(1)
Publicity Department, Ministry of Water Resources(MWR), China
2#, Lane 2,Baiguang Road, Xuanwu District, Beijing,100053 China
Tel:+86-10-63202019, Fax:+86-10-63548250, Email: hwguo@mwr.gov.cn
Wu Congcong
Institute of Water Resources & Hydro Power Planning and Designing, MWR
Haihe River Water Conservancy Commission,MWR
Chen Zhaohe
Beijing Postgraduates School, North China Institute of Water Resources & Hydropower
Abstract: On the basis of precious researches[1~3],
a mathematical model developed by the authors to simulate the gradual process of
earth-fill dam break due to overflow and flood routing downstream is described
in this paper. Considering both erosion and sloughing of the breach side slope
due to instability, the model may simulate the process of gradual erosion of
earth-fill dam by inputting such physical parameters as the median particle
diameter, cohesion, angle of internal friction and porosity of the construction
material of the dam body as well as the initial breach dimension. Then the model
applies complete unsteady hydrodynamic equation to route the breach outflow.
Having discussed the basic mechanism of overflow dam breach, an engineering
application with the model presented herein is reported. The results are shown
in the figures and tables, which demonstrates the practicability of the gradual
dam break model. It is believed that the model presented in this paper can give
a prediction on dam break and flood routing and provide a scientific basis for
flood control decision making. In the meanwhile, this case study also shows that
the gradual dam-break model is more actual than that based on assumption that
the final dimensions of the breach may be specified in advance of simulation and
the breach will develop linearly from the initial to the final dimensions
specified.
Keywords: dam break, flood routing, mathematical model, prediction
On the basis of precious researches[1~3], a mathematical model GDBFR(Gradual Dam Break and Flood Routing) was developed by the authors to simulate the gradual process of dam breach and its flood routing downstream, Its principle and engineering application is described in the paper. The prediction provided by GDBFR is more actual and practicable than the model of instantaneous and full failure dam-break.
2 GRADUAL DAM BREAK
Simulation of the earth-fill dam breach process should consider the hydrological, hydrodynamical, sediment transport and geotechnical aspects. The physical phenomena of the gradual dam breach occurs as the following: (1) water from the reservoir overflows through the breached section of the dam, causing enlargement of the breach either by erosion or sloughing. The process continues until the reservoir is emptied or the dam can resist further erosion; (2) the duration for the breach process depends upon the velocity and discharge of overflow eroding the materials of the dam, the dam height, the type of construction material, and the compactness of the material; (3) the initial breach will be enlarged in the horizontal and vertical direction during overflow. Considering both erosion and sloughing of the breach side slope due to instability, the model may simulate the process of gradual erosion of earth-fill dam by inputting such physical parameters as the median particle diameter, cohesion, angle of internal friction and porosity of the construction material of the dam body as well as the initial breach dimension. Then the model applies complete unsteady hydrodynamic equation to route the breach outflow.
(1)
where H reservoir water level measured from a datum plane; As surface area of water stored in reservoir; Io inflow; Qb breach outflow discharge; Qo outflow over the crest of the dam; Qsp outflow through the spillway; and Qou outflow through the hydropower sets and/or outlet works in the dam body.
Qsp=Qsp(H,t) and Qou=Qou(H,t) mentioned above may be specified accurately for prediction of dambreak flood. On the other hand,
Qb=[C1*b+ C2*(H-Z)tanq](H-Z)3/2 (2)
Qo=C1*(BD-B)(H-Z0)3/2 (3)
Where C1* , C2* dimensional coefficients; b bottom width of the breach; Z bottom elevation of the breach; q angle between vertical and the breach side; BD top width of the dam ( i.e. crest length); and B top width of the breach .
Combining Equation(1)(2)(3)yields the reservoir water volume depletion equation as follows:
As(H)dh/dt=Io-[C1*b+C2*(H-Z)tanq](H-Z)3/2-C1*(BD-B)(H-Z0)3/2-Qsp(H,t)-Qou(H,t) (4)
Equation(4) is a nonlinear ordinary differential equation with two unknowns: water elevation H and breach bottom elevation Z.
After development of an initial breach on the dam, the hydrodynamic forces continue to enlarge the breach by eroding the soil material. The Einstein-Brown formula was selected to compute the breach bottom erosion rate.
(1) Einstein-Brown bed-load formula.
The basic idea of the Einstein-Brown theory is
that initiation and cessation of sediment motion depend on the probability that
relates instantaneous hydrodynamic lift forces to the submerged weight of a
particle. The final results are presented in the following dimensionless
expression:
(5)
where f sediment transport rate function; and y inverse of Shields dimensionless shear stress.
Explicitly, the quantities f and y are given as
(6)
and
(7)
where qbw bed load discharge, weight per unit width; gs specific weight of soil;
g
specific weight of water; Ds representative size of bed sediment; t bed shear stress; g2 specific weight of submerged soil; and n kinematic viscosity of the water.
Usually, the median size D50 is taken as Ds, while bed shear stress t is estimated as
（8）
where u eroding velocity of the water flow; Ch Shield coefficient; Rh hydraulic radius; and Sf friction slope.
(2) Breach bottom erosion rate
Once the bed-load discharge qbw has been estimated, the rate of erosion of the bottom of the breach can be directly calculated. Indeed, scouring of the breach, DZ, during time interval Dt can be given as
(9)
where p is the soil porosity.
