changes in river bed around the fukawa...
TRANSCRIPT
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 1
CHANGES IN RIVER BED AROUND THE FUKAWA CONTRACTION AREA
BY FLOODS AND CHANNEL IMPROVEMENT WORKS
IN THE LOWER TONE RIVER
Naoki IWAYA1 and Shoji FUKUOKA
2
ABSTRACT
In the lower Tone River, channel dredging and widening have been conducted for the river
improvement to respond to the flood discharge increase. Especially, the flood discharge capacity in
the Fukawa contraction area has been an important problem of flood control in the lower Tone
River. In the Fukawa contraction area, the river bed elevation lowered below the design bed
elevation, and therefore channel dredging stopped in 1966. Nevertheless, the river bed degradation
has continued until around 1998. In this study, we investigated the causes of the river bed
degradation in the Fukawa contraction area in relation to flood flows, the channel dredging and
widening by both observed data and analysis of flood flow and bed variation considering channel
widening and dredging. Observed data demonstrated that the river bed degradation in the Fukawa
contraction area has occurred due to floods, channel dredging and widening in the upstream and
downstream of the Fukawa contraction area. Numerical analysis provided a good explanation for
mechanics of river bed degradation and scouring during floods.
1. INTRODUCTION
In the lower Tone River, river improvement works including channel dredging and widening have
been conducted to respond to the flood discharge increase. The Fukawa contraction area is located at
76km from the river mouth, and the flood discharge capacity in the Fukawa contraction area has
been an important issue of flood control in the lower Tone River. Because there were many houses
along the Fukawa contraction area, channel dredging had been main countermeasures for
improvement works. In the Fukawa contraction area, the river bed elevation lowered below the
design bed elevation, and therefore channel dredging stopped in 1966. Nevertheless, the river bed
degradation has continued until around 1998.
The objective of this study is to clarify the causes of the river bed degradation and scouring in the
Fukawa contraction area in relation to the channel dredging, widening and flood flows. First, we
investigated the causes from the observed data of the channel dredging, widening and flood flows.
Secondly, we conducted the unsteady quasi 3-D flow analysis and bed variation analysis using
1 Graduate Student of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan,
Email: [email protected] 2 Professor, Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan, FL
Email: [email protected]
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 2
observed temporal changes in water surface profiles in order to understand the bed variation in and
around the Fukawa contraction area from 1981 to 1983.
2. CHANGES IN RIVER BED IN AND AROUND THE FUKAWA CONTRACTION
AREA IN THE LOWER TONE RIVER
Figure 1 shows study area and location of observation points in and around the Fukawa
contraction area in the Lower Tone River. Figure 2 shows the maximum annual discharge at Toride
(85.3km) and Fukawa (76.5km) observation point from 1962 to 2007, and the broken lines indicate
the design high water discharge at the observation points. The dredging volume in the upstream and
downstream of the Fukawa contraction area from 1972 to 2007 is shown in Figure 3. Since 1999, the
dredging has not conducted in the upstream of the Fukawa contraction area and the dredging volume
has been decreasing year by year in the downstream.
Figure 4 shows the changes in main channel width in the study area. Main channel has been
widened longitudinally from 1961 to 1983 in the upstream and from 1961 to 1998 in the
Figure 1 Study area and location of observation points in and around the Fukawa contraction area
in the Lower Tone River.
.
Figure 2 Maximum annual discharge.
Figure 3 Dredging volumes in the upstream and downstream of the Fukawa contraction area.
0
2,000
4,000
6,000
8,000
10,000
12,000
1962 1967 1972 1977 1982 1987 1992 1997 2002 2007
the design high water discharge (Toride)
the design high water discharge (Fukawa)
0 2,000 4,000 6,000 8,000
10,000 12,000
1962 1967 1972 1977 1982 1987 1992 1997 2002 2007
Toride(85.3km) Fukawa(76.5km)
0
10
20
30
40
50
1972 1977 1982 1987 1992 1997 2002 200701020304050
1972 1977 1982 1987 1992 1997 2002 2007
The downstream of the Fukawa contraction area (76km~61km)
The upstream of the Fukawa contraction area (85.5km~78km)
Dis
charg
e (m
3/s
)D
redgin
g v
olu
me
(×10
4m
3)
85.5km
66.5km
5.0km
Tone River
76.0km
78.0km
Water level observation point
Discharge observation point
Toride
SugaFukawa(76.5km)
Oshitsuke(78.5km)
Todai(0.5km)
Toyota weir(1.0km)
Nakago
Takasu(3.5km)
Yahoo Japan map
Koka River
Levee alignment
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 3
downstream for increasing the capacity of the river channel. In this area, large floods occurred at
Sep.1983, Sep.1998 and Sep.2008. Figure 5 shows flood marks and observed maximum water level
of each flood. Figure 6 and Figure 7 are the changes in average and deepest elevation of river bed in
the study area. In the Fukawa contraction area, although the dredging has not conducted since 1966,
the average and deepest river bed has greatly degraded from 1961 to 1983. On the other hand, from
1983 to 1998, the degradation of the river bed has caused mainly near the exit of the Fukawa
contraction area (from 77.0km to 76.0km). And, the river bed elevation has not much degraded since
1998.
