# Percolation Threshold of Red-Bed Soft Rock during Damage and Destruction
## 3 Results and Discussion  
### Theoretical Determination of Percolation Threshold for Damage of Red-Bed Soft Rock  
The renormalization transformation rule corresponds to the nonlinear iteration. According to the renormalization model of red-bed soft rock established in Section 2.1, the probability of failure of a basic unit can be calculated. A basic unit is composed of 8 skeleton members and 12 cementing members. Therefore, there are 13 cementing member failure situations, namely, 0, 1, 2,..., 12 cementing member failure cases, and a total of 2765 cases that can lead to unit failure. If the probability of damage for the first-order cementing member is set at \(P\), then the probability of non-damage of the cementing member is \(1-P\). Then, the failure probability for each unit given the 13 types of failure conditions is shown in Table 1.  
According to Table 1, the probability of the first-level unit failing is:  
\[ \begin{split} P_{1}=12P^{3}(1-P)^{9}+99P^{4}(1-P)^{8}+360P^{5}(1- P)^{7}+216P^{6}(1-P)^{6}+792P^{7}(1-P)^{5}+495P^{8}(1-P)^{4}\\ +220P^{9}(1-P)^{3}+66P^{10}(1-P)^{2}+12P^{11}(1-P)+P^{12}\\ =-27P^{12}+108P^{11}-144P^{10}+64P^{9}-27P^{8}+72P^{7}-48P^{6}-9P^{ 4}+12P^{3}\\ \end{split}  \tag{18}\]  
Under renormalization transformation, according to renormalization theory, the probability of the second level unit failing can be obtained as follows:  
\[P_{2}=-27P_{1}{}^{12}+108P_{1}{}^{11}-144P_{1}{}^{10}+64P_{1}{}^{9}-27P_{1}{}^{ 8}+72P_{1}{}^{7}-48P_{1}{}^{6}-9P_{1}{}^{4}+12P_{1}{}^{3} \tag{19}\]  
When renormalization reaches all the way to level n:  
\[P_{n}=-27P_{n-1}^{12}+108P_{n-1}^{11}-144P_{n-1}^{10}+64P_{n-1}^{9}-27P_{n-1}^{ 8}+72P_{n-1}^{7}-48P_{n-1}^{6}-9P_{n-1}^{4}+12P_{n-1}^{3} \tag{20}\]  
According to renormalization theory, when n is large enough (with a sufficient number of renormalization iterations), a fixed point will be reached which satisfies the following:  
\[P_{n+1}=P_{n}. \tag{21}\]  
Substituting Equation (20) into Equation (21), we obtain:  
\[P_{n}=-27P_{n}^{12}+108P_{n}^{11}-144P_{n}^{10}+64P_{n}^{9}-27P_{n}^{8}+72P_{n }^{7}-48P_{n}^{6}-9P_{n}^{4}+12P_{n}^{3} \tag{22}\]  
To calculate Equation (22), there are three solutions in the 0-1 range: 0, 0.3636 and 1, that is, there are three fixed points as shown in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Damaged Number** & **Probability of** & **Damaged Number** & **Probability of** \\
**of Cementing** & **Unit Failure** & **of Cementing** & **Unit Failure** \\
**Members** & & **Members** & **Members** \\ \hline
0 & 0 & 7 & \(792P^{7}\) (\(1-P\))\({}^{5}\) \\
1 & 0 & 8 & \(495P^{8}\) (\(1-P\))\({}^{4}\) \\
2 & 0 & 9 & \(220P^{9}\) (\(1-P\))\({}^{3}\) \\
3 & \(12P^{3}\) (\(1-P\))\({}^{9}\) & 10 & \(66P^{10}\) (\(1-P\))\({}^{2}\) \\
4 & \(99P^{4}\) (\(1-P\))\({}^{8}\) & 11 & \(12P^{11}\) (\(1-P\)) \\
5 & \(360P^{5}\) (\(1-P\))\({}^{7}\) & 12 & \(p^{12}\) \\
6 & \(708P^{6}\) (\(1-P\))\({}^{6}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Failure probability of a unit under the 13 types of failure conditions for cementing members. The derivative with respect to \(P_{n}\) on the right side of Equation (22) can be obtained as follows:  
\[g^{\prime}(P_{n})=-324P_{n}^{11}+1188P_{n}^{10}-1440P_{n}^{9}+576P_{n}^{8}-216P_{n }^{7}+504P_{n}^{6}-288P_{n}^{5}-36P_{n}^{3}+36P_{n}^{2}. \tag{23}\]  
By substituting the three solutions into Equation (23), the first derivative value of the transformation function at the fixed point can be obtained as \(g^{\prime}(0)=0\), \(g^{\prime}(0.3636)=2.241\) and \(g^{\prime}(1)=0\). Since the critical point is the unstable fixed point among the fixed points, and the absolute value of the derivative of the transformation function at the critical point is greater than 1, the percolation threshold of red-bed soft rock under the failure conditions for loads is 0.3636. In other words, when under load conditions the inner structural unit of the red-bed soft rock is destroyed by 36.36%, the perforation failure occurs.