## 2 Materials and Methods
### Comparison  
In this part, the method proposed in this paper is compared with the existing physical mechanism model method to highlight the innovation of this paper. To calculate the breach outburst flow, existing models usually use the weir formula. However, the integrative model is proposed in this paper, the water depth and flow velocity of the previous time are used to drive the evolution of the outburst in the next time step, and then the flow of the breach is calculated. In the judgement of initial erosion, existing models are mostly based on the flow velocity or shear stress. In this paper, the critical flow velocity formula, with and without seepage force, is derived by analyzing the force of a soil particle on the slope. To calculate the breach lateral evolution, existing models are mostly assumed in a linear relationship between the lateral expansion and vertical erosion of the breach. In this paper, the lateral evolution formula is derived from the theoretical analysis of the initial sediment movement in the dam slope. At the same time, the breach side slope stability module is considered in this paper, which is not considered in some existing models. The comparison between the integration model and other common simplified physical models is shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Models** & \begin{tabular}{c} **Breach Flow** \\ **Calculation** \\ \end{tabular} & **Breach Vertical Erosion** & \
\begin{tabular}{c} **Breach Lateral** \\ **Evolution** \\ \end{tabular} &
\begin{tabular}{c} **Breach Side Slope** \\ **Stability** \\ \end{tabular} \\ \hline
1 & DAMBRK [53] & Weir formula & Assumed linear erosion & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} No consider breach \\ stability \\ \end{tabular} \\ \hline
2 & HR BREACH [16] & \begin{tabular}{c} Variable weir plus 1D \\ steady nonuniform \\ equation \\ \end{tabular} & \begin{tabular}{c} Various equations, \\ noncohesive and \\ cohesive soils \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline
3 & WinDAM [18] & Weir formula & \begin{tabular}{c} Headcut advance, \\ bottom and lateral \\ erosion \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} No consider breach \\ stability \\ \end{tabular} \\ \hline
4 & MIKE DB [28] & Weir formula & \begin{tabular}{c} Engelund-Hansen bed \\ load formula \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline
5 & DLBreach [19] & Weir formula & \begin{tabular}{c} Exponential erosion \\ model \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline
6 & DB-IWHR [22] & Weir formula & \begin{tabular}{c} The hyperbolic erosion \\ model \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline
7 & NHRI-DB [20] & Weir formula & \begin{tabular}{c} Exponential erosion \\ model \\ \end{tabular} & \
\begin{tabular}{c} Linearly related to \\ vertical erosion \\ \end{tabular} &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline
8 & Integration model & \begin{tabular}{c} Direct calculation by \\ integration model \\ \end{tabular} & \
\begin{tabular}{c} The hyperbolic erosion \\ model \\ \end{tabular} & Newly proposed &
\begin{tabular}{c} Consider breach \\ stability \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: The comparison between the integration model and existing models.