The 7th Int. Conf. on Hydroscience and Engineering (ICHE-2006),Sep 10 –Sep 13,Philadelphia,USA

1

EVALUATION OF DISCHARGE HYDROGRAPHS AND FLOOD STORAGE
IN THE MAIN RIVER AND TRIBUTARY CONFLUENCE
Shoji Fukuoka1 and Hiroaki Sato2
ABSTRACTRivers possesses the function of storage and peak discharge attenuation of flood water.
Author's recent studies have deepened basic understanding of these important characteristics of theflood flow. At the river confluence, downstream boundary conditions of a tributary change withtemporary change in water level of the main river. Waters flowing down the tributary andtransmitted back into the tributary from the main river are stored in the tributary section.
Furthermore, at the river confluence, a water level changes with time and adverse water
surface slope occurs under the temporal change in the downstream boundary conditions. To estimate
accurate discharge hydrographs under such complicated hydraulic conditions is important for theriver planning of the main river and tributary. The authors, focusing on the confluence between theTone and Watarase Rivers—for which temporally detailed flood level hydrographs were available—
decided to use two-dimensional unsteady-flow analysis to re-create the temporal changes in water
surface profile with the goal of creating highly precise estimated discharge hydrographs and storage
rate for the main river–tributary confluence.
Key Words : water surface profile, flood discharge hydrograph, flood storage volume

1. INTRODUCTION
It has long been known that as a flood flow having a temporally variable discharge travels
downstream, storage of the floodwaters occurs according to the planform and cross-sectional formof the channel, resulting in the attenuation of peak discharge and hydrograph transformation(Nomitsu 1964, Fukuoka 2005). Fukuoka and Watanabe (2003) arrived at a fundamental
understanding of this storage phenomenon by conducting a series of experimental investigations
from the viewpoint that estimating both the transformation of flood discharge hydrographs and theattenuation of peak discharge is of great significance, and that storage volume should bequantitatively assessed and incorporated into river planning. In addition, Fukuoka and Watanabe(2004a) using the two-dimensional unsteady-flow analysis that emphasizes temporal change inwater surface profile, have made highly precise estimates of discharge hydrographs and channel
storage in the Edo and Maruyama Rivers on the basis of longitudinally detailed observed waterlevel hydrograph data. Fukuoka and Watanabe (2004b) have also investigated the issue of1

Professor, Research and Development Initiative, Chuo University，1-13-27 Kasuga Bunkyo-ku Japan(sfuku@tamacc.chuo-u.ac.jp)
2
Director, Foundation of River Basin Integrated Communications, Japan, 1-3 Koujimachi Chiyoda-ku Japan(hi-sato @ river.or.jp)

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measuring water level at practical longitudinal intervals, and have found that an interval of 3 to 4
km would yield a discharge hydrograph of sufficient precision.
One-dimensional unsteady-flow analysis, which is widely used in such cases, is capable ofincorporating the effects of storage arising from the unsteadiness of flood flows but cannot
adequately account for two-dimensional behavior of flood. For instance, it is difficult for thetechnique to assess, with a high degree of precision, flood discharge hydrographs and storage
volume arising from irregularity in a channel’s profile and cross-section. Consequently, at the very
least, two-dimensional unsteady-flow analysis is needed to describe vividly changes in flood
discharge hydrograph and storage in a watercourse with a complex planform. (Fukuoka 2005)
At the confluence of a main river and tributary whose downstream-end boundary conditions
change temporally, flow retention occurs in the section of the tributary affected by confluencewater level, resulting in overlapping of the flow from the tributary’s upstream reaches and the flowtransmitted back into the tributary from the main river. Because tributary water level at such a
confluence changes temporally (including taking on a negative water surface slope) due to thedownstream-end boundary conditions, high-precision estimated discharge hydrographs for suchtributaries were considered difficult to generate. Therefore, The authors, focusing on theconfluence between the Tone and Watarase Rivers—for which temporally detailed flood level
hydrographs were available—decided to use two-dimensional unsteady-flow analysis to re-create
the temporal changes in water surface profile with the goal of creating highly precise estimateddischarge hydrographs and storage rate for the main river–tributary confluence.
2. METHOD OF ANALYSIS
Velocity, water level, and other quantities used in the analysis were determined using
equations of motion (1) and (2) and continuity equation (3) below in a physical component
representation form based on a general coordinate system. The method of calculation is the sameas that used by Fukuoka and Watanabe (2004a).
~
~
~
∂ζ
∂ζ
∂θ ξ
∂θ ξ
∂U
~ ∂U ~ ~ ~~ ∂U ~ ~ ~h + Uh{ ~ − J (V − U cos θ ηξ ) ~ } + V h{ ~ − J (V − U cos θ ηξ ) ~ } = − gh{ ~ + cos θ ηξ ~ } − τ zξ
∂η
∂η
∂η
∂t
∂ξ
∂ξ
∂ξ
+

