# IJEGE-11_BS-Yoshino-et-alii

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-059*

**APPLICATION OF A TWO LAYER MODEL WITH THE AID OF A SLOPE**

**COLLAPSE MODEL TO THE NATURAL LANDSLIDE DAM OUTBURST**

**PROCESS**

**INTRODUCTION**

downstream area (e.g., C

*et alii.*, 1988; k

stream area requires accurate prediction of the over-

topping erosion of the landslide dam and the down-

stream flood runoff.

rapid changes in the sediment movement pattern, in-

cluding the transition from debris flow to sediment

sheet flow and bed load. Previously, a method based

on one-dimensional simulation was proposed for the

prediction and assessment of degree of erosion; this

model was used to estimate the peak discharge rate

and changes in the riverbed after the overtopping ero-

sion of the landslide dam (s

*et alii*, 2007).

the discharge rate and changes in a riverbed in the

event of a landslide dam outburst (e.g., m

*et alii*,

*et alii*, 2010). However, since this

not be used for obtaining data on the extended width

of the water channel during the overtopping erosion

of a landslide dam. Hence, a twodimensional simula-

tion model must be used for a detailed assessment of

the flow around a landslide dam and for predicting the

erosion process.

*et alii*(1993) proposed a two-dimen-

**ABSTRACT**

or landslide. Landslide dams can result in catastrophic

outburst floods or debris flows when the dam breaches

with overtopping erosion. Therefore, it is important to

determine the discharge rate and area of flooding due

to overtopping erosion to mitigate disasters triggered

by landslide dams. This study applied a two-layer

model that incorporates a slope-collapse model to

the erosion process at landslide dams. The two-layer

model was proposed by Takahama and deals with the

process of deposition-erosion with fully dispersed de-

bris flow and sediment sheet flow. A two-layer model

is derived to unify these two kinds of flow. Moreover,

Takahama applied this model to a two-dimensional

numerical simulation of debris flow, and found that

the model can analyze phenomena in which the veloc-

ity direction of the upper-layer water flow differs from

that of the lower layer sediment-water mixture flow.

Furthermore, we introduced the slope-collapse model

proposed by Sekine to express gradual collapse of the

slope due to erosion in the riverbed. The model shows

the process that maintains a constant angle at the side

bank. We examined the erosion and deposit process of

a landslide dam with a numerical simulation.

**K**

**ey**

**words**

**:**landslide dam, outburst flood, numerical simula-*tion, and slope-collapse*

*k. YOSHINO, J. TAkAHAMA, T. MIZUYAMA, k. OGAwA & T. UCHIDA*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

t

*et alii*(2004) and performed a two-dimen-

landslide dams.

**CALCULATION MODEL**

*EQUATION FOR TwO-LAYER MODEL*

*et alii*(2004) is used.

the essential differences in the constitutive laws for

the sediment moving layer and the water flow layer in

the sediment sheet flow, as shown in Figure 1. Further,

a governing equation based on the volume conser-

vation law and momentum conservation law is used

in this model. The conservation law is proposed by

taking into account the water flow flux (s

two fluxes are dependent on the velocity vector at the

interface. When this model is extended to two dimen-

sions in the horizontal plane by using the X-Y coor-

dinate system, the dominant equations can be given

as follows.

method involves the classification of sediment trans-

portation patterns and the application of the friction

law to each pattern. This model takes into account the

side bank erosion caused by the shear stress. With this

method, the shear stress is estimated to be one-half of

that in the direction of the riverbed, and the erosion

velocity is assumed to decrease uniformly across the

calculated area because of the relative elevation of the

collapsed part of the dam on the side bank. On the

other hand, t

*et alii*(2003) assumed that the

layer under unsteady conditions is reflective of the

constitutive law for each layer and proposed an analy-

sis method based on a two-layer model. In this model,

a governing equation is considered for each layer, on

the basis of the volume conservation law and momen-

tum conservation law; further, it is assumed that the

sediment sheet flow comprises a low-concentration

layer (water flow layer) and a high-concentration lay-

er (sediment moving layer) separated by a interface.

t

*et alii*(2004) extended this model to a two-

of the sediment transportation patterns is not neces-

sary and direct calculation of the unsteady flow in the

sediment moving layer and water flow layer can be

carried out. s

degree of erosion, resulting in the same gradient of

slope; this method could be used for the accurate pre-

diction of the collapse of the side bank.

