# Study on the Development Rule of Mudstone Cracks in Open-Pit Mine Dumps Improved with Xanthan Gum
## 3 Results
### Image Analysis
#### 3.1.3 Fractal Dimension and Probability Entropy  
The fractal dimension is a concept that describes the shape, contour, or forms in nature without considering their complexity. Generally, the fractal dimension gives the autocorrelation of the profile or the rate of change in length (perimeter) in response to a change in the measurement scale (area). It essentially reflects the change in the morphological factor (complexity of crack boundaries) when the crack area increases, which can be calculated by the Form factor (_ff_) [31]. The form factor is often used to describe the shape of features and is defined by the perimeter (\(C\)) and area (\(S\)) as follows:  
\[ff=4\cdot\pi\cdot S/C^{2} \tag{2}\]  
The formula gives the relationship between fractal dimension and form factor:  
\[Df=1-log_{s}ff\cdot a^{-1} \tag{3}\]  
where \(a\) is a constant, and \(Df\) is the fractal dimension.  
The expression for probability entropy, commonly used to describe the directionality of pores, can be used in this paper to describe the directionality of crack development. The expression is as follows:  
\[H=-\sum_{i=1}^{n}P_{i}log_{n}P_{i} \tag{4}\]  
\(H\) represents the probability entropy; \(P_{i}\) indicates the percentage of cracks within a specific range, where it is the identifier for the direction range division, and \(n\) is the number of direction ranges divided. The fractal dimension and probability entropy of the specimen obtained by image analysis are shown in Figure 5.  
The value of fractal dimension generally lies between 1 and 2, and the complexity of the boundary increases with the increase in fractal dimension. The larger the fractal dimension,  
Figure 5: Average fractal dimension and probability entropy of modified specimens.  
the rougher and more complex the crack boundary. The experimental results show that adding a small amount of XG (0.5-1%) does not change the complexity of the mudstone specimen's crack boundary, which remains above 1.3. After adding an appropriate amount of XG, the fractal dimension of the specimen decreases significantly, indicating that the cracks produced by the specimen's drying shrinkage become more straightforward and regular. The change in fractal dimension can affect the permeability of the rock and soil body, and many studies have shown that the fractal dimension of cracks is negatively correlated with the permeability [32,33]. The addition of XG can improve the permeability of mudstone to a certain extent, and the importance of the impact needs to be proven by subsequent experiments.  
According to the formula, probability entropy is between 0 and 1. When the crack direction is the same, H = 0; when the crack direction is entirely random, H = 1. The more chaotic and random the pore direction, the greater the H value. The experimental results show that with the increase in the mixing ratio, the probability entropy of the specimen shows a trend of first decreasing and then increasing. The unmodified U1 specimen has a probability entropy that remains above 9.6 in multiple wet-dry cycles, essentially in a state of random direction. Among the modified specimens, the M3 specimen has a probability entropy that decreased to 0.879 in the 5 th cycle, which is the lowest value in the entire experiment; the M4 specimen has the lowest average probability entropy of 0.943, and the crack development direction is still highly random. The decrease in probability entropy can affect the specimen's crack development disorder. Through image comparison, the cracks of the modified specimens are relatively fixed in direction, but no obvious directionality is observed. This study's slight decrease in probability entropy is due to the reduced crack ratio and the decline in effective cracks.