Calculation of Overtopping Risk Probability and Assessment of Risk Consequences of Cascade Reservoirs

Abstract

In the case of extreme disasters such as local rainstorm and excessive flood, the safety risk analysis and prevention and control of cascade reservoirs face new challenges. Therefore, this article conducted a risk analysis based on typical watersheds and proposed a method for calculating the risk rate of overtopping in cascade reservoir groups, dynamically simulated the evolution process of overtopping floods in cascade reservoirs under different scenarios, delineated the scope of flood inundation, and evaluated the risk of overtopping of cascade reservoirs under different scenarios. Research has shown that dam failure floods in cascade reservoirs have both cumulative and cumulative effects, with scenario 3 being the most unfavorable. In scenario 3, the peak flow rates at the dam sites of each reservoir reached 24,500, 19,200, and 20,100 m³/s. According to the comprehensive risk assessment criteria, scenarios 1 and 2 are classified as moderate risks, while scenario 3 is classified as mild risk. Research has found that although the probability of dam overflow is extremely low, the high vulnerability calculated for each scenario indicates that a breach will cause significant social losses. This study can provide reference for the risk assessment of overtopping in cascade reservoirs and flood control and disaster reduction.

1. Introduction

Since the 1930s, global watershed hydropower development has transitioned from single reservoirs to cascade reservoir systems. However, this development model still harbors significant safety risks. Under the increasing frequency of extreme weather events and geological disasters driven by climate change, these dams may experience failures, triggering systemic risks at the watershed scale and ultimately leading to unpredictable losses in human lives and property. In May 2020, a cascading dam failure occurred at the Edenville and Sanford dams in Michigan, USA, resulting in the forced evacuation of tens of thousands of residents. The event caused significant disruptions to downstream industrial and agricultural activities, leading to substantial economic losses and environmental hazards [1]. In September 2023, extreme rainfall triggered the failure of two dams in Derna, Libya, resulting in the catastrophic loss of nearly 100,000 lives, including those missing or deceased [2]. Therefore, accurately predicting the failure probability and flood inundation extent of cascade reservoir systems, as well as assessing the associated risk consequences, is crucial.

Currently, the probability calculation methods for cascade reservoir systems predominantly rely on stochastic simulation approaches. Gabriel-Martin et al. [3] introduced gate failure scenarios and proposed a stochastic method to assess the failure probability of dams. Meanwhile, uncertainties in hydrological, hydraulic, and other factors can also influence the results of risk probability calculations. Zhang et al. [4], based on the Monte Carlo method and the verification point method, proposed a failure probability model for earth-rock dams under the influence of flood randomness and initial water levels. El-Awady et al. [5] considered the impact of variable water levels and used Bayesian networks to predict the failure probability of the Mountain Chute dam and its power station. Dong et al. [6] took into account the spatial variability of material parameters and the randomness of reservoir water levels, and ultimately assessed the risk of dam overtopping failure induced by surge waves caused by landslides. With the maturation of risk probability calculation methods, researchers have gradually recognized that solely calculating risk probabilities cannot comprehensively reflect the actual impacts of dam failure events. Therefore, since 2010, an increasing number of studies have incorporated the comprehensive assessment of dam failure consequences into the risk analysis framework [7,8]. However, in cascade reservoir systems, the occurrence of cascading failures exhibits significant risk propagation and accumulation effects, which complicates the risk assessment process. Therefore, Wang et al. [9] proposed a reservoir system dam failure risk assessment method that considers risk transmission and accumulation effects. This method utilizes the HEC-RAS model for dam failure flood simulation and integrates multidimensional factors, such as economic losses and environmental damage, for comprehensive evaluation. However, current research insufficiently addresses uncertainties and fails to effectively quantify the impacts of inflow floods, wind and wave factors, and upstream reservoir floods on downstream reservoirs. Further research is still required in this regard.

In risk quantification, due to the different units of measurement for risk probability and consequences, direct cumulative calculations cannot be performed. Therefore, the United Nations Office for the Coordination of Humanitarian Affairs defines risk as the product of dam breach flood hazard and vulnerability, with the final risk being termed “risk degree”. Vulnerability represents the potential maximum negative impacts that a disaster event could cause in terms of time and spatial extent, including casualties, economic losses, and ecological/environmental damage, among other consequences. Zhou et al. [10], based on field survey data from eight dam failures in China, proposed a dam breach life loss assessment model tailored to China’s actual conditions. Based on the flood evolution characteristics of dam breaches and their destructive impact on economic assets, Wang [11] proposed an assessment method for dam breach economic losses based on inundation zone classification. Wang [12] established a social and environmental impact assessment index system for dam breaches. The assessment of hazard usually relies on a comprehensive analysis of historical disaster data, environmental characteristics, and disaster occurrence mechanisms, using probability statistics or numerical simulation methods to determine the frequency and intensity distribution of specific disaster events. The hazard degree assignment function is less studied in dam breach risk assessment. Currently, the flood overtopping hazard degree assignment function proposed by Jiang [13] is widely used, and it provides detailed classification criteria that comprehensively consider various factors influencing hazard evaluation.

