217

Floods: From Risk to Opportunity (IAHS Publ. 357, 2013).

Bed variation analysis using the sediment transport formulaconsidering the effect of river width and cross-sectional form in
the Ishikari River mouth
SEIJI OKAMURA1, KAZUNORI OKABE2 & SHOJI FUKUOKA3

1 IDEA Consultants, Inc., 3-15-1, Komazawa, Setagaya-ku, Tokyo, Japanseijiokamura5@gmail.com
2 Hokkaido Bureau, Ministry of Land, Infrastructure, Transport and Tourism, 2-1-3, Kasumigaseki, Chiyoda-ku, Tokyo,
Japan
3 Research and Development Initiative, Chuo University, 1-13-31214, Kasuga, Bunkyo-ku, Tokyo, Japan

Abstract For river management, it is important to estimate the sediment discharge rate and the degree of bed
variations during floods. Many of the previous sediment transport formulae have been proposed based on
results by experimental channels. Therefore, they cannot estimate well the sediment discharge rate in rivers
because of the complicated channel shapes and bed forms. It is necessary to calculate the sediment discharge
rate appropriately to improve the accuracy of bed variation analysis. Fukuoka (2010) thought that stablecross-sectional scales of alluvial rivers (such as width and depth) are determined by physical quantities
which indicate characteristics of basins (such as discharge, river bed slope and river bed material). He thenderived formulae between dimensionless quantities of width, depth and discharge using field observed databy dimensional analysis. Based on the above analysis, he also derived the bed load formula considering theeffect of river width and cross-sectional form using field observed data. In this study, we develop a bed
variation analysis with the Fukuoka’s bed load formula to calculate the 1981 flood of the Ishikari River thatcaused large bed scouring at the river mouth. We compare calculated results between the bed load formulaof Fukuoka (2010) and the previous formulae of Ashida & Michiue (1972) and Sato et al. (1958) for theamount of bed load discharge rates during the flood and the bed forms after the flood. As a result, the crosssectional bed forms at meandering channel are well reproduced when we use the bed load formulae of
Fukuoka (2010) and Sato et al. (1958). This indicates that these bed load formulae can calculate the amountof bed load discharge rate appropriately in the case of the Ishikari River mouth.
Key words Fukuoka’s bed load formula; bed variation analysis; Fukuoka’s equation; river mouth

INTRODUCTION
For river management, it is important to estimate the sediment discharge rate and the degree of bed
variations during floods. But it is difficult to measure temporal variations of bed elevations andsediment discharge rate during floods. Many of the previous sediment transport formulae havebeen proposed based on results by simplified experimental channels. Therefore, they cannot
estimate well the sediment discharge rate in rivers because of the complicated channel shapes andbed forms.
Fukuoka (2010) thought that stable cross-sectional scales of alluvial rivers (such as width anddepth) are determined by physical quantities which indicate characteristics of basins (such asdischarge, river bed slope and river bed material). Then, he derived formulae between
dimensionless quantities of width, depth and discharge by the dimensional analysis using field data
observed in natural rivers and experimental channels whose width vary with bank erosion(equations (1) and (2)).
 Q

B

d
= 4.25
5 dr gId r 
 Q

h

d
= 0.13
5dr gId r 

0 .40

(1)

0 .38

(2)

where B = surface width, h = section-averaged depth, Qd = dominant discharge, g = gravitational
acceleration, I = energy gradient, and dr = representative grain diameter.
Copyright  2013 IAHS Press

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Seiji Okamura et al.

Fukuoka (2010) thought that sediment discharge rate should have a relationship withdimensionless discharge because sediment transports and bed variations during floods occur in thestable river channels which have been formed by dominant physical quantities of basins. Then, he
indicated that dimensionless bed load discharge rate is determined by the product of dimensionlessdischarge and energy gradient by using field observed data, as shown in Fig. 1 and equation (3),using field observation data of Nakato et al. (1977, 1981), USGS (1989)and Fukuoka (2010). Dataof bed load discharge rates in Fig. 1 are observed at stable river sections which satisfy the
relationships in equations (1) and (2).
 Q
 I 
= 0.02
I 1 − c  5
I 
 gId 5 
sgId r
r


