partial T}{\partial x}+V_{y}^{*}\frac{\partial T}{\partial y}\bigg{)} \tag{12}\]  
\[S^{*}\frac{\partial H}{\partial t}=\frac{\partial}{\partial x}\bigg{(}K_{x}^{* }\frac{\partial H}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}K_{ y}^{*}\frac{\partial H}{\partial y}\bigg{)}+Q \tag{13}\]  
where  
\[S^{*}=\left\{ \begin{array}{ll}S_{\rm f}&T\leq T_{\rm m}\\ S_{\rm u}&T>T_{\rm m}\end{array} \right. \tag{14}\]  
\[V_{x}^{*}=-K_{x}^{*}\frac{\partial H}{\partial x},\ V_{y}^{*}=-K_{y}^{*}\frac {\partial H}{\partial y} \tag{15}\]