CALCULATION METHOD OF INUNDATION DISCHARGEHYDROGRAPH FROM THE TEMPORAL CHANGES IN
WATER SURFACE PROFILE DUE TO LEVEE BREACH
Shoji Fukuoka1, Tomonori Abe2 and Yosuke Tsukamoto3
1

Professor, Research and Development Initiative, Chuo University
1-13-27, Kasuga Bunkyo-ku, Tokyo, 112-8551, Japan, e-mail: sfuku@tamacc.chuo-u.ac.jp
2
Tsukaba Central Research Institute
1-6, minamihara, Tsukuba City, Ibaraki, 305-8516, Japan
3
CTI Engineering, co, Ltd
1-14-6, Kamikizaki Urawa-ku, Saitama, 330-0071, Japan, e-mail: tsukamoto@ctie.co.jp

ABSTRACT
For predicting discharge hydrograph and progress of inundation flow caused by thelevee breach, it is important to evaluate inundation discharge precisely. In this study, utilizingthat the change in hydraulic and geometrical conditions during the flood appear in thetemporal changes in water surface profiles, we developed a calculation model to estimate
inundation discharge hydrograph using unsteady two-dimensional flow equations and thetemporal change in water surface profiles including effects of the levee breach. To validate
the model, we carried out field experiments of levee breach of a channel constructed in the
Joganji River. Compared with the experimental results, the simulated inundation discharges
showed acceptable accuracy. Finally, several issues were raised during appllying this method
to actual rivers.
Keywords: levee breach, inundation discharge hydrograph, field experiment, numericalcalculation, temporal change in water surface profile

1. INTRODUCTION
The plan-scale floods happen frequently, and the levee-breach tends to be problem inJapan in recent years. The levee-breach occurs, the flood flow goes into landside andcatastrophic damage is caused to a lot of human, social properties. It is necessary to estimate
the inundation discharge that considers effective measures to the inundation according to thelevee-breach. The inundation discharge is a boundary condition when the inundation analysis
is done, and the accuracy of the inundation analysis is tightly related with the accuracy of theboundary condition. Therefore, it is important to estimate inundation discharge accurately andto make the progress of the flood flow in landside predictable to minimize damage by thelevee-breach.
Homma’s formula (1962) has been used as a conventional method to calculate theinundation discharge. However, this method is not capable evaluating the phenomenon oflateral overflow, since it is front overflow formula. PWRI (Public Works Research Institute)proposed a method corrected from the Homma's formula which considers the inflow angle tothe levee-breach point and bed slope by the hydraulic model experiment that reproduces thecharacteristic of lateral overflow (1996).
The changes in hydraulic and geometrical conditions of river channel during the flood
can be reflected by the temporal changes in water surface profiles. Fukuoka (2004, 2005,
2005b, 2006) suggested a two dimensional flow analysis method using measured changes ofwater surface profiles to estimate the river discharge and verified its effectiveness. Meanwhile,
field experiments of levee-breach were carried out in the Joganji River.
Based on the results from the experiments, the following three aspects are presented
in the paper. Firstly, a calculation method for the inundation discharge hydrograph isdeveloped, which uses the changes of water surface profiles which reflects effects of levee
breach, and is validated through the comparison between the simulated and observed
discharges. Secondly, the applicability of this method in the actual river systems is examined.
Thirdly, the conventional method is applied in the experiment, and its accuracy is examined.
2. FIELD EXPERIMENTS OF LEVEE BREACH IN THE JOGANJI RIVER
In 2005 and 2006, the field experiments were carried out for simulating the leveebreach in the Joganji River (2006b). In this paper, 2006 experiment is mainly examined. The
experimental channels (11.1-km point, 120-m-long, 6-m-wide, side slope 1:1) were made by
digging sand bar in the river. Figure 1 shows the planform of the experimental waterway and
Figure 2 shows the photograph of the experimental way and inundated water after leveebreach. Inundated waterway was set in W13 point at flow attack point. Inflow discharge waschanged by adjusting inlet width of main waterway in upstream end, where the sand andWater level (level survey)
Cross-sectional shape
Discharge observation

Inundated waterway W16
W14

W18 W20

Water level (water gauge)