QB

(3)

where QB = bed load discharge rate of full cross-section, Q = observed discharge (when bed load
discharge rates were measured), s = specific gravity of sediment in water, and Ic = critical gradientfor sediment movement.
Joganji River Experiment

Tanana River at Fairbanks

Wisconsin River at Muscoda

Black River near Galesville

Chippewa River near Caryville

Chippewa River at DurandChippewa River near Pepin

Mississippi River

Des Moines River108
1.0E+08
Eq.3

106
1.0E+06
4

10
QB 1.0E+04
5

sgId1.0E+02
r 102
1
1.0E+00
10-2
1.0E-02

1.0E-01
10-1

1.0E+0110

1.0E+03103

1.0E+05105

 I 
I 1 − c 
gId r  I 
Q

1.0E+07107

1.0E+09
109

5

Fig. 1 Relationship between dimensionless bed load discharge rate and the product of dimensionlessdischarge and gradient of full cross-section.

In this study, we develop a numerical analysis of unsteady flow and bed variation with the
Fukuoka’s bed load formula to calculate the 1981 flood of the Ishikari River which caused a large
bed scouring at the river mouth. We compare calculated results between the bed load formula of
Fukuoka (2010) and the previous formulae of Ashida & Michiue (1972) and Sato et al. (1958) for
the amount of bed load discharge rates during the flood and the bed forms after the flood. Then we
investigate the effect of bed load formula for the accuracy of bed variation analysis.
AUGUST 1981 FLOOD IN THE ISHIKARI RIVER MOUTH AND THE PREVIOUS
STUDIES
Figure 2 shows a discharge hydrograph of August 1981 flood of the Ishikari River mouth. The
peak discharge of the flood exceeded the design discharge of the time. During the flood, temporal
data of water levels were measured at many observation points in the reach 15 km upstream fromthe river mouth, as shown in Fig. 3.

219

Bed variation analysis of the Ishikari River mouth
14000

Discharge（m3/s ）

12000
10000
8000
6000
4000
2000
0
8/4 0:00

8/5 0:00

8/6 0:00

8/7 0:00

8/8 0:00

8/9 0:00

Fig. 2 Discharge hydrograph of August 1981 flood of the Ishikari River mouth.

Bed elevation(T.P.m)Observation point of water level

Fig. 3 Planform of the Ishikari River mouth and observation points of 1981 flood.

There has been considerable research on the bed variation of the 1981 flood in the Ishikari
River Mouth. Mori et al. (1985) developed 3D flow and 2D bed variation analysis to themeandering compound channel of the Ishikari River mouth under constant discharge conditions.
He indicated that scouring areas shift to the centre of the main channel under the influence ofinflow from the flood plain. Shimizu et al. (1986) focused on the role of suspended load toestimate the bed variation by the flood. He developed 1D quasi-steady flow and bed variation
analysis considering the concentration of suspended load whose vertical distribution is assumed to
be an exponential curve. Inoue et al. (2004) extended the above analysis to 2D quasi-steady flow
and bed variation analysis considering the effect of secondary flow, where, the flux of suspended
load is calculated by concentration of suspended load and horizontal velocity whose vertical
distributions are assumed to be exponential and polynomial curves, respectively.
Okamura et al. (2010) elucidated the bed variation during the flood by the quasi-3D unsteadyflood flow and 2D bed variation analysis using observed temporal changes in water surface
profiles of the flood. This computational method is based on the idea that the influences of channel
shape, bed resistance, vegetation, bed variation during flood, etc. are reflected in temporal changes
in water surface profiles. In the analysis, the concentration of suspended load is calculated by 3D
advection-diffusion equations where the vertical distributions of velocities are calculated by quasi3D unsteady flow analysis. But a problem to be solved remains that the cross-sectional bed slopeof calculation was steeper than that of observation. It is considered as one of the factors that theamount of lateral sediment discharge rate is not estimated properly.
In this study, we calculate the amount of bed load discharge rate using the bed load formula of
Fukuoka (equation (5)) and the previous formulae of Ashida & Michiue (1972) and Sato et al.
(1958). Then we investigate the effect of sediment transport formula for the accuracy of bed
variation analysis.

