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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
523
DOI: 10.4408/IJEGE.2011-03.B-058
SIMULATING TRIGGERING AND EVOLUTION
OF DEBRIS-FLOWS WITH SPH
G
iaComo
VICCIONE
(*)
& v
ittoRio
BOVOLIN
(*)
(*)
University of Salerno, Department of Civil Engineering - Via Ponte Don Melillo - 84084 - Fisciano (Italy)
Email: gviccion@unisa.it
Results show how different are the conditions
of motion, by varying the location of the triggering
area, the pressure threshold plim and the shear stress
τ
bed
. In order to measure the capability of SPH into
simulating such events, some comparisons are made
with corresponding Flo 2D results.
K
ey
words
: SPH, debris-flow initiation, debris-flow propagation
INTRODUCTION
Debris flow usually take place on steep slopes after
heavy or long rainfall events, mobilizing loose material,
such as soil, vegetation, debris ranging from clays to boul-
ders. They may represent a threat for people living nearby
such areas and for buildings, i.e. bridges, facilities, etc.
Therefore, understanding the movement mechanism is
of considerable interest, particularly in the evaluation of
potential mitigation policies (t
akaHasHi
, 1991; i
veRson
,
1997). In addition to precipitations, landslide mechan-
ics depend on several factors, such as type of weather,
morphology, geology, land use and plant growth. In this
work, we only investigate about the ability of simulat-
ing debris-flow initiation and subsequent movement with
the Smoothed Particle Hydrodynamics (SPH) technique.
Triggering is here settled randomly, making free to move
a particle located in the upper part of the slope being con-
sidered. The others are all initially frozen. Motion of re-
maining particles is related to the achievement of a pres-
sure threshold p
lim
(Fig. 1). The resulting process is like a
domino effect or a cascading failure.
ABSTRACT
Recently, debris-flow kinds of phenomena have
been reproduced by means of Lagrangian methods,
such as Distinct Element Method (DEM) or Lagrang-
ian Finite Element Method (LFEM). Among the oth-
ers, meshless, Lagrangian numerical method, known
as Smoothed Particle Hydrodynamics (SPH), is here
applied to simulate debris-flow initiation and propaga-
tion over the slope of a mountain located in the city of
Nocera Inferiore (Southern Italy). Debris-flows have
been simulated since long time for hazard mitigation
assessment or deposit evaluation via Eulerian-based
methods. Since they may feature mesh distortion as
the computational domain evolves, heavy grid refine-
ment algorithms are sometimes necessary, especially
for those problems characterized by large deforma-
tions. SPH overcomes such difficulties since no mesh
is needed over the physical domain. Spatial discretiza-
tion is indeed carried out with a collection of parti-
cles without connectivity bonds among them. While
boundary particles are fixed over time, computing
particles are free to move in response of external and
internal forces such as gravity and pressure.
More in detail, computing particles are all initially
frozen. Once a particle located in the upper region of
the slope is set free, the others close to it move if a
pressure threshold p
lim
is reached. Other particles are
subsequently triggered where the mentioned condition
occur, as a domino effect. Runout velocity is control-
led by handling the shear stress τ
bed
with the fixed bed.
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G. VICCIONE & V. BOVOLIN
524
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
With this technique, non-linear partial differential
equations of Navier-Stokes (b
atCHeloR
, 1974) are
adapted as follows:
where ρ is the density, v the velocity, p the pressure, τ
int
the internal viscous shear stress, τ
bed
the internal viscous
shear stress, f the external forces per unit of volume.
The above equations of conservation are then dis-
cretized on particle position (l
iu
& l
iu
, 2003), yield-
ing the following new set of equations:
in which viscous terms are modelled with the follow-
ing artificial viscosity model:
where
The notation ā
ij
= (a
i
+ a
j
) / 2, b
ij
= b
i
- b
j
has been
used above. Sums appearing in Equations 2 refer to the
neighbouring particles within a short range “r
c
” (Fig-
ure 2) respect to the particle “i”. “W” (Figure 3) is the
weighing or kernel function (m
onaGHan
& l
attanzio
,
1985), whose expression is given by the following:
in which q = |x
i
- x
j
| / 2 • r
c
is the relative dimensionless
distance between “i” and “j” particles, A = 10/7 • π or
A = 1/π respectively for 2-dimensional (n
d
= 2) or 3-di-
mensional (n
d
= 3) problems.
“α
k
” is the linear viscosity coefficient (m
ona
-
GHan
, 1994) taken to be fixed (α
k
= α
int
= 0,1) when
particle “i” interacts with another computing particle
“j”, or variable (α
k
= α
bed
) when particle “j” is a bound-
ary one; “β” is always set to be 0, i.e. the nonlinear
term in Π
ij
is not taken into account.
