## 3 Results and Discussion  
### Results Analysis  
The erosion resulting from water flowing through the discharge channel widens the break, while the initial discharge is minimal. As the breach becomes bigger, the amount of discharge gradually increases. Simultaneously, the outlet's erosion capacity contributes to  
Figure 8: Study area and topographic of Tangjiashan barrier dam. the swift enlargement of the breach. When the breach's depth reaches a critical threshold, the breach slope collapses due to instability, promptly causing further expansion of the breach and accelerating the evolution process. The Taum Sauk dam break is selected as an example in this paper. Firstly, the vertical comparison with the TRITON is used to verify the influence of the integrated model after adding the breach evolution module on the simulation accuracy of the dam break flood. Secondly, it is compared with the measured data and physical models DLBreach to verify the accuracy and efficiency of the newly proposed physics-based integrated model. Table 3 summarized and compared the observed values and simulated values of the five important parameters of the Taum Sauk dam break event, including the peak discharge of the breach (\(Q_{p}\)), the top width of the final breach (\(B_{f}\)), the depth of the breach (\(D_{f}\)), the arrival time of the maximum peak discharge (\(T_{p}\)), and the completion time of the final breach (\(T_{b}\)). Figure 9 shows the results of the breach flow processes of different models. It can be seen that the integration model has the best fit with the measured data, the peak discharge simulated by the TRITON distributed model is larger, and the arrival time of the maximum peak discharge simulated by the DLBreach model is delayed. As shown in Table 3, among the five important parameters, the relative errors of the other four parameters are within \(\pm 10\%\), except for the dam break completion time (\(T_{b}\)), which has a large relative error. The simulation accuracy of the peaking flow of the integrated model is higher than that of the distributed model, and the arrival time of the peaking flow is better than that of other models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & **Measured** & **Distributed** & **Relative** & **Integration** & **Relative** & **DLBreach** & **Relative** \\  & **Data** & **Simulation** & **Results** & **Error** & **Results** & **Error** & **Error** \\ \hline \(Q_{p}\) & \multirow{2}{*}{8180} & \multirow{2}{*}{9362.7} & \multirow{2}{*}{21.11\%} & \multirow{2}{*}  
Figure 10 shows the measured shape of the final breach after the Taum Sauk dam failure and the calculated breach development by the integrated simulation model. As shown in Table 3 and Figure 10, the length and shape of the final breach calculated by the simulation model fit the actual breach well, and the relative error of the top width of the final breach is 5.38%. The formation time of the final breach simulated by the integrated model is 32 min, which is 6 min longer than the actual formation time, and the relative error is 28.00%. The relative error is within the acceptable range, indicating that the integrated model is acceptable. The Tangjiashan landslide dam is selected as another example in this paper. The performance of the integrated model was assessed by comparing and analyzing the simulation results of breach discharge, breach breadth, and breach bottom elevation with both the actual data and the simulation results of DB-IWHR [22]. Figure 11 displays the discharge of water flowing out of the breach, as well as the variations in the width and bottom elevation of the breach. The integrated model is capable of accurately representing the discharge process of the breach and the changes in breach and bottom elevation of the breach. When comparing the DB-IWHR model and actual data, the integrated model demonstrates superior ability to accurately mimic the regression process of the flow, as shown in Figure 11a.