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Seiji Okamura et al.

THE ANALYSIS OF BED VARIATION DURING 1981 FLOOD IN THE ISHIKARI
RIVERComputational method
We develop the unsteady numerical analysis of 1981 flood flow and bed variation using observed
temporal changes in water surface profiles. The analysis consists of quasi-3D unsteady flow
analysis (Uchida & Fukuoka, 2009) and 2D bed variation analysis for graded sediment (Fukuoka,
et al., 1998), employing the general coordinate system. In order to calculate the suspended load
affected by the secondary flow of meandering channel, the concentration of suspended load is
calculated by 3D advection-diffusion equations. The pick up rate of suspended load from bed
surface is calculated by the formula of Itakura & Kishi (1980).
Fukuoka’s bed load formula
Equation (3) is a relationship between cross-sectional bed load discharge rate and discharge over afull cross-section. This equation is not applicable to a horizontal 2D bed variation analysis,
because the analysis requires local bed load discharge. In this study, equation (3) is transformed toequation (4) for the 2D bed variation analysis. Equation (4) is a relationship between bed load
discharge rate per unit width and discharge per unit width. This equation also corresponds withobserved field data as shown in Fig. 4. These are same data with those of Fig. 1.
 q I 
= 0.02
I 1 − c  3
I 
 gId 3 
sgId r
r


qB

(4)

where qB = bed load discharge rate per unit width, and q = discharge per unit width. Equation (4) is
the relational expression for dr (representative grain diameter).
Joganji River Experiment
Tanana River at Fairbanks

Wisconsin River at Muscoda

Black River near Galesville

Chippewa River near Caryville

Chippewa River at DurandChippewa River near Pepin

Mississippi River

Des Moines River

グラフタイトル

1.0E+055

10

Eq. 4

1.0E+033

10

qB 1.0E+01
10sgId r3

1.0E-01-1

10

1.0E-03-3

10

10-5
1.0E-0410-4

1.0E-05-2
1.0E-0210

1

1.0E+002
1.0E+0210

 I 
I 1 − c 3
gId r  I q4
1.0E+04106
1.0E+06

10

Fig. 4 Relationship between dimensionless bed load discharge rate per unit width and the product ofdimensionless discharge per unit width and gradient.

In this study, in order to apply Fukuoka’s bed load formula to graded sediment condition, we
assume that equation (4) can be extended for dk (arbitrary grain diameter), as shown in equation (5).
q
 I 
(5)= 0.02 Pk 
I 1 − ck  3
I 
 gId 3 
sgId kkwhere qBk = bed load discharge rate per unit width of dk, and Pk = ratio of dk on bed surface.
Critical gradient for sediment movement Ick is obtained by the follow equation:
q Bk

Bed variation analysis of the Ishikari River mouth

I ck = τ *ck

sd k
h

221

(6)

where τ*ck = dimensionless critical shear stress. We use the value of τ*ck = 0.05 in this study.
Fukuoka’s bed load formula is based on the field data (shown in Figs 1 and 4). These data areobserved in stable river sections which satisfy the relationship in equations (1) and (2). Therefore,
before applying Fukuoka’s bed load formula, we check the relationships between dimensionlesswidth, depth and discharge of the section in question.
Figure 5 shows the relationships between dimensionless width, depth and discharge of section
from 1 km to 15 km during the 1981 flood. In this section the channel shape is not uniform. As
shown in Fig. 3, it is straight channel upstream of 9 km, wide compound meandering channel from4 km to 9 km section and narrow single cross-section downstream of 3 km. The relationships
between dimensionless width, depth and discharge correspond well with equations (1) and (2) ineach section.
108

106

B h 4
, 10
dr dr

15k

15k

11k

11k

8k

8k

5k

5k

1k

1k

B
dr

Eq.1

102

1

103

Eq.2

105

107

109

h
dr

1011

Q

1013

1015

1017

5

gId r
Fig. 5 Relationships between dimensionless width, depth and discharge during 1981 flood.