1 ∂ Jh ~∂θ ξ ~∂θ ξ∂ Jh ~~[ ( τ ξξ ) +( τ ξη )] − J h( −τ~ξξ cos θ ηξ + τ~ξη ) ~ − J h( −τ~ξη cos θ ηξ + τ~ηη ) ~∂η∂η dηJ ∂ξ dξ
∂ξ

(1)

~
~
~
∂V
∂θ η
∂θ η
∂ζ ∂ζ
~ ∂V ~ ~ ~~ ∂V ~ ~ ~h + Uh{ ~ + J (U − V cos θ ηξ ) ~ } + V h{ ~ + J (U − V cos θ ηξ ) ~ } = − gh{cos θ ηξ ~ + ~ } − τ zη
∂t
∂
η
∂
η
∂ξ
∂ξ
∂ξ ∂η
+

J

1 ∂ Jh ~∂ Jh ~∂θ η ~∂θ η~[ ( τ ηξ ) +( τ ηη )] − J h(−τ~ξξ + τ~ξη cos θ ηξ ) ~ − J h(−τ~ξη + τ~ηη cos θ ηξ ) ~∂η dη∂ηJ ∂ξ dξ
∂ξ

(2)

~
~
∂h ∂ JUh∂ JV h+()+()=0∂t ∂ξ dξ
∂η dη

(3)

∂
∂
~=
∂ξ dξ∂ξ∂∂=~∂η dη∂η(τ ,τ ) = ⎛⎜⎜ gn + gh ⎞⎟⎟ u + v (U~,V~ )
2

zη

zη

a

⎝h

1/ 3

K2 ⎠

ha = min(h, htree)

2

2

~ ~
~~
~
u 2 + v 2 = J 2 (U 2 − 2UV cos θ ηξ + V 2 )

(4)

3

where ( ξ ,η ) is the general curvilinear coordinate system, ( dξ , dη ) is the contravariant distance, ~~ ~is a physical component, ( U , V ) is contravariant velocity, h is depth, J =[ xξ yη − xη yξ = 1 /(ξ xη y − ξ yη x ) ] is the Jacobian, ( θ ξ , θ η ) is the angle of the x-axis in thecontravariant coordinate system, θ ηξ is the angle of intersection in the contravariant coordinate
system, and [ u 2 + ν 2 ] is absolute velocity. The shear stress terms [ τ zξ ,τ zη ] indicate bottom

Downstream-end
water-level time series

3. ANALYSIS CONDITIONS FOR
THE TONE AND WATARASE
RIVERS

Figure 2 shows the planformand ground cover of the analyzed
sections of the Tone and Watarase
Rivers. The flood in these sections areprimarily grassland, with generally
sparse growth of taller vegetation,although the flood plain of the 128.0–
129.5 km section of the Tone River
contains dense vegetation. In addition,
public-use areas have been establishedin the flood plain of both rivers,
including downstream from the 130.0
km point of the Tone River, on the leftbank. There is also a golf course on theleft bank of the Watarase River,
between the 3.5 and 5.0 km points.

Downstream-end weirheight adjustment

Results of analysis

Discharge hydrograph
Water level profile

Comparison withobserved discharge

Adjustment of
absolute resistanceComparison withobserved water
surface profile

Adjustment of resistance
distribution

Figure 1. Flowchart of analysis.

