Dimensionless width and depth and sediment transport rate
in stable rivers
Shoji Fukuoka
Research and Development Initiative, Chuo University
E-mail: sfuku@tamacc.chuo-u.ac.jp

Abstract
The author, using data from Japan’s natural River in the1880’s, natural rivers in Canada, and basic
policy channels for river improvement of Class A Japanese rivers, demonstrates that a dynamic
relation equation between a river’s dimensionless width, dimensionless depth, and dimensionlesschannel-forming discharge obtained with dimensional analysis using flood flow and channel
properties in the river basin is a guideline for river management that is sound in terms of both flood
control and environmental considerations; that the dimensionless bed load equation is determinedwithin the dynamic relations in which stable dimensionless river width and dimensionless depth, etc.,
are determined; and that these relation equations play an important role in close-to-nature rivermanagement that achieves harmony between flood control and the environment.

1.

Introduction

A river is a natural object in which natural phenomenon occur and therefore requires proper
maintenance in order to carry out the functions desired of it. This paper will consider, from a goal ofmanaging rivers comprehensively in terms of flood control, water utilization, and the environment,
how a river as a natural object should be viewed and managed, how to minimize the flooding damage
feared from global warming, and how to resolve the issues of river planning in properly implementing
river management.
River width and surface width are particularly important aspects of a river’s cross-sectional form:
these control flood flows, including sediment transport at the river bed. Despite the dynamic andtopographic means by which river width, water surface width, depth, and other river cross-section
variables are determined being fundamental issues in river studies and river engineering, the author
knows of almost no research that addresses river channel formation dynamically and systematically1).
Instead, domestic and foreign river engineering research in general discusses flood flows, sedimenttransport, and other hydraulic phenomena with river width treated as a given.
Ikeda et al. have proposed an expression for determining channel width and depth that is based on
hydraulic considerations, but it cannot account for broadly changing flood flows and river conditions2).
Numerous empirically derived regime theories have been published on the stable channel crosssectional forms of irrigation channels and alluvial rivers. However, equations derived from regime
theories have inadequate dynamic consideration and are empirical with no general applicability.
Yamamoto, assuming the channel-forming discharge in a Japanese alluvial river to be mean annual
maximum discharge, has considered main channel width, river cross-sectional area, and velocity using
national river survey data3) However, the Yamamoto equation cannot represent dimension properly.
Theses reflect that when discussing channel scale and bed stability, external forces and the responses
to them must be discussed in terms of dimensionless external forces and dimensionless river width, forinstance. In addition, although relatively small scales of external force such as annual mean maximum
discharge are used as the main channel’s channel-forming discharge , in view of the coming increasein flood discharges, channel-forming discharges that determine the entire river width and entire crosssectional area must be discussed in the context of river management, rather than just the channelforming discharge determining the main channel’s width.

1

The transport of water and sediment in a channel involves many phenomena relating to flood
control, water utilization, and the environment, and is the foundation of river planning. The workingsof sediment transport during flooding is far less understood compared to flood flows because of therelative difficulty of measuring it, necessitating the use of sediment discharge equations to estimate theamount of sediment transported. Current sediment discharge equations do not account for river width.
As stated above, river width is the most important determinant of flood flow and sediment discharge ina river. Furthermore, current sediment discharge equations are woefully imprecise for rivers withcomplex channel topography. This has resulted in a lag in technically addressing sediment transport inthe context of preventing river disasters and protecting riverine environments, calling for efforts todevelop, from a new perspective, a method for estimating sediment transport. One effective method of
estimating is to utilize the fact that sediment transport volume is determined by the samemechanism—discussed later—that determines river width and cross-sectional form.

2. Learning from Nature: Close-to-Nature River Development and Maintenance from a
Basin-Wide Perspective
In principle, river development and management should be based on the concept of “close tonature”. Thus , close-to-nature river development means managing rivers in such a way as to achieve
harmony between flood control and the environment, taking the river’s natural functions into account.
Current close-to-river development, however, does not achieve harmony between the environment andflood control based on the realities of flooding. Although research inside and outside of Japan inrecent years has studied rivers’ natural flow regimes and the effects of flood discharge change on
ecosystems, true close-to-nature river development cannot be said to be in place without a discussionof near-natural river width and cross-sectional form and design discharge from a basin-wide
perspective.
Achieving a river channel in harmony with flood control and the environment requires placing at
the forefront the issue of river width, water depth, and cross-sectional form. Such a channel crosssection should be close to that of a natural river. The characteristics of a river made by nature are
evident in such aspects as planform, profile, and cross-sectional form; river width and cross-sectional
form are determined by nature’s laws of dynamics. When considering, for instance, what channel
cross-section would be stable and achieve harmony between flood control and the environment, it istherefore important to keep in mind the structure of natural rivers and consider the current structure ofa river being impacted by human activity. Such an approach can also lead to river design andmanagement techniques that are widely applicable for a broad range of channel and flood scales.
In the next section, in the context of close-to-nature river development from a river basin perspective, I
discuss, using findings from studies of natural Japanese and foreign rivers, the dynamic relationshipbetween flood flows and channel characteristic quantities in a river basin that determine the important
close-to-nature river parameters of river width, water surface width, water depth, and cross-sectional
form1).

