ANALYSIS OF TWO-DIMENSIONAL RIVERBED VARIATION IN
CHANNELS WITH STONY BEDS

1

2

Kengo OSADA , and Shoji FUKUOKA
1. Research and Development Initiative, Chuo University, Tokyo, ON, Japan(osada@tamacc.chuo-u.ac.jp)
2. Research and Development Initiative, Chuo University, Tokyo, ON, Japan(sfuku@tamacc.chuo-u.ac.jp)

ABSTRACTIn a stony-bed river with wide grain-size distribution, boulders that would not move during a flood act
as strong fluid-resistance element. In addition, sand and gravel often remain in the bed irregularities
formed by large stones. A stony-bed river will remain in a static equilibrium condition when the river
bed surface grain size distributions corresponding to tractive force of the flow are reached. However,conventional riverbed variation analysis does not account for these essential mechanisms in stonybed rivers. Therefore, we had developed a one-dimensional riverbed variation analysis method byconsidering the mechanism of riverbed variation in stony bed rivers. This method is generally capableof explaining water surface profile, quantitative bed variation, and bed material size distribution instony-bed rivers. In this study, we developed the method of two-dimensional riverbed variation
analysis by considering the mechanism of riverbed variation and bank erosion in stony bed rivers. We
verified the applicability of the model with the field experimental results which was conducted in themeandering channel in the Jyoganji River.

1.

INTRODUCTION

In stony bed rivers with wide grain size distribution, flow resistance is determined by the drag force
acting on large stones. In addition, sand and gravel often remain in the bed irregularities formed bylarge stones. A stony-bed river will remain in a static condition (Fukuoka et al., 2008). However, it isdifficult to simulate riverbed variations in stony-bed rivers by the conventional river bed variation
analysis methods, which does not account for these essential mechanisms by large stones in stonybed rivers. To improve channel planning in stony-bed rivers, a accurate technique for riverbed
variation is required (Fukuoka, 2008).
The authors have conducted large scale field experiments in the Jyoganji River to elucidate themechanism of bed variation in stony-bed rivers with wide grain size distributions (Fukuoka and Abe,2007). We had developed the new method of one-dimensional riverbed variation analysis based on
the following important mechanism of bed variation in stony-bed rivers (Osada and Fukuoka, 2008).
First, sediment transport of small size materials depend on large stones. The amount of sedimenttransport of the small size materials is difficult to estimate from equilibrium sediment transport formulawith the critical tractive force. The sediment transport model is composed of pick-up rate from theriverbed and deposit rate to the riverbed. Second, large voids are formed by exposed large stones on
the riverbed surface. The large voids on the riverbed surface are important to analysis of bed variationin the stony bed rivers. To take account for the large voids, we calculated the heights of each sizegroup on the riverbed surface. Third, to consider the form resistance of large stones, we developed
the resistance formula using d90 in the bed surface. It was demonstrated that water surface profile, bedvariation and grain size distribution in straight channels with stony bed were estimated by the onedimensional analysis.

1

In this study, we develop a numerical method of two-dimensional bed variation by considering
mechanism of sediment transport and bank erosion in stony bed rivers. It is important for the twodimensional analysis method to estimate bank erosion and sediment transport in the transverse
direction. In general, bank erosion rate in sand and gravel bed rivers is calculated with the
Hasegawa’s method (Hasegawa, 1981). In this method, river bank is collapsed when the side slope ofthe bank exceeds the angle of repose. In the stony bed rivers, cobbles and boulders deposits into theriverbed near the bank due to the bank erosion, and the riverbed become stable condition.
Hasegawa’s method by using angle of repose estimates the bank erosion to be large. Therefore, thismethod is difficult to apply to bank erosion in stony-bed rivers. We propose pick-up rate formula toaccount for the bank erosion in stony bed rivers. We verified the applicability of the model by the field
experimental results conducted by the meandering channel in the Jyoganji River (Fukuoka and Osada,2009).