Back calculation of resistance distribution

shear stress and fluid resistance of vegetation and are determined with equation (4) using
Manning's roughness coefficient n, vegetation permeability coefficient K, and vegetation height
htree. Convection terms are discretized as an upwind differential and indicated with first-orderaccuracy, as it includes the flood front inundating over the flood channel. The overall procedure ofanalysis is shown in the flowchart in Figure 1. Conventional unsteady-flow analysis is used to
calculate temporal changes in water level and discharge from observed discharge and water level
for the upstream and downstream ends, respectively, under the assumption that observed discharge
is accurate. However, observed flood discharge is thought to contain more error than water
surface profile determined from water level data simultaneously obtained at multiple locations.
Consequently, in this analysis, theauthors, assuming water profile to be
Topography data
Water-level data
more accurate than observed discharge,
Ground cover
Planform
Discharge data
information
Cross-section
and using the same methodology
employed previously by Fukuoka andBoundary conditions:
Watanabe (2004a), back-calculated
Analytical grid
Upstream-end water-level time
resistance coefficient distribution soseriesDownstream-end water-level
Water level andthat temporal changes in calculated
time series
Roughness distribution
discharge analysis
discharge and water surface profile
Vegetation distribution
generally agree with the observed data,
Upstream poolthen used this estimated resistance
Upstream-end waterwater-level
coefficient distribution to determine thelevel time series
adjustment
discharge hydrograph.

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(km)

●：Discharge measurement station

7

Watarase.River○：Water level observation point

Flow

6

5

Koga
5

4
3

Public use area

4
2
Flow

3

Public use area

1

Dense vegetation

Tone River
2
139

134

136

138

1

132

Glider landing strip

Saitama Bridge

Kurihashi

130

128

0
0

2

4

6

8

10

12

(km)

Figure 2 Planform and channel conditions of the studied sections.

Table-1 Roughness coefficient and vegetation permeability coefficient of the Tone River

Location
Main-channel
Flood-plain
public use area
128.0-129.0
Right
Left
129.0-130.0
Right
Left
130.0-131.0
Right
Left
131.0-132.0
Right
Left
132.0-133.5
Right
Left
133.5-135.5
Right
Left
135.5-136.5
Right
Left
136.5-139.0
Right
Left

Roughness
Coefficient
0.029
0.034
0.039
0.048
0.042
0.052
0.042
0.045
0.048
0.046
0.048
0.048
0.038
0.040
0.040
0.036
0.040
0.038

Location
128.0〜129.0 Right bank129.0〜131.5 Right bank131.5〜133.5 Right bank133.5〜135.0 Right bank
135.0〜139.0 Right bank
128.0〜129.5 Left bank129.5〜130.5 Left bank130.5〜132.0 Left bank132.0〜134.0 Left bank134.0〜136.0 Left bank136.0〜137.0 Left bank137.0〜139.0 Left bank

Vegetation
permeability
coefficient (m/s)
42.0
48.0
52.0
50.0
52.0
33.5
42.0
38.0
48.0
46.0
48.0
44.0

5

Table-2 Roughness coefficient and vegetation permeability coefficient of the Watarase River

Location
Main-channel
Flood-plain
public use area
0.0-2.0
Right
Left
2.0-3.0
Right
Left
3.0-4.0
Right
Left
4.0-5.0
Right
Left

Roughness
Coefficient
0.029

Location
0.0〜2.0 Right bank
2.0〜3.0 Right bank
3.0〜4.0 Right bank
4.0〜5.0 Right bank
0.0〜2.0 Left bank
2.0〜3.0 Left bank
3.0〜5.0 Left bank
0.0〜2.0 Right bank
2.0〜3.0 Right bank
3.0〜4.0 Right bank
4.0〜5.0 Right bank
0.0〜2.0 Left bank

0.034
0.042
0.040
0.038
0.036
0.039
0.038
0.038
0.036

Vegetation
permeability
coefficient (m/s)
48.0
52.0
49.5
53.0
46.0
54.0
44.0
48.0
52.0
49.5
53.0
46.0