3. Dimensionless Channel-Forming Discharge that Determines Dimensionless River
Width and Dimensionless Water Depth
River width, water depth, and other aspects of a river’s cross-sectional form are thought to be a
product of the effects of the external factors of channel-forming discharge, river basin topography andgeology, channel slope, and bed material (size distribution) experienced over the course of countlesslarge floods over the river’s history (Figure 1).
A channel’s cross-sectional form and water surface width can be considered essentiallydynamically stable toward flood flows below the channel-forming discharges that created the river as
it exists today. Here, a dynamically stable channel is defined as one that returns to its original stablestate even after changes caused by the external forces of flooding. Once it becomes stable, a channel’s
cross-sectional form remains in equilibrium even as it changes and fluctuates because of theinteraction between floods, the channel’s planform and longitudinal and cross-sectional form, and2

sediment transport, as indicated by the arrows in Figure 1. Normal bed change is a phenomenon that
results from this interaction.
Basin features・channel-forming discharge ・topography, geology
・channel slope ・bed material

Planform, longitudinal and cross-sectional form
river width, depth

Sediment discharge

Flow・river bed variation
・bank erosion anddeposition

・water surface profile
・resistance law(sand waves)Figure 1 Dynamic relation of stabe channel formation
Before achieving this stability, the channel cross-sectional form changes because of theinterdependence between river width, water depth, discharge, slope, and bed material size, but theseparameters become mutually independent once channel-forming discharge results in stability.
Furthermore, the effects of geology and topography on channel formation can, except in special
circumstances, be represented as bed material size and bed slope.
When a channel’s cross-sectional form is represented with river width and water depth, stablecross-sectional form is determined by eight physical quantities, including channel-forming discharge,
bed slope, and bed material properties, as follows:f (Q, B, h, I , d r , g ,  ,  )  0(1)

where Q = discharge, B = river width, h = water depth, I = riverbed slope, dr = representative
diameter, g = gravitational acceleration, ρ = water density, and σ = bed material density. Using the π
theorem, we derive the following dimensionless relationship:





Q

 gId 5
r


,

B h

, , I,   0
dr dr



(2)

The first term is dimensionless discharge, which indicates the state of the flood; the second isdimensionless river width; and the third dimensionless water depth. For the representative material
diameter dr, I used 60% (d60) in this research. Thus, nondimensionalization allows a dynamicallystandardized explanation of stable channel cross-section regardless of the scale of the channel or of theflooding. This is discussed in greater detail in the next section.

4.
The Relationship among Dimensionless River Width, Water Depth, and ChannelForming Discharge in a Natural RiverLet us first consider the relationship among dimensionless river width, water depth, and channelforming discharge (equation 2) in light of data from Japanese and Canadian natural rivers and data onchannel-forming discharge from channel-widening experiments conducted in the field and in thelaboratory.
For the Japanese natural river, I used flood flow data from channel sections in the upper andmiddle reaches of the Tone River in the Meiji Period (1868–1912).4) Figure 2 is a rapid survey map of
the river’s upper reaches obtained from surveying done in 1884. The Tone River channel of that timeretained the cross-section it had in the preceding Edo Period (1603–1868) and, having only
discontinuous levees (where any were present), was essentially in its natural state, with overtoppingfrom the channel occurring during floods. Dashed lines in the figure indicate the channel as of 1885,

3

Figure 2 Upper reach Tone river by 1884 surveying (rapid survey map)
Table. 1 Flood discharge and probability scale in the Meiji period (1868-1912)
Discharge(m3/s)
Reference point
Probability scale
3780
Menuma2-3 years
3750
Menuma2-3 years
6960
Downstream Menuma
Fig.
dimensionless water surface5-10
width
and water depth to55704 Correlation ofYattajima
years

Meters above the sea level

Year
1895
1898
1910
1911

dimensionless channel-forming discharge

28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
-100

Design high water level (1924) 20.864m
Design high water level (1911) 18.062m
high water level (1883) 20.864m

3

1883-channel明治18年計画高水量3,780m3/s
1885 design flood 3,780m /s
明治16年河道1936-channel 1911 design flood 5,570m 3/s

昭和11年河道 明治44年計画高水量5,570m3/s
1959-channel 1949 design flood 14,000m 3/s

昭和34年河道 昭和24年計画高水量14,000m3/s
1999-channel 1980 design flood 17,000m 3/s

平成11年河道 昭和55年計画高水量17,000m3/s

0

100

200

300

400

500

600

700

800

Transverse distance

Figure 3 Channel cross-sections at Kurihashi point in each flood year4)
Table-2 Data used for the investigation(mmused)(m)h (m)Table-2 dData
forBinvestigation
Tone river in Meiji period
0.3~40.0
340~840
4.0~6.6
Natural river in Alberta
19.0~145.0
14~545
0.4~6.9
Joganji river field experiment80~125
7.8~9.6
0.5~1.2
Channel-widning experiment23)
0.83
0.84
0.02
Channel-widning experiment24)
0.67
0.34
0.02

