## 4 Analysis of Simulation Results
### Temperature Field Calculation Results
By the 22 nd day of freezing, the \(-1\)\({}^{\circ}\)C isotherms of both freezing zones' outer and inner walls intersect. However, the contour has noticeable irregularities and an uneven temperature distribution. A clear temperature gradient exists between the strong and weak freezing zones. Additionally, due to the obstruction from the pipe curtain structure, the soil within the gaps of the pipe curtain cools more slowly, preventing the full formation of a permafrost structure. The strong freezing zone forms \(-10\)\({}^{\circ}\)C permafrost associations, while the weak freezing zone shows a gradual expansion of \(-10\)\({}^{\circ}\)C permafrost columns. By the 25 th day of freezing, the inner and outer freezing walls of \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C intersect. By the 28 th day, the outline of the curtain in both the strong and weak freezing zones becomes smoother, the temperature gradient difference between the two zones decreases, and a complete \(-1\)\({}^{\circ}\)C permafrost structure is formed. On the 30 th day of freezing, a stable and solid permafrost curtain structure is achieved with the thickness of the permafrost curtain in both the strong and weak freezing zones meeting the design standards. By the 45 th day, the irregularities of the permafrost curtain nearly disappear, the transition area between the two freezing zones becomes smoother, and the curtain expands further outward. Figure 4: Temperature cloud maps and equivalence surface maps at different freezing times. The freezing temperature in the soil body is an important basis for judging whether the permafrost curtain is formed. The cooling curve in the region can provide important data for its permafrost curtain development and the law of freezing and expansion effect; 1-5 monitoring points are extracted from the strong freezing area above, 6-7 monitoring points are extracted from the weak freezing area in the corner, and 8-9 monitoring points are extracted from the weak freezing area on both sides, as shown in Figure 6.  
The temperature change curve for each monitoring point throughout the freezing process is illustrated in Figure 7. As seen in Figure 7, points 1 and 3 in the strong freezing zone are less directly affected by the cold source than other monitoring points. Consequently, the heat exchange process at these points takes longer. However, the cooling rate at point 3 changes drastically on the 28 th day of freezing. This phenomenon, when considered alongside the isothermal analysis, is attributed to the continuous development of the freezing front. During this period, a significant temperature gradient exists at both ends of point 3, prompting bidirectional heat transfer and an acceleration in the cooling rate. Figure 5: Isotherm plots of \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C for cross-sections at different freezing times. Figure 6: Schematic of cross-section temperature monitoring points. By the 35 th day of freezing, point 1 reaches \(-10\,^{\circ}\)C, indicating that the inner edge of the stable frozen soil curtain formed by this freezing project has developed to point 1. Points 2, 4, and 5 in the strong freezing zone are directly affected by the cold source, resulting in a faster cooling process. The cooling rate slows significantly when the soil temperature reaches the phase transition temperature. This deceleration is due to the latent heat release required for the water in the soil to freeze, keeping the temperature near the freezing point during this stage. The soil temperature gradually decreases until most water completes the phase transition. This phenomenon is more pronounced closer to the cold source. After this phase, the cooling rate of the soil changes due to variations in parameters such as thermal conductivity. Additionally, compared to monitoring points 2 and 4, the cooling rate at point 5 is slower after the phase transition.