IAHR-APD 2020,Sapporo, Japan

EFFECTS OF PARTICLE-SIZE DISTRIBUTION AND STREAM GRADIENT ON SEDIMENT SORTING OF A DEBRIS FLOWTAKASHI WADA, JUMPEI MAEDA and HIROSHI MIWA

Department of Social Systems and Civil Engineering, Tottori University, Japan
3. APPLICATION WITH ONE-DIMENSIONAL NUMERICAL MODEL CONSIDERING
Falling volume of k-th particles
SEDIMENT SORTING OF A DEBRIS FLOWr in p-th layer

1. INTRODUCTION
A debris flow is characterized by coarser particles being concentrated toward the flow front duringdown flow in mountainous streams. (Figure 1)kp3.1 Outline of our developed modelThis characteristic implies that sediment sorting occurs in the flow’s interior.

Sediment concentration ofThe devastation and loss of life caused by a debris flow on flooding areas can be extensive as thedestructive force at the flow front become enlarged by this characteristic.
A practical method is required to predict sediment sorting in the debris flow’s interior, including
the concentration of coarser particles at the flow front during down flow in mountainous streams.

1) We conducted flume experiments with a tilted straight flume using sediment mixtures composed ofparticles of two different sizes, in order to gain insight into the mechanism of the sediment sorting and theeffects of particle-size distribution and stream gradient on the sediment sorting.

Notations:where a = the coefficient related to the falling of particles, sp= the interstice between particles in the p-th layer, whichis evaluated on the equidistant particle arrangement of the flow including highly-concentrated particles proposed byBagnold (1954), dk = the diameter of k-th particles, dmp= the mean volume diameter of all particles in the p-th layer,
ke = the number of particle classes, Du /Dz = the velocity gradient (shear strain) between the p-th and p-1th layers.

2. FLUME EXPERIMENTS

The momentum equation for the flow mixture in the total flow layer

FirstFourth Third
Secondsampler sampler samplersampler

Sediment flux concentration

7891011上流端給水開始からの経過時間(sec)0.40

Initial proportion; 0.50

0.20
0.00
1.00 0 Smaller
12Particle;
3 d=3.0mm
456789上流端給水開始からの経過時間(sec)
0.80
0.60
0.40
0.20

Initial proportion; 0.50

101112Particle;
3 d=10.7mm
456Larger

789101112

上流端給水開始からの経過時間(sec)

0.80
0.60
0.40
0.20

Initial proportion; 0.200.00
12 1.00 00.80
0.60
0.40
0.20

12 Particle;
34d=7.1mm
56Smaller

78

上流端給水開始からの経過時間(sec)Initial proportion; 0.80

9101112Proportion of coarser particles at the debris flow front

12Particle;
3 d=7.1mm
456Larger

0.60

Proportion of each sized particle

Sediment flux concentration

Proportion of each sized particle

0.80

[20°]

0.80.80.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.00
103) Measurements were performed to determine
the temporal changes in the total flow discharge,
sediment discharge, sediment flux concentration,and proportion of each size in each sample.

Case 2.1
Case 3.1
Case 4.1
Case 5.1
Case 6.1

Case 2.2
Case 3.2
Case 4.2
Case 5.2
Case 6.2

Calculated results

Initial proportion; Case 1.1-6.1

15

Case 1.1Caseand 1.2

Case 2.1Caseand 2.2
Case 3.1Caseand 3.2
Case 4.1Caseand 4.2
Case 5.1Caseand 5.2
Initial proportion; Case 1.2-6.2
Case 6.1Caseand 6.2
The sizes ofthe materials
2025 are coarser.

0.00
Slope gradient (deg.)
012345678
9 10 11 12012345678
9 10 11 12Figure 4. Proportion of each sized particle at the debrisTime from the start time of supplying water (sec)Time from the start time of supplying water (sec)Figure 3. Temporal changes in proportions of each sized particle of the debris flows flow fronts at downstream end of the flume in all cases.
at the downstream end of the flumes. (Experimental and Calculated results)(Experimental and Calculated results)
0.00

k4

k2

k4

k2

k3

k1

k3

k4

k2

k4

k3
k4
k4

up

k4

k1

k3

k4

k4

k3

k4

k2
k4

unp

k4

k4

k2

k3

k2

k3

k1
k4

k4

k3

k4

u2
u1

k3

Figure 5. Outline of our developed 1-D model

Notations:where t = the time, x = the coordinate axis of the flow direction, b = the momentum coefficient (= 1), M = the momentum flux for the total flow
layer (=uh), zb = the height of the movable bed, tb = the shear resistance of the river bed, ibk = the sediment erosion/deposition velocity of k-th(4) particles, qbkp = the transportation volume of k-th particles per a unit width in the flow direction in the p-th layers (=upCkph/np), rkp’ = thetransportation volume of k-th particles from the p-1th layer to the p-th layer, which is the surplus volume for the maximum volume in the p-1th
layer obtained by the theoretical equation for distributions of sediment concentration by Takahashi et al. (1996).