*Fig. 1 - Pattern diagram for thet wo layer model*

**APPLICATION OF A TWO LAYER MODEL WITH THE AID OF A SLOPE COLLAPSE MODEL TO THE NATURAL LANDSLIDE DAM**

**OUTBURST PROCESS**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*t*

*AKA*

*-*

*hAmA*

*et alii*., 2004) obtained by extending the erosion

*et alii.*(1988)

at the interface.τ

respectively, both of which are obtained from the con-

stitutive law proposed by e

*et alii*(1997) un-

are correction coefficients based on the flow velocity

and concentrationdistribution and are set to 1 for the

all analyses in this study.

locity used in this study can be found in t

*et*

*alii*(2003). Note that in this study, we introduce the

following approximation and assumptions (t

*et alii*, 2003):

value of this parameter is zero. Thus, at equilibrium,

the stress in the riverbed is the same for the approxi-

mation solution and exact solution. The ratio of p

(e

*et alii*, 1997): p

the sediment moving layer is considered as total sedi-

ment moving layer, with c

distribution and friction coefficient of the sediment

moving layer, the friction coefficient is calculated by

using the equilibrium gradient corresponding to the

average concentration in the total layer (c

coefficient obtained by m

*et alii*(2002), who

der the condition that the yield stress is greater than

the external force.

proposed by m

sediment moving layer and the water flow layer, re-

spectively. M = uh ; N = vh ;where u , and v denote the

average layer flow velocity in the x- and y-directions,

respectively. h denotes the flow layer thickness; ρ ,

the average layer density; and P , the integrated pres-

sure. p

and sediment concentration of the deposit layer. z

*k. YOSHINO, J. TAkAHAMA, T. MIZUYAMA, k. OGAwA & T. UCHIDA*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

rections (denoted by tan α and tan ω , respectively):

the gradient exceeds tanφ, which is determined from

the repose angle. In this study, we obtain the average

gradient by estimating the bank slope of the water

channel generated by the overtopping erosion of an

actual landslide dam.

creased by a factor of ε. The collapsing surface is

shown by the shaded ∆TAN in Figure 2 (a); ε denotes

the extent of decrease in the vertical direction at point

O and is given by

form during one calculation time step is similar to the

collapse that goes to completion within a time step

of the same duration. Here, the amount of sediment

to the scenario where the momentum value calculated

by excluding the yield stress lies in a circle whose

radius is a product of the yield stress and pitch time

and whose center is the point of origin (m

time step adjacent to the stopping step exceeds the

maximum stiction, remigration occurs in this model

(t

*SLOPE COLLAPSE MODEL*

dimensional simulation, such changes do not occur in

the region where there is no flow at the horizontal sur-

face. For example, if a water channel, as in a natural

landslide dam, causes riverbed decrease (flat head of

landslide dam erosion) to such an extent that the rela-

tive height difference becomes very large and there

is no flow into the nearby mesh, no further decrease

occurs in the mesh, and a very steep gutter is gener-

ated in the water channel. For a simple representation

of the side-bank collapse caused by erosion in the flat

head of the landslide dam, we use the slope-collapse

model proposed by s

point of origin (O), as shown in Figure 2. When the

steepest slope is directed from point O to point S, the

vector for the point of appearance of this slope is as

shown below.

*Fig. 2 - Conceptual diagram for slope-collapse model (S*

*eKiNe*

*, 2003)*

**APPLICATION OF A TWO LAYER MODEL WITH THE AID OF A SLOPE COLLAPSE MODEL TO THE NATURAL LANDSLIDE DAM**

**OUTBURST PROCESS**

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

occurs, as shown in Figure 3. In other words, if a mesh

in which no flow occurs is lowered in height owing to

the side-bank collapse and a new flow is generated in

the process of calculation, the adjacent mesh is taken

into account for investigating the side-bank collapse

**CONDITIONS FOR NUMERICAL CAL-**

**CULATION**

*DETERMINATION OF ANGLE OF THE SIDE*

BANk

BANk

of the actual landslide dam formed during the Iwate-Mi-

yagi inland earthquake (y

*et alii*, 2010). Figure 4

ter channel of the landslide dam and the slope of the side

follows:

discharge is toward the steepest gradient; the vector

corresponding to the discharge rate is as shown be-

low.:

discharge rate (shown in formula (13) determined from

the sediment discharge function. We use this concept

to calculate the equilibrium of sediment caused by the

side-bank collapse from the summation ( Q

( Q

lowing equations of continuity. Note that the sediment

amount on the side bank is considered to be com-

pletely included in the nearby mesh in the collapse

direction.