Water level/elevation (m)Water-level measuring points are indicated in the figure with hollow circles. Discharge is measured
at three cross-sections: Saitama Bridge, Kurihashi, and Koga.
A trial-and-error approach was employed to determine the roughness coefficients andvegetation permeability coefficients used for the main channel and flood plain in the analysis. The
values are listed in Table 1. The analytical grid is divided longitudinally into 372 cells for the Tone
River and 180 cells for the Watarase River, and divided laterally into 20 cells for both rivers.
The flood studied was one thatlasted for several days beginning on
Lateral distance (m)September 10, 2001 (see Figure 3).
150
300
0
450
900
750
600
Discharge was measured at three25
Koga(automated water-level meter)
locations—Koga and Saitama Bridge at
Koga(water level at discharge observation station)
the upstreamend and Kurihashi at the
Saitama Bridge(water level at discharge observation station)
downstream end—approximately every
20
2 hour over a 48-hour period, from 8:00
p.m. September 10 to 8:00 p.m.
September 12. Water level was
15
measured approximately every hour at 1km intervals in both rivers: on the rightbank of the 130.0–138.5 km section of
10
the Tone River and on the left bank of
Cross-section at Saitama
Bridg
the 0–3.5 km section of the Watarasee
時間(hour)River. To improve the accuracy of
Cross-section
at Koga5calculations, the authors extended the
-20
0
20
40
60
80
100
target river section beyond the locations
Time (hour)of the upstream (Watarase River) anddownstream (Tone River) discharge
measurement stations.
Figure 3(a). Observed water levels and cross-sections
Analysis focused on the 48-hour
at discharge observation stations.
period from 8:00 p.m. September 10 to6

Water level/elevation (m)8:00 p.m. September 12. The former
Lateral distance (m)
300
600
900
150
450
750
time, which marked the start of25
observations, was defined as the
Kurihashi (automated water-level meter)
reference time.
Kurihashi (water level at discharge observation station)
Cross-section at Kurihashi
Figure 3(a) and 3(b) show
20
water level recorded by automatedwater-level meters, water level atdischarge observation stations, and15
river cross-sections at the upstreamend (Saitama Bridge and Koga) andthe downstream end (Kurihashi).
10
The automated water-level meterdata showed good agreement withthe discharge observation station
5
data at Koga and Kurihashi and were
80
100
-20
0
20
40
60
Time (hour)Time (hour)
therefore used as the upstream anddownstream
endboundary
Figure 3(b). Observed water levels and crossconditions. Although no automatedsections at discharge observation stations
water-level meter was in place at
Saitama Bridge, because of theagreement between the aforementioned
data, we judged observation error to be slight and therefore used the Saitama Bridge discharge
observation station data as the Tone River upstream end boundary condition(water level). Whenintensive observation began at 8:00 p.m. on the 10th, water level at Kurihashi was 15.5 m, alreadyinundating the flood plain to a depth of roughly 1.0m. The flood plains throughout the observation
section were similarly inundated.

4. ANALYSIS RESULTS AND CONSIDERATIONS
4.1 Water Level Profile
Figures 4(a) and 4(b) and Figures 5(a) and 5(b) show the temporal change in observed andcalculated water level profile in the Tone and Watarase Rivers, respectively. Figures 4(a) and 5(a)show the rising-water period, Figures 4(b) and 5(b) the receding-water period. The calculated
results for the studied section of the Tone River show good agreement with the observed results.
Early in the rising-water period, as seen in Figure 4(a), water level remains essentially uniform near
the 132.0 km point. This is believed to be due to the effect of outflow from the Watarase River, as
the point in question is the confluence with the Watarase River. For the Watarase River section, too,
the calculated results show good agreement with the observed data. The Watarase water surface
profile changes due to the effect of the Tone River’s water level. As Figure 5(a) shows, thecalculations re-create the change from a negative water surface slope in the rising-water period to a
gentle positive slope. Agreement is also good between water levels at the confluence—Tone River
point 132.0 km and Watarase River point 0.0 km—indicating good connectivity at the confluence.
4.2 Comparison of Discharge Hydrographs
Figure 6 compares observed and calculated discharge at the Tone River’s upstream- anddownstream-end discharge observation points (Saitama Bridge and Kurihashi, respectively). Other
than during the rising-water period, the discharge hydrographs exhibit good agreement.

