Journal of JSCE, Vol. 7, 15-21, 2019
(Originally published in Journal of Japan Society of Civil Engineers, Ser. B1, Vol. 73, No. 4, I_859-I_864, 2017 in Japanese)

THE INFLUENCE OF BED SLOPE CHANGE
ON EROSIONAL MORPHOLOGY
Tamaki SUMNER1, Takuya INOUE2 and Yasuyuki SHIMIZU3
1Member of JSCE, Engineer, Suiko Research Co., Ltd. (3-3-2-7, Hiragishi, Toyohira, Sapporo 062-0933, Japan)

E-mail: t-sumner@suiko-r.co.jp

2Member of JSCE, Researcher, River Eng. Research Team, Civil Eng. Research Inst, for Cold Region(1-3-1-34, Hiragishi, Toyohira, Sapporo 062-8602, Japan)
E-mail: inoue-t@ceri.go.jp
3Fellow of JSCE, Professor, Graduate School of Engineering, Hokkaido University(North 13, West 8, Kita, Sapporo 060-8628, Japan)
E-mail: yasu@eng.hokudai.ac.jp

New waterway projects such as cutoffs and gravel extractions have been carried out all over the country.
The shortcuts have lowered river water surface elevation during flooding, and they have been shown tosuppress these flooding events. On the other hand, the riverbed elevations rapidly decreased as bed slopechanged artificially and canyons were formed in several rivers. Little is known about how riverbed degradation occurs or how the canyon is formed by change in the bed slope. In this study, we conducted numerical simulations to evaluate the deformations of alluvial bed and peat bed. The results suggest that the bed
slope becomes constant on the alluvial bed and the erosion proceeds at the bare peat bed. The results also
show that the upstream migration of the peat bed slope forms cyclic steps morphology. Also, we carriedout numerical simulation to reproduce bed degradation of the Yubari River caused by cutoffs.
Key Words : new waterway project, erosional morphology, numerical simulation, cyclic step, peat layer

1. INTRODUCTION

of flow and deformation of river bed at the groundsill installation5), 6), etc.
The river bed at the downstream of the YubariRiver where the canyon appeared, is composed ofpeat mostly. The previous studies on river bed configuration analysis dealt with the cases of alluvial
bed, but hardly explored the changes in the peat bed.
According to the Geotechnical Terms Glossary7),

River improvement works, such as shortcut constructions (cut-off channels), new waterway projects,
and gravel extractions have been conducted all overthe country. Especially, shortcut constructions of rivers have the effect of improving the discharge capacity during floods and reducing the flooding, which
contribute greatly to easing our lives today. On theother hand, although the original riverbed slope ischanged by the river improvement, the riverbed may
degrade rapidly. For example, downstream of the
Yubari River1), a canyon resembling Niagara Falls(Fig. 1) appeared, requiring the installation of groundsills as countermeasures (Fig. 2).
Machida2) stated that man-induced changes in rivers impact the fluvial geomorphic system and the sediments try to achieve an equilibrium in the upstream
and downstream sides of the river. Besides, as a characteristic phenomenon at the slope changing point, it
has been confirmed that coarse sediment accumulates
at the upstream of the change point3). The impact ofshortcuts on rivers has already been studied, such as
the habitat prediction of the river environment after
shortcut in the Satetsu River4), numerical calculation

Fig. 1 Canyon at the new waterway (From History of Ishikari
River).

15

Fig. 2 Kiyohoro Groundsill in September 2016.

