## 2 Spatial Variability of Hydrothermal Parameter
### Variability Distribution Characteristics  
The spatial variability of thermodynamic parameters in soft clay rock is caused by a complex interplay of several underlying mechanisms. One primary factor contributing to this variability is the heterogeneous nature of clay at both micro and macro scales. At the microscale, clay minerals exhibit intricate structures and compositions, resulting in variations in thermal properties such as thermal conductivity and specific heat capacity. Variations in mineral composition, including the presence of different types of clay minerals and accessory minerals, can significantly influence thermal conductivity. Additionally, the arrangement of clay particles and their orientation within the clay matrix can create anisotropic behavior, further contributing to spatial variability. At the macroscale, geological processes such as sedimentation, weathering, and diagenesis play a crucial role in shaping the spatial distribution of clay properties. Variations in depositional environments, including factors like sedimentary facies and depositional conditions, can lead to spatial heterogeneity in clay properties across different geological formations. Furthermore, post-depositional processes such as compaction, cementation, and deformation can further modify the spatial distribution of soft clay rock properties, introducing additional variability. Environmental factors also influence the spatial variability of thermodynamic parameters. Variations in moisture content, temperature gradients, and chemical composition of pore fluids can affect thermal conductivity and specific heat capacity. Additionally, hydraulic gradients and flow pathways within clay formations can lead to preferential fluid flow and transport, influencing permeability distribution [35,36]. The spatial variability of clay thermodynamic parameters is further influenced by scale-dependent phenomena. At smaller scales, such as the pore scale, interactions between clay particles, pore fluid properties, and surface chemistry play a significant role in determining thermal and hydraulic properties. At larger scales, geological heterogeneity, structural features, and boundary conditions become more pronounced, influencing the overall spatial distribution of clay properties. Therefore, the spatial variability of thermodynamic parameters in soft clay rock is governed by a combination of factors, including mineralogical composition, geological processes, environmental conditions, and scale-dependent phenomena. Understanding these mechanisms is essential for accurately characterizing and predicting the behavior of soft clay rock in various engineering and environmental applications. Based on test data and random field theory, the COV, scale of fluctuation (SOF), and ACD can be calculated. The variability results of hydrothermal parameters are shown in Table 1.  
Random field theory is a powerful framework for characterizing the spatial variability of hydrothermal parameters [37]. In this theory, soft clay rock properties are treated as random fields, exhibiting inherent variability across space. The essence of this approach lies in recognizing that soft clay rock properties vary stochastically due to diverse geological