## 3 Results and Discussion
#### 3.1.2 Analysis of Yield Stress and Apparent Viscosity  
Figures 6 and 7 present the rheological properties of CSFMs with varying fractal dimensions. Gagne filling slurry belongs to non-Newtonian fluid. There are generally three types of rheological models describing non-Newtonian fluid, namely the power-law fluid model, the Bingham fluid model, and the Herschel-Bulkley (H-B) model. Among them, the power-law fluid model is suitable for fluids without a power-law relationship between the yield stress, shear stress, and shear rate. Because the mass fraction of the slurry selected in this paper is 75%, it belongs to high-concentration slurry. Compared with the Bingham model, the H-B model fits the rheological curve better. For high-concentration slurry, the Herschel-Bulkley (H-B) model is utilized to fit the relationship between the shear stress and shear rate [47; 48]. The fitting results and rheological parameters are shown in Table 2. The H-B model can be expressed as [49]:  
\[\tau\ =\ \tau_{0}+K\gamma^{n} \tag{3}\]  
The formula \(\tau\) represents the shear stress, \(K\) is the consistency coefficient, \(\tau_{0}\) is the yield stress, \(\gamma\) denotes the shear rate, and \(n\) is the rheological index. When \(n<1\), the slurry exhibits shear-thinning behavior; when \(n=1\), the slurry conforms to the H-B model; and when \(n>1\), the slurry exhibits shear-thickening behavior [50; 51]. It can be seen from Table 2 that the flow index \(n\) of CSFMs with a fractal dimension of 2.3 to 2.7 is greater than 1. The rheological test results of slurry under different gradation conditions are shown in Figure 6. The results indicate that appropriately optimizing the particle size distribution of gangue particles in CSFMs can significantly reduce the yield stress of the slurry. As the fractal dimension \(D\) of CSFMs increases from 2.3 to 2.7, the initial yield stress \(\tau_{0}\) of the slurry decreases first and then increases. This phenomenon occurs because, at smaller fractal dimensions, large-particle gangue settles and comes into contact with the rotating blade. This interaction causes significant fluctuations in the unstable beating value. The resulting increased shear stress leads to a higher initial yield stress of the slurry [52]. As the branch dimension \(D\) of the slurry continues to increase (a decrease in the particle thickness ratio), the proportion of fine particles in the slurry increases. Compared with coarse particles, fine particles better water absorption, resulting in an increase in the concentration of slurry and an increase in the initial yield stress \(\tau_{0}\) of the slurry. The rheological characteristic curves of CSFMs with \(D=2.45\), \(D=2.50\), and \(D=2.55\) have the same trend, indicating that the flocculation structures of CSFMs with \(D=2.45\), \(D=2.50\), and \(D=2.55\) are similar. In addition, [PERSON] et al. [33] also obtained the same conclusion on the influence of aggregate fractal series on the yield stress of paste, which is highly consistent with the trend of the fractal dimension on the yield stress of paste in this paper. In addition, some scholars have discussed how rheological parameters depend on different solid mass fractions and standing time, which are also factors affecting slurry rheology [53]. It is pointed out that the yield stress of slurry increases with the increase in mass fraction. The yield stress and viscosity of slurry increase with the increase in standing time. Figure 7 shows the change in the apparent viscosity with shear rate. The results demonstrate that the plastic viscosity of CSFMs initially decreases and then increases as the shear rate increases, showing the shear thickening phenomenon. When the apparent viscosity of the slurry decreases in the early stages, it may be because the shear action destroys the original network structure, resulting in a significant change in the arrangement direction of the solid particles and reducing its apparent viscosity.