## 2 Spatial Variability of Hydrothermal Parameter
### Statistical Distribution Characteristics  
The results of thermodynamic parameter testing for soft clay rock reveal a distinct statistical distribution pattern, highlighting the inherent variability in clay properties. Factors such as mineral composition, pore structure, and moisture content contribute to this variability, leading to diverse values of hydrothermal parameters. Rather than displaying a singular fixed value, these parameters exhibit a range of values distributed probabilistically. This statistical distribution reflects the heterogeneous nature of soft clay rock, where differ  
Figure 1: Hydrothermal parameter for 10 different sampling points: (**a**) Thermal conductivity of frozen state, (**b**) thermal conductivity of unfrozen state, (**c**) heat capacity of frozen state, (**d**) heat capacity of unfrozen state, (**e**) hydraulic conductivity of horizontal direction, (**f**) hydraulic conductivity of vertical direction. ent regions within a sample may possess varying properties [33; 34]. Understanding this statistical distribution is essential for accurate modeling and prediction of clay behavior in engineering applications, as it allows for the incorporation of uncertainty and variability into predictive models. Advanced statistical techniques are often employed to analyze and interpret these results, providing valuable insights into the spatial variability and complexity of soft clay rock. Recognizing the statistical distribution of thermodynamic parameters in clay enhances our understanding of its behavior and facilitates more informed decision-making in engineering and environmental contexts. Based on the test results of the hydrothermal parameters of soft clay rock for ten different sampling points, the statistical value was calculated. The results are shown in Figure 2. For the thermal conductivity of the frozen state, the maximum is 1.907 W/m/\({}^{\circ}\)C while the minimum is 1.294 W/m/\({}^{\circ}\)C. The mean and standard deviation (SD) are 1.640 W/m/\({}^{\circ}\)C and 0.134 W/m/\({}^{\circ}\)C, respectively. For the thermal conductivity of the unfrozen state, the maximum is 1.526 W/m/\({}^{\circ}\)C while the minimum is 0.947 W/m/\({}^{\circ}\)C. The mean and SD are 1.241 W/m/\({}^{\circ}\)C and 0.124 W/m/\({}^{\circ}\)C, respectively. For the heat capacity of the frozen state, the maximum is 2.336 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C while the minimum is 1.445 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C. Mean and SD are 1.893 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C and 0.1764 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C, respectively. For the heat capacity of the unfrozen state, the maximum is 2.755 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C while the minimum is 1.697 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C. Mean and SD are 2.227 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C and 0.230 \(\times\) 10\({}^{6}\) J/m\({}^{3}\)/\({}^{\circ}\)C, respectively. For the hydraulic conductivity of horizontal direction, the maximum is 2.869 \(\times\) 10\({}^{-6}\) m/s, while the minimum is 1.504 \(\times\) 10\({}^{-6}\) m/s. Mean and SD are 2.203 \(\times\) 10\({}^{-6}\) m/s and 0.266 \(\times\) 10\({}^{-6}\) m/s, respectively. For the hydraulic conductivity of vertical direction, the maximum is 2.999 \(\times\) 10\({}^{-6}\) m/s while the minimum is 1.471 \(\times\) 10\({}^{-6}\) m/s. Mean and SD are 2.204 \(\times\) 10\({}^{-6}\) m/s and 0.265 \(\times\) 10\({}^{-6}\) m/s, respectively. It can be seen that the statistical distribution characteristics of hydraulic conductivity in the horizontal direction and vertical direction are approximate. Thus, it can be concluded that the permeability of soft clay rock in different directions is the same. In general, the statistical characteristics of the hydrothermal parameters show a normal distribution.