## 5 Application Case
### Results of the Simulation
#### 5.3.2 Simulation Results of Hydraulic Head and Leakage  
For verifying the validity of the simulations and the reasonableness of the calculated results, the leaching parameters determined by the four algorithms were substituted into the finite element model. The variations in the hydraulic head at measurement point \(P_{4}^{\theta}\) and dam base leakage between 2013 and 2010 were calculated. The monitoring data documented the actual hydraulic head and dam foundation leakage changes in these eight years. The position of the piezometer \(P_{4}^{\theta}\) is presented in Figure 1.  
The leakage quantity of the dam foundation is roughly estimated by Equation (9). \[q_{b}=\iint vL_{0}dxdz \tag{9}\]  
where \(q_{b}\) is the leakage quantity of the dam foundation; \(v\) denotes the flow velocity perpendicular to the overflow surface; and \(L_{0}\) represents the total length of the dam, taken as 645 m.  
Figure 4(a) shows the comparison between the calculated and monitored heads at the piezometer \(P_{4}^{\theta}\). Linear fitting was carried out according to the observed values of the hydraulic head. The hydraulic heads determined by PSO show the best consistency with the monitoring data, while the results simulated by GA give the worst performance. The poor performance of GA may be due to the easy convergence of the algorithm to the local optimal solutions. The results of PSO embody the high accuracy in this analysis and demonstrate the validity of the simulation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inversion Parameter** & \(\Omega_{0}\) & \(n\) & \(\mathbf{k_{r}}\) \\  & & & (\(\mathbf{m/s}\)) \\ \hline GA & 9334 & 1029 & \(5.36\times 10^{-7}\) \\ SA & 13,583 & 1036 & \(1.53\times 10^{-7}\) \\ SSA & 12,756 & 1225 & \(1.48\times 10^{-7}\) \\ PSO & 12,253 & 1594 & \(8.83\times 10^{-8}\) \\ Range of parameter & 5000–17,000 & 500–1700 & \(1.0\times 10^{-8}\)–1.0 \\  & & & \(\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of the simulated parameters for four algorithms. Figure 5: Comparison of measured and simulated values at piezometer \(P_{4}^{\theta}\): (**a**) hydraulic head; (**b**) hydraulic head relative error. The relative error of the hydraulic head is calculated by Equation (10). \[\delta_{P}=\frac{P-\overline{P}}{\overline{P}}\times 100\% \tag{10}\]  
where \(\delta_{P}\) is the relative error of hydraulic head; \(P\) represents the monthly average of predicted hydraulic head; and \(\overline{P}\) means the monthly average of measured hydraulic head. The annual average head at the piezometer \(P_{4}^{9}\) was applied to calculate the relative error values. Figure 4(b) shows the relative error line graph of the four algorithms at the piezometer \(P_{4}^{9}\) from 2003 to 2010. The maximum value of relative error in the hydraulic head is \(-5\%\) in 2007. Among the four algorithms, PSO reflects the slightest relative error fluctuation. The hydraulic head relative error corresponding to PSO is between \(-3\%\) and \(3\%\). Figure 5(a) presents the comparison between the calculated and measured dam foundation leakage by the four algorithms. Similarly, a fitting curve was made according to the observed leakage of the dam foundation. The simulated results by PSO agree well with the measurements, illustrating the accuracy of the simulation. The results determined by SA and SSA show medium accuracy in the analysis. The simulated results by GA deviate furthest from the monitoring data. The relative error of leakage quantity is calculated by Equation (11). \[\delta_{Q}=\frac{Q-\overline{Q}}{\overline{Q}}\times 100\% \tag{11}\]  
where \(\delta_{Q}\) is the relative error of hydraulic head; \(Q\) means the annual average of predicted leakage; and \(\overline{Q}\) denotes the annual average of measured leakage. Relative error values were calculated using the annual average dam base leakage from the water measuring weir. Figure 5(b) shows the histogram of the relative error in dam base leakage determined by these algorithms from 2003 to 2010.