Fig. 3 Peat in the Yubari River.

soil containing 50% or more of organic components
are generally called high organic soil, while decomposed and fibrous soil is classified as peat. Ninety
percent of peatlands are distributed in cold regions of
the northern hemisphere, and focusing on Japan,
most of them are distributed in Hokkaido. In the
Peaty Soft Ground Countermeasure Manual8), peat
has a higher water content ratio, a higher void ratio,
and a higher compressibility compared to soft clay.
On the right bank of KP6.5 in Yubari River, we confirmed that the peat consisted of organic substanceand soft fluffy soil mixed with adhesive materials(Fig. 3). KP stands for the distance (km) from theconfluence point of main stream.
In this study, numerical simulations were conducted to elucidate two questions: how the riverbed
deformation process of the peat bed differed from thealluvial bed, and how the canyon was formed and migrated upstream. Also, we investigated if the model
can reproduce the peat bed erosional morphology that
occurred in Yubari River after the river shortcut construction.

erosion dependent on shear velocity, proposed by
Nishimori and Sekine10), could be applied. Sekine etal. has confirmed that the erosion rate of the cohesive
soil bed depends on the cube of the friction speed, thewater temperature, and the water content of 2.5th
power, applied to the peat bed as follows:
𝐸𝐸𝑝𝑝 = 𝛩𝛩𝑢𝑢∗3𝛩𝛩 = 𝛼𝛼(𝑇𝑇)𝑅𝑅𝑤𝑤𝑤𝑤 2.5

(1)
(2)

where 𝐸𝐸𝑝𝑝 denotes the erosion rate of the cohesive soil
converted to peat bed; 𝛩𝛩 denotes a coefficient that
depends on the type of clay mineral, water temperature, etc.; 𝑢𝑢∗ represents the shear velocity; 𝛼𝛼 is a coefficient of water temperature; 𝑇𝑇 stands for watertemperature; and 𝑅𝑅𝑤𝑤𝑤𝑤 denotes the water content.
Sekine et al. stated that the surface of the cohesive
soil is not always smooth, waves on river bed may beformed, and sediment transported as bedload may bedeposited in the recesses. Therefore, assuming thatthe erosion rate of the peat bed was influenced by the
alluvial cover rate, the following equation was developed:

2. METHOD OF NUMERICAL SIMULATION𝜕𝜕𝜂𝜂𝑝𝑝= −𝛩𝛩𝑢𝑢∗3 (1 − 𝑃𝑃𝑐𝑐 )𝜕𝜕𝜕𝜕(1) Erosional rate of the peat bed
We adopted the calculation of unsteady two-dimensional plane flow similar to the solver Nays2DH
of iRIC software (http://i-ric.org/en/software/18/) for
the numerical simulation model of this study. Explanations of basic formulas of the flow and the bed deformation method of the alluvial bed are described on
the Nays2DH home page, but only the part related to
the river bed deformation method of the peat bed is
described in this paper. In addition, for the continuous equation of sediment, the method of the numerical calculation of Inoue et al.9) is used.
In this study, we assumed that the erosion would
occur even with a small flow velocity on the peat bed,
and it was speculated if the prediction formula of(3)

where 𝑡𝑡 denotes time; 𝜂𝜂𝑝𝑝 denotes the elevation of thepeat bed; and 𝑃𝑃𝑐𝑐 denotes the alluvial cover rate. The
alluvial cover rate is expressed using the ratio of themacroscopic roughness height of the peat bed 𝐿𝐿 tothe alluvial thickness 𝜂𝜂𝑎𝑎 11):
𝑃𝑃𝑐𝑐 = �𝜂𝜂𝑎𝑎
�𝐿𝐿 for 0 ≤ 𝜂𝜂𝑎𝑎 < 𝐿𝐿�
1 for 𝜂𝜂𝑎𝑎 ≥ 𝐿𝐿

(4)

According to the study of Sekine et al., the height ofthe surface roughness corresponds to three to seventimes the grain size. Therefore, in this study, 𝐿𝐿 was
set to three times the grain size10).

16(2) Total height of the riverbed deformation
Total height of the riverbed deformation 𝜂𝜂 was calculated using the following formula, which combined
the variation in the alluvial thickness and the erosionof the peat bed:𝜕𝜕𝜕𝜕 𝜕𝜕𝜂𝜂𝑎𝑎 𝜕𝜕𝜂𝜂𝑝𝑝
(5)
=
+
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

Fig. 4 Longitudinal profile of the initial riverbed.

3. COMPARISON BETWEEN ALLUVIAL
BED AND PEAT BED

Table 1 Numerical conditions of case1.

