Numerical Simulation and Risk Assessment of Cascade Reservoir Dam-Break

Abstract

Despite the fact that cascade reservoirs are built in a large number of river basins nowadays, there is still an absence of studies on sequential embankment dam-break in cascade reservoirs. Therefore, numerical simulations and risk analyses of cascade reservoir dam-break are of practical engineering significance. In this study, by means of contacting the hydraulic features of upstream and downstream reservoirs with flood routing simulation (FRS) and flood-regulating calculation (FRC), a numerical model for the whole process of cascade reservoir breaching simulation (CRBS) is established based on a single-embankment dam-break model (Dam Breach Analysis—China Institute of Water Resources and Hydropower Research (DB-IWHR)). In a case study of a fundamental cascade reservoir system, in the upstream Tangjiashan barrier lake and the downstream reservoir II, the whole process of cascade reservoir dam-break is simulated and predicted under working schemes of different discharge capacities, and the risk of cascading breaching was also evaluated through CRBS. The results show that, in the dam-break of Tangjiashan barrier lake, the calculated values of the peak outflow rate are about 10% more than the recorded data, which are in an acceptable range. In the simulation of flood routing, the dam-break flood arrived at the downstream reservoir after 3 h. According to the predicted results of flood-regulating calculations and the dam-break simulation in the downstream reservoir, the risk of sequential dam-break can be effectively reduced by setting early warnings to decrease reservoir storage in advance and adding a second discharge tunnel to increase the discharge capacity. Alongside the simulation of flood routing and flood regulation, the whole process of cascade dam-break was completely simulated and the results of CRBS tend to be more reasonable; CRBS shows the great value of engineering application in the risk assessment and flood control of cascade reservoirs as an universal numerical prediction model.

1. Introduction

Reservoir dams have played a momentous role in flood control, irrigation, electricity generation, shipping, water supply, aquaculture and the improvement of the ecological environment, and create considerable economic value and social benefits [1]. Nowadays, in order to make full and reasonable use of abundant water resources and achieve comprehensive water resource regulation of river basins, the exploitation pattern of cascade hydropower is becoming an international tendency. For example, the Dadu River cascade hydropower station, one of the thirteen hydropower bases in China, has developed about eighty large-scaled cascade reservoir dams on its main stem and tributaries, of which seventeen are high-embankment dams over 100 m [2].

However, the huge potential energy embedded in cascade reservoir storage also poses a threat to the downstream basin. Whether the downstream reservoir could resist the upstream dam-break flood is directly related to the safety of the whole river basin when the upstream embankment dam collapses under extreme floods [3]. As huge economic and social benefits are produced through the construction of cascade reservoirs, the safety of cascade reservoir under extreme floods attracts more and more attention of the researchers [4]. The Dam-Break Flood Forecasting model (DAMBRK), a numerical model proposed by Fread in 1988 [5] which can be used for simplified calculations of cascade reservoir failure, needs to input parameters about the final shape of the breach and the starting time of dam-break. However, these parameters are very difficult to preset when lacking measured data. Shi et al. [6] simulated the cascading dam breaching of the Tangjiashan landslide dam and two smaller virtual dams downstream based on the Dam Breach Analysis model (DABA), and evaluated the influences of geometric factors on the breaching process. However, the process of flood regulation in reservoirs and the flood routing in the riverway were ignored in this study, and it would greatly influence the simulating results. Xue et al. [7] conducted an experiment on rectangular glass flume to simulate the sequential embankment dam-break. This experiment indicated that the hydraulic features of dam-break water flow will be influenced by the initial water level and the distance between upstream and downstream dams. Niu et al. [8] researched the cascade and single dam failure modes triggered by overtopping floods through a series of physical model experiments in a sloping flume, and they found each part of the dam’s cross section broke uniformly instead of forming a breach when enlarging the inflow discharge. Cao et al. [9] concluded that the dam-break flood would be influenced by the dam-to-dam distance and height difference in a cascade reservoir through the shallow water hydrodynamic model. They also found the cascading dam-break is more dangerous than a single dam-break when suffering from catastrophic floods. Zhou et al. [10] proposed two extra risk control standards based on the existing Chinese dam design criterions to classify the single and cascade embankment dams over 200 m, and verified the feasibility of them through structural reliability analysis and a virtual case study of cascade reservoir dam-break. Bouchehed et al. [11] evaluated the potential cascading dam-break flood risks through the two-dimensional hydrodynamic model Telemac-2D between Mexa Dam and Bougous Dam; however, the physical process of dam-break was ignored and the simulations was only based on the flood routing theories. From what has been discussed above, the existing studies on cascade reservoir dam-break can be divided into two types. The first type mentioned focused on the characteristics of dam-break through experimental studies, which are subject to the model size and the test condition; the latter established oversimplified numerical models to simulate the process of cascade reservoir dam-break, which are irrational in principle and hard to apply in the design stage of a cascade reservoir project. Therefore, as the most direct and effective research tool, it is of great scientific significance and applied engineering importance to establish a reasonable numerical prediction model to simulate the sequential embankment dam-break for a cascade reservoir.