The term “c” represents the speed of sound whose
order of magnitude is ten times greater than the maxi-
mum estimate of the velocity field, “v” and “x” repre-
Computing particles are initially set with a pres-
sure being equal to the atmospheric value. The relative
spatial allocation is obtained with a specific mesh gen-
erator, which guarantees a distribution of points form-
ing triangles approximately equilaterals. While some
particles are moving, they may approach others ini-
tially still, to the point for which the relative distance
yields a pressure greater than a threshold value. Once
reached such point, those neighbouring particles, pre-
viously fixed, are then set free to move.
NUMERICAL APPROACH
Debris-flow initiation and subsequent evolution
is here simulated with the Smoothed Particle Hydro-
dynamics technique. Introduced three decades ago for
astrophysical applications (G
inGold
& m
onaGHan
,
1977; l
uCy
, 1977), it has been subsequently applied
in many other and different research areas such as
multi-phase flows (m
onaGHan
& k
oCHaRyan
, 1995),
flows through porous media (m
oRRis
et alii, 1999;
z
Hu
et alii, 1999), high explosive detonation and ex-
plosion (s
weGle
& a
ttaway
, 1995; l
iu
et alii, 2000),
hyper velocity impact and penetration (z
ukas
, 1990;
R
andles
& l
ibeRsky
, 1996).
Such as a meshless, Lagrangian, particle method,
the SPH method presents some special advantages
over the traditional grid-based methods. Among the
others, the most appealing feature is the adaptive na-
ture which means that is not affected by arbitrariness
of particle distribution. Indeed, there is no need to pre-
scribe the connectivity between the moving particles.
With such technique, computing particles carry physi-
cal properties such as velocity or density. Advection
is exactly computed without numerical errors. More
consideration has been subsequently devoted to SPH
as good choice for geomechanical problems.
Fig. 1 - Neighbour particle destabilization. a) Particle “i
is approaching the neighbour particle “j”. b) De-
spite the relative distance “|r
ij
|” is decreased, par-
ticle “j” is still fixed because p
ij
< p
lim
. c) Particle
j” is set free to move because the interstitial pres-
sure “p
ij
” has reached the threshold value “p
lim
(1.a)
(1.b)
(2.a)
(2.b)
(3)
(4)
(5)
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SIMULATING TRIGGERING AND EVOLUTION OF DEBRIS-FLOWS WITH SPH
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
525
extremely small. In the present work a stiff equation
of state is assumed (b
atCHeloR
, 1974):
with γ = 7, ρ
0
= 2600kg/m
3
is the reference density
of the superficial soil within the area of investiga-
tion and “B” is given by:
in which the reference speed of sound “c
0
” is chosen
considering a small Mach number M ac = 0.1 ÷ 0.01
(m
onaGHan
, 1994).
DEBRIS FLOW SIMULATION
The SPH based code has been used for a preliminary
study of debris-flow triggering and propagation over a
slope located in the area of Nocera Inferiore (Italy). The
area, object of study, is shown in the next Figure 4.
The available domain has been discretized with
GiD
®
software. (GID Reference manual, 2009) The
resulting grid mesh is well performed, with a refer-
ence distance being d
0
= 2.5 m and particles form-
ing triangles as equilateral as possible. A single layer
of moving particles has been laid on the upper part
of the slope (blue region in the Figure 5). The total
number of moving and boundary particles are re-
spectively N
f
= 5000 and N
bound
= 11000.
All moving particles are initially fixed. Basically
the motion is made by unfreezing a particle located in
the upper part of the slope. Such region is schematically
shown in the above figure with red circles. The subse-
quent mobilization along the slope is subordinated by
the achievement of a particle pressure greater than a lim-
sent the velocity and position respectively, η = 0.05 • r
c
is employed to prevent numerical divergences when two
particles are colliding.
It should be noted that the artificial viscosity is
both function of the relative position and velocity, that
is Π
ij
= Π
ij
(x
ij
, v
ij
).
Since no connectivity is established among the parti-
cles, Neighbourhood identification is carried out by means
of a special search algorithm (v
iCCione
et alii, 2008).
The correction XSPH (m
onaGHan
, 1994) on the
time derivative of particle position
has been adopted, by adding the following quan-
tity to the right hand side of Equation 5:
doing so, each particle is locally constrained to
move with a velocity depending on the average val-
ue of its neighbourhood. This is useful in the case of
high velocity or impact problems, because it avoids
unphysical flow separation.
Closure is assured by adding an equation of state,
describing the relationship between density and pres-
sure. As a matter of facts, for near incompressible me-
dia, the real equation of state determines time steps
Fig. 2 - Short range interactions. Properties of particle
“i” are computed on the basis of those particles
within a cut-off distance “r
c
” (bold marked). Out-
er particles give no contribution
Fig. 3 - Smoothing kernel adopted with its gradient
(5.bis)
(6)
(7)
(8)
Fig. 4 - Map location of the area, object of the investigation
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G. VICCIONE & V. BOVOLIN
526
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
CONCLUSIONS
As can been seen from the above Figures, varying
the location of the triggering area, the limit pressure
p
lim
and the shear stress τ
bed
, the condition of motion are
quite different. More specifically, the area been mobi-
lized become larger when decreasing the isotropic pres-
sure p
lim
. At the same time, by increasing the shear stress
with the bed, the front wave propagates slower, accord-
ing to the common experience. A comparative analysis
of SPH results with Flo-2D commercial code has been
carried out, obtaining a good agreement in terms of both
mobilized volumes and front propagation.