Q (m3/s)

1/I

2240~6960
6~8212
3.2~14.4
0.00480.002

470~5000
67~4545130
60
400

the orange lines the planned channel in 1891, and the solid blue line the planned channel alignment in
1948. The river experienced large floods in 1885, 1898, 1910, and 1911; Table 1 lists each flood’s
discharge and probability scale. During the Meiji Period, the 2- to 10-year probability discharges wereroughly equivalent to the design scale discharge, indicating a discharge that formed the main channelof the time. Figure 3 is channel cross-section at the Kurihashi Point on the Tone River in each flood
year. Bed slope and representative bed material size at that point is assumed to be unchanged today
from the Meiji Period, and so the current channel’s slope and bed material representative size are used.

4

As the foreign natural rivers, let us consider 67 river data from Bray et al. for rivers in Canada’s
Alberta Province.5),6) These natural rivers are comprised of a floodplain and a main channel, which issaid to be formed by the 2-year probable discharge, i.e., the main channel bankfull discharge. This
1.0E+08△

1.0E+07カナダ（アルバータ州）
5),6)△ Naturalrivers in Alberta11)12)
10),1

×

1.0E+06

B h
dr dr

21)
4)
明治時代の利根川上・中流部の河道
Toneriver in Meiji period21)

＊

B
dr

常願寺川現地実験

1)
1)
Joganji river’s field experiment

平野(1973)

23)

7)
Hirano’s experiment23)

福岡・山坂(1984)24)

24)● Fukuoka and Yamasaka’s experiment8)

1.0E+05
1.0E+04
1.0E+03
1.0E+02hdr

1.0E+01

 Q B 4.25
 gId 5 
dr
r 


0.40

 Q 
h

 0.13
5 dr gId r 

0.38

1.0E+00
1.0E+03

1.0E+05

1.0E+07

1.0E+09

1.0E+11

1.0E+13

Q
gId r5

1.0E+15

1.0E+17

Figure 5 Correlation of dimensionless water surface width and water depth

Figure 4 Correlation of dimensionless water surface width andto dimensionless channel-forming discharge
water depth to dimensionless channel-forming discharge
work also uses the 2-year probable discharge. For details on the data, see references 5 and 6. The data
used are listed in Table 2. Table 2, in addition to data on the Tone River and Canada’s natural rivers,
also gives the results of three experiments: one by the author and others with a large, stony-bed
compound waterway (length 190 m, total width 8 m, main channel width 3 m, water surface slope
1:130, representative bed material size 8 cm) built in the bed of the Joganji River1) and having straightand meandering sections, and channel-widening experiments conducted by Hirano7) and by Fukuoka
and Yamasaka8) with straight channels.
Figure 4 plots the correlation of dimensionless water surface width and water depth todimensionless channel-forming discharge in the Meiji Period Tone River, Canadian natural rivers, andriver-widening experiments in actual and experimental waterways. As the figure shows, within the
broad dimensionless range of dimensionless river width and water depth (102–106 and 100–104,
respectively) on the vertical axis and dimensionless discharge (103–1014) on the horizontal axis, thesevariables are determined by dimensionless channel-forming discharge, which in turn is determined by
the combination of channel-forming discharge, bed slope, and representative bed material size, andcan be represented with equations (3) and (4), the results of which are indicated by straight lines in the
figure.

 Q B 4.25
 gId 5 
dr
r 

 Q 
h

 0.13
 gId 5 
dr
r 


0.40

(3)

0.38

(4)

Equations (3) and (4) are called “Fukuoka equations”, hereafter. This indicates that thedimensionless width and water depth of a stable channel are governed by dimensionless channelforming discharge Q/ gIdr 5 as represented with channel-forming discharge, bed slope, andrepresentative bed material size.

5. The Relationshipamong Dimensionless River Width, Water Depth, and ChannelForming Discharge at Japanese Class A River Reference Point
5

Figure 5: Relationship of dimensionless width and depth in design level of Japanese rivers