This study aims to establish a numerical model that could simulate the whole process of sequential embankment dam-break occurring from upstream to downstream. The numerical model is verified through a case study of sequential embankment dam-break: the upstream Tangjiashan barrier dam and the downstream embankment dam II. Moreover, we analyze the risk of cascading dam-break under different operating conditions: enlarging the discharge capacity and releasing the flood in advance.

2. Methods

2.1. Simulation of Single-Embankment Dam-Break

The whole process of cascade reservoir dam-break can be divided into three parts: 1. A series of individual dam-breaks occur successively from upstream to downstream. 2. Flood routing in the riverway. 3. Flood regulating in the downstream reservoir. Therefore, the accurate simulation of hydraulic elements over time in the process of a single-embankment dam-break, such as dam-break outflow rate, water level and the geometrical shape of the breach, is the key to establishing a numerical model for cascade reservoir dam-break [12]. Up to now, various numerical models have been developed, providing theoretical tools for risk assessments of single-embankment dam-break [13]. The first mathematical model for single-embankment dam-break was established by Cristofano in 1965 [14]. Afterwards, Brown, Singh and Fread proposed the BRDAM model [15], Breach Erosion of Earthfill Dams model (BEED) [16], Dam-Break Flood Forecasting model (DAMBRK) [5] and BREACH model [17], respectively, which have been widely used in practice so far. In these numerical models, the broad-crested weir flow formula is adopted to calculate the dam-break outflow rate, and the Schoklitsch formula, Einstein–Brown formula, linear predetermined erosion formula and Meyer-Peter and Muller formula are taken to simulate sediment transportations, respectively. In recent years, Hydrologic Engineering Center’s River Analysis System (HEC-RAS) [18] and MIKE 11 [19], famous software developed by the United States Army Corps of Engineers and Denmark Hydraulic Institution, have been coupled with the DAMBRK model and also have the capability to simulate the process of single-embankment dam-break. However, in the BEED model, the forming process of the breach is ignored and the initial shape of the breach needs to be preset; in the BRDAM model, the slope of the breach is fixed at 45° and the lateral enlargement of the breach is ignored; in the DAMBRK model, the final shape of the breach and the holding time of the dam-break need to be preset; in the BREACH model, it tends to be difficult to have stable computations when the velocity of dam-break flow exceeds 5 m/s or the gradient of the downstream slope is smaller than four [20]. Apart from these unreasonable defects, through sensitivity analyses, researchers have also proved that the computed results of these numerical models are sensitive to input parameters [21,22].

To solve these problems, based on the programming of Excel’s Macro and VBA, Chen et al. [23] established Dam Breach Analysis—China Institute of Water Resources and Hydropower Research (DB-IWHR) to simulate the single-embankment dam-break, which is a physically-based mathematical model based on the theories of hydrodynamics and sediment dynamics. Compared with other dam-break models, DB-IWHR does not merely avoid redundant parameters presetting the process of dam-break, but the computed results were also proven to possess low parameter sensitivity and high numerical stability [24]. DB-IWHR is to be adopted as the single-embankment dam-break model, and the simulation results will be verified in this paper. In this section, the fundamentals of three computational components in DB-IWHR will be introduced briefly: dam-break outflow, vertical erosion and the horizontal enlargement of the breach.

2.1.1. Dam-Break Outflow

When collapse triggered by overtopping occurs in the embankment dam as shown in Figure 1, Equation (1), the broad-crested weir flow formula, can be used to figure out the outflow rate of dam-break floods, supposing that the outlet boundary condition is the free discharge.

$$Q=C_1{C}_{2}B{(H-z)}^{3/2}$$

(1)

where Q is the discharge flow rate; C₁ is discharge coefficient ranging from 1.3–1.7 m^0.5/s [25]; C₂ is correction coefficient; B is breach bottom width; H is water level in front of the dam; h is water depth in the breach; z is breach bottom height; The coefficient C₁ and C₂ are determined according to experience in general [6,15,16,17]. However, to improve the accuracy of the simulation, through tuning the value of C₁ and C₂ and matching the simulative relationship curve between the discharge per unit width (Q/B) and the head (H−z) to the measured data based on Equation (1), the coefficients C₁ and C₂ can be calibrated for this numerical model [23].

