## 4 Case Study
### Prediction of Water Gushing Based on the Effective Radius of the Dynamic
#### 4.3.2 Effective Radius of the Dynamic  
Substituting the recorded data into Equation (9) to Equation (13), the variation pattern of the \(R_{n}\) corresponding to each segment was obtained, as shown in Figure 7.  
Figure 6: Distribution of parameters by segments. (a) Ln and Hn; (b) \(\upmu\) and \(\upalpha\).  
Figure 7: Variation pattern of radius of influence Rn for each segment.  
As can be seen from Figure 7, \(R_{n}\) increased as \(T\) and \(H_{n}\) increased. The rate of increase of \(R_{n}\) was decreasing with an overall non-linear increase. In order to reflect the mathematical relationship better between the above three factors, a power function model was chosen for the regression analysis of \(R_{n}\) and \(H_{n}\), with the condition of a defined duration of water gushing \(T\). Here, \(R_{n}\) could be set as follows:  
\[R_{n}=a(H_{n})^{b}. \tag{21}\]  
The regression results of Equation (21) were shown in Figure 8. In order to present the relationship better between \(Rn\), \(T\), and \(H_{n}\) through mathematical expressions, power function representations and quadratic equations were used to represent the parameters \(a\) and \(b\). At this point, they could be set as follows:  
\[R_{n}=R(T,H_{n})=a_{1}T^{b_{1}}H_{n}^{a_{2}T^{2}+b_{2}T+c_{2}}. \tag{22}\]  
Based on the data in Figure 8, Equation (23) was derived by regression, as follows:  
\[\left\{\begin{array}{l}a_{1}=12.16\\ b_{1}=0.94\\ \end{array}\right.,R^{2}=0.99 \tag{23}\] \[\left\{\begin{array}{l}a_{2}=4.63\times 10^{-7}\\ b_{2}=-4.32\times 10^{-4}\\ c_{2}=0.15\end{array}\right.,R^{2}=0.99.\]  
Additionally, Equation (22) could be set as follows:  
\[R_{n}=R(T,H_{n})=12.16T^{0.9}H_{n}^{4.63\times 10^{-7}T^{2}-4.32\times 10^{-4}T+0.15},(T\leq 350). \tag{24}\]  
The relationship between \(R_{n}\), \(T\), and \(H_{n}\) was reflected by Equation (24). As the storage of water from bedrock fractures was limited, \(R_{n}\) did not increase indefinitely. Additionally, Equation (24) was used to calculate \(R_{n}\) only for one rainfall cycle after the occurrence of the water gushing. Equation (24) was substituted into Equation (9) to (13) and simplified to obtain Equation (25), as follows:  
\[Q=\sum_{n=1}^{N}\left(W_{n(1)}+W_{n(2)}\right)=\sum_{n=1}^{N}\left(12.16\mu H_ {n}T^{-0.1}+0.06576 ahT^{0.9}\right)L_{n}\times H_{n}^{4.63\times 10^{-7}T^{2}-4.32 \times 10^{-4}T+0.15}. \tag{25}\]  
The results of calculation, derived from Equation (25), were shown in Figure 9 as a comparison with dynamic monitoring data.  
Figure 8: \(R_{n}\) regression results.  
From Figure 9, it can be seen that the front end of the curve had a plummeting trend, with a relatively flat middle and a slight rise at the end. According to the amount of water gushing, the period of water gushing could be divided into three stages. The first stage was a time period of 1 to 6 days, and the stage had a negative growth rate of greater than 10% for a single day. The second stage was 7 to 170 days into a period of larger water gushing, with a negative growth rate of less than 10% in single day of water gushing. The third stage was 170 days after. This stage of water gushing increased very slowly, with a positive single day growth rate of greater than 0.01%.