# Artificial Neural Network-Based Model for Prediction of Frost Heave Behavior of Silty Soil Specimen
## 3 Evaluation of Frost Heave Ratio  
Using the governing equations described above, we performed a THM analysis to predict frost heave for a saturated soil specimen. The material properties used in the numerical model are presented in Table 1. The governing equations are highly non-linear, and thus, the commercial finite element (FE) software COMSOL Multiphysics [47] was used to solve the complex differential equations. Furthermore, the numerical simulation was implemented for a one-dimensional freezing test. The depth of the soil specimen was set as 100 mm, and the initial temperature of the entire specimen was set as 5 \({}^{\circ}\)C. The analysis was conducted until thermal equilibrium was achieved while maintaining constant bottom and top boundary temperatures (Top boundary temperature was set as 5 \({}^{\circ}\)C and bottom boundary temperature was set as \(-\)5 \({}^{\circ}\)C and hence the temperature gradient was 1 \({}^{\circ}\)C/cm). The groundwater level (GWL) was fixed at the bottom boundary to allow for a continuous water supply during freezing, and the overburden pressure applied on the top boundary was set to atmospheric pressure (101.3 kPa). The frost heave ratio was obtained using Equation (21). \[\xi(\%)=\frac{\Delta H_{f}}{H_{0}} \tag{21}\]  
where \(\zeta\) is the frost heave ratio (\(\%\)). \(\Delta H_{f}\) is the amount of total heave (mm), \(H_{0}\) is initial specimen height (mm), and \(\Delta t\) is the elapsed time (h). Figure 4 shows the variation of frost heave ratio (\(\zeta\)) and the position of the freezing front over time obtained from the simulation model. The propagation rate of the freezing front gradually slowed down as time passed: the freezing front propagated rapidly during the initial stages of freezing but came to a halt as it approached thermal equilibrium. The amount of frost heave also steadily increased until thermal equilibrium was achieved. After reaching thermal equilibrium (approximately after 60 h), the freezing front no longer moved, and no further severe frost heave occurred. Overall, the amount of frost heave increased in a nonlinear manner with time, a tendency that was also observed in the experimental results of [PERSON] and [PERSON] [11]. In Figure 4, the calculated frost heave ratio at 60 h was approximately 8%. To verify the reliability of these simulation results, we compared the results with those of the previous numerical studies for the same freezing conditions. As shown in Figure 5, the predictions of the model for the pore pressure and temperature with specimen depth showed good agreement with the results of [PERSON] and [PERSON] [21]. This suggests that the numerical model used in this study is reliable. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Density of porewater, \(\rho_{w}\) & 1000 & kg/m\({}^{3}\) \\ Density of ice \(\rho_{i}\) & 917 & kg/m\({}^{3}\) \\ Density of solid particle, \(\rho_{s}\) & 2600 & kg/m\({}^{3}\) \\ Heat capacity of water at constant pressure, \(C_{\text{\tiny{E}}}\) & 4180 & J/kg/K \\ Heat capacity of ice at constant pressure, \(C_{i}\) & 2044 & J/kg/K \\ Heat capacity of solid particle at constant pressure, \(C_{s}\) & 831 & J/kg/K \\ Latent heat of fusion, L & 334.5 & kJ/kg \\ Thermal conductivity of water, \(\lambda_{w}\) & 0.56 & W/m/K \\ Thermal conductivity of ice, \(\lambda_{i}\) & 2.24 & W/m/K \\ Melting point, \(T_{0}\) & 0 & \({}^{\circ}\)C \\ Young’s modulus, \(E_{s}\) & 1.2 & MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties used in FE model. Figure 4: Frost heave ratio and frost front position with time. Figure 5: Comparison results with those of the previous numerical studies [21] for the 1D freezing. A parametric study was subsequently conducted to quantitatively analyze the effects of geotechnical properties on the frost heave ratio. This study considered the thermal conductivity and initial hydraulic conductivity of the soil particles as crucial parameters.