Next let us see whether the correlation among dimensionless river width, water depth, anddischarge in equations (3) and (4) that holds for natural rivers also holds for flooding at a reference
point along a Class A Japanese river. Reference points are important locations for devising high- andlow-water plans of Class A Japanese rivers. In the case of a Class A river, various data are collected atreferences points over a long period, including discharge and water level hydrographs of flood flows,
channel cross-sectional form, and bed material size distribution. For bed slope, mean bed slope at thereference point is used. For sections where the bed is nearly flat, such as at the mouth, water surface
slope is used instead of bed slope.
The basic policy channel for river improvement is the river planned so as to carry the design highwater discharge below the design high-water level. Cross-sectional area, water surface width, andwater depth in the basic policy channel are determined from the design high-water level profile, i.e.,
the profile at design high-water discharge. Equations (3) and (4) are applied for the design high-water
discharge passing each reference point of basic policy channels for river improvement of 109 Class A
river systems. Mean depth is used for the water depth of channels with a compound form.1)
Figure 5 shows the correlation of the dimensionless values for discharge to river width and water
depth. However, the correlation between the data is not as strong as with the Tone River or the Alberta
natural rivers. This is because while reference points on Class A Japanese rivers represent a variety of
topographical conditions—e.g., delta marshes, lowlands, and mountainous region. Furthermore, crosssectional shape (i.e., simple or compound cross-section) is not accounted for in the calculation ofdimensionless river width and water depth. In the figure, most of the compound cross-section channels
are found above the approximation curve for dimensionless river width, while most simple crosssection channels are found below the curve. Conversely, compound cross-section channels are found6below the dimensionless water depth curve and simple cross-section channels above it. Respective
approximate equations for the compound channel and the simple cross-sectional channel are shown infigure 51).
From the above discussion, it is highly significant that Fukuoka equations hold similarly forfloods and channels for a wide range of sizes, including many natural rivers and Japan’s basic policy
channels for river improvement.

6. Calculating Dimensionless Bed Load Transport Rate and Designing Channel CrossSections for Sustainable Sediment TransportThis section discusses a method estimating the bed load transport rate as a dynamic phenomenon
determined within the stable channel cross-section, that is based on an approach unlike previous ones,
and accounts for the mechanism (discussed in the previous section) that determines river width andother stable cross-section parameters. Bed load transport rate is a quantity determined by there being afixed cross-sectional form and so was not incorporated into equation (1) as an independent quantity for
determining river width, water depth, and cross-section. Of course, bed load transport rate is closely
involved in the process by which the stable cross-section is created, as large amounts of sediment aretransported into the river from the river basin. Once the channel stabilizes, however, bed load transport
rate becomes a dependent variable of cross-sectional form and the parameters that determine it.
Thus bed load transport rate is determined by the dynamic interrelation between mutuallyindependent quantities that form the stable channel cross-section, and so it can be calculated from theinterrelations in equation (5):QB  f Q, B, h, I , d r , g,  ,  (5)where QB is bed load transport rate. Applying the dimensional analysis, we derive the dimensionlessinterrelationship of equation (6):

 Q
B h
 


, , , I,

5
5 d

r dr
sgId r
gId
r


QB

(6)

as s = ( – )/. Equation (6) accounts only for bed load transport, not for suspended sediment, whichis strongly affected by factors such as bank erosion and sediment yield in the mountain area. Because
the 2nd and 3rd terms on the right side of equation (6) can be represented as the dimensionless form of
the 1st term on the right side of equations (3) and (4), equation (6) can be rewritten as follows:

 Q

 
,I
5
 gId 5 
sgId r
r


QB

(7)

The bed load data used to derive the functional relationship of equation (7) is listed in Tables 3through 5. The variable dm in the tables represents mean grain diameter and here is used as dr. Except
for data from Japan’s Civil Engineering Research Institute for Cold Region, all of the data are direct
measurements made with bed load transport measurement instruments. Table 3 contains 50 data points
collected by Nakato for America’s Mississippi River9)10). Of the bed load data in Table 4, which
Williams et al.11) collated from U.S. Geological Survey data collected with a Helley-Smith bed loadsampler, we used 127 data that were usable with equation (7). Table 5a contains data obtained with alarge-scale experimental waterway (length114m, width 1.76m, 0.78m) at Japan’s Public Works
Research Institute (PWRI)12) while Table 6b gives bed load data from a large-scale experimental
waterway (length 24m, width 1.0m) at the Civil Engineering Research Institute for Cold Region

7

Table-3. Bed load data measured in the Mississippi river9)10)River

d (mm)

B (m)

h (m)

Q (m3/s)

1/I

QB (m3/s)

Mississippi R.

0.25~0.85

86~8801.4~7.046~4500

1400~5500000

4.37×10-7~2.15×10-2

Table-4. Bed load data of USGS11)riverdm (mm)

B (m)

h (m)

Q (m3/s)

1/I

QB (m3/s)

Tanana River at Fairbanks40

107~469

1.4~2.9345~2020

1900~2400

2.3×10-4~3.4×10-2

Wisconsin River at Muscoda

0.5

219~310

0.7~3.4

87~1240

1900~4500

4.0×10-4~1.8×10-2

Black River near Galesville

0.6

72~1220.55~1.9

13~2562800~9000

7.1×10-5~1.5×10-3

Chippewa River near Caryville

8.0

124~2470.89~2.8

31~7794000~11000

2.0×10-5~5.1×10-3

Chippewa River at Durand

0.8

153~244

0.61~3.2

51~8842800~4300

2.2×10-4~1.1×10-2

Chippewa River near Pepin

0.5

171~2770.75~1.8

70~3991700~5900

6.4×10-4~5.5×10-3

Table-5. Bed load data obtained by large-scale model tests(b) Experiment by Civil enfineering Research Institutefor Cold Region13)(a)Experimental data by Public Works Research Institute12)3664