The calculated results are quantitatively consistent with the experimental results on the proportions of each sized particle of the flow front at thedownstream end of the flumes. Our model can quantitatively describe the temporary changes in the flux concentrationof coarser and finer particles
00Proportion
10 20in the30 4th40layer
50 60 70 80 90 100
at the flow front (Figures 3 and 4).
60
81.0
Table 2.

Experimental cases and conditions

Parameters/Variables
Value
Unit
Total simulation time
30sec.
Time step Dt
0.0001
sec.
Diameters of particles dk10.70, 7.10,mm(two of four particle diameters)
3.00, 1.40
Mass density of material s
2635
kg/m3
Mass density of interstitial fluid rm
1000
kg/m3
Concentration in the static sediment bed C*
0.558–
Internal friction angle of material f
32.85
deg.
Gravity acceleration g
9.8
m/s2
Coefficient of erosion rate de0.03–
Coefficient of accumulation rate dd0.05–
Interval of calculation points Dx0.01
mCoefficient to be related to the falling of500–
particles a
Number of the divided layers np4–

Flow height, Mean volume diameter

507Initial mean diameter; 5.77 mm

40630

52010014

Fixed bed
Movable bed
Flow sarface
Mean diameter

010203034050

4th layer

6070

8090

2100

distance (cm)
Total sediment Horizontal
3rd layer
2nd layer
concentration
1st layer

0.8
0.6
0.4
0.200
0.0

0

10203040506070
Horizontal distance (cm)

80

90100

Mean volume diameter (mm)Black material; d=3.0mm

Decreasing the debris-flow velocities enlarges the movement of materials in the depth direction, Middleton’s suggested mechanism (1970), which
is a falling mechanism of finer particles through the interstice between the materials in the flow’s interior (dynamic sieving), may explain thisphenomenon.
0.60
Case 4.1 [20°]―Run 1
0.60
Case 1.2 [15°]―Run 1
1.01
All Particle
All Particle
Case 4.1 [20°]―Run 2
The sizes ofCase 1.2 [15°]―Run 2
0.50
0.50
Case 4.1 [20°]―Run 3
the materials Experimental results
Case 4.1
Case 1.2 [15°]―Run 3
Case 1.2
Calculation Result by our model
0.9
0.9
are coarser.
0.40
0.40
Case 1.1 Case 1.2Calculation Result by our model
0.30
0.20
0.10
0.00
12 1.00 0k1

Horizontal distance (cm)

0.8
0.6
0.4
0.2
0.0

Proportion of each sized particle

Blue material; d=10.7mm

As the sizes of the debris-flows materials became coarser or the flume gradient became lower, the sediment sorting at the flow front progressedmore remarkably (Figure 4). Decreasing the debris-flow velocities due to lowering the flume gradient and enlarging the materials’ sizes might havecaused the sediment sorting to progress more remarkably.

[15°]

k3

k3k2(5)

Vertical distance (cm)15, 20, 25

The proportions of coarser particles increased more as being closer to the flow front, whereas the proportions of finer particles decreased (Figure 3).

0.30
0.20
0.10
0.00
1.00 0k4

k3

k4

The continuity equation of k-th particles in the p-th layer

Sediment concentration

Figure 1. Frontal part of debris flow
15, 20, 25 occurred at valley Kamikamihorizawa ofMt.Yakedake in 1976 (Okuda et al., 1977)

2) When the debris flow arrived at the downstream end of the flume, theflow encountered the sampler moving at a constant speed in atransverse direction. The debris flow front was separated by the sampler
into the four boxes over a constant time interval in the range of 1–2 s.

1) The materials were set on the flume.
Water was supplied at the upstream end of the flume.

k3

k4

k3

Our model can explain that coarser particles exist relatively in the upper layer and concentrate at the flow front during downflow (Figure 6).

15, 20, 25

Moving in a transverse direction

k4

k1

3.2 Calculation resultFlowdirection

Blue material; d=10.7mmYellow material; d=7.1mm

4k2

(3)
The equation of bed variation

15, 20, 25

Procedurek2

k4
k4k2k4k3

k4k2k3
k4

k1

k4

(2)

Particle-size distribution of Mean volume Flumematerial
diameter
gradient(Diameter; ratio)(mm)(°)
Case 1.1 10.7 mm; 50%, 7.1 mm; 50%
9.25
15, 20, 25
Case 1.2 10.7 mm; 20%, 7.1 mm; 80%
8.10
Case 2.1 10.7 mm; 50%, 3.0 mm; 50%
8.55
15, 20, 25
Case 2.2 10.7 mm; 20%, 3.0 mm; 80%
6.44