From 1961 to 1983, the dredging has been conducted actively in the upstream and downstream of
the Fukawa contraction area. And, the amount of sediment from the upstream has reduced due to the
erosion and sediment control management and the construction of dams in the upper Tone River in
the 1960s. Therefore, the average and deepest elevation of river bed has degraded severely in the
upstream and downstream of the Fukawa contraction area (See Figure 6 and Figure 7). And, the
-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
1961 1983 1998 2008
-10.0 -9.0 -8.0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
1961 1983 1998 2008
6.0
7.0
8.0
9.0
10.0
11.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
Figure 5 Flood marks and observed maximum water level of each flood.
Figure 6 Changes in average elevation of river bed.
Figure 7 Changes in deepest elevation of river bed.
6.0
7.0
8.0
9.0
10.0
11.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
Sep.1983 flood marks Sep.1983 observed water level
Sep.1998 flood marks Sep.1998 observed water level
Sep.2008 flood marks Sep.2008 observed water level
6.0
7.0
8.0
9.0
10.0
11.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
Sep.1983 flood marks Sep.1983 observed water level
Sep.1998 flood marks Sep.1998 observed water level
Sep.2008 flood marks Sep.2008 observed water level
Longitudinal distance (km)
Longitudinal distance (km)
Longitudinal distance (km)
Bed e
levati
on (Y
.P.m
)B
ed e
levati
on (Y
.P.m
)W
ate
r ele
vati
on (Y
.P.m
)
0
100
200
300
400
500
66.5 67.5 68.5 69.5 70.5 71.5 72.5 73.5 74.5 75.5 76.5 77.5 78.5 79.5 80.5 81.5 82.5 83.5 84.5 85.5
1961 1983 1998 2008
Longitudinal distance (km)
wid
th (
m)
The Fukawacontraction area
Figure 4 Changes in main channel width.
Dredging stopped in 1966 Dredging stopped in 1999
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 4
river bed elevation in the Fukawa contraction area has also lowered due to flood flows and the river
bed degradation in the upstream and downstream of the Fukawa contraction area from 1961 to 1983.
From 1983 to 1998, the dredging has continued in the upstream and downstream as shown in
Figure 3. And, the amount of the river bed degradation in the downstream is greater than in the
upstream (See Figure 6 and Figure 7). Figure 8 shows cross sectional form at 76.5km after Sep.1983
and Sep.1998 flood, and broken lines indicate the peak water levels of the floods. Table 1 indicates
the cross section area at the peak water levels. As shown in Figure 8 and Table 1, the water level in
Sep.1998 flood is lower than in Sep.1983 flood. However, the cross section area in Sep.1998 flood
is almost as large as in Sep.1983 flood. In the downstream of the Fukawa contraction area, because
the river bed elevation decreased more greatly than in the upstream and the main channel has been
widened, the water level has degraded. It is supposed that the scouring near the exit of the Fukawa
contraction area progressed to get the almost same cross section area of Sep.1983 flood. From the
above discussions, the dredging and widening in the downstream of the Fukawa contraction area are
closely related to the river bed degradation near the exit of the Fukawa contraction area from 1983
to 1998.
After 1998, although the floods comparable to the design high water discharge have occurred (See
Figure 2), the river bed elevation has not much degraded as shown by Figure 6 and Figure 7. There
are two reasons for this. One is the decrease of dredging volume and the other is the geological
factor in the Fukawa contraction area. Figure 9 shows the geologic cross section at 76.5km. The
Figure 8 Peak water levels and cross sectional form at 76.5km.