Table-6. Bed load dataobtained by large-scale model tests
3dm(mm)

h(cm)

1.038

19.3～43.7

2.21

Q (l/s)43～20011.4～51.5

28～3501/I

QB (m /s)613~4545

0.10~25.25

63~15380.01~66.94dm(mm)h(cm)Q (l/s)

1/I

QB (m3/s)

Run 1

28

57.1

1000

214

165.49

Run 2

27.6

55.3

1000

214

386.21

Run 3

22

59.3

1000

214

230.52

Run 4

23.3

56.8

1000

214

183.60325.1063

2.62

19.6～51.3

90~325340~1667

0.21~54.31

52

3.76

18.1～51.8

80~400352~1111

0.22~57.66

Run 5

28.4

59.2

1000

214

31

4.58

19.4～42.3

80~350431~800

0.39~18.97

Run 6

24.6

58.6

1000

214

83.87

172~2450.80~192.80

Run 7

11.3

65.2

1500

214

610.31

Wisconsin River at Muscoda
Black River near GalesvilleRiver near CaryvilleRun 8Chippewa
30.3River near Caryville
64.2
Wisconsin Riverat Muscoda 1.53~270.31
Black River near Galesville
Chippewa
540~1630
133~1831500

214

616.21H-S
H-SJoganji
sampler
Fieldsampler
experimentRiver
掃流砂捕捉箱掃流砂捕捉箱2110.0
21.0～50.5in the395~1443
14

Tanana River at Fairbanks
Tanana23.1～50.2
River at Fairbanks

18.1

Chippewa River at DurandChippewa River at DurandChippewa River near Pepin
Mississippi River
Chippewa River near Pepin
Mississippi River

Des Moines River
Des Moines River

1.0E+08
1.0E+08

1.0E+06
QB 1.0E+06
sgId r 5
1.0E+04
1.0E+04

1.0E+08

1.0E+02
1.0E+02
1.0E+08

1.0E+08

1.0E+08

1.0E+06
1.0E+00
1.0E+00

1.0E+06
1.0E+06
1.0E+04
1.0E+03
1.0E+05 1.0E+08 1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+08
1.0E-02
1.0E+06
1.0E-02
1.0E-01
1.0E+08
1.0E-01
1.0E+04
1.0E+04
1.0E+02
1.0E+06
1.0E+06
1.0E+04

1.0E+08

1.0E+08

1.0E+08

1.0E+08

1.0E+06

1.0E+06

掃流砂捕捉箱掃流砂捕捉箱H-S
H-SJoganji
sampler
Fieldsampler
experiment in theRiver
掃流砂捕捉箱掃流砂捕捉箱H-S
H-SJoganji
sampler
Fieldsampler
experiment in theRiver
掃流砂捕捉箱Chippewa River at DurandChippewa River near Pepin

Tanana River at Fairbanks
TananaRiver at Fairbanks
1.0E+04

1.0E+06

Chippewa River at DurandChippewa River at DurandRiver at DurandTanana Chippewa
River at Fairbanks
Tanana River at Fairbanks

Mississippi River
Chippewa
near Pepin Black River
Mississippi
River
Wisconsin
River atRiver
Muscodanear Galesvill
Wisconsin River at Muscoda
Black River near G

Chippewa River at DurandChippewa River near Pepin
Mississippi River
Chippewa River near Pepin
Mississippi River

Chippewa River at Durand
H-S
H-S
sampler
Fieldsampler
experiment in the
Joganji
River
掃流砂捕捉箱1.0E+08

1.0E+08River at Muscoda
Wisconsin
Black River near GalesvilleWisconsin River at Muscoda
Black River near GalesvilleTanana River at Fairbanks
Tanana River at Fairbanks

1.0E+08
Chippewa River at DurandChippewa River near Pepin
Mississippi River
1.0E+08
Chippewa River at DurandChippewa River near Pepin
Mississippi River
H-S
H-SJoganji
sampler
Fieldsampler
experiment in theRiver
掃流砂捕捉箱掃流砂捕捉箱TananaRiver
River at Fairbanks
1.0E+08
H-S
H-SJoganji
sampler
Fieldsampler
experiment in the
掃流砂捕捉箱掃流砂捕捉箱1.0E+06
1.0E+06
Tanana River at Fairbanks
1.0E+04

Tanana River at Fairbanks
Wisconsin River at Muscoda
Black River near Galesvill
TananaRiver
at Fairbanks
Wisconsin River at Muscoda
Black River near G
Williams
andRosgen(1989)

