# Numerical Simulation of Water Migration during Soil Freezing and Its Resulting Characterization
## 4 Results and Analysis  
Figure 4 illustrates the variation in the amount of freezing expansion relative to the depth of soil column freezing over time. The results indicate consistent trends and patterns despite differences in initial freezing temperatures and initial volumetric water content. Initially, the volume of freezing increases linearly with time. In the early stages of freezing, the depth of freezing accelerates, then the rate diminishes gradually, and eventually, the depth of freezing stabilizes. The experimental results show a linear increase in frost heave changes, whereas the model results indicate that frost heave initially changed more rapidly before increasing linearly at a slower rate. However, the final outcomes from both approaches closely align. For instance, as demonstrated in the results for samples b and c, the discrepancy between the experimental and modeled results for the amount of freezing and swelling was approximately 2 mm, which represents a relatively minor error margin. When the soil water content was higher, as in sample a, the experimental data exceeded the simulation results by about 25%. This discrepancy arose because the highly saturated soil during the experiment formed thicker ice lenses, with all unfrozen water collecting at the lens front and subsequently freezing. In contrast, the infiltration model used in the simulations suggests the presence of residual unfrozen water content regardless of temperature. Consequently, the ice content in the modeled frozen zone was less than that observed experimentally, resulting in a smaller amount of soil freezing swelling. Overall, the model results were consistently smaller than the experimental results. Currently, measuring pore water pressure in low-temperature permafrost is challenging, and there is limited literature on the topic in China. Based on existing experimental data from clay samples, the pore water pressure after ice lens formation is approximately 12 times greater than before the formation of the second lens. According to Figure 5, the value of pore water pressure was at its lowest at the cold end surface, increasing gradually until it reached zero at the freezing front. The duration of freezing did not influence the peak value of the pore water pressure. However, the formation of ice lenses caused the pore water pressure to increase by a factor of 12, aligning with findings reported in the literature [34]. The model further accounts for the matrix suction of ice and considers the pore pressure as a weighted average of the water and ice pressures. This adjustment addresses the shortcomings of the Clapeyron equation when applied to soil freezing, particularly its failure to accurately reflect water pressure at lower temperatures [35]. Figure 3: Modeling flowchart. (**a**) Experimental photo of frozen soil column. (**b**) Schematic diagram of soil column structure. (**c**) Modeled temperature contour. As shown in Figure 6, we conducted experimental validation of the model using pore water pressure data from the literature [34]. Unidirectional freezing and thawing cycle experiments were performed on Qinghai-Tibet clay over approximately five cycles, each lasting 24 h. The experimental samples were taken from the powdery clay along the Qinghai-Tibet Railway, and the main mineral compositions were silica and calcium carbonate. The pore pressure probe was positioned in the middle of the specimen, which had a height of 10 cm. During the freezing phase, the upper surface temperature was  
Figure 4: Comparison between simulation results and experimental data: the relationship between frost heaving and freezing depth over time.