عنوان مقاله [English]
Introduction: Dam failure and its flooding is one of the destructive phenomena today. Therefore, estimating the peak outflow (QP) with reasonable accuracy and determining the related flood zone can reduce risks. Qp of dam failure depends on important factors such as: depth above breach (Hw), volume of water above breach bottom at failure (Vw), reservoir surface area (A), storage (S) and dam height (Hd). Various researchers have proposed equations to estimate QP. They used the regression method to obtain an appropriate equation. Regression is a mathematical technique that requires initial test and diagnosis. These researchers present a new regression model for a better estimation of Qp.
Materials and Methods: The data used in this study are related to 140 broken dams in the world for 34 of which sufficient data are available for analysis. Dam failure phenomenon is a rapidly varied unsteady flow that is explained by shallow waters equations. The equations in the one-dimensional form are known as Saint-Venant equations and are based on hydrostatic pressure distribution and uniform flow under rectangular steep assumption. Although hydraulic methods to predict the dam failure flood have been developed by different software, due to the complex nature of the problem and the impossibility of considering all parameters in hydraulic analysis, statistical methods have been developed in this field. Statistical methods determine the equations that can approximate the required factors from the observed parameters. Multiple regression is a useful technique to model effective parameters in Qp, which can examine the statistical aspects of the model. This work is done by different tests, such as the model coefficients necessity test, analysis of variance table and it creates confidence intervals. Data analysis in this paper is done by SPSS 16 software. This software can provide fit model, various characteristics and related tests in the Tables.
Results and Discussion:This paper proposes a new relationship with better estimation of discharge peak (Qp) based on Hw and Vw factors. Results showed how to choose the appropriate model (fitting the model) and the initial required tests, according to the diagnostic model. And it compares the estimated error (relative efficiency) of the researchers’ models with the proposed models. The number of models can be classified to three convenient linear, multiplicative and transformed bases on Vw, Hw and Qp (nonlinear terms Qp). The best models for each of the three models were selected. Their corrected determination coefficients (Adj R2) are close together and are between 0.86 until 0.864. The relative efficiency criteria based on the root mean square error (RMSE) was used to determine the best model. This standard was also used for other researchers’ models. RMSE of the three models presented in this article is lower than that of other models (from 745 to 759). Diagnostics analysis of the three models is not possible due to the large volume, so some statistical analysis for the model 2 are presented in detail. The results are given in the following Tables. Test level has been assumed to be 5%. From the point view of hydraulics, it can be said that the final equation for Qp should be proportional to Hw 1.5. So although the model (2) has the lowest RMSE, but the model (3) of the hydraulics viewpoint seems more logical and its RMSE is not very different from the model (2), so this model can be selected as the best model. Figure 1 show diagnostics diagrams of model (3). The right Figure shows the homogeneity of residuals (follow the normal law) as a histogram. This homogeneity is confirmed by the crouch graph (center Figure). The left graph shows the stabilization of residual variance. According to the preliminary and diagnostics tests results, the model (3) has been selected. Its determination coefficient (0.864) also shows good strength.
Table 1- Top models presented in this research
Table 2- Statistical characteristics of the proposed models
model Adjusted R Square Durbin
Watson F VIF Std.
Residual Cook's Distance Centered Leverage
1 0.862 1.716 104.383 1.283 [-1.975 , 2.908] [ 0,0.569] [0,0.363]
2 0.860 1.744 102.545 1.283 [-1.824 , 2.834] [0,0.608] [0,0.363]
3 0.864 1.687 211.048 1 [-2.202 , 2.699] [0,0.527] [0,0.335]
Figure 1- Model 3 diagnostics pattern diagrams: histogram (right), crouch diagram (middle) the estimated residuals (left)
Conclusion: In this study, data from 140 broken dams were used to provide an appropriate model for estimating the peak outflow of dam failure. Standard statistical principles including preliminary tests, diagnostic and the efficiency of the models are the innovations of this paper. Analysis showed that the three models are competitive, and that the best of them was selected. The determined coefficient of these models was from 0.86 to 0.864 ranges. Relative efficiency was calculated by the RMSE index. The results showed that these models are more accurate than the models presented by other researchers. The model (3) was presented in this research, the best result was estimated for Qp and its error was less than the other models.