This study proposes a method for calculating the probability of overtopping risks of single reservoirs and cascade reservoir groups under different scenarios, and evaluates the overtopping risk of cascade reservoir groups in conjunction with the dynamic simulation of flood inundation areas. First, based on the considerations of the randomness of incoming floods, the uncertainty of wind load effects, and the dynamic changes in reservoir capacity, a probability calculation method for cascade reservoir groups is presented. Second, under different scenarios, a dynamic simulation of the overtopping evolution process of cascade reservoir groups is conducted, revealing the flood overflow processes of each reservoir and delineating the flood inundation areas. Finally, using danger degree, vulnerability, and risk degree as evaluation indicators, a quantitative assessment of the overtopping risk of cascade reservoir groups is carried out.

2. Methods

This study first proposes a method for calculating the probability of overtopping risk in cascade reservoirs. A model for analyzing the risk of overtopping floods in a two-dimensional cascade reservoir group was mentioned, and three simulation scenarios for the evolution process of overtopping floods in the cascade reservoir group were presented, along with evaluation indicators for the severity of flood inundation. Finally, an evaluation method for the risk level of overtopping was provided.

2.1. Overtopping Risk Probability Calculation for Reservoirs

2.1.1. Risk Probability Calculation Process for Cascade Reservoir Systems

Overtopping is considered the primary mode of dam failure [14,15]. The occurrence of overtopping events is primarily influenced by both flood and wind-wave effects, while uncertainties in the reservoir scheduling process also exacerbate the risk of such events. The overtopping risk rate model for a single reservoir is constructed as shown in the following Equation (1):

$$P={\displaystyle \frac{1}{N}}\left\{{\displaystyle \sum _{j=1}^N\left[(H_{\mathrm{j}}+e_j+R_j)-D>0\right]}\right\}$$

(1)

where P represents the probability of a single reservoir experiencing an overtopping event; H is the reservoir water level in front of the dam corresponding to the j-th simulation after a flood, in meters (m); e_j is the wind-induced wave height at the reservoir surface corresponding to the j-th simulation, in meters (m); R_j is the wave surge height at the reservoir corresponding to the j-th simulation, in meters (m); D is the height of the reservoir dam, in meters (m); and N is the total number of random simulations.

The assessment of overtopping risk probability for cascade reservoir systems not only focuses on the overtopping risk of individual reservoirs but also considers the interactions between upstream and downstream reservoirs, such as the impact of upstream reservoir discharge on the water level of downstream reservoirs. This approach provides a more comprehensive reflection of the safety and reliability of the entire cascade system. In this study, the impact of uncertain factors such as floods, wind waves, and upstream floods on the water level in front of the dam was comprehensively considered. The process of risk probability calculation for cascade reservoir systems is shown in Figure 1.

The risk probability calculation method for cascade reservoir overtopping is shown in the following Equation (2):

$$P_{\mathrm{total}}={\displaystyle \frac{1}{N}}\sum _{j=1}^N\prod _{k=1}^m{1}_{\left\{H_k^{(j)}+{\displaystyle \sum _{i=1}^{k-1}}{\Psi }_{i\to k}^{(j)}\cdot \Delta Q_i^{(j)}>D_k\right\}}$$

(2)

$$H_k^{(j)}=H_{\mathrm{base},k}^{(j)}+e_k^{(j)}+R_k^{(j)}$$

(3)

$${\Psi }_{i\to k}={\displaystyle \frac{e^{-{\lambda }_{i\to k}{L}_{i\to k}}}{C_k\sqrt{2g{D}_k}}}\cdot {\displaystyle \frac{1}{1+{\beta }_{i\to k}{t}_{p,i\to k}}}$$

(4)

$$t_{p,i\to k}=L_{i\to k}/{\nu }_w$$

(5)

$$v_w=\sqrt{g{D}_k}/n_k{D}_k^{1/6}$$

(6)

$$\Delta Q_i^{(j)}={[(H_i^{(j)}+e_i^{(j)}+R_i^{(j)})-D_i]}^{+}\cdot C_i$$

(7)

where P_total represents the overtopping risk rate of the cascade reservoir system; N is the total number of random simulations; m denotes the reservoir index; $1_{\left\{.\right\}}$ is an indicator function, which equals 1 when the specified condition is met and 0 otherwise, representing the simultaneous overtopping event of all reservoirs; $H_k^{(j)}$ refers to the total local head of the k-th reservoir, including the natural water level, wind setup, and wave run-up (m); $H_{\mathrm{base},k}^{(j)}$ is the randomly simulated maximum pre-dam water level after flood regulation for the k-th reservoir (m); $e_k^{(j)}$ denotes the wind setup height of the k-th reservoir (m); $R_k^{(j)}$ represents the wave run-up height of the k-th reservoir (m); ${\Psi }_{i\to k}$ is the transmission coefficient between reservoirs; ${\lambda }_{i\to k}$ is the river attenuation coefficient, given by the empirical formula β = 0.02 + 0.0003S; S is the riverbed gradient (‰); $L_{i\to k}$ represents the reservoir spacing (km); $t_{p,i\to k}$ is the flood propagation time (s); $v_w$ denotes the flood wave velocity (m/s); $\Delta Q_i^{(j)}$ is the dam-break discharge (m³/s); $C_i$ is the peak discharge coefficient calculated using the Xierenzhi formula; and D_k indicates the dam crest height of the k-th reservoir.

