## 3 Uncertain Analysis Method of Thermodynamic Properties
### Hydrothermal Parameters and Boundary Conditions  
Boundary and initial conditions significantly influence the thermal characteristics analysis of freeze curtains in underground space engineering. Boundary conditions dictate the exchange of heat between the soft clay rock and the surrounding environment, affecting temperature distribution and freeze front progression. Similarly, initial conditions determine the starting temperature profile within the ground, impacting the rate and extent of freezing. Accurate consideration of these conditions is vital for assessing the effectiveness of soft clay rock in controlling ground temperatures and preventing settlement during underground space construction [41]. Understanding their influence aids in optimizing underground space resource development design and operational parameters to ensure efficient heat extraction and ground stabilization. Moreover, it facilitates the prediction of potential thermal impacts on adjacent structures and utilities, enhancing safety and sustainability in underground infrastructure projects. According to the coupling equation of the soft clay rock, the content of unfrozen water in soft clay rock is an important index to evaluate the water transport characteristics, and it is also a common parameter in thermal calculation. The experimental fitting curve equation can be expressed as  
\[W_{u}=\left\{ \begin{array}{ll}P|T|_{Q}&T\leq T_{\text{m}}\\ W&T>T_{\text{m}}\end{array} \right. \tag{18}\]  
Based on field test data, parameters \(P\) and \(Q\) are 0.348 and \(-\)0.172, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Autocorrelation Functions** & **Parameter Relationship** \\ \hline DSNX & \(\rho(\mathbf{\tau})=\exp\left[-\left(\frac{\Delta x}{\theta_{x}}+\frac{\Delta y}{ \theta_{y}}\right)\right]\) & \(\theta_{x}=\frac{\delta_{y}}{2},\theta_{y}=\frac{\delta_{y}}{2}\) \\ \hline DSQX & \(\rho(\mathbf{\tau})=\exp\left\{-\left[\left(\frac{\Delta y}{\theta_{x}}\right)^{2 }+\left(\frac{\Delta y}{\theta_{y}}\right)^{2}\right]\right\}\) & \(\theta_{x}=\frac{\delta_{x}}{\sqrt{\pi}},\theta_{y}=\frac{\delta_{y}}{\sqrt{ \pi}}\) \\ \hline DSMK & \(\rho(\mathbf{\tau})=\exp\left[-\left(\frac{\Delta x}{\theta_{x}}+\frac{\Delta y}{ \theta_{y}}\right)\right]\left[\left(1+\frac{\Delta x}{\theta_{x}}\right) \left(1+\frac{\Delta y}{\theta_{y}}\right)\right]\) & \(\theta_{x}=\frac{\delta_{y}}{4},\theta_{y}=\frac{\delta_{x}}{4}\) \\ \hline DCSX & \(\rho(\mathbf{\tau})=\exp\left[-\left(\frac{\Delta x}{\theta_{x}}+\frac{\Delta y}{ \theta_{y}}\right)\right]\cos\left(\frac{\Delta y}{\theta_{x}}\right)\cos\left( \frac{\Delta y}{\theta_{y}}\right)\) & \(\theta_{x}=\delta_{x},\theta_{y}=\delta_{y}\) \\ \hline DBIN & \(\rho(\mathbf{\tau})=\left\{\begin{array}{ll}\left(1-\frac{\Delta x}{\theta_{x}} \right)\left(1-\frac{\Delta y}{\theta_{y}}\right)&\Delta x\leq\theta_{x}, \Delta y\leq\theta_{y}\\ 0&\Delta x>\theta_{x},\Delta y>\theta_{y}\end{array}\right.\) & \(\theta_{x}=\delta_{x},\theta_{y}=\delta_{y}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Autocorrelation functions for characterizing the spatial variability of hydrothermal properties. Water head boundary condition:  
\[H=H_{0} \tag{19}\]  
Flow boundary condition:  
\[K_{x}\frac{\partial H}{\partial x}\cos(n,x)+K_{y}\frac{\partial H}{\partial y}\cos (n,y)=-q_{0} \tag{20}\]  
Temperature conditions:  
\[T(r_{0},t)=T_{c} \tag{21}\]  
\[T(\infty,t)=T_{0} \tag{22}\]  
where \(r_{0}\) is outer diameter; \(T_{c}\) is surface temperature; \(T_{0}\) is initial temperature. Initial temperature conditions:  
\[T(r,0)=T_{0} \tag{23}\]  
In this paper, underground space engineering is implemented in soft clay rock layers with three rings of freezing pipes. The inner ring has a diameter of 3.5 m, the middle ring has a diameter of 4.1 m, and the outer ring has a diameter of 4.7 m. There are 18 freezing pipes in both the inner and middle rings, while the outer ring has 32 freezing pipes.