W21

W22

flow
W12
W10
W08

W01

W02 W03 W04

W06
Main waterway 120m

Figure 1. Planform of the experimental waterway

Inundated waterway

Flow
Flow

Main waterway

Figure 2. Photograph of the experimental way and inundated water after levee-breachMeasured upstream discharge

Measured downstream discharge

Measured inundation discharge

18

Average bed heightF5

14

F4 Water level

W20

W22

W16

12

F3

10
8

F4

Elevation (m)Discharge (m3/s)

F3 Water level

F5 Water level

60

16

F6

F2

6
4

F2 Water level

F7
F1

W04 W08
W01

58

Breach point

Time of levee-breach

2
0
13:30

59

W12

57
14:00

14:30

15:00

15:30

16:00

0

20

Time

Figure 3.

Temporal change in discharge in theexperiment

40

60

80

100

120

140

Longitudinal distance (m)Figure 4.

Bed height and F2 to F5 water surface
profilesgravel were moved by heavy equipment. Firstly, a flow discharge same as the peak discharge
in the levee-breach experiment was released into the experimental channel, and the stableriver bed was then made according to the peak discharge in the experiments. Secondly, theunsteady flow was released into experimental channel and the levee breached before the peakdischarge was reached. The items, such as water level, discharge, shapes of channel andgrain-size distribution of bed materials were measured. And the temporal changes in water
surface profiles were observed in detail. Discharge was observed at three cross-sections:
upstream and downstream from the breach point and the inundated waterway.
Results of the experiments
Figure 3 shows the temporal change in the experimental discharges. The peakdischarge of the experiment is 14.7m3/s, the inundation discharge is 4.6m3/s. Figure 4 shows
the longitudinal average bed height and F2 to F5 water surface profiles. Comparison between
before and after levee-breach, the water surface slope is growing in levee-breach pointneighbourhood because of discharge of main waterway flowed out to the inundated waterway.
From results of the experiment, it is found that the change in the river conditions caused bylevee-breach is clearly reflected by the temporal changes in water surface profiles.
3. METHOD OF ANALYSIS
Analytical outline
The levee-breach phenomenon has changed the main channel condition complexly
because of changes in flow regime before and after the levee-breach, the variation of river bed
and transportation of the earth and sand in breach point. However, the changes in the riverconditions caused by levee-breach can be clearly reflected by the temporal changes in water
surface profiles.
The accuracy of the observed temporal variation of river water surface profiles ishigher than that of the observed discharges. Fukuoka (2004, 2006) calculates accurately
discharge hydrograph using a two dimensional flow analysis method together with measured

Temporal change in Water
surface profile data

Discharge timeseries data
(Reference)

Topography data
Planform
main channel
Cross-section

Boundary conditions
Upstream-end water-level time series
Downstream-end water-level time series
Width of levee-breach point

River conditions

Roughness distribution

Water level analysis (main channel)
Upstream water-level time series

Upstream pool water-level adjustment

Downstream water-level time series

Downstream pool water-level adjustment

water-level time series of leveebreach point neighbrhood

Inundated waterway-end pool waterlevel adjustment

Result of analysis
Temporal change in water surface profile
Temporal change in velocity distribution

Adjustment of resistance
distribution

Comparison with measured water
surface profile
Discharge hydrograph (main cannel)

Operation after
levee breachInundation discharge hydrograph

Figure 5.

Flowchart of the analysis.

changes of water surface profiles. From this logic, this analytical method can apply thecalculation of inundation discharge hydrograph from the temporal change in the water surface
profiles in main channel which reflects the effects of levee-breach. This method yields theinundation discharge hydrograph without using overflow weir formulas.
Analysis method
The analysis method was developed by Fukuoka and Watanabe (2004). The governing
equations are two dimensional shallow water equation and continuity equation based on
general coordinate system. Figure 5 shows the flowchart of the analysis. Before levee-breachoccured, ponds and weirs were installed hypothetically in the computation at the upstream anddownstream end in main channel, and weirs in upstream and downstream were used to adjust
the water levels of ponds. From the comparison between the measured and calculated waterlevel profiles, the spatial distribution of roughness coefficients is adjusted so that thecalculated water surface profiles can represent measured profiles. After the levee-breachoccured, inundation discharge hydrograph is determined by adjusting a weir in downstream

Upstream boundary condition
Downstream boundary condition
W20 W21
Width of levee breach W16 W18

W22

Table 1. Main channel's Manningroughness coefficient

Hypothetically
Inundation waterway
W14
W12

Area
W01〜W03
W03〜W06
W06〜W11
W11〜W17
W17〜W23

flow

W10
W08
W06

Elevation (m)Figure 6.