(1) Numerical conditions
In order to elucidate the longitudinal erosion process, the numerical experiments were conducted under narrow river width conditions. Figure 4 shows
the longitudinal profile of the initial riverbed of numerical experiments. A channel shape was set to arectangular cross-section. The flume had a length of
30m, width of 0.1m, bed slope of 1/300 in the upstream for 10m and 10m of downstream, and bed
slope of 1/50 in the middle, which was assumed to be
the section of cut-off. The grid size was set to
0.1m×0.1m. The upstream and downstream ends
were extended by 10m each as an approach section.
Sediment grain size of 0.77mm was employed, thewater discharge per unit was 0.0125m2/s, and Manning’s roughness coefficient was 0.012.
In the study of Sekine et al.12), the flume experiments showed that the value of coefficient Θ wasfrom 0.78×10-5 to 3.89×10-5 when 𝑅𝑅𝑤𝑤𝑤𝑤 =1. In thisstudy, Θ was set to a larger value of 0.5 to save on
calculation time. However, we confirmed that thesame topography was formed between 0.005 and 0.5
for the value of Θ. The values of water temperature
and water content could be constant as we wanted tocompare the peat bed and alluvial bed. We also assumed that the peat bed had a more significant influence on friction velocity than the cohesive bed. The
validity of the value of Θ is an issue to be concerned
about in the future. The dimensionless critical shear
stress of the alluvial bed calculated from Iwagaki’s
formula13) is 0.034 and the dimensionless critical
shear stress of the peat bed adapted to Inoue’s equation, 14), 15) which changes with riverbed roughness is0.025.
Table 1 shows the numerical conditions of the experiments. The type of riverbed was calculated using
case1_a for the alluvial bed and case1_p for the peat
bed. The sediment feed rate was discharged at
5.71×10-6m2/s, which was the equilibrium of the sediment transport rate per unit of the upstream end condition where the bed slope was 1/300 for both
case1_a and case1_p, and the calculations for 100
hours were performed.

CaseType of bed

Sediment feed rate
per unit (m2/s)

case1_acase1_p

Alluvial bed
Peat bed

5.71×10-6
5.71×10-6

Fig. 5 Height of riverbed deformation by alluvial bed (case1_a).

Fig. 6 Height of riverbed deformation by peat bed (case1_p).

(2) Calculation results
Figures 5 and 6 show the calculation results ofcase1_a and case1_p, respectively. In the results, thebed slopes eventually became constant, and the river
bed degradation remarkably occurred in all cases.
Comparing the calculation results of case1_a andcase1_p, the final riverbed slopes approached theequilibrium state of slope 1/300; however, riverbed
degradation processes were different. The alluvial
bed became constant from the early stage, and theslope shifted to 1/302 towards the end of the experiment. Meanwhile, in the case of peat bed of case1_p,
the erosion occurred from a transition point from the
sub-critical flow to super-critical flow by slope

17

1)

4)

2)

5)

3)

6)

Fig. 7 Process of canyon formation by calculation.

change on the upstream side of the middle section.
The incision progressed and formation of a cyclic
step topography occurred. This step topography repeatedly eroded while moving towards the upstreamside with the lapse of time, and finally the riverbed
became constant due to bed degradation in all sections. Once the degradation happened, after 30 hours
the bed slope became 1/384, which was milder than
1/300. As sediment gradually accumulated and approached the equilibrium, the slope was 1/357 in 100
hours.

step topography of cohesive soil that cyclic step formation occurs in steep rivers. Calculation resultshows that cyclic step topography is formed in steep
slope, which is in agreement with the theoretical analysis. 3) As Parker and Izumi stated, it was also observed that cyclic steps migrated upstream whileeroding. The bed slope gradually becomes mild, andthe flow velocity slows down as the vertical erosion
progresses to deposit layer of sediment. Therefore,
the sediment begins to deposit on the peat bed. 4) By
repeating the upstream migration of transition to alluvial bed, the height of the river bed drops increase.
5) This is because the difference between the slope of
the alluvial bed shifts and the initial bed height increases toward the upstream. The bed slope eventually becomes constant owing to sediment deposition.
6) Finally, the river bed reaches a state of equilibrium
to the sediment feed rate. The slope of erodible bed
can be changed to constant or dynamic equilibrium
state because the slope attempts to sustain equilibrium corresponding to the sediment feed rate by feeding a constant amount of sediment. This calculation
result suggests that the peat bed is different from thealluvial bed in terms of the process of river bed degradation. However, eventually both have a constantslope and approach the same eroded landform.