To effectively quantify the impact of initial water levels, this study employs the Monte Carlo method, which is widely used in uncertainty analysis. This method offers strong flexibility, ease of implementation, and the capability to comprehensively quantify uncertainties, thereby providing reliable risk assessment and decision support [16]. Certain regional regulations require considering the relationship between return periods and watershed characteristics to determine flood frequency, incorporating this factor into the safety assessment of hydrological dams [17]. Therefore, to ensure that the number of random simulations satisfies the flood return period requirements of the watershed, Wu et al. [18] established the relationship between the number of random simulations N and the failure probability. Based on the return period of the reservoir, the required N water levels for the cascade reservoir system during both flood and non-flood seasons were ultimately determined, as shown in the following Equation (8):

$$N\ge 100/P_f$$

(8)

where N represents the number of random simulations, and P_f denotes the corresponding flood control failure probability, which is equivalent to the reservoir flood control return period.

2.1.2. Development of Reservoir Breach and Flood Routing Models

In cascade reservoir dam breach simulations, the focus is primarily on peak discharge and the flood hydrograph. In probabilistic analysis, simplified equations can be used for related calculations. These equations offer higher simulation efficiency and can ensure a certain level of accuracy when applied across multiple random simulations. The Xierenzhi formula is widely used for calculating the dam breach discharge process. The peak discharge can be approximated as a quartic parabolic form to obtain the flood hydrograph [16,19], as shown in the following Equation (9):

$$Q_{\mathrm{m}}=\lambda B_0\sqrt{g}{H}_0^{1.5}$$

(9)

where Q_m represents the peak discharge (m³/s); $\lambda$ is the flow parameter; B₀ denotes the breach width (m); g is the gravitational acceleration, taken as 9.8 m/s²; and H₀ represents the upstream water depth (m).

The flood routing model primarily describes the temporal variations of hydraulic parameters such as discharge and water level, ultimately determining the inflow hydrograph for downstream reservoirs. Based on hydraulic theory, flood routing is mainly based on the kinematic wave approximation of the one-dimensional Saint-Venant equations, with the governing equations as shown in the following Equation (10):

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial A}{\partial t}}+{\displaystyle \frac{\partial Q}{\partial x}}=q\\ S_f=S_0-{\displaystyle \frac{\partial h}{\partial x}}-{\displaystyle \frac{v}{g}}{\displaystyle \frac{\partial v}{\partial x}}-{\displaystyle \frac{1}{g}}{\displaystyle \frac{\partial v}{\partial t}}\end{array}\right.$$

(10)

where A is the cross-sectional area of flow (m²); Q is the discharge (m³/s); t is time (s); x is the spatial coordinate along the flow direction, in meters (m); q is the lateral inflow or outflow per unit length (m²/s); S_f is the friction slope; S₀ is the bed slope; h is the water depth (m); v is the average flow velocity at the section (m/s); and g is the gravitational acceleration (m/s²).

2.1.3. The Uncertainty of Inflow Flood and Wind-Wave Factors

To accurately assess the overtopping failure probability of cascade reservoir groups, the annual maximum wind speed is considered, as it reflects the regional extreme meteorological conditions and exhibits good statistical properties. The annual maximum wind speed is therefore used as the basis for calculating wind speed. Based on extreme value theory and engineering practical experience, the maximum wind speed is typically assumed to follow the Type I extreme value distribution (Gumbel distribution), with its probability density function as shown in the following Equation (11):

$$f(v)={\displaystyle \frac{1}{{\sigma }_v}}{e}^{-(z+\exp (-z))}$$

(11)

$$z={\displaystyle \frac{v-{\mu }_v}{{\sigma }_v}}$$

(12)

Random wind speed samples following the Gumbel distribution are generated using the inverse transform method, as follows:

$$v_i={\mu }_v-{\sigma }_v\ln (-\ln U_i)$$

(13)

where v represents the sustained wind speed at the water surface (m/s), ${\mu }_v$ and ${\sigma }_v$ denote the mean and standard deviation of v, and $U_i\sim \mathrm{Uniform}(0,1)$.

The wind setup height e is the water surface elevation caused by the sustained action of wind force. The calculation of its random variable is given by the following Equation (14):

$$e_i={\displaystyle \frac{K{v_{\mathrm{w}i}}^2{D}_w}{2g{H}_m}}\cos \beta$$

(14)

where e_i represents the wind setup height at the water surface for the i-th random variable (m); K is the composite friction coefficient, taken as 3.6 × 10⁻⁶; v_wi is the wind speed for the i-th random variable, in meters per second (m/s); D_w is the length of the wind zone (m); H_m is the average water depth at the dam forebay, in meters (m); and $\beta$ is the angle between the wind direction and the normal to the dam axis (°).