Condition for analysis

62.5

W16 beforeW16
levee
breach
破堤前61.5

W16 after levee
breachW16 破堤後

62.0
Elevation (m)W01 W02 W03 W04

Roughness coefficient
0.018
0.040
0.018
0.040
0.018

60.5
59.5
58.5

W08 before
levee
breachW08
破堤前
W08 after W08
levee破堤後
breach

61.0
60.0
59.0
58.0
57.0

57.5
-5

-3

-1

1

3

Lateral distance (m)Figure 7.

5

-5

-3

-1
1
Lateral distance (m)35Cross-sectional shape of breach point neighborhood

end of inundated waterway to represent temporal changes in water surface profiles in main
channel under the influence of levee breach.
4. LEVEE BREACH EXPERIMENTATION AND CONSIDERATION ON THE
RESULTS OF ANALYSIS
Conditions for analysis
Figure 6 shows the condition for analysis. The experiment was carried out for 3-hours
from 13:00 to 16:00 October 18, 2006. The same time was used in the analysis. The analyticalreach is 120-meter-long from W01 to W22 point. Water levels were measured every 2
seconds using the pressure type water gauges placed at an 8-meter interval. Simultaneously,water levels were also measured according to level surveying every 20-minutes at a 4-meterinterval. The boundary conditions in upstream side and downstream side are W22 and W02
points respectively. Water levels of four (W08, W10, W12, and W14) are used asconsiderable influence part of levee-breach point neighborhood. The width of breach point ina hypothetically inundated waterway uses the width after the experiment ends. The spatial
distribution of roughness coefficients used for the analysis is shown in Table 1. The value of
0.040 is used in the section of W03-W06 and W11-W17 in meandering part. 0.018 is used in
the section of W01-W03, W06-W11, and W17-W23 in straight part. In the meandering part, alarge roughness coefficient is used because big power exerted from the wall when the streamline is bent by the curvature of the waterway. Figure 7 shows the cross-sectional shape of
breach point neighbourhood. The analysis is treated as a fixed bed by using the measured

13:40 Measured water level

13:40 Calculated water leve

14:40 Calculated water level

14:40 Measured water level

14:00 Measured water level

14:00 Calculated water level

14:50 Calculated water level

14:50 Measured water level

14:20 Measured water level

14:20 Calculated water level

15:00 Calculated water level

15:00 Measured water level

59.6

59.0
W04

W08

W02

58.1

59.0
58.7
58.4

W02

57.5
15

35

55

75

95

115

135

15

Longitudinal distance (m)

75

95

115

135

Figure 8(b). Measured and calculated water-levelprofile ( Immediately after levee breach )15:20 Measured water level

Calculated upstream discharge

Measured upstream discharge

15:40 Calculated water level

15:40 Measured water level

Calculated downstream discharge

Measured downstream discharge

15:50 Calculated water level

15:50 Measured water level

Calculated inundation discharge

Measured inundation discharge

16:00 Calculated water level

16:00 Measured water level

59.0

W22

W20

W16

16
Discharge (m3/s)

59.3
W12
W04

W08

W02

12

8

4

57.8
57.5
15

55

15:20 Calculated water level

59.6

58.7

35

Longitudinal distance (m)Figure 8(a). Measured and calculated water-levelprofile ( rising-discharge phase )Water level (m)

W08

W22

W20

57.8

57.5

58.1

W04

W12

58.1

57.8

58.4

W16

59.3

W12

58.7
58.4

W20

W16

Water level (m)Water level (m)

59.3

59.6

W22

35

55

75

95

115

135

Longitudinal distance (m)Figure 8(c). Measured and calculated water-levelprofile ( Falling-discharge phase )

