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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
435
DOI: 10.4408/IJEGE.2011-03.B-049
DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL
STUDIES OF BOULDERY DEBRIS FLOWS
K.M. HILL
(*)
, y
oHannes
BEREKET
(*)
, w
illiam
E. DIETRICH
(**)
, & l
eslie
HSU
(**)
(*)
St. Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55414
(**)
Department of Earth and Planetary Sciences, University of California-Berkeley, Berkeley, CA 94720
INTRODUCTION
Bouldery debris flows –consisting of particles
ranging from boulders to fine particles with a variety
of potential interstitial fluids – are dramatic features in
steep upland regions (e.g., i
veRson
, 1997) and refer-
ences within). They play an important role in sculpt-
ing the landscape in steep upland regions and have the
potential for causing tremendous loss of damage and
property (e.g., s
toCk
& d
ietRiCH
, 2006 and references
within). Of additional interest is the wide variety of
complex behaviors exhibited by debris flows. They
exhibit a rich variety of dynamics including complex
solid-like and fluid-like behaviour and dynamic spon-
taneous examples of pattern formation. Debris flows
often start to flow under conditions such as a large
rainfall event, but the initiation point is difficult to pre-
dict. Once they start to move, they exhibit a variety of
behaviours from those similar to a shallow fluid flow,
to that of an energetic granular material. Segregation
of particles by size mediates the behaviour while the
debris flow travels and also in the manner in which it
comes to rest. Like a granular material, debris flows
stop flowing over a bed of nonzero slope; in other
words, they resist macroscopic shear. However, the
angle of the slope at which they stop is significantly
lower than the measured angle of repose of the debris
flow giving rise to a so-called long-runout avalanches
(P
HilliPs
et alii, 2006; l
inaRes
-G
ueRReRo
, 2007). This
is likely due in part to a dynamic pore pressure ef-
fect giving rise to complex fluid-particle interactions
ABSTRACT
Bouldery debris flows exhibit a rich variety of
dynamics including complex fluid-like behaviour and
spontaneous pattern formation. A predictive model for
these flows is elusive. Among the complicating factors
for these systems, mixtures of particles tend to segre-
gate into dramatic patterns whose details are sensitive
to particle property and interstitial fluids, not fully cap-
tured by continuum models. Further, the constitutive
behaviour of particulate flows are sensitive to the par-
ticle size distributions. In this paper, we investigate the
use of Discrete Element Model (DEM) techniques for
their effectiveness in reproducing these details in debris
flow. Because DEM simulations individual particle tra-
jectories throughout the granular flow, this technique is
able to capture segregation effects, associated changes
in local particle size distribution, and resultant non-uni-
formity of constitutive relations. We show that a simple
computational model study using DEM simulations of
a thin granular flow of spheres reproduces flow behav-
iour and segregation in an experimental model debris
flows. Then, we show how this model can be expanded
to include variable particle shape and different intersti-
tial fluids. Ultimately, this technique presents a manner
in which sophisticated theoretical models may be built
which consider the evolving effects of local particle
size distribution on debris flow behaviour.
K
ey
worDS
: debris flow, segregation, simulations, rotating
drum, dense granular flows
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k.M. HILL, Y. BEREkET
, w.E. DIETRICH & L. HSU
436
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
size distribution at the front is notably higher than the
particle size distribution at the tail. Field and experi-
mental studies of bedrock erosion associated with de-
bris flows indicate that the particle size distribution in
the segregated course front or snout plays a significant
role in the rate of erosion (s
toCk
& d
ietRiCH
, 2006,
H
su
et alii, 2008, H
su
, 2010).Segregation of the larger
particles outward laterally, results in lateraldeposits or
levees that have a high average particle size compared
tothat of the body of the debris flow.
Even the particle size in a relatively monosized
system affects certain rheological relationships such as
the relationship between applied stress and flow rate;
segregation affects the local average particle size, so the
dynamics of the flow must vary across the debris flow.
In others words, it is becoming increasingly clear that
one needs to understand the effect of an evolving par-
ticle size distribution to develop a predictive model for
the dynamics of debris flows when some of the particles
are large enough so that interparticle interactions plays
an important role in their constitutive behaviours.
There have been a number of modeling tech-
niques used for segregation in dense systems such as
that which occurs in debris flows. (See, for example,
s
avaGe
& l
un
, G
Ray
& t
HoRnton
(2005), and k
Ha
-
kHaR
et alii (1997).) The most successful predictive-
formulation involves an empirical segregation model
superposed ona continuum model for the average
flow. Examples include themodel first proposed G
Ray
& t
HoRnton
(2005) (expanded toinclude diffusion
by G
Ray
& C
HuGunov
, 2006) applied to thinflows in
G
Ray
& a
nCey
(2008) and the model of k
HakHaR
et
alii (1997) applied to chaotic flows in a rotating drum
by H
ill
et alii (1999). However, these models are not
fully predictive in that they employ empirical rules for
particle segregation. One of thecomplicating factors to
the application of these models is the degreeto which
segregation behaviours are sensitive to particle prop-
erty (H
ill
& f
an
, 2008), interstitial fluid (J
ain
et alii,
2003), andboundary conditions of granular flows (f
an
& H
ill
, 2010).Currently, there are no physical models
that account for thesedetails, only those that account
for them in an empirical manner.
