Research Article Risk Analysis for Cascade Reservoirs ...when the water in Changheba reservoir is at...

Research ArticleRisk Analysis for Cascade Reservoirs Collapse Based onBayesian Networks under the Combined Action of Floodand Landslide Surge

Ping Li and Chuan Liang

College of Water Conservancy and Hydropower Engineering, Sichuan University, Chengdu 610065, China

Correspondence should be addressed to Chuan Liang; [email protected]

Received 7 August 2016; Revised 16 October 2016; Accepted 7 November 2016

Academic Editor: Gerardo Severino

Copyright © 2016 P. Li and C. Liang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A method based on a Bayesian network (BN) combined with stochastic Monte Carlo (MC) simulation is used in this researchto calculate the probability and analyze the risk of a single reservoir dam overtopping and two reservoirs collapsing under thecombined action of flood and landslide surge. Two adjacent cascade reservoirs on the Dadu River are selected for risk calculationand analysis.The results show that the conditional probability of a dam overtopping due to flooding in a single reservoir is relativelysmall; the conditional probability of a dam overtopping due to landslide surge in a single reservoir is relatively large; a combinationof flooding and landslide surge greatly increases the risk of the dam overtopping. The conditional probability that the dam in(downstream) Changheba reservoir overtops as a result of a dam-break flood from (upstream) Houziyan reservoir is greater than0.8 when the water in Changheba reservoir is at its normal level. Under the combined action of flooding and landslide surges, thejoint probability that the two cascade reservoirs collapse in a variety of typical situations is very small.

1. Introduction

According to the available statistics, over half of the cases ofembankment dam break are caused by a dam overtopping;that is, dam overtopping is the main reason that dam breaksoccur. As is well known, flooding is a major factor thatmay result in dam overtopping. In addition, a landslideinto a reservoir can arouse a huge surge which may alsolead to dam overtopping. If these two factors (flooding andlandslide surge) occur at the same time, then it may beexpected that the risk of the dam overtopping will be greatlyincreased. There have been many studies aimed at analyzingthe risk that a dam will overtop due to flooding, but little hasbeen done in respect of landslide surge. Stochastic modelshave been established for flood risk analysis and comparedwith a Bayes method [1]. Another approach used was toanalyze the mechanism responsible for dam failure from thepoint of view of hydraulics [2]. The latter authors used anintegrationmethod to directly calculate the probability of thedam overtopping. Other scholars have used (several) otherdifferent methods to calculate and analyze landslide surges

in the Three Gorges Reservoir [3]. An ISPH model has alsobeen used to simulate landslide surges in a reservoir and thesubsequent process of a dam overtopping [4].

In recent years, Bayesian networks (BNs) have been oneof the most successful artificial intelligence methods used todeal with uncertainty. For example, they have been used toanalyze the reliability and risk management of earthquakedisasters [5]. Other researchers have used BNs to studythe risk assessment surrounding warship retirement [6],proposed BNs to model the risk of natural disasters [7], anddiscussed the application of BNs to the analysis of fire risk [8].

As far as dam overtopping is concerned, the BN studiesundertaken so far mostly encompass just a single factor thatmay cause a dam to overtop; there are hardly any risk analysesfor dam overtopping which consider the combined action offlooding and landslide surges. In addition, the BN studiesaimed at analyzing the risk of dam overtopping must beconsidered as preliminary attempts. Such risk analyses onlyencompass a single reservoir (or a single system) withoutgiving consideration to multiple-reservoir collapse. To date,there is still very little research in the literature focusing on

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 2903935, 13 pageshttp://dx.doi.org/10.1155/2016/2903935

2 Mathematical Problems in Engineering

X1 X2

X3

Figure 1: A “V”-type Bayesian network.

analyzing the risk of collapse of multiple reservoirs usingBNs. Therefore, this study is based on BN theory and aims toestablish a BN model to calculate the probability of collapsefor two reservoirs under the combined action of floodingand landslide surge. One notable feature of the work is thata stochastic Monte Carlo (MC) simulation method is usedto estimate the probability of each node event in the BNstructures. We also consider some examples to show theeffectiveness of the combined BN/MC method as applied toanalyzing the risk of reservoir collapse.

2. Methods

2.1. Bayesian Networks. A Bayesian network is a conditionalprobability table that contains a directed acyclic graph modeland a complex joint-probability distribution of the compactrepresentation. A BN can be used to express and analyze theuncertainty or probability of events and allows one to makeinferences using incomplete or uncertain knowledge andinformation. A BN assigns the joint-probability distributionfor 𝑛 random variables 𝑋 = {𝑋1, 𝑋2, . . . , 𝑋𝑛}. The BN isexpressed via its conditional independence and conditionalprobability relationships and the joint-probability distribu-tion can be written in the compact form:

𝑃 (𝑋1, 𝑋2, . . . , 𝑋𝑛) = ∏𝑃(𝑋𝑖 | 𝜋𝑋𝑖) ,𝑃 (𝑋𝑖) = ∑𝑃(𝜋𝑋𝑖) 𝑃 (𝑋𝑖 | 𝜋𝑋𝑖) .