Tanana River at Fairbanks

Des Moines River
Des Moines River

1.0E+06

Wisconsin River at Muscoda
BlackRiver near Galesville
Chippewa River near Caryville
1.0E+06
Wisconsin River at Muscoda
Black River near Galesville
Chippewa River near Caryville
B
Wisconsin River at Muscoda 1.0E+08
Black River near Galesville
Chippewa River near Caryville
Chippewa
River at River
Durand
River
near
Pepin
Mississippi
River River near CaryvilleDes Moines River
Wisconsin
at Muscoda Chippewa
Black
River
near
Galesville
Chippewa
1.0E+06
5
1.0E+08
Chippewa River at DurandChippewa River near Pepin
Mississippi River
Des Moines River
1.0E+06r Moines
Chippewa River near Pepin
Mississippi River
River
NakatoandKennedy (1977)
B Des
1.0E+04
Chippewa River near Pepin
Mississippi River
Des Moines River

Q

sgId
Q

H-S
H-SJoganji
sampler
Fieldsampler
experimentRiver 掃流砂捕捉箱 掃流砂捕捉箱
1.0E+04
Nakatoand Vadnal (1981)
51.0E+04
 in theQ
Q Wisconsin
1.0E+08
 I RiverTananaRiverc   at MuscodasgId r Black River near Galesville
B at Fairbanks 1.0E+06
H-S
sampler
H-SJoganji
sampler
1.0E+08
Field
experiment in theRiver
掃流砂捕捉箱 0.at02Fairbanks
I 1 Wisconsin  River at Muscoda
1.0E+02掃流砂捕捉箱
1.0E+02
Tanana RiverBlack River near Galesville
1.0E+04
1.0E+06
5
1.0E+04
 gId
 RiverIRiver near
1.0E+08
QBr 5Chippewa
Tanana River at Fairbanks
Wisconsin
River
MuscodaBlack
near
Chippewa River
sgIdRiver
Chippewa
Pepin
Mississippi
Rivernear Caryviller atatDurand
1.0E+08 Galesville
1.0E+04

Chippewa River near Cary
Chippewa River n

WisconsinRiver
RiveratatDurandMuscodaChippewa

Black RiverRivernearnear
Galesville
1.0E+02
Chippewa
Pepin

Chippewa River near Caryville
Chippewa River near Caryville

Des Moines River
Chippewa River
Mississippi
Rivernear Caryville
Des Moines River

1.0E+04
1.0E+02
1.0E+02
1.0E+025
1.0E+09
Chippewa River at DurandChippewa River near Pepin sgId Mississippi River
Des Moines River
1.0E+06
Chippewa River at DurandChippewa River near Pepin r
Mississippi River
Des Moines River
1.0E+06
1.0E+04
1.0E+00
1.0E+00
1.0E+02Q
1.0E+04
1.0E+02B
1.0E+01
1.0E+03
1.0E+05 1.0E+08 1.0E+07
1.0E+09
1.0E+06
1.0E+01
1.0E+03
1.0E+05 1.0E+08 1.0E+07
1.0E+09
1.0E+06
Q
1.0E+00
Q Ic  5
Q1.0E+08
1.0E+00
1.0E+00
1.0E+02
1.0E+00B
I 1  sgId r I 
1.0E+08c
5 Q I1.0E+04
sgI 11.0E+04
 
gId r
1.0E-025
1.0E-02Q
1.0E+00
1.0E+02
sgId
5
1.0E+00r
1.0E+02
I1.0E+01
 1.0E+06
1.0E+03
1.0E+06
1.0E+06
gId
1.0E+04r
1.0E-01
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09
1.0E+02
1.0E+02sg
1.0E+04
1.0E-02Q
1.0E-02
1.0E-02B
1.0E+00
1.0E+00
1.0E-02
1.0E+06
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1
1.0E+06
1.0E+04
1.0E+04
1.0E+025 1.0E+01
1.0E-01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09
1.0E+02 1.0E+05
QB1.0E-01
1.0E-01
1.0E+01
1.0E+03
1.0E-02
sgId r1.0E+02
QB Q
1.0E+00Q
1.0E-02  Ic 
1.0E+00

0
.
02
I
1


1.0E+04
1.0E+04
I1.0E+05
1  5  1
1.0E-01
1.0E+01
1.0E+03
1.0E+02 5
5
1.0E+04
1.0E+00
1.0E+0013)
5   1.0E+05
sgId gId
1.0E-01
1.0E+01
1.0E+03
IgId
sgId1.0E+02
rrr
1.0E-02
1.0E-02
Q r   I c 
1.0E+04
1.0E+02
1.0E+02
I
1 

I
Q
1.0E+00Q
1.0E+04
1.0E+00c  5B
1.0E-01
1.0E-01
1.0E+01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09
1.0E+00I 
 0.02
I 1  gId