(3) Erosional morphology in the riverbed withslope change
As a result of case1_p, a formation process of erosional morphology in the riverbed with slope change
is assumed as follows (Fig. 7): 1) First, sediment deposits on the river bed where the slope is mild, andthe peat bed is exposed where the slope is steep. Erosion occurs at places where the flow velocity is large
in the erosion exposed section. 2) Due to the erosionof the river bed, the flow velocity is distributed in theflow direction. The flow transits continuously fromsub-critical flow to super-critical flow and from super-critical flow to sub-critical flow to form the steps
topography, and it creates a waterfall. Parker andIzumi16) confirmed in the theoretical analysis of the18

(2) Numerical conditions
Calculation was performed to reproduce thechange in bed slope of Yubari River due to the construction of a new waterway section (KP2 to KP16 of
the Kurisawa Weir) at the confluence of Ishikari

4. REPRODUCTION OF RIVERBED DEGRADATION AT THE DOWNSTREAM
OF YUBARI RIVER(1) Overview of the Yubari RiverThe Yubari River is a tributary of the Ishikari Riverlocated in the central area of Hokkaido. The low-lying area in the downstream part of the Yubari River,which was the subject of this study, was a soft landwith a wetland spreading along the peat distribution
area in the early Meiji period. Moreover, it wasplaced in an unstable and wet environment where inundation occurred every year due to snow melt runoff
and floods. As a countermeasure to prevent floodingin the low-lying area, the Yubari River New Channel
Project, which directly switched the Yubari River(which had merged with the Chitose River to the Ishikari River), was completed in 1951. In spite of thepassage of water, one year later, a scouring like Niagara Falls of 4m in height near KP2.2 appeared. The
scouring migrated upstream at a speed of 5–7m perday. In the new waterway project, a groundsill was
installed at KP8 for the agricultural siphon and scouring prevention of the Kiyohoro Bridge. Therefore, asecond groundsill was installed downstream from thefirst due to the unexpected scouring. The river channel implemented at the slope of 1/1200 gently became 1/4000, and the height of the drop was larger at
the second groundsill, the Kiyohoro Groundsill.

Yubari River

Ishikari River

Fig. 8 Yubari River (Google Map).

Fig. 9 Initial bed slope of the Yubari River.

18hours

10hours

Canyon topography migrates upstream

Super-critical flow

Drops by super-critical flow
Initial bed elevation
Total bed elevation

Initial bed elevation
Total bed elevation

15hours

20hours
Drop height is larger

Canyon topography is formed

Initial bed elevation
Total bed elevation
Drops by super-critical flow

Initial bed elevation
Total bed elevation

Fig. 10 Longitudinal profile of riverbed elevation and alluvial cover rate.

19

Kiyohoro Groundsill

The drop height increased to 8.79m at KP13.84 after
20 hours. As a result, the migrating speed of the canyon was accelerated to 1.59km/h in 18 hours, and1.94km/h in 20 hours. The characteristics of the river
bed deformation, such as the increased canyon’s dropas it migrated upstream, was represented by the numerical calculation.
There is no record of river bed elevation of the canyon formation time in the Yubari River (in 1937).
Therefore, we compared the measured average bed
elevation from 1953 to 200317) and the average bed
elevation of the calculation result when the canyon
reached the Kiyohoro Groundsill (Fig. 11). Compared with the oldest average bed elevation in 1953,
the results of calculation show that the elevation was
1m higher than the measurement between KP6.5 anddownstream of the Kiyohoro Groundsill. However,
the results suggest that we could reproduce the deformation caused by canyon formation occurring due tothe erosion at downstream and migration to upstream, which was characteristic of river bed degradation at downstream of the Yubari River. Moreover,unlike the peat bed calculation result (case1_p) described above, the canyon topography migrated upstream rather than the step topography forming
eroded morphology. It can be inferred that the river
bed slope at downstream of the Yubari River was
mild and the flow was unlike the condition the cyclic
steps formed. In this numerical calculation, the coefficient depending on the type of clay mineral and water temperature and the parameters of water discharge
were given values from inference, and adequate consideration is needed for validation.

