# Effect of Thick-Hard Main Roof Fracturing on the Spatiotemporal Evolution of Overburden Fractures
## 5 Fissure Ratio-Permeability-Fractal Dimension Theoretical Analysis
### Calculation of Fractal Dimension of Soft Rock Fracture Based on Box-Dimensional Method  
A fractal dimension is a quantity used to describe the complexity of a fractal graph. It is an extension of the traditional Euclidean concept of geometric dimensions. In Euclidean geometry, points are 0 dimensions, lines are 1 dimension, surfaces are 2 dimensions, and volumes are 3 dimensions. But fractal graphs have complex properties such as self-similarity, and their dimensions are not integers. The fractal dimension can represent the degree to which a fractal object is filled in space, and it is also a valid indicator of the complexity of a fractal object. The higher the fractal dimension of an object, the more complex and irregular its structure becomes.  
The box-counting dimension method is an important tool to describe the geometric features of fractal structures. Based on the counting of fractal objects covered by boxes, it approximates the dimensionality of fractal objects by constantly adjusting the size of boxes and observing the changes in the number of boxes. This method is suitable for various types of fractal structures, including topography, tree branches, urban development patterns, etc., which is of great significance for understanding the geometric characteristics and dynamic behavior of complex systems. The fractal dimension of the two-dimensional image fracture networks is between one and two. That is, the closer the fractal dimension is to two, the more fully developed the overburden fracture is, which is more conducive to gas migration.  
The box-counting dimension method calculates the dimensionality of a fractal object through a series of steps, which can be briefly summarized as follows:  
1. Select box size and box cover: select a series of square boxes or cubes of different sizes to cover or enclose the fractal object.
2. Count the number of fractal objects covered by boxes: place a box of the selected size on the fractal object and count the number of boxes covered by the fractal object.
3. Calculate the fractal dimension: repeat step 2 by adjusting the size of the box, and then calculate according to the relationship between the number of boxes and the size of the box. The calculated data are linearly fitted, and the slope of the fitting line is the fractal dimension \(D_{r}\).  
\[D_{r}=-\lim_{r\to 0}\frac{\lg N(r)}{\lg r} \tag{10}\]  
The square box with an edge length \(r\) covers the binary map of the development of the overlying soft rock fractures when the working face advances by 80 m and 100 m. By counting the number of square boxes containing fractures and denoting it as \(N(r)\), different fractal dimensions can be calculated by continuously changing the value of the box side length \(r\) to different non-empty boxes. Then, linear fitting of all the results is performed, which leads to the conclusion that the slope of the straight line is the fractal dimension of the fracture network diagram.  
In the four schemes, the corresponding fractal dimensions of the overburden fractures were 1.101, 1.317, 1.43, and 1.298 when the working face advanced by 80 m, and 1.326, 1.356, 1.473, and 1.353 when the working face advanced by 100 m, respectively. The calculation results show that when the working face advanced from 80 m to 100 m, the fractal dimension corresponding to the soft rock fracture network of the four schemes increased to varying degrees, among which the non-fracturing scheme witnessed the most significant increase of 20.44%. The horizontal fracturing scheme yielded the largest fractal dimension of 1.473. This indicates that the fracture network under this scheme was more complex, the soft rock fractures were more fully developed, and the permeability was more prominent, which is more conducive to gas migration. Under the four schemes, the corresponding fractal dimension calculation results when the working face advanced by 80 m and 100 m are shown in Figure 12.