Table 1 Cross section area at peak water levels.
Peak dischrge (m3/s) Cross section area (m2)Sep.1982 flood 7700 2500Sep.1998 flood 7600 2550
-10.0
-6.0
-2.0
2.0
6.0
10.0
14.0
-100 0 100 200 300 400
S36
S58
H10
H19-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
1961
1983
1998
2008
Lateral distance (m)
Alluvial cohesive soil
Alluvial sandy soil
Diluvial cohesive soil
Diluvial sandy soil
Embankment
Figure 9 Geologic cross section at 76.5km.
Bed e
levati
on (Y
.P.m
)
Sheet pile
-10.0
-6.0
-2.0
2.0
6.0
10.0
14.0
-100 0 100 200 300 400
Lateral distance (m)
Bed e
levati
on (Y
.P.m
)
-10.0
-6.0
-2.0
2.0
6.0
10.0
14.0
-100 0 100 200 300 400
1983
1998
the peak water level in Sep.1983 flood
the peak water level in Sep.1998 flood
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 5
geologic cross section indicates that the river bed surface has been covered with the diluvial sandy
soil since 1998, which is more difficult to erosion than alluvium. For this reasons, it is considered
that the river bed degradation near the exit of the Fukawa contraction area almost stopped.
3. BED VARIATION ANALYSIS CONSIDERING THE CHANNEL DREDGING AND
WIDENING DURING THE FLOODS
3.1 Analysis Method
The foregoing section showed that flood flows, channel dredging and widening had an effect on
change in the river bed in and around the Fukawa contraction area. But, flood flows and bed
variation in the study area during the floods is not discussed. Okamura, Fukuoka et al. (2010)
clarified bed variation and flood flows in the Tone River mouth during the flood by analysis method
using time series of the observed water surface profiles. The analysis is conducted based on the idea
that the effects of river conditions and bed variation during flood appear in time series of the water
surface profiles. We conducted the numerical analysis of flood flow and bed variation using
observed temporal changes in the water surface profiles for the five large floods in Aug. 1981, Aug.
and Sep. 1982 and Aug. and Sep. 1983. In this period, the river bed elevation in the Fukawa
contraction area lowered severely.
The analysis consists the unsteady quasi 3-D flow (Uchida and Fukuoka, 2011) and bed variation
analysis. The bed variation analysis was considered both bed load and suspended load because the
grain diameter is relatively so small as shown in Figure 10 that the amount of suspended load is not
negligible. The rate of bed load transport was calculated by Ashida and Michiue formula (Ashida
and Michiue, 1972). The vertical distribution of suspended load concentration, fall velocity and
amount of suspended load were calculated by using Lane-Kalinske formula (Lane and Kalinske,
1941), Rubey formula (Rubey, 1933) and Kishi-Itakura formula (Itakura and Kishi, 1980),
respectively. The suspended load concentration was calculated by depth averaged planar 2-D
convective diffusion equation.
The initial bed forms were measured in 1980. The upstream and downstream boundary conditions
were given by the time series data of observed water level at Toride (85.3km), Suga (66.5km)
observation point in the lower Tone River and Nakago (5.0km) observation point in the Kokai river
(See Figure 1). The mean grain diameter is 0.25mm in the study area. In the Fukawa contraction
area, we assumed the river bed degradation does not occur at the place of diluvium in the analysis.
Manning's roughness coefficients were set to 0.020 for the main channel and 0.038 for the flood
channel.
In this calculation, flood flows and bed variation were calculated during the 5 floods continuously.
The channel dredging and widening have been conducted in the study area after Aug. 1981 flood,
Sep. 1982 flood and Sep. 1983 flood, respectively. So, the bed variation analysis considered channel
dredging and widening after each flood in the following manner. In this study, we defined dredging
as excavation of river bed and widening as extension of main channel width.
Grain size (mm)
Volu
me p
erc
enta
ge fin
er
than (
%)
0
10
20
30
40
50
60
70
80
90
100
0.01 0.1 1 10
0
10
20
30
40
50
60
70
80
90
100
0.01 0.1 1 10
69.0km71.0km72.0km73.0km75.0km76.0km78.0km80.0km81.0km82.0km83.0km84.0km
The downstream
The Fukawacontraction area
The upstream
Figure 10 Grain size distribution in 1996.