(1)

As an example, for the network diagram shown in Figure 1,the joint probability is

𝑃 (𝑋1, 𝑋2, 𝑋3) = 𝑃 (𝑋1) 𝑃 (𝑋2) 𝑃 (𝑋3 | 𝑋1, 𝑋2) ,𝑃 (𝑋3) = 𝑃 (𝑋1) 𝑃 (𝑋3 | 𝑋1)

+ 𝑃 (𝑋2) 𝑃 (𝑋3 | 𝑋2) .(2)

The Bayesian formula indicates that

𝑃 (𝜋𝑋𝑖 | 𝑋𝑖) = 𝑃 (𝑋𝑖) 𝑃 (𝑋𝑖 | 𝜋𝑋𝑖)∑𝑛𝑗=1 𝑃 (𝑋𝑗) 𝑃 (𝑋𝑗 | 𝜋𝑋𝑗) , (3)

where 𝑃(𝑋1, 𝑋2, . . . , 𝑋𝑛) is the joint probability, 𝑃(𝑋𝑖) is theprior probability, 𝑃(𝑋𝑖 | 𝜋𝑋𝑖) is the conditional probability,and 𝑃(𝜋𝑋𝑖 | 𝑋𝑖) is the posterior probability.2.2. Establishing the BN Model. Suppose that there are twoadjacent cascade reservoirs A and B. In addition to flooding,landslide surges may also occur and pose a risk of causingdam overtopping to a single reservoir. That is, these twofactors together constitute the causes leading to a singlereservoir dam overtopping. For two adjacent reservoirs, inaddition to flooding and landslide surge, dam-break floodingfrom upstream reservoir A is another main factor that canlead to the dam overtopping in downstream reservoir B. Thatis, these three factors together constitute the causes of thedam in reservoir B overtopping. As dam overtopping is acrucial factor that can result in an earth-rock dam break, itcan be considered that “dam overtopping” means the sameas “dam break.” Therefore, in this study, the effects of otherfactors, for example, failure of the structure of the dam, arenot considered.

The BN structure for two reservoirs collapsing under thecombined action of flooding and landslide surge is shown inFigure 2(a). According to the conditional independence ofthe BN, the reason that dam break in reservoir A has a directeffect on reservoir B is only via the dam-break flood. Thus,all the causal nodes resulting in dam break in reservoir A canbe condensed down to one causal node (dam-break flood).Subsequently, the BN structure can also have the form shownin Figure 2(b). The definitions of the various parametersappearing in the BNs are given in “Abbreviations.”

According to the BN structures described above we havethe following.

For a single reservoir,

𝑃 (𝑂𝐴) = 𝑃 (𝐹𝑂)= 𝑃 (𝐹𝐴) 𝑃 (𝑂𝐴 | 𝐹𝐴) + 𝑃 (𝑆𝐴) 𝑃 (𝑂𝐴 | 𝑆𝐴) ,

𝑃 (𝑂𝐵) = 𝑃 (𝐹𝐴) 𝑃 (𝑂𝐴 | 𝐹𝐴) + 𝑃 (𝑆𝐴) 𝑃 (𝑂𝐴 | 𝑆𝐴) .(4)

For two reservoirs,

𝑃 (𝑂𝐵)= 𝑃 (𝐹𝐵) 𝑃 (𝑂𝐵 | 𝐹𝐵) + 𝑃 (𝑆𝐵) 𝑃 (𝑂𝐵 | 𝑆𝐵)

+ 𝑃 (𝐹𝑂) 𝑃 (𝑂𝐵 | 𝐹𝑂) ,𝑃 (𝐹𝐴𝑆𝐴𝐹𝑂𝐹𝐵𝑆𝐵𝑂𝐵) = 𝑃

= 𝑃 (𝐹𝐴) 𝑃 (𝑆𝐴) 𝑃 (𝐹𝐵) 𝑃 (𝐹𝑂) 𝑃 (𝑆𝐵) 𝑃 (𝑂𝐵) .

(5)

2.3. Probability of Overtopping due to Flooding

2.3.1. Probability of Flooding. InChina, it is generally believedthat the peak flow obeys a type P-III probability distribution.Our main calculation method is based on observed flooddata and uses the moment method to estimate statisticalparameters for the mean, 𝑥, coefficient of variation (𝐶V), andcoefficient of skewness (𝐶𝑠). The data was fitted to the P-III

Mathematical Problems in Engineering 3

P(FA)

P(OA|FA)

P(OA)

P(OA|SA)

P(SA)

FA OA SA

P(O

B|O

A)

FBP(OB|FB) OB SB

P(FB) P(OB) P(SB)

P(OA|SB) P(OB|FB) OB SB

P(FB) P(OB) P(SB)

P(FO)

FO

FBP(OA|SB)

P(O

B|F

O)

(a) (b)

Figure 2: BN structures for two-reservoir collapse.

distribution curve using eye-estimation, and the probabilitiesthat the peak flows occurredwere obtained.That is, the valuesof 𝑃(𝐹𝐴) and 𝑃(𝐹𝐵) were derived.2.3.2. Conditional Probability of Overtopping due to Flooding

(1) Stochastic Method of Simulating Flood Peak Flows. Here,we use an acceptance-rejection sampling method based onstochasticMonte Carlo simulation to estimate the value of theconditional probabilities 𝑃(𝑂𝐴 | 𝐹𝐴) and 𝑃(𝑂𝐵 | 𝐹𝐵). Theacceptance-rejection method is applied to convert the P-IIItype probability distribution function. Thus, we have

𝑥𝑡 = 𝑎0 + 1𝛽 (1 −[𝛼]∑𝑘=1

ln 𝜉𝑡𝑘 − 𝐵𝑡 ln 𝜉[𝛼]+3) , (6)where 𝛼, 𝛽, and 𝑎0 are three parameters that are used todescribe the P-III type curve. During the simulation, we let𝑡 = 1, 2, . . . , 𝑁, so that𝑁 purely random values are generatedwhich obey the P-III type distribution using random numbergeneration.

