Modeling Flood Peak Discharge Caused by Overtopping Failure of a Landslide Dam

: Overtopping failure often occurs in landslide dams, resulting in the formation of strong destructive ﬂoods. As an important hydraulic parameter to describe ﬂoods, the peak discharge often determines the downstream disaster degree. Based on 67 groups of landslide dam overtopping failure cases all over the world, this paper constructs the calculation model for peak discharge of landslide dam failure. The model considers the inﬂuence of dam erodibility, breach shape, dam shape and reservoir capacity on the peak discharge. Finally, the model is compared with the existing models. The results show that the new model has a higher accuracy than the existing models and the simulation accuracy of the two outburst peak discharges of Baige dammed lake in Jinsha River (10 October 2018 and 3 November 2018) is higher (the relative error is 0.73% and 6.68%, respectively), because the model in this study considers more parameters (the breach shape, the landslide dam erodibility) than the existing models. The research results can provide an important reference for formulating accurate and effective disaster prevention and mitigation measures for such disasters.


Introduction
Landslide is a common mountain hazard, which is widely distributed in the steep mountain gorge area [1,2]; it often has the characteristics of high location, high speed, long distance of movement [3]. In particular, the frequency of large-scale landslides is higher under the action of heavy rainfall or high-intensity earthquake [1]. When there are rivers in the direction of landslide movement, the landslide is easy to accumulate into the river and form a dam to block up the upstream water level. Once a dam failure occurs, it will also cause a huge flood disaster in the downstream and enlarge the scope and scale of the hazard [2,[4][5][6][7][8][9][10][11]. For example, in 1786, a strong M = 7.75 earthquake in Luding-Kangding area, Sichuan Province, southwestern China, formed a landslide dam and the flood caused by the landslide dam failure killed more than 100,000 people [12]. In addition, Yigong Landslide dammed lake (9 April 2000) [13], Tangjiashan landslide dammed lake (12 May 2008) [14] and Baige landslide dammed lake (10 October 2018 and 3 November 2018) [15] occurred in recent years, which caused serious life and property losses of upstream reservoir inundation and downstream extraordinary outburst flood.
Landslide dam is a kind of natural earth rock dam without special design and specific spillway. Its geometry, material composition and internal structure are significantly different from those of artificial earth rock dam [16] and its failure probability is much higher than that of artificial earth rock dam [17][18][19]. According to statistical data [19], the service life of landslide dams ranges from a few minutes to thousands of years, with 87%, 83%, 71%, 51% and 34% of them less than one year, half a year, one month, one week and one

Landslide Dam Failure Data
Based on the extensive collection of relevant literature, this paper obtains 67 groups of historical landslide dam failure data all over the world (Appendix A Table A1), including 45 groups of historical landslide dam failure data summarized in Peng et al. [19]. In Table A1, dam erodibility indicates the resistance of dam materials to the erosion action of water flow. Briaud (2008) [47] proposed six erosion categories (very high, high, medium, low, very low and non-erosive) according to erosion rate and breach velocity or flow shear stress. The higher the dam erodibility is, the greater the peak discharge is. However, the data of erosion rate, velocity and shear stress cannot be obtained before the dam break. Therefore, this paper refers to Zhang et al. (2019) [9] and Cui et al. (2008) [48] and divides the dam erodibility into three categories (high, medium, low) according to the particle composition of the landslide dam. The classification criteria are: when the landslide dam is dominated by large stones, it is low erodibility (L); when the landslide dam is dominated by soil, it is high erodibility (H); when the landslide dam contains a large number of stones and soil, it is medium erodibility (M). The meanings of other parameters are shown in Figure 1.

Landslide Dam Failure Data
Based on the extensive collection of relevant literature, this paper obtains 67 groups of historical landslide dam failure data all over the world (Appendix A Table A1), including 45 groups of historical landslide dam failure data summarized in Peng et al. [19]. In Table A1, dam erodibility indicates the resistance of dam materials to the erosion action of water flow. Briaud (2008) [47] proposed six erosion categories (very high, high, medium, low, very low and non-erosive) according to erosion rate and breach velocity or flow shear stress. The higher the dam erodibility is, the greater the peak discharge is. However, the data of erosion rate, velocity and shear stress cannot be obtained before the dam break. Therefore, this paper refers to Zhang et al. (2019) [9] and Cui et al. (2008) [48] and divides the dam erodibility into three categories (high, medium, low) according to the particle composition of the landslide dam. The classification criteria are: when the landslide dam is dominated by large stones, it is low erodibility (L); when the landslide dam is dominated by soil, it is high erodibility (H); when the landslide dam contains a large number of stones and soil, it is medium erodibility (M). The meanings of other parameters are shown in Figure 1.