Water level (m)

7

21

21

20

20

19

19

18

18

17

17

16

Observed water level
Calculated water level(4h)Calculated water level(12h)Calculated water level (20h)15Calculated water level(0h)Calculated water level(8h)Calculated water level(16h)16Observed water level
Calculated water level(28h)Calculated water level(36h)Calculated water level(48h)

15

14
130

132

136

134

14

138

130

Longitudinal distance (km)Water level (m)

132

134

136

138

Longitudinal distance (km)Figure4(b) Observed and calculated water levels(receding-water period; Tone River)

Figure4(a) Observed and calculated water levels

20

20

18

18

Water level/elevation(m)Calculated water level(24h)Calculated water level(30h)Calculated water level(42h)1616Observed
level
Observed water
water level
Calculated
water
level
Calculated waterlevel(4h) (4h)Calculated
water
level
Calculated waterlevel(12h) (12h)Calculated
water
level
Calculated waterlevel(20h) (20h)Observed water level
Calculated water level(28h)Calculated water level(36h)Calculated water level(48h)Calculated
water level (0h)Calculated
water level (0h)Calculated
water level (8h)Calculated
water level (8h)Calculated
water level (16h)Calculated
water level (16h)14Calculated water level(24h)Calculated water level(30h)Calculated water level(42h)

14

0

1

2

3

4

0

1

2

3

4

Longitudinal distance (km)Longitudinal distance (km)Figure5(a) Observed and calculated water levels(rising-water period; Watarase River)

Figure5(b) Observed and calculated water levels(receding-water period; Watarase River)Observed discharge in the rising-water period was measured using floats—a techniquewhose poor accuracy compared to results produced in receding-water periods has been seen in
research on the Edo and Maruyama Rivers by Fukuoka and Watanabe (2004), suggesting that thesame phenomenon is reflected in the authors’ Tone River data. The poor accuracy in the risingwater period of a flood is because vegetation in the flood plain impedes the floats’ flow, resulting
in poor discharge accuracy. Starting around peak discharge, however, when the vegetation issnapped or bent by the flow and submerged, the movement of the floats becomes more uniform,
leading to improved accuracy in observed discharge. Therefore, when computationally obtaining
the discharge hydrograph, we emphasized —and attempted to re-create—observed discharge
between its peak in the rising-water period through the receding-water period. Figure 7 graphs
observed and calculated discharge for the 3.5km point (upstream-end discharge observation point)of the Watarase River, as well as calculated discharge at the 0.0 km point (downstream end) and the
1.0 km point. The data indicate a forward flow in the Watarase River during the rising-water period,
despite a negative water surface slope due to the strong effects of the Tone River.

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8000

Discharge (m3/s)

7000
6000
5000
4000
3000
Saitama Bridge observed discharge
Saitama Bridge calculated discharge
Kurihashi observed discharge
Kurihashi calculated discharge

2000
1000
0

0

10

20

30

50

40

Time(hour)

Figure 6. Observed and calculated discharge in
the Tone River.
2500
2000

Discharge (m3/s)

4.3 Comparison of Storage Volumes
This section concerns storage
volume in a section (130.0–138.5 km) of
the Tone River. Figure 8 shows that thecalculations of the average water level inthe river section accurately re-create theobserved average water level at 2-hourintervals. Multiplying the change inwater level in Figure 8 by the water
surface area yields storage volume S foreach point in time. Time-differentiating
S yields storage volume per unit time
(dS/dt). Figure 9 shows temporal change
in average storage rate in the section at
2-hour intervals as determined from thedata in Figure 8; also shown are thedifferences between observed andcalculated discharge at the upstream anddownstream ends. As can be seen in
Figure 9, storage rates (dS/dt)
determinedfromobserved
andcalculated water surface profile are
essentially the same as storage rates
determined from calculated discharge
differences (Qin – Qout). Storage rates
determined from observed discharge
differences (Qin – Qout) differ
considerably from storage rates (dS/dt)
determined from water surface profile
because of the error in observed
discharge, which results in large error inboth differences.
To determine the decrease in peakdischarge, methods that track temporalchange in water surface profile produce
better accuracy than methods that use
observed discharge at the upstream anddownstream ends. In an 8.5 -km sectionof the Tone River containing theconfluence with the Watarase River,

9000

1500

1000
Koga (3.5 km point) observed discharge
Koga (3.5 km point) calculated discharge

500

1.0 km point calculated discharge
0.0 km point calculated discharge

0
0

10

20

30

40

50

Time(hour)

Figure 7. Observed and calculated discharge
in the Watarase River.
20

Average water level (m)This indicates that the energy gradient of
the flood flow in the Watarase River was
positive. At hour 0, discharge at the 0.0
km point was low—the flow was nearlystagnant. Thus, because of the effects of
the Tone River, discharge decreased at aslow rate from peak discharge through thereceding-water period, lengthening theflood duration.

