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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
457
DOI: 10.4408/IJEGE.2011-03.B-051
QUASI-THREE DIMENSIONAL TWO-PHASE DEBRIS FLOW MODEL
ACOUNTING FOR BOULDER TRANSPORT
C.E. MARTINEZ
(*)
, F. MIRALLES-WILHELM
(**)
& R. GARCIA-MARTINEZ
(***)
(*)
Department of Civil and Environmental Engineering, Florida International University
(**)
Department of Earth and Environment, Florida International University
(***)
Department of Earth and Environment, Florida International University and FLO-2D Software, Inc
K
ey
words
: debris flow, boulders accumulation, finite
element method, discrete element method, lagrangian
formulation
INTRODUCTION
Debris flow is a frequently occurring phenom-
enon in mountainous regions. It occurs when masses
of poorly sorted sediments, rocks and fine material,
agitated and mixed with water, surge down slopes in
response to water flow and gravitational attraction. A
typical surge of debris flow has a steep front or “head”
with the densest slurry, the highest concentration of
boulders and the greatest depth. A progressively more
dilute and shallower tail follows this head.
Reviews presented by i
veRson
(1997), exhaus-
tively describe the physical aspects of debris flow mo-
tion and clearly divide previous debris flow research
into two distinct categories. The first, based upon the
pioneering work of J
oHnson
(1965), assumes that de-
bris flow behaves as a viscoplastic continuum. This
model describes a single-phase material that remains
rigid unless stresses exceed a threshold value: the
plastic yield stress. Various rheological models have
been proposed, derived from experimental results or
from theoretical considerations, such as the Bingham
model (b
inGHam
& G
Reen
, 1919), the Cross model
(b
aRnes
et alii, 1989) and the quadratic model pro-
posed by O’b
Rien
& J
ulien
(1985). The Bingham
plastic model is the most commonly used in practice.
The second approach has focus on the mechanics
ABSTRACT
We present a quasi three-dimensional numeri-
cal model to simulate debris flows accounting for
a continuum non-Newtonian fluid phase composed
by water and fine sediments, and a non-continuum
phase for large particles such as boulders. Particles
are treated in a Lagrangian frame of reference us-
ing the 3D Discrete Element Method. The fluid
phase flow equations are solved by the RiverFLO-
2D computational model which is based on the 2D
depth-averaged shallow water approximation and
uses the Finite Element Method on a triangular
non-structured mesh. The model considers particle-
particle and wall-particle collisions, and considers
that particles are immersed in a fluid and subject to
gravity, friction and drag forces. Bingham and Cross
rheological models are used for the continuum phase
providing very stable results, even in the range of
very low shear rates. Results show that the Bing-
ham formulation proves better able to simulate the
stopping of the fluid when applied shear stresses are
low. Results from numerical simulations comparison
with analytical solutions and data from flume-exper-
iments, show that the model is capable of reasonable
approximating the motion of large particles moving
in the fluid flow. An application to simulate a debris
flow event that occurred in Venezuela in 1999 shows
that the model can model the main boulder accumu-
lation reported for the alluvial fan in that event.
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C.E. MARTINEZ , F. MIRALLES-wILHELM & R. GARCIA-MARTINEZ
458
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Assuming non-Newtonian and incompressible
fluid phase, the depth averaged continuity and mo-
mentum equations in Cartesian coordinates can be
written as follows:
where H is the water depth, η is the free-surface el-
evation, and and are the depth averaged velocities in
x and y directions respectively; g is the gravitational
acceleration and is fluid density. FD represents the
fluid-solid interaction force exerted on the fluid by
particles through the fluid drag force, this force is
evaluated as:
where F
FD
i is the fluid drag force on each particle i,
V is the volume of the computational cell and n is the
number of particles in the cell.