Considering the velocity of overtopping flow is relatively small, the flow velocity and water head loss in front of the breach are assumed to be ignored. Therefore, the flow velocity of the breach can be derived according to the energy balance equation as shown in Equations (2) and (3). Through introducing the drop coefficient m, the calculation for h through the Manning equation can be avoid, and the high parameter sensitivity between the computed results and the slope J is also avoided. Equation (4) illuminates the relationship between reservoir storage and water level.

$$V=C_1{C}_2{m}^{-1}\sqrt{H-z}$$

(2)

$$m=\frac{h}{H-z}\approx \frac{n^{0.6}{C}_1{}^{0.6}{C}_2{}^{0.6}}{J^{0.3}}$$

(3)

$$W=[a{(H-H_r)}^2+b(H-H_r)+c]\times {10}^6$$

(4)

where m is drop coefficient ranging from 0.4–0.9 [23], and the value of m can be tuned through matching the simulative relationship curve between the water depth (h) and the head (H − z) to the measured data based on Equation (3); J is the slope of the channel; n is roughness coefficient; W is the reservoir storage; H_r is the basis flooding level; a, b and c are fitting coefficients for the storage-capacity curve which can be obtained from the hydrogeological data of the reservoir;

Based on Equations (1)–(4), the relationship between outflow rate and inflow rate in front of the breach can be summarized as in Equation (5), according to the principle of water balance.

$$Q=C_1{C}_{2}B{(H-z)}^{3/2}=\frac{\mathrm{d}W}{\mathrm{d}H}\frac{\mathrm{d}H}{\mathrm{d}t}+q$$

(5)

where q is the upstream inflow rate including initial river flow and flood.

2.1.2. Vertical Erosion of the Breach

The key to the calculation of vertical erosion is to establish the relationship between erosion rate and shear stress. Shear stress along solid boundaries triggered by water flow can be solved according to the equations of open channel flow as in Equation (6).

$$\tau =\gamma RJ=\frac{\gamma n^2{V}_{}^2}{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$$

(6)

where τ is shear stress; γ is the density of water; R is hydraulic radius; J is the slope of the spillway channel; n is roughness coefficient.

Based on the linear model and the exponential model, Chen et al. [23] proposed a hyperbolic erosion model to describe the relationship between the erosion rate and the shear stress as shown in Equations (7) and (8).

$$s=\Phi (\tau )=\frac{v}{a_2+b_{2}v}$$

(7)

$$v=k(\tau -{\tau }_c)$$

(8)

where s is the erosion rate; a₂, b₂ are the fitting coefficients; k is the unit conversion factor and the value of k is 100; τ is the shear stress; τ_c is the critical shear stress; ν is the intermediate variable.

In the hyperbolic erosion model, the coefficient b₂ represents the reciprocal of the maximum possible erosion rate and 1/a₂ represents the tangent of this curve at ν = 0, which means that the hyperbolic model sufficiently takes into account that the strength of soil has a limit when resisting erosion. Through tuning the value of a₂ and b₂ and matching the simulative relationship curve between the erosion rate (s) and the shear stress (ν) to the measured data, the coefficients a₂ and b₂ can be calibrated for this hyperbolic erosion numerical model. In this way, compared with the empirical coefficient in other erosion models which are only related to dimensions, the reasonable value of a₂ and b₂ in the hyperbolic model can be determined according to their physical meanings and the measured data, and the sensitivity of input coefficients are reduced.

2.1.3. Lateral Enlargement of the Breach

The vertical erosion and horizontal enlargement caused by overtopping floods usually occur at different rates, which would lead to the slope instability on both sides of the breach. The simulation of lateral enlargement of the breach surface is carried out by STAB [26,27], a procedure for slope stability analysis. In STAB, the simplified Bishop slice method and the total stress analysis method are applied, and the undrained shear strength parameters of the dam body, the soil shear strength C_u and internal friction angle ϕ_u, are employed in the stability analysis. Through STAB, all the potential slip surfaces of the breach would be searched out through trial calculations until the breach bottom reaches a safe elevation to ensure slope stability as shown in Figure 2, and then the breach bottom elevation in each trial step would be calculated.