(2) Calculation of 𝑃(𝑂𝐴 | 𝐹𝐴) and 𝑃(𝑂𝐵 | 𝐹𝐵). For reservoirs𝐴 and 𝐵, theMonte Carlo method is used to simulate𝑁𝐴 and𝑁𝐵 peak flow data points, respectively. Then, by amplifyingthe typical flood hydrograph using the same magnification,we can get the 𝑁𝐴 and 𝑁𝐵 flood hydrographs. The reservoircapacity and discharge flow curves are combined. Using theseand the flood hydrographs, we can obtain𝑁𝐴 and𝑁𝐵 highestwater levels in front of the dam according to the reservoirdispatching rules for flood routing through the reservoir.Thenumber of times (𝑚𝐴, 𝑚𝐵) wherein the water level exceedsthe height of the dam in 𝑁𝐴 and 𝑁𝐵 highest water levels iscounted, respectively. The value of m/N can be consideredto be the conditional probability of the dam overtopping (or,alternatively, the dam break) due to flooding. That is, 𝑃(𝑂𝐴 |𝐹𝐴) = 𝑚𝐴/𝑁𝐴, and 𝑃(𝑂𝐵 | 𝐹𝐵) = 𝑚𝐵/𝑁𝐵.2.4. Probability of Overtopping due to Landslide Surge. If theslope of a reservoir bank has poor stability, it may slideunder the disturbing action of an earthquake or heavy rain.A landslide often provokes a huge surge, which may, in turn,

lead to the damovertopping.Therefore, if wewant to calculatethe conditional probability of a dam overtopping due tolandslide surge, the first thingwe need to do is calculating andanalyzing the stability of the slope. Here, we assume that theprobability of a landslide occurring is equal to the probabilityof a landslide surge occurring. Then, we can calculate theconditional probability of the dam overtopping due to alandslide surge according to the calculated probability of alandslide surge occurring combined with a stochastic MCsimulation.

2.4.1. Calculation of the Stability with respect to Landslide.Landslide stability is generally measured by the coefficient ofstability 𝐾. If 𝐾 > 1, it indicates that the slope is relativelystable, and landslide will not occur. If 𝐾 < 1, the opposite istrue.

In this work, the Bishop method is adopted to calculatethe landslide stability coefficient. Ignoring the effect of porewater pressure, the stability coefficient is given by

𝐾 = ∑𝑟𝑖=1 (1/𝑚𝛼𝑖) [𝑐𝑏𝑖 + 𝑊𝑖 tan𝜑]∑𝑟𝑖=1𝑊𝑖 sin𝛼𝑖 ,𝑚𝛼𝑖 = cos𝛼𝑖 + tan𝜑𝐾 sin𝛼𝑖,

(7)

where 𝛼𝑖, 𝑏𝑖, and 𝑊𝑖 are, respectively, the average slope angle,width, and weight of the 𝑖th soil slice, r is the total number ofslices, 𝑐 is the cohesion, and 𝜑 is the internal friction angle.2.4.2. Calculation of the Landslide Surge. Sliding velocity hasa great impact on a landslide surge. Therefore, the first stepin the landslide surge calculation is to calculate the slidingvelocity of the landslide.

(1) Sliding Velocity. Of the various methods available tocalculate the landslide sliding velocity, the energy method isthe most widely used method and the concepts involved areclear. The sliding velocity (V) is given by

𝑉 = √2𝑔ℎ ⋅ √1 − cot𝛼 ⋅ tan𝜑 − 𝑐𝐿𝑀𝑔 sin𝛼 . (8)


In this expression, 𝑔 is the acceleration due to gravity(9.8m/s2), h is the height of the center of gravity of thelandslide body with respect to the surface of the water in thereservoir (m), 𝛼 is the average dip angle of the sliding surface(∘), c is the cohesion (kPa), 𝜑 is the internal friction angle (∘),L is the length of the landslide body (m), and 𝑀 is the massof the landslide body (kg).

(2) Landslide Surge. After comparing several different meth-ods (including the American Society of Civil Engineersrecommended method, the method of the China Institute ofWater Resources and Hydropower Research, and the Panji-azheng method), Hu et al. pointed out that the Panjiazhengmethod is relatively consistent with the actual situation[3]. This method is therefore adopted here to calculate thelandslide surge occurring in the reservoir.

The surge height on the other side of a landslide, 𝜂 (m), isgiven by

𝜂 = 2𝜂0𝜋 (1 + 𝑘)𝑛∑

𝑛=1,3,5,...

{{{𝑘2(𝑛−1)

⋅ ln[[

𝑙(2𝑛 − 1) 𝐵 + √1 + [ 𝑙(2𝑛 − 1) 𝐵]2]]}}}

(9)

and the surge height in front of the dam, 𝜂𝑥 (m), is given by𝜂𝑥 = 𝜂0𝜋

𝑛∑𝑛=1,3,5,...

(1 + 𝑘 cos 𝜃𝑛) 𝑘(𝑛−1)

⋅ ln[[[

√1 + (𝑛𝐵/ (𝑥0 − 𝐿))2 − 1(𝑥0/ (𝑥0 − 𝐿))√1 + (𝑛𝐵/𝑥0)2 − 1

]]]

,(10)

where 𝜂0 is the initial surge height caused by the landslide(m), k is the wave reflection coefficient (𝑘 ≈ 1), 𝑙 = 𝐿/2 isthe half length of the landslide along the bank (m), B is theaverage breadth of reservoir’s water surface (m), and 𝑥0 is thedistance from the calculation point to the landslide (m).

The angles 𝜃𝑛 are given bytan 𝜃1 = 𝑥0𝐵 ,tan 𝜃3 = 𝑥03𝐵 ,tan 𝜃𝑛 = 𝑥0𝑛𝐵 .

(11)

2.4.3. Calculation of the Probability of a Dam Overtoppingdue to Landslide Surge. As a result of the perturbation fromexternal factors, the cohesion (c), internal friction angle (𝜑),andweight (W) of the landslide will be uncertain; that is, theyshould be viewed as random variables. Hence, the values ofthe stability coefficient (K) and sliding velocity (V) are alsouncertain. In view of this, we use a stochastic Monte Carlomethod to simulate these random variables in order to obtainvalues for 𝐾 and 𝑉. Then, the prior probability of a landslide

Potentiallandslide

Water levelReservoir

Dam

Trun

k str

eam

b

bili

mih

ai

x0

Figure 3: A simple analysis model for the landslide surge simula-tions.

surge occurring and the conditional probability of the damovertopping due to the landslide surge can be computedaccording to the simulation results. The assumptions madeand simulation steps involved are as follows.

(1) Assume that there is a potential landslide at a distance𝑥0 from the dam. The breadths of the landslide (b) andlandslide height h are random variables that obey normaldistributions. The breadth of each soil slice of the landslidein the vertical direction is divided into 1m pieces; that is, thetotal number of soil slices is b.

(2) Assume that the length (𝑙𝑖), thickness (𝑡𝑖), slope angle(𝛼𝑖), cohesion (𝑐𝑖), and internal friction angle (𝜑𝑖) of soil slice𝑖 are all random variables which obey normal distributions.We also have 𝑊𝑖 = 𝑙𝑖𝑡𝑖𝛾, where 𝛾 is the volumetric weigh ofthe soil. Figure 3 shows a simplemodel used in the simulationanalysis.

(3) Following theMCmethod,N group randomnumbersobeying a normal distribution were obtained after 𝑁 simu-lations for each random variable (the calculations here wereperformed using MATLAB�). Then, the 𝑁 group randomnumbers were substituted into the formulae for 𝐾 and 𝜂𝑥to give 𝑁 values of 𝐾 and 𝜂𝑥. The number of 𝐾 values lessthan 1 was counted as was the number of the values of 𝜂𝑥 thatexceeded the dam height when added to the reservoir waterlevel.This yielded values for𝑚𝐾 and𝑚𝜂, respectively. Finally,the probabilities 𝑃(𝑆𝐴) (or 𝑃(𝑆𝐵)) = 𝑚𝐾/𝑁 and 𝑃(𝑂𝐴 | 𝑆𝐴)(or 𝑃(𝑂𝐵 | 𝑆𝐵)) = 𝑚𝜂/𝑁 were determined.2.5. Conditional Probability of a Dam Overtopping due to aDam-Break Flood. The risk analysis for the occurrence of thedam overtopping in (downstream) reservoir B due to a dam-break flood from (upstream) reservoir A involves two mainprocesses: (i) the dam break in reservoirA itself and (ii) floodrouting of the dam break from reservoir A to reservoir B. Weconsider each in turn.

2.5.1. The Dam-Break Process in Upstream Reservoir A(1) Dam Break Maximum Flow. We consider the worst casescenario; that is, the dam undergoes instantaneous failure.

Mathematical Problems in Engineering 5Q/Q

m

Q0/Qm0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

t/T

Figure 4: A typical profile for the flow resulting from an instan-taneous dam-break flood (𝑄0 is the inflow before dam break, 𝑄𝑚is the dam break maximum flow, and 𝑇 = 5𝑉/𝑄𝑚, where 𝑉 is themaximum capacity).

The maximum flow 𝑄𝑚 can be found using the Xierenzhiformula [9]:

𝑄𝑚 = 𝜆𝐵√𝑔𝐻1.50 ,𝜆 = 𝑚𝑚−1 [2√𝑚 + V0/√𝑔𝐻01 + 2𝑚 ]

2𝑚+1 . (12)

In these expressions, B is the dam crest length (m), 𝐻0 is thewater depth in front of the dam (m), 𝜆 is a traffic parameter,M is the valley cross section shape index, and V0 is the averageflow velocity in the channel before dam break (m/s).

(2) Dam Flood Process. This consists of three parts: (i)reservoir discharge before dam break, (ii) flow during dambreak, and (iii) natural runoff after the reservoir dischargeends.The shape of the instantaneous dam-break floodprocessis akin to four parabolic processes [10] (Figure 4).

2.5.2. Flood Routing Process. The routing process of a dam-break flood from an upstream to a downstream reservoir canbe enumerated using the Muskingum method [11]:

𝑄2 = 𝐶0𝐼2 + 𝐶1𝐼1 + 𝐶2𝑄1,𝐶0 + 𝐶1 + 𝐶2 = 1,

𝐶0 = 0.5Δ𝑡 − 𝑇𝑥𝑇 − 𝑇𝑥 + 0.5Δ𝑡 ,𝐶1 = 0.5Δ𝑡 + 𝑇𝑥𝑇 − 𝑇𝑥 + 0.5Δ𝑡 ,𝐶2 = 𝑇 − 𝑇𝑥 − 0.5Δ𝑡𝑇 − 𝑇𝑥 + 0.5Δ𝑡 ,

(13)

where 𝐼1 and 𝐼2 are the upper-section flow (m3/s) at thebeginning and end of the process, respectively, 𝑄1 and 𝑄2are the lower-section flow (m3/s) at the beginning and end ofthe process, respectively, Δ𝑡 is the time step used (s), T is thepropagation time of the flood in the reach of the river understeady flow (s), and 𝑥 is the tank storage coefficient.

2.5.3. Calculation of 𝑃(𝑂𝐵 | 𝐹𝑂). Suppose reservoir A isput into 𝑋 different water level conditions which cause dambreak to occur. Then, X maximum flood peak flows can beobtained after the dam-break flood routes to the downstreamreservoir B by using the Muskingum method. Assume thatthis peak flow series obeys a P-III distribution, then, tocalculate the statistical parameters (mean (𝑥), coefficientof variation (𝐶V), and coefficient of skewness (𝐶𝑠)) fromthe flood series, combined with a Monte Carlo method forstochastic simulation to get 𝑁 peak flow data points, and𝑁 flood hydrographs will be obtained after amplifying theselected typical flood hydrograph. Finally, in accordance withthe capacity and discharge curves and the flood dispatchingrules of the downstream reservoir B, the 𝑁 highest waterlevels in front of the dam can be calculated according to thereservoir flood regulating calculation. If there are 𝑚 highestwater levels in front of the dam that exceed the dam height ofreservoir B, thenm/N can be considered to be the conditionalprobability of the dam overtopping in reservoir B due to adam-break flood from upstream reservoir A; that is, 𝑃(𝑂𝐵 |𝐹𝑂) = 𝑚/𝑁.3. Results and Discussion3.1. Project Overview. The Houziyan and Changheba reser-voirs are adjacent in the Dadu River trunk stream in Sichuan,China, and will be analyzed using our method. The riverfrom the Houziyan to Changheba reservoirs runs throughmountainous terrain in deep ravines. Deep, narrow valleyscontain water that is fast-flowing and, thus, complex.

Houziyan Hydropower Station is located in KangdingCounty, Sichuan. It forms a 9-cascade hydropower station onthe Dadu River which is subjected to 22 development pro-gram recommendations. The reservoir’s dam is an embank-ment dam, and the dam site above the controlled river basinarea covers 54,036 km2. The design standard for the reservoiris 𝑃 = 0.1% and the checking standard is the probablemaximum flood (PMF). The design flood level is 1841.55m,the flood check level is 1845.41m, and the normal storagelevel is 1842m.The dam’s total capacity is 706 millionm3 andthe normal storage level capacity is 662 millionm3. The damcrest elevation is 1848.5m, the dam is 223.5m high, and thecrest length of the dam is 283m. The dead level is 1802.00mand the regulating capacity is 3.87 millionm3, with seasonalregulation of performance applied.

Changheba Hydropower Station is a 10-cascade hydro-power station on the Dadu River scheme which is down-stream of the Houziyan Hydropower Station. The site islocated in Kangding County, Sichuan. It is located on thetrunk stream of Dadu River about 4–7 km below the Jintangriver estuary. The reservoir dam is an embankment dam.Theriver basin area above the dam site measures 56,648 km2,accounting for 73.2% of the total basin area. Its designstandard is 𝑃 = 0.1% and the checking standard is thePMF. The design flood level is 1690m, the checking floodlevel is 1694.6m, and the normal storage level is 1690m. Thetotal capacity is 1.075 billionm3, the normal storage levelcapacity is 1.015 billionm3, and the regulating capacity is 4.15millionm3.The dam crest elevation is 1697m, the dam height


Table 1: Designed flood results for the Houziyan and Changheba reservoirs.

Reservoir Statistical parameter Designed valueMean (m3/s) 𝐶V 𝐶𝑠/𝐶V Criterion 𝑃 Value (m3/s)

Houziyan 3300 0.26 5 Check PMF 9940Design 0.1% 7550

Changheba 3550 0.24 5 Check PMF 10400Design 0.1% 7650

Table 2: Simulation parameters and calculated probabilities.

Reservoir Simulationtime, 𝑁 Probabilityinterval, Δ𝑃Adjusted water

level Water level overdam (times),mDam overtopping

frequency, 𝑃 = 𝑚/𝑁 Overtopping probabilityCase Value (m) Probability Δ𝑃 ∗ 𝑃

Houziyan 1000 0∼0.0001 Normal 1842 17 0.017 𝑃 (𝑂𝐴 | 𝐹𝐴) 1.70 × 10−61000 0∼0.0001 Check 1845.41 21 0.021 𝑃 (𝑂𝐴 | 𝐹𝐴) 2.10 × 10−6

Changheba 1000 0∼0.0001 Normal 1690 14 0.014 𝑃 (𝑂𝐵 | 𝐹𝐵) 1.40 × 10−61000 0∼0.0001 Check 1694.6 17 0.017 𝑃 (𝑂𝐵 | 𝐹𝐵) 1.70 × 10−6

Table 3: Statistical parameters for the random variables.

Parameter 𝑙𝑖 (m) 𝑡𝑖 (m) c (kPa) 𝜑 (∘) 𝛼𝑖 (∘) 𝑟 ℎ (m)Mean 1500 20 35 35 45 150 30Standard deviation 880 12 5 4.5 27 85 18

Table 4: Calculated prior probabilities for the occurrence of a landslide surge.

Dam-landslidedistance, 𝑥0 (km)

Soil-slice parameter Simulation and calculation results

Width, 𝑏𝑖 (m) Volumetric weight, 𝛾(kN/m3) Simulation time,𝑁 Number of times𝐾 < 1, 𝑚𝐾 Landslide surge probability,𝑃(𝑆𝐴)/𝑃(𝑆𝐵)3 1 20 10000 387 0.0387

is 240m, and its crest length is 497.94m. The river channelfrom the Houziyan reservoir to the downstream Changhebareservoir is about 50 km long.

3.2. Risk Analysis for a Single Reservoir Overtopping due toFlooding. The statistical parameters relating to the inflowflood peak series for the Houziyan reservoir and the inter-val flood peak series for the Changheba reservoir areshown in Table 1 together with the corresponding designvalues.

After selecting an appropriate probability range for thesimulations and calculations, the conditional probabilities offlooding leading to dam overtopping for different reservoirwater levels were obtained (according to the methods out-lined above for calculating the probability of peak floodsoccurring and conditional probability of flooding leading todam overtopping). The simulation parameters and results ofthe probability calculations are shown in Table 2.

The results shown in Table 2 indicate that the conditionalprobability that flooding will result in the dam overtoppingin either of the two reservoirs is of the order of 10−6.This shows that the risk of flooding leading to a singlereservoir dam overtopping is very small. It also shows thatthe difference between the risks for the two reservoirs issmall. It is also apparent that the higher the water level in a

reservoir, the greater the risk of the dam overtopping due toflooding.

3.3. Risk Analysis for a Single Reservoir Overtopping due toLandslide Surge. With reference to landslide investigationsin the Dadu River basin (and other related literature),values were determined for the statistical parameters of eachrandom variable (Table 3).Then, 10,000MC simulations weremade (using MATLAB software) to calculate the stabilitycoefficients and other characteristics of the landslide surge.Subsequently, the prior probability that a landslide surgewill occur and the conditional probability that the damwill overtop due to the landslide surge were calculated fordifferent water levels in the reservoir. The parameters andsimulation results are shown in Tables 4 and 5.

The results in Tables 4 and 5 indicate that the prior prob-ability of occurrence of a landslide (surge) in the reservoir is0.0387. The conditional probability that the dam will overtopdue to a landslide surge when either the Houziyan reservoiror Changheba reservoir is at normal or check water levelis about 0.2. Clearly, landslide surge has a relatively largerisk of leading to dam overtopping compared to flooding. Inaddition, the higher the water level in the reservoir is, thelarger the conditional probability that a landslide surge willlead to dam overtopping; that is, the risk of dam overtoppingis greater.


Table5:Con

ditio

nalprobabilityresults

ford

amovertopp

ingas

aresulto

flandslid

esurge

ford

ifferentw

ater

levelsin

ther

eservoirs.

Reservoir

Water

level

Meanwater

Surfa

cewidth,𝐵(

m)

Meanwater

depth,

d(m

)Coefficientk

Volumetric

weight,

𝛾(kN/m3)

Simulation

time,N

Timeo

fdam

overtopp

ing,

𝑚 𝜂Overtop

ping

prob

ability

Case

Value(m)

Prob

ability

Value

Hou

ziyan

Normal

1842

580

86.59

120

1000

01936

𝑃 (𝑂𝐴

|𝑆 𝐴)

0.1936

Check

1845.41

630

901

201000

02352

𝑃(𝑂𝐴

|𝑆 𝐴)

0.2352

Changh

eba

Normal

1690

620

95.4

120

1000

01884

𝑃(𝑂𝐵|𝑆 𝐵

)0.1884

Check

1694.6

660

100

120

1000

02126


)0.2126


1,821 1,823 1,825 1,827 1,829 1,831 1,835 1,839 1,842 1,844 1,847 1,849 1,852

Maximum flow of dam-break flood at HouziyanMaximum flow of dam-break flood at Changheba

Water level of reservoir (m)

0

10000

20000

30000

40000

50000

60000

70000

Q(m

3/s

)

Figure 5: Maximum flows associated with the dam-break flood (at the Houziyan reservoir and after routing to the Changheba reservoir).

Q(104

m3/s

)

0

0.5

1

1.5

2

2.5

3

3.5

4

Qm

Q0

2 4 6 8 10 12 14 16 18 20 22 240

t (h)

Figure 6: Typical process line for an instantaneous dam-break floodat the Houziyan reservoir.

3.4. Risk Analysis for Dam Overtopping due toDam-Break Flooding

3.4.1. Dam-Break Flooding Process

(1) Dam-Break Flood Maximum Flows. We assume that theHouziyan reservoir undergoes a dam break under the actionof excessive flooding. The calculated maximum flows areshown in Figure 5.

(2) Dam-Break Flow. A profile can be drawn to illustrate thetypical flow process resulting from a dam break (Figure 6)based on the maximum flow results for a dam-break floodgiven above and the generalized flooding profile for aninstantaneous dam break. In addition, by combining theupstream flow process from the Houziyan reservoir and thedischarge process during reservoir flood regulation, morecomprehensive flood profiles, before and after dambreak, canbe constructed, as shown in Figure 7.

(3) Dam-Break Flood Routing. The Muskingum method cannow be used to calculate the flood routing process basedon the flooding process lines before and after dam breakat the Houziyan reservoir. Then, the maximum flows and

Table 6: Calculation parameters for the Muskingum method.

Parameter 𝑥 T (h) Δt (h) C0 C1 C2Value 0.25 3 3 0.2 0.6 0.2

the flood process can be obtained after the dam-break floodroutes from the Houziyan reservoir to the inflow section ofthe downstream Changheba reservoir. The maximum flowsat the inflow section of the Changheba reservoir are shown inFigure 5, and a typical flood process line is shown in Figure 7.The calculation parameters for the Muskingum method canbe found in Table 6.

Figure 5 shows that if the Houziyan reservoir undergoesa dam break, then the higher the water level in the Houziyanreservoir, the larger the maximum flow. Furthermore, Fig-ure 7 shows that the reservoir discharge increases suddenlywhen the dam break occurs. After the dam-break flood prop-agates to downstream Changheba reservoir, the maximumflows are reduced but the shape of the flood process line isbasically unchanged.

