River Flow 2014 – Schleiss et al. (Eds)© 2014 Taylor & Francis Group, London, ISBN 978-1-138-02674-2

New computation method for flood flows and bed variations
in a low-lying river with complex river systems
K. TabataChuo University, Tokyo, Japan

S. Fukuoka

Research and Development Initiative, Chuo University, Tokyo, Japan

ABSTRACT: The lower Shinanogawa River is known as a typical low-lying river in Japan. This river
system is comprised of several tributaries, a branched river and a no embankment area. In the recordbreaking flood observed in 2011, backward flows around confluences and large inundation from the noembankment area were caused due to large inflows from main tributaries. Flood flows and bed variations in such a complex river system have not been studied so far. In this study, we developed a numericalmodel applying the Bottom Velocity Computation (BVC) method and 2D bed-variation analysis using
observed water surface profiles in the lower Shinanogawa River. Finally the mechanisms of the complex
flows around river confluences and functions of the flood flow storage in the complex river system wereclarified by using the numerical model developed.
1

INTRODUCTION

The lower Shinanogawa River has a complex
river system, which is comprised of several tributaries, branched rivers and a no embankmentarea (see Fig. 1). The Ikarashigawa River and­Kariyatagawa River are main tributaries of thelower ­Shinanogawa River. The Nakanokuchigawa
River and the Sekiya diversion Channel bifurcatefrom the ­Shinanogawa River at the Nakanokuchigawa Sluice Gate and Shinanogawa Sluice Gate,
respectively and the Nakanokuchigawa River joins
again at the downstream of the lower Shinanogawa
River. No embankment area is located at just
upstream of the inflow point of the ­Kariyatagawa
River. In the record-breaking flood observed in2011, large floods from the Ikarashigawa Riverand Kariyatagawa River caused a complex flood
flow and large inundation in the no embankmentarea. Flood flows and bed variations in such acomplex river system have not been clarified inspite of occurrence of frequent flood disasters.
In the lower Shinanogawa River having theconfluences of several tributaries and large longitudinal changes of plan forms and cross-sections,multi-scale flows are developed. They influenceeach other and generate the complex flood propagation and bed variation. Therefore, it is necessary
to establish a numerical model elucidating the small
and large scale hydraulic phenomena and bed variations in the whole reach from the river mouth to
the Arai weir.

While simulations of complex flows have been
made possible by recent advanced 3D turbulencemodels (Huang et al. 2002; Wu 2008), applicationsof the 3D model are still limited to small scale phenomena in experimental channels because of large
computational load. Therefore, some kind of depth
integrated simulation method should be applied to

Figure 1. Plan form of the lower Shinanogawa Riverand water level observation points.

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complex river fields. The depth integrated model
taking into a helical flow in a curved channel(Nishimoto 1992) using well developed secondary flow model (Engelund 1974) has ­proposed.
A number of Quasi-3D models taking into vertical distribution of horizontal velocities have been
proposed (Ishikawa 1989, Jin 1993). In these models, a vertical distribution of horizontal velocities
is solved by 3D momentum equations using aweighted residual method. In recent years, Uchida
and Fukuoka (2011; 2012) developed the Bottom
Velocity Computation (BVC) method. The BVC
method is an integrated multi-scale simulation offlows and bed variations in rivers, which is capable of evaluating vertical distribution of horizontal velocities and bottom velocities by introducing
depth-averaged horizontal vorticity equations andhorizontal momentum equations on water surface
to shallow water equations.
Fukuoka (2011) proposed the numerical method
for flood flows and bed variations using time seriesof observed water surface profiles during a flood.
This method is based on the idea that effects ofplan forms, cross-section, hydraulic structuresand bed variations appear in time series of water
surface profiles observed during a flood. It has
been shown that the numerical method using timeseries of observed water surface profiles applying
the BVC method can elucidate flood flows andbed variations in short reach like a river mouth­(Okamura 2013). However, the applicability of the
BVC method to river fields with a complex river
system like the lower Shinanogawa River has notbeen investigated enough.
The objectives of this study are to develop thenumerical model applying the BVC method and 2D