Yubari River

Numerical Result

Longitudinal Distance (km)Fig. 11 Comparison of mean riverbed elevation of measured
and calculation results.

River (Fig. 8.) The calculation model adopted theerosion rate formula of the peat bed used in case1_p.
Coefficient depended on the type of clay mineral, andwater temperature was set to 0.5 as in case 1_ p performed in previous experiments. The numerical
flume has a width of 72m according to the plan specifications of the Yubari River Shortcut Construction1)
and total length of 18km. One grid size is
100m×72m. The riverbed slopes of the waterway are
set as 1/1200 from KP0 to KP11, 1/1500 from KP11
to KP16, 1/360 from the confluence point of the Ishikari River KP-1 to KP0, and 1/4000 from the mainstream of Ishikari River KP-2 to KP-1 (Fig. 9), whichis in accordance with the bed elevation data obtainedat the time of installation of the new waterway.
From 1973 to 2014, 40m3/s of average annual water discharge was noted, producing sediment of 2.67×
10-4m2/s, which was an equilibrium sedimenttransport rate per unit with a slope of 1/1200.The uniform sediment grain of 1mm from the river bed material sieving test table (1964) and Manning’s roughness coefficient of 0.025 were applied.

5. SUMMARY
The findings obtained in this study are as follows:
We carried out numerical simulations to understand the difference in the bed deformation process between the alluvial bed and the peat bed
when river bed slope changes. As a result, in thecase of the alluvial bed, the slope smoothed andbecame constant from the beginning of the calculation to maintain the balance of the bed. Conversely, in the case of the peat bed, the river bed
was balanced by the sediment and tended to approach a constant slope similar to the alluvial
bed; however, the process of bed deformation
was different. We observed that the peat bed topography like cyclic steps due to the influenceof the friction velocity. The topography also
formed an erosional morphology with the dropas it migrated upstream.
2) We carried out a numerical simulation of thedeformation in the Yubari River leading to the1)(3) Results and discussion
Figure 10 shows the calculation results of riverbed
height and the alluvial cover rate 𝑃𝑃𝑐𝑐 after 10 hours
from the start of the calculation. After the exposureand erosion of the peat bed near KP0, the step-like
topography was observed in the section of the transition of super-critical flow to sub-critical flow (10hours). The graph of 𝑃𝑃𝑐𝑐 shows that the place the sediment deposited and the place the peat bed exposed
are mixed in the downstream section of KP3 eroding
the peat bed. After 15 hours, a drop formed by thesuper-critical flow gradually became larger, and acanyon-like topography of 3.64m in height wasformed near KP5.3. The bed in the upstream of thecanyon near KP5.3 was not eroded and flat because
of sediment deposition. After 18 hours, the canyon
headed to KP10 and the drop height became 5.04m.

20

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formation of a canyon, by using the erosion
model of the peat bed. As a result, the erosion
caused by super-critical flow at the point of bed
slope changed near the confluence of Ishikari
River. Furthermore, the erosional morphology
migrated upstream with gradual acceleration
and became a canyon with a large drop inheight. The erosional morphology indicates thatthe degradation process caused by the construction of a shortcut of downstream of the YubariRiver can be reproduced roughly. However, thelack of elevation data and water discharge datafrom the time of formation of canyon in 1937
must be considered. Validating various conditions such as the coefficient depending on thetype of clay minerals and water temperature isalso important.
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