The relationship between wave height H_w and wind speed v_w is given by the Putian formula (15), with the explicit value of H_w being solved using Newton’s iterative method, as follows:

$${\displaystyle \frac{g{H}_w}{{v_w}^2}}=0.13\tanh \left[0.7{\left({\displaystyle \frac{gd}{{v_w}^2}}\right)}^{0.7}\right]\cdot \tanh \left[{\displaystyle \frac{0.0018{\left(\frac{gF}{{v_w}^2}\right)}^{0.45}}{\tanh \left[0.7{\left(\frac{gd}{{v_w}^2}\right)}^{0.7}\right]}}\right]$$

(15)

where H_w is the significant wave height (m); v_w is the average wind speed above the water surface (m/s); F is the length of the wind zone, which is consistent with D_w in the wind setup formula (m); d is the average water depth, which is consistent with H_w in the wind setup formula (m); and tanh is the hyperbolic tangent function, used to adjust for the effects of water depth and wind zone on wave height.

Under the same wind speed conditions, the wave run-up induced by wind load follows a Rayleigh distribution, with its probability density function given by the following:

$$f(R)={\displaystyle \frac{R}{{{\mu }_R}^2}}\exp (-R^2/2{{\sigma }_R}^2)$$

(16)

$${\sigma }_R=\sqrt{2/\pi }\cdot H_w$$

(17)

where R represents the wave run-up, m; ${\mu }_R$ and ${\sigma }_R$ denote the mean and standard deviation of R; and H is the significant wave height (m).

For each wave height H_wi, the corresponding wave run-up R_i is generated following a Rayleigh distribution, as expressed in the following Equation (18):

$$R_i=\mathrm{Rayleigh}({\sigma }_R)$$

(18)

$${\sigma }_R=\sqrt{{\displaystyle \frac{2}{\pi }}}\cdot H_{wi}$$

(19)

The cumulative total elevation Z_i due to wind setup and wave run-up is ultimately obtained.

$$Z_i=e_i+R_i$$

(20)

2.2. Overtopping Flood Routing in Cascade Reservoir Systems

The HEC-RAS software (https://www.hec.usace.army.mil/software/hec-ras/) simulation can meet the refined modeling requirements of large-scale watersheds. Therefore, the HEC-RAS software is employed to simulate the overtopping flood inundation of cascade reservoir systems by establishing a two-dimensional hydrodynamic overtopping flood model. When using the HEC-RAS model to calculate the breach outflow process, the breach discharge can be assumed to follow the broad-crested weir flow pattern. Under this assumption, the discharge calculation formula is given in the following Equation (21) [20,21,22]:

$$Q_n=C{L}_n{Z_n}^{3/2}$$

(21)

where Q_n represents the discharge at the cross section of the n-th reservoir (m³/s), C is the weir flow coefficient, L_n denotes the weir crest length of the n-th reservoir (m), and Z_n represents the upstream water head above the weir of the n-th reservoir (m).

When simulating the two-dimensional dam overtopping and breach flood routing process, the HEC-RAS software employs a simplified two-dimensional form of the Navier–Stokes equations—namely, the shallow water equations—for unsteady flow computations. Assuming that the flow is incompressible, the continuity equation and momentum equations are expressed as follows [23]:

Continuity equation:

$${\displaystyle \frac{\partial H}{\partial t}}+\nabla ·hV+q=0$$

(22)

Momentum equation:

$${\displaystyle \frac{\partial V}{\partial t}}+V·\nabla V=-g\nabla H+v_t{\nabla }^{2}V-c_{f}V+fk\times V$$

(23)

where H, h, V, q, t, and v_t represent the water surface elevation (m), water depth (m), velocity (m/s), lateral inflow (m³/s), time (s), and horizontal eddy viscosity coefficient (m²/s). c_f, g, f, and k denote the bed friction coefficient, gravitational acceleration (m/s²), Coriolis parameter, and unit vector in the vertical direction.

2.3. Risk Assessment of Cascade Reservoir Systems

By establishing a systematic flood overtopping disaster risk assessment framework, theoretical foundations and decision support can be provided for mitigating disaster losses. To this end, this section introduces the concept of risk index, where a disaster risk index can be expressed as the product of hazard and vulnerability [24]. The risk index of an overtopping event can be formulated as follows:

$$RI=HI\times VI$$

(24)

where RI represents the risk index, which requires normalization or classification for final assessment. HI denotes the hazard index, associated with the probability of cascade reservoir failure, representing the dam breach probability within the range of 0 to 1. VI signifies the vulnerability index, related to the maximum potential loss caused by flood inundation following reservoir failure, and must be normalized to the 0–1 range.

Bradford’s Law, as a classical classification method, effectively identifies key risk factors and quantifies their impact, making it suitable for constructing multidimensional risk assessment frameworks. In this study, the regional analysis method based on Bradford’s Law is employed to classify overtopping vulnerability and hazard levels, categorizing them into five distinct grades. The specific classification criteria are presented in

Table 1. Risk value ranges and corresponding levels.

