## 4 Discussion
### Three-Dimensional Fractal Dimension of Red-Bed Soft Rock Fracture Structure  
The deterioration of rocks is characterized by irregular and disordered generation and the evolution of fractures. Therefore, the irregular evolution of rocks in the deterioration process can be characterized in terms of three-dimensional fractal dimensions to study the complexity of the arrangement of fractures. The single fractal dimension, as a quantitative index, is a fundamental parameter used to quantitatively measure self-similar random shapes and phenomena. The box-counting method is a widely used approach in fractal calculations in nature. Therefore, this method is adopted in this paper to calculate the fractal dimension of the fracture distribution in the red-bed soft rock specimens. This method offers a more profound understanding of the fracture distribution law and enables an accurate interpretation of the self-similar characteristics of fracture extension in red-bed soft rock. This, in turn, provides a more scientific foundation for the assessment of the stability of red-bed soft rock structures (Wu et al. [25]). The single fractal dimension of the fracture distribution on the surface of a cylindrical specimen of red-bed soft rock can be expressed as  
\[N(r)=C\cdot r^{-D} \tag{2}\]  
where \(r\) is the mesh edge length after squaring the fractures, \(N(r)\) is the number of meshes to mesh the fractures, \(C\) is the scale constant and \(D\) is the fractal dimension. Taking logarithmic treatment for Equation (2) yields  
\[\log N(r)=-D\log r+\log C \tag{3}\]  
The red-bed soft rock specimens have a 3D structure, and a single fractal dimension is not able to represent the distribution and development process of 3D fractures. Therefore, we introduce the concept of multiple fractals to characterize the statistical self-similarity with different local properties. The 3D fractal structure is similar to that in 1D. While, in 1D fractals, a mesh is used to cut the clefts, in 3D fractals, a cubic mesh is required to cut the irregular clefts. The cubic mesh constitutes a succession of small boxes, each of which covers only one cleft. The fractures throughout the specimen are wrapped by numerous  
Figure 6: Microporous fracture statistics. small boxes, which are used to describe the irregular geometric objects of the 3D structure (Xie [26]). Its 3D structure is calculated by  
\[\lg V=\frac{3}{D}\lg A-\lg(C_{0}^{3}) \tag{4}\]  
where \(V\) denotes the volume of the geometric object, \(D\) denotes the 3D fractal dimension, A denotes the surface area of the geometric object and \(C_{0}\) is the scale constant. The soft rock is saturated with water and produces fractures, which are numerous and messy, but the new fractures have self-similar characteristics to the original fractures after different saturation times. Therefore, the Label Analysis algorithm can be applied to calculate the target volume, and the information of the fractures in different areas can be extracted. Calculations were performed using the extracted surface area and volume, and the fracture volume and surface area were separately logarithmically processed. The coordinate corresponding points of different small boxes were plotted separately in double logarithmic coordinates, and the principle of least squares was utilized to fit the different points into a straight line. Meanwhile, we define the slope of the fitted straight line segment as the fractal coefficient \(K\), which is further generalized to the 3D coordinates. The fractal dimension \(D\) is calculated by  
\[D=\frac{3}{K} \tag{5}\]  
In accordance with the aforementioned fractal theory, VGStudio MAX was utilized to superimpose the fracture pixels for each class of red-bed soft rock samples under varying water saturation time conditions. The number of small boxes was 50 when the fissures are the most numerous, and thus 50 small boxes were selected to calculate the fractal coefficients and fitted as a straight line using the least squares method.