bed-variation analysis using time series of observed
water surface profiles during the 2011 flood in thewhole of the lower Shinanogawa River including
the several river confluences and branches and to
elucidate the flood propagation and bed variationin the complex river system. At last we clarify the
propagation mechanism of the flood flow aroundriver confluences, flood flow storages and bed variations in the lower Shinanogawa River by using thedeveloped numerical model.
2

Objective Area and Flood

Figure 1 shows the plan form of the lower­Shinanogawa River and water-level observation
points. No inflow enters from the Arai Weir locatedat the upstream end of the lower Shinanogawa
River because the weir gates are completely closed
during floods to prevent inflows of flood waterfrom the upstream area. Therefore, the flood discharge hydrograph in the lower Shinanogawa Riveris almost determined by discharges from tributaries of the Ikarashigawa River and Kariyatagawa
River. The discharge of the Nakanokuchigawa
River is controlled by operating the Nakanokuchigawa Sluice Gate at the branch point of the­Shinanogawa River and Nakanokuchigawa River.
Figure 2 is the enlarged figure of the dash
box frame in Figure 1. As shown in Figure 2, noembankments are located at right bank of thereach from 47 km to 50.7 km and upstream of
the inflow points of the Ikarashigawa River and­Kariyatagawa River and the branch point of the
Shinanogawa River and Nakanokuchigawa River.
The area of the flood plain at no embankment

Figure 2. Plan form in the reach including the confluences of Ikarashigawa River and Kariyatagawa River andbranch point of the Nakanokuchigawa River.

1792area is about 2.8 km2 and the averaged elevation isabout T.P. + 10.5 m. T.P. denotes Tokyo Peil, whichis mean sea-level of Tokyo Bay.
Figure 3 shows the discharge hydrographs converted by H-Q curves at Aramachi located downstream of the confluence with the Ikarashigawa
River, Teisekibashi located downstream of the confluence with the Nakanokuchigawa River, and the
Nakanokuchigawa Sluice Gate. The figure showsthat the discharge hydrographs with two peaks
lasted for three days. The peak discharge almost
reached the design flood discharge at Teisekibashi
and Aramachi and exceeded the design flood discharge at the Nakanokuchigawa River.

As shown in Figure 1, the Shinanogawa Riverand Nakanokuchigawa River flowing through the
low-lying basin are equipped with many drainage
pump stations and head works. Water levels havebeen recorded for operations. Therefore, we could
get time series of observed water surface profiles at
1 to 5 km intervals by using water levels measured
at the drainage pump stations and head works inaddition to water-level gauging stations.
Figure 4 shows the time series of observed water
surface profiles in the 2011 flood and longitudinal distributions of elevation on the levee crest
and averaged bed elevation in the lower Shinanogawa River. Water levels at the peak discharge
exceeded the High Water Level (H.W.L.) in the section located 26.0∼42.0 km from the river mouth.
­Moreover, water levels exceeded flood plain elevations at the no embankment area and caused thevast inundation.
At the confluence of the Shinanogawa Riverand Ikarashigawa River indicated by dash box
frame in Figure 4, adverse slopes of water surface
profiles were seen because large inflows from the
Ikarashigawa River caused backward flows. This
fact shows that inflows from the Ikarashigawa
River obviously had significant effects on the 2011
flood flow in the Shinanogawa River.
3

Analysis of flood FLOWs and bed
variations during the 2011 floodin the lower Shinanogawa river

3.1 Numerical analysis method

Figure 3. Discharge hydrographs converted by H-Q
curves during the 2011 flood.

