## 5 Numerical Simulation of Temperature Field of Pemafrost Subgrade
### Model Description
#### 5.1.2 Mathematical Model  
The subgrade is simplified as a planar structure, and for the two-dimensional hydrothermal coupling problem of the subgrade, according to [PERSON]'s law and considering the latent heat of the phase transition as a heat source, obtain the temperature field control equation under the phase transition [23]. \[\rho C_{(\theta)}=\frac{\partial\mathrm{T}}{\partial\mathrm{t}}=\lambda_{( \theta)}\
abla^{2}\mathrm{T}+\mathrm{L}\rho_{\mathrm{I}}\frac{\partial\theta_{ \mathrm{I}}}{\partial\mathrm{t}} \tag{3}\]  
where \(\rho\) is the density of the soil; \(\mathrm{C}\) is the volumetric heat capacity; \(\lambda\) is the thermal conductivity of the soil; \(\
abla\) is the differential operator, representing the differentiation of the space; \(\mathrm{L}\) is the latent heat of the phase transition of ice and water (355 j/g); \(\rho_{\mathrm{I}}\) is the volumetric weight of ice; \(\theta_{\mathrm{I}}\) is the volumetric ice content. According to [PERSON]'s equation, and combined with the hindering effect of pore ice on the migration of unfrozen water, the migration law of water permafrost can be expressed as the following equation:  
\[\frac{\theta_{\mathrm{u}}}{\partial\mathrm{t}}+\
abla(-\mathrm{D}_{(\theta_{ \mathrm{u}})}\
abla-\mathrm{K}_{g(\theta_{\mathrm{u}})})+\frac{\rho_{\mathrm{ I}}}{\rho_{\mathrm{w}}}\frac{\partial\theta_{\mathrm{I}}}{\partial\mathrm{t}}=0 \tag{4}\]  
where \(\mathrm{D}_{(\theta_{\mathrm{u}})}\) is the soil's hydraulic conductivity; \(\mathrm{K}_{g(\theta_{\mathrm{u}})}\) is the soil permeability coefficient in the direction of gravitational acceleration; and \(\rho_{\mathrm{I}}\) is the density of ice. The blocking effect of pore ice reduces the migration of unfrozen water, and pore ice makes the original diffusion rate and infiltration rate of water I times that of the same unfrozen water content. The blocking coefficient can reflect the difference between the diffusion of water in frozen and unfrozen soil. \[\mathrm{D}=\mathrm{k}\cdot\mathrm{I}/\mathrm{c} \tag{5}\]  
\[\mathrm{I}= 10^{-10\theta_{\mathrm{I}}} \tag{6}\]  
where I is the impedance factor. In this numerical simulation, the relative saturation S is used as the solution variable instead \(\theta_{\mathrm{u}}\), and the expression is as follows:  
\[\mathrm{S}= \frac{\theta_{\mathrm{u}}-\theta_{\mathrm{r}}}{\theta_{\mathrm{s}}- \theta_{\mathrm{r}}} \tag{7}\]  
where \(\theta_{\mathrm{s}}\) is the saturated moisture content; \(\theta_{\mathrm{r}}\) is the residual moisture content of the soil. Figure 7: Geometric model and calculation unit division for ordinary subgrade. Equation transformations can be obtained as follows:  
\[\theta_{\rm u}=(\theta_{\rm s}-\theta_{\rm r})S+\theta_{\rm r} \tag{8}\]  
The parameters included in the set of control differential equations for the temperature and moisture fields of permafrost are temperature, pore ice volume content, and unfrozen water volume content. [PERSON] [24] concluded that the unfrozen water content is related to the initial moisture content of the soil, freezing temperature, and soil temperature, and by introducing empirical coefficients, an empirical expression for the unfrozen water content was given:  
\[\frac{\omega_{0}}{\omega_{\rm u}}=\left(\frac{\rm T}{\rm T_{f}}\right)^{\rm b} \rm,T<\rm T_{f} \tag{9}\]  
where \(\omega_{0}\) is the initial moisture content; \(\rm T_{f}\) is the freezing temperature of the soil (\({}^{\circ}\)C); \(\rm b\) is the empirical coefficient related to the soil quality, which can be determined by test or taking the empirical value  
The volume ratio of unfrozen water to pore ice in permafrost is the solid-liquid ratio of permafrost, denoted as \(\rm B_{f}\):  
\[\rm B_{f}=\frac{\theta_{1}}{\theta_{u}}=\left\{ \begin{matrix}1.1\left(\frac{ \rm T}{\rm T_{f}}\right)^{\rm b}-1.1&\rm T<\rm T_{f}\\ 0&\rm T>>\rm T_{f}\end{matrix} \right. \tag{10}\]  
where the constant \(1.1\) represents the ratio belonging to the density of ice, \(\frac{\rho_{\rm L}}{\rho_{\rm u}}\).