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Article

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

by 1,2, 1,2,3,*, 1,2, 1,2,3 and 1,2
1
Key Laboratory of Mountain Hazards and Surface Processes, CAS/Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, China
2
University of Chinese Academy of Science, Beijing 100000, China
3
CAS Center for Excellence in Tibetan Plateau Earth Sciences, Beijing 100101, China
*
Author to whom correspondence should be addressed.
Academic Editors: Francesco Fiorillo and Samuele Segoni
Water 2021, 13(7), 921; https://doi.org/10.3390/w13070921
Received: 2 March 2021 / Revised: 19 March 2021 / Accepted: 24 March 2021 / Published: 27 March 2021
(This article belongs to the Section Hydrology)

Abstract

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.
Keywords: peak discharge prediction; landslide dams; floods; overtopping failure; calculation model peak discharge prediction; landslide dams; floods; overtopping failure; calculation model

1. 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 day, respectively. Most of the cases (91%) were overtopping and only a few landslide dams were piping failure (8%) and slope instability. This kind of “short life, multiple overtopping failure” nature creates the high risk of a landslide dam, which is closely related to the shape, material composition, internal structure, main river shape and hydraulic characteristics of a landslide dam [2,8,10,19]. Therefore, the failure mechanism of landslide dams has become an important scientific problem.
Several numerical models of overtopping dam break based on physical processes have been proposed, such as DAMBRK model [20], BREACH model [21], BEED model [22], BRES model [23], Tingsanchali and Chinnarasri’s model [24], HR-BREACH [25], SIMBA model [26,27], Wang and Bowles’s model [28], BRES-Zhu model [29], Faeh’s model [30], Wu and Wang’s model [31] and Gradual dam breaches model [32]. However, the above model is almost for artificial earth rock dam and it is difficult to be applied to landslide dam with complex natures. In addition, Zhong et al. [8] established a water soil coupling mathematical model to simulate the overtopping failure process of Tangjiashan dam. Chang and Zhang [33] proposed a landslide dam overtopping failure model based on physical process to simulate the effect of soil erodibility changing with depth on the erosion process. Shen et al. [2] considered not only the change of soil erodibility with depth, but also the erosion mode (unilateral dam failure and bilateral dam failure) and different material composition of landslide dam. Cao et al. [34] also established a two-dimensional model of landslide dam failure based on the traditional shallow water dynamic equation. Even so, the computational efficiency is greatly reduced and the reliability of the results is low due to the difficulties in obtaining the parameters required by the model, controlling the boundary conditions and over simplification of the model. Therefore, a rapid dam failure risk assessment method is needed.
As one of the most important parameters in the process of dam failure, the peak discharge of dam failure often determines the downstream disaster degree and the reliability of its prediction results determines the accuracy and effectiveness of disaster prevention and mitigation measures [35]. The simulation of flood peak discharge has been studied as early as the 1980s. At present, a large number of simulation models have emerged, such as Kirkpatrick’s model [36], Macdonald and Langridge-Monopolis’ model [37], Singh and Snorrason’s model [38], Costa’s model [39], Costa and Schuster’s model [17], Froehlich’s model [40], Webby’s model [41], Walder and O’Connor’s model [42], Pierce et al. model [43], Peng and Zhang’s model [19] and Hakimzadeh et al. model [44]. Among them, most of the models are only applicable to earth rock dams and most of the formulas are single independent variables, so the calculation results are uncertain and the existing models do not consider the impact of the breach shape on the peak discharge. In addition, because the breach flow is a strong unsteady flow when the dam failure, the upstream water level is complex and difficult to describe quantitatively, the height of water level drop (d) and the potential energy of the water body in the process of dam break (PE) is difficult to be measured, so the formula containing this parameter still has strong limitations. In fact, the breach shape determines the cross-section area of the outburst flood; the dam erodibility determines the duration of the outburst flood [45,46], so these two parameters still have an important impact on the peak discharge of the dam break flood.
The purpose of this study is to solve the shortcomings of the existing researches, such as the low simulation accuracy of the peak discharge of landslide dam failure and the incomplete consideration of the parameters. According to 67 groups of historical landslide dam failure cases with relatively complete data, the calculation model of the peak discharge of landslide dam failure is constructed on the basis of considering the final breach shape, dam body shape, dam erodibility and reservoir capacity and compared with the existing models. The Baige dammed lake (98°41′52.15″ E, 31°4′54.91″ N) outburst event in 2018 is used to further verify it. The model can provide reference for disaster prevention and mitigation of landslide dam failure.

2. Materials and Methods

2.1. 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.

2.2. Modeling Dam-Break Peak Discharge

2.2.1. 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:
W b = λ V l 1 / 3
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:
h b H r = ξ ( H d H r ) α ( V l 1 / 3 H d ) β
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 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.