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 6
Table 2 shows dredging range, dredging volume and implementation year. But, there was no data
of dredging depth and area of each dredging location. We determined the dredging depth and area so
that calculated bed profiles after the floods reproduce the observed bed profiles. Concretely, Figure
11 shows the observed river bed contour in 1983 and the calculated river bed contour after the
floods and the area enclosed by black line indicates the determined channel dredging area. The
channel dredging depth and area is estimated to correspond with the dredging volume in the location
which the calculated river bed elevation is higher than the observed river bed elevation. If the
calculated bed profiles after the floods do not sufficiently reproduce the observed bed profiles, we
adjust the channel dredging depth and area by trial and error so that the calculated bed profiles
reproduce the observed bed profiles.
dredging location (m3) dredging location (m3) dredging location (m3)
81.25k right bank 30,465 81.5k right bank 32,535 81.0k right bank 41,460
78.75k left bank 6,850 74.5k left bank 37,119 75.0k left bank 43,440
74.5k right & left bank 21,844 73.75k left bank 12,321 73.5k left bank 19,44674.0k left bank 18,800 71.5k right bank 17,100 69.5k left bank 5,020
68.5k right & left bank 35,060 68.0k right bank 24,560 69.0k right bank 49,74068.0k right bank 35,830 68.75k right bank 35,229 68.0k right bank 40,550
1981 1982 1983
Table 2 Dredging range, volume and implementation year.
Figure 11 Observed river bed contour in 1983 and calculated river bed contour after the floods.
Elevation (Y.P.m)
The dredging area
75.0km 74.5km
The dredging area
75.0km 74.5km
a) Observed river bed in 1983 b) Calculated river bed after the floods
1981 1982 1983
Toride Toride 72.0km
Inba Inba 79.0km
79.0km
Table 3 Widening place name, kilometer post and implementation year.
Figure 12 Cross sectional form before and after the river widening at 72.5km.
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
10.0 12.0 14.0
-200 0 200 400 600
Bed e
levati
on (Y
.P.m
)
Lateral distance (m)-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
10.0 12.0 14.0
-200 0 200 400 600
1980 (before widening)
1983 (after widening)
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 7
In the study area, the widening place name, kilometer post and implementation year have been
recorded as shown in Table 3. But, widening area has not been recorded. In order to determine the
widening area, we used the observed cross sectional form before and after the floods and the data
shown in Table 3. First, we set the widening depth by comparing the cross sectional forms every
point which cross-sectional leveling was conducted (e.g., Figure 12). From this, the planar area of
channel widening is determined as indicated in Figure 13.
Conclusively, we set the planar area of channel dredging and widening of each year as shown in
figure 14.
3.2 Analysis Results
Toride
Inba
After 1981 flood
After 1982 floods
After 1983 floods
Dredging area
After 1981 flood
After 1982 floods
After 1983 floods
Widening area
Figure 14 Planar area of channel dredging and widening of each year.
Widening area
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
10.0 12.0 14.0
-200 0 200 400 600
1980 (before widening)
1983 (after widening)
Figure 13 Planar change in main channel width from 1981 to 1983.
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
0 20 40 60 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60
Aug.1981 Sep.1982 Aug.1983 Sep.1983
Wate
r ele
vati
on (Y
.P.m
)
23456789
1011
0
Toride(85.3km) Obs. Oshitsuke(78.5km) Obs. Fukawa(76.5km) Obs. Suga(66.5km) Obs.
Toride(85.3km) Cal. Oshitsuke(78.5km) Cal. Fukawa(76.5km) Cal. Suga(66.5km) Cal.
Figure 15 Comparison between observed and calculated water level hydrographs of the floods
in the Tone River.
Aug.1982
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 8
Figure 15 shows the comparison between observed and calculated water level hydrographs of the
floods in Tone River. Figure 16 shows the comparison between observed and calculated discharge
hydrographs of the floods in the Tone River and the Kokai River. The calculated water level and
discharge hydrographs correspond well with the observed water level and discharge hydrographs.
Figure 17 is the comparison between observed and calculated water surface profiles at the peak of
each flood. The calculation coincides with observed water surface profiles. Figure 18 shows the
comparison between observed and calculated river bed elevation before and after the floods. The
numerical computation provided a good explanation for the observed bed elevation after the floods.