1.0E-02
5
1.0E-02I   r gId 5 
1.0E+02
1.0E+02r
I r  1.0E+05
QB1.0E+01  Q 1.0E+03
1.0E+00
1.0E-02
1.0E-02
 sgId1.0E-01
1.0E+07
1.0E+09
1.0E+02
1.0E+00
1.0E+01
0.02
I 1.0E+03
1 c 
1.0E-01
1.0E+05
1.0E+071.0E+
51.0E-01 1.0E+00
1.0E-01
1.0E+01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09I Q
 Ic 
1.0E+02 gId 5 
1.0E+00
sgId
1.0E-02rr
1.0E+02
1.0E-02 I 1  
1.0E-025  I 
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+07r
QBgId1.0E+07
1.0E-01
1.0E+01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09
1.0E+00
Q 1.0E+09
 I 
 1.0E+09
1.0E+00
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E-02
1.0E+00 0.02
I 1  c  Q Ic 1.0E-02
5
5
I 

 gId

I 1  1.0E+09
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+07r
I r 
QB 1.0E+07
1.0E+00
 sgId
1.0E-02
1.0E-02 Q 1.0E+09
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+00
I 1  c  
gId r 5 0.02 I 
Q 1.0E+09
 Ic 
55 
1.0E-01
1.0E-01
1.0E+01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09IsgId rgId r
I 1  
1.0E-02
1.0E-02
1.0E-02
I 
gId r 5 
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+07
1.0E+09-01
1.0E-01
1.0E+01
1.0E+01
1.0E+03
1.0E+03
1.0E+05
1.0E+05
1.0E+07
1.0E+07
1.0E+09
1.0E+09
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+07
1.0E+09
1.0E-02Q
 Ic 
1.0E-02
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+075 I 1  1.0E+09
I 
1.0E-01
1.0E+01
1.0E+03
1.0E+05
1.0E+09

gId1.0E+07

Figure 6: Relationship of dimensionless sediment transport rate and dimensionless discharge(CERICR)Of the latter’s bed load transport data, those on changes in bed height and sedimenttransport rate from upstream during running water were used in the calculations.

Figure 6 shows the correlation between dimensionless discharge and dimensionless bed loadtransport rate determined using the American river bed load data and experimental channel data fromthe two aforementioned Japanese national organizations. Bed load transport rate is affected by water
surface slope (I), which is a local hydraulic quantity. To account for this effect, dimensionless bed
load is expressed as the product of dimensionless discharge (a determinant of channel cross-section)
Q

gId r

8

5

 I 
I 1  c 
I 


r

and water surface slope (a local hydraulic quantity of the channel). Here, dimensionless discharge ismultiplied by I, which is the value that best fits the dimensionless sediment transport rate data.
Ordinarily, a slope (I) that is less than the critical slope (Ic) is thought to allow sediment to stay inplace. It would therefore be appropriate to express the horizontal axis (I) in Figure 6 as I – Ic. With adimensionless critical tractive force (τ*c) of 0.05, Ic can be approximated as 0.083 dr/h. The value of Ic
should be accounted for with swiftly flowing rivers where the grain size/water depth ratio is relatively
large but is essentially zero in gently flowing rivers. The solid line in Figure 6 is a relation equation(equation (8) below) of dimensionless bed load transport rate.

 Q
I 
 0.02
I (1  c ) 5 gId 5I 
sgId r
r


QB

（8）

Equation (8) is a dimensionless function incorporating dimensionless discharge and bed slope,which determine dimensionless river width and water depth. Conventional dimensionless bed loadtransport equations incorporate a dimensionless tractive force (τ *) for the center of a watercourse andcalculate bed load without taking river width into account. Equation (8), in contrast, yields a
dimensionless bed load that comprises the same parameters that determine dimensionless riverwidth—discharge, bed slope, and grain size—and in this regard is a practical equation for determining
dimensionless bed load transport rate. Thus dimensionless discharge not only determines a channel’s
dimensionless river width and cross-section but is also an important quantity for determining
dimensionless bed load transport rate.
A river that satisfies nearly all flood-control and riverine environmental requirements is a channel
that looks like a natural river. A natural river, in turn, is one that satisfies equations (3) and (4)—i.e.,
has a channel with a dimensionless river width and water depth determined by dimensionless channelforming discharge—and whose flow cross-section roughly complies with Fukuoka equations even forwater surface widths resulting from other discharges and water levels. A channel with its channelforming discharge carries bed load corresponding to cross-sectional form and river width, with theresult being that the cross-sectional form is maintained. This suggests that even when flood discharges
or water levels are lower, dimensionless bed load corresponding to the water surface width and water
depth represented with equation (8) occurs within the cross-section formed by the channel-forming
discharge, resulting in a stable channel in which the channel-forming cross-section is essentiallymaintained. A channel cross-section that provides a desirable relationship between flood flows andbed load transport gives thus a standard of a channel cross-sectional form in harmony between flood
control and the environment. Such a channel cross-section is a ship-bottom shaped cross-sectional
channel having the sort of continuous wetted perimeter seen in natural rivers1).
Thus for a channel with a ship-bottom shaped cross-section, it is possible to define thedimensionless river width, water surface width, and water depth for the dimensionless discharge and,with equation (8), to calculate the resultant dimensionless bed load transport rate.
The ability to calculate dimensionless river width and bed load transport rate for a standard channel
cross-section such as a ship-bottom shaped cross-section provides a basic tool for creating a soundcross-sectional form in difficult-to-manage channels with a fixed water path or overgrown vegetation.
In many rivers, flows and sediment movement in a cross-section are extremely non-uniform during
flooding, while at normal discharges, flow concentrates in the fixed water path; aquatic wildlife exists
in locations limited by localized hydraulic and sediment-movement phenomena, meaning that it is notwidely applicable to discuss sound riverine ecosystems. Instead, a river section requiring flood control
and environmental improvement to correct fixed water path and vegetation overgrowth must be
reformed into a ship-bottom shaped cross-section to achieve sound, sustainable flow and sediment
movement and a correlation between desirable dimensionless river width (i.e., water surface width),discharge, and bed load transport, after which factors such as desirable bed shape, habitats, andvegetation management can be considered.

