Proceedings of the 22nd IAHR-APD Congress 2020, Sapporo, JapanNON-EQUILIBRIUM BED VARIATION ANALYSIS IN STONY-BED RIVERSCONSIDERING CHANGES IN THE BED SURFACE STRUCTURE: APPLICATION TO
ENLARGEMENT MECHANISM OF THE MONOBE RIVER MOUTH DUE TO FLOODSYOSHIHARU TAKEMURA
Research and Development Initiative, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo, Japan, E-mail: takemura@tamacc.chuou.ac.jp
SHOJI FUKUOKA
Research and Development Initiative, Chuo University, Kasuga 1-13-27, Bunkyo-ku, Tokyo, Japan, E-mail: sfuku@tamacc.chuou.ac.jp

ABSTRACTThe bar at the Monobe River mouth composed of gravel particles and cobbles narrows the width at the outlet tothe sea. The second-largest flood occurred in July 2018 greatly enlarged the opening width of the river mouthand formed a spit-like depositional landforms at the coastal area. The sediment transport of gravel particles andcobbles are essentially non-equilibrium phenomena due to the influence of the unevenness of bed surface
configurations and the local bed gradients. This paper proposes an analysis method that can demonstrate threedimensional flood currents and non-equilibrium motion of gravel particles and cobbles by improving the bed
variation analysis method for stony-bed rivers and combining it with the Q3D-FEBS. Moreover, it is shown thatthe proposed analysis method can explain the enlargement mechanism of the river mouth opening and theformation process of the depositional landforms at the coastal area than the conventional analysis method basedon the equilibrium bed load formula.
Keywords: Flood flows, river mouth, Monobe River, Q3D-FEBS, non-equilibrium sediment transport

1.

INTRODUCTION

Bars formed at a river mouth increase the risk of flood inundation and bank erosion by narrowing the passage
of flood flows. River mouth bars are also related to the various problems such as intrusion of saltwater andwaves into river channels, sediment supply in coastal zones, and fish migration. Therefore, it is necessary to
explore the enlargement mechanism of a river mouth opening and the deposition process of the fluvial sedimentat coastal areas during floods.
The bar composed of gravel particles and cobbles in the range 2–100 mm is formed in the Monobe River mouth.
The gravel bar narrows the opening width of the river mouth as Figure 1 (a). The flood occurred in July 2018 is
the second-largest flood in the history of the observation of the Monobe River. The July 2018 flood greatly
enlarged the opening width of the river mouth and formed a spit-like depositional landforms at the coastal area
(Figure 1 (b)).

(b) 2018/7/18 10:52(a) 2018/6/13 10:47
Figure 1. Satellite images of the Monobe River mouth (source: EO Browser).
1

Kadota et al. (2008) numerically investigate the enlargement mechanism of the Monobe River mouth opening
by using the 3D flow model and the equilibrium bed load formula (Rijn, 1984). However, the Rijn’s formula
was proposed for sediment particles in the range 0.2-2.0 mm. The applicability to the Monobe river mouth isunknown. The transport mechanism of gravel particles and cobbles are essentially non-equilibrium phenomena
because the threshold of sediment motion and transport process are largely affected by the unevenness of bed
surface configurations (e.g., imbrication, cluster) and the local bed gradients.
Recently, we developed a new non-hydrostatic quasi three-dimensional model (Q3D-FEBS) that can calculate
three-dimensional velocities and pressure distributions due to the complex riverbed topography with highcomputational efficiency compared to the 3D flow model (Takemura and Fukuoka, 2019). This paper proposes
an analysis method that can demonstrate three-dimensional flood currents and non-equilibrium motion of gravel
particles and cobbles by improving the bed variation analysis method for stony-bed rivers proposed by Osadaand Fukuoka (2012) and combining it with the Q3D-FEBS. We analyze the enlargement mechanism of the rivermouth opening and the formation process of the depositional landforms at the coastal area for the July 2018flood in the Monobe River (see Figure 1(b)). Moreover, the importance of the non-equilibrium sediment motion
to explain these phenomena was discussed from the comparison with the analysis results using the conventional
bed variation analysis.
2.