Unfortunately, the models for segregation in
dense flows are rather limited, and there is precious
little investigation as to how theparticle size affects
important rheological relationships. Analternative
approach to a continuum model for understanding-
(e.g., P
HilliPs
et alii, 2006) but also possibly due to
the complex dynamics exhibited by particle mixtures
(l
inaRes
-G
ueRReRo
, 2007).
Over the last several decades, there have been a
number of approaches to modelling different aspects
of debris flows. Examples include non-Newtonian
fluid-like models to capture certain details of the con-
stitutive behavior such as the generalized Bingham
or Herschel-Buckley model (e.g., i
veRson
, 1997 and
k
aitna
et alii, 2007). These models don’t take into
account the effect of particle size in determining be-
haviors and therefore are probably most appropriate
for debris flows comprised primarily of finer particles
or even for the interior of bouldery debris flows where
finer particles are most highly concentrated (e.g., i
veR
-
son
, 1997). For debris flows where particles are large
enough that interparticle collisions become important,
interparticle interactions including Coulomb stress
and collisional stress appears important for determin-
ing internal stress and other details. Collisional stress
in sheared granular flows, first modelled by b
aGnold
(1954), has been shown important for determining the
behaviour of a wide variety of granular flows (e.g.,
s
ilbeRt
et alii, 2001, b
RewsteR
et alii, 2008). Exam-
ples for flow in the idealized case where the internal
stresses are dominated by Coulomb friction include
those by s
avaGe
& H
utteR
(1989, 1991). i
veRson
&d
enlinGeR
(2001, 2004), and others have general-
ized this to three dimensional flow over rocky terrain
In addition to the constitutive behaviour of de-
bris flows, recent focus has included the segregation
behaviour of mixtures of debris flows consisting of
gravels and larger particles where segregation be-
comes increasingly important in determining local
and global constitutive behavior. For example, an
apparent feedback mechanism between local particle
size distribution and the constitutive behavior ap-
pears to be a primary driving mechanism for some
pattern formation problems such as a fingering phe-
nomenon observed in some large scale pyroclastic
flows (P
ouliQuen
& v
alanCe
, 1999).
In channelized debris flows, segregation of the par-
ticles by size gives rise to sorted patterns at a variety of
length scales that shapes the behaviours of the debris
flows. Large particles tend to segregate to the free sur-
face. They also segregate to the front and sides of de-
bris flows. The segregation of large particles to the front
gives rise to a course front or “snout” where the particle
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DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL STUDIES OF BOULDERY DEBRIS FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
437
ticle interaction are of critical importance for dense
granular systems where particles undergo enduring
interaction with one another, often with more than
one particle at a time.
As is typical, in our soft sphere DEM model,
when two particles come into contact, their deforma-
tions are modelled with an effective overlap between
the particles (Fig. 1). The force exerted by the parti-
cles on one another is estimated based on the effective
deformation or overlap of each particle in contact. The
resulting interparticle force is dependent on the mag-
nitude and rate change of the overlap tangential and
normal to the line connecting the particles. This may
be thought of as a relative movement of the contact
points between the particles normal and tangential to
the plane of overlap, δ
n
and δ
t
, respectively.
The force model consists of two parts in each
direction: a nonlinear restorative spring, dependent
only on the amount of deformation or overlap, and a
damping mechanism, dependent on the rate change of
deformation or overlap:
Here, the subscripts n and t indicate normal com-
ponents and tangential components of each parameter
respectively. F
n
and F
t
represent the normal and tan-
gential components of the forces as previously de-
segregation and other details of particle interac-
tions in debris flowsis the Discrete Element Method
(DEM). First proposed in the late1970’s by Cundall
and Strack, the DEM approach has the capabilityof
tracking individual particles throughout a simula-
tion. Rather thana bulk model for rheological rela-
tionships, the DEM approach relieson simple models
describing interactions between particles and the-
nobtains the resulting motion through numerical in-
tegration of theaccelerations of the particles derived
from the total force on eachparticle, as detailed in
the next section. The technique therefore hasthe ca-
pability of capturing segregation effects and subse-
quentchanges in the local rheological behaviour of a
particulate flow andeven forces and stresses between
the particles and the boundaryrather than imposing
rules based on, for example, empirical data. Inthis
paper, we first provide an overview of the DEM for
dry granularflows. Then we present results compar-
ing data from these DEMsimulations with model
experimental debris flows and discuss howthe DEM
simulations can be made to more realistically repre-
sent debris flows.