S
fx
and S
fy
are the depth integrated stress terms
that depend on the rheological formulation used to
model the slurry. Assuming a Bingham rheologi-
cal model and Manning’s formula, as proposed by
o’b
Rien
& J
ulien
(1985), the stress terms for the fluid
can be expressed as
of granular materials. Based on b
aGnold
(1954), two-
phase models have been developed by several authors,
such as t
akaHasHi
(1991) and i
veRson
(1997). These
models explicitly account for solid and fluid volume
fractions and mass changes respectively.
Despite of the considerable progress over the
past few years, the flow dynamics and internal proc-
esses of debris flows are still challenging in many
aspects. In particular, there are many factors related
to the movement and interaction of individual boul-
ders and coarse sediments that have not been fully
addressed in previous works. a
smaR
et alii (2003)
introduced the Discrete Element Method (DEM) to
simulate the motion of solid particles in debris flows.
DEM is a numerical method to simulate dry granular
flows whereeach particle is traced individually in a
Lagrangian frame ofreference by solving Newton’s
equation of motion.
This paper describes the development of a
quasi threedimensional model to simulate stony
debris flows, considering a continuum fluid phase,
and large sediment particles, such as boulders, as a
non-continuum phase. Large particles are treated in
a Lagrangian frame of reference using DEM, and
the fluid phase composed by water and fine sedi-
ments is modeled with an Eulerian approach using
the depth-averaged Navier–Stokes equations in two
dimensions. Bingham and Cross rheological models
are used for the continuum phase. Particle’s equa-
tions of motion are fully threedimensional. Numeri-
cal simulations have been compared with analytical
solutions, with data from laboratory experiments and
with a real debris flow event.
GOVERNING EQUATIONS
The flow domain is divided in computational cells
with triangular base and depth H, as shown in Figure 1.
Fig. 1 - Schematic representation of debris flow with
large solid particles
(1)
(2)
(3)
(4)
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QUASI-THREE DIMENSIONAL TWO-PHASE DEBRIS FLOW MODEL ACOUNTING FOR BOULDER TRANSPORT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
459
The external force F
E
is given by
The expression to compute the net force acting on
the particledue to gravitational effects is
(14)
where R is the particle radius and ρ
p
is the particle
density.
The expression for the drag on particles in viscous
fluid is given by
where C
d
is the drag coefficient, u is the fluid veloc-
ity vector at the particle location, and v is the particle
velocity vector.
The last two terms in equation (12) represent the
collision forces or contact forces among particles.
Based on the simplified model that uses a spring-
dashpot-slider system to represent particle interac-
tions (a
smaR
et alii, 2003), the normal contact force
and the tangential contact force are evaluated as
The normal contact force F
NC
is calculated using a
Hook’s linear spring relationship,
where N is the Manning roughness coefficient.
The fluid dynamic viscosity μ and yield stress y,
are determined as functions of the volume sediment
concentration Cv, using the relationships proposed by
o’b
Rien
& J
ulien
(1988):
in which α
1
, α
2
β
1
and β
2
are empirical coefficients ob-
tained by data correlation in a number of experiments
with various sediment mixtures.
Using a quadratic formulation combined with the
Cross rheological model, the stress terms for the fluid
are expressed as
where m
eff
is the effective viscosity of the fluid de-
fined by:
In the solid phase, spherical particles of differ-
ent diameters are considered. Particle trajectories are
tracked using Newton’s second law; considering grav-
ity, buoyancy, fluid drag and collision forces.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(11 b)
(12
(13)
(15)
(16)
(17)
(18)
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C.E. MARTINEZ , F. MIRALLES-wILHELM & R. GARCIA-MARTINEZ
460
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
to zero in the formulation, as well as the velocity in
this direction, w. Then, torque in x and y directions
are not detectable by the numerical model. Last com-
ponent is the torque in direction perpendicular to the
sloping surface, which is essentially negligible.
Fluid governing equations (1-3) are solved by
the Galerkin Finite Element method using three-node
triangular elements. To solve the resulting system of
ordinary differential equation, the model applies a
four-step time stepping scheme and a selective lump-
ing method, as described by G
aRCia
-m
aRtinez
et alii.
(2006).
