# Inverse Modeling of Seepage Parameters Based on an Improved Gray Wolf Optimizer
## 4 A Casebook Study
### Results of the Simulation
#### 4.4.3 Leakage Volume of the Dam Foundation  
The seepage volume is estimated by the flow velocity and the overflow surface. The formula for calculating leakage volume is given in Equation (15).  
\[Q_{i}=\iint vBdxdz \tag{15}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Material**} & **Hydraulic Conductivity** & **Search Range** \\  & **(m/s)** & **(m/s)** \\ \hline Grout curtain & 3.00 \(\times\) 10\({}^{-5}\) & 1.00 \(\times\) 10\({}^{-6}\)–1.00 \(\times\) 10\({}^{-4}\) \\ The top cover layer & 5.33 \(\times\) 10\({}^{-2}\) & 1.00 \(\times\) 10\({}^{-3}\)–1.00 \(\times\) 10\({}^{-1}\) \\ The second cover layer & 2.67 \(\times\) 10\({}^{-3}\) & 1.00 \(\times\) 10\({}^{-4}\)–1.00 \(\times\) 10\({}^{-2}\) \\ Moderately-weathered zone & 5.50 \(\times\) 10\({}^{-4}\) & 1.00 \(\times\) 10\({}^{-5}\)–1.00 \(\times\) 10\({}^{-3}\) \\ Slightly weathered zone & 1.10 \(\times\) 10\({}^{-4}\) & 1.00 \(\times\) 10\({}^{-5}\)–1.00 \(\times\) 10\({}^{-3}\) \\ Fresh bedrock zone & 3.10 \(\times\) 10\({}^{-5}\) & 1.00 \(\times\) 10\({}^{-6}\)–1.00 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Hydraulic conductivity of each medium determined by IGWO.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Monitoring**} & **Measured** & **Simulated** & **Absolute Error** & **Relative Error** \\  & **Hydraulic Head** & **Hydraulic Head** & **(m)** & **(\%)** \\  & **(m)** & **(m)** & **(m)** & **(\%)** \\ \hline \(P_{DB-13}\) & 2736.17 & 2735.74 & \(-\)0.43 & 0.29 \\ \(P_{DB-24}\) & 2718.67 & 2717.51 & \(-\)1.16 & 0.78 \\ \(P_{DB-25}\) & 2718.28 & 2717.16 & \(-\)1.12 & 0.76 \\ \(P_{DB-26}\) & 2718.21 & 2716.80 & \(-\)1.41 & 0.95 \\ \(P_{DB-27}\) & 2716.61 & 2716.24 & \(-\)0.37 & 0.25 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison between the calculated and measured hydraulic head values.  
where \(Q_{i}\) is the simulated seepage volume; \(v\) means the flow rate at the vertical spillway surface; \(B\) represents the length of the dam taken as 355 m.  
The relative error for the leakage volume is determined by Equation (16).  
\[\delta_{Q}=\frac{|Q_{i}-Q|}{Q}\times 100\% \tag{16}\]  
where \(\delta_{Q}\) is the relative error of the dam body leakage; \(Q_{i}\) and \(Q\) are the simulated and measured values of the dam body leakage, respectively.  
Table 6 shows the comparison of the calculated and measured values of the dam body leakage during the stable upstream water level. The simulated leakage values are in good agreement with the actual measurements, showing the accuracy of the IGWO strategy and the reliability of the simulation.  
The leakage measurements are averaged over the simulation period. The relative error of the calculated seepage volume is 5.00%, demonstrating the positive performance of IGWO in the simulation.  
Figure 8 presents the contour map of the hydraulic head at the dam site. The distribution of contours is in accordance with seepage characteristics. The results show the accuracy of the simulation and the reasonableness of the calculated hydraulic conductivity.