ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-8

Evaluation Method of Positive and Negative Retarding Storage
Volumes for Unsteady Two Dimensional Flows and Propagation
Mechanisms of Peak Discharge and Peak Water Level
Y. Takemura & S. Fukuoka

Research and development initiative, Chuo University, Japan

ABSTRACT: Understanding of the storage effects of river channels, such as attenuation of peak discharge, delay in propagation and difference between arrival times of peak discharge and peak water levelat river cross sections during flood events, has been important for river management and improvement.
This paper defines the parts of flood water stored in river channels during the rising period and flood water released from river channels during the descending period that cause deformation of discharge hydrographs as positive retarding storage and negative retarding storage of flood water, respectively. The evaluation method of the positive and negative retarding storage volumes in river channels is proposed byderiving the advective equation for flood discharge from the unsteady two dimensional flow equations. In
addition, the mechanism causing the difference between arrival times of peak discharge and peak waterlevel at river cross sections is discussed with relation to the positive retarding storage around the flood
peak time. Finally, the evaluation method is applied to a large flood in the valley of the upper KitakamiRiver to clarify the storage effects of the valley.
Keywords: Flood propagation, Storage, Retarding storage, River geometries, Discharge hydrograph,
Water level hydrograph, Valley
1 INTRODUCTION
When floods occur in river reaches, channel areas are temporally filled with flood water and they are
gradually released to downstream. Discharge hydrographs of flood flows passing through river channels
change in shapes through this process and it causes attenuation of peak discharge, delay in propagationand difference between arrival times of the peak discharge and the peak water level at river cross sections(Takahashi (1971) and Menedez et.al. (1982)). They are known as the storage effects of river channels
and largely affected by unsteadiness of flood flows and channel geometries such as cross sectional shapes
or its longitudinal changes (Henderson (1966) & Fukuoka (2003)).
Understanding of the storage effects of river channels is a great interest of river engineers or researchers and many fundamental studies are proposed (e.g. Lighthill and Whitham (1955), Ponce and Simons(1977)). However, applications of these studies are limited to the specific problems in rivers because themost of studies are proposed under the ideal conditions (e.g. uniform and quasi-steady flow condition or
linear stability analysis). Now, flood propagations in river channels can be analyzed with high accuracy
for the development of the observation technique and numerical analysis method (e.g. Fukuoka and
Watanabe (2004)). How to evaluate the storage effects of river channels during flood events by using thenumerical analysis results becomes an important problem. Mishra et.al. (1997) adopts the area of the loopof the rating curve (discharge - water level curve) in non-dimensional form as the parameter to explain the
propagation characteristics of flood flows passing through river reaches in their study. However, the rating curve cannot represent the amount of the deformation of discharge hydrographs in the river channels
which is the dominant factor to evaluate the storage effects of the river channels quantitatively because it
represents only the discharge - water level relationship at each river cross section during flood events.
Figure 1 shows the discharge hydrographs at upstream and downstream ends of a river reach. The discharge hydrograph deforms from the black broken line to the gray solid line in the reach. Areas surround303

ed by the discharge hydrographs of black solid line and gray solid line (colored areas and hatched areas in
Figure 1) are equal to storage volume of flood water in the river reach during the rising period and release
volume of flood water from the river reach during the descending period, respectively. Takemura and Fukuoka (2012, 2014) characterized the parts of the storage and release volume that cause the deformationof the discharge hydrographs (hatched areas in Figure 1) as positive and negative retarding storage volumes to evaluate the deformation of discharge hydrographs in river channels qualitatively. In addition,
reasonable method to evaluate the positive and negative retarding storage volumes is proposed based on
the advective equation for flood discharge derived from the unsteady one dimensional flow equations.
However, the proposed method was not enough to evaluate the amount of the deformation of dischargehydrographs accurately.
This paper derives the advective equation for flood discharge from the unsteady two dimensional flow
equations and propose the more precise evaluation method of the positive and negative retarding storage
volumes. In addition, the cause of the occurrence of the difference between arrival times of peak discharge and peak water level at river cross sections is discussed with relation to the positive retarding storage around the flood peak time. Finally, the evaluation method is applied to the results of the unsteady 2D flow analysis for a large flood in the valley of the upper Kitakami River, Japan.
Qou
Discharge

Qin
Rsr: Positive retarding storage volumeof flood water
Rising period

Rsd: Negative retarding storage
volume of flood water
Qout (unchanged in shape)++:Storage volume of flood water:Release volume of flood water:Positive retarding storage volume of flood water:Negative retarding storage volume of flood water

Descending period

Time
Figure 1. Relationship between deformation of the discharge hydrograph and positive and negative retarding storage
volumes in a river reach.