DISCRETE ELMENT METHOD SIMULA-
TIONS
Modelling granular materials using the DEM
method was first proposed by C
undall
& s
tRaCk
(1979). The DEM method is in many ways similar
to Molecular Dynamics (MD), (e.g., a
llen
, 2004),
though in contrast to MD simulation, individual mac-
roscopic particles rather than molecules are treated as
distinct objects; in other words, the interaction between
two particles is modelled as a single force (rather than
a sum of all the molecular forces associated with atoms
and molecules composing each particle). Additionally,
in DEM simulation of macroscopic dry granular sys-
tems, only particles in direct contact with one another
interact and may repel and rotate one another.
DEM models vary in how they treat the forces
between particles. In our simulations, we use what
is called a “soft sphere” model (e.g., C
undall
and
s
tRaCk
, 1979, t
suJi
et alii, 1992) which takes into
consideration the deformation that occurs over time
during particle-particle interactions. While this detail
is not always important for relatively sparse granular
materials where interparticle interactions are relative-
ly rare and typically binary, the details of interpar-
Fig. 1 - Illustration of the interaction of two particles in
a soft sphere DEM model. (a) Two particles ap-
proach each other at velocities of V1 and V2 and
rotational speeds of ω1 and ω2. (b) During colli-
sion the particles the deformation is represented
by the normal overlap (δn) between the two cir-
cular particles. The plane of contact is assumed
to be flat and perpendicular to the line joining the
centers of the two particles. n is the axis perpen-
dicular (normal) to the contact plane and t is the
axis parallel (tangential) to the contact plane in
the direction of relative movement normal to n. δt
is not shown
(1)
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k.M. HILL, Y. BEREkET
, w.E. DIETRICH & L. HSU
438
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
translational and rotational accelerations using New-
ton’s and Euler’s second laws. We use the Fourth Order
Runge Kutta numerical scheme to integrate the particle
accelerations to determine the rotational and transla-
tional velocities and displacement of the particles at
each instant in time. To assure numerical stability we
use a time step of approximately 1.4 μs for these calcu-
lations (m
unJiza
, 2004).
PRELIMINARY RESULTS: DRY SPHERI-
CAL PARTICLES IN A DRUM
For this paper, the boundary conditions of our
simulations were modelled after an experimental drum
used for studying model debris flows at the University
of California at Berkeley described in detail in H
su
et
alii, (2008). The drum diameter D = 0.56 m and its
width w = 0.15m. The front vertical wall is made of
clear acrylic, so quantities such as surface profile and
particle location for those particles adjacent to the wall
are accessible. The flat vertical sidewalls are relatively
smooth. For the experiments described here, the outer
curved boundary over which the particles move – i.e.,
the bed – was roughened with attached sandpaper but
otherwise is not bumpy.
The drum has the capacity to house erodible pan-
els at the “bed” of the model debris flows, though for
this paper we focus instead on the kinematic behav-
iour of the particles. For this, we model thebehaviour
of 13.8mm spherical glass marbles that were sheared
in the experimental drum and also the behaviour of
very simple mixtures – a single larger particle among
the 13.8mm particles. The 13.8 mm “matrix” particles
had a measured polydispersity of 10%, so in reality
they ranged from approximately 12.4 mm – 15.2 mm.
The simulated particles were given the properties
of quartz, similar to those typical of glass. The proper-
ties are shown in Table I. In all cases, the particles may
be considered in the hard sphere limit as we discuss in
Hill and Yohannes (2010, under review). The particle
scribed. k
n
and k
t
are the stiffness factors of particles,
δ
n
and δ
t
are the overlaps between particles and γ
n
and
γ
t
are the damping factors. μ is the coefficient of fric-
tion between the interacting particles
The exponents and stiffness factors of particles kn
and kt are based on Hertizian contact theory (H
eRtz
,
1896) and m
indlin
& d
eResiewiCz
(1953):
Where R
eff
is the effective radius of the two parti-
cles in contact, E
eff
is their effective modulus of elas-
ticity, and G
eff
, is their effective shear modulus:
Here, the subscript refers to one of the two parti-
cles in contact, and r
i
, v
i
, E
i
are the radius, Poisson’s ra-
tio, and modulus of elasticity of particle i, respectively
There is no simple theoretical description for the
damping component. We use the following form simi-
lar to t
suJi
et alii (1992):
where m
eff
is the effective mass of two particles
in contact: m
eff
= (1⁄m
1
+ 1⁄m
2
)
-1
, and α varies with the
coefficient of restitution according to the numerical
solution by t
suJi
et alii (1992).
The force model for interactions between parti-
cles and other objects (such as container boundaries
or walls) is similar to the inter-particle force model
described by Equations 1(a) and 1(b). The dimensions
of the walls are usually much larger than the particle
sizes, so wall mass and radius are considered infinite.
Therefore, the effective radius during a particle-wall
contact is considered equal to the radius of the particle.