Forces on each particle are evaluated at each time
step, and the acceleration of the particle is computed
from the particle governing equation, which is then
integrated to find velocity and displacement of each
particle.
RESULTS
ANALYTICAL SOLUTIONS
The first modeling step was the implementation
of different rheological models for the simulation
of mud flows. This modeling would account for the
representation of the fluid phase of the debris flow.
The numerical model was run using RiverFLO-2D
software, a finite modeling system for detailed analy-
sis of river hydrodynamics, sediment transport and
bed evolution (G
aRCía
-m
aRtínez
et alii, 2006). In
the software, two rheological quadratic formulations
were implemented, the first, including Bingham the-
ory and Manning’s formula, as proposed by o’b
Rien
& J
ulien
in 1985, and the second, combining Cross
formulation and the Manning’s formula as proposed
in m
aRtinez
et alii (2007).
In order to compare with simple results, an ana-
lytical solution, proposed by H
uanG
& G
aRCia
(1997),
was studied and implemented in a computer program.
This implementation provided enough data for verifi-
cation and testing of the new rheological formulations
proposed (m
aRtinez
et alii, 2007).
LABORATORY EXPERIMENTS
A series of experiments were carried out in a
laboratory flume, using homogeneous fine sediment
mixtures for the continuum phase and spherical mar-
bles for the discrete phase. The experiments were
performed in a 1.9 m long, 0.19 m wide, Plexiglas
walled flume, with adjustable slope. The downstream
where k
N
is the normal contact stiffness and δ
N
is the
displacement overlap) between particles i and j. The
maximum overlap is dependent on the stiffness k
N
.
Typically, average overlaps of 0.1-1.0% are desirable,
requiring stiffness of the order 105-107 N/m.
The normal damping force F
ND
is also calculated
using a linear relation given by
where ν
N
is the normal component of the relative ve-
locity between particles and C
N
is the normal damp-
ing coefficient. This constant C
N
is chosen to give a
required coefficient of restitution b, defined as the ra-
tio of the normal component of the relative velocities
before and after collision.
The tangential contact force, F
TC
, represents the
friction force and it is constrained by the Coulomb
frictional limit, at which point the particles begin to
slide over each other. Prior to sliding, the tangential
contact force is calculated using a linear spring rela-
tionship,
where k
T
is the tangential stiffness coefficient, and
δ
T
is the total tangential displacement between the
surfaces of particles i and j since their initial contact.
When k
t
δ
T
exceeds the frictional limit force
μ
f
F
NC
,
particle sliding occurs. The sliding condition is de-
fined as
where μ
f
is the dynamic friction coefficient.
The tangential damping force F
TD
is not included
in this model, since it is assumed that once sliding oc-
curs, damping is accounted for from friction. Particle
rotation is not considered, since it its impact on boul-
der transport is assumed to be of much less impor-
tance than the friction and drag forces.
Since the fluid phase governing equations are
depth integrated, gradients along z direction are equal
(19)
(20)
(21)
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QUASI-THREE DIMENSIONAL TWO-PHASE DEBRIS FLOW MODEL ACOUNTING FOR BOULDER TRANSPORT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
461
EXPERIMENT 1
In this experiment the flow of a mixture of 23.5%
volume concentration was studied. The flume bottom
slope was set to 4° and the initial volume released was
6.3 L. For t = 3 s the wave practically stopped flowing
as shown in Fig. 2. The propagation of the wave was
recorded for different times t to construct the spread-
ing diagram shown in Fig. 3.
Figure 3 compares the experimental data with the
numerical solution using Bingham rheological formu-
lation. Numerically, the condition of stopping the fluid
is not easy to achieve; however, it is possible to appre-
ciate how the maximum velocity in the fluid decreases
with time and it becomes very close to zero about the
time the fluid must stop. This fact shows that velocity
criteria could be used numerically to stop the fluid.
