## 5 Conclusions  
In this study, the spatial variations of hydrothermal parameters of soft clay rock were calculated based on experimental data. The statistical variability characteristics of variable hydrothermal parameters are estimated. A stochastic coupling model of soft clay rock with heat conduction and porous flow is proposed. According to the relationship between anisotropic spatial variations and statistical variability characteristics for the different random field correlation models, the effects of the ACF, COV, and ACD of variable hydrothermal parameters on underground space engineering are analyzed. The main conclusions obtained are as follows:  
1. The average temperature calculation results of the freezing soft clay rock are close to the experimental results. When the mean temperature approaches 0 \({}^{\circ}\)C, there is a noticeable plateau in the curve. The average thickness and temperature of the freezing curtain show an overall increasing trend with increasing freezing time;
2. From the second month to the tenth month, both the average thickness and temperature increase at a faster rate. When the freezing time reaches twelve months, it tends to stabilize. The overall trend of the SD of the freezing curtain increases with increasing freezing time;
3. The greater the COV of hydrothermal parameters, the larger the SD of the freezing curtain temperature. For the ACF, the most significant impact comes from DBIN, followed by DSQX, with DSNX having the smallest effect. For COV, the most significant impact comes from thermal conductivity, followed by permeability coefficient, with specific heat capacity having the smallest effect;
4. Different ACD parameters have varying impacts on the temperature of the freeze-soft clay rock. The biggest influence is the thermal conductivity, the second is the permeability coefficient, and the least influence is the specific heat capacity. The larger the ACD of the hydrothermal parameter, the smaller the SD of freezing soft clay rock temperature. The variable hydrothermal parameter of soft clay rock holds profound implications for urban underground space sustainable development. As cities continue to expand, the construction and maintenance of underground space infrastructure play a pivotal role in enhancing transportation efficiency, reducing carbon emissions, and fostering economic growth. However, the effectiveness of freeze curtains in controlling water ingress and stabilizing underground space structures depends on the accurate characterization of thermodynamic parameters such as thermal conductivity, permeability coefficient, and specific heat capacity. In the future, we can focus on the dynamic evolution relationship between parameter spatial variability and temperature. By considering the variability in these parameters, urban planners and engineers can make informed decisions that contribute to the sustainability of underground space engineering. Understanding the variable hydrothermal parameters enables the design of freeze curtains that are resilient to changing environmental conditions and geological characteristics. Furthermore, considering variability fosters innovation in sustainable infrastructure development. This aligns with the broader goals of sustainable urban development, which seek to create cities that are livable, equitable, and environmentally conscious. Therefore, the consideration of variability in thermodynamic parameters of soft clay rock is instrumental in promoting urban sustainable development. Formal analysis, T.W. ; Investigation, D.W. and H.L. ; Resources, J.G. ; Data curation, T.W. and K.R. ; Project administration, H.L. and J.G. ; Funding acquisition, T.W. All authors have read and agreed to the published version of the manuscript. This research was supported by the National Natural Science Foundation of China (Grant No. U20B6001, 92255302), the Open Fund of State Key Laboratory of Coal Mining and Clean Utilization (China Coal Research Institute) (Grant No. 2021-CMCU-KF019). Not applicable.