The relationship between the breach bottom width and elevation can be established according to the geometric features of the breach, combining with the breach bottom elevation calculated by STAB, the breach bottom width at each moment during the dam-break can be calculated through Equations (9) and (10).

$$B=B_0+2\left(z_0-z\right)+2h\tan \left[{\alpha }_1+\frac{z_0-z}{z_0-z_{end}}\left({\alpha }_2-{\alpha }_1\right)-90°\right]$$

(9)

$$B_0=\frac{q_0{\left(C_1{C}_2\right)}^2}{{\left(m{V}_c\right)}^3}$$

(10)

where B₀ is the initial breach bottom width; z is the breach bottom elevation; z₀ is the dam crest elevation; z_end is the final value of z, which is obtained from STAB; α₁ is the initial lateral inclination; α₂ is the final lateral inclination; q₀ is the initial inflow rate; V_c is the initial erosion velocity.

In the conventional computing method such as BREACH and BEED, the integral computation is on the basis of time integration scheme and it requires presetting the initial expanding time and time step. However, in DB-IWHR, the integral computations are based on velocity integration scheme and it only needs to input the initial flow velocity and velocity increment. Then, the increment of water level, flow velocity and breach bottom elevation would be calculated. In this way, the nonlinear iterative calculation is avoided so that the process of simulation is with faster speed and higher accuracy, and the redundant parameter-presetting are also avoided in the meantime.

Through the analysis above, the closed equation set, composed of Equations (1), (3)–(5), (7) and (9), can be solved directly and linearly, and DB-IWHR model is established. The dam-break begins when the flow velocity on the dam crest reaches the initial erosion velocity Vc, therefore, it is necessary to activate DB-IWHR to judge and simulate the process of embankment dam-break when overtopping occurs.

2.2. Flood Routing Simulation (FRS)

The upstream dam-break flood would flow over into the downstream reservoir through the middle watercourse, and this process requires flood routing simulation (FRS) to figure out the hydraulic elements such as water level and flow rate over time. The routing flood in a complex natural watercourse is generally regarded as unidimensional steady nonuniform flow in the open channel. Under this assumption, the outflow rate and water level of the downstream cross-section can be deduced from those of the upstream cross-section through Equations (11) and (12), which are known as Saint-Venant equations when taking water depth h and flow rate Q as variables.

$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q_{\mathrm{l}}$$

(11)

$$\frac{\partial }{\partial t}\left(\frac{Q}{A}\right)+\frac{\partial }{\partial x}\left(\frac{\beta Q^2}{2A^2}\right)+g\frac{\partial h}{\partial x}+g(S_f-S_0)=0$$

(12)

where A is the area of river cross-section; t is time coordinate; x is space coordinate; q_l is the later flow rate per unit width; β is the momentum correction factor; S₀ is the gradient of channel slope; S_f is the friction slope; g is the acceleration of gravity.

The numerical algorithm for Saint-Venant equations include the Finite Difference method, Characteristics method, Finite Element method and Finite Volume method, among which the Finite Difference method is the most widely used. In this paper, the Newton–Raphson method is adopted to linearize the nonlinear equations set, and then it can be solved iteratively with Preissmann four-point linear implicit finite-difference scheme [28].

2.3. Flood-Regulating Calculation (FRC)

When the upstream reservoir has remaining capacity for discharge control, part of the flood can be stored, and the downstream reservoir dam-break can be delayed effectively. The process above needs flood-regulating calculation (FRC) to figure out the water level and outflow rate overtime based on storage capacity curve, discharge capacity curve, flood hydrograph [29]. The fundamental principle of FRC is the water balance equation as shown in Equation (13), and the most common numerical solutions for Equation (13) is the fourth-order Runge–Kutta method, which will be adopted in this study [30].

$$\frac{dV}{dz}\frac{dz}{dt}=Q_{in}-q_{out}$$

(13)

where Q_in is the inflow rate; q_out is the outflow rate, which is affected by dam-break and water-release structures: Before the dam-break, q_out is the discharge of the release structures; After the dam-break, q_out is the discharge of the release structures and the breach. Supposing that the release structures only include spillway and discharge tunnel, q_out can be calculated by Equations (14)–(17). Equations (15) and (17) are derived from the broad-crested weir flow formula as shown in Equation (1), and Equation (16) is derived by Torricelli’s theorem.

$$q_{out}=q_s+q_g+q_b$$

(14)

$$q_s=C_s{L}_s{(z-z_s)}^{1.5}$$

(15)

$$q_g=C_g{A}_g{(z-z_g)}^{0.5}$$

(16)

$$q_b=CB{\left(z-z_d\right)}^{1.5}$$

(17)

where q_s, C_s, L_s and z_s are, respectively, the outflow rate, discharge coefficient, width and elevation of the spillway; q_g, C_g and z_g are, respectively, the outflow rate, discharge coefficient and elevation of the discharge tunnel; A_g is the open area of water gate; q_b, C, B and z_d are, respectively, the outflow rate, discharge coefficient, width and elevation of the breach.