Based on the individual contact forces between all
particles (and, where appropriate, the walls) and the
masses of those particles, we calculate the net force
and net moment on each object and, from this, the
(2)
(3)
(4 a)
(4 b)
Tab. 1 - Properties of the simulated particles
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DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL STUDIES OF BOULDERY DEBRIS FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
439
the side walls occurs at the front of the flow, analogous
to the motion associated with levy formation in debris
flows in nature. Individual particle trajectories in this
general circulation pattern vary depending on particle
size, as we describe shortly.
Since the flow in both the experimental and com-
putational drum is shallow, the entire depth of the flow
is sheared, and there is no plug-like section that is com-
mon in deeper flows. From the side view, there is a
steady, spatially varying velocity field from the top to
the bottom of the flow between the upstream and down-
stream moving particles. The movement of the particles
in the drum is analogous to that the flow of particles
down a hillslope, with two chief exceptions. First, the
hillslope is stationary, so none of the particles are mov-
ing backwards except in a relative sense, for example,
to the direction of motion of the particles on the top of
the debris flow. Second, the bed at the back of the flow
in the drum reaches unreasonably high slopes com-
pared to that of real debris flows. We limit ourselves to
the front and middle of the flow in the drum for making
analogies to real debris flows.
In Figures 3 and 4 we compare some data from the
physical and computational experiments. Fig. 3 shows
sample surface profiles derived experimentally (a) and
computationally (b) and (c) for the same mass of 13.8
mm spherical glass particles (3.13 kg). As qualitatively
apparent from the snapshots in Fig. 2, the model de-
bris flows are bulky at the front and diminish toward
the back. The similarity between the results from the
physical and computational models over the whole
field of particles gives credence to the effectiveness of
the DEM simulations in modelling these systems. The
sizes we use in the simulations are again identical to
those in the physical experiments: d = 13.8 mm ±10%
for most of the experiments. The values of the numeri-
cal parameters for the nonlinear force model between
individual particles [Equation 2(a) and 2(b)] were de-
rived from the formulas detailed in the previous section.
The model drum has dimensions identical to that
of the experimental drum: D=0.56 cm and W = 0.15
cm. The walls of the drum are physically smooth (i.e.,
without bumps), and the wallparticle frictional coeffi-
cient is slightly greater than that between particles to
account for the additional friction associated with the
sandpaper. All other material properties of the walls are
the same as that for the particles. Snapshots from one of
these computational experiments are shown in Fig. 2.
Particle trajectories in the experimental and com-
putational drums follow similar 3-dimensional circula-
tion patterns. Particles in contact with the bed and walls
are dragged upstream in the direction of the walls,
though there is some slip between the walls and the par-
ticles so that the particles immediately adjacent to the
walls are not dragged as quickly upstream as the bed is
moving. The particles adjacent to the bed are carried to
the back where they then flow down the top surface of
the particles. In other words, the top and bottom sur-
faces move like two sides of a conveyor belt relative to
one another. From a plan view, lateral motion toward
Fig.2 - Snapshots from two different perspectives from an
instant in time from a simulation performed mod-
eled after physical experiments in the drum de-
scribed in the text with 3.13 kg of 13.8 mm spheri-
cal glass particles. Rotation direction indicated
by the vectors outside of the drums. Movement of
the particles indicated by arrows drawn over the
beads
Fig. 3 - Surface profiles of the particles in the physical and
computational experiments described in the text.
(a) the longitudinal profile from the physical ex-
periments at one instant in time. (b) the same from
the computational experiments. (c) the average
over 20 rotations from the computational experi-
ments
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k.M. HILL, Y. BEREkET
, w.E. DIETRICH & L. HSU
440
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
similarity of the profiles indicate the internal stresses
of the bulk granular flow are well-represented by the
interparticle force model within the DEM simulation.
In a mixture, the behaviour of the particles is simi-
lar to that described above and shown in Figs. 2 and
3, but there are marked differences in the behaviour of
individual particles that vary with particle size. For ex-
ample, while larger particles roughly follow the circula-
tion of the general flow at the front, once they start to
move back, they quickly segregate upward toward the
free surface, as is typical of mixtures of different sized
particles (e.g., k
HakHaR
et alii, 1997). Once at the top,
the large particles follow the trajectories of their new
neighbors at the top of the flow and return to the front.
This results in a higher concentration of large
particles at the front of the flow than their representa-
tive concentration in the bulk. Particles that are only
moderately larger than their neighbors take longer to
segregate to the free surface and therefore, they typi-
cally have a higher concentration slightly further back
in the flow. In other words, while the segregation be-
haviour is similar for all larger particles, it is most
dramatic for the largest particles. The relationship
between this segregation pattern and particle size is
complicated (e.g., f
élix
&t
Homas
, 2004).
We performed simple experiments and simula-
tions to validate sorting effects in the simulation using
“mixtures” consisting primarily of 13.8mm spherical
“matrix” particles and a single larger “intruder” parti-
cle. We kept the mass constant for each mixture so that
in each case a quantity of matrix particles whose mass
equalled that of the intruder particles was removed.