Hazard Level		Vulnerability Level		Risk Level
Hazard Index Range	Classification	Vulnerability Index Range	Classification	Risk Index Range	Classification
0.00–0.20	None	0.00–0.20	Extremely low	0.00~0.04	Extremely Low
0.20–0.40	Mild	0.20–0.40	Mild	0.04~0.16	Low
0.40–0.60	Moderate	0.40–0.60	Severe	0.16~0.36	Moderate
0.60–0.80	Serious	0.60–0.80	High	0.36~0.64	High
0.80–1.00	Extremely Serious	0.80–1.00	Very High	0.64~1.00	Extremely High

.

2.3.1. Vulnerability Estimation

In this study, to assess the vulnerability of cascade reservoirs under overtopping conditions, the flood inundation extent derived from the two-dimensional HEC-RAS model is integrated with the population and GDP data of the affected areas. The assessment includes calculations of loss of life, direct and indirect economic losses, and various social and environmental impact factors. Loss of life is estimated using the dam failure fatality assessment method proposed by Zhou et al. [10] for China. Direct and indirect economic losses are evaluated following the dam failure economic loss assessment approach by Wang et al. [11]. The social and environmental impact factors are quantified using the evaluation index system developed by Wang et al. [12]. The Analytic Hierarchy Process (AHP) is employed to determine the weight of each index [25], and a Delay function is applied to normalize the losses related to life, economy, and social–environmental factors. Finally, the vulnerability of cascade reservoir overtopping is estimated based on the linear weighted method.

2.3.2. Hazard Estimation

The probability of a specific disaster event occurring is a key indicator for assessing the likelihood of disaster occurrence. In this study, to evaluate the hazard level of cascade reservoirs under overtopping conditions, the method proposed by Jiang Qingling [13] is adopted to assign hazard values to cascade reservoir overtopping. The corresponding formula is presented in the following Equation (25):

$$HI=\left\{\begin{array}{cc}1\hfill & P>10P_d\\ {\left[{\displaystyle \frac{\lg \left(P/P_s\right)}{\lg \left(10P_d/P_s\right)}}\right]}^b\hfill & P_s\le P\le 10P_d\\ 0\hfill & P<P_s\end{array}\right.$$

(25)

where HI, P, and P_d represent the overtopping hazard level, overtopping risk rate, and dam flood control design standard. P_s is the socially acceptable overtopping risk rate, where P_s = 10⁻⁶; b is a coefficient, which is taken as 0.8 in this study.

2.4. Project Overview

As a case study, it is considered the cascade of 3 reservoirs on Yalong River for which the geographical location of the reservoir is shown in Figure 2, and the main parameters are listed in

Table 2. Main reservoir parameters.

Reservoir	RE1	RE2	RE3
Dead water level (MASL)	1321.0	1155.0	1012.0
Normal reservoir level (MASL)	1330.0	1200.0	1015.0
Design flood level (MASL)	1330.44	1203.5	1018.67
Crest elevation (MASL)	1334.0	1205.0	1020.0
Crest length (m)	516.0	774.69	468.7
Total reservoir capacity (108 m3)	7.6	61.3	0.91
Verification flood standard	5000 years	5000 years	1000 years
Verification flood peak flow (m3/s)	18,880	23,900	23,600

.

Among the cascade reservoirs RE1, RE2, and RE3, the RE2 reservoir has the largest storage capacity of 6.13 billion m³, resulting in a greater breach hazard. The RE1 and RE3 reservoirs follow, with storage capacities of 760 million m³ and 91.2 million m³, respectively. The storage capacity of RE3 is less than 100 million m³, and the breach hazard is relatively smaller. The three dams studied are a concrete gravity dam, an arch dam, and a sluice dam. The likelihood of a sudden total breach is low; however, under extreme conditions such as exceeding design flood levels or earthquakes, the safety operation of the dam may be affected, potentially leading to partial dam failure. Therefore, this study assumes the occurrence of an extreme flood event in the study basin, affecting one of the sluice gates and resulting in partial failure. The bottom of the breach at RE1 extends to an elevation of 1292.0 MASL at the intake floor, the breach at RE2 extends to an elevation of 1188.5 MASL at the intake sill, and the breach at RE3 extends to an elevation of 973.0 MASL at the spillway floor, with the breach width at all three reservoirs set to 100 m.

When the dam encounters continuous heavy precipitation or extreme rainfall events, it is very easy to trigger the chain failure of a single or multiple reservoirs. For example, in 2023, the Mediterranean storm in Derna, Libya, triggered extreme rainstorms, causing the dams of the two reservoirs of Al Bilad and Abu Mansour to burst one after another [2]. In 1975, more than 10 reservoirs, including Banqiao and a rocky floodplain in Henan Province, collapsed one after another [26]. In order to compare the impact of flood evolution and inundation extent in cascade reservoirs, the following three overtopping scenarios for the cascade reservoir group are designed:

Scenario 1: Local breach occurs at RE1, while RE2 and RE3 experience overtopping without breach.