START
Volume of sediment transport

Initial condition

VS i , j ,kUnsteady two dimensional flood flow analysis
continuity equation and equation of motion(1)~(5)

t

z

 2

CD 2 d k2u 2f x

  1.0 : pick-up

Z B i , j ,k

(6)

t

Pick-up rate

VP i , j ,k   P i , j ,k PP i , j ,k

Pi , j ,kTP i , j ,kt

 2 VP i , j ,k  VD i , j ,k 
3Pi , j ,k

(14)

k 1

 2 VP i , j ,k  VD i , j ,k 
3
dk

(15)
(16)

Average of riverbed
elevationn(9)

Deposition rate

VD i , j ,k  PC i , j ,kVS i , j ,k

(13)

n

AP i , j ,k 

N P i , j , k d VP i , j ,k  VD i , j ,k (12)  AP i , j ,k  P0i , j ,k  APi , j ,k

(8)
3
3 k



Each grain size fraction on the bed surface

Judgment of stationary equilibrium condition

ghi I e   Fxqb i , j ,kBed elevation of each grain size  1.0 : no pick-up
qb i , j ,k  u p i , j ,kVS i , j ,kqb i , j ,k  v p i , j ,kVS i , j ,k

C L 2 d k2u 2f2 x     g 3d k3

qb i , j ,kSediment transport rate

Two dimensional riverbed variation analysis
Judgment of pick-up



Z B i, j 
(11)

 P
k 1

i , j ,k

 Z B i , j ,k n

 Pi, j ,k



di, j
2

(17)

k 1

t  t  t

END

Figure 1 Procedure of the two-dimensional riverbed variation analysis

2.

TWO-DIMENSIONAL RIVERBED VARIATION ANALYSIS FOR STONY-BED RIVERS

Figure 1 shows the procedure of the two-dimensional riverbed variation analysis. In this method, weuse combination of the unsteady two-dimensional flow analysis and the two-dimensional bed variation
analysis. The flood flow is calculated by equations (1)~(3).

  h    q     q 
 
 0
 
t  J    J    J (1)2 y   q y   y
  q     u  q     u  q   q x    x      u u  x    u  u
t  J    J    J  J   J 

 













  x2   y2 H  x x   y y H  FD90  x2    gh u x2 h  x x
 u x2 h
 J
 J


J


J


J




2
y 
 y y  x y   y x 
2 x y  u y2 h  u y2 h  u x u y h  u x u y h  S
J 
J 
J

J 











(2)



 y 
 x  q y    y
  q    u  q    u q  q x    x      u   u u
 ut  J    J    J  J   J 

 









  x x   y y H  x2   y2 H  FD90  x2    gh u x2 h  x x
 u x2 h


J


J



J
J


J




2
y 
 y y  x y   y x 
2 x y  u y2 h  u y2 h  u x u y h  u x u y h  S
J 
J 
J

J 















(3)



Where h : water depth, q  , q : contravariant components of flow discharge in the  and  direction,u  , u : contravariant components of velocity in the  and  direction, J : Jacobian of transformation,

q x , q y : flow discharge per unit width in the x and y direction,  ux2 ,uy2 ,ux2uy2 : Reynolds stress

tensors, and FD 90 , FD 90 : The bed resistance. These are calculated by equations (4) and (5).

 FD90   x Fx   y Fy  FD 90   x Fx   y Fy

 D 90ux
2 2
 Fx  N D 90 2 C D 2 d 90u f  2
u x  u y2uy
 F  N  D 90 C  d 2 u 2 yD 90
D 2 90 f
2
u x2  u y2


(4)

(5)

Where N D 90 : the number of d90 per unit area,  D 90 : the shielding coefficient of d90,  2 , 3 : 2-D and
3-D shape factors of the particles, C D : drag coefficient, u f is the velocity acting on d90 which isdetermined by logarithmic velocity distribution. And, we add the secondary flow term ( S , S )proposed by Nagata et al. (1999) into equation of motion(2),(3).
Next, we explain the two dimensional riverbed variation analysis method. Pick-up judgment of eachparticle is evaluated by the moment balance equation (equation (6)). The concept of pick up judgment
is shown in Figure 2. The angle  k is calculated with equation (7).


 1 Z B i , j ,k  Z d 80i , j   C i , j
2
d


80
i
,
j



 k  90.0  45.0  exp 

3(7)

Where subscript k : the number of each grain size material, subscript i, j indicates the computational
grid number, Z B i , j ,k : height of diameter dk, Z d 80i , j : height of d80.  C i , j : Riverbed and side bankslope of right direction for velocity. This equation represents the relationship between the height of d80
and the height of each particle. For instance, when the height of each particle is lower than the heightof d80, angle θk is greater than 45. This means that the particle cannot move easily. On the other hand,when the height of each particle is higher than the height of d80, angle θk is smaller than 45. In
addition, the effect of the side bank slope is considered by  C i , j .
Next we explain the method of determining bed stability. In the calculations, when d80 and d90 do not
move, fluid force act on those particles. In this instance, the pick-up quantities of all particles from the
bed become zero.
After the process outlined up to this point, equation (9) is used to determine the pick-up rate. Here, P i , j ,k is a coefficient that accounts for the shielding effect of particles larger than dk, PP i , j is the ratio
between the fluid field force acting on the bed and the total drag force acting on the particle, and is
calculated as follows:

PP i , j ghi , j I e(10)

 N P i , j ,kCD 2 d k2u 2f 2
k 1 
n

TP k is defined as the time required for a particle to be transported as shown in Figure 2. Figure 3
show the computational grid. Sediment deposition rate VD, sediment volume VS and bed height ofeach particle size ZBk are calculated at the center of the grid. And, pick-up rate VP is calculated at
center of the side line.

TP

Z Bk



uf

0.7 Z Bk  Z B

 k

ZB

z
x
Fixed particle

Figure 2 Concept of the pick-up particle

VD i , j ,k ,VS i , j ,k
, Z B i , j ,k , Pi , j ,k

VPU i , j ,k , qB i , j ,k , uP i , j ,k
VPR i , j ,k , qB i , j ,k , uP i , j ,kFigure 3 Calculation grid

4

FLOW

y

y

x

x
Figure 4 Concept of the saltation analysis
We assume that sediment deposition in stony bed rivers is occurred by collision event between
transport particles and riverbed materials. Therefore, sediment deposition rate VD is calculated byequation (11) using sediment volume VS and rest ratio PC that is discussed below. Sediment volume iscalculated by equation (12). Sediment transport rate is calculated by equation (13). Here, u p , v p areparticle velocity. Particle velocity is determined by dividing the calculated time (5 seconds) into theparticle’s movement distance determined by the saltation time calculations. The author’s method
includes calculating 5-second saltation using the particle equation of motion, then determining the bed
particle collision rate, which is wed to evaluate the rest rate.
The heights of each particle size group are calculated using each size group’s pick up quantity fromthe bed, deposition quantity to the bed, and the bed surface ratio of dk particles with equation (14). As
seen in equations (15) and (16), the surface ratio of areas occupied each particle size group is
calculated by each size group’s pick up and deposition quantity. Here, P0i,k is the subsurface particle
diameter ratio. Average bed height, which is needed to calculate the flow field, is calculated by theequation (17). The first term on the right side of equation (17) is the average particle height in thecomputational grid, and the second term on the right is radius of the average-diameter particle.

3. APPLICATION TO FIELD EXPERIMENTAL RESULTS IN MEANDERING CHANNEL IN THE
JYOGANJI RIVER

3.1 Conditions of Calculations
We apply the bed variation analysis method to the results of field experiments conducted in the
Jyoganji River in 2006. Figure 5 is a top view of the meandering channel. The channel contained two
meanders, it was dug into a sandbar at the 11.1 km point in the Jyoganji River. The channel was a
120m long, 3m wide and side slope 1:1. Meander 1 contained natural riverbanks, while revetments
were built along the outer bank of meander 2. We measured water level by the pressure gauges andlevel surveying, discharge, bed cross-section, and bed grain size distribution. Table 1 shows the flow
discharges of the experiment. Peak discharge of the experiment was 11m 3/s.
The computational grid is made by 2.5m intervals in the downstream-direction and 0.35m intervals in
the across-direction. We use observed water level hydrographs at W02 and W21 as the boundary
conditions.
Figure 6 shows the grain size distributions using this calculation. The grain size distribution was basedon the dimensionless particle size distribution determined by Kuroda and Fukuoka et al. (2005). Seven
particle sizes were used in the calculations—350mm, 270mm, 200mm, 120mm, 80mm, 50mm, and 25
mm.

5meander 1

W20 W21

W16

W12
10m

W01 W02

W22

FLOW
water level（level surveying）
and channel cross section
water level（pressure- type gauge）
discharge measurement
sectionbank revetment

W08

meander 2

W04

Figure 5 Plan view of the experimental channel

Table 1 Observed discharge

Percent finer (%)

100
90
80
70
60
50
40
30
20
10
0

Surface grain size
distribution
Substrate grain size
distribution

1

10

100

1000
Grain size (mm)Figure 6 Initial conditions of grain size distribution

6

3.2 Result of Calculation
Figure 7 and Figure 8 show the calculated and observed temporal change in water level profiles. And,
Figure 9(a)~(c) shows the temporal changes in calculated bed variation contour. Looking at the results
for water levels (Figure 7, Figure 8), we can see that the calculated results differ about 10cm from theobserved water surface profile in the section of meander 1. But, the calculation results almost
reproduce the observed water surface profiles. In the Figure 9(b) of the maximum discharge in thehydrograph, bank erosion in the outer bank of the meander 1 and aggradations by sedimentdeposition in the inner bank of meander 2 become large, comparing with Figure 9(a) of the earlier
stage. After the maximum discharge, the bed elevation is almost maintained. Figure 9(d) show theobserved bed variation. The calculated result (Figure 9(c)) is agreement with experimental result.
However, we have to improve the evaluation method for the amount of bank erosion and sedimentdeposition in meander 1. Figure 10 shows the bed cross-section profiles. In the W20 located in thestraight channel section, calculated bed variation reproduce riverbed aggradations and bank erosionof the experiment. In the W14 where located in meander 1, calculated bed variation cannot explain
well for sediment deposition in the inner bank.
Figure 11 shows the comparison between observed and calculated discharge hydrographs. The
calculated discharge hydrograph reproduces the observed discharge hydrograph, because calculated
water surface profile and bed elevation are good agreement with observed values. Figure 12 shows
the transverse distribution of sediment discharge per unit width for each particle size at W19 and W14.
In the W19 located in straight channel section, sediment discharges excluding grain size 80mm
around center of the channel are lower than. On the other hand, sediment discharges around bank arelarge than that of the channel center. Around the channel center, stable riverbed is formed byarmoring due to bed scour caused by low flow discharge. Around bank, sediment discharge becomeslarger than around the channel center due to bank erosion caused by large flow discharge. The data in
Figure 12 for the sediment transport of each bed material indicate that 80mm and 120mm particles
move actively. The shielding effect of the large size material reduced the sediment discharge of 25mm.
Sediment discharge express as following characteristics in the section W14 (Figure 12(b)). Sediment
discharge is large around outer bank, whereas it is low around inner bank. Figure 13 show thecomparison between observed and calculated grain size d80 longitudinal distribution. In the meander
section, the calculated d80 distribution cannot reproduce the experimental d80 distribution. However, asa whole, the calculated results generally agree with the observed results. Figure 14 shows calculated
results of d80 cross-sectional distribution. Mild slope and armoring near the bank are formed by the
supply of cobbles and boulders due to bank erosion. The calculation result reproduces the mild slope
and armoring near the bank.
As discussed above, the two-dimensional model is generally capable of explaining water surface
profile, bed variation, and bed material size distribution in stony-bed rivers with a wide range of bed
material sizes.

7

Z(m)

11:00 （EXP)

11:15 （EXP)

11:30 （EXP)

11:45 （EXP)

59.5

11:00 （CAL)

11:15 （CAL)

11:30 （CAL)

11:45 （CAL)

59.0
58.5

W20

58.0

Meander 1

W04 Meander 2

57.5

0

20

40

60

80

100 X(m)Figure 7 Comparison between observed and calculated water surface profile (Left bank)
Z(m)

59.5
59.0

10:50 （EXP)

10:50 （CAL)

11:10 （EXP)

11:10 （CAL)

11:30 （EXP)

11:30 （CAL)

