## 3 Uncertain Analysis Method of Thermodynamic Properties  
### Coupled Governing Equations  
The mechanism of temperature-permeability coupling in frozen soft clay rock elucidates the intricate interplay between thermal dynamics and fluid flow. At its core, the mechanism involves the manipulation of temperature to induce freezing in rock pores, altering their permeability and hydraulic conductivity. As temperature decreases, pore water freezes, leading to a reduction in pore size and an increase in rock matrix strength. Consequently, the permeability of the soft clay rock decreases, restricting fluid flow. However, the coupling is not solely determined by temperature. Factors such as ice formation rate, soil composition, and boundary conditions also influence the process. Understanding this mechanism is critical for optimizing freezing techniques, designing effective containment systems, and predicting the behavior of soft clay rock under various conditions. By comprehending the temperature-permeability coupling mechanism in underground space, engineers can develop sustainable solutions for infrastructure development in cold regions,  
Figure 3: Variability characteristics of hydrothermal properties: (**a**) Discrete distribution of COV, (**b**) discrete distribution of SOF, (**c**) discrete distribution of ACD, (**d**) statistical characteristics of COV, (**e**) statistical characteristics of SOF, (**f**) statistical characteristics of ACD. mitigate environmental impacts, and ensure the long-term stability of engineered structures in frozen soft clay rock environments [38]. The coupled equations can be expressed as follows:  
In the frozen area \(\Omega_{\mathrm{f}}\):  
\[C_{\mathrm{f}}\frac{\partial T_{\mathrm{f}}}{\partial t}=\ \ \frac{\partial}{ \partial x}\bigg{(}\lambda_{\mathrm{f}}\frac{\partial T_{\mathrm{f}}}{ \partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}\lambda_{\mathrm{f}} \frac{\partial T_{\mathrm{f}}}{\partial y}\bigg{)}-C_{\mathrm{m}}\rho_{w} \bigg{(}V_{x}^{\mathrm{f}}\frac{\partial T_{\mathrm{f}}}{\partial x}+V_{y}^{ \mathrm{f}}\frac{\partial T_{\mathrm{f}}}{\partial y}\bigg{)} \tag{1}\]  
\[S_{\mathrm{f}}\frac{\partial H_{\mathrm{f}}}{\partial t}=\ \ \frac{\partial}{ \partial x}\bigg{(}K_{x}^{\mathrm{f}}\frac{\partial H_{\mathrm{f}}}{\partial x }\bigg{)}+\frac{\partial}{\partial y}\bigg{(}K_{y}^{\mathrm{f}}\frac{\partial H _{\mathrm{f}}}{\partial y}\bigg{)}+Q \tag{2}\]  
\[V_{x}^{\mathrm{f}}=-K_{x}^{\mathrm{f}}\frac{\partial H_{\mathrm{f}}}{\partial x },\ V_{y}^{\mathrm{f}}=-K_{y}^{\mathrm{f}}\frac{\partial H_{\mathrm{f}}}{ \partial y} \tag{3}\]  
In the unfrozen area \(\Omega_{\mathrm{u}}\):  
\[C_{\mathrm{u}}\frac{\partial T_{\mathrm{u}}}{\partial t}=\ \ \frac{\partial}{ \partial x}\bigg{(}\lambda_{\mathrm{u}}\frac{\partial T_{\mathrm{u}}}{ \partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(}\lambda_{\mathrm{u}} \frac{\partial T_{\mathrm{u}}}{\partial y}\bigg{)}-C_{\mathrm{m}}\rho_{w} \bigg{(}V_{x}^{\mathrm{u}}\frac{\partial T_{\mathrm{u}}}{\partial x}+V_{y}^{ \mathrm{u}}\frac{\partial T_{\mathrm{u}}}{\partial y}\bigg{)} \tag{4}\]  
\[S_{\mathrm{u}}\frac{\partial H_{\mathrm{u}}}{\partial t}=\ \ \frac{\partial}{ \partial x}\bigg{(}K_{x}^{\mathrm{u}}\frac{\partial H_{\mathrm{u}}}{\partial x }\bigg{)}+\frac{\partial}{\partial y}\bigg{(}K_{y}^{\mathrm{u}}\frac{\partial H _{\mathrm{u}}}{\partial y}\bigg{)}+Q \tag{5}\]  
\[V_{x}^{\mathrm{u}}=-K_{x}^{\mathrm{u}}\frac{\partial H_{\mathrm{u}}}{\partial x },\ V_{y}^{\mathrm{u}}=-K_{y}^{\mathrm{u}}\frac{\partial H_{\mathrm{u}}}{ \partial y} \tag{6}\]  
In Equations (1)-(6), f and u represent the frozen and the unfrozen states, respectively. \(T_{\mathrm{f}}\), \(C_{\mathrm{f}}\), and \(\lambda_{\mathrm{f}}\) are the temperature, volumetric heat capacity, and thermal conductivity of foundation soils in the frozen area \(\Omega_{\mathrm{f}}\), respectively.