Figure 4 illustrates the longitudinal position of relative-
ly large intruder particles using a probability distribu-
tion (pdf) of the intruder particle in each case. Similar
to large-scale bouldery debris flows with a much wider
particle size distribution, the large particles are found
toward the front of these physical (Fig 4.a) and com-
putational (Fig. 4.b) model debris flows. The top figure
in each pair shows the pdf of the intruder particle loca-
tion for the experiments using a single 50 mm spherical
particle among the 13.8 mm particles. The next three
rows show results the same total mass of particles, but
the size of the intruder particle decreases in each subse-
quent panel to 40mm, 34mm, and then 25mm.
For the 50 mm intruder particle experiments and
simulations, the peak is near θ = -5°, i.e., near the front
of the flow, and relatively narrow. For smaller intruder
particles (though still larger than the matrix particles),
the peak broadens, and the tail of the distribution thick-
ens toward the back of the flow. For the 25 mm intruder
particle experiments and simulations, the front peak is
notably farther back and broader, and there appears to
be a secondary peak two thirds of the way back in the
flow. (The latter is likely related to the complications
that arise from the anomalously high slopes and not
relevant for real debris flows).
Admittedly, there are also a few differences be-
tween experiments and simulations. The experimental
results are somewhat noisier and the peaks are broader.
We believe these differences are primarily due to sub-
tle differences between the physical experiments and
the simulations. First, the simulated particles do not
capture all details of the experimental particles which
are slightly aspherical and have slight asperities. The
experimental drum also is not perfectly true as it is in
the simulations. Finally, while we can follow a particle
throughout the simulations, no matter where it is lo-
cated, we can only see the particles in the physical ex-
periments if they are immediately adjacent to the wall.
However, we feel the trends are similar enough to say
that trends associated with particle size distribution are
Fig.4 - Results from (a) physical and (b)computational
experimentsusing particles of total mass 3.13 kg
rotated in a drum of dimensions D = 0.56m, w =
0.15m. The particles in each experiment consisted
of a single large particle (size indicated in the fig-
ures) in a matrix of 13.8 mm particles. The plots
are the probability distribution function (pdf) of
the location of the large intruder particle. Please
see Figs. 2 and 3 for definition of θ
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DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL STUDIES OF BOULDERY DEBRIS FLOWS
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441
Other DEM modellers used polygons to generate
rougher form of asphericity, such as those described
in u
llditz
(2001). However, the singularity problem
at the corners of the polygon makes this method dif-
ficult to implement efficiently. Further, both methods
for representing aspherical particles in DEM simula-
tions take significant additional computational time.
The second common approach used for modelling
nonspherical, irregularly shaped particles involves
what one might call “computationally gluing” spheri-
cal particles together (t
Homas
& b
Ray
, 1999; z
eGHal
,
2001; C
alantoni
et alii, 2004; y
oHannes
, 2008). In
other words, multiple spherical particles are attached
to one another into a rigid-body cluster. When a sphere
in one cluster contacts a sphere in another cluster, the
forces between them are calculated as if they were in-
dividual spheres using Equations 1(a) and 1(b). The
primary difference between simulations involving sin-
gle spherical particles and these aspherical particles
comprised of clusters of spherical particles arises from
the calculation of the kinematics of the aspherical par-
ticle clusters. To calculate the movements of the parti-
cle clusters at each time step, first all the forces on all
particles in a cluster are used to calculate the resultant
force for the cluster and the resultant force moment
about the mass center of the cluster. Then, the rota-
tional and translational acceleration of each individual
particle cluster is determined from the force and force
moments using the cluster mass and moment ofinertia
of each cluster particle.
There are several benefits to this last method of
generating aspherical particles. Gluing spherical par-
ticles offers flexibility without significant additional
computational cost. Any number of particles with dif-
ferent sizes can be glued together to form a wide vari-
ety of particle shapes. (A few are illustrated in Fig. 5.)
C
alatoni
et alii (2004) showed that by using aspheri-
cal particles consisting of as few as 3 glued spheres
per cluster, one can reproduce such macroscopic
captured well by the simulations
While the results obtained show promise for
capturing certain qualitative details of model debris
flows, there are several simplifications in the simula-
tions compared with real debris flow systems. We are
currently working to build the level of sophistication
of our model to address some of the more complicated
effects. We describe our efforts in the next section..
ADDITIONAL EFFECTS: PARTICLE
SHAPE AND INTERSTITIAL FLUIDS
The simulations described in the previous two sec-
tions are performed using dry spherical particles. How-
ever, particles in natural debris flows are aspherical,
real debris flows involve particles that are distributed
over a wide range of particle size and involve millions
of particles, and typically interstitial fluid influences
the behaviour of debris flows through effects such as
cohesivity and pore pressure. While current compu-
tational power limits the number of particles that we
can simulate, we can address some of the other issues
through various computational techniques. We de-
scribe our preliminary efforts to simulate debris flows
that have aspherical particles and interstitial fluids that
alter the dynamics of the flow briefly in this section
PARTICLE ASPHERICITY IN DEBRIS FLOwS
In reality, unlike the particles in the model de-
scribed above, the particles that compromise the
granular materials in debris flows are, in general, as-
pherical. Compared with aspherical particles, spheri-
cal particles tend to roll and slide past other spheri-
cal particles relatively easily. This difference leads to
discrepancies in bulk macroscopic properties and is
generally believed to cause an unphysically low stiff-
ness in some DEM models (t
anaka
et alii, 2001 and
t
inG
et alii, 1995), subsequently, an unreasonably low
angle of repose (C
alantoni
et alii, 2004). A number
of different techniques have been used to represent the
effect of the asphericities.
There are two primary techniques used for model-
ling aspherical particles. One is through the modifica-
tion of the shape of the primary entities of the parti-
cles from spheres to aspherical particles. This requires
changing the algorithm and force contact model de-
scribed above depending on the orientation of the par-
ticles. The most widely used aspherical particle shape
for DEM simulations is an ellipse (t
inG
et alii, 1995).
Fig. 5 - Particles that may be created from glued spheri-
cal particles that (a) may not or (b) may overlap,
and (c) may even be of different sizes
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k.M. HILL, Y. BEREkET
, w.E. DIETRICH & L. HSU
442
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
& C
laRk
, 1965; m
elRose
, 1966; G
illesPie
& s
et
-
tineRi
, 1967; H
o
ita
et alii, 1974). The liquid forms
what is referred to as a ‘liquid bridge’ between pairs
of closely-spaced particles, as sketched in Fig. 6. Liq-
uid bridges are responsible for an apparent attractive
force between neighboring particles associated with
two effects: (1) the surface tension of the interstitial
fluid and (2) the pressure gradient between the fluid
in the liquid bridge and the “void space,” occupied by
air (H
o
ita
et alii, 1974). The resulting force Fc can be
modelled as detailed in H
o
ita
et alii (1974):
Where Γ is the surface tension of the interstitial
fluid, d is the particle diameter, a is the half filling
angle and β is the contact angle formed between the
liquid and a particle (please see Fig. 6). ΔP
c
is the pres-
sure difference between the inside of the liquid bridge
and the atmosphere (determined by the radius of cur-
vature of the liquid bridge), and is computed from the
Laplace-Young equation:
where R1 is the meridian radius of curvature, and
R2 is the minimum width of the liquid bridge. Y′ is the
spatial derivative of Y with respect to X, and Y′′ is the
spatial derivative of Y′ with respect to X. The size of
each liquid bridge is determined by the level of satura-
tion and the number of particles close enough for liq-
uid bridges to form. The effect of the liquid bridges is
more pronounced on smaller size particles (maximum
size of about 2 mm). Therefore for bouldery debris
flows this method is primarily helpful for understand-
ing the effect of cohesivity on debris flows. Previously,
measures of the internal stress of the materials such as
the angle of repose. In addition, inter-particle contact
detection can be performed easily, and calculating in-
terparticle forces is simply a matter of using the equa-
tions described above for spherical particles, though
in the case of the clusters, one must consider contacts
between individual particles in a one cluster with
those in another. Admittedly, there is additional com-
putational time associated with the number of parti-
cles in a cluster. Nevertheless, the computational time
requirement is significantly less than other methods
for simulating aspherical particles described above.
We have found simulating aspherical particles
by using glued clusters of particles in pairs or in tri-
plets to sufficiently reproduce tests of the strength of
materials, as described in y
oHannes
(2008). We are
currently using this approach to investigate the ef-
fect particle asphericity on the kinematics of gravity-
driven debris flows.
INTERSTITIAL FLUIDS IN DEBRIS FLOwS
In addition to particle shape, it is well known
that interstitial fluid plays an important role in most
granular flows. The presence of a fluid introduces a
cohesivity between particles, provides lubrication
allowing particles to more easily slide past one an-
other, and can be the source of anomalous effects as-
sociated with pressure in the fluid that is greater than
the hydrostatic pressure. Modelling these effects are
somewhat more complicated than modelling dry par-
ticles alone as they involve longer-range interactions
between particles.
In this section we present two methods we are
investigating for their effectiveness in modelling the
presence of an interstitial fluid in debris flows and
some preliminary results from simulations employing
these methods. In each case, we investigate a method
for reproducing the effect of a fluid at the scale of par-
ticle-particle interactions and investigate the resulting
change in dynamics at the scale of the system.
MODELING COHESION ASSOCIATED wITH
INTERSTITIAL FLUIDS
When the pore spaces between the particles are
not entirely filled with water, the effect of an intersti-
tial fluid at the scale of individual particles is prima-
rily associated with a cohesivity between particles that
is not present in dry granular materials (e.g., m
ason
Fig. 6 - Liquid bridge between two spherical particles. R1
and R2 are the longitudinal and meridian radius
of the curvature of the surface of liquid
(6)
(7)
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DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL STUDIES OF BOULDERY DEBRIS FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
443
fects are apparent from pore pressure effects as we
address in the next subsection.