We develop a numerical model applying unsteadyflow and 2D bed-variation analyses using observed
water-surface profiles in the lower Shinanogawa
River. The numerical model consists of ­quasi-3D

Figure 4. Time series of observed water surface profiles in the 2011 flood with the longitudinal distribution ofelevation on the levee crest and averaged bed elevation in the lower Shinanogawa River.

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unsteady flow analysis by the BVC method­(Uchida and Fukuoka, 2012) and 2D bed-­variation
analysis.
The BVC method is able to evaluate vertical
distributions of horizontal velocities and bottom
velocities by introducing depth-averaged horizontal vorticity equation, horizontal momentum equation on water surface to shallow-water equation.
A set of governing equations (1)∼(5) is solved forthe following unknown quantities: water depth h,
Depth Averaged (DA) horizontal velocities Ui, DA
horizontal vorticities Ωi, horizontal velocities onwater surface usi and bottom velocities ubi according to the process as shown in Figure 5. The bottom velocity is derived by integrating horizontal
vorticities respect to depth (equation (1)).
ubi = usi − Ω j h(1)where, ubi: bottom velocity, usi: water surface velocity and Ωj: DA horizontal vorticity.
To evaluate the bottom velocity by equation (1),
BVC method is composed of depth-integrated
horizontal vorticity equation (4) and water surface

velocity equation (5) in addition to shallow waterequations (2), (3). In this study, equation (2)∼(5)
are converted to the general curvilinear coordinate
system.
∂h ∂Uj h
=0
+
∂xj
∂t

(2)

∂z
∂ρU i h ∂ρU iU j h= − ρ gh s − τ 0i
+
∂xi
∂x j
∂t
+

(

∂h τ ij − ρ u ′ i u ′ j
∂x j

)

(3)

where, h: water depth, Ui: depth averaged horizontal velocity, ρ: density, g: gravity, zs:water level, τ0:
bed shear stress, τij: horizontal shear stress due toturbulence and u ′i u′j : correlation of vertical distributions of horizontal velocities ui, uj.
∂hDω ij∂Ω i h= ERσ i + Pω i +
∂xj
∂t

(4)

where, ER: rotation term of vertical vorticity, Pωi:
production term of vorticity from the bottom thinvortex layer and Dωij: horizontal vorticity flux due
to convection, rotation, dispersion and turbulencediffusion.
∂z
∂u
∂usi
+ usj si = − g s + Psi
∂xi
∂xi
∂t

(5)

where, Psi: production term due to shear stress acting on thin water surface layer.
Vertical distributions of horizontal velocity are
expressed by the cubic function (6) by using depthaveraged velocity Ui, velocity differences δui and∆ui and dimensionless depth η, as;

(

(

)

ui = U i + ∆ui 12η 3 − 12η 2 + 1 − δ ui 4η 3 − 3η 2

) (6)where,∆ui = usi − U i ,δ ui = usi − ubi , η = ( zs − zb ) h
The correlation of vertical distributions of horizontal velocities included in equation (3) is solved
by equation (7) which is derived by integrating
equation (6) respect to depth.
u′i u′ j =133∆ui ∆u j + δ uiδ u j
35
35
2
−
∆uiδ u j + ∆u j ∆ui
35(

Figure 5. Flow of simulation method.

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)

(7)

The bed variation is computed by the continuity equation for sediment and grain sizes (Hirano
1971) using bed load formula (Ashida and Michiue1972) and suspended load formulas (Itakura andKishi 1980; Lane and Kalinske 1941). The transport of the suspended sediment is computed by 2D
advection and diffusion equations.
3.2