NUMERICAL ANALYSIS METHOD

2.1 Q3D-FEBS and the bed variation analysis method for stony-bed rivers (Osada and Fukuoka 2012)
The definition sketch of the Q3D-FEBS is shown in Figure 2. We define a bottom surface 𝑧 slightly above amean bed surface 𝑧 . A vertical coordinate is given as 𝜂 = (𝑧 - 𝑧)/ℎ and approximates vertical distributions of
the horizontal velocities by the third-order polynomial. In addition to the depth-average flow velocities, flow
velocities at the free surface and bottom surface are solved by the equations of motion.
𝑢
𝑣

𝑧

⋅ 12𝜂

12𝜂

1

𝑢,

𝑢 ,

⋅

4𝜂

3𝜂

𝑈

𝑢 , : free surface velocities, 𝑢 , : bottom surface velocities
𝑈 : depth-average velocities, ℎ: water depth𝑧 : free surface, 𝑧 : bottom surface, 𝑧 : mean bed surface,𝛿𝑧 : distance between bottom surface and mean bed surface (= 0.03 ℎ)

𝑢
𝑧
𝑣

𝑧

𝑢,
𝑈
𝑧
𝑧
ℎ

𝜂

ℎ
𝛿𝑧

𝑢

Figure 2. Definition sketch of the Q3D-FEBS.

The bed variation analysis method for stony-bed rivers (Osada and Fukuoka, 2012) is performed by nonequilibrium sediment motion by the continuity equation of bed load at non-equilibrium condition and theequation of motion for a sediment particle as follows.
𝜕𝑞 ,
𝜕𝑥

𝜕𝑉
𝜕𝑡
𝑞

,

𝑃
𝑢

,

𝐷

(1)

∙𝑉

(2)

Where, 𝑖 1,2 𝑥
𝑥, 𝑥
𝑦 , 𝑉 : bed load volume of particles of size 𝑑 per unit area, 𝑞 , : bed load rateof particles of size 𝑑 per unit width in 𝑥 direction, 𝑢 , : mean velocity of particles of size 𝑑 in 𝑥 direction,
𝑃 : pick up rate of particles of size 𝑑 per unit area, 𝐷 : deposition rate of particles of size 𝑑 per unit area.
𝜌

𝜌𝐶

𝛼 𝑑

𝑑𝑢 ,
𝑑𝑡

𝜌 𝛼 𝑑 𝒈𝒙𝒊

𝛼 𝑑

𝜌𝑔

𝜕𝑧
𝜕𝑥

𝜕𝑝
𝜕𝑥

𝜌𝐶 𝛼 𝑑
𝑢
2

,

𝑢

,

(3)

Where, 𝑖 1,2,3 𝑥
𝑥, 𝑥
𝑦, 𝑥
𝑧 , 𝒈𝒙 𝒊0,0, 𝑔 , 𝑔: acceleration of gravity, 𝜌: density of water, 𝜌 :density of sediment, 𝑢 , : velocity of particles of size 𝑑 in 𝑥 direction, 𝑢 , :relative velocity between fluid
and particles of size 𝑑 in 𝑥 direction, 𝑝 : non-hydrostatic pressure at the bottom surface, 𝐶 : drag coefficient(= 0.4), 𝐶 : added mass coefficient (= 0.5), 𝛼 , 𝛼 :2D and 3D shape factor of particles (= 𝜋/4, 𝜋/6). The
pressure gradients in Eq. (3) are evaluated from the Q3D-FEBS. 𝑢 , shown in Eq. (2) are calculated from thesaltation analysis using Eq. (3). 𝑃 and 𝐷 are calculated by considering the height distribution of stationary
particles in a bed surface layer in order to incorporate the effects of the unevenness of bed surface configurations
in stony-bed rivers. The height distribution of the particles is calculated by the following manner. ➀: Calculate2the mean height of the centroid of particles of each size in a bed surface layer 𝑧̅ . ➁: Assume that the height ofthe centroid of particles of each size is normally distributed around 𝑧̅ as Figure 3.

