## 4 Analysis of the Numerical Simulation Results
### Thickness and Average Temperature on Typical Continuous Sections  
The average temperature of the frozen wall has two meanings. One refers to the average temperature distribution along the thickness direction of the frozen soil curtain on a certain cross-section of the frozen soil curtain, and the other refers to the average temperature of the entire frozen wall. The former varies due to the difference between the calculated section position and the relative position of the freezing pipe. The average temperature of the whole frozen soil body is generally used when the freezing wall is assumed to be homogeneous in the mechanical calculation. And the average temperature refers to the average temperature distribution along the thickness direction of the frozen soil curtain on a certain cross-section when a specific cross-section needs to be examined. This paper discusses the average temperature on the path of the characteristic section of the frozen wall. The thickness of the frozen wall is obtained by using the finite element software programming expression to extract the length less than \(-1\)\({}^{\circ}\)C on the path of the characteristic section, and then the integral of the product of the unit length and the average temperature of the unit is carried out for all the finite element units in the thickness of the frozen wall on the path. The calculated integral value is divided by the thickness of the frozen wall on the path, and the average temperature of the frozen wall under the path can be calculated. The thickness and average temperature of the frozen wall on path 1 on the seven continuous sections develop with time, as shown in Figures 5 and 6. Figure 4: Development of frozen curtain at different freezing time. Figure 5: Curve of frozen wall thickness changing with time. It can be seen from Figure 5 that the thickness of the frozen wall in path 1 of each section increases rapidly and then gradually flattens with the increase in freezing time. The development trend of the final thickness of the frozen wall in sections of B-E is similar; the thickness of the frozen wall increased by about 5 m from 0 to 50 d, and the thickness of the frozen wall only increased by 1 m from 50 to 150 d, which is because the frozen wall expands outward after the internal circle of the frozen wall is intersected. As the thickness increases, the more contact it has with the external soil, the greater the heat exchange, leading to a slower growth rate. Because the freezing tube of section A has only 12 freezing tubes in the outer ring and is located at the end, it begins to intersect to form a freezing wall at 90 days. Section G is located in the contact between the shield wall and the air; therefore, convective heat transfer exists in the interface. The thickness of the freezing wall of the two sections is small, which is about 2 m at 150 days. It can be seen from Figure 6 that the average temperature of path 1 in the A-F section decreases logarithmically, and the number of frozen tubes in the E and F sections is the largest and is not affected by end effect and convective heat transfer with the increase in freezing time. The average temperature is the lowest, which is \(-20\)\({}^{\circ}\)C at 150 d. The average temperature of path 1 of section G rises from \(-13\)\({}^{\circ}\)C to \(-9\)\({}^{\circ}\)C after 30 days due to convective heat transfer. The thickness and average temperature of the frozen wall on the continuous section path 1 develop with space at 150 days of freezing, as shown in Figure 7.  
Figure 6: The curve of average temperature changing with time. Figure 7: Curves of average temperature and thickness changing with space. It can be seen from Figure 7 that the thickness of the frozen wall increases first and then decreases in the direction of shield advance, and the average temperature decreases first and then increases. The extreme values of the two are basically between sections D and E, which are 6 m and \(-20\)\({}^{\circ}\)C, respectively.