….The following two figures compare longitudi-
nal profiles and maximum velocity for two different
mesh sizes. Figures 4 and 5 show how mesh refine-
ment contributes to improve substantially the
solution in the advancing front. The wet-dry al-
gorithm proposed by G
aRCia
at alii (2009) was im-
plemented to eliminate dry elements from the calcula-
tion, and then there is a well defined interface between
dry and wet elements.
However, there is a numerical tendency to form
a spurious low depth front that that could be reduced
part of the flume was connected to a wood horizontal
platform, 0.75 m long and 0.95 m wide. A dam-break
type of flow was initiated by an abrupt removal of a
gate, releasing mixtures from a 0.40 m long reservoir
situated on the upstream part of the flume. Water-clay
mixtures were used in all the experiments, with vol-
ume sediment concentration 23.5 % and 26.5 %. For
preparation of the mixtures, kaolinite clay with spe-
cific unit weight of 2.77 was used. Fluid density was
measured in the laboratory and rheological parameters
μ and τ
y
were determined using equations (7) and (8)
in which parameters are α
1
= 0.621x10
-3
, β
2
= 17.3, α
2
=
0.002 and β
2
= 40.2.
Tab. 1 - Rheological properties of experimental fluids
Fig. 2 - Experiment 1, fluid stops flowing over the slop-
ing channel
Fig. 3 - Experiment 1, spreading relation
Fig. 4 - Final free-surface longitudinal profile and
U
max
- mesh size 0.03 m
Fig. 5 - Final free-surface longitudinal profile and
Umax - mesh size 0.01 m
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C.E. MARTINEZ , F. MIRALLES-wILHELM & R. GARCIA-MARTINEZ
462
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
decreasing the element size as well as reducing the
minimum depth parameter that determines the distinc-
tion between dry and wet elements. Best results were
found with a minimum depth parameter equal to 0.01
times the average fluid depth as shown in Fig. 5.
EXPERIMENT 2
In this experiment the same mixture as in experi-
ment 1 is used. In this case, the flume bottom slope
was set to 9.54° and the initial volume released was
6.4 L. The objective of this test was to study the
spreading of the fluid in the fan and to study particle
movement into the fluid.
Fourteen particles, with diameter 2.5 cm and
density 2500 kg/m
3
, were placed in two rows over a
small piece of wood in the mud reservoir, just behind
the gate. By the time the fluid was released, the piece
of wood was quickly removed, so the particles could
start their movement along the channel with the fluid
Figure 6(a) shows the particles resembling the
velocity parabolic distribution across the channel
at t = 0.5 s. Blue particles represent those particles
placed initially in the first row, orange particles are
those placed in the second row. In Figure 6(b) can
be noticed how particles in the center tend to move
forward to reach the front of the wave, particles in
the second row displace particles in the first row to
the sides and these are then left behind because of the
fluid velocity gradient. By the time the flow reaches
the fan, particles move to the sides of the flow as it is
shown in Figure 7.
Figure 7 compares final position of particles ob-
tained experimentally (final position of particles was
measured at the lab) with the numerical results. In the
numerical solution can be seen that there exist some
delay on the particles positioned close to the walls;
this is due to the velocity boundary condition at the
walls. In practical applications of the model, it is
necessary to allow slip at the walls, since the no slip
condition formulated in finite elements becomes very
restrictive. However; it is not possible a total slip con-
dition, since for this case no velocity profile would be
created across the channel. In this simulation 90% of
slip at the wall was considered.
EXPERIMENT 3
In this experiment, a mixture of volumetric con-
centration of 26.5% was studied. In this case, the
flume bottom slope was increased to 10.7° and the
initial volume released was 11.1 L. The objective of
this test was to study the spreading of the fluid and
study particle movement into a mixture with higher
clay concentration.
Figure 8 shows the spreading relation in the lon-
Fig. 6. - Experiment 2, numerical simulation (a) t = 0.5
s, (b) t = 1.6 s
Fig. 7 - Experiment 2: Final position of particles (a)
experimental data(b) numerical solution
Fig. 8 - Experiment 3, spreading relation
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QUASI-THREE DIMENSIONAL TWO-PHASE DEBRIS FLOW MODEL ACOUNTING FOR BOULDER TRANSPORT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
463
the distal end of the fan near the coastline. Data was
used to map the distribution and thickness of depos-
its and to draw contours of maximum boulder size, as
shown in Figure 11.