2.4. Cascade Reservoir Breaching Simulation (CRBS)

If the flood-regulating capacity of the downstream reservoir is insufficient, the dam-break flood will exceed the dam crest. When the overtopping flow velocity reaches the initial erosion velocity V_c, the downstream dam starts to break, and DB-IWHR should be invoked again to simulate the downstream sequential dam-break. Thus far, the whole process of a fundamental cascade reservoir dam-break has been simulated, which includes upstream reservoir Ⅰ and embankment dam Ⅰ, downstream reservoir II and embankment dam II and the middle riverway as shown in Figure 3. Any kind of complex cascade reservoir system can be decomposed into fundamental cascade reservoirs and simulated circularly.

In CRBS, the results of the previous calculation module can be directly imported into the next calculation module as initial conditions. Through inputting relevant parameters and initial conditions and invoking three calculation modules in turn, the whole process of cascade reservoir dam-break can be simulated in CRBS. The calculation flow of CRBS is shown in Figure 4.

3. Results and Discussion

Tangjiashan barrier lake was located in Beichuan county, Sichuan province. It was formed in a landslide triggered by the Wenchuan earthquake in 2008, and after that it stored water about 316 × 10⁶ m³ [31]. In order to eliminate the risk of dam-break in Tangjiashan barrier lake, artificial destruction of the Tangjiashan barrier lake was conducted on 10 June 2008 by means of excavating a discharge channel. The artificial discharge channel was initially with a depth of 12 m, a bottom elevation of 741 m and a gradient of 1:1.5 on both side of the channel. After the dam-break, the breach is about 600 m long; the top width ranges from 145–235 m and the bottom width ranges from 80–100 m [32]. The artificial destruction of Tangjiashan barrier lake is the only embankment dam-break which was detailed recorded, providing valuable measured data for risk assessment and back analysis of embankment dam-break [33]. Figure 5 is a location diagram of Tangjiashan barrier lake.

Considering the measured data can be taken as comparisons, Tangjiashan barrier lake is selected as upstream reservoir Ⅰ. Reservoir II, a fictitious reservoir which is assumed to be 85.6 km downstream, forms a fundamental cascade reservoir system together with reservoir Ⅰ. The storage-capacity curve and the characteristic parameters of reservoir II is shown in Figure 6 and

Table 1. Characteristic parameters of reservoir II.

Dam Crest Elevation (m)	Dam Height (m)	Check Flood Level (m)	Normal Pool Level (m)	Dead Water Level (m)	Total Storage (×108 m3)	Dead Storage (×108 m3)
465	113.5	462.6	461	445	2.763	1.615

. Taking this fundamental cascade reservoir as a calculating example, the numerical simulation and risk assessment of cascade reservoir dam-break is conducted through the numerical model CRBS in this study.

3.1. Dam-Break of Tangjiashan Barrier Lake

BREACH, one of the most commonly used numerical model in the simulation of dam-break, has been applied and verified in the simulation and prediction of Teton Dam, Lawn Dam and Mantaro Landslide Dam [17]. Therefore, the results of BREACH can be regarded as standards in this study. In the simulation of the upstream dam-break, the recorded data and the results of BREACH and DB-IWHR are compared as shown in Figure 7. In this way, selecting DB-IWHR as the numerical model for the simulation of single-embankment dam-break can be validated. The main input parameters of DB-IWHR and BREACH are referred to the measured data in [31,32,33], which are also compared to analyze the usability of the numerical models in actual engineering as shown in

Table 2. The main input parameters of Dam Breach Analysis—China Institute of Water Resources and Hydropower Research (DB-IWHR).

Category	Parameters	Sign (Unit)	Value
Parameters of dam body	Dam crest elevation	z0 (m)	740
	Initial inflow	qi (m3/s)	80
	Basis flooding level	Hr (m)	700
	Initial width of the breach	B0 (m)	14.4
	Initial slope of the breach	J	1.2
Parameters of reservoir capacity	Fitting coefficients of water level-storage	a, b, c	0.06, 1.96, 44
Parameters of weir flow	Discharge coefficient	C1	1.43
	Submergence coefficient	C2	0.94
	Correction ratio for head loss	m	0.80
Parameters of erosion	Initial erosion velocity	Vc (m/s)	2.7
	Fitting coefficients of erosion rate	a2, b2	1.1, 0.0005
	Shear strength	Cu (kPa)	25
	Internal friction angle	ϕu (°)	26

and

Table 3. The main input parameters of BREACH.