2.2.2. Calculation of Peak Discharge of Dam Failure

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).
q = W b M l + M r
where q is combinative parameter of the cross section of breach, which is a parameter reflecting the shape of the breach. Wb is the bottom width of landslide dam breach. Ml and Mr are the slope ratio of left and right side of the breach, respectively; when landslide dam is overtopping failure, Ml = Mr = 1.0.
Then, the characteristic parameter (Wu) of final breach water depth is calculated by Equation (4).
W u = h b 2 q
Then, according to Equation (5), the characteristic parameter (W) of the breach water depth is calculated.
W W u = G ( W u ) x * G ( W ) + ( 2 W 2 + 1 W 2 + 1 ) 1 / 2
where G(Wu) and G(W) are two functions of Wu and W, respectively. x* is the dimensionless distance, x* = 0 at the breach.
The expressions of G(Wu) and G(W) are as follows:
G ( W u ) = 2 2 [ 1 275 2 10 arctan W u W u 19 2 10 ( 1 W u 2 + 1 ) 1 2 9 ( 1 W u 2 + 1 ) 2 ]
G ( W ) = 2 2 [ 1 275 2 10 arctan W W 19 2 10 ( 1 W + 1 ) 1 2 9 ( 1 W + 1 ) 2 ]
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).
{ W = 0.6339 W u 0.9809 exp ( 0.1551 W u ) 0.00036 0.02 W u 0.45 W = 0.7513 W u 1.1650 exp ( 0.0475 W u ) + 0.01960 0.45 < W u 3.00
In order to illustrate the accuracy and rationality of Formula (8), a point is taken every 0.01 interval in the interval Wu ∈ [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 Wu 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 (Qp) of landslide dam failure is calculated.
Q p = 1.64 2 g q 5 ( M l + M r ) W 3 ( W 2 + 1 ) 3 2 W 2 + 1
where Qp is the peak discharge of landslide dam failure, m3/s; g is the acceleration of gravity, m/s2.

3. Results and Discusions

3.1. 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.
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 m3/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 m3/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 (R2) 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 R2 reflects the correlation between the calculated value and the measured value. The simulation will be more accurate with smaller MRE and RMSE and larger R2. MRE, RMSE and R2 are as follows:
M R E = 1 N i = 1 N | Q c a l , i Q o b s , i | Q o b s , i
R M S E = 1 N i = 1 N ( Q c a l , i Q o b s , i ) 2
R 2 = 1 i = 1 N ( Q c a l , i Q o b s , i ) 2 i = 1 N ( Q o b s , i Q o b s m ) 2
Q obsm = 1 N i = 1 N Q o b s , i
where Qcal,i is the simulation peak discharge of the ith landslide dam failure case (i = 1, 2, 3); Qobs,i is the measured peak discharge of the ith landslide dam failure case (i = 1, 2, 3,...), N = 67 in this paper; Qobsm is the mean of the measured peak discharges. According to the calculated results (Table 3), MRE, RMSE and R2 of this model are smaller than those of other models, indicating that the calculated effect of this model is the best.

3.2. Example Application

On 10 October 2018 and 3 November 2010, a large-scale landslide occurred in Baige (98°41′52.15″ E, 31°4′54.91″ N) within the boundaries of Jiangda County in Tibet and Baiyu County in Sichuan Province, blocking the main stream of Jinsha River and forming a barrier lake (Figure 3). After that, overtopping failure occurred (Figure 4), causing serious threat and loss to the upstream and downstream affected areas. Zhang et al. [9] investigated the two dam failure events of Baige landslide and obtained detailed dam failure data (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.

4. 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.

Author Contributions

Conceptualization, H.R., H.C. and T.W.; methodology, H.C., T.W. and J.C.; validation, H.R., H.C. and H.L.; formal analysis, H.R. and H.C.; investigation, H.R., J.C., T.W. and H.L.; writing—original draft preparation, H.R.; writing—review and editing, H.C. and T.W.; supervision, H.C.; project administration, H.C.; funding acquisition, H.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Key Research Program of Frontier Sciences, CAS (grant QYZDY-SSW-DQC006), the National Natural Science Foundation of China (grant 41731283, 41771045). and Foundation of Youth Innovation Promotion Association, CA S (grant 2017425).

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.

Conflicts of Interest

All authors declare no conflict of interest.

Abbreviations

HdDam height
V1Volume of dammed lake
WdDam width
VdVolume of landslide dam
hbBreach depth
WtBreach top width
WbBreach bottom width
QpPeak discharge of landslide dam failure
PEPotential energy of water
dHeight at which the water level drops during a dam outbreak
gAcceleration of gravity
aErodibility coefficient of landslide dam
HHigh erodibility of the landslide dam
MModerate erodibility of the landslide dam
LLow erodibility of the landslide dam
HrUnit length, Hr = 1 m
qCombinative parameter of the cross section of breach
MlSlope ratio of left side of the breach
MrSlope ratio of right side of the breach
WuCharacteristic parameterof initial breach water depth
WCharacteristic parameter of the breach water depth
λParameter determined by the erodibility of landslide dam
ξParameter determined by the erodibility of landslide dam
αParameter determined by the erodibility of landslide dam
βParameter determined by the erodibility of landslide dam
MREMean relative error
RMSEStandard deviation
R2Coefficient of determination

