River Flow 2016 – Constantinescu, Garcia & Hanes (Eds)© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02913-2

Quasi-three dimensional numerical and scale-model studies on flood
propagation and water storage in a dam reservoir
Y. Tsukamoto

Chuo University, Tokyo, Japan

S. Fukuoka

Research and Development Initiative Chuo University, Tokyo, Japan

ABSTRACT: The flood management in dam reservoirs has been performed by the estimation of inflow
discharge using the assumption that a water level observed near the dam body holds everywhere in thereservoirs. However, we are not sure whether the water level in dam reservoirs rise horizontally or notduring a flood because propagation characteristics of flood water level and discharge hydrographs are different. For efficient and floods and sediment management in dam reservoirs, it is important to clarify themechanism of flood propagation and three-dimensional flow structures. Hydraulic model test was carriedout in order to investigate the characteristics of reservoir storage and flow mechanism in dam reservoirs.
Moreover, the quasi-three dimensional flood flow analysis method considering main stream layer andnon-equilibrium vortex layer was developed to understand dynamic characteristics of flood flows in dam
reservoirs and applied to the hydraulic model tests. From the analysis result, characteristics of flood flow
dynamics and flood propagation in dam reservoirs were clarified.
1

INTRODUCTION

The flood management in dam reservoirs is important for the improvement of the safety level control
in downstream rivers of the dam. Figure 1 presents
the dam reservoir flow during a large flood. The
inflow discharges to dam reservoirs have been evaluated by using rate of storage volume dS/dt fromthe H-V relation assuming the horizontal rise ofreservoir water level and outflow discharges estimated by reservoir water elevation near the dam
body. This provides a simple estimation method
for the outflow and inflow discharges in dam reservoirs. Assuming the horizontal rise of reservoir
water level means that the flood flow propagateswith very large velocity within the reservoir. However, floods in dam reservoirs flow down transforming a water-level and discharge due to three
dimensional reservoir flows. But, the present reservoir management conducted by assuming thehorizontal rise of water levels has not considered
flood propagation mechanism and flow dynamics
of flood water in reservoirs.
The reservoir water levels were observed in orderto understand the mechanism of translatory waves
during floods in Wheeler dam reservoir in America(Wilkinson et al. 1944). In this study, It was shownthat the water surface profiles in reservoirs weregenerally horizontal. The hydraulic model test andFigure 1. The dam reservoir flows during a flood.

analysis have been performed in order to understand characteristics of the flood propagation ina simple reservoir model channel with the uniform
bed-gradient and width (Yano et al. 1965). Theirstudy clarified three typical flow areas such as riverarea, transition area and calm area as shown in­Figure 1 and provided the fundamental information
about the flood propagations in a dam reservoir,
but it was not clear the dynamics of flood flow in
dam reservoirs. The one-dimensional flow analysis
model was applied to the system of reservoirs andthe network of tributaries (Garcia-­navarro et al.
1993). They investigated the applicability of themodel. The effects of the gate operation in smallhydropower dams on flood propagation and water
storages were clarified using unsteady one-dimensional flow ­analysis using observed water-level data

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(Takemura et al. 2010). The ­accuracy of inflow
and outflow discharge hydrographs were investigated by two-dimensional unsteady flow analysis
using observed temporal changes in water surface
profiles in the Kusaki Dam reservoir in Japan(­Tsukamoto et al. 2014). Their results suggested
that the unsteady two-dimensional flow analysis
in dam reservoirs could not explain sufficiently the
dynamics of flood flows because of three dimensional features of reservoir flows.
For efficient and safe management in dam reservoirs, it is necessary to verify the estimation accuracy of inflow and outflow discharge hydrographs
and establish a proper analysis model for explaining three-dimensional flow features such as verticaland horizontal velocity distributions and transformation of water level and discharge. To predict thebottom velocity is very important for estimationof the sediment movement in the reservoir. For
these purpose, we measure characteristics of flood
flows in dam reservoir model by using large-scale
hydraulic model test and analyze observed flow
data by newly developed quasi-three dimensional
numerical model.
Generation mechanism of three-dimensionalflows in dam reservoirs was considered by introducing non-equilibrium vortex layer near the bed andclarified the three-dimensional flow dynamics andflood propagation mechanism in dam reservoirs.
2

Scale-model experiment

It is hard to measure flood flow structures inlarge reservoir such as water levels, flood velocity and discharges over time in detail. The large-

scale model test was carried out to investigate theflow dynamics in dam reservoirs. Figure 2 shows
the plan form and longitudinal distributions of
the average and the lowest bed elevations of thereservoir model. The model scale is 1/75 of the
Kusaki Dam in Japan and the total length is about
60 m. Table 1 shows the conditions adopted in thehydraulic model test. Inflow discharge hydrographis the 2013 flood discharge of the Kusaki Dam. The
peak discharge of the 2013 flood was 1,000 m3/s,
but it was extended to 1,300 m3/s (converted about
26.6 l/s in scale model) because the flow velocityof 1,000 m3/s flood was too small to be measured in the hydraulic model. The initial reservoir
water elevation is set to 0.74 m of the flood season
control level. The outflow discharge hydrographis given according to the operation rules of the
Kusaki dam. The items measured are water levels,
inflow and outflow discharges, and flow velocity
distributions.
Table 1. Conditions used in hydraulic model test.
Model
scale 1/75

Field

Model

Discharge

2013 flood
1,000 m3/s 20.5 l/s
Extension
1,300 m3/s 26.7 l/s
Roughness River
0.050 m-1/3s 0.024 m-1/3s
Coefficient Reservoir
0.025 m-1/3s 0.012 m-1/3s
Model size Total length
4,500 m60 mWidth (Reservoir) 300 m
4m
Width (River)
50 m
0.67 mExperiment
11 hr
1.27 hrtime

Figure 2. Plan form of the experimental channel and longitudinal distributions of the average and lowest bed
elevation.

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3.2

Figure 3. Diagram of the flow regime and bed elevation
contour in the hydraulic model test.

Figure 3 shows the diagram of the flow regime
visualized by the dyestuff and river bed elevation
contour in the hydraulic model test. This diagramvisualizes the flow after 18 minutes from beginning of the test. Flood flows entering from the
upstream river formed multiple large horizontal
eddies at the right bank of around 3.6 km point.
In the downstream section from 3.2 km point, thevelocity was decreased and nearly stagnant since
the cross-sectional area was increasing with thedownstream section in dam reservoirs. Horizontal
eddies were affected each other by the increase inwater depth and river width and shape of microlandform in river bank (see Figure 3). Thus, theflood flow had three-dimensional characteristics byvertical and cross sectional geographical changes in
the reservoir.
3

Framework of the dam reservoir
flow analysis model

The quasi-three-dimensional flow analysis method(the Bottom Velocity Computation (BVC method))
was developed by Uchida & Fukuoka (2011, 2013).
This method can estimate three-dimensional horizontal and vertical velocity distributions using
observed temporal changes in flood water surface
profiles in various rivers.
Figure 4 shows the framework of dam reservoir flow analysis model. This model consists of amain stream layer and a vortex layer. The former
is a layer of flood flows with large momentum.
The BVC method is applied for this layer of dam
reservoirs.
The latter is a thin vortex layer on bed surface where the streamwise velocity near the reservoir bed is non-equilibrium. This is caused bylongitudinal change in the flow with steep slopeand large depth in transition area. The continuity and momentum equations in the vortex layer
are developed and the whole flow area is examinedconsidering the exchange of mass, momentum andvorticity between main stream layer and vortex
layer.
3.3

Governing equations

The governing equations of the BVC method are
given by Uchida et al. (2014) and applied to themain stream layer of reservoirs. In this chapter,
the analysis method of flows in the vortex layer ismainly explained.
The continuity equation and the equation of
motion in the vortex layer are given as follows:
wσ b = −∂δ zb uvi
∂xi∂(dpb + ρ gzs )
∂u
∂uvi
τ
τ
+ bi − 0i+ uvk vi = −
∂ xk
∂t
ρ∂xi
ρδ zb ρδ zb

Quasi-three dimensional
numerical CalCULATiON in dam
reservoirs

3.1 Necessity of three-dimensional flood flow
analysis in dam reservoirs
Generally, dam reservoirs have the characteristic shapes of cross sectional areas since the water
depth and width become larger in the downstream
direction. So, flood flows in reservoirs change not
only horizontally but vertically and vary waterlevel and discharge hydrographs. It is importantto analyze three-dimensional flows in dam reservoirs with complex three-dimensional shape forproper dam reservoir management. Therefore, thequasi-three-dimensional flood-flow analysis model
is developed for the estimation of complex flow
fields.

Figure 4.

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Dam reservoir flow analysis model.

(1)
(2)

where, k = 1,2,3, wσb: velocity normal to the surfaceof vortex layer, δzb: the thickness of vortex layer,
uvi: i direction velocity in the vortex layer, dpb: pressure deviation from hydrostatic pressure distribution at the bottom, zs: water level, τ bi, τ 0i: shear
stress acting on the upper boundary and bottomof the vortex layer, respectively.
The shear stress in the equation (2) for the vortex layers is given by using the eddy viscosity coefficient as follows:τ bi  ∂ui (u − uvi )= vtb ⋅ Ab bi= vthρ  ∂z  b
Ab =1
α (cb − cv )

(3)
(4)

The coefficients cb and cv are expressed by equation (5) by the logarithmic velocity distribution.
cb =1  zb 
1  δ z / 2 + δ z0 
ln   + Ar , cv = ln  b + Ar (5)ksκ  ks κ The eddy viscosity coefficient at the bottom of
the vortex layer is expressed by equation (6).
vtb = α bδ ub h, δ ub 2 = (ubi − uvi ) (ubi − uvi )(6)

where, vtb: eddy viscosity coefficient, ubi: i direction bottom velocity, h: water depth, zb: bed level,α = κ/6, κ = 0.41, Ar = 8.5. The bottom eddy viscosity coefficient vtb is presented in the previous
papers (Uchida & Fukuoka, 2011, 2013):
vtb =ακω b h22 ln( zs / zb )(7)

We use the average value of equation (6) and (7)for vtb in this model.
The non-equilibrium bottom vorticity is given
by equations (8) and (9) which are derived from theequilibrium vorticity in the production term of thevorticity equation in the vorticity layer.
ubi − uvi
h

(8)

z 
1
ln  s κ (cb − cr )  zb (9)ω bej = 2ε ij 3AωAω =where, ω bei: equilibrium vorticity on the bottom.
3.4

Analysis condition

We applied the quasi-three dimensional reservoir
flow analysis model consisting of the main stream

Figure 5.
reservoir.

Downstream boundary conditions in dam

layer and vortex layer to the results of hydraulic model test. The analysis was performed from
2.0 km to 83.725 km shown in Figure 1. The computational mesh size is set to dx = dy = 5 m in the
upstream river, and dx = dy = 20 m in reservoir
areas. The upstream boundary condition is givenby the observed inflow discharge hydrograph anddownstream boundary condition by the observed
reservoir water level hydrograph. Figure 5 shows
the method determining the downstream boundary condition. The outflow discharge hydrograph
was set by adjusting the gate opening so as tocoincide calculated water levels near the dam bodywith observed discharge hydrograph. We assumed
that the vorticity fluxes were discharged out by theheight of gate open out of vorticity fluxes in meshes
in front of the dam body. Moreover, the dynamic
pressure and the static pressure are consideredon the dam body. The roughness coefficients are
determined so that calculated temporal changes inwater surface profiles may agree with observed one
during the flood. As a result, temporal roughness
coefficients were set to n = 0.053 to 0.087 (n = 0.026
to 0.042 in scale model) for transition area and the
upstream river, respectively, n = 0.025 (n = 0.012 inscale model) for the calm area.
3.5

Calculation results

Figure 6 shows the comparison between observedand calculated water surface profiles during theflood. The velocity of the flood flows is reduced
and become nearly stagnant by the dam body, since
the inflow discharge is small as compared with avolume of the reservoir. Longitudinal water surface
profiles in the calm area are generally horizontal inrising and falling water stages. In the transition area
at 4.0 km to 4.4 km section, longitudinal changes
in water surface profiles occurrs in discharge rising
stage (experiment time 20 minutes to 32 minutes).
At 4.4 km to 5.0 km section, calculated water surface profiles cannot explain the observed ones, since
the vorticities due to flow separation generated in

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Figure 6. Comparison between observed and calculated water surface profiles.

Figure 7. Comparison between observed and calculated
outflow discharge and reservoir elevation.

narrow pass are not described enough in this model.
In the upstream river, temporal longitudinal water
surface profiles are made mainly depending on river
bed-gradient. Calculated water surface profiles arefound to reproduce mostly the observed ones in thereservoir.
Figure 7 shows the temporal changes in observedand calculated outflow discharges and reservoir
water levels in the calm area. The inflow dischargehydrograph at the upstream boundary condition
is described for reference. The observed outflow
discharges is calculated by outflow discharge formula based on gate opening and reservoir waterlevel. Although the calculated outflow discharge isa little larger than observed one around the timeof peak discharge, the calculated results reproduce
well observed ones. Both observed and calculated
water levels in the dam reservoir go up horizontallyin totality.
Figure 8 shows the comparison between
observed and calculated horizontal velocities at
the water surface and the bottom. In discharge rising stage, the water depth is small and flow velocities are large in reservoirs. The flood flows enter
into the reservoir with large inertial force from the
upstream river area. The area around 4.0 km is the

Figure 8. Comparison between observed and calculated
horizontal velocity.

occurrence area of water storage dS’ (see Figure 1
and the section of 4.0 km to 4.4 km in Figure 6),
which causes local changes in water surface andresulting change in flow dynamics of the reservoir.
Horizontal and vertical mixing happens by large
velocity difference between water surface and bottom in this area.
The flow which enters into the calm area fromthe transition area is curred to the right at around3.6 km point, and is divided into flows of the downstream and upstream. Multiple horizontal eddies
are formed and their occurrence points depend on
the reservoir water level and flood discharge scale.
The inflow discharges enter into the calm area withsmall velocity, reservoir water levels go up generally horizontally. At the time of peak reservoir
water level, longitudinal water surface profiles tend1759

oped model and depth averaged 2D calculation
model to large-scale hydraulic model test.
Figure 10 shows the horizontal velocity at thewater surface and the bottom by using the dam
reservoir flood analysis model and 2D calculation
model in discharge rising stage. The calculated
result of flow dynamics by the new model can
reproduce the observed one. However, the calculated result of flow dynamics by the 2D calculation
model cannot explain the observed one around2.8 km point.
Figure 11 shows the difference of bottom vorticity between equilibrium and non-equilibrium
flow. The non-equilibrium bottom vorticity wasestimated by the difference of velocity between
bottom and vortex layer as shown in equation (8).
The equilibrium bottom vorticity was estimated by
the velocity assuming the logarithmic distribution

Figure 9. Comparison between observed and calculated
vertical distributions of streamwise velocity.

to be horizontal by the 5.0 km point. After then,dynamic flow movement in the reservoir occurrs
again when the outflow discharge is greater thaninflow discharge and reservoir water level goes
down. Calculated horizontal velocity explainsobserved one fairly well in dam reservoirs.
Figure 9 presents observed and calculated vertical distributions of streamwise velocity. The reservoir width and depth around 3.6 km point aresmaller, calculated and observed vertical distributions of velocity are similar to those of rivers. In
3.2 km point, calculated vertical velocity distributions generally reproduce observed ones except foraround bottom. Velocities at 2.8 km are smaller
than those of the upstream area.
3.6

Figure 10. Horizontal velocity at the water surface andthe bottom.

The applicability of the dam reservoir flood
analysis model

In order to clarify the applicability of the dam reservoir flood analysis model, we applied the devel-

Figure 11. Difference of bottom vorticity between
equilibrium and non-equilibrium flow.

1760low in vortex layer. Around 2.8 km point, nonequilibrium flow was caused because the difference of bottom vorticity between equilibrium andnon-equilibrium is large. Thus, non-equilibrium
flow changes in the velocity distributions temporally and spatially, and the three-dimensional flows
were formed in dam reservoirs.
These results demonstrate that the quasi-three
dimensional reservoir flow analysis model can
explain characteristics of rather small velocity fieldin dam reservoirs with three-dimensional shape,and provides a prediction tool for sediment movement in reservoirs.
4

FLOOD Propagetion IN DAMReSERVOIRs

Figure 12 shows calculated longitudinal change inwater level and discharge hydrographs. Dischargehydrograph is the value averaged for 1 minute ineach cross section. The propagation mechanism ofwater level and discharge is greatly different fromeach other. While the peak of water level and discharge occurs at nearly same time in the upstream
river, the occurrence time of peak water level is
11 minutes behind the time of peak discharge inthe calm area. In the upstream river, the dischargehydrograph is hardly transformed in shape. In thecalm area, the peak discharge is decreased clearlywith distance.
Figure 13 shows the longitudinal distributionsof calculated flood discharge. The discharges indischarge rising stage (dashed line in Figure 13)
reduce remarkably in the downstream section of
4.0 km point which is the boundary of the calm
area and transition area. The rate of storage water
volume dS/dt is evaluated by the difference of

Figure 13. Longitudinal distributions of calculated
flood discharge with time.

inflow discharges Qin and outflow discharges Qoutor temporal changes in water surface elevation in
the target area.
∂AdS= Qin − Qout = ∫
dx
∂t
dt
L

(10)

where, S : storage water volume, A: cross sectional
area of flood flow, L: length in the target section.
As shown in equation (10), the longitudinal difference of flood discharges is equal to the storage
water volume. The shape of discharge hydrographschanges in the longitudinal direction since theflood flow is stored in the calm area and transition area. Moreover, in the discharge falling stage(solid line in Figure 13), storage water volume is
reduced gradually since inflow discharge into dam
reservoirs is smaller than outflow one from the dam
gate. As shown in Figure 12, water level hydrographgreatly transforms in shape in the transition area,and becomes the same shape in the calm area. Thisis because the inflow discharge in this model test issmall enough as compared with the reservoir capacity and the reservoir water levels go up almost horizontally in mid scale floods. Thus, the difference
from the horizontal water level is not so large totransform the shape of discharge hydrograph. Thismeans that current estimation method assuming
the horizontal rise of reservoir water level is fairly
accurate for the estimation of the inflow dischargehydrograph of mid scale floods.
5 dynamics of flood flows
in a reservoir where large
floods inflow

Figure 12. Longitudinal change in water level and discharge hydrographs.

When larger floods flow into a dam reservoir, it
is imagined that flows in the transition and calm
areas are different from mid scale floods. The
quasi-three dimensional numerical analysis was
conducted about the flow and storage in the reser-

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Figure 14. Depth averaged velocity and increment water-level per unit time.

Figure 15. Longitudinal distribution of reservoir storage volume per unit time and per unit distance.

voir during the large-scale flood. Figure 14 shows
the increase in water-level elevations per unit timeat the peak discharge. The peak discharge of largescale flood was extended to 2,200 m3/s (discharge
probability about 1/200). At the time of peakdischarge, horizontal eddies were created around3.5 km to 4.0 km points. Then, the local water levelchanges in this area.
Figure 15 shows the longitudinal distribution ofreservoir storage volume per unit time and unit distance at the time of peak discharge. Similar figures
can be drawn in each time. Figure 16 shows the rateof storage volume during large-scale flood flow. The
reservoir storage volume estimated by the water level
hydrograph at the dam front is compared with one
calculated by temporal water surface elevations andvelocity in each analysis mesh size. The storage volume estimated by assuming horizontal rise of waterlevel used in current dam reservoir management islarger than the storage volume considering local
depth and flow velocity distributions during floods.
In the downstream end of the calm area, the water
surface elevation nearly equals to the energy head,because the outflow discharges is small enough andthe kinematic energy is sufficiently small. Therefore,
the water level at this point is regarded as the highest
within the reservoir. Therefore, the storage volume
estimated by the present method may be overes-

timated a little. It can be interpreted as giving theclearance to the capacity of reservoir storages in thepresent dam reservoir management during a flood.
It suggests that the inflow discharges may be estimated larger than actual one into dam reservoirs.
To obtain accurate inflow discharge hydrographs
and understand three-dimensional flow dynamics in dam reservoirs are essential in order to perform efficient and safe dam reservoir management
against the large-scale flood and to mitigate sedimentation problems in reservoirs. It is also important to establish the flood observation systems in
dam reservoir and upstream and downstream rivers during floods in order to assess properly theflow dynamics by dam reservoir flow analysis.

Figure 16. Rate of storage volume during large-scale
flood flow.

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6

CONCLUSIONSIn this paper, we developed the dam reservoir flow
analysis model and checked the validity of thismodel by applying to large-scale hydraulic model
test. Main conclusions are drawn as follows.
1. The hydraulic model test were carried out andthe flood propagation and flow mechanism inthe dam reservoir were investigated in detail.
2. The dam reservoir flow analysis model wasdeveloped considering observed temporalchanges in water surface profiles. We applied
this model to the hydraulic model test and clarified the mechanism of flood propagation andflow dynamics in dam reservoir.
3. This paper proved that to obtain inflow discharge hydrographs with a high accuracy and toclarify flow dynamics in dam reservoirs as wellas understanding of the occurrence mechanismof the flood retarding volume were essential forthe reliable dam reservoir management.
4. The developed model presents detailed velocity
distributions near reservoir beds. This model
gives a useful tool for mitigating reservoir sedimentation problems.
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