## 2 Hydro-Thermal Coupling Mathematical Model
### Coupling of Temperature Field to Seepage Field  
The temperature field during soil freezing is a transient heat transfer problem.it is assumed that ice, soil particles and water exist simultaneously in the freezing soil during the freezing process, and have different heat transfer characteristics.  
According to the heat transfer and seepage principles [23], the control equations for the temperature and seepage fields considering the phase change are  
\[\theta_{i}C_{i}\rho_{i}\frac{\partial T}{\partial t}+\varphi L\rho_{l}\frac{ \partial\chi}{\partial T}\frac{\partial T}{\partial t}+\theta_{l}C_{l}\rho_{ l}\frac{\partial T}{\partial t}+\rho_{l}C_{l}(\mathbf{u}\cdot\
abla T)+\theta_{s}C_{s} \rho_{s}\frac{\partial T}{\partial t}=\
abla[(\theta_{i}k_{i}+\theta_{l}k_{l} +\theta_{s}k_{s})\
abla T]+Q \tag{3}\]  
\[\rho_{l}[\alpha_{l}\varphi+\alpha_{s}(1-\varphi)]\frac{\partial p}{\partial t }+\
abla\cdot(\rho_{l}\mathbf{u})=\rho_{l}q \tag{4}\]  
\[\mathbf{u}=-\frac{Kt}{\rho_{l}g}\
abla p \tag{5}\]  
\[Kt=K_{f}+\Big{(}K_{0}-K_{f}\Big{)}\chi \tag{6}\]  
where \(t\) is time; \(C_{i}\), \(C_{l}\) and \(C_{s}\) denote the specific heat capacity of water, ice and soil particles; \(K_{i}\), \(K_{l}\) and \(K_{s}\) are their thermal conductivity; \(\rho_{i}\), \(\rho_{l}\) and \(\rho_{s}\) are their density; \(L\) denotes ice-water phase change latent heat; \(\mathbf{u}\) denotes the relative velocity of water; \(Q\) denotes the total amount of equivalent heat source; \(\alpha_{s}\) is the basic equivalent compression ratio; \(\varphi\) is porosity; \(q\) denotes the volume of flow; \(p\) is the osmotic pressure; \(K^{\prime}\) is hydraulic conductivity; and \(K_{f}\) and \(K_{0}\) are the hydraulic conductivity of completely frozen area and unfrozen area respectively.  
The change of absolute porosity during freezing can be expressed by the change of volume content of unfrozen water, therefore, Equation (4) can be expressed as  
\[\rho_{l}[\alpha_{l}\theta_{l}+\alpha_{s}(1-\theta_{l})]\frac{\partial p}{ \partial t}+\
abla\cdot(\rho_{l}\mathbf{u})=\rho_{l}q \tag{7}\]  
The temperature of the outer wall of the freezing tube is the freezing temperature, where the temperature boundary at the inlet of the freezing tube is the inlet temperature, and the convection condition at the outlet of the freezing tube is set, as shown  
\[\mathbf{n}_{d}\cdot(-k\
abla T)|_{x=a_{2}}=0 \tag{8}\]  
where \(a_{2}\) denotes the horizontal coordinate of the export of the freezing pipe, \(\mathbf{n}_{d}\) denotes the axial direction vector at the export.  
Setting the other boundaries to be adiabatic and not considering the heat transfer effect between the boundaries, the boundary conditions are expressed as  
\[-\mathbf{n}\cdot\left(-k\
abla T\right)\rvert_{y=b_{1},b_{2}}=0 \tag{9}\]  
where \(b_{1}\) and \(b_{2}\) are the ordinates of the boundary, and \(\mathbf{n}\)\(\mathbf{n}\) denotes the direction vector of the normal direction within the boundary.  
The temperature boundary considering other boundary heat transfer effects can be expressed as  
\[T=T_{0} \tag{10}\]  
where \(T_{0}\) is the temperature over time.  
The initial conditions are  
\[p\rvert_{t=0}=p_{0} \tag{11}\]  
where \(p_{0}\) denotes the soil pressure distribution in the initial state.  
Therefore, the mathematical model of hydro-thermal coupling during freezing process can be composed of Equation (3), Equation (7) and the above boundary conditions.