Figure 3. Height distribution of each particle.

However, the calculation method of 𝑧̅ proposed by Osada and Fukuoka (2012) has a problem that the change
in volume fraction of each particle in a bed surface layer is neglected. The paper corrects this problem andproposes the new calculation method of 𝑧̅ . Moreover, based on this correction, the threshold for the sedimentmotion and the pickup rate of nonuniform sediment are reformulated.
2.2 Calculation method of the average height of each particle
We define the bed surface layer and subsurface layer as Figure 4.

(2) Coarse particles move actively.

(1) Coarse particles hardly move.

Figure 4. Definition sketch of a bed surface layer and a subsurface layer.

The subsurface layer is defined as a layer in which the porosity of sediment can be regarded as substantiallyconstant. The bed surface layer is defined as a layer above the subsurface layer in which the stationary particles
can exist on average. 𝑧 is the elevation of the interface of the bed surface layer and the subsurface layer. The
average height of the centroid of the stationary particle of size 𝑑 relative to 𝑧 in the bed surface layer can be
defined as
𝛿𝑧

1
𝑛

𝑧

𝑧

(4)

where, 𝑛 : number of the stationary particles of size 𝑑 in the bed surface layer, 𝑧 : height of the centroid ofparticles of size 𝑑 . Thus, the temporal changes in 𝛿 𝑧𝑘 can be written as
𝛿𝑧 𝜕𝑛
𝑛 𝜕𝑡

𝜕𝛿𝑧
𝜕𝑡

1
𝑛

𝜕𝑧
𝜕𝑡

𝜕𝑧
𝜕𝑡

(5)

The second term of the right-hand side of Eq. (5) can be written as
1
𝑛

𝜕𝑧
𝜕𝑡

𝛼
𝛼

𝑃

𝐷

(6)

𝑝

where, 𝑝 : volume fraction of the stationary particle of size 𝑑 in the bed surface layer. 𝑛 is expressed by using
the volume of the stationary particles of size 𝑑 in the bed surface layer 𝑉 and 𝑝 as 𝑉 ∙ 𝑝 / 𝛼 𝑑 . We canobtain Eq. (7) from Eq. (5) by assuming 𝑉 has a constant value.
𝜕𝛿𝑧𝜕𝑡

The temporal changes in 𝑧

𝛿𝑧 𝜕𝑝
𝑝 𝜕𝑡

𝛼
𝛼

𝑃

𝐷

𝑝

𝜕𝑧
𝜕𝑡

(7)

and 𝑝 are calculated by
3

𝛼
𝛼

𝜕𝑧
𝜕𝑡
1α
𝐿 α

𝜕𝑝
𝜕𝑡
𝜕𝑧
𝜕𝑡

𝑃

𝐷

0 𝑝 |

𝑝

𝑃

𝐷

(8)

𝑝 | 𝜕𝑧
𝐿
𝜕𝑡
𝜕𝑧
0 𝑝 |
𝜕𝑡

(9)
𝑝

where, 𝑘: number of different particle sizes, 𝑝 :volume fraction of particles of size 𝑑 in the subsurface
layer, 𝐿 : thickness of the bed surface layer. Substituting Eq. (8) and Eq. (9) into Eq. (7), the temporal changesin 𝛿 𝑧𝑘 is calculated by
𝜕𝛿𝑧
𝜕𝑡

1

𝛿𝑧
𝐿

𝛼
𝛼

𝑃

𝐷

1

𝑝

𝛿𝑧 𝑝 |
𝑝
𝐿

𝛼
𝛼

𝑃

𝐷

(10)

The first term of the right-hand side of Eq. (10) represents the effects of sediment exchange with the bed load
and the second term represent the effects of sediment exchange with the subsurface layer.
The height distributions of the centroid of particles of size 𝑑 are given as
𝑓 𝑧
𝜎

1
√2𝜋

𝑚𝑖𝑛 𝛿𝑧̅ , 0.5𝐿

𝑒𝑥𝑝

𝑧
2

𝑧

,𝑧

∙ 𝑚𝑖𝑛 0.435

𝑧̅

(11)