19

18

17

Average observed water level
Average calculated water level
16

0

10

20

30

40

50

Time (hour)

Figure 8. Temporal change in section-averaged
water level in the Tone River studied section.

9

5. CONCLUSION

1200

dS/dt observed
Qin – Qout observed
dS/dt calculated
Qin – Qout calculated

Storage rate (m3/s)

800
400
0
-400
-800

-1200
-1600
-2000
0

10

20

30

40

50

Time (hour)

Figure 9. Temporal changes in average storage
rate in the observed section (130.0-138.5km) of
the Tone River.
400

dS/dt observed
dS/dt calculated
Qin – Qout calculated

300

Storage rate (m3/s)

calculated dS/dt was approximately
1,030 m3/s at peak, meaning that roughly
20% of the discharge entering thissection at this time was being stored.
Fig. 10—the Watarase River storage rateas calculated from Qin – Qout, along withstorage rate calculated from observedand calculated water surface profile—shows that dS/dt calculated fromobserved and calculated water surface
profile is roughly equal to the storage
rate calculated from the computationallyobtained Qin – Qout. In short, theproperly
re-create
calculations
storage in a confluence where the water
surface profile exhibits a negative water
surface slope. At peak flooding, thecalculated value of dS/dt in the 3.5 -km
section of the Watarase River was
roughly 330m3/s, indicating that an
amount of water equivalent to 51% of
the discharge entering the section at thistime was stored in the channel. Although
the September 2001 flooding exhibited
such storage properties, other floods
occurring in the Tone and Watarase
Rivers must be similarly studied.

200
100

-

0
-100

The authors used a combination-200
of field data and computations to derive
10
40
50
0
20
30
-10
discharge hydrographs and channel
Time (hour)storage volumes for the confluence
Figure 10. Temporal changes in storage rate
between the Tone and Watarase Rivers.
in the observed section (0-3.5km) of the
The primary conclusions reached in this
Watarase River
research are as follows.
(1)Using detailed observed water surface profile data, the authors were able to estimate, with a highdegree of accuracy, the discharge hydrograph for a confluence having a negative water surface
slope in an early stage of rising water period by performing two-dimensional unsteady-flow
analysis that conformed to the observed water surface profile. In addition, the calculated dischargehydrograph made it possible to estimate changes in storage volume in the channel. Furthermore, theparticular Watarase River flood examined was characterized by a negative water surface slope in
the rising-water period due to the large flood discharge, although the flow itself was a forward one.
(2) In the Tone River section—a straight section that confluences with the Watarase River—storage
peaked at approximately 20% of the discharge entering the section at that time. In addition, storage
was greater in the section of the Watarase River affected by the Tone River.

10

ACKNOWLEGEMENTS

The authors express our appreciation of the assistance rendered us by Mr. S, Nagai, graduate studentof Hiroshima University.
REFERENCES

Fukuoka, S., Watanabe, A., Seki, K., Kurisu, D., and Tokioka, K. (2003). “Assessment andMechanism of Flood Flow Storage in Channels”, Journal of Hydraulic, Coastal andEnvironmental Engineering, JSCE, No.740/II-64, pp.31-44.
Fukuoka, S., Watanabe, A., Hara, T., and Akiyama, M. (2004a). “Highly Accurate Estimation of
Hydrograph of Flood Discharge and Water Storage in Rivers by Using an Unsteady TwoDimensional Flow Analysis Based on Temporal Change in Observed Water Surface Profiles”,
Journal of Hydraulic, Costal and Environmental Engineering, JSCE, No.761/Ⅱ-67, pp.45-56.
Fukuoka,S., Watanabe,A.,and Nagai,S. (2004b). “A Water Level Observation for a Flood Storage
Estimation in Rivers and Influence of Unsteadiness of Flow on Roughness Coefficient”, Journalof Advances in River Engineering, JSCE, Vol.10, pp71-88.
Fukuoka, S. (2005). Flood Hydraulics and Channel Design, Morikita Publishing Company.
Nomitsu,K. (1969). New Potamology, Chijin Shokan Publishing Company.