HIGH MOISTURE CONTENT
When the pore spaces between the particles are
entirely filled with liquid, the effect the liquid has on
the interparticle interactions is not limited to cohesiv-
ity. The large-scale mechanics of the particle-fluid
interactions in this case are not fully understood, but
they are generally linked to the effect of the variability
of pore pressure (e.g., s
assa
, 1984; G
abet
& m
udd
,
2006; d
eanGeli
, 2009; i
veRson
et alii, 2000).
To understand this, we first consider a static sys-
tem of macroscopic particles where the spaces among
particles are mutually connected and entirely filled
with a liquid. In this case, the pressure in the liquid,
the pore pressure, is typically equal to what it would
be in a static fluid, its hydrostatic pressure: p
h
=ρ
L
gh,
where ρ
L
is the density of the liquid, g is gravitational
acceleration, and h is depth beneath the free surface.
However, in a sheared (or in some case, even slightly
disturbed) fluid-particle flow the pore pressure can in-
crease in part due to relative fluid and particle motion.
In addition to their movement associated with the av-
erage flow, particles tend to approach and recede from
one another. The presence of liquid in the pores damp-
ens this somewhat, particularly when the liquid is un-
able to move through the pores rapidly enough to ac-
commodate relative particle movement. This tends to
reduce contact between particles. When this happens,
the interstitial liquid rather than interparticle contacts
supports the particles, resulting in pore pressure in ex-
cess of hydrostatic pressure.
This can result in dramatic differences in the be-
haviour of the debris flow including increased mobil-
ity leading to long run-out avalanches (C
asaGRande
,
in other contexts, cohesivity has been shown to have
a strong effect on details such as particle segregation
(l
i
& m
C
C
aRtHy
, 2005) and shear strength of granular
materials (l
iu
& s
un
, 2002). We investigate the ef-
fect for thin granular flows by simply considering the
force associated with Equation (6) (with the solution
for pressure difference described by Equation (7)) as
an additional normal force between separated particles.
Figure 7 shows a snapshot from a mixture of three
different sized particles when there is no interstitial
fluid and when there is 10% water in among relatively
small particles. The picture is representative of a near
steady state for both systems. Two differences are ap-
parent. First, the segregation is clearly affected by the
presence of liquid. In the case of the system shown in
Fig. 7(b), the larger particles no longer segregate all the
way to the front of the flow. Second, the longitudinal
profile is altered, suggesting the internal stresses of the
granular assembly as a whole are affected by the local
interactions between individual particles.
Figure 8 shows the steady state velocity profiles
for the model systems depicted in Fig. 7. Figure 7(a)
is the profile of the particles alone, while Fig. 7(b)
shows the profile for the case with some fluid. The
fluid in this case reduces the maximum velocity and
the average shear rate by approximately 1/3. All of
this information is useful in considering the possible
modifications one could make to continuum models
for particle/fluid mixtures. However, additional ef-
Fig. 7 - Snapshots from simulations of relatively small
particles (a) without interstitial fluid and (b)
with 10% interstitial fluid. In both case, the drum
diameter D = 140 mm; the drum thickness: w =
37.5 mm; and the particle diameters are: 1.25
mm, 3.0mm and 6.25mm; total mass of particles:
0.0524 kg; rotational speed: 48 rpm
Fig. 8 - Velocity profiles from two different locations
in the model debris flows depicted in Fig. 7(a)
without and (b) with 10% interstitial fluid.
Results from two different positions in the drum
are shown for values of θ indicated in the legend.
(Please see Figs. 2 and 3 for definition of θ)
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k.M. HILL, Y. BEREkET
, w.E. DIETRICH & L. HSU
444
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
1971; w
anG
& s
assa
, 2003; s
assa
& w
anG
, 2003;
o
kada
& o
CHiai
, 2008; d
eanGeli
, 2009). There are a
number of details that moderate the degree to which
excess pore pressure occurs and influences the behav-
iour of a debris flow, particularly factors that modify
the permeability including void ratio, particle size
distribution of the granular material and fine particle
content (w
anG
& s
assa
, 2003). A reduced perme-
ability impedes both the movement of fluid and, cor-
respondingly, the dissipation of excess pore pressure
generated due to some deformation in saturated granu-
lar materials. To better understand the effects of the
pore-pressure on flow properties, models that account
specifically for the modified particle-particle and par-
ticlefluid interactions are helpful.
