## 4 Finite Element Analysis Method and Results
### Thaw-Settlement Scenario  
In the thaw-settlement scenario, the internal temperature of the pipeline is increased, thereby leading to melting and settling of the frost-susceptible soil and resulting in sinking behavior of the pipeline. After setting the initial temperature of the frost-susceptible soil model to \(-15\)\({}^{\circ}\)C, the Fourier boundary condition is applied to the inside surface of the pipeline, where the temperature and convective heat transfer coefficient are set to 60 \({}^{\circ}\)C and 55 W/m\({}^{2}\) K, respectively. The analysis is conducted until the temperature of the entire model converges by gradually increasing the soil temperature around the pipeline. In addition, for the top surface of the soil boundary condition of the model, the Fourier boundary condition is identical to that applied in the frost-leave scenario, and solely the temperature is set differently by changing it to \(-15\)\({}^{\circ}\)C. In this scenario, the initial conditions for the frozen soil are applied, i.e., the initial porosity is set to 0.457. Accordingly, the initial unfrozen water content and pore ice ratio of the soil are set to 0.07859 and 0.5895, respectively. The reference point of the temperature convergence for the entire case is set based on the measurement point \"S-1-\", the farthest point from the thermosiphons and pipeline. Figure 15a depicts the temperature change of the soil with time measured at the point \"S-1-\". Among the analysis cases, CASE_T--the case in which thermosiphons are not utilized--exhibits the largest temperature change and the slowest temperature convergence rate, corresponding to a convergence time of approximately 680 h. Conversely, for CASE_TC, which features the maximal number of thermosiphons, the change in temperature is minimal, and the temperature convergence rate is the fastest, corresponding to a convergence time of approximately 350 h.  
This temperature trend induces the trends of unfrozen water content and pore ice ratio, as illustrated in Figure 15b,c, respectively. The convergence time trends of unfrozen water content and pore ice ratio for all cases are identical to the convergence time trend of the temperature. In addition, the change ratio of unfrozen water content and pore ice ratio decreases as the number of applied thermosiphons increases. These trends of convergence time show that thermosiphons perform well as the ground stabilization method and it can be confirmed that the performance increases as the number of thermosiphons increases  
Additionally, when the temperature of the measurement point \"S-1-\" converges, the temperature of the soil around the pipeline increases due to the high temperature of the  
Figure 15: Results obtained at measurement point “S-1-1” for all cases: (**a**) temperature, (**b**) unfrozen water content, and (**c**) pore ice ratio. pipeline. Therefore, the thaw bulb, which refers to the thawed shape of the soil above 0\({}^{\circ}\)C, forms completely. Figure 16 demonstrates the evolution of the thaw-penetration depth, which is the maximum depth of the thaw bulb in the x-z symmetry plane. Figure 16a illustrates the result of CASE_T. Notably, an even thaw-penetration depth (0.23 m) is formed according to the length direction. Figure 16: Time progression of thaw-penetration depth at x–z symmetry plane: (**a**) CASE_T, (**b**) CASE_TA, (**c**) CASE_TB, and (**d**) CASE_TC. However, in the cases featuring thermosiphons, the thaw-penetration depth of the soil adjacent to the thermosiphon has a shallow depth due to the effect of the thermosiphon, as indicated in Figure 16b-d. The maximum thaw-penetration depth is observed at No. 1 in the cases featuring thermosiphon application along the length for the point that is furthest from the thermosiphons. Accordingly, the minimum thaw-penetration depth is formed at No. 7. Moreover, the maximum thaw-penetration depths of CASE_TA, CASE_TB, and CASE_TC are 0.205, 0.163, and 0.139 m, respectively, with minimum thaw-penetration depths of 0.129, 0.118, and 0.107 m, respectively.