## 4 Results and Discussion  
### Frost Heave Simulation  
Figure 6 shows a schematic diagram of the temperature field over time during the freezing of the tunnel. By the 5 th day of freezing, the temperature of the part of the soil in direct contact with the freezing tube dropped below 0 \({}^{\circ}\)C, and the frozen soil wall was initially formed with the freezing tube as the center of the circle. By the 10 th day of freezing, the radius of the circular frozen soil wall formed by the freezing tube gradually increased, and the soil temperature reached below 0 \({}^{\circ}\)C except for the part of the soil within the middle ring of the freezing tube. By 20 days of freezing, the soil inside the frozen pipe reached below \(-5\)\({}^{\circ}\)C. The main body of the freeze began to change from the frozen soil between the pipes to the soil outside the pipes, and the freeze impact area began to increase. After 40 days of treatment, the soil inside the frozen tube reached below \(-10\)\({}^{\circ}\)C, and the overall temperature of the soil outside the outer ring of the frozen tube dropped. The frozen soil wall thickness met the freezing design requirements, and the positive freezing work was completed. Figure 5: Flow chart of soil frost heave calculation. To further verify the correctness of the development of the freezing temperature field in the DEM simulation of the soil, the temperature change curves of the middle and inner ring temperature measurement holes in the simulation were analyzed and compared with the actual temperature change curves. Figure 7 shows the simulated temperatures of the middle ring temperature measurement holes compared with the measured temperatures. Figure 8 shows the simulated temperatures of the inner ring temperature measurement hole compared with the measured temperatures. The simulated values and measured values are generally in good agreement. However, the heat transfer method in the numerical simulation only considers contact heat transfer, ignoring radiation heat transfer and convection heat transfer; therefore, some calibration corrections that are needed within the simulated values remain. Figure 6: Soil temperature field development: (**a**) freezing for 5 days; (**b**) freezing for 10 days; (**c**) freezing for 20 days; (**d**) freezing for 40 days. Figure 7: Temperature comparison chart for center ring temperature measurement hole. The computed ground displacement fields of soil frost heave are shown in Figures 9 and 10. From the beginning of freezing to the 5 th day of freezing, the main body of freezing is mainly the soil in direct contact with the freezing pipe, and the soil starts to change from the original ground temperature at this stage, so the total amount of frost heave is small and the vertical displacement grows slowly. From day 6 to day 20 of freezing, most of the soil between the frozen pipes freezes. This is the main stage when the soil freezes, and at this moment, the frost heave displacement of the soil grows more quickly. From the 21 st day of freezing to the completion of freezing, a small amount of soil swelling occurred outside the outer ring of the frozen pipe, with a relatively slow increase in displacements. After freezing was completed, a maximum vertical soil frost heave displacement of 23 mm and a maximum vertical ground surface frost heave displacement of 8 mm occurred directly above the tunnel. Figure 8: Temperature comparison chart for inner ring temperature measurement hole. Figure 9: Vertical displacement of the soil: (**a**) freezing for 10 days; (**b**) freezing for 20 days; (**c**) freezing for 30 days; (**d**) freezing for 40 days. The ground surface frost heave displacement distributions are shown in Figures 11 and 12. The ground surface displacement distributions are roughly symmetrical about the center of the tunnel. The highest vertical displacement occurs right above the tunnel center, and the displacement of the surface diminishes with increasing transverse distance from the tunnel center.