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IJEGE-11_BS-Fan-et-alii

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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
409
DOI: 10.4408/IJEGE.2011-03.B-046
SIMULATION BASED ON FINITE VOLUME METHOD
OF THE ENTRAIMENT OF DEBRIS FLOW
y
unyun
FAN
(*)
, s
iJinG
WANG
(**)
, e
nzHi
WANG
(**)
& z
HonGGanG
LIU
(**)
(*)
Northeastern University-China - College of Resources and Civil Engineering; email: fyun1982@gmail.com
(**)
Tsinghua University-China, State Key laboratory of Hydroscience and Engineering
ures of prevention and reduction of disasters can be
offered. Therefore, the dynamic simulation of debris
flow is significant and valuable.
In the development of debris flow, the surface
bulk material along the way would have a shear fail-
ure and slide with debris flow during the motion proc-
ess so that the entrainment is formed which increase
the volume of debris flow and change the composition
of motion material. For this reason, the surface mate-
rial is an important factor in the development of debris
flow and should be considered in its dynamic simula-
tion. At the moment, many scholars have begun their
studies on the entrainment problems during the mo-
tion of debris flow and some progress has been made.
In order to realize the dynamic simulation of de-
bris flow entrainment and study the influences of this
entrainment on dynamic simulation, the back analy-
sis is made to study a typical historical debris flow,
Nomash River in Canada, by combining entrainment
dynamic model theory which considers the entrain-
ment effects and the method of finite volume discre-
tization based on the approximate Riemann solver of
HLL Scheme. The calculation also takes the variation
of terrain slope caused by entrainment into account.
The calculation results which are identical to actual
disaster-caused area confirm the effectivity of the dy-
namic model theory and the numerical solution. Com-
pared with the results which have no consideration of
entrainment, the influence of entrainment action on
dynamic process of debris flow is analysed.
ABSTRACT
The mountain hazards like snow avalanches,
landslides, rock falls, debris flows all have strong de-
structive power which seriously threaten human lives
and belongings. Therefore, it is necessary to study
more about the happenings and developments of these
disasters. Among the important and common features
of the debris flow, the entrainment is the one that can
increase the volume of debris flow and affects signifi-
cantly the flow motion. These influences usually re-
sult in a more harmful and stronger destructive power.
With the method of finite volume discretization, the
numerical simulation of the entrainment process of
debris flow is achieved in this paper. The influences
of entrainment capabilities on the motion and deposi-
tion of debris flow are mainly studied. The typical en-
gineering examples show and prove the stability and
effectivity of numerical methods.
K
ey
words
: debris flow, entrainment, finite volume, dynamic,
simulation
INTRODUCTION
The dynamic simulation of debris flow can not
only improve human cognition in the regularity and
characteristics of disasters by inversing and reappear-
ing the development law of debris flow disasters, but
also predict the disaster-caused area and intensity of
debris flow, and consequently, scientific references
of industry planning and construction, and the meas-
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y. FAN, S. wANG, e. wANG & Z. LIU
410
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
where v=(u,v) is velocity vector.
(2) Voellmy resistance
If there exists water during the motion process of
debris flow, its resistance effect should also be consid-
ered, in this case, Voellmy resistance is usually used as
the resistance model (H
unGR
& e
vans
, 2004; m
C
d
ou
-
Gall
& H
unGR
, 2005; H
unGR
2009):
where sgn(a) is sign function which can be described as:
In eq. (5), f is friction coefficient, ξ is hydraulic
parameter, the first item from the right hand represents
friction resistance which has the same form with eq.
(4), f corresponds to tan δ, and the second item repre-
sents other resistances, which are experientially used
to express the items relative to velocity in the analysis.
ENTRAINMENT VELOCITY
The entrainment velocity form of Hungr is used
in this paper (H
unGR
& e
vans
, 2004; m
C
d
ouGall
&
H
unGR
, 2005), so the relationship between entrain-
ment velocity E
t
and rate of increase of entrainment
E
s
is as follows:
A reasionable E
s
can be obtained only if it is ajust-
ed according to the actual condition repeatedly. One
feasible method is to use the average rate of increase
Ē
s
, obtained by use of the total volume before and af-
ter entrainment, which can be describes as:
where V
0
is the total volume before entrainment, and
V
f
after entrainment, and d is the approximate average
length in entrainment area.
FINITE VOLUME NUMERICAL METHOD
FINITE VOLUME DISCRETIZATION
The computational domain is discretized by us-
ing the layout form of the triangle unstructured tri-
angular meshes and Cell Center which have a great
adaptability.