To develop such a model, we consider Fig. 9
which illustrates the dynamics that can lead to in-
creased pore pressure for a single particle approach-
ing or departing from a group of other particles. A
particle approaching a group of other particles needs
to push out a pocket of interstitial fluid and thus that
particle and the group of particles feels a mutually
repellent force. DEM models that incorporate this ef-
fect of pore pressure have been proposed by t
aRumi
& H
akuno
(1988) and o
kada
& o
CHiai
(2007). These
models are based on tracking the change in the pore
spaces between particles. For example, if volume of a
pore space decreases, the pressure gradient forces the
fluid to flow to adjacent pore spaces. These models
were capable of demonstrating the evolution of pore-
pressure associated with applied stresses in saturated
granular materials. However, tracking the pore spaces
in DEM models is computationally very intense and
these two DEM models are limited to 2D set up or to a
very few particle in 3D. This makes their implementa-
tion into debris flow models impractical.
We suggest a simpler model to capture the es-
sence of the effect of pore pressure without as much
additional computational intensity. This model cap-
tures the “action at a distance” for particles moving
relative to one another as suggested by the sketches
in Fig. 9. As we noted above, the presence of fluid
dampens relative particle motion for particles in close
proximity. A model that captures this basic idea can be
written as follows:
where in suggesting this, we propose that F
12
, the
fluid-mitigated force associated with two nearby parti-
cles moving relative to one another, increases as their
relative velocities increase and decreases with dis-
tance to one another. v
n
is their relative velocity in the
direction connecting their two particle centres (this is
similar to the definition of δ
n
of Equation 1(a), though
here the particles are not necessarily touching); r
12
is
the distance between particle centres, and k can be
thought of as a “pore pressure coefficient” that varies
with fines content and other details that influence the
permeability of the particle network. We expect k to
vary with details of the system including fines content
and more general considerations of the particle size
distribution.
The model described by Equation 8 is completely
phenomenological, and we expect it to be modified
with more detailed comparisons with experimental
and field data. We consider it a first order attempt to
represent the action at a distance brought about by the
saturated interstitial fluid in a DEM that facilitates in-
vestigations of this effect on larger systems.
Figure 10 shows data comparing the velocity pro-
file from a dry system with one where the fluid satu-
rates the system as modelled by Equation 8. Figure
10(a) shows the velocity profile of a system that does
not include consideration of any fluid effects and is
thus fully represented by Equation 1 and Fig. 10(b)
includes the additional normal force described in (8)
added to Equation 1(a). For the system where an ex-
tra term is added to account for pore pressure effects,
the flow is slowed by approximately 30%. More no-
tably, however, there is a qualitative difference with
the additional pore pressure effect. The flow profile
concavity has changed and it resembles slightly more
the plug flow expected by excess pore pressure effects
and observed in experimental flows in large scale ex-
perimental drums.
Fig. 9 - Sketches illustrating the effect of a pore-filling
liquid on damping relative particle motion. The
liquid can either work to create a repulsiveforce
between particles if they are approaching or to
effectively attract them if they are moving away
from one another
(8)
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DISCRETE ELEMENT MODELING AND LARGE SCALE EXPERIMENTAL STUDIES OF BOULDERY DEBRIS FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
445
terstitial fluids more common in natural debris flows.
Ongoing work includes the following:
Investigation of the effect of particle shape on
flow behaviour
Investigation of the influence of the effects of
cohesivity and action at a distance associated with
interstitial fluids Initial comparison of these results
with large scale experiments with aspherical particles
and different interstitial fluid is ongoing. Preliminary
results indicate the simple modifications suggested in
this paper have potential for reproducing some of the
complicated effects in natural debris flows. In the long
run, results from Fig. 10 Velocity profiles from model
debris flows where particle interactions are modelled
as (a) dry, completely described by Equation (1) and
(b) saturated, where the additional long range force
described by Equation (8) is added to the normal
force. (a) (b) simulations such as those described here
can inform the development of sophisticated con-
tinuum models for debris flows that better represent
evolving particle size distributions and solid and fluid
concentrations.
ACKNOWLEDGEMENTS
Support for this research was provided in part by
the University of Minnesota and the University of
California – Berkeley, and by the National Center for
Earth Surface Dynamics (NCED), a NSF Science and
Technology Center funded under agreement EAR-
0120914.
DISCUSSION AND CONCLUDING RE-
MARKS
The first results of this preliminary model for de-
bris flow capture a number of important details from
debris flows including the following:
Individual particles segregate according to size
from front to back in a thin sheared granular flow;
Larger particles segregate to the front. This trend
decreases systematically with particle size relative to
the size of particles in the bulk;
The longitudinal profiles from the simulations
were similar tothat from experiments comprised of
dry spherical particles, suggesting that the constitu-
tive behaviour is well-represented by the DEM model;
While the results of this model to date are primarily
from dry flow of spherical particles, implementation
of additional modelling techniques show promise for
simulating more sophisticated flows. These include
more details associated with realistic particles and in-
Fig. 10 - Velocity profiles from model debris flows whe-
re particle interactions are modelled as (a) dry,
completely described by Equation (1) and (b)
saturated, where the additional long range force
described by Equation (8) is added to the normal
force
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Statistics