𝜎
𝑒𝑥𝑝

6

𝑑
𝑑

, 0.6

(12)

where, 𝑓 𝑧 : probability density function for the height of the centroid of particles of size 𝑑 , 𝑑 : mean
diameter of the stationary particles in the bed surface layer.
2.3 Threshold for the sediment motion and the pickup rate for nonuniform sediment
The pickup rate of particles of size 𝑑 in the bed surface layer is calculated by Eq. (14).
𝑃

𝜀

𝑝
𝛼 𝑑

𝛼 𝑑
𝑇

(14)

Where, 𝜀 : volume fraction of the stationary particles of size 𝑑 in the bed surface layer that start to move, 𝑇 :time required for the stationary particles of size 𝑑 in the bed surface layer to be entrained into the bed load.
The stationary particle at a height of 𝑧 = 2.2𝜎 (see Figure 3) are used for determining the threshold of sedimentmotion. Assume that if the fluid dynamic forces acting on the particle exceed the threshold, the particles existing
above 𝑧 = 2.2𝜎 are entrained into the bed load. Then the value of 𝜀 can be obtained as 0.014 from Eq. (11).
The fluid dynamic forces acting on the particle are assumed as Figure 5.

Figure 5. Fluid dynamic forces acting on a particle.

Driving the equation for rotation angle 𝜃 according to Nakagawa et al. (1991) as
𝑑 𝜃
𝑑𝑡
𝐹
𝐹

1
𝜀𝜌 𝐶 𝛼 𝑑 𝑢21𝜀𝜌 𝐶 𝛼 𝑑 𝑢
2

𝑊 𝑠𝑖𝑛𝜃

𝐹 𝑐𝑜𝑠𝜃
𝐹 𝑠𝑖𝑛𝜃
7𝜌𝑑𝜌𝐶 𝛼 𝑑
𝑑
10
𝜕𝑧
𝜕𝑝
𝜌𝑔
𝛼 𝑑 𝑐𝑜𝑠𝜃
𝜌𝑔
𝜕𝑠
𝜕𝑠
𝜕𝑝
𝜕𝑧
𝜌𝑔
𝛼 𝑑 𝑐𝑜𝑠𝜃
𝜌𝑔
𝜕𝑧
𝜕𝑠
𝑢
𝑧
𝑧
1
𝑙𝑜𝑔 30.2
𝑢∗ 𝜅
𝑑

(15)
𝜕𝑝
𝛼 𝑑 𝑠𝑖𝑛𝜃𝜕𝑧𝜕𝑝
𝛼 𝑑 𝑠𝑖𝑛𝜃
𝜕𝑠

(16)
(17)
(18)

where, 𝑊𝜌 𝛼 𝑑 𝑐𝑜𝑠𝜃, 𝑧 : height of particles at 𝑧 = 2.2𝜎 in the bed surface layer. 𝜀: shielding coefficient(= 0.4), 𝐶 : drag coefficient (= 0.4), 𝐶 : lift coefficient (= 0.4), 𝑑 : representative diameter of sediment particles
in the bed surface layer. The pressure gradients including Eq. (16) and Eq. (17) are evaluated from Q3D-FEBS.
4

𝑇

can be calculated by solving Eq. (15) numerically until the 𝜃 becomes 0 under the conditions that 𝜃𝜃 and 𝑑𝜃 /𝑑𝑡 0 at 𝑡 0. However, we use the theoretical solution for the simplicity and computational
efficiency. Eq. (19) can be obtained by assuming the fluid dynamic forces acting on the particle unchanged until
the particle entrained into the bed load.
2𝜃 ⁄𝐴

𝑇
𝐴

(19)

𝑊 𝑠𝑖𝑛𝜃

𝐹 𝑐𝑜𝑠𝜃
𝐹 𝑠𝑖𝑛𝜃
7𝜌𝑑𝜌𝐶 𝛼 𝑑
𝑑
10
0.32𝑑
0.5𝑑
cos
𝜃
0.5 𝑑
𝑑

(20)
(21)

The threshold for sediment motion of each particle size can be investigated by putting the left-hand side of theEq. (20) as 0 (𝐴 = 0). Figure 6 shows an example of the relationship between the dimensionless critical shear
stress 𝜏∗ (= 𝜌𝑢∗ /( 𝜌𝑠 𝜌 𝑔𝑑 )) and the particle size ratio 𝑑 /𝑑 , and 𝛿𝑧̅ (mean height of the particle ofsize 𝑑 relative to 𝑧 in the surface layer). 𝑑 is the mean diameter of the particles in the surface layer and 𝑢∗
is the bed shear velocity. The thickness of the surface layer 𝐿 is given by the representative diameter 𝑑 .
Therefore, 𝛿𝑧̅ /𝑑 (or 𝛿𝑧̅ /𝐿 ) has a value between 0 to 1 for each particle size. 𝜏∗ decreases as the values of𝑑 /𝑑 and 𝛿𝑧̅ /𝑑 increases, and vise vasa. In the stony bed river, the fine particles can be hidden at the lower
part of the bed surface layer in the situation that the coarse particles can stay on the upper part of the bed surface
layer (see Figure 4 (a)). Contrary, fine particles are exposed on the upper part of the bed surface layer and easily
entrained into the flows when coarse particles actively move (see Figure 4 (b)). Such mechanisms are importantto analyze the bed variations in the stony-bed rivers, but it will be difficult to explain from the conventional
methods in which 𝜏∗ has a specific value against the particle size ratio such as 𝑑 /𝑑 . Our method candetermine 𝜏∗ by considering the relative height of each particle in the bed surface layer in addition to theparticle size ratio as shown in Figure 6. Therefore, it is more suitable for the bed variation analysis in the stonybed rivers than the conventional methods.

𝐿 : thickness of the bed
surface layer𝑑particle of size 𝑑𝛿𝑧̅ 1.6
𝑑 1.4
1.2
1

mean height of the particleof size 𝑑𝛿𝑧̅ /𝑑𝑧

0~10.8: 𝑑 ⁄𝑑: 𝑑 ⁄𝑑: 𝑑 ⁄𝑑finecoarse

3
1
1/3

apper

0.6
0.4
0.2

lower

0
0.01

0.05

0.1

𝜏∗

1

10

Figure 6. Relationship between the dimensionless critical shear stress 𝜏∗ and the particle size ratio,
and the average height of particles relative to 𝑧 in the bed surface layer 𝛿𝑧̅ .

3.

ENLARGEMENT MECHANISM OF RIVER MOUTH BAR IN THE MONOBE RIVER DUE
TO THE JULY 2018 FLOOD

3.1 The July 2018 flood and the observation system of the studied reach
The Monobe River is in the Kochi Prefecture, Japan. Figure 7 shows the observation system of the studied reach.
Pressure type water level gauges have been installed at the point of ●.

Studied reachTosa Bay

Figure 7. Observation system at the studied reach.

Figure 8 shows the observed discharge hydrograph at Hukabuchi (3.65km), sea level and significant wave heightat the Tosa Bay during the July 2018 flood. The opening width of the river mouth was greatly enlarged, and aspit-like depositional landforms were formed in the coastal area during the flood as shown in Figure 1. The
flood water surface profiles were observed by the water level gauges installed at the location shown in Figure
7. Besides, bed topography around the river mouth was measured just after the flood (see Figure 8).
5

5000

8

Discharge (m3/s)

4000
3500
3000

Satellite image of the river mouth
was obtained (Figure 1 (b)).

2000

6
5
4

Bed topography around theriver mouth was measured.

2500

7

3
2

1500

1

1000

0

500

-1

0

-2

Sea level (T.P.m)
Significant wave height(m)Discharge(Fukukabuchi 3.65k)
Significant wave height
Sea level4500

7/19 0:00

7/18 0:00

7/17 0:00

7/16 0:00

7/15 0:00

7/14 0:00

7/13 0:00

7/12 0:00

7/11 0:00

7/10 0:00

7/9 0:00

7/8 0:00

7/7 0:00

7/6 0:00

7/5 0:00

7/4 0:00

7/3 0:00

7/2 0:00

Figure 8. Observed discharge hydrograph at Hukabuchi (3.65km), sea level and significant wave heightat the Tosa Bay during the flood in July 2018.

