## 4 Case Study
### Fuzzy Clustering Method for Identification of the Source of Water Gushing  
Due to the high pressure and high flow of water gushing in the excavation phase of the tunnel, the source of water gushing in the tunnel needs to be understood urgently. It is important that the source of water gushing is accurately identified for predicting the amount of water gushing. Data from water samples can be analysed and processed efficiently by fuzzy clustering methods. Samples with a high degree of similarity between them will be classified into one category. These samples from around and inside the tunnel will be differentiated as to whether they originate from the same source. The igneous rocks are relatively rich in minerals. The physicochemical properties of the samples can be derived from tests. The relationship between the samples inside and around the tunnel can be determined with the use of fuzzy clustering analysis [37]. During construction, 14 water samples were taken for testing indoors and analysis, including 10 samples of water gushing in the tunnel for the first time, as well as 4 samples of surface water in the tunnel site area for the second time. The specific implementation steps of this study are as follows:  
Figure 3: Water gushing at the tunnel face. (**a**) Tunnel face; (**b**) the vicinity of tunnel face. Figure 2: Distribution of fracture structures in the tunnel site area (1:10,000). Step 1. Ten samples of water gushing were analysed by the hydro-geochemistry method. Testing indicators included redox potential, pH, TDS content, conductivity, and water temperature (sample water temperature was measured during field sampling). There were also quantitative elemental characteristics, such as Cl\({}^{-}\), SO\({}_{4}\)\({}^{2-}\), HCO\({}_{3}\)\({}^{-}\), Ca\({}^{2+}\), K\({}^{+}\), Mg\({}^{2+}\), and Na\({}^{+}\). Some test results for the 10 samples from the tunnel are shown in Table 1.  
Step 2. The test results were classified using the fuzzy clustering method. The data in Table 1 was initially processed by Equations (1) and (5) to obtain \(Z_{1}^{\prime\prime}\), as expressed in the following Equation (15):  
\[Z_{1}^{\prime\prime}= \begin{bmatrix}0.659&0.611&0.433&0.430&0.263\\ 0.993&0.035&0.933&0.959&0.474\\ 1.000&0.000&0.967&0.974&0.526\\ 0.764&0.399&1.000&1.000&0.368\\ 0.855&0.257&0.933&0.933&0.474\\ 0.798&0.340&0.600&0.605&0.000\\ 0.400&0.670&0.700&0.735&0.368\\ 0.168&0.833&0.000&0.000&0.421\\ 0.000&1.000&0.767&0.753&0.368\\ 0.349&0.677&0.800&0.809&1.000\\ \end{bmatrix} . \tag{15}\]  
Step 3. According to Equation (7) to Equation (8), \(Z_{1}^{\prime\prime}\) was processed for the first time to give the original matrix \(C_{1}\), which was shown in the following Equation (16):  
\[C_{1}=\left[ \begin{array}{cccccc}1.0000&0.5880&0.5641&0.6551&0.6556&0.8083&0.7 974&0.6616&0.6322&0.6081\\ 0.5880&1.0000&0.9701&0.8164&0.8928&0.6859&0.6190&0.2800&0.4228&0.5634\\ 0.5641&0.9701&1.0000&0.7993&0.9100&0.7252&0.7324&0.5692&0.5503&0.6761\\ 0.6551&0.8164&0.7993&1.0000&0.9100&0.7243&0.7530&0.3486&0.5796&0.6524\\ 0.6556&0.8928&0.9100&0.9100&1.0000&0.7252&0.7164&0.3483&0.5248&0.6482\\ 0.8083&0.6859&0.7252&0.7243&0.7252&1.0000&0.7324&0.4930&0.5424&0.5176\\ 0.7974&0.6190&0.7324&0.7530&0.7164&0.7324&1.0000&0.5692&0.7865&0.7364\\ 0.6616&0.2800&0.5692&0.3486&0.3483&0.4930&0.5692&1.0000&0.5503&0.4698\\ 0.6322&0.4228&0.5503&0.5796&0.5248&0.5424&0.7865&0.5503&1.0000&0.6761\\ 0.6081&0.5634&0.6761&0.6524&0.6482&0.5176&0.7364&0.4698&0.6761&1.0000\\ \end{array} \right]. \tag{16}\]  
The transitive closure \(t(C_{1})=C_{1}^{4}=C_{1}^{8}\) could be found for the 4-th time. Then, \(t(C_{1})\) was given in Equation (17), as follows: