DB-IWHR manual for the analysis of dam breach

1 Introduction

The breaking of natural or synthetic dams often causes significant disasters, and the related

research is in high demand. Estimation of the dam-break flood is of prime importance in such research,

especially when dam safety emergency responses are concerned. Early examples of analytical models

for peak breach outflow can be attributed to Cristofano (1965), followed by the works by Harris and

Wagner (1967), Brown and Rogers (1977, 1981), Ponce and Tsivoglou (1981), MacDonald and

LangridgeMonopolis (1984), Costa (1985), Fread (1988), Froehlich (1995), Walder and O’Connor

(1997), Singh and Scarlatos (1988), Wang and Bowles (2006), Macchione (2008), Chang and Zhang

(2010), and Wu (2013), among many others. Stateof-the-art reviews on dam breach (Morris and Hassan

2002; Zhu et al. 2004; ASCE Task Committee 2011; Wahl 2010; Wu and Wang 2010) generally agree

that the ability to predict the breach outflow is still far from advanced, demonstrating the following

limitations and deficiencies

This uses a dam breach model to reproduce the well-monitored outflow

hydrograph obtained during the dam breaching process of the Tangjiashan barrier

lake, which was formed by a landslide triggered by the Wenchuan earthquake on

May 12, 2008 in China. The key parameters that affect the model results, such as

soil erosion and breach lateral enlargement, are reviewed by using field

measurements followed by extensive sensitivity studies. This advocates a

hyperbolic model for soil erosion rate and a circular slip surface approach for

breach lateral enlargement, which contribute to more reliable model results. The

governing equations are solved using a numerical method that allows

straightforward calculations coded in an Excel 2010 spreadsheet. This provides an

easy, transparent, and robust tool that could enable practicing engineers to

perform dam breach analyses with a comprehensive understanding of the

uncertainties involved.

2 Dam-Breach Analysis Model

2.1.1 The calculation of dam breach hydraulics

（1） breach hydraulic conditions

1) calculate the breach discharge

Like in most of dam breach models, the outflow through the breach is

estimated using the hydraulics of a broad-crested weir. In general, it is expressed

by the following equation for a channel with a rectangular cross section:

11\* MERGEFORMAT ()

where C = discharge coefficient whose theoretical value is 1.7 m 1/2/ s (Singh 1996).

Previous researchers have adopted values of C ranging from 1.3 to 1.7 (Jack 1996).

When the breach process approaches its end and the reservoir water level is close

to that of the downstream tailwater, a coefficient that accounts for the effect of

submergence is introduced (Fread 1988; Singh et al. 1988). A number of

researchers adopt a lumped coefficient for C in Eq. (1) in their calculations (Harris

and Wagner 1967; MacDonald and Langridge-Monopolis 1984; Chang and Zhang

2010). This value can be determined based on experience and calibrations.

(2)

where h = flow depth

H'

H

z

h V

Fig 1Hydraulic relations at entrance of channel

2）The calculation of breach velocity

When the water flow from the reservoir into the breach, there will be a water-surface drop. The

water level reduced from H to h. It is assumed that the reservoir flow velocity V0 and entrance head loss

can be neglected, there are:

(3)

(4)

According to (2) and (4), then:

(5)

Where, V is velocity.

（2） water balance equation

According to historical data, river section measurement and remote sensing method, the relation

between reservoir capacity and water level can be get:

(6)

where W is water storage capacity, H is reservoir water level, Hr is Base level

Then： (7)

(8)

where, a1、b1、c1 is the relation between reservoir capacity and water level, respectively.

The outflow discharge can be determined by the reservoir capacity loss per unit time

(9)

Where, q is natural inflow into the reservoir.

According to mass conservation, the equation of water balance can be got:

(10)

(11)

2.1.2 Initial erosion conditions of breach

Under a constant flow, the effective shear stress is equal to the effective weight of water acting on

the bottom of the water discharge groove, then:

(12)

Where γ is density of water; R= hydraulic radius, S = slope of the channel;

The shear stresses are calculated using the following widely used equation

(e.g., Macchione 2008; Gaucher et al. 2010)

(13)

where γ = density of water; n = roughness coefficient (0.025 m-1/3s in this

case); J = slope of the channel; and R0 = hydraulic radius that can be

approximated by h if the channel width B is sufficiently larger than the average

flow depth h (Guo and Jin 1999).

The following calculation method of different starting methods is then:

（1） Determining the initial erosion condition according to the velocity

Table 1 Determining the initial erosion condition according to the velocity in cohesive soil

Authors Equation parameters

Tan

g （ 1963

）m=6(Natural channel) or m=4.7(h/D)0.06

C=2.9×10-4g/cm

Wuha

n （ 1900

）

h is flow depth；D is particle diameter

is bulk density, is water bulk density

Do

u （ 1999

）hd is atmospheric pressure in water column height

δ=3×10-8cm

Table 2 Determining the initial erosion condition according to the velocity in sand soil

Authors Equation Parameters

岗 恰 洛 夫（1962） H is flow depth

沙莫夫（1952） D is particle diameter

张 瑞 锦（1900） s is bulk density, is water bulk density

（2） Determining the initial erosion condition according to the shear stress

Table 3 Determining the initial erosion condition according to the shear stress in cohesive

soil

Authors Equation Parameters

Julian and Torres

(2006)P：Fine content (%)<0.063

Otsubo and Muraoko

(1988)C is cohesion

Smerdon and

Beaseley(1961)PI is Plasticity index；D50 is median diameter

Table 4 Determining the initial erosion condition according to the shear stress in sand soil

Authors Empirical expression Main parameters

Schoklitsch

（1914）

= shape factor ， （ =1 ， spherical particle; =4 ， flat

particle）;

d = average particle size （m）; γs = unit weight of sediment

（N/m3）;γ = unit weight of water （N/m3）.

Shields

（1936）

d: average particle size（m）;

d u¿

ν=Re*=11.6

dδ

= shear Reynolds number;

δ= Laminar sublayer.

Egiazaroff

（1965） = average diameter of grain for both gradation curves ， for

grains in movement， and for total sediments

Van Rijn

（1984）d = average particle size（m）;

s = relative density;

= kinematic viscosity.Soulsby

（1997）Annandale

（2006） = friction angle （ °）；ρs = mass density of soil （kg/m3）;

ρw = mass density of water （kg/m3）.

2.1.3 The test to determine the erosion rate based on the conditions of breach

The relationship between erosion rate and shear stress is the erosion model. Erosion model is an

important problem in the study of dam breach, which includes two aspects: the erosion velocity and the

erosion rate. The erosion velocity determines the initial time and the end time of dam breach, and

erosion rate. On the basis of summing up the results of previous experiments, combined with the actual

engineering information, the erosion model can be divided into three categories: linear model;

exponential model; hyperbolic model. The hyperbolic model is a combination of Tangjiashan measured

data and summarizing the previous data.

（1） Exponential model

A large number of research works have dealt with the relationship between the soil erosion rate and shear stress for both cohesive and noncohesive materials. In general, an exponential expression has been proposed for noncohesive materials (Roberts et al. 1998; Gaucher et al. 2010)

(14)

where is the erosion rate in mm/s, is in Pa, and time t is in seconds. a1 and b1 are

coefficients either regressed from the test results or based on experience.

（2） linear model

In the exponential model, but b1 takes 1, the erosion rate and shear stress is a simple linear

relationship between:

(15)

Cook Hanson- (1992) and Briaud (2008) have proposed a linear model.

（3） hyperbolic model

In this paper, a hyperbolic model is suggested, which takes the following form.

(16)

Where v is the shear stress with reference to its critical component

(17)

and k = unit conversion factor that allows to approach its asymptote

within the working range of τ

Here, k is taken to be 100 with a unit of Pa for and 10-3 mm/s for . The hyperbolic curve has an

asymptote represented by =1/b as v approaches infinity, and 1/a represents the tangent of this curve

at v=0. This model is established based on the understanding that like a structural material, soil should

not have unlimited ‘strength’ against erosion. With reference to the measured data, a set of parameters

‘a=1.1, b=0.0007 and c=30 Pa’ is proposed, which results gives mm/s. This set of

parameters will be used in the back analysis followed by sensitivity studies.

2.1.4 The erosion rate model based on the theory of sediment dynamics

（1） Meye peter- Muller Model

Meyer Peter &Muller formula is widely used in sediment transport formula, Fread in breach

(1984, 1988) model, Bechteler&Broich (1991), both use the formula calculation of sediment

erosion rate. Its formula is calculated by the (18).

(18)

where ， — the bedload discharge per unit width ； — proportion ， ； —

acceleration of gravity； —Water density； —shear stress； —critical shear stress

The Tangjiashan calculated results compared with the measured value are shown in Figure 2.

（2） Eintein-Brown model

Einstein-Brown formula is one of the more widely used in the sediment transport formula, Singh

used the formula for the calculation of sediment transport in the BEED (1989) model.The formula can

be calculated in the formula (23).

(19)

(20)

(21)

(22)

(23)

(24)

where —the bedload discharge per unit width ； — acceleration of gravity ； — D50

diameter；The value of 、 is calculated according to the formula (20) ~ (23), respectively. Shear

stress is calculated according to the formula (24).

The Tangjiashan calculation results are shown in Figure 2 compared with the measured values.

（3） Du Boys model

Du Boys formula is Fread in breach (1988) model of sediment transport equation, one of its clear

formula for calculating the reduction as shown in equation (25).

(25)

is According to table 5。The comparison between Tangjiashan calculated results and the measured value are as shown in

Fig 2.

Table 5 and in Du boys

Diameter

(mm)1/8 1/4 1/2 1 2 4

(m3/N-m) 0.852 0.5051 0.3061 0.1796 0.1056 0.06327

=2600N/m3

(Pa) 0.7644 0.8134 1.0584 1.5288 2.4402 4.132

（4） Englund-Hense model

Englund-Hansen formula is one of the methods used in the calculation of sediment transport by

MIKE11 software, and the bedload discharge per unit width can be sorted as the formula (26).

(26)

where ， — the bedload discharge per unit width ； — proportion ， ； —

acceleration of gravity, Ds— D50 diameter； —Water density； —shear stress； —critical shear

stress

The comparison between Tangjiashan calculated results and the measured value are as shown in

Figure 2.

0 15 30 45 60 75 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Measured dataMeyer-Peter & MullerEinstein-BrownDu BoysEnglund-Hensen

剪应力 (Pa)

dz/d

t(mm

/s)

Fig 2The comparison chart between Tangjiashan measured data and sediment erosion model

2.1.5 Breach Lateral Enlargement Model

As the base of the breach is cut deep to a certain elevation, the banks on both

sides of the channel may collapse, resulting in a wider channel. This is known as

lateral enlargement. The lateral enlargement due to the collapse of the channel

wall is the main mechanism of breach widening. In most of the existing breach

models, this process is usually modeled using a wedge failure analysis with a

straight line slip surface subjected to gravity and seepage forces (Fread 1988;

Singh 1996; Wu 2013). However, the geotechnical profession has widely accepted

more rigorous analytical methods with circular slip surfaces, such as Bishop’s

simplified method (1955) and the method proposed by the U.S. Army Corps of

Engineers (1970). The procedure for calculating the factor of safety F is repeated

among a variety of possible slip surfaces until a critical one associated with the

minimum factor of safety Fm is found. This can be achieved by use of optimization

methods (Chen and Shao 1988; Duncan 1996), then followed by a procedure to

find the critical depth of toe-cutting that makes Fm = 1. The calculations can be

performed by a computer program, for example, SLOPE/W (Krahn 2004) or STAB

2007 (Chen and Wang 2000).

In the lateral enlargement model, soil shear strength and internal friction

angle are needed. Because the permeability of the landslide material could not

allow free drainage during the rapid drawdown of the channel water surface and

the pore water in the dam could hardly be determined by either analytical or

empirical approaches with reasonable accuracy, the total stress analysis method,

employing undrained shear strength parameters, is commonly used (Sherard et

al. 1963; Lowe and Karafiath 1959; Johnson 1974).

Since the slope stability analysis contains tedious procedures (e.g., modeling

of a vertical toe cutting and stepped collapsing, searching for the critical slip

surface and applying a total stress analysis method), a spreadsheet for computing

the lateral enlargement process, namely, the DBS-IWHR, in conjunction with the

DB-IWHR, has been developed.

Then, the information on the stepped failures becomes the input to DB-IWHR.

However, if the details of many steps of failure are input into DB-IWHR, it becomes

too tedious for a dam breach analysis. On the other hand, there is not much loss

in accuracy if the information of the intermediate steps is approximated using

linear interpolation. Further, it requires more computation effort to determine the

flow surface width. By remaining the original approach for circular channel sides

based on Chen et al. (2015), the new version of DBS-IWHR adds an option that

simplifies the circular sided breached channel by a series of trapezoidal cross-

sections. The inclination of the straight line channel side β is taken to be the

average value of the inclinations of the chord and the tangent of the circle at the

toe, as illustrated in Fig. 3.

In the DB-IWHR 2015, it can calculate water depth according to . The surface width

can be obtained that is the breach width, according to the triangle.

(X c,Yc)

R

(0,740)

B0

x

y

727

758

Scale0 10 20

(Xc,Yc)

R

(0,740)

B0

x

y

727

scale0 10 20

758

Fig 3Equivalent simplification of lateral enlargement process

On the lateral enlargement simulation process and the using method of DBS-IWHR, and other

special paper.

2.2 Numerical Method

Conventional approaches (e.g., Fread 1988; Singh et al. 1988; Chang and

Zhang 2010) start the calculation from an initial time t0 with a given step t, for which the increments ΔH, Δz, and ΔV are obtained iteratively. By examining these equations, it can be found that once V is given, the solutions to ΔH, Δz, and Δt can be obtained by straightforward calculations without need of iteration. Therefore, a new approach is herein proposed that starts from an initial velocity V0 with an interval of ΔV. The formulations and procedures of this new approach are described in the following subsections.

2.2.1 Formulations

At a velocity step from V0 to V0+V, the average velocity is

(27)

The average values of H and z are

(28)

(29)

From Eq. (5), the average velocity can be found by

(30)

where

(31)

Once V is given, is obtained with Eq. (27) and then s is obtained by reformulating Eq. (31)

as

(32)

Eqs. (10) and (14) or (16) can be expressed in finite difference forms, respectively, as

(33)

(34)

where is determined by Eq. (13), in which V and h are replaced by their average values

and determined by Eqs. (27) and (2), respectively.

Eliminating t and H in Eqs. (33) and (34), one obtains

(35)

where

(36)

(37)

(38)

(39)

2.2.2 Procedures

The calculation in a velocity step starts with a given

V based on the known values of H0, z0, and

V0 determined in the previous step. It includes the following computations:

(1) Calculate by Eq. (30)

(2) Calculate s by Eq. (31)

(3) Calculate z by Eq. (35)

(4) Calculate H and t based on the known values of , s, and z from Eqs. (31) and (34),

respectively.

The above procedures are straightforward. However, special treatments are required to make the

calculation smoothly pass the point at which V attains its maximum Vm and V transits from a positive

to a negative value. The details are presented in the Appendix.

A spreadsheet entitled DB-IWHR 2014 is coded in Microsoft Excel 2007 with its VBA

programming facilities. This program is simple, iteration-free, and transparent, allowing for quick

prediction of the peak discharge of the breach flow. The spreadsheet and detailed information are

available for download at the following website: www.geoeng.iwhr.com/geoeng/download.htm.

2.2.3 Program interface

ProjectName

SerialNumbe

Calculated by Checked by Date Company

Z 0 740.00 Elevations of channel bed,m

Q in 80.00 Inflow flow,m3/s

H r 700.00 Elevations of dead Water,m B 0 14.40Channel Width,beginning

B end 40.40 Channel Width, ending

а 1 144.00 Channel side inclination, beginninga 1 0.06 W~H curve coefficient a2 170.00 Channel side inclination, endingb 1 1.96 W~H curve coefficient Z end 727.00 Elevation endingc 1 44.00 W~H curve coefficient n 207 Row number

Note: obtained by water storage capacity curve worksheet

D50 0.0050 mm平均粒径（ ）ρ s 2650.0000 kg/ m颗粒密度（ 3)

m q 0.36 Broad-crested weir flow coefficient υ 0.4300 孔隙比m b 0.90 Lateral shrinkage coefficient Gs 2.6500 颗粒重度m 0.80 Ratio of tailwater L 36.0000 m)冲刷长度（

n 0.0250 河道粗率

V c 2.70 The incipient velocity,m/s

a 2 1.1000 Erosion coefficient

b 2 0.0007 Erosion coefficient

z c 0.00 Erosion coefficient

Qp 7469.96 m3/s Magnitude

tp 4.85 h Time

Error 0.13% error

图2图1

Dam Breach Analysis DB-IWHR 2014 Copy Right：Chen ZuYu（Email：chenzuyu@cashq.ac.cn）

Disclaimer: The author assumes No responsibility and makes no guarantees, expressed or implied, on the quality, reliability, orany other characteristic of this software

690

700

710

720

730

740

750

0 5 10 15

h-t

z-t

elevation （m

)

Time（h）

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20

Q-t

Dam

flow（

m3/s)

Time（h）

There are several candidate methods to consider the erosion rate of sediments,The User can swtich the methods by click the Drop Down box

Defaul t val ue

Automati cal l y

Manual l y

Peak Fl ow

Defaul t val ues Manual l y

Breach extensi on curve

Rel ati on between Storage Capaci ty and Water Level

Coeffi ci ent of Broad-crested wei r

Coeffi ci ent of erosi on rate

Aval anche Lake I nformati on

Automati cal

Manual l y

Export Figs

Manual l y

calculation

sedi ment transport parameter

cal cul ati on buttom

(a)

(b)

Fig 4 DB-IWHR Spreadsheet: (a) input (b) DB-IWHR Spreadsheet

2.2.4 Determination principle of initial width

In the case of single breach, according to the natural runoff q0

The initial condition of the dam breach is assumed as

(40)

Then:

(41)

(42)

其中 Vc 为起动流速。Where Vc is the starting velocity.

(43)

(44)

(45)

For tangjiashan landslide dam

The elevation of initial channel bottom is:

(46)

3 Program instructions

3.1 Basic functions

The basic function of the DB-IWHR spreadsheet program is: (1) simulation of breach flow

process and the breach lateral enlargement development process; (2) program can use a variety of

erosion rate calculation formulas.

Conventional approaches (e.g., Fread 1988; Singh et al. 1988; Chang and Zhang 2010) start the

calculation from an initial time t0 with a given step t, for which the increments H, z, and V are

obtained iteratively.

By examining these equations, it can be found that once V is given, the solutions to H, z, and t

can be obtained by straightforward calculations without need of iteration. Therefore, a new approach is

herein proposed that starts from an initial velocity V0 with an interval of V in DB-IWHR.

And has been given the six available erosion rate formulas: hyperbolic form erosion rate formula,

exponential form erosion rate formula; Meye peter- Muller sediment bedload formula; Englund-Hense

sediment bedload formula; Du boys sediment bedload formula; Eintein-Brown sediment bedload

formula.

3.2 User Interface

The calculation program contains the following seven tables:

（1） Calculation： The main interface for the input of the calculation parameters, as well as the choice of calculation method;

（2） W.H curve ： is the Calculation sub interface, for regression analysis of reservoir parameters.

（3） erosion model： is the Calculation sub interface, select the erosion model, fill in the erosion parameters;

（4） User Manual： Basic introduction to the whole processCalculation table is the main operating table, including the following 7 areas:

（1）Dam parameters (including high dam, flow as the basic parameters)

（ 2 ） Parameters of wide crest weir (including the correction coefficient of flow rate and the

coefficient of the submerged coefficient)

（3）Capacity parameters

（4）Erosion parameters (based on the above formula and the actual dam material)

（5）The breach side wall parameters (Reference Principle Introduction)

（6）Calculation method selection

（7）Parameters in the bed load formula

（8）Calculate and export buttons

Fig 5 Basic parameter input area

（1）

（2）

（3）

（4）

（5）

（6） （7）

（8）

3.3 The program Settings before using

steps：（1）click “Microsoft Office file”，then click“Excel options”.

（2） as seen in Fig 6， in“Excel options”，click“add-ins”，then choose “solver add-in”，the

click“go”。（ 3 ） In“add-ins”” ， choose“Analysis Toolpak”,“Analysis Toolpak-VBA”, “Solver Add-

in”，then click “OK”，as seen in Fig 7.

Fig 6choose add-in

Fig 7start solver add-in

2. Macro settings

“Microsoft Office file” —— “Excel options” —— “Trust center settings” —— “Macro

settings”（Fig 8），choose“enable all macros”.

Fig 8 macro settings

After this settings ， when open this spreadsheet, Pop-up security warning may appear. Click

“enable content”

Fig 9 security warning

3、if “Can't find the project or library”

If there is“Can't find the project or library”，as seen in Fig 10. This reason lies in the “Solver” is

not properly reference.

Steps：(1) click the “ok” button, the VBA program is running state.

(2) Stop running: first, input a space at the end of a line in the code window (in order to stop

running fast), and then click the “reset” button on the toolbar, or the “reset” button in “run” menu

command, as shown in Fig. 12.

(3) Click the “reference” command in “tools” toolbar, in the pop-up box, remove the check box in

front of “丢失 Solver”, as seen in Fig 11.

Fig 10 编译错误

Fig 11 SOLVER 丢失

Fig 12 中断 VBA 的运行

3.4 Steps

1. Data input

Input directly according to the parameters, be easy to modify and understand.

2: The results of calculation

The program can calculate automatically by the Excel powerful computing system, and the below

of calculation page is the calculation results page figure.

The calculation results can also get by the export button “figs export”. When press the button, it

will get a "result" results EXCEL documents, contains flow curve and data.

3.5 The results of calculation

After completing the parameters according to the above description, the program will give the

results automatically, as shown in Fig 13.

Qp 7469.96 m3/s Magnitude

tp 4.85 h Time

Error 0.13% error

图2图1

690

700

710

720

730

740

750

0 5 10 15

h-t

z-t

elevation （m

)

Time（h）

0

1000

2000

3000

4000

5000

6000

7000

8000

0 5 10 15 20

Q-t

Dam

flow（

m3/s)

Time（h）

Peak Fl ow

Fig 13 Calculation result

Click the "Export Figs" button can get the results file "result".

Fig 14 Calculation results display area

4 Example of Tangjiashan breach

Test topic: compared with the measured value of Tangjiashan barrier lake process.

Test objective: to verify the program feasibility and accuracy of the calculation results.

Test content: select reasonable parameters using the measured data, compared the calculated

results with the measured data, and analyze the sensitivity of the parameters.

4.1 Simulation results

The calculated results of the reservoir water level H, channel bed elevation z, water surface width

B, flow discharge Q, and velocity V are plotted in Fig 15. For quantitative comparison, Table 8 gives

various characteristic values obtained by field measurement and this back analysis, together with the

results from subsequent sensitivity studies.

The back analysis predicts a peak outflow of 7610 m3/s, compared with the measured value of

6500 m3/s. It can be found from Table 7 that all the calculated characteristic values agree with the

measured data well before the peak outflow. After that moment, the calculated elevations of reservoir

water and channel bed keep lowering while the field measured data presented almost unchanged

values. This may be explained by the sedimentation of a large amount of scoured material in the

downstream river bed after the peak outflow. The coupled erosion and sedimentation effects should be

considered if the model is expected to simulate the entire dam breaching process. Another reason could

be the heterogeneity of the landslide materials. The erosion stopped at the level of large rocks that had

not been disintegrated and had much higher critical shear stress (Chang and Zhang, 2010).

Table 6 The input parameters for the back analysis case

Item Parameters Notes

Natural inflow q 80 m3/s

Initial breach

widthBo 16 m

Determined based on the draining

channel geometry and a flow height of 3 m

Broad crested

weir

C 1.35Parameters involved in Eqs (1) and (2)

m 0.8

Reservoir water p1 0.063 The relationship between the pool water

storage

level and storage for Eq. (10) can be found in

Liu et al. (2010) and is approximated by

in m3

p2 196.6

p3 44

Hr 700 m

Erosion rate

Vc 2.7 m/s

Parameters involved in Eq.(16)a 1.1

b 0.0007

Lateral

EnlargementBased on 2.1.5

(a)

(b)

(c)

(d)

(e)

Fig 15 Calculated results versus time compared with the measured data : (a) reservoir water

level, (b) flow discharge, (c) velocity , (d) flow surface width,(e)channel bed level.

4.2 Sensitivity Studies

Determining the size and growth rate for breaches is not a precise exercise (Gee 2009). The

concerted action on dam-break modelling (CADAM) Project report states that an estimate of 50% for

predicting peak discharge is suggested, with the accuracy of predicting the time of formation being

considerably poor (Morris and Hassan 2002). Therefore, a sensitivity study should be a part of dam-

breach analysis.

Sensitivity studies are conducted for the Tangjiashan dam breaching based on the back analysis case

by changing one of the model parameters each time while keeping the others unchanged. The

considered parameters and some of the model results are summarized in Table 7, with details being

described below. They represent two more sets of erodible soils. The hydrograph of Case B-2 adopts an

ultimate erosion rate of =1/b=3.3 mm/s, which is three times that of the maximum measured rate

1.19 mm/s. Referring to Table 7 and Fig 17 the peak discharge of Case B-2 is 13000 m3/s, which is

nearly double that of the value of the field measurement, and of the back analysis case. This indicates

that the hyperbolic model could handle a large range of possible parameter inputs.

Table 7 Summaries of the characteristic values for various cases

Peak flow Peak velocity

Occurring Time Water level Channelbed level Discharge Peak velocity Erosion rate

tm H z Qm Vm dz/dt

Hour m m m3/s m/s mm/s

Field measurement 12:30 732.25 720.9 6500 4.96 1.19

Back analysis case 11:02 735.21 723.41 7609.97 5.78 1.16

A

A-1 m=0.6, C=1.35 11:12 730.04 717.36 7829.65 7.60 1.31

A-2 m=0.5, C=1.35 11:16 729.49 716.40 7858.80 6.54 1.35

A-3 m=0.5, C=1.69 10:27 731.31 719.52 8300.19 5.89 1.38

BB-1 a=1.0,b=0.0005 10:57 730.55 717.06 9475.62 6.18 1.58

B-2 a=0.9,b=0.0003 10:26 724.61 707.10 13524.99 7.05 2.54

C

C-1 a1=8,b1=1.2 16:53 730.04 718.25 7512.91 5.78 1.24

C-2 a1=10,b1=1.2 15:31 724.96 710.53 10357.93 6.39 1.94

C-3 a1=8,b1=1.3 15:11 718.77 699.59 15192.20 7.37 3.35

D D-1 Table 4 13:26 727.36 712.54 6740.95 6.48 1.20

4.2.1 Case A: Parameters related to the broad-crested weir flow

Cases A-1 and A-2 investigate the influence of taking different values of m, namely m=0.6 and

0.5, respectively, compared with the value of m=0.8 used in the back analysis case.