Scenario 2: Local breaches occur at both RE1 and RE2, while RE3 experiences overtopping without breach.

Scenario 3: Local breaches occur at RE1, RE2, and RE3.

3. Results

3.1. Model Validation

There have been no dam break events in the history of the RE1, RE2, and RE3 reservoirs in the study basin, and there is a lack of measured data on dam break floods. Therefore, the flood process that occurred during the period of 7/1 to 8/29 in the study basin within 2 years was simulated and verified. A comparison between the simulation results and the measurement results of the hydrological station is shown in Figure 3.

According to Figure 3a, during the selected time period, the measurement results of the hydrological station showed a peak water level of 1655.34 MASL. The simulated peak water level is 1655.50 MASL. In Figure 3, the difference between the measured and simulated peak water levels is only 0.16 m, which is relatively small. According to Figure 3b, during the selected time period, the measurement results of the hydrological station showed a peak value of 1654.33 MASL. The simulated peak water level is 1654.70 MASL. In Figure 3b, the difference between the measured and simulated peak water levels is only 0.37 m, which is relatively small. In addition, in Figure 3a,b, the simulated and measured trends of water level changes are relatively close. Therefore, the model established in this article can be used for subsequent research on dam break flood simulation.

3.2. Flood Inundation Scenario of Cascade Reservoir Systems

3.2.1. Water Level-Discharge Process

(1) Scenario 1

In Scenario 1, a local breach occurs at RE1 within the study area, while RE2 and RE3 experience overtopping without breach. The variations in flow and water level at the dam sites of each reservoir are shown in Figure 4.

As shown in Figure 4, after the local breach at the RE1 reservoir, the peak discharge at the dam site reached 24,500 m³/s, and the maximum water level was 1329.05 MASL. Due to the considerable distance between RE2 and RE1, as well as the larger storage capacity of RE2, the required storage time for RE2 was longer. The flow at the RE2 dam site initially increased and then decreased, with a peak discharge of 15,300 m³/s and a maximum water level of 1209.03 MASL. Approximately 5 h later, RE3 reservoir experienced overtopping, with a peak discharge at the dam site of 7380 m³/s and a maximum water level of 1023.4 MASL.

(2) Scenario 2

In Scenario 2, local breaches occur at RE1 and RE2 within the study area, while RE3 experiences overtopping without breach. The variations in flow and water level at the dam sites of each reservoir are shown in Figure 5.

As shown in Figure 5, following the local breach of the RE1 reservoir, the peak flow at the dam site reached 24,500 m³/s, with a maximum water level of 1329.05 MASL. Approximately 42 h after the local breach of RE1, the RE2 reservoir also experienced a local breach. As the breach continued to develop, the peak flow at the RE2 dam site reached 19,200 m³/s, with a maximum water level of 1200.69 MASL. Following the local breach of RE2, the flood propagated downstream toward the RE3 reservoir, where the peak flow at the dam site reached 7690 m³/s, with a maximum water level of 1023.31 MASL.

(3) Scenario 3

Under Scenario 3, local breaches occurred at all three reservoirs—RE1, RE2, and RE3—within the study area. The flow and water level variations at each reservoir dam site are presented in Figure 6.

As shown in Figure 6, following the partial breach of the RE1 reservoir, the peak flow at the dam site reached 24,500 m³/s, with a maximum water level of 1329.05 MASL. Due to the greater distance between RE1 and RE2 and the larger storage capacity of RE2, a longer time was required for water accumulation in RE2. After the partial breach of the RE2 reservoir, the flow at the dam site exhibited an initial increase, followed by a decrease, with a peak flow of 19,200 m³/s and a maximum water level of 1200.69 MASL. At the RE3 reservoir dam site, the peak flow reached 20,100 m³/s, with a maximum water level of 1019.82 MASL.

As shown in

Table 3. Summary of peak flow and maximum water level for each reservoir in the cascade reservoir system under different scenarios.

Reservoir	Scenario 1		Scenario 2		Scenario 3
	Peak Flow (104 m3/s)	Maximum Water Level (MASL)	Peak Flow (104 m3/s)	Maximum Water Level (MASL)	Peak Flow (104 m3/s)	Maximum Water Level (MASL)
RE1	2.45	1329.05	2.45	1329.05	2.45	1329.05
RE2	1.53	1209.3	1.92	1200.69	1.92	1200.69
RE3	0.738	1023.4	0.769	1023.31	2.01	1019.82

, in both Scenario 1 and Scenario 2, the RE1 reservoir experienced a partial breach, resulting in a peak flow of 24,500 m³/s, which altered the water storage process of the RE2 reservoir. In Scenario 2, the peak flood discharge of the RE2 reservoir reached 19,200 m³/s, exceeding the 15,300 m³/s observed in Scenario 1. In Scenario 3, the peak flow values at the dam sites of the RE1, RE2, and RE3 reservoirs were 24,500 m³/s, 19,200 m³/s, and 20,100 m³/s, respectively, representing the most critical scenario. Therefore, in the event of dam failure within the RE1, RE2, and RE3 cascade reservoir system, it is crucial to avoid a situation where all three reservoirs experience partial breaches simultaneously.

