# Influence of Fracture Geometric Characteristics on Fractured Rock Slope Stability
## 4 Sensitive or Parametric Stability Analyses
### Influence of Fracture Trace Length  
Table 4 shows the values of \(\mu_{F_{S}}\) for nine different cases. Note that the parameter values in Case 0 are selected from Table 1. From the line chart in Figure 7a, it can be seen that the \(\mu_{F_{S}}\) gradually decreases with the increase in \(\mu_{T_{1}}\) and \(\mu_{T_{2}}\). This is mainly because a larger \(\mu_{T_{1}}\) and \(\mu_{T_{2}}\) leads to more extended horizontal and inclined fractures for slippage, which can facilitate slope slip.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \(\mu_{T_{1}}\)**(m)** & \(\mu_{T_{2}}\)**(m)** & \(\varepsilon\) & \(\mu_{F_{S}}\) \\ \hline Case 0 & 8.5 & 5.9 & 0.23 & 1.78 \\ \hline Case 1 & 15 & 3 & 0.12 & 2.05 \\ Case 2 & 9 & 3 & 0.12 & 2.09 \\ Case 3 & 3 & 3 & 0.12 & 2.36 \\ \hline Case 4 & 15 & 9 & 0.35 & 1.64 \\ Case 5 & 9 & 9 & 0.35 & 1.69 \\ Case 6 & 3 & 9 & 0.35 & 1.76 \\ \hline Case 7 & 15 & 15 & 0.58 & 1.58 \\ Case 8 & 9 & 15 & 0.58 & 1.62 \\ Case 9 & 3 & 15 & 0.58 & 1.7 \\ \hline \end{tabular} Note: \(\varepsilon\) is the ratio of \(\mu_{T_{2}}\) to slope height \(H\).  
\end{table}
Table 4: Statistical table of the mean trace length and mean stability.  
The change in \(\mu_{F_{S}}\) for the nine cases with different \(\varepsilon\) values (the ratio of \(\mu_{T_{2}}\) value to slope height \(H\)) is illustrated in Figure 7b. Notably, the \(\mu_{F_{S}}\) rapid declines when \(\varepsilon\) ranges from 0.12 to 0.35, while the \(\mu_{F_{S}}\) changes slowly when the \(\varepsilon\) values increase from 0.35 to 0.58. This result is because with \(\mu_{T_{2}}\) increases, these inclined fractures near the slope can easily be connected into some more extensive fractures, which is not conducive to slope stability.  
In addition, it can be found that the values of \(\mu_{F_{S}}\) change a little when \(\mu_{T_{1}}\) or \(\mu_{T_{2}}\) increases from 9 m to 15 m (Figure 7). This result indicates that the sensitivity of the mean trace length to the slope stability decreases with the increase in fracture trace length. That is because when the trace length increases to a certain extent, the rock mass gradually breaks into discrete blocks, and then the increasing trace length has little influence on the slope stability. According to this analysis, the longest horizontal and inclined fracture trace lengths (e.g., 15 m) attained the smallest mean Fs (e.g., 1.58) of the slope as well as the highest inclined trace-length-to-slope-height ratio (e.g., \(\varepsilon\) = 0.58). In contrast, the lowest horizontal and inclined fracture trace length (e.g., 3 m) as well as the lowest ratio of inclined trace length to slope height (e.g., \(\varepsilon\) = 0.12) result in the highest mean Fs (e.g., 2.36).  
Then, four x-displacement contours with different rock structures (i.e., Case 3, 5, 6, and 7) are shown in Figure 8. The \(F_{S}\) values are obtained through one-time Monte Carlo simulation. As shown in Figure 8a, both the horizontal and inclined fractures in Case 3 have a short trace length, resulting in only minor damage to the rock mass. Hence, the slopes with such a structure usually have better stability, such as \(F_{S}\) = 2.86 in Figure 8a. When the trace length of inclined fractures increases in Case 6, it is easy to form some larger through fractures near the slope (Figure 8b), causing a significant decrease in the slope stability (\(F_{S}\) = 1.93). Further, when the trace length of two sets of fractures increases in Case 5, the rock will be more fragmented (Figure 8c) and the slope stability will be worse than that in Case 6. Moreover, the slope stability is the worst (\(F_{S}\) = 1.29) when the trace length of two sets of fractures increases to 15 m (Figure 8d).  
Figure 8: X-displacement contours with different fracture trace lengths.