Conditions of the numerical analysis

We set the scope of the numerical model for the reachfrom the river mouth to the Arai Weir of the lower
Shinanogawa River including the Nakanokuchigawa River, the Sekiya Diversion Channel and theconfluences with the Ikarashigawa River, Kariyatagawa River, Gejyogawa River, Kamogawa Riverand Koaganogawa River. In addition, we included
inundation areas due to no embankment in thescope of analysis, considering landforms of theflood plain using aerial laser profiler data.
The upstream boundary condition of the­Shinanogawa River was given by no-flux condition because the Arai Weir gates at located around
51 km were completely closed during floods.
The downstream boundary conditions of the­Shinanogawa River and the Sekiya Flood Control
Channel were given by the observed water level
hydrographs at the river mouth.
Inflow discharge hydrographs from the several tributaries were provided so that the computed water-surface profiles in the whole of the­Shinanogawa River agree with the observed ones.
The discharge hydrograph through the
Nakanokuchigawa Sluice Gate was estimated so as
to minimize the difference between observed andcomputed water-surface profiles in the upstream
and downstream sections of the sluice gate by taking account of hydrostatic pressure and hydrodynamic pressure acting on the gate. Similarly, thedischarge hydrograph through the Shinanogawa
Sluice Gate was estimated by this method.
In the 2011 flood, a lot of waters in the river
basin were pumped out by many drainage pumpstations along the Shinanogawa River andNakanokuchigawa River. Figure 6 shows temporal

Figure 6. Temporal changes of observed total discharges from many drainage pump stations in the Shinanogawa River and Nakanokuchigawa River.

changes of observed total discharges from drainage pump stations in the Shinanogawa River andNakanokuchigawa River. Maximum discharges
in the Shinanogawa River and Nakanokuchigawa
River were 480 m3/s and 190 m3/s, respectively andnot negligible compared to the flood discharge.
Therefore, observed discharge hydrographs fromeach pumping station were given for the analysis.
Figure 7 shows grain size distributions used forthe analysis. The bed materials in the Shinanogawa
River and Nakanokuchigawa River are composedof silt and sand and average diameters are about
0.5 mm.
The Manning roughness coefficients and vegetation permeability coefficients were determined so
as to explain the time series of observed water-surface profiles in the 2011 flood by the BVC method.
Manning roughness coefficients and vegetation
permeability coefficients used in the analysis areshown in Table 1.
3.3 Results and considerations
3.3.1 Water-surface profiles and flood dischargehydrographs in the lower Shinanogawa River
Figure 8 shows the comparison between thecomputed and observed water-surface profiles
and longitudinal distributions of the averaged
bed elevations in the lower Shinanogawa Riverand Nakanokuchigawa River. In Figure 8(a),the computed water-surface profiles around theconfluences of the several tributaries and branch
point of the Nakanokuchigawa River could elucidate the observed ones in the ­Shinanogawa
River. In addition, the computed water-surface
profiles could explain the observed ones in the
Nakanokuchigawa River (see Fig. 8(b)), while
the computed water levels were lower thanobserved ones around directly downstream of
the Nakanokuchigawa Sluice Gate. The water
surface profiles at the peak time almost exactly
matched flood marks in the Shinanogawa Riverand Nakanokuchigawa River.

Figure 7.

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Grain size distributions used for the analysis.

Table 1. Manning roughness coefficient and vegetation permeability coefficient.
Manning roughness coefficient (m-1/3 ⋅ s)
SectionMain channel

Flood plain

Vegetation permeability
coefficient (m/s)

Sekiya flood control channel
Sinanogawa River (1.8∼29 km)
Shinanogawa River (29∼44 km)
Shinanogawa River (44∼50.7 km)
Nakanokuchigawa River
Ikarashigawa River
Kariyatagawa River
Kamogawa River, Koaganogawa River,
Gejyogawa River

0.013
0.020∼0.028
0.028
0.025
0.022
0.027
0.025
0.020∼0.030

0.030
0.025∼0.040
0.025∼0.050
0.030∼0.050
0.030∼0.050
0.035∼0.055
0.030∼0.040
0.030∼0.070

–
40∼90
20∼90
40
20∼90
–
–
–

Figure 8. Comparison between the computed and observed water-surface profiles and longitudinal distributionsof the averaged bed elevations in the lower Shinanogawa River and Nakanokuchigawa River.