3.2 Computational conditions

Volume percentage finer than（%）

The boundary conditions at the upstream and downstream ends are given by the observed water level
hydrographs at Fukabuchi (3.65 km) and the sea level hydrograph, respectively. Figure 9 shows the particle size
distributions of sediment material used for the bed variation analysis. The maximum diameter of the sedimentparticle is 150 mm. The particle size distributions of the river channel and the gravel bar at the river mouth are
determined based on the measurement, respectively. We determine the 𝑑 as the representative diameter 𝑑and the thickness of the bed surface layer 𝐿 . 𝜀 shown in Eq. (14) is determined as 0.018. Further study isneeded for the value of 𝜀 .
100
90
80
70
60
50
40
30
20
10
0

River channel
River mouth gravel bar
1

10

100

1000

Sediment diameter（mm）

Figure 9. Particle size distributions of sediment material used for the bed variation analysis.

3.3 Enlargement mechanism and deposition process at the coastal area during the July 2018 flood
Figure 10 shows the comparison of the observed and the calculated water surface profiles in the rising periodof the July 2018 flood.
10

Elevation (T.P.m)

8
6
4
2
0
-2
-0.4

-0.2

0

0.2

0.4

0.6
0.8
1
Longitudinal distance (km)

1.2

1.4

1.6

1.8

Average bed level (before flood)
Flood mark elavation (right bank)
Flood mark elevation (left bank)Calculated water level 7/4 19:00 (right bank)Calculated water level 7/5 19:00 (right bank)Calculated water level 7/6 1:00 (right bank)Calculated water level 7/6 7:00 (right bank)Calculated water level 7/4 19:00 (left bank)Calculated water level 7/5 19:00 (left bank)Calculated water level 7/6 1:00 (left bank)Calculated water level 7/6 7:00 (left bank)Observed water level 7/4 19:00 (right bank)Observed water level 7/5 19:00 (right bank)Observed water level 7/6 1:00 (right bank)Observed water level 7/6 7:00 (right bank)Observed water level 7/4 19:00 (left bank)Observed water level 7/5 19:00 (left bank)Observed water level 7/6 1:00 (left bank)Observed water level 7/6 7:00 (left bank)Figure 10. Comparison of observed and calculated water surface profiles in the rising period of the July 2018 flood.

The calculated water surface profiles can almost explain the observed water surface profiles and flood markprofile. The difference of the water levels between the left bank and right bank becomes extremely large due tothe flood flow concentrated near the right bank around the river mouth. Figure 11 shows the comparison of theobserved and the calculated discharge hydrographs at the Fukabuchi (3.65km) in the July 2018 flood. The
calculation results show good agreement with the observation results.
Figure 12 shows the comparison of the observed and calculated bed topography around the river mouth after
the July 2018 flood. The calculation results are shown in two cases. Figure 12 (c) is the result using the analysis
method proposed by this paper (proposed model) and Figure 12 (d) is the result using the Ashida-Michiue
formula (1972) that is the equilibrium bed load formula commonly used in Japan (conventional model). As
shown in Figure 12 (a) and Figure 12 (b), the opening width of the river mouth at -0.2km point is enlarged about
480m and a spit-like depositional landforms are formed. The proposed model (Figure 12 (c)) shows that the
opening width is slightly smaller and the deposition height of the fluvial sediments at the coastal area is lower
than the observation results (Figure 12 (b)). However, the proposed model shows the promising result in the
opening width of the river mouth and the depositional landforms at the coastal area than the conventional model.
6

Total amount of the sediment discharged from the river mouth to the coastal area estimated from the proposedmodel is 58764m3.
5000

Discharge (m3/s)

4000

Observed discharge
Calculated discharge

3000
2000
1000
0
7/4 0:00 7/4 12:00 7/5 0:00 7/5 12:00 7/6 0:00 7/6 12:00 7/7 0:00 7/7 12:00 7/8 0:00 7/8 12:00 7/9 0:00

Figure 11. Comparison of observed and calculated discharge hydrograph at the Fukabuchi (3.65 km)
in the July 2018 flood.