Appendix A

Table A1. Historical case data of landslide dam failure. “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.
Table A1. Historical case data of landslide dam failure. “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.
No.NameLocationData of
Formation
Dam Height
Hd (m)
Dam Width
Wd (m)
Dam Volume Vd (×106 m3)Lake Volume Vl (×106 m3)Dam
Erodbility
Breach Depth hb (m)Breach Top Width Wt(m)Breach Bottom Width Wb(m)Peak Discharge
Qp (m3/s)
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Figure 1. Schematic diagram of dam shape.
Figure 1. Schematic diagram of dam shape.
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Figure 2. Comparison of calculated results of different models. (a,b) are separated from the same figure, because if all models are put into the same graph with too many data points, it is difficult to distinguish the calculation accuracy of each model.
Figure 2. Comparison of calculated results of different models. (a,b) are separated from the same figure, because if all models are put into the same graph with too many data points, it is difficult to distinguish the calculation accuracy of each model.
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Figure 3. Location of Baige landslide. The figure comes from Zhang et al. (2019) [9].
Figure 3. Location of Baige landslide. The figure comes from Zhang et al. (2019) [9].
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Figure 4. Baige landslide blocking the river on October 10, 2010. (a) is the panorama before the landslide dam break, and (b) is the top view of the dam crest when the landslide dam break.
Figure 4. Baige landslide blocking the river on October 10, 2010. (a) is the panorama before the landslide dam break, and (b) is the top view of the dam crest when the landslide dam break.
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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.
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.
Dam Erodibilityλξαβ
H0.274.5540.2830.433
M0.155.6410.0470.532
L0.070.72470.4040.612
Table 2. Empirical formula for peak discharge of landslide dam burst flood.
Table 2. Empirical formula for peak discharge of landslide dam burst flood.
Empirical FormulaNoteReferences
Q p = 6.3 ( H d ) 1.59 Hd: Dam height (m).Costa (1985) [39]
Q p = 672 ( V l ) 0.56 V1: Volume of dammed lake (m3).Costa (1985) [39]
Q p = 181 ( V l H d ) 0.43 Hd: Dam height (m). V1: Volume of dammed lake (m3).Costa (1985) [39]
Q p = 1.60 ( V 0 ) 0.46 Vl: Volume of dammed lake (m3).Walder and O’Connor (1997) [42]
Q p = g 1 2 H d 5 2 ( H d H r ) 1.371 ( V 1 1 / 3 H d ) 1.536 e a g: The acceleration of gravity (m/s2). Hd: Dam height (m). Hr = 1 m; V1: Volume of dammed lake (m3). a: Erodibility coefficient. When the dam erodibility is high, medium and low, a is 1.236, −0.380 and −1.615 respectively.Peng and Zhang (2012) [19]
Table 3. Comparison of calculated effects of flood peak discharge calculation model.
Table 3. Comparison of calculated effects of flood peak discharge calculation model.
ModelReferencesMRERMSE (m3/s)R2
This study - 0.48 4387 0.995
Q p = g 1 2 H d 5 2 ( H d H r ) 1.371 ( V 1 1 / 3 H d ) 1.536 e a Peng and Zhang (2012) [19]0.5012,2270.963
Q p = 181 ( V l H d ) 0.43 Costa (1985) [39]1.4944,4790.510
Q p = 672 ( V l ) 0.56 Costa (1985) [39]1.3429,9430.778
Q p = 6.3 ( H d ) 1.59 Costa (1985) [39]3.1262,4120.035
Q p = 1.60 ( V l ) 0.46 Walder and O’Connor (1997) [42]1.3248,2780.423
Table 4. Parameters of Baige landslide dam on Jinsha River in 2018.
Table 4. Parameters of Baige landslide dam on Jinsha River in 2018.
Dam Failure EventsDam ErodibilityDam Height hd (m)Volume of Dammed Lake Vl (×106 m3)Data Sources
10 October 2018 M6149Zhang et al. [9]
3 November 2018 M-H81494
Table 5. Simulated results of peak discharge of two dam breaks in Baige of Jinsha River in 2018.
Table 5. Simulated results of peak discharge of two dam breaks in Baige of Jinsha River in 2018.
Prediction ModelPeak Discharge (m3/s)Relative Error (%)
10 October 20183 November 201810 October 20183 November 2018
This study 992731,6340.736.68
Q p = g 1 2 H d 5 2 ( H d H r ) 1.371 ( V 1 1 / 3 H d ) 1.536 e a 799930,52820.019.95
Q p = 181 ( V l H d ) 0.43 11,36917,24413.6994.91
Q p = 672 ( V l ) 0.56 14,76521,67047.6593.61
Q p = 6.3 ( H d ) 1.59 4345682156.5597.99
Q p = 1.60 ( V 0 ) 0.46 11,65115,96716.5195.29
Measured peak discharge10,00033,900----
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