2 ADVECTIVE EQUATION FOR DISCHARGE AND EVALUATION METHOD OF POSITIVE
AND NEGATIVE RETARDING STORAGE VOLUMES FOR UNSTEADY TWO DIMENTIONAL
FLOWS
2.1 Derivation of the advective equation for discharge
The advective equation for discharge discussed in this paper is derived from the continuity equation for
two dimensional flows∂h ∂uh ∂vh++=0      
∂t ∂x
∂y

(1)

where h = water depth, u and v = depth average velocity components of x and y directions. Since a flow
direction in river channels is usually defined at right angle to river cross sections, difference of the flow
direction between river cross sections should be taken into account for the derivation of the advective
equation for discharge. However, the flow direction is assumed to be unchanged at each river cross section and equal to x direction in this section to understand the physical meanings of the equation moreclearly.
Multiplying Equation (1) by u and adding v∂uh/∂y to the both sides of the equation, we obtain
Duh  ∂uh
∂uh
∂uh 
∂u
∂v
∂u
+u
+v
=
 =h − uh + vh               Dt  ∂t
∂x
∂y 
∂t
∂y
∂y

(2)

The left side of Equation (2) is substantial derivative respect to unit discharge uh and the three terms on
the right side of the equation represent the effects of temporal changes in u, changes in streamline spacing
and lateral advection of flood water on the rate of change in uh, respectively. Integrating Equation (2)with respect to a river width by using the Leibniz integral rule and kinematic boundary conditions at water edges, we obtain the advective equation for discharge, as304

DQ  ∂Q
∂Q 
+U
=
=
Dt  ∂t∂x −∂y( u − U ) uh
∂x

+

yr

∫h
yl

( u − U ) ∂uh dy∂u∂u+ vh − ( u − U )
∂t
∂y
∂x

∂y
( u − U ) uh
∂x

(3)

l
r
y y=
y y1
=
2

where
Q
=

yr

yr

yl

yl

Q

hdy, U    =, A ∫=
∫uhdy
A

(4)

Q = discharge, A = cross sectional area, U= mean velocity, yl and yr = positions of water edges at rive
cross sections. As the result, second term of the right side of Equation (2) is canceled out and third - fifthterms are appeared on the right side of Equation (3). These three terms represent the effects of velocity
distributions in river cross sections on the rate of change in Q. Equation (3) can be written in the more
simplified form
DQ  ∂Q
∂Q 
∂U
A
     
 = +U=
Dt  ∂t
∂x 
∂t

(5)

The rate of change in Q can be represented by only the product of partial derivative ∂U/∂t and flow crosssectional area A when we follow the mean motion of a fluid. This indicates that deformation of dischargehydrographs in river channels is controlled by temporal changes in mean velocity at river cross sections.
2.2 Evaluation method of the positive and negative retarding storage volumes
Rate of storage in a river reach is represented by the difference between inflow and outflow discharge ofthe reach or
∂A
dS
=∫
dx
L
∂t
dt

(6)

where S = storage volume in the river reach, L = longitudinal distance of the river reach. Differentiating
discharge Q (=UA) respect to time, ∂A/∂t on the right side of Equation (6) can be written∂A 1 ∂Q A ∂U=−
  ∂t U ∂t U ∂t(7)

It can be easily understood with relation to Equation (5) that the second term of the right side of Equation(7) (A/U)∂U/∂t represents the rate of storage per unit distance due to the deformation of discharge hydrographs. To assign a meaning of the first term (1/U)∂Q/∂t, we think the case that mean velocity does not
temporally change throughout river reaches; i.e., discharge hydrographs are unchanged in shape as it
moves downstream since the right side of Equation (5) A∂U/∂t is 0. In this case, the difference of arrival
time of the discharge hydrograph between two river cross sections is only the factor causing storage inriver channels. Then rate of storage per unit distance ∂A/∂t is equal to (1/U)∂Q/∂t since the second term ofEquation (7) (A/U)∂U/∂t is 0. It allows us that the first term of the right side of Equation (7) (1/U)∂Q/∂t
can be taken as the rate of storage per unit distance due to the difference of the arrival time of the discharge hydrograph.
From the above discussions, it can be understood that the positive and negative storage volumes defined as the parts of the storage and release volumes in the river reaches that cause the deformation of thedischarge hydrographs (hatched areas in Figure 1) are produced by the second term of the right side ofEquation (7). The rate of the positive and negative retarding storage volumes per unit distance are writtenas  ∂U
1   ∂t ≤ 0 
∂rsr
A ∂U
 

=
−δ1
, δ1 =
at the rising period

∂t
U ∂t
∂
U


0
> 0


  ∂t305(8)

Discharge

Qx-dx/2:Discharge hydrograph at x-dx/2 section, Qx+dx/2:Discharge hydrograph at x+dx/2 section, Q’x+dx/2:Discharge hydrograph at x+dx/2 section (unchanged in the shape), TpQ: Arrival time of the peak discharge, Tph: Arrival time of the peak water levelaQx − dx / 2

∂Q
=0
∂t

c
b

∂A  ∂Q 
=0=∂t  ∂x 

1 ∂Q
dx
U ∂t
1 ∂Q
A ∂U
Qx + dx / 2 = Qx − dx / 2 −
dx +
dx
U ∂t
U ∂t

Qx′ + dx / 2 = Qx − dx / 2 −

∂A
dx
∂rsr
∂t
dx
∂t
Difference between arrival times of the peak discharge
and the peak water level at the x section

Rising period++:Positive retarding storage
volume of flood water:Negative retarding storage
volume of flood water

Descending period

TpQ

Tph

:Storage volume of flood water:Release volume of flood water

Time

Figure 2. Discharge hydrographs around the flood peak time at river cross sections infinitesimal space dx away from x section
in upstream and downstream directions.

  ∂U
1   ∂t ≥ 0 
∂rsd
A ∂U
 

=
−δ2
, δ2 =
at the descending period

∂t
U ∂t
 0  ∂U < 0   ∂t

(9)where rsr = positive retarding storage volume per unit distance, rsd = negative retarding storage per unit
distance, δ1 and δ2 = response functions.
Figure 2 shows the discharge hydrographs around the flood peak time at river cross sections infinitesimal space dx away from x section in upstream and downstream directions. The times TpQ and Tph are thearrival times of peak discharge and peak water level at x section, respectively. The rate of storage and therate of positive retarding storage in the space dx are represented by ∂A/∂tdx, ∂rsr/∂tdx, respectively. Considering the right side of Equation (7), the first term is less than 0 (i.e., (1/U)∂Q/∂t < 0) and second term isequal to the rate of positive retarding storage (i.e., -(A/U)∂U/∂t = ∂rsr/∂t) from the time TpQ to the time Tph.
Therefore, the rate of positive retarding storage is larger than the rate of storage (i.e., ∂rsr/∂tdx > ∂A/∂tdx)
in this time as shown in Figure 2. This indicates that the volume of the positive retarding storage obtainedby integrating Equation (8) respect to time during the rising period overestimates the volume surroundedby the points a, b and c in Figure 2. The overestimated volume can be eliminated by using Equation (10)instead of Equation (8).
  ∂U
1   ∂t ≤ 0 
∂rsr
 

 A ∂U ∂A =
δ1min  −,  , δ1 =∂t
 U ∂t ∂t 
 0  ∂U > 0   ∂t

(10)Generally, it can be neglected because the overestimated volume would be considered sufficiently smaller
than the total volume of the positive retarding storage in river reaches. However, Equation (10) should be
used to discuss the positive retarding storage around the flood peak time.
Integrating Equation (8) (or Equation (10)) and Equation (9) respect to the length of river reaches L,
we obtain the rate of the positive retarding storage and negative retarding storage in the river reaches∂rdRsr= ∫ sr dx
L
∂t
dt

(11)

∂r
dRsd
= ∫ sd dx
L
∂t
dt

(12)

where Rsr = positive retarding storage volume in river reaches, Rsd = negative retarding storage volume inriver reaches.

306

2.3 Consideration about the occurrence of the difference between arrival times of the peak discharge
and the peak water level
Unlike the kinematic wave (Lighthill and Whitham (1955)), there is a difference between arrival times ofthe peak discharge and the peak water level at river cross sections as shown in Figure 2 due to the deformation of a discharge hydrograph. In Figure 2, the peak discharge occurs at x section at the time TpQ eventhough the greater discharge still exist upstream of x section (since ∂Q/∂x < 0) since the greater discharge
will attenuate to less than the peak discharge when they reaches x section. The peak water level occurs at
x section when the greater discharge no longer exists upstream of x section (i.e., ∂Q/∂x = 0). It is the causethat occurrence of the peak water level is late for the occurrence of the peak discharge at river cross sections. In other word, the attenuation of discharge hydrographs after the occurrence of the peak discharge
postpones the occurrence of the peak water level by increasing the storage volume in river cross sections.
This implies that the more the volume of the positive retarding storage between the times TpQ and Tph increases, the more the difference between the arrival times of the peak discharge and the peak water level
increases at river cross sections.
3 STORAGE EFFECTS OF THE VALLEY OF THE UPPER KITAKAMI RIVER
3.1 Study area and method
The valley of the Upper Kitakami River is located on the border of Iwate and Miyagi prefectures, Japan(see Figure 3). Figure 4 shows the air photograph of the valley. The channel width is about 150m in average and the bed elevation changes only about 10 meters in the channel reach of about 80km from the uppermost of the valley to the river mouth. Since the capacity of the valley is not enough to convey the large
amount of flood water downstream at a time, the flood water is temporally stored on large flood plain
along the channel seen at the downstream of the junction of the Satestu River (see Figure 4) when large
flood occurs in the valley. It may decrease the flood discharge and delay flood propagation times to thedownstream. In order to make clear the storage effects of the valley, we have investigated flood flows
passing through the river reach from Suwamae (67.6 km) to Ooizumi (48.9) shown in Figure 4 by using
the unsteady 2-D flow analysis with the general curvilinear coordinates for the 2009 large flood (Takemura and Fukuoka, 2012).
This paper investigates temporal and special distributions of the positive and negative retarding storage
volumes from the calculation results of the unsteady 2-D flow analysis and by rewriting the Equation (8)(12) for the general curvilinear coordinates to clarify the storage effects of the valley of the upper
Kitakami River.

Shijyushida dam
75km：Observation points

Iwate

Miyagi:Kitakami River basin: Study area

Ohizumi(48.9km)(Drainage area: 7800 km2)
55km

Japan Sea

Suwamae(67.6km)(Drainage area: 7509 km2)
70km
60km

65km

Satetsu River(Drainage area:375km2)

50km

Figure 3. Locations of the valley of the upper
Kitakami River.

Figure 4. Air photograph of the valley of the upper KitakamiRiver.

307

3.2 Temporal and special distributions of the positive and negative retarding storage volumes in thevalley of the upper Kitakami River
Figure 5 shows the calculated discharge hydrographs at the Suwamae (67.6 km) and Ooizumi (48.9) during the 2009 flood. Figure 6 shows rate of storage and rate of the positive and negative retarding storage
in the river reach from the Suwamae (67.6 km) to the Ooizumi (48.9). The hatched areas in Figure 6 areequal to the volumes of the positive and negative retarding storage in the river reach, respectively. The
rate of storage has a maximum value in the early time of the rising period and gradually decreases toward
the flood peak. On the other hand, the rate of the positive and negative retarding storage have significant
values after the flood discharge rises about 3000m3/s. Flood water begins to fill the flood plain areas
when the flood discharge rises about 3000～4000m3/s in the river reach. Figure 7 shows temporal changes
in mean velocity at 63.8 km section. The mean velocity decreases in the rising period when the water level is above the flood plain level due to the increase of flow resistance as the result of momentum exchanges between high velocity flow in the channel and slow velocity flow in the flood plain and the mean
velocity increases in the descending period as the flood water stored on the flood plain returns into thechannel (Fukuoka et.al. (2003)). These mechanisms produce the positive and negative retarding storage inthe river reach. The total volume of the positive retarding storage in the river reach is about 2,500,000 m3.
It is almost equal to the flood control volume of the Shijyushida dam reservoir locating on the uppermostof the Kitakami River (see Figure 3) during the 2009 flood.
Figure 8 shows the longitudinal distribution of the positive and negative retarding storage volumes perunit distance in the river reach. They have relatively high values in the upstream region of 61km section
because the flood plain level is relatively lower than the downstream region. The positive and negative retarding storage volumes per unit distance becomes maximum around 62km section where the curvature of
the river channel is extremely large and flood plain exists along the outer bank of the river channel (seeFigure 4).

Discharge(m3/s)

5000

67km

4000

49km

3000
2000
1000
0
0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

Time (hour)

Rate of storage volume (m3/s),
Rate of positive and negative
retarding storage volumes(m3/s)

Figure 5. Calculated discharge hydrographs at the Suwamae (67.6 km) and Ooizumi (48.9) for 2009 flood.

++:Storage volume of flood water:Release volume of flood water:Positive retarding storage
volume of flood water:Negative retarding storage
volume of flood water

Time (hour)

Figure 6. Rate of storage volume in the channel reach from the Suwamae (67.6 km) to the Ooizumi (48.9), and rate of positive
and negative retarding storage volumes in the channel reach.

308

0.8

U: mean velocity
U0: mean velocity before theflood
Umax: maximum mean velocity

0.6
0.4
0.2

Elevation (T.P.m)(U-U0)/(Umax-U0)

10

Flood plain

20

River channelwater level15

Bed elevation

10

100m

5
0
0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Time(hour)

Positive and negative retarding storage volumes
per unit distance (m3/m)

Figure 7. Mean velocity hydrograph in non-dimensional form, water level hydrograph and crosssectional shape at 63.8km section.

:Positive retarding storage
volume of flood water:Negative retarding storage
volume of flood waterLongitudinal distance (km)Figure 8. Longitudinal distribution of positive and negative retarding storage volumes per unit distance.

3.3 Propagation characteristics of the peak discharge and the peak water level in the valley of the upper
Kitakami River
Figure 9 shows the longitudinal distributions of the arrival time of the peak discharge TpQ and arrival timeof the peak water level Tph in the river reach from 67km section to 49km section in Figure 4. The verticalaxis shows the arrival time and it has 0 value when the peak discharge occurs at 67km section. Brokenlines in Figure 9 indicate the arrival times of the peak discharge and peak water level when they moves
downstream with the propagation speed of the kinematic wave (C=5/3U). As shown in Figure 4, the arrival time Tph is late for the arrival time TpQ at each river cross section and their difference gradually increases toward downstream. Finally, the difference between the arrival time Tph and the arrival time TpQ
reaches for two and a half hours at 49km section. Figure 10 shows the longitudinal distribution of the positive retarding storage volume from the arrival time TpQ to the arrival time Tph per unit distance. It increases toward downstream even though the total volume of the positive retarding storage per unit distance shown in Figure 8 has smaller values in downstream region of the river reach compared with that in
the upstream region. It means that the amount of the deformation of the discharge hydrograph around theflood peak time increases toward downstream and it postpones the occurrence of the peak water level at
each river cross section in the river reach. In addition, longitudinal distribution of the arrival time of peakwater level Tph significantly deviates from that of the Kinematic wave Tph(KW) compared with the peakdischarge. The reason is considered below. Since the flood flows has large amount of momentum in flow
direction, flood discharge cannot change in short distance. On the other hand, the water level changes easily under the influence of flow conditions or channel geometries compared with the discharge (Takemera
and Fukuoka (2012)).

