## 4 Results and Analyses  
### Verification of Stochastic Coupling Model  
Validating analytical models for analyzing the thermal characteristics of soft clay rock in underground space engineering is essential for several reasons. Firstly, it ensures the reliability and accuracy of predictions concerning temperature distributions and the progression of the freezing front within the surrounding soil. Accurate predictions are critical for optimizing freeze designs and operational parameters to effectively control ground temperatures and prevent soil settlement during underground space engineering, ensuring the safety and stability of the infrastructure. Secondly, validation enables the assessment of the model's applicability across various geological and climatic conditions. Subsurface conditions can vary significantly from one location to another, and validating the model across different scenarios helps ensure its robustness and generalizability. Thirdly, the validation process helps identify any limitations or discrepancies between model predictions and real-world observations. This feedback loop guides improvements and refinements to the analytical model, enhancing its accuracy and reliability over time. Ultimately, validated analytical models serve as valuable tools for engineers and planners, providing insights into the thermal behavior of soft clay rock and supporting informed decision-making in underground space engineering. Points A, B, and C represent three sample points vertically spaced 1 m apart within the first ring of freezing pipes. Figure 4a shows the results of the experimental and numerical temperatures of soft clay rock for Points A, B, and C. From Figure 4a, it can be observed that the calculated mean temperature results of the uncertain analysis model of thermodynamic properties for the soft clay rock are close to the experimental results at Points A, B, and C. This is consistent with the law of large numbers in statistics, indicating that the Monte Carlo random simulation yields mean temperatures that are approximately equal to the statistically measured temperatures. As freezing time increases, the temperature of Points A, B, and C gradually decreases. Whenthe mean temperature approaches 0 \({}^{\circ}\)C, there is a noticeable plateau in the curve. This is due to the phase transition between ice and water, releasing latent heat, which slows down the cooling trend. Similarly, Points D, E, and F represent three sample points horizontally spaced 1 m apart within the first ring of freezing pipes. Figure 4b shows the results with the experimental temperatures and numerical temperatures of soft clay rock for Points D, E, and F. From Figure 4b, the calculated mean temperature results of the uncertain analysis model of thermodynamic properties for the soft clay rock are close to the experimental results at Points D, E, and F. This is consistent with the Law of Large Numbers in statistics, indicating that the average temperature generated by random simulation is approximately equal to the statistically measured temperature. As freezing time increases, the temperature of Points D, E, and F gradually decreases. When the mean temperature approaches 0 \({}^{\circ}\)C, there is a noticeable plateau in the curve. This is due to the phase transition between ice and water, releasing latent heat, which slows down the cooling trend. Therefore, the proposed stochastic analysis model for the thermal characteristics of soft clay rock, considering the spatial variability of variable hydrothermal parameters, is scientifically reasonable.