3.4.2. Conditional Probability Calculation. Based on themax-imum inflow flood peak series at the Changheba reservoircoming from the dam-break flood from the Houziyan reser-voir, the statistical parameters for the inflow flood, and theflow results subject to various probabilities, can be obtainedafter fitting the data to a P-III curve (Table 7). Then, furtherMATLAB simulations and calculations can be carried outto yield the conditional probability that the Changhebareservoir suffers damovertopping due to the dam-break floodfrom the Houziyan reservoir (results in Table 8).

As can be seen from Table 8, when the Changhebareservoir is at a normal (check) water level, the conditionalprobability of the dam overtopping due to a dam-break floodfrom the Houziyan reservoir is 0.8075 (0.8981). It is clear thata dam break from the Houziyan reservoir induces a large riskthat the dam in the Changheba reservoir will overtop; that is,the risk that both reservoirs collapse is relatively large. Also,we can see that the higher the water level in the Changheba


Dam-break flood at upstream reservoirDam-break flood at downstream reservoir

Dam-break flood at upstream reservoirDam-break flood at downstream reservoir

10th flooding 20th flooding

Q(m

3/s

)

Q(m

3/s

)

0

10000

20000

30000

40000

10 20 30 40 50 60 70 80 900

t (h)

0

10000

20000

30000

40000

50000

60000

10 20 30 40 50 60 70 80 900

t (h)

Figure 7: Typical flood process lines before and after dam break at the Houziyan reservoir and at the inflow section of the Changhebareservoir.

Table 7: Statistical parameters for the dam-break flood at the Changheba reservoir and the designed flood results.

Item Statistical parameter Designed P valueMean 𝐶V 𝐶𝑠/𝐶V 1% 2% 5% 10% 20% 50%𝑄max (m3/s) 31740 0.4 1 64650 60200 53770 48320 42020 30910

Table 8: Simulation parameters and conditional probabilities that the Changheba reservoir dam will overtop due to a dam-break flood fromthe Houziyan reservoir.

Reservoir Simulationtime, 𝑁 Probabilityinterval, Δ𝑃 Adjusted water level Water level overdam, 𝑚 (times) Overtoppingfrequency, 𝑃 = 𝑚/𝑁 Overtopping probabilityCase Value (m) Probability Δ𝑃 ∗ 𝑃Changheba 10000 0∼1 Normal 1690 8075 0.8075 𝑃 (𝑂𝐵 | 𝐹𝑂) 0.8075

10000 0∼1 Check 1694.6 8981 0.8981 𝑃 (𝑂𝐵 | 𝐹𝑂) 0.8981

reservoir, the greater the risk of the dam overtopping due tothe dam-break flood.

3.5. Risk Analysis for Reservoir Collapse under the CombinedAction of Flood andLandslide Surge. Here,we choose two setsof typical cases for the Houziyan and Changheba reservoirs.The upstream natural flood peak flow is at the designedflood flow, and the reservoir is either at the normal or checkwater level. Both reservoirs are considered to be at risk ofovertopping due to landslide surge.The probability results fordam overtopping in the Houziyan and Changheba reservoirsare shown in Table 9. The two typical cases mentioned abovefor the two reservoirs are combined into four cases (labeledI–IV). The probability results for two reservoir collapses forthe various combinations of typical cases, and the posteriorprobability results, are shown in Table 10 and Figure 8.

From Table 9, we can see there is little difference betweenthe probabilities that a single reservoir damwill overtop in thefour cases when the two reservoirs are under the combinedaction of flood and landslide surge (all are of the order of10−3). Once again, the higher the water level in the reservoiris, the greater the probability that a single reservoir dam willovertop.Obviously, the probability of damovertopping underthe combined action of flooding and landslide surge is much

larger than that resulting from flooding alone. As shown inTable 10 and Figure 8, dam-break flooding from theHouziyanreservoir increases the probability of theChangheba reservoirdam overtopping by a factor of almost two. This shows thatthe chance that a dam-break flood results in the Changhebareservoir dam overtopping is relatively large. In addition,for the various combinations of typical cases, the higher thewater level in a reservoir, the greater the probability of theChangheba reservoir dam overtopping and the greater thejoint probability. The joint probabilities of the two reservoirscollapsing under the combined action of flood and landslidesurge shown in Table 10 are very small (of the order of 10−15).The posterior probability results indicate that there is littledifference between the risk of the Changheba reservoir damovertopping due to dam-break flooding from the Houziyanreservoir and that from the combined action of intervalflooding and landslide surge in the Changheba reservoir.

4. Conclusions

We conclude the following:

(1) The probability that the dams in the Houziyan andChangheba reservoirs overtop due to a single flood


Table9:Prob

abilitie

sfor

asingler

eservoirdam

overtopp

ingas

aresulto

fthe

combinedactio

nof

flood

andland

slide

surge.

Reservoir

Reservoirw

ater

level

Prob

abilityof

flood

causingdam

overtopp

ing

Prob

abilityof

dam

overtopp

ingdu

eto

land

slide

surge

Prob

abilityof

asingle

reservoird

amovertopp

ing

𝑃(𝑂𝐴)/𝑃(

𝑂 𝐵)Ca

seVa

lue(m)

Prob

abilityof

occurrence

offlo

odCon

ditio

nalprobability

offlo

odcausingdam

overtopp

ing

Prob

abilityof

occurrence

ofland

slide

surge

𝑃(𝑆 𝐴)/𝑃(

𝑆 𝐵)Con

ditio

nal

prob

abilityof

dam

overtopp

ing𝑃(𝑂𝐴

|𝑆 𝐴)/


)Ca

seFlow

(m3/s)

Prob

ability

𝑃(𝐹𝐴)/𝑃(

𝐹 𝐵)𝑃(𝑂𝐴

|𝐹 𝐴)/𝑃

(𝑂 𝐵|𝐹 𝐵

)Hou

ziyan

Normal

1842

Designed

7550

0.001

1.70×10−06

0.0387

0.1936

0.0075

Check

1845.41

Designed

7550

0.001

2.10

×10−06

0.0387

0.2352

0.00

91

Changh

eba

Normal

1690

Designed

7650

0.001

1.40×10−06

0.0387

0.1884

0.0073

Check

1694.6

Designed

7650

0.001

1.70×10−06

0.0387

0.2126

0.0082


Table10:P

robabilityof

two-reservoirc

ollap

seforthe

vario

uscombinatio

nsof

typicalcases

(und

erthec

ombinedactio

nof

flood

andland

slide

surge).