Case A-3 takes into account a higher value of C, which is 1.69, as proposed as an upper limit by

Brater (1959) associated with m=0.5, This case may present a presumably highest peak flow as far as

the weir discharge coefficients are concerned.

From Fig 16 and Table 7, it can be found that different hydraulic weir parameters have limited

impact on the calculated peak discharges. Case A-3, which is presumed to be an upper bound of the

peak outflow, is 8300 m3/s, compared to 7610 m3/s of the back analysis case. From a practical point of

view, one may assume a lower value of m, and higher one for C, based on his experience as a

conservative approach to find the maximu m possible peak outflow.

Fig 16 Sensitivity studies of Case A : Curves of flow discharges versus time

4.2.2 Case B:Parameters related to the hyperbolic erosion model

Case B-1 takes a=1.0 and b=0.0005, while Case B-2 has a=0.9 and b=0.0003. They represent two

more sets of erodible soils. The hydrograph of Case B-2 adopts an ultimate erosion rate of =1/b=3.3

mm/s, which is three times that of the maximum measured rate 1.19 mm/s. Referring to Table 7 and Fig

17 the peak discharge of Case B-2 is 13000 m3/s, which is nearly double that of the value of the field

measurement, and of the back analysis case. This indicates that the hyperbolic model could handle a

large range of possible parameter inputs.

1

Conceptually speaking, the field measurement of 1.19mm/s could be presumed as a close estimate

to , which is the maximum possible erosion rate for the material in Tangjiashan, because the

reservoir still had sufficient energy at that time to increase this if the soil could have resisted it. The

maximum possible erosion rate has physical meaning, and the use of the hyperbolic model could

help experienced engineers reduce the risk caused by the adoption of inappropriate erosion parameters.

Fig 17 Sensitivity studies of Case B : Curves of flow discharges versus time

4.2.3 Case C: The exponential erosion model and parameters

Case C-1 investigates the exponential erosion model of Eq. (15) with the regressed parameters

a1=8 and b1=1.2. and a prediction of Qm of 7512.9 m3/s, as listed in Table 7. Case C-1 takes more time

to reach peak outflow compared with other cases adopting the hyperbolic model. This is not a problem,

because the time at which the channel erosion started cannot be identified exactly.

Similar to the investigations on the hyperbolic model, two more erodible parameters, namely,

‘a1=10 andb1=1.2’ and ‘a1=8 and b1=1.3’ have been assigned as Cases C-2 and C-3, respectively. The

peak discharges shown in Fig 18 exhibit large differences. Comparing the calculated peak discharges

with field measurements and those obtained in the back analysis case. Case C-3 shows that a slight

change of b1 from 1.2 to 1.3 would double the peak outflow. Thus, the use of the exponential erosion

model demonstrates the difficulties of assigning proper parameters for different materials.

2

Fig 18 Sensitivity studies of Case C : Curves of flow discharges versus time

4.2.4 Case D: Lateral enlargement

Studies on the uncertainties involved in the channel lateral enlargement analysis are focused on

the variance of shear strength parameters. Case D-1 investigates a higher set of strength parameters:

‘cu=25kPa andu=26’. Following the procedures for calculating the stepped landslides described in the

‘Modeling lateral enlargement’ section, the computation ended at the 4th step, resulting in an

enlargement cross section, which is small in size compared with the back analysis case shown in Fig

15. The calculated peak flow discharge in this case is 6740 m3/s, as shown in Table 7 and Fig 19,

compared with 7610 m3/s of the back analysis case. Fig 20 shows that the water surface width at the

channel bed elevation of 726 m is only 90 m, which is much smaller than the 140 m in the back

analysis case shown in Fig 15(d). However, the peak outflow does not reduce considerably. Thus, it

appears that the use of the formal geotechnical slope stability analysis approach could allow a wide

range of input shear strength parameters.

3

Fig 19 Sensitivity studies of Case D : Curves of flow discharges versus time

4

Fig 20 Sensitivity studies of Case D: Curves of water surface width versus time

5 References

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