ACKNOWLEDGEMENTS
The authors wish to thank for the support coming
from the University Consortium for Research on Ma-
jor Hazards (CUGRI ), with special regard to the di-
rector, Prof. Eugenio Pugliese Carratelli. The authors
are also in debt with Ing. Nicola Immediata from the
University of Salerno, for his precious contribution
and for helpful comments concerning this work.
it value “p
lim
”. Propagation velocity is handled by vary-
ing the bed shear stress whose expression is assumed to
be equal to the artificial viscosity (Eq. 3) introduced by
(m
onaGHan
, 1994). Different values of the linear coeffi-
cient α
bed
, are assumed. The following Table 1 shows the
simulations carried out, with specification of the trigger-
ing particles and of the parameters above introduced.
The next Figures 6 to 16 illustrate three instants for
each SPH based simulation, with the indication of the
area been mobilized. Following a previous study (v
iC
-
Cione
& b
ovolin
, 2010), the authors have also carried
out some simulation with Flo-2D (FLO-2D user manual,
2009) as shown from Figure 17 to Figure 22, willing to
perform a comparison for simulation times t = 50 sec and
t = 100 sec, with the corresponding SPH results shown
in Figure 6 (Simul. N. 1), Figure 11 (Simul. N. 6) and
Figure 14 (Simul. N. 9), showing a good agreement in
terms of both mobilized volumes and front propagation.
Fig. 5 - Spatial discretization of the study area. Red circles rep-
resent the region where a local triggering is imposed
Tab. 1 - List of simulations been carried out
Fig. 6 - Simul. N.1. Particle trig-
gered: PT1, limit pressure
p
lim
= 300 kgf /cm
2
, viscos-
ity coefficient α
bed
=0.1
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SIMULATING TRIGGERING AND EVOLUTION OF DEBRIS-FLOWS WITH SPH
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
527
Fig. 7 - Simul. N.2. Particle triggered:
PT1, limit pressure p
lim
= 200 kgf
/cm
2
, viscosity coefficient α
bed
=0.1
Fig. 8 - Simul. N.3. Particle triggered:
PT1, limit pressure p
lim
= 100 kgf /
cm
2
, viscosity coefficient α
bed
=0.1
Fig. 9 - Simul. N.4. Particle triggered:
PT2, limit pressure p
lim
= 300 kgf
/cm
2
, viscosity coefficient α
bed
=0.1
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G. VICCIONE & V. BOVOLIN
528
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Fig. 10 - Simul. N.5. Particle triggered:
PT2, limit pressure p
lim
= 200 kgf /
cm
2
, viscosity coefficient α
bed
=0.1
Fig. 11 - Simul. N.6. Particle trig-
gered: PT2, limit pressure
p
lim
= 100 kgf /cm
2
, viscosity
coefficient α
bed
=0.1
Fig. 12 - Simul. N.7. Particle
triggered: PT3, limit
pressure p
lim
= 300
kgf /cm
2
, viscosity co-
efficient α
bed
=0.1
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SIMULATING TRIGGERING AND EVOLUTION OF DEBRIS-FLOWS WITH SPH
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
529
Fig. 13 - Simul. N.8. Particle trig-
gered: PT3, limit pressure
p
lim
= 200 kgf /cm
2
, viscosity
coefficient α
bed
= 0.1
Fig. 14 - Simul. N.9. Particle
triggered: PT3, limit
pressure p
lim
= 100 kgf
/cm
2
, viscosity coeffi-
cient α
bed
= 0.1
Fig. 15 - Simul. N.10. Particle trig-
gered: PT3, limit pressure
p
lim
= 200 kgf /cm
2
, viscosity
coefficient α
bed
= 1
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G. VICCIONE & V. BOVOLIN
530
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Fig. 16 - Simul. N.10. Particle
triggered: PT3, limit
pressure p
lim
= 200
kgf /cm
2
, viscosity co-
efficient α
bed
=10
Fig. 17 - Comparison between Simul. N.1. and
Flo-2D results for t =50 secs
Fig. 18 - Comparison between Simul. N.1.
and Flo-2D results for t =100 secs
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SIMULATING TRIGGERING AND EVOLUTION OF DEBRIS-FLOWS WITH SPH
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
531
Fig. 19 - Comparison between Simul. N.6. and Flo-
2D results for t =50 secs
Fig. 20 - Comparison between Simul. N.6. and
Flo-2D results for t =100 secs
Fig. 21 - Comparison between
Simul. N.9. and Flo-2D
results for t =50 secs
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G. VICCIONE & V. BOVOLIN
532
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
Fig. 22 - Comparison between Simul. N.9. and
Flo-2D results for t =100 secs
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iu
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iu
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iu
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