Category	Parameters (Unit)	Value
Parameters of dam body	Initial water level (m)	742.5
	Dam crest elevation (m)	742.5
	Dam bottom elevation (m)	669.5
	Upstream slope	4
	Downstream slope	1.2
	Dam crest length (m)	400
	Manning coefficient	0.0177
Parameters of material property	Grain diameter (mm)	5
	Specific gravity	2.65
	Porosity	0.43
	Cohesive force (kPa)	25
	Internal friction angle (°)	26
	Coefficient of material discontinuity	30

.

The peak outflow rate computed by DB-IWHR is 7472.3 m³/s, 14.6% higher than the measured value 6500.5 m³/s, 3% higher than the calculated value of BREACH 7217.6 m³/s. The arrival time of peak outflow rate predicted by DB-IWHR and BREACH is 10:51 and 13:50, respectively. Obviously, there are still deviations compared with the measured data 12:30. The outflow rate computed by DB-IWHR is slightly larger than the measured data in the late stage of dam-break, which is closer to the measured data compared with BREACH. After the flood peak, the water level and the breach bottom elevation go down slowly until it becomes constant in the measured data, but the simulating value of DB-IWHR and BREACH go down constantly.

Through the analysis above, the peak outflow rate computed by BREACH and DB-IWHR are approximately 10% more than the measured data. In fact, the dam-break models can be divided into inversion models and prediction models. In inversion models such as BEED, BRDAM and DAMBRK, the simulation of the dam-break is actually interpolating calculations based on the parameters about the initial and final state of the breach; however, these parameters are difficult to preset if there are no recorded data of the dam-break. In prediction models such as BREACH and DB-IWHR, the simulations of the dam-break are based on iterative computations or trial calculations without presetting any parameters about the changing process of the breach so that they are more likely to generate errors. Therefore, as in the prediction model, both DB-IWHR and BREACH are of great guiding value in flood protection engineering, even though the approximate 10% errors exist in the predicting results of the peak outflow rate.

As to the errors in the simulations of the water level and breach bottom elevation in the two models, it is caused by the lack of simulation about the sediment accumulation in the later period of the dam-break. However, the complex simulation of sediment accumulation is remaining to further investigate and improvement, in which the accumulation of the gravel, the discontinuity of dam-body materials and the existence of large bedrock should be taken into account.

Compared with BREACH, the hyperbolic erosion model and simplified Bishop slice method are applied to simulate the vertical erosion and lateral expansion of the breach in DB-IWHR, avoiding iterative calculation and reducing parameter sensitivity. The input parameters of DB-IWHR about material properties are less than BREACH; however, the complex material properties of a natural embankment dam are difficult to determine in the early stage of engineering design and risk management. In summary, DB-IWHR is sufficiently practical in dam-break prediction, and it is reasonable to choose DB-IWHR as the calculation module for single-embankment dam-break in CRBS.

3.2. Flood Routing in The Middle Riverway

The downstream reservoir II is 85.6 km away from Tangjiashan barrier lake; the channel roughness is 0.025; The channel slope is 0.4; The geometric shape of the channel is inverted trapezoid; the bottom width is 30 m; the riverbank slope is 0.3; the time coefficient is 0.75; the space coefficient is 0.4, and the time step is 60 s. Considering that there are still inevitable deviations in the computed results of DB-IWHR, the measured data of the dam-break flood is directly used as the upstream boundary condition required by FRS.

According to the result of FRS as shown in Figure 8, the dam-break flood evolved to reservoir II at 09:00 with an outflow rate of 576.3 m³/s, and the peak outflow rate is 6325.4 m³/s appeared at 14:24.

It is obvious that the process of flood routing slowed the arrived time of the upstream dam-break flood for about 3 h. In this way, the discharge capacity of downstream reservoir can be adjusted in advanced according to the flood warnings from upstream reservoir, so that the sequential dam-break could be delayed. However, the process of flood routing would be ignored sometimes due to the short distance between the upstream and downstream reservoir such as the case study in [6], which would definitely accelerate the rising process of the water level in front of the downstream reservoir and cause calculation errors.

3.3. Flood Regulation in the Downstream Reservoir

The computed results of FRS are directly used as the boundary conditions required by FRC. It is supposed that the release structures only include the discharge tunnel, and the water gate of the discharge tunnel is fully opened to release flood at the elevation of normal pool level 461.0 m. After overtopping, the flood discharge is the sum of the outflow of the discharge tunnel and the overtopping flood. The water-level discharge capacity curve of reservoir II and the results of FRC are shown in Figure 9 and Figure 10, respectively.