The numerical simulation was performed using a
finite element mesh with 22,500 triangular elements.
The element characteristic size was 12 m on average.
The topography data used to define the finite element
mesh was interpolated from the original cartographic
information prior to the event (G
aRCia
, 2008).
A 500 year-return period hydrograph was used
as flow input at the fan apex, as shown in Figure
12, with an average volume sediment concentra-
tion of =0.3. The Manning coefficient considered
gitudinal direction for this experiment. This relation is
compared with numerical results obtained using Bing-
ham rheological model and using Cross rheological
model. Both rheological formulations produce very
similar results, they are not totally capable of resem-
bling the spreading of the flow; however, they show
a final fluid extend, when velocities in the fluid be-
come very close to zero, very similar to the real one.
Bingham formulation shows to be more effective in
decreasing the velocities oalong the fluid to zero.
In this experiment 14 particles were placed on the
fluid in a similar manner that was done in the previ-
ous experiment. Figure 9 compares the final particle
positions obtained numerically against final observed
particle location. Note that some particles lag behind
close to the flume wall and that the general location
of the particles on the alluvial fan is very close to the
observed locations.
.
APPLICATION: VENEZUELA’S 1999 ALLUVI-
AL FAN DEBRIS FLOODING EVENT
Heavy rainfall from a storm on December 14-16,
1999, triggered thousands of shallow landslides on
steep slopes of Cerro El Avila, north of Caracas, Ven-
ezuela, and caused flooding and massive debris flows
in the channels of major drainages that severely dam-
aged coastal communities along the Caribbean Sea.
The largest fan on this area is that of San Julián River
at Caraballeda, shown in Figure 10. This fan was one
of the most heavily damaged areas in the event. The
thickness of sediment deposition, maximum size of
transported boulders, and size of inundated area were
all notably larger in this drainage in comparison to the
other close watersheds.
The US Geological Survey studied the affected
area (w
ieCzoRek
et alii, 2001), measuring slope, de-
posit thickness, and boulder size from the fan apex to
Fig. 9 - Experiment 3: Final position of particles, (a)
experimental data(b) numerical solution
Fig. 10 - Caraballeda alluvial fan, Venezuela
Fig. 11 - Contours of maximum transported boulder size
on the Caraballeda alluvial fan, Venezuela.
From USGS, 2002
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C.E. MARTINEZ , F. MIRALLES-wILHELM & R. GARCIA-MARTINEZ
464
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
was equal to 0.065 in the whole fan area, in order
to take into account terrain irregularities. The same
value was used by G
aRCia
(2008), and was found a
good estimate for the area. The empirical relation-
ships (7) and (8) were selected for the calculation of
fluid rheological properties, using the parameters for
water-clay mixtures. As a result of the volume sedi-
ment concentration, = 0.3, ρ = 1531 kg/m
3
, m =
0.11 Pa.s, t
y
= 105 Pa.
During the simulation, 1600 boulders with sizes
ranging from 1 m to 6 m in diameter were included
in the event. The boulders were placed into the fan
during the first three hours of simulation, at a rate of
50 particles every 6 min. This amount of particles was
selected to ensure a manageable computational time.
Density for the boulders is r
p
= 2600 kg/ m
3
, equal to
the density of Gneiss boulders, the type of boulders
mostly found in the area by the USGS.
Figure 13 shows the flooded area at time t = 2.2
h, the time corresponding with the peak discharge in
Figure 12. Comparing this region with the post-event
aerial view shown in the background, it can be noted
that the model acceptably reproduces the extent of the
area affected by the debris flow.
Figure 14 shows the velocity field at time t = 2.2
h. It can be seen that major velocities occurs in the fan
apex, where the discharge of the river is simulated.