*Fig. 3 - Placement of the calculation mesh*

*Fig. 4 - Relation between the relative height of the water*

*channel and the angle of inclination of the side*

*shore*

*k. YOSHINO, J. TAkAHAMA, T. MIZUYAMA, k. OGAwA & T. UCHIDA*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

varies by around 20 degree even for the same relative

height and that the slope increases with relative height of

the water channel. Therefore, it is necessary to consider

this effect when calculating the slope of the side bank;

however, for a simple representation of the model in this

study, we perform calculations by setting the minimum

angle required for the occurrence of the side bank col-

lapse to φc = 36 degree, which is the average angle of

the water channel after overtopping erosion.

*CALCULATION CONDITIONS*

lowing three cases. The diagrams at the top in Figures

5and 6 show the cross-sectional profile of each side

bank, and the diagrams at the bottom show the longi-

tudinal profile. In addition, 1 shows the longitudinal

profile of the side bank, and 2 shows the longitudinal

profile of the water channel in each Figure.

on the upstream and 14° on the downstream. 2-cm-

high, 20-cm-wide water channel is form at the gradual

slope. In this case, the side bank collapse is taken into

account.

that the side bank collapse in the water channel is not

taken into account.

on the upstream and 3° on the downstream. 2-cm-

high, 20-cm-wide water channel is formed at the

steep slope. In this case, the side bank collapse is

taken into account.

height of the water channel, at the given supply dis-

charge rate; further, there is no flow over the water

channel. Therefore, flow-induced riverbed erosion

does not result in any increase in the width of the

water channel.

*Fig. 5. Case 1,3 (top: cross-sectional view bottom: longitu-*

*dinal section view)*

*Fig. 6 - Case 2 (top: cross-sectional view; bottom: longi-*

*tudinal section view)*

*Tab. 1 - Calculation conditions*

*Tab. 2 - Cases for calculation*

**APPLICATION OF A TWO LAYER MODEL WITH THE AID OF A SLOPE COLLAPSE MODEL TO THE NATURAL LANDSLIDE DAM**

**OUTBURST PROCESS**

water channel is maximum at the gradient changing

point. The water channel first becomes narrow in the

downstream direction and then becomes wide as we

proceed further downstream. As shown in Figure 10,

this is because riverbed erosion proceeds in the down-

stream direction, where the flow spreads over a wider

region in the transverse direction. On the other hand,

because side bank collapse is not taken into account in

**RESULTS OF NUMERICAL SIMULATION**

the riverbed level and water surface level determined

by our calculation.

Then, the riverbed erosion proceeds gradually to the

upstream to cause decrease at the upper end (flat head

*Fig. 7 - Variation of riverbed level with time (Case 1)*

*Fig. 8 - Variation of riverbed level with time (Case 2)*

*Fig. 9 - Variation of riverbed level with time (Case 3)*

*k. YOSHINO, J. TAkAHAMA, T. MIZUYAMA, k. OGAwA & T. UCHIDA*

**CONSIDERATIONS AND FUTURE CHAL-**

**LENGES**

lapse and used the model to carry out calculations by

assuming overtopping erosion of a landslide dam. We

simulated a side bank collapse process in which ero-

sion advances in the vertical direction. We also found

that the water flow layer is separated according to

the sedimentation pattern, resulting in a low-concen-

tration flow on the ground form after sedimentation;

this eventually leads to recurrence of erosion. Thus,

we could simulate the conditions under which a wa-

ter channel is formed. It has been reported that once

landslide dam breach occurs, a highconcentration flow

during which eroded sediment erosion occurs is usual-

ly induced; because of this type of flow, sediments are

often strong unsteady phenomenon property (t

*et alii,*2004). In addition, we considered the side

in the water channel width in the erosion area.

water channel formed by overtopping erosion in the

case of the landslide dam, which in turn was formed

during the 2008 Iwate-Miyagi Nairiku Earthquake.

We also provided a simple method for calculating the

average gradient. In reality, this angle may vary with

the relative height of the water channel, materials that

make up the landslide dam, apparent cohesion,etc. In

the future, it is necessary to study multiple cases of

landslide dam breach by carrying out measurements

with the help of a LiDAR data set; in this case, empha-

sis should be laid on various factors such as the relative

height of the water channel, angle of the side bank, and

materials that make up the landslide dam.

does not increase.

given time is greater than in Case 1. In Case 3, as

riverbed erosion advances from the upper end (flat

head of the landslide dam), the side bank collapses.

Then, the erosion advances gradually to the down-

stream. In addition to this, erosion occurs again in

the sediment deposited area, and the water channel

is gradually formed.

*Fig. 10 - The riverbed level and water surface level (top:*

*Case 1; middle: Case 2; bottom: Case 3)*

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**APPLICATION OF A TWO LAYER MODEL WITH THE AID OF A SLOPE COLLAPSE MODEL TO THE NATURAL LANDSLIDE DAM**

**OUTBURST PROCESS**

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