The water level in front of reservoir II showed a downtrend at first as a result of flood regulation. The water level reached the dam-crest elevation 465.0 m at 14:05 and it rose to its maximum value 470.3 m at 17:48. The peak inflow rate of 6325.4 m³/s appeared at 14:30, which is much later than the average annual runoff in the meantime. Even if part of the flood was released through the discharge tunnel, the water level still rose over the dam crest about 5 h after the upstream dam-break flood routed to the downstream reservoir, and the flood-carrying capacity of reservoir II is insufficient to eliminate the risk of sequential dam-break.

3.4. Sequential Break of Downstream Dam II

According to the results of FRC, the overtopping occurred and reservoir II is likely to starting breaching. Therefore, it is necessary to invoke DB-IWHR again to judge whether dam II would break sequentially. The input parameters of dam II are shown in

Table 4. The main input parameters of DB-IWHR for dam II.

Category	Parameters	Sign (Unit)	Value
Parameters of dam body	Dam crest elevation	z0 (m)	465
	Initial inflow	qi (m3/s)	232
	Basis flooding level	Hr (m)	445
	Initial width of the breach	B0 (m)	0.0
Parameters of reservoir capacity	Fitting coefficients of water level-storage	a, b, c	0.04, 4.94, 161.49
Parameters of weir flow	Discharge coefficient	C1	0.36
	Submergence coefficient	C2	0.90
	Correction ratio for head loss	m	0.67
Parameters of erosion	Initial erosion velocity	Vc (m/s)	3.0
	Fitting coefficients of erosion rate	a2, b2	1.10, 0.001

, and the computed results are shown in Figure 11.

Reservoir II began to burst at 14:48 when the overtopping flow velocity reached the initial scour velocity V_c, and the peak outflow rate was 8159.9 m³/s appeared at 22:06. After that, the outflow rate gradually decreased and the whole process of dam-break lasted for 24.6 h till the next day 15:28; the water level in front of reservoir II was 465.2 m when the dam-break began and it reached the maximum value 472.1 m at 15:37, after that it gradually declined to 388.4 m at the end of the dam-break; the breach bottom width increased rapidly before 21:10, after that it increased linearly with a velocity about 4.5 m/h, and it finally reached 144.6 m. The breach bottom elevation showed a linear decreasing trend with a velocity about −3.2 m/h, and it finally reached 387.4 m. Thus far, all the simulations in the fundamental cascade reservoir dam-break as shown in Figure 3 has been completed through CRBS.

3.5. Risk Analysis of Reservoir II under Different Operating Schemes

The cascade dam-break would cause more serious disaster to the downstream river basin compared with single-embankment dam-break. Therefore, it is necessary to reduce the risk of cascade dam-break in the downstream reservoir, and the most direct method is to increase the discharge capacity of the downstream reservoir through adding flood releasing structures. Besides that, according to the flood warning from the upstream reservoir, the discharge capacity of the downstream reservoir can be also enhanced by decreasing the water level through releasing floods in advance.

Through the analysis above, the risk of cascading breach should be analyzed under three other operating schemes: 1. Adding flood releasing structures. 2. Releasing floods in advance. 3. Taking both measures. Based on the calculation case in Section 3.3, the risk assessments under these operating schemes are also conducted in CRBS.

(a) Operating scheme 1

When the Tangjiashan barrier dam started to break at 06:00, through the flood warning, the downstream reservoir II immediately opened the water gate to release floods in advance. The FRC module is invoked to simulate the process of flood regulating in reservoir II, and the result is shown in Figure 12.

Due to the advanced flood regulation, the water level declined from 461.1 m (At 6:00) to 459.8 m at 9:00 when the dam-break flood routed to the downstream reservoir. After that, the water level still showed a downtrend until the inflow rate began to exceed the outflow rate at 11:24. The water level rose to 465.0 m at 14:35, which is the elevation of the dam crest. After that, the overtopping occurred, DB-IWHR is invoked to simulate the dam-break as shown in Figure 13. The dam-break of reservoir II started at 15:17, and the water level rose to its maximum value 471.3 m at 18:02 exceeding 5.3 m above the dam crest. Then, it declined to 388.5 m when the sequential dam-break finished at 15:58 the next day; the peak outflow rate 7830.7 m³/s appeared at 22:55; the final breach bottom elevation and width are 387.5 m and 144.7 m, respectively.

As can be seen from the simulation results, the sequential dam-break was delayed for 29 min. However, through releasing flood in advance according to the flood warning, the downstream reservoir II still broke sequentially. The analysis shows that the water level only decreased by 1.3 m as a result of the limited discharge capacity before the dam-break flood arrived at the downstream reservoir. Therefore, it is necessary to increase the discharge capacity of reservoir II to eliminate the risk of sequential dam-break.