Calculation of Breach Size
Previous studies have shown that the width of the breach is most closely related to the storage capacity of the barrier lake [49][50][51]. Therefore, this study assumes that the width of the breach bottom is proportional to the 1/3 power of the storage capacity and then the expression of the final width of the breach (Wb) with the same dimension can be obtained: where λ is a parameter determined by the erodibility of landslide dam; Vl is the capacity of barrier lake. Secondly, Peng et al. [19] pointed out that the breach depth is closely related to the height of the barrier dam and the capacity of the barrier lake. Therefore, this paper uses the formula given by Peng et al. [19] for reference and assumes that the breach depth can be expressed as: where, ξ, α and β are parameters determined by the erodibility of landslide dam, which reflects the influence of the erodibility of landslide dam on the depth of breach; Hr = 1 m; Vl is barrier lake capacity; hb is breach depth; Hd is landslide dam height. A total of 67 groups data of landslide dam failure in Table A1 are fitted manually in a graph and analyzed by using Formulas (1) and (2) and the corresponding λ, ξ, α and β  Previous studies have shown that the width of the breach is most closely related to the storage capacity of the barrier lake [49][50][51]. Therefore, this study assumes that the width of the breach bottom is proportional to the 1/3 power of the storage capacity and then the expression of the final width of the breach (W b ) with the same dimension can be obtained: where λ is a parameter determined by the erodibility of landslide dam; V l is the capacity of barrier lake.
Secondly, Peng et al. [19] pointed out that the breach depth is closely related to the height of the barrier dam and the capacity of the barrier lake. Therefore, this paper uses the formula given by Peng et al. [19] for reference and assumes that the breach depth can be expressed as: where, ξ, α and β are parameters determined by the erodibility of landslide dam, which reflects the influence of the erodibility of landslide dam on the depth of breach; H r = 1 m; V l is barrier lake capacity; h b is breach depth; H d is landslide dam height. A total of 67 groups data of landslide dam failure in Table A1 are fitted manually in a graph and analyzed by using Formulas (1) and (2) and the corresponding λ, ξ, α and β of three kinds of landslide dam erodibility are obtained, as shown in Table 1. In the future, new landslide cases should be added to further modify these parameters. Table 1. Calculation parameters of peak discharge of landslide dam with different erodibility. "H" refers to the high erodibility of the landslide dam, "M" refers to the moderate erodibility of the landslide dam, "L" refers to the low erodibility of the landslide dam. In order to consider the impact of breach shape on the peak discharge, this paper uses the semi analytical model proposed by Wang et al. [52] for reference. In this model, the cross-section shape of the breach is generalized as a trapezoid ( Figure 1b) and then combinative parameter of the cross section is calculated according to Equation (3).

Dam Erodibility
where q is combinative parameter of the cross section of breach, which is a parameter reflecting the shape of the breach. W b is the bottom width of landslide dam breach. M l and M r are the slope ratio of left and right side of the breach, respectively; when landslide dam is overtopping failure, M l = M r = 1.0. Then, the characteristic parameter (W u ) of final breach water depth is calculated by Equation (4).
Then, according to Equation (5), the characteristic parameter (W) of the breach water depth is calculated.
where G(W u ) and G(W) are two functions of W u and W, respectively. x* is the dimensionless distance, x* = 0 at the breach. The expressions of G(W u ) and G(W) are as follows: According to Equations (4)-(7), the characteristic parameter (W) of breach water depth can be obtained, but the solution process needs to be obtained by trial calculation method, so the implicit expression of characteristic parameter (W) of breach water depth is transformed into approximate explicit expression (Equation (8).
In order to illustrate the accuracy and rationality of Formula (8), a point is taken every 0.01 interval in the interval W u ∈ [0.02, 3.00] and then Formulas (4)-(7) and Formula (8) are used to calculate the characteristic parameter (W) of breach water depth and then the calculation error of Formula (8) is compared. The results show that the average relative error is 0.067% and the maximum relative error is only 0.259%. In addition, Formula (8) is used to calculate 67 groups of landslide dam failure cases in Table A1. The minimum value of initial breach water depth characteristic parameter W u is 0.28 and the maximum value is 1.53, which is far less than the calculation interval in Formula (8), so Formula (8) can meet the actual demand.
Finally, according to Formula (9), the peak discharge (Q p ) of landslide dam failure is calculated.
where Q p is the peak discharge of landslide dam failure, m 3 /s; g is the acceleration of gravity, m/s 2 .

