## 4 Case Study
### Fuzzy Clustering Method for Identification of the Source of Water Gushing
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & **Sampling Locations** & **Redox Potential** & **pH** & **TDS** & **Conductivity** & **Temperature** \\  & **(mV)** & **pH** & **(mg/L)** & **(μs/cm)** & **(\({}^{\circ}\)C)** \\ \hline S1 & Walls at K91 + 210 & \(-\)62.2 & 8.51 & 112 & 174.7 & 23.5 \\ S2 & Walls at K91 + 273 & \(-\)16 & 6.85 & 127 & 199.1 & 23.9 \\ S3 & Walls at K91 + 300 & \(-\)15 & 6.75 & 128 & 199.8 & 24 \\ S4 & Walls at K91 + 310 & \(-\)47.6 & 7.9 & 129 & 201 & 23.7 \\ S5 & Walls at K91 + 314 & \(-\)35 & 7.49 & 127 & 197.9 & 23.9 \\ S6 & Top of tunnel at ZK94 + 299 & \(-\)43 & 7.73 & 117 & 182.8 & 23 \\ S7 & Top of tunnel at ZK94 + 250 & \(-\)98 & 8.68 & 120 & 188.8 & 23.7 \\ S8 & Top of tunnel at ZK94 + 245 & \(-\)130 & 9.15 & 99 & 154.9 & 23.8 \\ S9 & Walls at ZK94 + 188 & \(-\)153.3 & 9.63 & 122 & 189.6 & 23.7 \\ S10 & Walls at ZK94 + 198 & \(-\)105 & 8.7 & 123 & 192.2 & 24.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Water quality characteristics of ten water gushing samples. \[t(C_{1})=C_{1}^{4}=C_{1}^{8}=\left[ \begin{array}{cccccccccccc}1.0000&0.7530&0.7530&0.753 0&0.7530&0.8083&0.7974&0.6616&0.7865&0.7364\\ 0.7530&1.0000&0.9701&0.9100&0.9100&0.7530&0.7530&0.6616&0.7530&0.7364\\ 0.7530&0.9701&1.0000&0.9100&0.9100&0.7530&0.7530&0.6616&0.7530&0.7364\\ 0.7530&0.9100&0.9100&1.0000&0.9100&0.7530&0.7530&0.6616&0.7530&0.7364\\ 0.7530&0.9100&0.9100&0.9100&1.0000&0.7530&0.7530&0.6616&0.7530&0.7364\\ 0.8083&0.7530&0.7530&0.7530&0.7530&1.0000&0.7974&0.6616&0.7865&0.7364\\ 0.7974&0.7530&0.7530&0.7530&0.7530&0.7974&1.0000&0.6616&0.7865&0.7364\\ 0.6616&0.6616&0.6616&0.6616&0.6616&0.6616&0.6616&1.0000&0.6616&0.6616\\ 0.7865&0.7530&0.7530&0.7530&0.7530&0.7865&0.7865&0.6616&1.0000&0.7364\\ 0.7364&0.7364&0.7364&0.7364&0.7364&0.7364&0.7364&0.6616&0.7364&1.0000\\ \end{array} \right] \tag{17}\]  
[PERSON] et al. [38] set \(\lambda=0.6173\) for the classification of hydraulic fissures. Referring to Section 2 for the definition of \(\lambda\), it was more reasonable to set the optimal classification threshold of \(\lambda=0.6616\) in this paper. The classification results were shown in Figure 4.  
From Figure 4, the following can be seen: \(\lambda_{min}>0.6\), the ten samples in the tunnel were relevant, according to Section 2. When \(\lambda=0.6616\), the 10 water samples from the pushing in the tunnel could be classified into the same category. This result indicated that the 10 water samples in the tunnel were similar and that they originated from the same source. For comparison with four samples from around the tunnel, samples S1 and S10 were selected to further analyse whether the water gushing in the tunnel originated from the surface water system. For the second time, four samples of surface water in the tunnel site area were taken for analysis and compared with samples S1 and S10. Repeating the above three steps, the samples were characterised for their elemental content of Cl\({}^{-}\), SO\({}_{4}\)\({}^{2-}\), HCO\({}_{3}\)\({}^{-}\), Ca\({}^{2+}\), K\({}^{+}\), Mg\({}^{2+}\), and Na\({}^{+}\). The test results are shown in Appendix A. Similarly, \(Z_{2}^{\prime\prime}\) was obtained by preliminary processing of the data in Appendix A through Equation (1) to Equation (5). It was represented as follows:  
\[Z_{2}^{\prime\prime}=\left[ \begin{array}{ccccccccc}0.898&0.905&0.908&0.269&1.000 &1.000&1.000&0.919&0.536\\ 1.000&1.000&1.000&0.077&0.505&1.000&0.917&0.178&1.000&1.000\\ 0.583&0.054&0.042&0.000&0.001&0.035&0.008&0.000&0.000&0.243\\ 0.417&0.239&0.185&0.385&0.271&0.255&0.223&0.368&0.475&0.160\\ 0.000&0.000&0.000&1.000&0.030&0.000&0.196&0.071&0.000\\ 0.032&0.016&0.013&0.462&0.000&0.005&0.001&0.258&0.061&0.062\\ \end{array} \right]. \tag{18}\]  
Figure 4: Fuzzy clustering analysis results for 10 water gushing samples in the tunnel. Furthermore, the raw matrix \(C_{2}\) was represented as follows:  
\[C_{2}=\left[