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
Toride(85.3km) Obs Fukawa(76.5km) Obs. Takasu(3.5km) Obs.Toride(85.3km) Cal. Fukawa(76.5km) Cal. Takasu(3.5km) Cal.
0 20 40 60 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60
Aug.1981 Sep.1982 Aug.1983 Sep.1983
Figure 16 Comparison between observed and calculated discharge hydrographs of the floods
in the Tone River and the Kokai River.
Tone River
Kokai RiverDis
charg
e (m
3/s
)
Aug.1982
Wate
r ele
vati
on (Y
.P.m
)
Longitudinal distance (km)
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
average bed elevation
before floods Obs.
after floods Obs.
after floods Cal.
-8.0
-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
deepest bed elevation
before floods Obs.
after floods Obs.
after floods Cal.
6.0
7.0
8.0
9.0
10.0
11.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 86.0
Aug.1981 flood marks (left bank) Aug.1981 flood marks (right bank) Aug.1981 flood peak Obs. AUg.1981 flood peak Cal.Aug.1982 flood marks (left bank) Aug.1982 flood marks (right bank) Aug.1982 flood peak Obs. Aug.1982 flood peak Cal.Sep.1982 flood marks (left bank) Sep.1982 flood marks (right bank) Sep.1982 flood peak Obs. Sep.1982 flood peak Cal.
The Fukawacontraction area
Figure 17 Comparison between observed and calculated water surface profiles
at the peak of each flood.
Longitudinal distance (km)
Figure 18 Comparison between observed and calculated river bed elevation
before and after the floods.
Bed e
levati
on (Y
.P.m
)
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 9
As a result, we reproduced the bed variation in and around the Fukawa contraction area from
1981 to 1983 by using numerical analysis as shown in Figure 19. The more the floods occurred, the
more bed scouring progressed at the Fukawa contraction area and sediment deposition became
noticeable at the exit of the Fukawa contraction area. Channel dredging was conducted around the
location in which sediment deposition was considerable at the end of the contraction area. However,
sediment deposition occurred again at the dredging location after the flood, in each year. In this way,
we think that respective influences of channel dredging, widening and flood flows on change in
river channel morphology can be analyzed by this method.
CONCLUSION
1. Observed data shows that river bed degradation and scouring in the Fukawa contraction area have
occurred due to flood flows, channel dredging and widening in the upstream and downstream of
Fukawa contraction area.
2. We estimated dredging area, depth and widening area so that calculated water surface profiles and
bed profiles reproduced the observed water surface profiles and bed profiles by using dredging
volume, implementation year and the cross sectional form before and after the floods, and the
analysis result provided a good explanation for the observed results of bed variation in and around
the Fukawa contraction area.
3. We evaluated the respective influences of channel dredging, widening and flood flows on change
in the river channel from the result of the analysis.
After Aug. and Sep. 1982
floods
After dredging and widening in 1982
Bed variation (m)Scouring Deposition
Before 5 floods(contour of bed elevation)
After Aug. and Sep. 1983
floods
After dredging and widening
in 1983
1981 1982 1983Dredging area
1982 1983Widening area
Bed elevation(Y.P.m)
After dredging and widening
in 1981(contour of bed variation)
Figure 19 Calculated bed variation contour in and around the Fukawa contraction area
from 1981 to 1983.
The 10th Int. Conf. on Hydroscience and Engineering (ICHE-2012), Nov. 4 – Nov. 7, Orlando, USA 10
REFERENCES
Ministry of Construction, Kanto Regional Construction Bureau. 1987. 100 year’s history of the
Tone River.
Moro, Y., Kazama, S. and Fukuoka, S. 2011. Change of river improvement works of the Lower
Tone River and effectiveness of channel dredging, Advance in River Engineering, JSCE, vol.17,
pp101-106. Okamura,S., Fukuoka,S. and Takemoto,T. 2010. Bed Form and Bed Variation during Floods of the
Tone River Mouth, Annual Journal of Hydraulic Engineering, JSCE, Vol.54, pp.751-756. Shirai, K. and Fukuoka, S. 2011. Changes of Tonegawa River channel by river improvement since
the Meiji era and hydroscience and hydraulic engineering analysis on its factor, Advance in River
Engineering, JSCE, vol.12, pp217-222.
Uchida, T. and Fukuoka, S. 2011. A bottom velocity computation method for estimating bed
variation in a channel with submerged groins. Proceedings of JSCE, vol.67, No.1, pp.16-29.