## 3 Model Test Results  
### Temperature Variation Analysis  
The temperature duration curves of each measurement point at three sections in the model test are shown in Figure 4.  
Figure 4 shows that the temperature drop curves of the measurement points suffer from three periods: steep drop, slow drop and tending to be stable. The temperature of the measurement point between the adjacent two freezing pipes decreases faster, while that outside the freezing pipe decreases slower. Figure 4a shows that the temperatures of measurement points are all below -1 \({}^{\circ}\)C after freezing for 55 min. The temperature drop rates of measurement point \(\backslash\) are 0.365, 0.182 and 0.024 \({}^{\circ}\)C/min at three periods; Figure 4b shows that the temperatures of measurement points are all below -1 \({}^{\circ}\)C after freezing for 70 min. The temperature drop rates of measurement point \(\backslash\) are 0.318, 0.120 and 0.011 \({}^{\circ}\)C/min at three periods; Figure 4c shows that the temperatures of measurement points are all below -1 \({}^{\circ}\)C after freezing for 60 min. The temperature drop rates of measurement point \(\backslash\) are 0.334, 0.148 and 0.019 \({}^{\circ}\)C/min at three periods. Figure 4: _Cont._  
Figure 4: Temperature variation curves of all measurement points at each section: (**a**) Section 1-1; (**b**) Section 2-2; (**c**) Section 3-3. Figures 5-7 show that after freezing for 4 min, a circular frozen soil column is formed around the freezing pipes. After freezing for 28 min, the diameter of the frozen soil columns further expands, the two frozen soil columns in the middle intersect first. After freezing for 44 min, the frozen wall basically intersects, the thickness of the frozen wall at the tunnel arch in three sections is 35.5 mm, 18.9 mm and 32.7 mm, respectively. After freezing for 60 min, the frozen wall completely intersects, and the thickness of the frozen wall at the tunnel arch in three sections is 45.7 mm, 39.1 mm and 47.3 mm, respectively. Moreover, the freezing temperature field can be calculated by the single pipe freezing theory within freezing for 28 min, while it can be calculated by the flat-panel freezing theory [31] after freezing for 28 min. After freezing for 60 min, the order of the freezing effect is Section 3-3 \(>\) Section 1-1 \(>\) Section 2-2. To quantitatively analyze the non-uniformity of temperature distribution at three sections, the temperature on two characteristic planes (the main and vice planes) of the horizontal frozen wall are selected to draw curves. Where the main plane refers to the characteristic plane passing through the center of the tunnel and each freezing pipe, the vice plane refers to the characteristic plane passing through the center of the tunnel and the midpoint of the connecting line between adjacent freezing pipes, while the axis plane refers to the arrangement circle of freezing pipes, as shown in Figure 8.  
Figures 9-11 show that the temperature curves on the main and vice planes of the frozen wall present a \"V\" type; specifically, the temperature on the axis plane is the lowest, while it gradually rises with the increase in distance to the axis plane. Moreover, the temperature curve near the inner and outer edges (R = 100 mm and R = 140 mm) of the designed frozen wall tends to be flat, and the temperature at the inner edge (R = 100 mm) is always lower than that at the outer edge (R = 140 mm) during freezing for 60 min. It indicates that the dissipation capacity of the cooling capacity inside the axis plane of the frozen wall is less than that outside the axis plane. Figure 7: Temperature field at Section 3-3: (**a**) 4 min; (**b**) 28 min; (**c**) 44 min; (**d**) 60 min. Figure 8: Characteristic planes of frozen wall. Figure 10: Temperature distribution on characteristic planes at Section 2-2: (**a**) main planes; (**b**) vice planes. Figure 9: Temperature distribution on characteristic planes at Section 1-1: (**a**) main planes; (**b**) vice planes.