3.2.2. Severity of Flood Inundation

In this section, the HEC-RAS hydrodynamic model and ArcGIS software (https://www.arcgis.com/index.html) were integrated to develop a severity map of flood inundation caused by dam overtopping in the cascade reservoir system. Along the river course, four representative flood inundation locations—A, B, C, and D—were selected based on their proximity to the dam and high population density, as shown in Figure 7. The flood inundation severity at each representative location was analyzed under Scenarios 1–3, enabling the visualization of the flood risk distribution associated with dam failures in the cascade reservoir system.

Flood depth and velocity are the primary indicators for assessing flood impact. Based on Zhou Kefa’s dam failure fatality assessment method for China [10], the severity of dam-break flooding can be classified using the product of water depth (D) and velocity (V). The flood severity is categorized into five levels: extremely low, low, moderate, high, and extremely high, as presented in

Table 4. Classification of flood severity.

Flood Severity	Depth·Velocity·(D·V) m2/s
Extremely	D·V ≤ 0.5 m2/s
Low	0.5 < D·V ≤ 4.6 m2/s
Moderate	4.6 m2/s < D·V ≤ 12 m2/s
High	12 m2/s < D·V ≤ 15 m2/s
Extremely high	D·V > 15 m2/s

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1.

Scenario 1

In Scenario 1, the severity of flood inundation at the representative flood locations A, B, C, and D is shown in Figure 8.

As shown in Figure 8, during the flood event, the inundation area at the representative locations A, B, C, and D gradually increased, reaching 0.262, 0.234, 0.248, and 0.309 km², respectively (the inundation area is the current inundated area minus the original riverbed area). This can be attributed to the fact that, generally speaking, the farther downstream one goes, the smaller the river slope and the wider the river channel. Therefore, the inundation area at location D is the largest. Moreover, based on the classification criteria outlined in Table 4, it can be observed that the majority of the inundation areas at these four locations exhibit an extreme flood severity level.

The detailed area range of the risk severity at each inundation representative location is presented in

Table 5. Area of each severity level in Scenario 1 (km²).

Representative Location	Extremely Low	Low	Medium	High	Extremely High	Total Area
A	0.006	0.022	0.025	0.01	0.199	0.262
B	0.028	0.035	0.018	0.011	0.142	0.234
C	0.058	0.042	0.012	0.011	0.125	0.248
D	0.048	0.076	0.051	0.017	0.117	0.309

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The area of the very high-risk zone at Location A is the largest, reaching 0.199 km², with the highest risk concentration, thus requiring priority control in the core area. On the other hand, the total inundated area at Location D is the largest, at 0.309 km². The risk structure at Locations B and C lies between the two extremes, with Location B having a very high-risk zone area of 0.142 km², second only to Location A. The very low-risk zone at Location C covers an area of 0.058 km².

2.

Scenario 2

In Scenario 2, the severity of flood inundation at the representative locations A, B, C, and D is shown in Figure 9.

As shown in Figure 9, in these four areas, the majority of the inundated area exhibits extreme severity. The detailed area ranges of risk severity at each inundation representative location are provided in

Table 6. Area of each severity level in Scenario 2 (km²).

Representative Location	Extremely Low	Low	Medium	High	Extremely High	Total Area
A	0.006	0.022	0.025	0.010	0.199	0.262
B	0.038	0.042	0.029	0.014	0.145	0.268
C	0.062	0.048	0.015	0.016	0.126	0.267
D	0.048	0.076	0.051	0.017	0.117	0.309

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The total extreme high inundation area at Location A reached 0.199 km², while Location D had the largest total inundated area (0.309 km²). However, its extreme high-risk area accounted for the smallest proportion, with the combined area of medium, low, and very low risk zones making up 57%, indicating that the risk is more widely dispersed. Therefore, long-term protection for the region’s infrastructure needs to be planned comprehensively. The extreme high-risk area at Locations B and C were 0.145 km² and 0.126 km², respectively, with Location C having the largest extremely low-risk area of 0.062 km².

3.

Scenario 3

In Scenario 3, the flood inundation severity at the representative locations A, B, C, and D is shown in Figure 10.

As shown in Figure 10, under Scenario 3, the majority of the inundated areas at the representative locations A, B, C, and D exhibit a flood severity level of extremely high. The detailed range of risk severity areas for each inundation representative location is presented in

Table 7. Area of each severity level in Scenario 3 (km²).

Representative Location	Extremely Low	Low	Medium	High	Extremely High	Total Area
A	0.006	0.022	0.025	0.010	0.199	0.262
B	0.038	0.042	0.029	0.014	0.145	0.268
C	0.076	0.034	0.018	0.016	0.134	0.278
D	0.024	0.085	0.056	0.019	0.136	0.320

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The extremely high-risk area at Location A is 0.199 km², while the total inundated area at Location D is the largest, measuring 0.320 km². The extremely high-risk area at Locations B and C are 0.145 km² and 0.134 km², respectively.