(b) Operating scheme 2

The discharge capacity of reservoir II is increased by adding a discharge tunnel but there is no flood warning in operating scheme 2. When the upstream dam-break flood evolves to reservoir II at 9:00, the flood regulation begins. The discharge capacity curve and the results of FRC are shown in Figure 14 and Figure 15.

The water level began to decrease at 9:00 when the water gates were opened to release the flood. The water level rose to the dam crest at 16:10, and it reached its maximum value 465.7 at 17:36, which is only 0.7 m above the dam crest. Compared with the maximum water level 470.3 m in the calculation case of Section 3.3, it is obvious that setting the second discharge tunnel to increase the discharge capacity is highly effective to reduce the risk of sequential dam-break. It also can be inferred that the simulation for the process of flood regulating makes a big difference to the results of the risk assessment. Therefore, simulating the whole process of the cascade reservoir dam-break through CRBS in this study would be more feasible compared with the case study of [6], in which the process of flood regulating is ignored.

The sequential dam-break was still likely to occur because of the overtopping. Through checking the computations of DB-IWHR, the flow velocity above the dam crest was always smaller than the initial erosion velocity V_c, so that the embankment dam II did not break sequentially. However, overtopping means that the dam crest received scouring of floods, and the potential safety hazard was still remaining. Therefore, only increasing the discharge capacity cannot completely guarantee the safety of the downstream reservoir after the upstream dam-break.

(c) Operating scheme 3

The flood warning and another discharge tunnel are both set in operating scheme 3. The flood-regulating process is shown in Figure 16. The water level kept decreasing at first, and it began to increase at 12:10 when the inflow rate began to exceed the outflow rate. After that, it rose to the maximum value 463.4 m at 17:54, which is 1.6 m below the dam crest. The results show that the flood overtopping did not occur in reservoir II, and the sequential dam-break was avoided under operating scheme 3.

4. Conclusions

This study aims to fill a gap in the research, where there is no numerical prediction model that cam simulate the whole process of cascade reservoir dam-break. In this paper, the upstream and downstream reservoirs in a fundamental cascade reservoir system are connected through flood routing simulation (FRS) and flood-regulating calculation (FRC); after that, the numerical model for cascade reservoir breaching simulation (CRBS) is established based on the single-embankment dam-break model DB-IWHR. Through CRBS, the whole process of cascade reservoir dam-break can be simulated with few hydrogeology parameters. Moreover, the flow rate, water level and the shape of the breach during sequential dam-break can be computed without presetting parameters about the initial and final shape of the breach, which is of great engineering applied significance.

Based on the case study of a fundamental cascade reservoir (in the upstream Tangjiashan barrier lake and the downstream reservoir II), the whole process of cascade reservoir dam-break is simulated by CRBS. Through comparison between the measured data and the computed results of BREACH and DB-IWHR, selecting DB-IWHR as the single-embankment dam-break model was verified. The upstream dam-break occurred at 6:00; the outflow rate calculated by DB-IWHR is closer to the measured data than BREACH; the peak outflow rate 7472.3 m/s² computed by DB-IWHR was 14.6% greater than the measured data, which is in an acceptable rage as results of a prediction model; the simulating value of breach bottom elevation and water level differed a lot from the measured data in the later stage of the dam-break, which is caused by the lack of the simulation of sediment accumulation; In the simulation of flood routing, the dam-break flood arrived at reservoir II at 9:00 and the peak outflow rate was 6325.4 m/s² appeared at 14:24; according to the result of FRC, reservoir II was overtopped at 14:05 and the maximum value of water level was 470.3 m appeared at 17:48. DB-IWHR was invoked again to simulate the sequential dam-break of dam II. The results show that the sequential dam-break started at 14:48 when it reached the initial erosion rate V_c and it lasted for 24.6 h until the water level decreased to 388.4 m in the second day. The peak outflow rate was 8159.9 m/s² at 22:06.

Under three different operating schemes, when changing the discharge capacity of the downstream reservoir, the risk assessment of sequential dam-break was conducted by CRBS. When only given flood warning, reservoir II broke sequentially; when only adding the second discharge tunnel, reservoir II was overtopped but not broken sequentially; with the addition of the flood warning and the second discharge tunnel at the same time, reservoir II was not overtopped. The results show that both early flood warning and extra releasing structures can effectively reduce the risk of cascade reservoir dam-break, and they also prove that the processes of flood routing and regulation are nonnegligible in a numerical model for cascade reservoir dam-break. In summary, CRBS is proved to be of great value in the risk assessment, planning and design of cascade reservoir projects as a numerical prediction model for cascade reservoir dam-break.