## 3 Risk Assessment of Water Inrush in the Grey Correlation Degree through Fault Zone
### Risk Assessment of Water Inrush in F\({}_{55-1}\) Through-Type Fault Zone
#### 3.3.2 Calculation of Correlation Degree  
(1) Dimensionless processing  
Use Formula (2) to homogenize each indicator, as shown in Table 6:  
(2) Calculation of correlation coefficient  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & Judgement matrix A–Bi (i = 1–3) & \\ \hline A & B1 & B2 & B3 & W (A/B) \\ B1 & 1 & 1/3 & 1/2 & 0.26195 \\ B2 & 3 & 1 & 1 & 0.24022 \\ B3 & 2 & 1 & 1 & 0.36522 \\  & & \(\lambda_{\text{max}}\) = 3.018 Cl1 = 0.009 Rl1 = 0.520 CR1 = 0.018 \\ \hline  & & & Judgement matrix B1–Ci (i = 1, 2) & \\ \hline B1 & C1 & C2 & W \\ C1 & 1 & 1/3 & 0.25 \\ C2 & 3 & 1 & 0.75 \\  & & \(\lambda_{\text{max}}\) = 2.000 Cl21 = 0 Rl21 = 0 CR21 = 0 \\ \hline  & & & Judgement matrix B2–Ci (i = 3, 4) & \\ \hline B2 & C3 & C4 & W \\ C3 & 1 & 1/2 & 0.3333 \\ C4 & 2 & 1 & 0.66667 \\  & & \(\lambda_{\text{max}}\) = 2.000 Cl22 = 0 Rl22 = 0 CR22 = 0 \\ \hline  & & & Judgement matrix B3–Ci (i = 5–9) & \\ \hline B3 & C5 & C6 & C7 & C8 & C9 & W \\ C5 & 1 & 1/3 & 1/4 & 1/2 & 2 & 0.10505 \\ C6 & 3 & 1 & 1/2 & 2 & 3 & 0.26848 \\ C7 & 4 & 2 & 1 & 1 & 4 & 0.34552 \\ C8 & 2 & 1/2 & 1 & 1 & 2 & 0.20210 \\ C9 & 1/2 & 1/3 & 1/4 & 1/2 & 1 & 0.07886 \\  & & \(\lambda_{\text{max}}\) = 5.196 Cl23 = 0.049 Rl23 = 1.120 CR23 = 0.044 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Consistency Test for Weight Calculation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**A/Ci** & **B1/0.26195** & **B2/0.24022** & **B3/0.36522** & **W (A/Ci)** \\ \hline C1 & 0.25000 & & & 0.0733 \\ C2 & 0.75000 & & & 0.1965 \\ C3 & & 0.33333 & & 0.0801 \\ C4 & & 0.66667 & & 0.1602 \\ C5 & & & 0.10505 & 0.0384 \\ C6 & & & 0.26848 & 0.0981 \\ C7 & & & 0.34552 & 0.1261 \\ C8 & & & 0.20210 & 0.0738 \\ C9 & & & 0.07886 & 0.0288 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Weights of Each Indicator to the Total Objective.  
The reference sequence is the number of four evaluation criteria indicators in Table 6. The comparison sequence is the level standard of evaluation indicators, and the correlation coefficient is determined using Formula (3). The four evaluation criteria indicators and their correlation coefficients are represented as follows:  
\[x_{1}=\begin{bmatrix}0.320&0.192&0.032&0.003\\ 1.255&1.046&0.532&0.209\\ 2.247&0.983&0.365&0.140\\ 2.143&1.190&0.714&0.238\\ 1.613&1.210&0.806&0.403\\ 1.651&1.238&0.413&0.413\\ 1.250&0.938&0.625&0.313\\ \end{bmatrix},\ \ x_{2}=\begin{bmatrix}0.227&0.568&1.023&1.477\\ 1.429&0.804&0.446&0.179\\ 0.496&0.609&0.727&0.840\\ 0.529&0.584&0.587&0.781\\ 0.920&1.000&0.950&0.859\\ 0.402&0.417&0.358&0.390\\ 0.736&0.701&0.670&0.769\\ \end{bmatrix}\]  
\[\xi_{1}=\begin{bmatrix}0.629&0.622&0.587&0.676\\ 0.662&0.606&0.652&0.786\\ \end{bmatrix},\ \ \xi_{2}=\begin{bmatrix}0.705&0.888&0.614&0.338\\ 0.852&0.778&0.750&0.647\\ \end{bmatrix}\]  
Based on the weight indicators determined by the Analytic Hierarchy Process (AHP), the weighted correlation degree is calculated using Formula (1): \(\tau_{1}=\begin{bmatrix}0.25&0.75\\ 0.6538&0.6100&0.6358&0.7590\end{bmatrix},\tau_{1\max}=0.759\). The maximum grey weighted correlation degree indicates that the risk level of hydrogeological conditions under this system is Level IV. Similarly, it can be concluded that:  
\(\tau_{2}=\begin{bmatrix}0.8030&0.8147&0.7047&0.5440\end{bmatrix},\tau_{3}= \begin{bmatrix}0.6513&0.6514&0.6874&0.7335\end{bmatrix},\tau_{2\max}\) = 0.8147, and \(\tau_{3\max}=0.7335\). The maximum grey weighted correlation degree indicates that the hazard level of fault conditions is Level II, and the hazard level of mining conditions is Level IV.  
(3) Calculation of water intrusion risk level in through-type fault zones  
The correlation degree of the risk level of water intrusion in the connected fault zone obtained is:  
\[\text{r}=\begin{bmatrix}0.26195&0.24022&0.36522\end{bmatrix}\times\begin{bmatrix} 0.6538&0.6100&0.6358&0.7590\\ 0.8030&0.8147&0.7047&0.5440\\ 0.6513&0.6514&0.6874&0.7335\end{bmatrix}=\begin{bmatrix}0.6020&0.5944&0.5869&0.5 974\end{bmatrix}\]