The bed load discharge rate is computed by the bed load formulae of Fukuoka as shown inequation (5), by Ashida & Michiue (1972) as shown in equation (7), and by Sato et al. (1958) asshown in equation (8).
3τ  u = 17 Pk τ *ek 2 1 − *ck 1 − *ck 3τ *k 
u* 

sgd k

q bk

q bk
sgd k

3

3

= ϕ(n)Pk τ *k 2 f(

n ≥ 0.025：ϕ(n) = 0.623
τ ck, f( τ ck ) = 1 − τ *ck) with 
− 3.5
τk
τ *k
τk
n < 0.025：ϕ(n) = 0.623( 40n)

(7)
(8)

where τ*k ,τ*ek = dimensionless shear stress and effective shear stress for dk, u*ck = critical shear
velocity for dk, n = Manning’s roughness coefficient.
Computational conditions
The initial bed forms before 1981 flood were surveyed in 1979 and are shown in Fig. 3. The
boundary conditions of upstream and downstream ends are given by the observed water level
hydrograph at the 15 km point and tide levels of the Otaru tidal observatory at sea 3 km from theriver mouth, respectively. The Manning’s roughness coefficients are determined so as to minimize
the difference between the observed and the calculated water surface profiles at each time point.

222

Seiji Okamura et al.

Furthermore, we also compare the calculated bed form after the flood with the observed one,because the water surface profiles are affected by the bed variations during the flood. In thisanalysis, the validity of the roughness coefficients and bed variations are checked by the temporalchanges in water surface profiles during the flood and the bed forms after the flood. As a result, n= 0.013 for the main channel downstream of 8 km, n = 0.021 for the main channel upstream of
8 km and n = 0.050 for the flood plain are given. This computational method has an advantage that
calculated bed variation during the flood may be verified by temporal changes in observed water
surface profiles.
Computational results
Figure 6 shows dimensionless bed load discharge rate calculated by the formula of Fukuoka(equation (5)), Ashida & Michiue (equation (7)) and Sato et al. (equation (8)). In the case of the
Ishikari River mouth, Equation (5) and equation (8) calculate similar amount of bed load discharge
rate, and equation (7) calculates more bed load discharge rate than the other equations. Bed load
discharge rate calculated by equation (5) (for dk arbitrary grain diameter) corresponds withequation (4) (for dr representative grain diameter).

105
103
Eq.4
10

qB
sgId r

3

10-1
10-3
10-5
10-4

10-2

Cal. Equation 5

15k

8k

5k

0k

Cal. Equation 7

15k

8k

5k

0k

Cal. Equation 8

15k

8k

5k

0k

102

1

q
gId r

3

104

106

 I 
I 1 − c I Fig. 6 Comparison of bed load discharge rate calculated by Equation 5, 7 and 8.

Figure 7 shows temporal changes in observed and calculated water surface profiles in theflood rising period. Figure 8 shows the comparison between observed and calculated longitudinal
bed forms before and after the flood. There are small differences between observed water levels
and calculated water levels of each calculation cases. In the Ishikari River mouth, suspended loadcauses the bed variations over a wide range and bed load causes local scours. The influences of
local scours on the water surface profiles are small. Therefore, calculated water surface profiles ofeach calculation cases are similar. For the same reason, calculated average bed elevations of eachcalculation case are similar and coincide with observed average bed elevation, as shown in Fig. 8.
However, clear differences between each calculation cases can be seen in the lowest bed elevationsof the meandering section from 3 km to 8 km (Fig. 8). The scour depths of calculation using
equation (7) are larger than those of the observation and the other calculation cases.
Figure 9 shows cross-sectional bed forms of observation and calculations in meandering
section from 3.5 km to 6.5 km before and after the flood. The cross-sectional bed slopes ofcalculation using equation (7) are steeper than those of the observation after the flood. It isconsidered that the amount of lateral sediment discharge rate is calculated excessively. However,223

Water level (T.P.m)

Bed variation analysis of the Ishikari River mouth
8
7
6
5
4
3
2
1
0

left

-3k

-2k

right

-1k

(Flood peak)
8/6 3:00

Obs.
Cal. Eq.5
Cal. Eq.7
Cal. Eq.8

0k

8/5 12:00
8/5 0:00

1k

2k

3k

4k

5k

6k

7k

8k

9k

10k 11k 12k 13k 14k 15k 16kFig. 7 Temporal changes in observed and calculated water surface profiles in flood rising period.
0