Num

bero

fcombinedcases

Hou

ziyanreservoir

Changh

ebar

eservoir

Overtop

ping

prob

ability

duetodam-break

flood

,𝑃(𝑂𝐵|𝐹 𝑂

)Prob

abilityof

Changh

ebar

eservoir

dam

overtopp

ing𝑃(𝑂𝐵)

Jointp

robabilityof

tworeservoirs

collapse𝑃

Upstre

amflo

odWater

level

Poste

rior

prob

ability

Intervalflo

odWater

level

Poste

rior

prob

ability

IDesigned

Normal

0.227

Designed

Normal

0.273

0.8075

0.0133

1.09×10−15

IIDesigned

Normal

0.225

Designed

Check

0.275

0.8981

0.0150

1.38×10−15

III

Designed

Check

0.251

Designed

Normal

0.249

0.8075

0.0146

1.46×10−15

IVDesigned

Check

0.249

Designed

Check

0.251

0.8981

0.0164

1.84×10−15

12 Mathematical Problems in EngineeringP

(10−15)

Joint probability P P(OB)

IVIIIIIINumber of combined cases

IVIIIIIINumber of combined cases

1

1.25

1.5

1.75

2

1

1.25

1.5

1.75

2

P(10−2)

Figure 8: The probabilities that the two reservoirs collapse for the four combined cases and for the combined action of flood and landslidesurge.

action is very small, with a magnitude of the order of10−6. The risk of overtopping due to a landslide surgeoccurring in one of the reservoirs is relatively large.The conditional probability that damovertoppingwilloccur due to landslide surge in the Houziyan andChangheba reservoirs (at their normal water levels)is about 0.2. If the reservoir level is higher, however,the overtopping probability will be greater.

(2) The conditional probability of the Changheba reser-voir dam overtopping due to dam-break floodingfrom the Houziyan reservoir is 0.808 when theChangheba reservoir is at its normal level (and 0.898if it is at its check level). Such dam-break floodingnearly doubles the probability of the dam in theChangheba reservoir overtopping. Thus, the risk ofthe dam overtopping in the Changheba reservoir dueto dam-break flooding from theHouziyan reservoir isrelatively large.

(3) The joint probability that the two reservoirs willcollapse under the combined action of flooding andlandslide surge is very small. For various combina-tions of typical cases, the probability of such an eventwas found to be of the order of 10−15.

(4) Combining Bayesian network theory with stochasticMonte Carlo simulation is an effective method forcalculating probabilities and analyzing the risk asso-ciatedwith a single reservoir damovertopping and forinvestigating whether multiple cascade reservoirs willcollapse under the combined action of various risksand working conditions. This study may constitute afirm basis for providing useful technical support withrespect to risk prevention and control in reservoirs inthe future.

Abbreviations

𝐹𝐴: Upstream natural flooding of reservoir 𝐴𝐹𝐵: Interval flooding, coming from the areabetween reservoirs 𝐴 and 𝐵𝐹𝑂: Dam-break flooding from reservoir 𝐴𝑆𝐴: Landslide surge in reservoir 𝐴𝑆𝐵: Landslide surge in reservoir 𝐵

𝑂𝐴: Dam overtopping in reservoir 𝐴𝑂𝐵: Dam overtopping in reservoir 𝐵𝑃(𝑂𝐴 | 𝐹𝐴): Conditional probability of damovertopping in reservoir 𝐴 due to flooding𝑃(𝑂𝐵 | 𝐹𝐵): Conditional probability of damovertopping in reservoir 𝐵 due to flooding𝑃(𝑂𝐵 | 𝐹𝑂): Conditional probability of damovertopping in reservoir 𝐵 due todam-break flooding from reservoir 𝐴𝑃(𝐹𝐴): Prior probability of occurrence of floodingin reservoir 𝐴𝑃(𝐹𝐵): Prior probability of occurrence of intervalflooding𝑃(𝐹𝑂): Probability of occurrence of dam-breakflooding (𝑃(𝑂𝐴) = 𝑃(𝐹𝑂))𝑃(𝑆𝐴): Probability of occurrence of landslidesurge in reservoir 𝐴𝑃(𝑆𝐵): Probability of occurrence of landslidesurge in reservoir 𝐵𝑃(𝑂𝐴): Probability of occurrence of damovertopping in reservoir 𝐴𝑃(𝑂𝐵): Probability of occurrence of damovertopping in reservoir 𝐵𝑃(𝑂𝐴 | 𝑆𝐴): Conditional probability of damovertopping in reservoir 𝐴 due to alandslide surge𝑃(𝑂𝐵 | 𝑆𝐵): Conditional probability of damovertopping in reservoir 𝐵 due to alandslide surge.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This study was sponsored by the National Key Basic Re-search and Development Program of China (Grant no.2013CB036401) and the Natural Science Foundation of China(Grant no. 20130181110045).The authors are also very gratefulto the Dadu River Basin Management Company for theprovision of data.


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