## 2 Frost-Heave Model and Thermosiphon Model
### TM Model: Constitutive Model  
For the constitutive model of the soil, the modified Cam-Clay model formulated by [PERSON] and [PERSON] [23] is employed. Figure 4 defines the normal compression lines (NCLs), unloading-reloading lines (URLs), and elliptical yield surface in the constitutive model when the soil freezes. The pore ice is the key factor changing the stiffness of the frozen soil. [PERSON] and [PERSON] [23] introduced the concept of pore ice ratio (\(e_{ip}\)) such that NCLs and URLs are defined according to the \(e_{ip}\). The pore ice ratio is defined as the ratio of the void ice volume (\(V_{ip}\)) to the soil volume (\(V_{s}\)), and this ratio is expressed in Equation (4) as  
\[e_{ip}=\frac{V_{ip}}{V_{s}}=(\overline{\omega}-\omega).\frac{\rho_{s}}{\rho_{w} }.109, \tag{4}\]  
where \(\rho_{s}\) and \(\rho_{w}\) represent the density of soil and water, respectively.  
In accordance with the pore ice ratio (\(e_{ip}\)), the NCLs and URLs are defined using Equation (5a) and Equation (5b), respectively, as  
\[\lambda_{f}=\ \lambda e^{(-\alpha_{1}e_{ip})}, \tag{5a}\] \[\kappa_{f}=\kappa e^{(-\alpha_{2}e_{ip})}, \tag{5b}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are constant values depending on the type of soil, which can be estimated via isotropic compression tests using a triaxial apparatus [30,31]. Notably, the value of \(e_{ip}\) is zero for the soil before freezing; therefore, the NCL is \(\lambda_{f}=\ \lambda\), and URL is \(\kappa_{f}=\kappa\).  
The elliptical yield surface is plotted in the \(q\) (deviatoric stress)-\(p^{\prime}\) (effective isotropic stress) plane, as depicted in Figure 4. This surface obeys the following law:  
\[f=q^{2}+M^{2}\big{(}p^{\prime}-p_{0t}\big{)}\,\big{(}p^{\prime}-p_{0}\big{)}=0, \tag{6}\]  
where \(M\) denotes the slope of the critical state line (CSL) in \(q\)-\(p^{\prime}\) space, which is constant regardless of soil freezing. The shape of the elliptical yield surface, as characterized using Equation (6), changes according to the \(p_{0t}\) and \(p_{0}\big{(}p_{0}^{\prime}\big{)}\). Equations (7) and (8a,b) define the various variables used in Equation (6) as follows:  
\[p_{0t}=p_{t}\big{(}1-e^{-\alpha_{3}e_{ip}}\big{)}, \tag{7}\]  
\[p_{0}^{f}=p^{r}\,e^{(\beta e_{ip})/(\lambda_{f}-\kappa_{f})}\,\left(\frac{p_{0} }{p^{\prime}}\right)^{(\lambda-\kappa_{f})/(\lambda_{f}-\kappa_{f})}, \tag{8a}\]  
Figure 3: (**a**) Tendency of porosity rate for different soils; (**b**) frost-heave amount by \(\dot{n}_{m}\).  
\[\beta e_{ip}=v_{0}^{f}-v_{0}-0.09e_{ip}, \tag{8b}\]  
where \(p_{0t}\) denotes the isotropic tensile strength and increases as the pore ice ratio (\(e_{ip}\)) increases; \(p_{0}\) depicts the pre-consolidation pressure, which is rendered \(p_{0}^{f}\) when the soil freezes; \(p^{r}\) represents the reference effective isotropic pressure; \(\beta\) denotes the relationship between the specific volume and pore ice ratio (\(e_{ip}\)); and \(v_{0}^{f}\) and \(v_{0}\) symbolize the specific volumes of frozen and unfrozen soils at the reference effective isotropic pressure, respectively. Further details on the constitutive model can be obtained in the literature, as reported by [PERSON] and [PERSON] [23].