DYNAMIC MODEL THEORY
BASIC EQUATION
Assuming that the density of debris flow is invari-
able during the motion process and ignoring the varia-
tion of physical variable in the direction of flow depth,
the dynamic model equation is as follows:
Where U is conservation variable, F=(E,H) is
calculation flux, S is source item, these items can be
expressed as follows:
In this expression, H is the depth of flow, u and
v are respectively the velocities in x direction and y
direction, g is acceleration of gravity, E
t
is the source
item of entrainment velocity, z
b
is elevation on bottom
surface in the coordinate system, and S
f
is the source
item of resistance, they can be used according to differ-
ent conditions, k
act/pass
is the coefficient of lateral stress
which can be expressed as (s
avaGe
& H
utteR
, 1989;
i
veRson
& d
enlinGeR
, 2001; P
iRulli
et alii, 2007):
where ϕ is internal friction angle and δ is bottom
friction angle. Different motion states correspond
with different cofficients of lateral stress. When
u/x+v/y>0, the motion is in active state, how-
ever when ∂u/x+v/y<0, the motion is in passive
state (s
avaGe
& H
utteR
, 1989; i
veRson
& d
enlinGeR
,
2001; P
iRulli
et alii, 2007).
RESISTANCE MODEL
Choosing a correct resistance model of the model
equation of dynamic theory is an important step in
the dynamic simulation of debris flow. In this paper,
analysis is achieved by use of friction resistance and
Voellmy resistance.
(1) Friction resistance
During the process of debris flow, the resistance
is mainly from the bottom friction, and the friction
resistance can be considered as (s
avaGe
& H
utteR
,
1989; i
veRson
& d
enlinGeR
, 2001; m
C
d
ouGall
&
H
unGR
, 2005; P
iRulli
et alii, 2007; H
unGR
, 2009):
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
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SIMULATION BASED ON FINITE VOLUME METHOD OF THE ENTRAIMENT OF DEBRIS FLOW
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
411
when the right side is dry, s
L
and s
R
can be calculated as
follows:
SPATIAL SECOND-ORDER LINEAR RECON-
STRUCTION
When the numerical discretization method of fi-
nite volume is employed, the variable of the control
volume of spatial second-order accuracy should be
distributed linearly. With the MUSCL method (o
sHeR
,
1996), the states of the both edges are concluded by
interpolating variables, the linear reconstruction on
unstructured mesh is as described in Fig.1.
In Fig. 1, U
i
, U
k
, U
l
, U
j
, U
s
and U
r
are centroid val-
ues of the element, U
lk
, U
lr
, U
ij
, U
ks
, and U
rs
are linear
interpolation, while U
L
ij
and U
R
ij
are the linear upwind
values at the both edges, obtained by extrapolating
variables, U
C
ij
is the central value at the edge.
The limited linear interpolation values are used to
calculated the variables at the both edges:
where the slope of linear interpolation can be ex-
pressed as:
In order to make the scheme satisfy the needs of
TVD, Φ(r) employs the scheme of Van leer:
Besides, with Runge-Kutta methods (H
ubbaRd
et
alii, 1999), the format of space-time second order ac-
curacy is obtained.
Changing eq. (1) into vector form, then find the
integral of this vector form:
Considering the U
i
and S
i
as the average values
of element variable and source item respectively, and
placing them in the centre of element, then:
By using the Green formula to transform eq. (9)
from surface integral to line integral, it is obtained that:
where L is the boundary of the ith element, A
i
is the
surface of element i, n=(n
s
,n
y
)=(cosθ,sinθ) is the direc-
tion of exterior normal, and θ is the angle between the
exterior normal and x-axis.
The following equation can be obtained by discre-
tizing eq. (11):
where ΔU
i
is the increment of variable, Δl
ij
is the side
length of element, F
ij
represents the normal numerical
flux at the j th edge of the i th element.
NUMERICAL FLUX CALCULATION
The present model employs the HLL Riemann solv-
er to compute the normal flux at the face of an element
as follows (H
aRten
et alii, 1983; a
liPaRast
, 2009):
where L and R are variables at the inner edge and out-
er edge respectively; (U
L
)
i,j
and (U
R
)
i,j
are the U value
on the left and right sides at the j th edge of the i ele-
ment, respectively; s
L
and s
R
are wave velocities of the
both edges, respectively, it can be expressed that:
where ,
u
*
and c
*
can be described as fol-
lows:
when the left side is dry, s
L
and s
R
can be calculated as fol-
lows:
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
Fig. 1 - Schematic plan of numerical reconstruction
(18)
(19)
(20)
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y. FAN, S. wANG, e. wANG & Z. LIU
412
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Before reaching the valley bottom, an additional
360 000 m
3
of saturated colluvium was impacte by the
entraiment, and the entrainment ration reached 0.96
which is really alarming number. After the entrain-
ment, a 100~150 m-wide erosion area was left, the
vestige and trace of entrainment are visible in Fig. 3
CALCULATION AND RESULT ANALYSIS
The topographic condition used in back analysis is
from the results of Hungr and Evans (m
C
d
ouGall
&
H
unGR
, 2005), by modifying the topography after the de-
bris flow. The volume before and after the entrainment is
V
0
=375.000 m
3
V
f
=735.000 m
3
and respectively, the aver-
age length of the erosion area is d=350 m, and according
to eq. (8), Ē
s
=0.0019 m
-1
is obtained; before the entrain-
ment, the friction resistance model is employed, where
the internal friction angle φ=35°, and the bottom friction
angle δ=30°, when the entrainment happened, Voellmy
CALCULATION OF TERRAIN SLOPE
Among the source items, the derivative term of
elevation on bottom surface should be especially
traited. On unstructured triangular meshes, through
the calculation with the coordinate values of the three
nodes of the triangle element, this derivative term can
be obtained as follows:
where (x
k
, y
k
) represents the coordinate value of the k th
node of the i th triangle, z
bk
represents the elevation on
bottom surface of the k th node in the coordinate system.