309

Arrival time (hour)

5Peak discharge
Peak water level
Peak discharge(KW)
Peak water level(KW)

Tph

4
3

Tph(KW)

2

TpQ(KW)TpQ

1
049 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

Positive retarding storage
volume from TpQ to Tph
per distance(m3/m)

Longitudinal distance (km)Figure 9. Longitudinal distribution of arrival times of peak discharge and peak water level.

:Positive retarding storage
volume of flood waterLongitudinal distance (km)Figure 10. Longitudinal distribution of positive retarding storage volume from TpQ to Tph per unit distance.

4 CONCLUSIONS
1.

The parts of flood water stored in river channels during the rising period and flood water released
from river channels during the descending period that cause deformation of discharge hydrographs
passing through river channels were defined as positive retarding storage and negative retarding storage of flood water, respectively, to evaluate the storage effects of river reaches with relation to thechannels geometries. Evaluation method of the positive and negative retarding storage volumes were
proposed by deriving the advective equation for flood discharge from the unsteady two dimensional
flow equations. In addition, the mechanism causing the difference between arrival times of the peakdischarge and the peak water level were considered with relation to the positive retarding storagearound the flood peak time.
2. Storage effects of the valley of the upper Kitakami River were evaluated quantitatively by investigating the temporal and spacial distributions of positive and negative retarding storages in the river
reach from the results of the unsteady 2-D flow analysis for the 2009 large flood. The followings
were clarified in this study.
(a) The positive retarding storage volume in the river reach from Suwamae(67.6) to Ohizumi(49.8km) is
about 2,500,00m3 during the rising period of the 2009 flood and large flood-plains existing along thechannels largely contribute to them.
(b) The arrival time of peak water level is late for the arrival time of peak discharge at each river crosssection and their difference increases toward downstream in the river reach.
(c) The greater amount of the deformation of discharge hydrograph around the flood peak time is, thegreater the difference between the arrival times of the peak discharge and the peak water level is.
REFERENCES
Fukuoka, S., Watanabe, A., Seki, K., Kurisu, D. and Tokioka, K. (2003): Assessment and Mechanism of Flood Flow Storage
in Channels, Proceedings of the Japan Society of Civil Engineers, JSCE, 740: II-64, pp. 31-44.
Fukuoka, S., Watanabe, A., Hara, T. and Akiyama, M. (2004): Highly accurate estimation of hydrograph of flood discharge
and water storage in rivers by using an unsteady two-dimensional flow analysis based on temporal change in observed water surface profiles, Proceedings of the Japan Society of Civil Engineers, JSCE, No.761: II -67, pp. 45-56.
Henderson, F. M. (1966): Open channel flow, Chapter 9, Flood Routing, Macmillan.

310

Lighthill, M. J. and Whitham, G. B. (1955): On kinematic waves. I. Flood movement in long rivers”. Proceedings of the Royal
Society A. pp. 281-316.
Menedez, A. N., and Norscini, R. (1982): Spectrum of shallow water waves: An analysis, J. Hydraul. Div. Am. Soc. Civ. Eng,
108(HY1), pp. 75-94.
Mishra, S. K. and Seth, S. M. (1997): Use of hysteresis for defining the nature of flood wave propagation in natural channels,
Hydrological. Sci., 41:2, pp. 153-170.
Ponce, V. M. and Simons, D. B. (1977): Shallow wave propagation in open channel flow, J. Hydraul. Div. Am. Soc. Civ. Eng,
103(HY12), pp. 1461-1478.
Takahashi, T. (1971): Study on propagation characteristics of flood flows in river channels, thesis, Kyoto University, Kyoto,
Japan.
Takemura Y. and Fukuoka S. (2012): Effects of river morphology on propagation and transformation of discharge and waterlevel hydrographs of flood flows through river valley, Journal of Japan Society of Civil Engineers, Ser. B1 (Hydraulic Engineering) Vol. 68, No. 1, pp. 35-54.
Takemura Y. and Fukuoka S. (2014): Effects of channel shape on propagation characteristics of flood flows through a valley,
Journal of Flood Risk Management, Volume 7, Issue2, pp. 152-158.

311