Averaged bed-2
Bed elevation (T.P.m)-4-6

Lowest bed

-8
-10
-12

Averaged bed elevation-14

Lowest bed elevation

before flood Obs.
after flood Obs.
after flood Cal. Eq.7
after flood Cal. Eq.8
after flood Cal. Eq.5

-16
-18

before flood Obs.
after flood Obs.
after flood Cal. Eq.7
after flood Cal. Eq.8
after flood Cal. Eq.5

-20
-3k

-2k

-1k

0k

1k

2k

3k

4k

5k

6k

7k

8k

9k

10k 11k 12k 13k 14k 15k 16kFig. 8 Comparison between observed and calculated bed elevations before and after the 1981 flood.
10

5
0
-5

before flood Obs.
after flood Obs.
after flood Cal. eq.7
after flood Cal. eq.8
after flood Cal. eq.5

-10
-15
0

100

Bed elevation (T.P.m)

10

200

300

400

0
-5
-10
-15
-20
100

200

300

10

400

500

600

700

0
-5
-10
-15
-20
100

200

300

400

-5

before flood Obs.
after flood Obs.
after flood Cal. eq.7
after flood Cal. eq.8
after flood Cal. eq.5

-10
-15
200

300

(m)

500

600

700

(m)

400

500

600

700

(m)

600

700

800

(m)

700

800

900

(m)

6.0km

5
0
-5
-10
-15
-20
200

300

400

10

5.0km

5

0

10

4.0km

5

5.5km

5

-20
100

500 (m)Bed elevation (T.P.m)

-20
-100

Bed elevation (T.P.m)

Bed elevation (T.P.m)

3.5km

Bed elevation (T.P.m)

Bed elevation (T.P.m)

10

500

6.5km

5
0
-5
-10
-15
-20
300

400

500

600

Fig. 9 Cross-sectional bed forms of observation and calculations in meandering section from 3.5 km to
6.5 km before and after the 1981 flood.

224

Seiji Okamura et al.

the cross-sectional bed slopes of calculations using equations (5) and (8) coincide with those of theobservation. This suggests that equations (5) and (8) calculate the amounts of bed load discharge
rates appropriately in the case of the Ishikari River mouth.
As shown in Fig. 4, the dimensionless bed load discharge rates measured in rivers varies
widely even if the products of the dimensionless discharges and energy gradients are same.
Therefore, Fukuoka’s bed load formula will not always calculate bed load discharge rate
appropriately. We have to examine for many rivers and floods which bed load formula evaluates
the bed variations appropriately. Then, we should use the bed load formula which evaluates the
bed variations appropriately, in each case.
CONCLUSION
In this paper, we develop the unsteady flood flow and bed variation analysis method with
Fukuoka’s bed load formula to calculate the 1981 flood of the Ishikari River mouth which causeda large bed scouring. We compare the amount of bed load discharge rates during the flood and the
bed forms after the flood calculated by Fukuoka’s bed load formula and the other formulae. Then,
we investigate the effect of bed load formula on the accuracy of bed variation analysis.
In the case of the Ishikari River mouth, the bed load formula of Ashida and Michiue calculates
more sediment discharge than the formulae of Fukuoka and Sato, Kikkawa and Ashida.
In the Ishikari River mouth, the wide range bed variation is mainly caused by suspended load,so the bed load has little effect on the average of bed variation. However, the local bed variations
such as scours at meandering channel are affected by bed load.
The bed forms after the flood calculated by using the formulae of Fukuoka and Sato, Kikkawa
and Ashida coincide with the observed bed forms. Especially, the cross-sectional bed forms after theflood in the meandering channel are well reproduced. This suggests that these formulae estimate theamounts of bed load discharge rates appropriately, in the case of the Ishikari River mouth.
Floods of many rivers for which bed load formula evaluates the bed variations properly need to
be examined. The bed load formula which evaluates the bed variations should be used in each case.
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