Model Verification and Comparison
As the height of water level drop (d) and the potential energy (PE) of water are difficult to be measured in the process of dam failure, the calculated formula of peak discharge of landslide dam failure in Table 2 is selected for comparison with the model of this paper. The calculated results of each calculated model are shown in Figure 2. Table 2. Empirical formula for peak discharge of landslide dam burst flood.

Empirical Formula
Note References Peng and Zhang (2012) [19] It can be seen from Figure 2 that the reliability of the model proposed in this study and the model proposed by Peng et al. [19] is significantly higher than that of other models; compared with the model proposed by Peng et al. [19], when the measured peak discharge of landslide dam failure is less than 10,000 m 3 /s, the calculated peak discharge of this study model is similar to that of Peng et al. [19]. When the measured peak discharge of landslide dam failure is greater than 10,000 m 3 /s, the calculated peak discharge of this study model is closer to the optimal line than that of Peng et al. [19].
The mean relative error (MRE), root mean square error (RMSE) and coefficient of determination (R 2 ) of the calculated peak discharge are further used to illustrate the calculated effect of the above model. The MRE reflects the average reliability or the average error rate of the calculated value. The RMSE reflects the precision of the calculated value because it is particularly sensitive to maximum and minimum values. The R 2 reflects the correlation between the calculated value and the measured value. The simulation will be more accurate with smaller MRE and RMSE and larger R 2 . MRE, RMSE and R 2 are as follows: where Q cal,i is the simulation peak discharge of the ith landslide dam failure case (i = 1, 2, 3); Q obs,i is the measured peak discharge of the ith landslide dam failure case (i = 1, 2, 3,...), N = 67 in this paper; Q obsm is the mean of the measured peak discharges. According to the calculated results (Table 3), MRE, RMSE and R 2 of this model are smaller than those of other models, indicating that the calculated effect of this model is the best. It can be seen from Figure 2 that the reliability of the model proposed in this study and the model proposed by Peng et al. [19] is significantly higher than that of other models; compared with the model proposed by Peng et al. [19], when the measured peak discharge of landslide dam failure is less than 10,000 m 3 /s, the calculated peak discharge of this study model is similar to that of Peng et al. [19]. When the measured peak discharge of landslide dam failure is greater than 10,000 m 3 /s, the calculated peak discharge of this    (Table 4).         This study model and models in Table 2 are used to simulate the peak discharge of two landslide dams. The simulated results are shown in Table 5. It can be seen from Table 5 that the simulated values of the peak discharge of this study model and Peng et al. [19] are more reasonable, but the simulated effect of this study model is still significantly better than that of Peng et al. [19], while there is a large gap between the simulated results of other models and the measured peak discharge, especially for the landslide dam failure event on 3 November, the simulated relative error is more than 90%. In addition, there are already some empirical simple models and physical or numerical complex models in landslide dam-failure [2,8,10,15,18,21,24,33,39,50,[52][53][54][55], while our proposed model is halfway, representing a promising compromise that needs to be further tested to new case studies.

Conclusions
The purpose of this study is to obtain the calculated model for peak discharge of landslide dam failure based on 67 historical landslide dam failure cases all over the world.
The key conclusions are: (1) The calculated model for peak discharge of landslide dam failure is proposed, which can consider the dam erodibility, the final shape of the breach, the shape of the dam, the lake volume and other parameters at the same time.
(2) Compared with other models, the calculated peak discharge of this model are more close to measured peak discharge. In addition, the model needs 12 parameters to calculate the breach depth, breach bottom width, breach top width and breach peak discharge, whereas the model proposed by Peng et al. needs 17 parameters. (3) The model proposed in this paper is used to simulate the peak discharge of two dam failure events of Baige landslide in Jinsha River (10 October 2018 and 3 November 2018). The relative error of peak discharge simulation of the two events is 0.73% and 6.68%, respectively, which is obviously better than other models, indicating that the simulated effect of the model is reasonable.
Finally, new cases of landslide dam failure should be added in the future study to further modify the model. This study can provide an important theoretical reference for disaster prevention and mitigation of landslide dam.

Institutional Review Board Statement:
The study did not involve humans or animals.

Informed Consent Statement:
The study did not involve humans.

Data Availability Statement:
The study did not report any data. All data used in the study are from Table A1.