Case 3.1 10.7 mm; 50%, 1.4 mm; 50%8.50
Case 3.2 10.7 mm; 20%, 1.4 mm; 80%
6.28
Case 4.1 7.1 mm; 50%, 3.0 mm; 50%
5.77
Case 4.2 7.1 mm; 20%, 3.0 mm; 80%
4.53
Case 5.1 7.1 mm; 50%, 1.4 mm; 50%
5.65
Movable sampler
Case 5.2 7.1 mm; 20%, 1.4 mm; 80%
4.19
Case 6.1 3.0 mm; 50%, 1.4 mm; 50%
2.46
Figure 2. Experimental setup
Case 6.2 3.0 mm; 20%, 1.4 mm; 80%
1.97※ ※These procedures were repeated three times for each case, and for each flume gradient.

k4k3

k3

The continuity equation for the flow mixture in the total flow layer

CaseWidth; 7 cmk3k3k
1stlayer DL

The governing equations of our developed model are briefly discussed as follows.

The flume experiments consisted of a tilted straight flume and a movable sampler with 4 boxes (Figure 2).
The sediment materials were composed of 2 particles chosen from 4 particles in the range of 1.4–10.7 mm; Boulder
the mass density of the materials (s) was 2.635 g/cm3, and the concentration in the static sediment bed(diameter about 3.1 m)(C*) was 0.558.
The experiments combined various conditions based on two key factors; the particle-size distribution of
the materials and the flume gradient (stream gradient) (Table 1). In these cases, the rate of supplied
water was set to a unit width of 67cm2/s.
Table 1. Experimental cases and conditions

Sediment mixtures
Pump (Thickness; 5 cm)
Flume gradient

2ndDlayer L

k4

k4

k4

k3

k1

k4

k1

k2

flow direction in p-th layer

k4

k3

k4

k4

up Migration velocity in thek3

k2

k4

k4

interior and the concentration of coarser particles at the flow front based on the model proposed by
Satofuka et al. (2007).

PCkp k–th particle in p-th layer
4k3

p-thDlayer L(1)2) We developed a numerical model to describe the changing particle-size distribution in the flow’s

Supply water; 67 cm2/s(a unit width)n p-thklayer DL

In the model, the debris flow depth (h) is divided into several layers (the number of layers = np)
with the same thicknesses (DL = h/np) (Figure 5). Based on the theoretical equations for
distributions of velocity and sediment concentration of a debris flow by Takahashi et al. (1996),
the migration velocity of materials (up) and the sediment concentration of k-th particles (Ckp)
were considered in each divided layer. The falling volume of downward movement of k-thparticles (rkp) was also incorporated.

Proportion in the 3rd layer

1.0
0.8
0.6
0.4
0.2
0.0

Proportion in the 2nd layer

1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
00.0

Proportion in the 1st (bottom) layer

7.10 mm)No.1 particles (d1 = 10.70
3.00 mm)No.2 particles (d2 = 7.10010203040506070
Horizontal distance (cm)8090100Figure 6. Calculated longitudinal results in 3.5 s after supplying waterat upstream end of the flumes in Case 4.1 [20°]4. CONCLUSIONSAs the sizes of the debris-flows materials became coarser or the flume gradient became lower, the sediment sorting at the flow front progressed more
remarkably.
Decreasing the debris-flow velocities due to lowering the flume gradient and enlarging the materials’ sizes might have caused the sedimentsorting to progress more remarkably.
Decreasing the debris-flow velocities enlarges the movement of materials in the depth direction, Middleton’s suggested
mechanism (1970), which is a falling mechanism of finer particles through the interstice between the materials in the flow’sinterior (dynamic sieving), may explain this phenomenon.
We developed a numerical model to describe the above mechanism in the flow’s interior based on the 1-D model proposed by Satofuka et al. (2007)
Our model can explain the inverse grading in the flow’s interior, and the concentration of coarser particles at the flow front
during downflow.
Our model can quantitatively describe the temporary changes in the flux concentration of coarser and finer particles at the flow front.
However, in the case that the sizes of coarser and finer particles were similar, the calculation results underestimated the experimental results for theproportion of coarser particles at the flow front.
In our model, incorporating the rising movements of coarser particles in the upper layer of a debris flow, which is caused by a collisionor contact between particles in the flow’s interior is needed to describe this more accurately.

REFERENCES

Middleton, G. V. (1970); J. ed., Flysch Sedimentology in North America, Geological Association of Canada Special Paper 7, pp.253–272.
Satofuka, Y., Iio, T. and Mizuyama, T. (2007); Proceedings of 4th International Conference on Debris-Flow Hazards Mitigation.
Takahashi, T., Satofuka, Y., and Chishiro, K. (1996); Annuals of Disaster Prevention Research Institute, Kyoto Univ., 39(B-2), pp.333–346 (in Japanese with English abstract).