During erosion process of the earth-fill dam, the situation of the breach slope is unstable and continuing developing. This happens when the hydrodynamic forces associated with seepage are greater than the soil friction and cohesion. The force balance equation is as follows:
FH+ Gtan(z-f)=CXp[1+tan z tan(z-f)] (10)
where FH horizontal seepage forces; G the total weight of sliding wedge; C cohesion; Xp horizontal projection of the shearing plane; and z angle between the shearing plane and the horizontal.
Failure of the slope occurs when the right-hand side of Eq.(10) is greater than the left-hand side.
Flood routing technique is based on the complete one-dimensional equations of unsteady flow, i.e. the Saint-Venant equations. The complete equations are the conservation of mass (continuity) and the conservation of momentum (motion).
The continuity equation is given as
（11）
The motion equation is given as
（12）
where A the active cross-sectional area of flow; Ao the inactive ( off-channel storage) cross-sectional area of flow; Sf friction slope; Se expansion-contraction slope; and q the lateral inflow or outflow per lineal distance along the channel ( inflow as positive and outflow as negative).
Where
（13）
where n Mannings roughness coefficient; R hydraulic radius defined herein as A/B where B is the top width of the active cross-sectional area; k expansion-contraction coefficient varying from 0.0 to 1.0 ( positive if contraction and negative if expansion); DX the distance between two adjacent cross-sections; and DV2 the difference between the square of the velocities at two adjacent cross-sections separated by a distance DX.
Xiaonanhai
Reservoir is located 35 km west of Anyang City, Henan province, China. It
controls a catchment area of 850 km2, creating a storage capacity of
1.07΄108m3.
Zhangwu Reservoir is constructed at 10km downstream of Xiaonanhai Reservoir, and
it controls a catchment area of 120km2 with a total storage capacity
of 0.78΄108m3.
Zhangwu reservoir is a homogeneous earth dam, its crest elevation is 137.6m, the
maximum height of the dam is 28.7 m, the normal water level is 128.7m and the
dead water level is 118m. Zhangwu reservoir was designed with a design flood of
frequency 1/50 and an extreme flood of frequency 1/1000.
It is the goal to simulate the dam break for Zhangwu Reservoir and routing the dam break flood downstream of Zhangwu Reservoir under the conditions that Xiaonanhai Reservoir occurs an inflow of frequency 1/2000, and Zhangwu Reservoir faces the outflow from Xiaonanhai Reservoir and the intervening flood with correspondence to Xiaonanhai reservoir inflow of frequency 1/2000.
(1) Xiaonanhai Reservoir
Occuring an inflow of frequency 1/2000, the Xiaonanhai dam is safe, and the superposition of its outflow and the intervening flood was taken as the inflow of Zhangwu Reservoir. The inflow hydrogragh of Zhangwu Reservoir is shown in figure 1.
Fig. 1 Inflow hydrogragh of Zhangwu reservoir
(2) Zhangwu Reservoir
With the inflow shown in Fig.1, the maximum water level of Zhangwu Reservoir reaches 137.62m, and overflow to be occurred. It is considered as the beginning of dam breach at once the water overflows the crest of the dam. The process of the breach was computed using the package GDBFR and the computed dam-break flood hydrogragh at the dam site is shown in figure 2 while the final dimension of the breach is listed in Table 1.
Table 1 The final geometric dimension of the breach of Zhangwu Dam
| Side slope | Bottom Elevation | Bottom Width | Top Width | Depth |
| 2.05 | 116.8 | 10.49 | 95.92 | 20.80 |
(3) Channel routing
Routing the dam break flood of Zhangwu dam (Fig.2) by the complete Saint-Venant equations is aimed at computing H, Q hydrograghs at specified cross sections along the downstream river channel as well as the unsteady water profile in the river. In addition, Figure 3 and 4 show the maximum. H, the maximum. Q and their arriving time at specified cross sections respectively, while the time is different for the peak discharge and the peak level at every cross section.
From Fig 2, it can be seen that the characteristics of the dam-break flood is occuring suddenly with high peak, it also can be seen from Fig. 3 and 4 that the level and discharge of the dam-break flood are all beyond the general record. For example, Anyang city located at the 26.5 km downstream of the Zhangwu dam, with the average ground elevation of 75m and historical flood peak of 970m3/s, was predicted with the dam-break flood peak and water level as shown in Table 2.
Fig. 2 Dam Break flood hydrogragh
Fig. 3 Peak Flood Elevation along the downstream channel
Table 2 GDBFR prediction value
|
Predicted flood peak |
Predicted flood level |
||
|
Numerical Value(m3/s) |
Ratio in Percent |
Numerical Value(m) |
Submerge Depth(m) |
|
1980 |
204% |
77.08 |
2.00 |
Fig. 4 Peak Discharge along downstream channel
(1) Considering reservoir water balance, particularly the overflow through dam crest, overflow erosion process and stability of side slopes of dam breach, a mathematical model for gradual dam break was developed and then complete Saint-Venent equations may be used to route the dam break flood predicted by the gradual dam-break model. So a computer package called GBDFR was developed.
(2) An engineering application with GDBFR to earth dam of Zhangwu Reservoir was described. The case study shows that a gradual dam-break model is more actual than that based on assumptions that the final dimensions of the breach may be specified in advance of simulation and the breach will develop linearly from the initial to the final dimensions specified.
References
[1] Handbook of Hydraulic Structure Design, vol 1, Water Power Press,1983 (in Chinese).
[2] Qian Jiahuan, Soil Mechanics, Hehai University Press, 1990 (in Chinese).
[3] Singh, Vi jay.P., Scarlatos, Panagiotis, Breach Erosion of Earth-Fill Dams and Flood Routing, Louisiana State University. 1989.