## 3 Three-Dimensional Freezing Temperature-Field Calculation Model  
### Mathematical Model of 3D Freezing Temperature Field  
The freezing temperature field of an inclined shaft is a 3D transient heat transfer problem with phase change [18]. Based on the heat transfer theory of the frozen soil, the control equation for the 3D freezing temperature field of inclined shafts takes the following differential form [19; 20]:  
\[C^{*}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(k^{*}\frac {\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(k^{*}\frac{ \partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(k^{*}\frac{ \partial T}{\partial z}\right) \tag{1}\]  
where \(T\) is the soil temperature, \({}^{\circ}\)C; \(t\) is time; \(C^{*}\) is the equivalent volumetric specific heat, J-g-1.K-1; \(k^{*}\) is the equivalent thermal conductivity, W-m-1.K-1.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Types of Freezing Holes** & **Depth/m** & **Diameter of Frozen Pipes/mm** & **Number of Holes** \\ \hline Plugging holes & 12.5–15.5 & \(\phi\)140 & 22 \\ Middle-row holes & 12.5–15.5 & \(\phi\)108/\(\phi\)140 & 16 \\ Side-row holes & 12.5–15.5 & \(\phi\)140 & 46 \\ \hline \end{tabular}
\end{table}
Table 1: Specific parameters of the freezing pipe.  
Figure 2: Layout of freezing holes and main plane pathways.  
Here:  
\[C^{*}=\begin{cases}C_{f}&(T<T_{d})\\ \frac{C_{f}+C_{u}}{2}+\frac{L}{T_{r}-T_{d}}&(T_{d}\leq T\leq T_{r})\\ C_{u}&(T>T_{r})\end{cases} \tag{2}\] \[k^{*}=\begin{cases}k_{f}&(T<T_{d})\\ k_{f}+\frac{k_{u}-k_{f}^{f}}{T_{r}-T_{d}}(T-T_{d})&(T_{d}\leq T\leq T_{r})\\ k_{u}&(T>T_{r})\end{cases} \tag{3}\]  
where \(C_{f}\) and \(k_{f}\) are the volumetric specific heat and thermal conductivity of the frozen soil, respectively; J\(\cdot\)g\({}^{-1}\cdot\)K\({}^{-1}\), W m\({}^{-1}\cdot\)K\({}^{-1}\); \(C_{u}\) and \(k_{u}\) are the volumetric specific heat and thermal conductivity of the unfrozen soil, respectively; J\(\cdot\)g\({}^{-1}\cdot\)K\({}^{-1}\), W m\({}^{-1}\cdot\)K\({}^{-1}\); \(T_{d}\) and \(T_{r}\) are the freezing and melting temperatures of the soil, respectively, \({}^{\circ}\)C; \(L\) is the latent heat of the phase change per unit volume of soil, J/m\({}^{3}\).  
Before freezing, the initial condition of the equation is [21,22]:  
\[T|_{t=0}=T_{0} \tag{4}\]  
where \(T_{0}\) denotes the initial temperature of the soil layer before freezing, \({}^{\circ}\)C.  
In the calculation process, the boundary conditions of the freezing pipe can be expressed as:  
\[T|_{(x_{p},y_{p},z_{p})}=T_{c}(t) \tag{5}\]  
where \(x_{p},y_{p},z_{p}\) is the coordinate of the freezing pipe; \(T_{c}(t)\) is the temperature of the brine in the tube, \({}^{\circ}\)C.  
The boundary conditions satisfy the Dirichlet boundary condition at an infinite distance from the wellbore freezing temperature field:  
\[T|_{(x=\infty,y=\infty,z=\infty)}=T_{0} \tag{6}\]  
After the excavation of the soil, convective heat transfer between the soil and the air at the exposed well position is applied as the boundary condition:  
\[q=h\times(T_{ext}-T_{s}) \tag{7}\]  
Here, \(h\) is the convective heat transfer coefficient between the soil and air, W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\); \(T_{ext}\) is the external environment temperature, \({}^{\circ}\)C; \(T_{s}\) is the soil temperature at the well-side position, \({}^{\circ}\)C.  
The aforementioned regulation differential equations, along with their boundary conditions, make up the definite solution of the 3D temperature area in the inclined shaft during the freezing and excavation processes.