0
13:30

Time of levee-breach

14:00

14:30

15:00
Time

15:30

Figure 9. Temporal change in measured
and calculated discharge hydrographs

results after the experiment ends because the variation of river bed is small without the breachpoint neighborhood. The computational grid consisteds of 120 divisions vertically and 10
division laterally. While for the hypothetical inundated way, the grid consists of 10 divisions
vertically and 10 division laterally.
Consideration on the results of analysis
Figure 8 (a)-(c) shows the measured and calculated longitudinal left-bank water levelprofiles in the rising-discharge phase, immediately after levee breach, and the fallingdischarge phase, respectively. It is found that the calculated water levels are generally in agood match with the observed values. In the meander part, calculated water levels are
relatively low reproduction. In immediately after levee breach, the difference between
calculated and measured water levels is growing in meander part, and the difference isremarkable as the discharge scale is growing.
In the meander part, it is a strong three dimensional flow-field, and it is difficult toreproduce the flow enough in the two dimensional shallow water equation to assume thehydrostatic pressure distribution because the pressure distribution is different from the

16:00

hydrostatic pressure distribution
Figure 9 shows the comparison between measured and calculated dischargehydrographs. Compared with the rising-discharge phase, the accuracy is relatively low near
the peak discharge during falling-discharge phase because the difference between measured
and calculated water surface profiles caused the difference of discharge in the meander part.
From the above examination, it was shown that this method can evaluate wellinundation discharge.
5. APPLICATION OF THIS METHODOLOGY TO ACTUAL RIVERS
From the field experiment and the analysis, the inundation discharge was evaluated byusing observed temporal change in water surface profiles. However, in actual rivers, waterlevel gauges are only set at an interval about several kilometres. Therefore, it is difficult tomeasure the temporal changes in water surface profiles accurately. This chapter describes theexamination of this analytical method concerning the application method to the river.
Evaluation of inundation discharge from difference of upstream and downstreamsides in main channel

Discharge (m3/s)

The influence of leveebreach appears in not only the
Calculated upstream discharge
Measured upstream discharge
breach point neighborhood but also
Calculated downstream discharge
Measured downstream discharge
Difference between up and down
Measured inundation discharge
the channel overall. Therefore, the16inundation discharge is appreciableby the separated analysis of
12
temporal change in water surface
profiles on the upstream side and8
the downstream side in main
channel. And discharge hydrographis estimated as the difference of4
calculated
upstream
anddownstream discharge in main
0
channel.
13:30
14:00
14:30
15:00
15:30
16:00
Time
The analysis method is thesame as that used by Fukuoka andFigure 10. Difference between upstream and
Watanabe (2004). Two dimensional
downstream sides calculated discharge hydrographs
analysis was carried out that
reproduced the observed temporal change in water surface profiles in each section. W08 andW02 point are boundary conditions in upstream side, W22 and W20 point are boundary
conditions in downstream side. The same Roughness coefficient is used as shown in Table 1.
Figure 10 shows the inundation discharge hydrograph calculated as the difference
between upstream and downstream sides. The difference of inundation discharge separately
calculated at upstream and downstream sides can reproduced measured discharge in main
channel as well as the analytical result shown in Figure 9.
It was shown to be able to evaluate the inundation discharge hydrograph with a good
accuracy by the separated analysis on the upstream and the downstream sides in main channel
where the levee-breach point was set.

6. EVALUATION OF THE CONVENTIONAL CALCULATION METHOD OFINUNDATION DISCHARGE HYDROGRAPH
The Homma’s over weir formula has been used to estimate the inundation dischargehydrograph in the locale. The river condition changes temporally in levee-breach
phenomenon. The breach section shape and the water level of the breach point neighborhood
are necessary to the application the formula. So it is impossible to obtain information withhigh accuracy especially near the breach point. Moreover, it is necessary to verify whether it
is good to apply the weir formula to the levee-breach with a different river condition. In thischapter, the problem when the conventional method for calculating inundation discharge isapplied to the field experiment, and its accuracy is examined.
Conventional calculation method of inundation discharge hydrograph
Homma’s front overflow formula calculating quantity of overflow from weir is written
by Eq. (1), (2) as condition of overflow
Complete overflow