In Scenarios 1–3, the RE1 reservoir experiences local breach due to overtopping. Location A, which is situated on the downstream side of the RE1 reservoir and the upstream side of the RE2 reservoir, consistently experiences the same total inundated area (0.262 km²) in all three scenarios. In Scenario 1, the maximum flood inundation area at Locations B, C, and D is relatively small. However, in Scenario 3, when all three reservoirs (RE1, RE2, and RE3) experience local breaches due to overtopping, the inundated area at Locations B, C, and D becomes significantly larger. Therefore, the most adverse scenario occurs when all three reservoirs experience local breaches, which should be avoided. For Area A, emphasis should be placed on the rapid response in high-risk zones, while for Area D, the focus should be on preventing the chain reaction effects of breaches. For Areas B and C, dynamic monitoring of medium- and high-risk areas should be strengthened.

3.3. Risk Loss Analysis

To assess the risk level in each scenario, the analysis combines publicly available population and economic data from the submerged areas to calculate life loss, direct and indirect economic losses, and social and environmental losses for each scenario. The resulting values are then converted into vulnerability. By integrating the cascade reservoir group’s risk probability from Section 2.1 and utilizing the risk hazard estimation method, the corresponding hazard values are derived. The vulnerability, hazard, and risk values for different scenarios are presented in

Table 8. Results of vulnerability, hazard, and risk levels under different scenarios.

Scenario	Vulnerability					Hazard		Risk
	Loss of Life (Person)	Direct Economic Loss (104 ¥)	Indirect Economic Loss (104 ¥)	Social and Environmental Loss	Adjusted Vulnerability	Risk Probability	Adjusted Hazard
1	2184	1176.74	712.04	1.44	0.804	1.6 × 10−5	0.361	0.290
2	2243	1210.62	726.37	13.75	0.843	4 × 10−6	0.207	0.175
3	2298	1229.28	737.57	42.30	0.862	4 × 10−6	0.184	0.159

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The calculated vulnerability values for dam overtopping in scenarios 1, 2, and 3 are 0.804, 0.843, and 0.862, respectively, all of which fall under the extremely high-risk category. The risk probability of the cascade reservoir group overtopping is converted into overtopping hazard through an assignment function. The calculated hazard values for scenarios 1, 2, and 3 are 0.361, 0.207, and 0.184, respectively, all close to the mild-risk level. Although the probability of overtopping is extremely low, the high vulnerability indicates that once a breach occurs, it will lead to significant social losses. The calculated risk values for scenarios 1, 2, and 3 are 0.290, 0.175, and 0.159, respectively. Scenarios 1 and 2 fall under moderate risk, while scenario 3 falls under low risk, but it is close to the critical state of moderate risk, indicating a potential risk increase in the event of an extreme flood.

Based on the above risk calculation results and considering the importance of the basin’s reservoir projects, the following recommendations are made to control the overtopping risk of the basin:

(1) Strengthen the joint scheduling and coordinated flood control of the cascade reservoir group. Dynamic adjustments to the storage and discharge strategies of each reservoir should be made based on upstream reservoir discharge, downstream river channel capacity, and reservoir storage capacity.

(2) Establish a risk transmission blocking mechanism. Emergency detention areas or diversion channels should be set up between the cascade reservoirs. When a reservoir approaches a breach threshold, flood diversion measures should be activated to prevent a cascading failure due to compounded flood effects.

4. Conclusions

This study focuses on the downstream RE1, RE2, and RE3 cascade reservoir group in a southwestern river basin. A Monte Carlo-based approach for calculating the overtopping risk probability of the cascade reservoir group is proposed, providing a quantitative assessment of the overtopping risk probabilities under three different scenarios. Using the HEC-RAS hydraulic model, the flood evolution process and inundation characteristics of the cascade reservoir group under various breach mode combinations are simulated. Finally, based on the risk degree model, the overtopping risk levels of the cascade reservoir group under the three scenarios are comprehensively evaluated. The main conclusions are as follows:

(1) Under scenario 3, where the RE1, RE2, and RE3 reservoirs all experience localized breaches, the peak flow at the dam site of each reservoir reaches 24,500, 19,200, and 20,100 m³/s, respectively, with maximum water levels of 1329.05 MASL, 1200.69 MASL, and 1019.82 MASL.

(2) In scenario 3, the inundation areas at the representative flood sites A, B, C, and D are 0.262, 0.268, 0.278, and 0.320 km², respectively. This result reveals the high vulnerability of the cascade reservoir group, providing crucial information for risk management.

(3) The risk assessment study found that the risk level of scenario 1 is 0.290, the risk level of scenario 2 is 0.175, and the risk level of scenario 3 is 0.159.

This study calculated the overtopping risk probability of the cascade reservoir group and conducted consequence analysis for a series of reservoirs under a single initial water level condition. However, this paper did not consider the risk scenarios under different initial water level conditions in the various reservoirs. Future research will further investigate the risk probabilities and consequences of the series of reservoirs under different initial water level conditions. The results of this study can provide references for the risk assessment in cascaded reservoir groups and regional flood prevention.