During the process of entrainment, elevation on bot-
tom surface changes with the reduction of the bottom
mass, at this time, the elevation on bottom surface recon-
structs linearly in space like other variables above-men-
tioned, so the derivative term of the source items can be
calculated with the reconstructed values of the element:
where (x
ij
, y
ij
) represents the coordinate value of the
middle point of the j th edge of the i th triangle ele-
ment, z
bLj
represents the internal elevation of bottom
surface of the j th edge which is obtained through the
spatial second-order linear reconstruction.
CALCULATION AND ANALYSIS OF AN
EXAMPLE
NOMASH RIVER DEBRIS FLOw
Nomash River lies in the west part of British Co-
lumbia, Canada, the area bedrock is mainly made of
crystalline limestone.
In April 1999, a debris flow occurred in the upper
course of the Nomash River during spring snowmelt,
as shown in Fig. 2. The source area located on a moun-
tain slope angled at 50° to the northeast of the U-shaped
river valley. At the beginning, a continuous shear plane
was formed after a series of crossed joints about 430
m above the river, then 300 000 m
3
of crystalline lime-
stone was seperated and suddenly collapsed. Assuming
25% bulking of the source failure, there was 375.000
m
3
of fragments produced by the slide, which fell into
the valley of the Nomash River perpendicularly (H
unGR
& e
vans
, 2004; m
C
d
ouGal
& H
unGR
, 2005).
(21)
(22)
Fig. 2 - Nomash River debris flow
Fig. 3 - Entrainment area of Nomash River debris flow
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SIMULATION BASED ON FINITE VOLUME METHOD OF THE ENTRAIMENT OF DEBRIS FLOW
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
413
is very significant in the dynamic process of debris flow.
The final results in Fig.5 obtained at the 40
th
second are
calculated without considering entrainment effects.
The above results show that without regard to en-
trainment, the deposit will stop at the valley bottom
because of lacking forces which can push it to climb
the slope, as a result, the disaster-caused area is much
smaller than that is shown in Fig.4. Thus it is known
that entrainment plays a very important role during a dy-
namic process of debris flow. It makes debris flow more
powerful and harmful and should be considered as a key
factor in the formation of large-scale debris flows.
resistance model is used, and the calculation parameters
are ξ=400 m/s
2
and f=0.05. With the above method, the
results at different time are obtained as shown in Fig.4.
It can be seen from Fig.4 that the thick real line rep-
resents the real disaster-caused area, the interval between
the geography contours is 20m, and the flow depth con-
tour is 1m. It can be seen from the above calculation that
the back analysis shows the whole motion process of the
debris flow: at the 30
th
s, the flow motion after the entrain-
ment almost arrived at the highest point of the opposite
mountain slope, at the 60
th
s, the flow motion went back-
ward after a second turning, and the deposit was gradually
formed. Certes, there still exists some differences between
the calculation results and actual situation, and the calcu-
lation results could be more exact if a more reasonable
modification of geography can be made and the calcu-
lation parameters of entrainment of path material which
can better conform to the actual situation are applicable.
Nevertheless, it is still satisfactory that the simulation re-
sults are relatively identical to actual disaster-caused area.
It can be seen from the calculation that the entrain-
ment effect which greatly increases the motion volume
Fig. 4 - Calculation results at different times
Fig. 5 - Calculation results without the consideration of
entrainment
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y. FAN, S. wANG, e. wANG & Z. LIU
414
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
CONCLUSIONS
Entrainment is one of the most important factors
that can not be ignored in the dynamic process of de-
bris flow, which should be considered in the dynamic
simulation of debris flows.
The dynamic simulation of the entrainment of de-
bris flow is realized by combining entrainment dynamic
model theory and the method of finite volume discreti-
zation. The variation of terrain slope caused by entrain-
ment is also considered in the simulation. The back
analysis is made to study Nomash River debris flow,
a typical historical debris flow happened in Canada.
The calculation results which are relatively identical
to actual disaster-caused area confirm the effectiv-
ity of the dynamic model theory and the numerical
solution. Compared with the results which have no
consideration of entrainment, it is can be seen that the
entrainment action can increase the motion volume of
entrainment, change the composition of motion mate-
rials and enhance the motility of debris flow, so that
the entrainment will be more destructive and harm-
ful. In fact, entrainment is a main cause of large-scale
debris flow, so it should be paid more attention in the
dynamic simulation of debris flow.
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aRten
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ax
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ubbaRd
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unGR
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vans
s.G. (2004) - Entrainment of debris in rock avalanches; an analysis of a long run-out mechanism. Geological
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unGR
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unGR
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