Opening width (about 480m)(a) Before the July 2018 flood.

Opening width of Figure 12 (b)(b) After the July 2018 flood (observation).

Opening width of Figure 12 (b)(c) After the July 2018 flood(d) After the July 2018 flood(calculation of the proposed model).
(calculation of the conventional model)
Figure 12. Comparison of the observed and calculated bed topography around the river mouth after
the July 2018 flood.

In the followings, we investigate the flood flows and sediment transport mechanisms around the river mouth
based on the calculation results of the proposed model. Figure 13 shows the water surface velocities and bottom
surface velocities around the river mouth opening and the velocity distributions in -1.0km and -2.0km crosssections at the peak of the July 2018 flood. The flood flows passing through the river mouth opening are greatly
accelerating in the longitudinal direction and high velocity zones are found near the gravel bar as seen in Figure
13 (b). In addition, a strong secondary current is formed at upstream of the gravel bar.
Figure 14 shows the trajectories of the sediment particles of 100 mm and 2mm and the longitudinal distributionsof the bed load around the river mouth opening at the peak of the July 2018 flood. As shown in Figure 14 (a),the motion of the coarse particles (100mm) is affected by local riverbed gradients compared to the fine particles(2mm). The motion of the coarse particles on the gravel bar (see circle Ⅰ in Figure 14 (a)) accelerates theenlargement of the opening width of the river mouth. On the other hand, the coarse particle cannot easily crossthe sediment deposited at the coastal area (see circle Ⅱ in Figure 14 (a)). This is considered to be one of thereasons that the spit-like depositional landforms were formed at the coastal area (Figure 12 (b)). As shown in
Figure 14 (b) and (c), the bed load increases sharply from the downstream of 0.0km point due to the influenceof the highly nonuniform flood flows as mentioned by Figure 13. The total bed load is the same order as theconventional model, but the longitudinal distribution is significantly different. From the above considerations,the sediment transport in the Monobe River mouth is largely affected by the three-dimensional currents and the
local riverbed gradients. Therefore, the non-equilibrium motion of the sediment particles plays an important7

role for the enlargement of the river mouth opening and the deposition process of the fluvial sediment at thecoastal area.

-0.1km-0.2km(a) Free surface and bottom surface velocities.

(b) Velocity distributions in -0.1 and -0.2 km cross-sections.

Bed load (m3/s)

Figure 13. Calculated water surface velocities and bottom surface velocities around the river mouth opening
and velocity distributions at -0.1 and -0.2 km cross-sections at the peak of the July 2018 flood.
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2mm
19mm
100mm

-1

-0.8

-0.6

-0.4

-0.2

Longitudinal distance (km)4.75mm
26.5mm
150mm

0

9.5mm
53mm

0.2

0.4

Bed load (m3/s)(b) Bed load of each particle size.
8
7
6
5
4
3
2
1
0

Propsed model
Cconventional model

-1

-0.8

-0.6

-0.4

-0.2

Longitudinal distance (km)

0

0.2

0.4

(c) Total bed load.
(a) Trajectories of particles of 100mm and 2mm.
Figure 14. Calculated free surface velocities and bottom surface velocities around the river mouth opening andvelocity distributions at -0.1 and -0.2 km cross-sections at the peak of the July 2018 flood.

4.

CONCLUSIONSThis paper proposed an analysis method that can demonstrate the three-dimensional flood currents and nonequilibrium motion of the gravel particles and cobbles by improving the bed variation analysis method for thestony-bed rivers (Osada and Fukuoka, 2012) and combining it with the Q3D-FEBS.
The proposed analysis method was able to explain the enlargement mechanism of the river mouth opening andthe formation process of the depositional landforms at the coastal area for the July 2018 flood in the Monobe
River than the conventional analysis method based on the equilibrium bed load formula. However, the proposed
analysis method showed that the opening width was slightly smaller and the deposition height of the fluvial
sediment at the coastal area was lower than the observation results.
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