Velocities decrease at the urban areas, ranging from
1.0 to 6 m/s at the time of the hydrograph maximum
value. Higher velocities are developed in the avulsion
zone, at the center of the fan,, reaching 10 m/s. The ve-
locities calculated by the model are in good agreement
with those estimated by USGS, which ranged from 1.3
to 13.6 m/s.
Figure 15 shows how boulders, are transported by
the flow along the main drainage paths.
It is interesting to note how the largest boulders
follow the path of the original concrete channel toward
the right side of the fan, while smaller boulders take
the central path. According to the USGS report, the
slope at the center was 4 degrees, while the concrete
channel direction, was steeper, with a slope gradient
of 5.5 to 6 degrees, then larger boulders were trans-
Fig. 12 - Inflow hydrograph for a 500 year-return pe-
riod, including solid concentration by volume
of 0.3
Fig. 13 - Flooded area at time t =2.2 h. Legend indicates
flow depth in m
Fig. 14 - Velocity field at t =2.2 h. Legend indicates ve-
locity in m/s
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QUASI-THREE DIMENSIONAL TWO-PHASE DEBRIS FLOW MODEL ACOUNTING FOR BOULDER TRANSPORT
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
465
CONCLUSIONS
This paper describes the development and appli-
cation of a quasi three-dimensional two-phase model
to simulate debris flows, considering large particles,
such as boulders. The continuum non-Newtonian
phase is solved by RiverFLO-2D finite element model
in two horizontal directions and the particle transport
with the Discrete Element Method in 3D.
The model is able to simulate fluid and particle
transport when compared against several experiments
in a laboratory flume-fan, including the effect of parti-
cle-particle and wall-particle collisions.
Bingham and Cross rheological formulations pro-
vide very stable results, even in the range of very low
shear rates. In the simulation of mud dam-break prob-
lems, Bingham formulation was better able to simu-
late the stopping stage of the fluid; however, the Cross
formulation proved more accurate for early stages of
the solution.
An application to the well documented debris flow
event that occurred in Venezuela in 1999 illustrates
the capability of the model to simulate large scale real
events. Results show that the model reasonably ap-
proximates the flood extent affected by the debris flow
and the observed boulder accumulation areas, including
distribution boulders sizes. Future work includes com-
parison with field events using larger number of boul-
ders and enhancement of the detecting particle contact
algorithm in order to improve computational time.
ported to this side. These values of mean nominal di-
ameter and slope steepness reflect USGS observations
that for the larger transported and deposited boulders
there was a proportional relationship between boulder
size and slope steepness.
Figure 16 shows boulder positions after 6 hours
of simulation. Smaller boulders continue taking the
central direction alignment, some of them reached the
shoreline or entered into the sea. Larger boulders were
deposited in the avulsion zone or took right direction
to the concrete channel. None of these large boulders
reached the shoreline. In Figure 16, it can be seen that
the model predicts reasonably boulder locations as
compared with the field data given in Figure 11.
According to USGS, the largest boulders were
found in the avulsion zone, within a thick matrix, evi-
dence that strongly supports transport by debris flow.
At other sites, the largest boulders were observed iso-
lated along the concrete channel, fact that suggests
that these boulders moved sliding along the bottom of
the channel in a dilute fluid until deposition occurred
(USGS Report 01-0144). There is no indication of big
boulders close to the shoreline at this site of the fan.
According to t
akaHasHi
(1991), during the proc-
ess of deposition, debris flows deposit the boulders
in order from bigger to smaller as it proceeds down-
stream on alluvial fans. This process was better ob-
served along the central direction and it was also rep-
licated in the numerical simulation.
Fig. 15 - Boulder positions at time t =1.8 h, Legend in-
dicates particle diameter in m
Fig. 16 - Boulder positions at time t = 6.0 h, Legend in-
dicates particle diameter in m
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C.E. MARTINEZ , F. MIRALLES-wILHELM & R. GARCIA-MARTINEZ
466
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
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