7.

Afterword
9This paper has, from the new perspective of learning from natural rivers, derived Fukuoka
equations for river development that achieves harmony between flood control and the environment
throughout the river basin, such as dimensionless river width, water depth, cross-sectional form, andbed load transport rate. Fukuoka equation demonstrated that an alluvial river’s determinants such as
dimensionless river width, water depth, and cross-sectional form were determined by the natural lawsof river dynamics, and that the equations are universal expression equations for channel formation in ariver basin. Close-to-nature river development must fully recognize a river’s naturality and understandthat a river is the product of interaction between macrostructures—river width, water surface width,
cross-sectional form, and other parameters determined by the river basin—and microtopographicalfeatures and other elements sculpted by the flood flows and sediment movement within thatmicrostructure.
Fukuoka equations can estimate the required river width, water surface width, water depth, andother factors -even for rivers for which data are sparse- as long as discharge, bed slope and bed
material representative grain size are known, and so the equations are highly applicable to riverplanning and river maintenance.
This paper has discussed dimensionless river width ,water depth, and sediment transport andpresented a practical and effective equation for calculating dimensionless bed load transport. Despite
its close connection to flood control and riverine environmental management, and despite its
importance, bed load transport has been the subject of little research in recent years. Based on the ideathat bed load transport is determined in the framework of the mechanism by which flood flows
produce a stable river width, the author has studied published results on bed load measured in actual
rivers and observed experimentally and has created a practical dimensionless bed load equation that
explains these data. The author hopes that this research will renew awareness of the importance of
proactively measuring sediment discharge as was done in Japan in the 1950s and 1960s, and will thus
lead to widespread measurement of a quantity so important in channel planning.

REFERENCES
1) Fukuoka, S. (2011) River Engineering Adaptations against the Global Warming-Towards
Generalization of Close-To –Nature Rivers, Journal of JSCE, JSCE,F, Vol.66,No.4, pp.471-489 .
2) Ikeda, S., Parker, G., Chiyoda, M. and Kimura, Y. (1986) Stable Cross-Section of Gravel Rivers
with Actively Transported Bed Materials, Proceedings of JSCE, JSCE, Vol.375, pp.117-126.
3) Yamamoto, K.(2004) Alluvial Rivers, Sankaido.
4) Shirai, K. (2010) Study on Channel Changes, Revetments and Spurdikes in the Tone River,Doctoral Thesis, Chuo University.
5) Bray, D.I. (1979) Estimating Average Velocity in Gravel-Bed Rivers , J. Hydraulics Div. ASCE,
Vol. 105, HY9, pp. 1122-1130.
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pp.13-20.
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Widening Process of a Straight Alluvial Channel, Proceedings of JSCE, JSCE, Vol.351/II-2, pp.8796.
9) Nakato,T., and Kennedy, J. F. (1977) Field Study of Sediment Transport Characteristics of theMississippi River near Fox Island (RM 355-6) and Buzzard Island (RM 349-50), IIHR Report, No.
201, Iowa Institute of Hydraulic Research.
10) Nakato, T., and Vadnal, J. L. (1981) Field Study and Test of Several One-Dimensional SedimentTransport Computer Models for Pool 20，Mississippi River, IIHR Report, No. 237, Iowa Institute ofHydraulic Research.
11)Williams,G.P.and Rosgen,D.L.1989, Measured Total Sediment Load for 93 United Streams,
U.S.Geological Survey, Open-File Report, 89~67.

1012) Sato, S., Kikkawa, H. and Ashida, K. (1958) Study on the Bed-Load Transportation (1), Report of
PWRI, Vol.98, pp.1-18.
13) Ogawa, T. and Watanabe, Y. (2003) Experimental Study on Bed Load Transportation Using
Gravel Sand Collected at Upper Reach of River, Annual Journal of Hydraulic Engineering, JSCE,
Vol.47, pp.535-540.

11