58.5

W20

58.0

Meander 1

W04 Meander 2

57.5
0

20

40

60

80

100 X(m)Figure 8 Comparison between observed and calculated water surface profile (Right bank)Z  0.4

0.4  Z  0.2
0.2  Z  0.0

ΔZ(m)Z  0.0
0.0  Z  0.2
 0.2  Z  0.4
Z  0.4

Meander 1

Meander 1

W14
W18
W03

Flow

Flow
W14

W21

W10
W06

W18
W03

W10
W06
Meander 2

Meander 2(a) Calculated bed variation (11:15)(c) Calculated bed variation (after the calculation)

Flow

Flow

Meander 1

Meander 1

W14
W18
W03

W21

W14

W21

W18

W10

W10
W06

W03

W06
Meander 2

Meander 2(b) Calculated bed variation (11:45, Peak discharge)(d) Experimental result
Figure 9 Comparison between observed and calculated bed variation contour

8

W21

Initial bed elevation
Observed bed elevation (after the experiment)
Calculated bed elevation (after the experiment)

Z(m)

60.5
60.0
59.5
59.0
58.5
58.0
57.5
Z(m)

W20

-5

-3

3

5

Transverse distance (m)

60.5
60.0
59.5
59.0
58.5
58.0
57.5
Z(m)

初期河床高
通水後河床高（実測）
通水後河床高（解析）
-1
1
W14

-5

-3

初期河床高
通水後河床高（実測）-11
通水後河床高（解析）

60.0
59.5
59.0
58.5
58.0
57.5
57.0

3

5

Transverse distance (m)

W08

-5

-3

-1

1

3

5

Transverse distance (m)Figure 10 Comparison between observed and calculated bed cross-section profile

Q(m3/s)12.0

obsarved upstream end discharge
calculated upstream end discharge
observed downstream end discharge
calculated downstream end discharge

10.0
8.0
6.0
4.0
2.0

0.0
10:30

11:00

11:30

12:00

Figure 11 Comparison between observed and calculated discharge hydrographs9Grain size 350mm
Grain size 120mm
Grain size 25mm

2

qB(m /s)

0.0003Grain size 200mm
Grain size 80mm
Bed elevation

Z(m)

60.0

0.000259.0

0.000158.0

0.0000

57.0
-4

-2

0

2

4

(a) W19
Grain size 350mm
Grain size 120mm
Grain size 25mm

2

qB(m /s)

0.0008Grain size 200mm
Grain size 80mm
Bed elevation

Z(m)

60.0

0.000659.0

0.000458.0

0.00020.0000

57.0

-4

-2

0

2

4

(b) W14
Figure 12 Sediment discharge cross-section profile for each particle size
Grain size (mm)
300
250
200
150
100

Experimental result50

Calculation result

0

Figure 13 Comparison between observed and calculated grain size d80
calculation riverbed elevation（11:15）
calculation riverbed elevation（11:45）
calculation water surface（11:15）
calculation water surface（11:45）
D80（11:15）
D80（11:45）

Z(m)

60.5

59.5

Grain size (mm)

300

250
200
150

58.5

100
50

57.5

0
-4

-2

0

2

4 Transverse distance (m)Figure 14 Calculated result of d80 cross-sectional distribution

10

4. CONCLUSIONS

We developed the two-dimensional riverbed variation analysis method focused on the mechanism of
sediment transport in stony-bed rivers. This method was applied to field experiments in the Jyoganji
River. As a result, it is capable of explaining water surface profile, bed variation, grain size distribution
and flow discharge hydrograph in stony-bed rivers. Moreover, we showed that sediment transport rateof the cobble-class material becomes larger than the others size group. Further investigation isrequired to make clear the reproducibility of the riverbed scouring near the bank and channel
deformation during a flood by the two-dimensional model.

REFERENCES

Fukuoka,S. (2008). Flood Flow and Bed Variation Analysis Considering the Change in Sediment
Environment -Practical Issues and Perspectives of the Sediment Research-, Advance in River
Engineering, JSCE, Vol.14, pp.1-6.
Fukuoka,S. and Abe,T. (2007). Mechanism of Low Water Channel Formation and the Role of GrainSize Distribution in Gravel-bed Rivers, 10th International Symposium on River Sedimentation, Moscow,
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