## 3 Establishment of Numerical Model
### Control Equation of Freezing Temperature Field  
The freezing process of soil involves transient heat conduction in the early stage. In the later stage of freezing, due to the low heat conduction rate, the freezing front moves slowly. At this time, the temperature field of frozen soil can be regarded as a steady state; that is, the heat absorbed by the freezing pipe does not change with time. In general, the transient evolution of freezing temperature in the early stage is used to analyze the characteristics of temperature field change in the process of active freezing and to judge the temperature development and development trend in the process of on-site construction. Therefore, it is necessary to analyze the transient evolution of temperature.  
The heat transfer control equation of a three-dimensional transient temperature field can be obtained according to the Fourier law and the law of conservation of energy as follows:  
\[C*\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}(k*\frac{\partial T }{\partial x})+\frac{\partial}{\partial y}(k*\frac{\partial T}{\partial y})+ \frac{\partial}{\partial z}(k*\frac{\partial T}{\partial z})+Q \tag{1}\]  
where \(T\) is the soil temperature (\({}^{\circ}\)C) and \(t\) is the time (s), \(C\) is called equivalent volumetric heat capacity \(\left(\text{kJ}/\left(\text{m}^{3,\circ}\text{C}\right)\right)\), and \(k\) is called equivalent thermal conductivity \(\left(\text{W}/\left(\text{m}^{\cdot\circ}\text{C}\right)\right)\). \(Q\) represents the heat absorbed from external space.  
\(C\) and \(k\) are also given by  
\[C=C_{u}-\left(w*Ehen1/\left(1+w\right)/T_{f}\right)*flc2 hs\left(T-T_{f}\right) \tag{2}\]  
\[k=\begin{cases}k_{f}\left(T<T_{f}\right)\\ k_{u}\left(T>T_{f}\right)\end{cases} \tag{3}\]  
where \(k_{f}\) and \(k_{u}\), respectively, are called the thermal conductivity \(\left(\text{W}/\left(\text{m}^{\cdot\circ}\text{C}\right)\right.\) of frozen soil and unfrozen soil. \(C_{u}\) is called the volumetric heat capacity \(\left(\text{kJ}/\left(\text{m}^{3,\circ}\text{C}\right)\right)\) of unfrozen soil. \(Ehen1\) is \(334.56\)\(\left(\text{kJ}/\text{kg}\right)\). \(w\) is the moisture content. And \(T_{f}\) is the freezing temperature (\({}^{\circ}\)C) of soil. And where the parameter of \(C_{u}\) is given by  
\[C_{u}=c_{u}\rho_{u} \tag{4}\]  
In Equation (4), \(c_{u}\) is the specific heat \(\text{kJ}/\left(\text{kg}{\cdot}^{\circ}\text{C}\right)\) of unfrozen soil. \(\rho_{u}\) is the density \(\left(\text{kg}/\text{m}^{3}\right)\) of unfrozen soil.  
The initial condition of Equation (1) is  
\[T|_{t=0}=T_{0} \tag{5}\]  
where \(T_{0}\) is the initial temperature (\({}^{\circ}\)C) of soil.  
In the mathematical model, each freeze pipe could be simplified into one line, so the boundary condition for each freeze pipe is  
\[T|_{(x_{p},y_{p},z_{p})}=T_{c}(t) \tag{6}\]  
where \((x_{p},y_{p},z_{p})\) is the Cartesian coordinates of each point on the line (freeze pipe), and \(T_{c}(t)\) is the brine temperature (\({}^{\circ}\)C) of brine in the pipe.  
The boundary condition for the soil, where is infinite distance away from frozen soil around cross passage, is given by  
\[T|_{(x=\infty\ or\ y==\infty\ or\ z=\infty)}=T_{0} \tag{7}\]  
So, the governing partial differential equation, initial conditions, and boundary conditions given above make the problem of the transient freezing temperature field of cross-passage's freezing process determinate.