In Figure 8(a), the computed results almost
agreed with observed average bed-elevations, while
the bed elevations were estimated lower than theobservations in the section located 39.0∼42.0 km
where a large separation area was formed due tolarge inflow rates from the Ikarashigawa River.

Figure 9 shows the discharge hydrographs andthe temporal changes of the inundation volume
at the no embankment area. Plots of the inundation volumes indicate values evaluated by using theobserved water-level hydrograph at the Arai Weirand ground levels of the no embankment area. In

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Figure 9. Discharge hydrographs and temporal changes
of inundation volume at the no embankment area.

Figure 10. Computed horizontal velocity vectors at
the peak of the first flood (18:00 on July 29).

this figure, we can see that the computed dischargehydrographs at Aramachi and Nakanokuchigawa
Sluice Gate were in good agreement with theobserved results. From our simulation, the inundation volume at the peak was estimated about
4,000,000 m3. Although this is slightly higher thanthe estimated value, the computed results could
estimate the storage and drainage of the inundation water in a flood period. The computed inundation volume hydrograph also coincided well withthe estimated one.
3.3.2 Flow and bed variation at the confluence of
the Shinanogawa River and Ikarashigawa
River
Figures 10 and 11 shows the computed horizontal velocity vectors and bed variation contour at
the peak of the first flood (18:00 on July 29) atthe confluence of the Shinanogawa River andIkarashigawa River. Here, red and black arrows
in Figure 10 indicate the depth averaged velocity
and bottom velocity respectively. Figure 12 shows
computed discharge hydrographs in the section
including the confluences of the Ikarashigawa
River and Kariyatagawa River (see Fig. 2) withtemporal changes of the inundation volume at theno embankment area.
The flow attacking point is seen at the left bankin 41.1 km point and the secondary flow is developed at the first peak as shown in Figure 10 since
the inflow discharge from Ikarashigawa River isvery large compared with that from the Shinanogawa River. Flow attacking to the river bankcaused bank erosions as shown in Figure 11.
The bed scouring occurred along the left bankof the main channel around 41 km point (seeFig. 11) where inflow discharges from the ­Ikarashi
River concentrated. At the confluence of the­Shinanogawa River and Ikarashigawa River indicated by dash box frame in Figure 8(a) the adverse

Figure 11. Computed bed variation contours at thepeak of the first flood (18:00 on July 29).

Figure 12. Computed discharge hydrographs in the section including the confluences of the Ikarashigawa Riverand Kariyatagawa River with temporal changes in inundation volume at the Nishino area.

slopes of water surface ­profiles are computed aswell as the observed ones (see Fig. 4). The backward flow occurred at the confluence with the
Ikarashigawa River and backward flow discharges
reached to 580 m3/s (see Fig. 12).

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At the second flood, since inflow discharges
from the Kariyatagawa River increased in addition to the inflow from the Ikarashigawa River,
the inundation volume at the no embankment area
increased and reached to maximum value during
the flood. As above, the present numerical model
could elucidate the flood flows and bed variation
including large and small scale hydraulic phenomena in the lower Shinanogawa River with the complex river system.
4 conclusion
In this study, we developed the numerical model
consisting of the BVC method and 2D bed­variation analysis using observed water-surface profiles ­during the 2011 flood in the whole of the lower
Shinanogawa River with the complex river system.
The present model explained well the observed
discharge hydrographs and inundation volume
hydrograph and averaged bed elevations in thewhole of the lower Shinanogawa River during theflood. Furthermore secondary flow and bed variation at the confluence of the Shinanogawa Riverand Ikarashi River were described well, by thisnumerical model.
It was concluded that the present calculating
method could elucidate the flood flows and bed
variation including large and small scale hydraulicphenomena in the lower Shinanogawa River withthe complex river system.
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