## 4 Finite Element Analysis Method and Results
### Soil Behavior Scenarios and Modeling
\begin{tabular}{c} - \\ **0.019** \\ **0.0008** \\ **0.0019** \\ **0.0008** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} - \\ **0.019** \\ **0.0008** \\ **0.0032** \\ **0.0666** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} - \\ **0.015** \\ **and** \\ **5** \\ **and** \\ **5** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} - \\ **and** \\ **and** \\ **and** \\ **5** \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} - \\ **case of the frost-heave scenario, the heave of the buried pipe occurs due to the pressure caused by the frost heave of the soil; by contrast, in the thaw-settlement scenario, pipe subsidence is primarily caused by the weight of the pipe itself and the soil weight. This allows for a further increase in the pipeline subsidence under the thaw-settlement scenario. Accordingly, the buried depth of the pipeline is also set deeper in the thaw-settlement scenario such that it can be affected more by the soil thaw settlement. The element used in this numerical analysis is the coupled temperature-displacement trilinear element (8-node). Figure 7 illustrates that the elements are set densely in the vicinity of the pipeline and thermosiphon and the total number of nodes in the model ranges from 28,488 to 77,868, depending on the scenario. The thermosiphons are modeled with a height of 0.5 m, which is identical to that reported by [PERSON] et al. [12]. Accordingly, half the height of the thermosiphon (0.25 m) is modeled as being buried in the soil. Figure 8 illustrates the top view of the thermosiphon arrangement for each case. In both scenarios, the number of thermosiphons in cases A, B, and C are identical: (2 pair \(\times\) 1), (2 pair \(\times\) 2), and (2 pair \(\times\) 3), respectively. The measurement points set in this study are illustrated in Figure 9. A measurement point is named such that its object and location can be determined. For example, the first letter of the measurement point \"S-3-B\", \"S\", implies soil, whereas \"P\" indicates pipeline. The second term, the number \"3\", represents the number corresponding to the x-direction in Figure 9a. Notably, the six points are set at an interval of 0.0875 m in the frost-heave scenario. Similarly, seven points are set at an interval of 0.175 m in the thaw-settlement scenario. The letter \"B\" denotes a point corresponding to the position of the letter in Figure 9b. For the soil, nine points are set ranging from A to I. Additionally, for the pipeline, 0\({}^{\circ}\), 90\({}^{\circ}\), and 180\({}^{\circ}\) points of angle are set instead of letters. Therefore, a total of 54 and 18 measurement points are obtained for the soil and pipeline, respectively, under the frost-heave scenario. Similarly, 63 and 21 measurement points are manifested for the soil and pipeline under the thaw-settlement scenario, respectively. The measured values, such as the unfrozen water content, displacement, and stress, are acquired at these points. Figure 8: Arrangement of thermosiphons in different scenarios and cases: (**a**) frost heave and (**b**) thaw settlement. Table 2 lists the material properties and parameters of the soil mixture utilized in this study [23; 25; 33]. The corresponding parameters of unfrozen water contents and porosity rate function model are calibrated to the experimental results reported by [PERSON] et al. [34]. Among the calibrated parameters, compared to that reported in the previous study, the maximum porosity growth rate is reduced (\(\dot{n}_{m}\)= 0.33 \(\times\) 10\({}^{-5}\) s\({}^{-1}\)) to prevent excessive frost heave. [PERSON] and [PERSON] [23] also performed their analysis using reduced parameters. They compared the frost-heave amounts of the soil using the existing parameters and reduced parameters featured in this study with the experimental case reported by [PERSON] et al. [34].