Q0

0.35h1 B 2 gh1

Submerged overflow

Q0

0.91h2 B 2 g ( h1

(1)

h2 )

(2)

In which h1 and h2 is water level, Q0 is quantity of overflow, B is width of breach point, g is
gravitational acceleration
The PWRI (Public Works Research Institute) proposes calculation method ofinundation discharge (written by Eq (3), (4), (5)) that corrects Homma's formula. This method
considers the inflow angle to the levee-breach point and bed slope by the hydraulic modelexperiment in order to reproduces the characteristic of lateral overflow.
When bed slope is I 1 / 1580 thenQ / Q 0 (0.14 0.19 log 10 (1 / I )) cos( 48 15 log 10 (1 / I ))(3)

When bed slope is 1 / 1580 I 1 / 33600 thenQ / Q0 (0.14 0 .19 log 10 (1 / I ))(4)

When bed slope is 1 / 33600

Q / Q01

I then(5)

In which Q is inundation discharge, Q0 is Homma’s quantity of overflow, I is bed slope.
Result and consideration of application
Figure 11 shows the comparison between measured total inundated volume andcalculated one by the overflow weir formula. Horizontal axis shows the measured total
inundated volume and longitudinal axis shows the calculated total inundated volume by
Homma’s formula, PWRI method, and this method from water surface profiles. The reason touse not inundation discharge (m3/s) but total inundated volume (m3) is that total inundated
volumes are essentially important for the inundation phenomena. The calculated total
inundated volume is calculated by time integrating the calculated discharge from Homma’s
formula, PWRI method, and this method from water surface profiles by using hydraulic andCalculated total inundated volume (m3)geometrical value at each observation
time. And the measured total inundated
15000
volume is calculated in similar way. The
calculated total inundated volume
obtained by this method expresses the
12000
measured total inundated volume
accurately. Moreover, the calculated
9000
total inundated value from the Homma’s
formula in 2006 experiment seems to be
6000
a result of explaining the observation
2006 Homma’ formula2005 Homma’ formulavalue. But, it should be thought a
2006 PWRI method
3000
coincidence in this result. The temporal
2005 PWRI method
2006 This method from water surface profiles
change in water surface profiles and2005 This method from water surface profiles
0
width of breach point neighborhood is
0
3000
6000
9000
12000
15000
necessary to evaluate the inundation
Measured total inundated volume (m3)discharge by using Homma's formula
Figure 11. Comparison between measured andand the PWRI method. These values
overflow weir formulawere observed in the experiment, and it
was possible to use it to calculate.
However, in a actual flood, it is almost impossible to obtain these values with good accuracy.
From this examination, it is found that this method to calculated inundation discharge
from temporal change in water surface profiles is corrected physically and holds highreliability compared with conventional method using width and water levels in breach pointthat measurement is difficult. However, Homma’s formula and PWRI method are useful forthe estimation simply.
7. CONCLUSIONS
The authors carried out the levee-breach experiment to develop a calculation model toestimate inundation discharge hydrograph accurately and examined its validity. Secondly, theapplication method to the river of this analysis method was examined. Thirdly, theconventional method to calculate inundation discharge was applied to the experiment, and theapplicability and the problem were examined. The conclusions in this research are presented
below.
(1) Calculation method of inundation discharge hydrograph and results of analysis
Inundation discharge hydrograph was calculated by using temporal change in water
surface profiles that reflects effects of levee breach and two dimensional flow analysis.
Comparison with experimental results, it was showed that this method can evaluate
inundation discharge precisely.
(2) Application of this methodology to actual rivers
It was shown to be able to evaluate the inundation discharge hydrograph as thedifference of discharge between upstream and downstream sides of the levee-breach point in
main channel with a good accuracy.
(3) Evaluation of the conventional calculation method of inundation discharge hydrograph
The conventional method using the overflow weir formula was applied to the levee
breach experiment, and the problem was shown. It was shown that this method to calculate
inundation discharge from temporal change in water surface profiles were useful.

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pp.215-241.
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Formula of Flooding Discharge due to Dike-Breach under Consideration of the
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