## 2 Mathematical Model of the Freezing Temperature Field and Related Physical Parameters
### Geometric Model  
The east ventilation shaft (494.6 m) of the Chengjiao coal mine (located in Yongcheng city, Henan Province, P. R. China, with an east longitude of 116\({}^{\circ}\)39\({}^{\prime}\) and a north latitude of 33\({}^{\circ}\)94\({}^{\prime}\)) passes through an alluvial layer with a thickness of 358.2 m, as shown in Figure 3. The net diameter of the east ventilation shaft is 5.0 m. A rupture disaster occurred in the east ventilation shaft wall in May 2015, and the company decided to use the freezing method to repair the shaft wall to prevent accidents with sudden water and sand gushing.  
The damaged depths of the shaft wall (below ground level) in the east ventilation shaft are \(-199.5\) (minus 2 m) and \(-329.5\) m (plus/minus 17.5 m), respectively, where the concrete is badly damaged, and a large amount of steel reinforcement is twisted and exposed. The lithological profile of the shaft is shown in Table 1 (the damaged depths of the shaft wall has been bolded) and the photograph appearance of the damaged wall is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Accumulative** & **Layer Thickness/m** & **Lithologic Classification** & **Accumulative Thickness/m** & **Layer Thickness/m** & **Lithologic Classification** \\ \hline
8.7 & 8.7 & Clays & 240.1 & 3 & Fine sands \\
14.2 & 5.5 & Clays & 247.5 & 7.4 & Baucite clay \\
21.5 & 7.3 & Silty sands & 252.5 & 5 & Fine sands \\
53.9 & 32.4 & Clays & 257.6 & 5.1 & Clays \\
57.5 & 3.6 & Clays and silty sands & 260.1 & 2.5 & Fine sands \\
74.5 & 17 & Clays & 262.6 & 2.5 & Silty clays \\
77.5 & 3 & Fine sands & 272.7 & 9.4 & Clays' \\
83.6 & 6.1 & Clays & 277.5 & 5.5 & Fine sands \\
85.8 & 2.2 & Silty sands & 285 & 7.5 & Clays \\
98.3 & 12.5 & Clays & 298 & 13 & Clays \\
118.5 & 20.2 & Sandy clays & 302 & 4 & Baucite clay \\
121 & 2.5 & Silty clays & 307.5 & 5.5 & Fine sands \\
129.5 & 8.5 & Clays & **312.5** & **5** & **Clays** \\
133.4 & 3.9 & Silty clays & **319.8** & **7.3** & **Baucite clay** \\
152 & 18.6 & Baucite clay & **332.6** & **12.8** & **Clays** \\
161 & 9 & Clays & **358.2** & **25.6** & **Kaolinite** \\
191.2 & 30.2 & Fine sands & 361.6 & 3.4 & Mudstone \\
**202.5** & **11.3** & **Silty clays** & 366.5 & 4.9 & Fine sandstone \\
215.4 & 12.9 & Silty clays & 379.7 & 13.2 & Mudstone \\
220.4 & 5 & Clays & 404.5 & 24.8 & Sandy mudstone \\
235.5 & 15.1 & Fine sands & 414.7 & 10.2 & Fine sandstone \\
237.1 & 1.6 & Clays & & & \\ \end{tabular}
\end{table}
Table 1: The layer conditions in the topsoil around the east ventilation shaft.  
Figure 3: Location of Chengjiao coal mine.  
According to the previous computational research results [44; 45], the radius of the affected area (the cooling zone) in the outer side of the shaft wall would not exceed 5-8 times the whole thickness of the frozen soil wall in the frozen soil wall development process. In the freezing, engineering, and construction, the whole thickness development of the frozen soil wall in soil generally does not exceed 3 m; thus, the radius of the cooling zone is approximately 15-24 m. The layout radius of the freezing tube is 5.5-6 m, so the radius of the affected area is 6 + (15-24) m = 21-30 m; therefore, the influence radius in the calculation model is set as 30 m. The freezing time nodes were taken to be 30 d, 60 d, 90 d, 120 d, and up to 180 d. The geometric model is shown in Figure 5.  
Figure 4: The photograph appearance of the damaged wall of the east ventilation shaft: (**a**) \(-\)199.5 m, (**b**) \(-\)339.5 m.