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Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
153
DOI: 10.4408/IJEGE.2013-06.B-12
MODELING OF RUNOUT LENGTH OF
HIGH-SPEED GRANULAR MASSES
F
rancesco
FEDERICO & c
hiara
CESALI
University of Rome “Tor Vergata” - Rome, Italy
of high speed granular mass often depends on the in-
terstitial pressures at the base of the mass, that can
vary between null and higher than hydrostatic values
(i
verson
, 1997), due to possible water pressure ex-
cess, related to very rapid changes of pore volumes,
often localized along a thin layer in proximity of the
sliding surface. Experimental observations showed in
fact the growth of a basal “shear zone” where initially
great deformations and then dilation and collisions oc-
cur, differently from the top (h
ungr
, 1995).
Field observations denote the dependence of the
runout length on the debris flow volume, but the usual
Mohr-Coulomb (M-C) shear resistance criterium
doesn’t allow to obtain this result.
Thus, more complex resistance laws must be
developed to describe the rapid sliding of a granular
mass because high speed relative motion and colli-
sions between solid grains take place within the basal
shear layer, causing a fluidification effect (h
ungr
&
e
vans
, 1996) coupled with energy dissipations.
To this purpose, B
agnold
(1954) defined “dis-
persive pressure” the stress component normal to the
boundary that encloses a set of particles in grain-in-
ertial regime:
(1)
being:
a
i
, the “Bagnold coefficient”; B
agnold
(1954) and
T
akahashi
(1981) suggest the value 0.042;
ρ
s
, the solid fraction mass density;
ABSTRACT
The power balance of a high speed granular mass
sliding along planar surfaces is written by taking into
account its volume, the slopes of the surfaces (runout
and runup), an assigned basal fluid pressure and dif-
ferent possibilities for the energy dissipation. In par-
ticular, collisions acting within a thin layer (“shear
zone”) at the base of the mass and shear resistance
due to friction along the basal surface induce the dis-
sipation of energy. The solution of the ODE describ-
ing the mass displacements vs time is numerically ob-
tained. The runout length and the speed evolution of
the sliding mass depend on the involved geometrical,
physical and mechanical parameters as well as on the
rheological laws assumed to express the energy dis-
sipation effects. The well known solutions referred
to the Mohr-Coulomb or Voellmy resistance laws are
recovered as particular cases. The runout length of a
case is finally back analysed, as well as a review of
some relationships expressing the runout length as a
function of the volume V of the sliding mass.
K
ey
words
: sliding granular mass, granular temperature,
shear layer, collisions
INTRODUCTION
The analysis of the complex mechanisms of the
chaotic movement of high speed granular masses
sloping along mountain streams till their arrest is nec-
essary to identify hazardous areas. The runout length
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a comparison among solutions of G-M, r
ickenmann
’s
empirical formula (1999) and c
orominas
s
results
(1994) of analyses are finally developed.
BASIC ASSUMPTIONS
• The granular sliding body is composed by two
masses of equal basal area Ω and length l; they rep-
resent respectively the "shear zone" (thickness s
s
) and
the superimposed mass (thickness s
b
,“block”). The to-
tal height of the sliding mass is H = s
b(
t) + s
s
(t). The
global geometry (Ω, l and H) does not change; erosion
or deposition processes are neglected.
• The “shear zone” is composed by particles that,
moving at high velocity and colliding each with oth-
ers, induce appreciable fluctuations of their velocities
(granular temperature); the “block” is dominated by
inertial forces and quasi-static stress.
• The sum of the masses of the shear zone (m
s
)
and the overlying block (m
b
) equals the total sliding
mass m
0
. Both masses vary during the sliding and may
change their volume:
ρ
s
, ρ
b
simply assume constant values although it is pos-
sible to define their dependence upon the sliding rate.
• The thicknesses s
b
(x(t)), s
s
(x(t)) are not a priori
known along the travelled distance x, at time t. Their
values may be obtained by imposing the equilibrium
in the direction orthogonal to the sliding planes: the
resulting N
tot
(= Wcosζ, ζ=θ or α, Fig. 1) must be bal-
anced by the lithostatic stresses, σ
lit
, as well as by the
dispersive pressures, p
dis
, introduced by B
agnold
(1954). The equilibrium equation is therefore written
as follows:
(4)
being:
(5)
H=s
0
b
, block initial thickness;
the functions r and are defined to allow a rational
splitting of the force N
tot
between the lithostatic force
and the resultant of the colliding forces. They are writ-
ten as a function of the rate of the sliding masses():

(6)
(7)
being:
λ, the “linear concentration”, that is a function of solid
fraction v
s
;
d
p
, the characteristic diameter of the grain;
(du/dy)
2
, the square of the velocity gradient;
Φ, the internal dynamic friction angle of granular
bulk.
If a linear change of velocity, along the orthogo-
nal direction of the motion, is assumed, the following
dependence is obtained between the Bagnold’s defini-
tion of dispersive pressure and the rate x ̇ of the sliding
mass: p
dis
~x
2
.
o
gawa
(1978) defined “granular temperature” as
the mean square deviance of the relative velocities of
sliding and colliding particles with respect to the mean
value
(2)
Several Authors apply the Voellmy law (V-M): a
turbulent resistance is added to the M-C resistance, to
estimate the runout length of debris flows.
To overcome these limitations, the rapid sliding
of a granular mass along planar surfaces is analyti-
cally modelled in the paper, by taking into account the
effects of granular temperature and dispersive pres-
sure
, acting within the basal ‘shear zone’. An original
model is firstly proposed, based on some simplifying
hypotheses. The governing equations are formulated
by introducing the parameters describing the granu-
lar temperature
and the dispersive pressure. After an
evaluation of the model parameters, some paramet-
ric results are developed. The comparisons among
solutions obtained according to the General (G-M),
Coulomb (C-M) and Voellmy Models (V-M) are then
shown. The schematic back analysis of a well de-
scribed avalanche is carried out through the G-M and
F
ig. 1 - Problem’s setting and reference systems. The ori-
gin (x = 0) of each reference system coincides with
the projection of the position of the gravity centre
of the sliding mass along the
sliding surfaces
(3)
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155
(17)
(18
)
Counterslope:
(19)
(20)
Transition zone:
(21)
(22)
(23)
(24)
(25)
(26)
In the proposed model, to simplify the numerical
solution, it is directly assumed a linear combination
on powers of energies:
(27)
being
(28)
(29)
The kinetic energy and the power of the kinetic
energy are expressed as follows:
(30)
(31)
(32)
(33)
EFFECTS OF COLLISIONS
E
gt
, E
coll.
The energy E
gm
transferred to the “shear
η, parameterϵ[0.005,0.5];
v
crit,
the critical value of the speed for which the re-
gime dominated by the inertial forces turns towards a
regime governed by the collisions.
By recalling the expression [4], it is obtained:
(8)
(9)
being:
(10)
• The angle ζ of the slope may assume only two
values: ζ = θ, runout; ζ = α, runup. Therefore, the total
length traveled by a high speed sliding granular mass
is obtained through the analysis of three sliding phas-
es. (I): the granular mass runs along the first slope (θ
and length L are assigned) and progressively acceler-
ates; (II) intermediate section: the granular mass runs
at the same time along both slopes (θ and α, see Fig.
1); (III): the granular mass runs only along the coun-
terslope (α); its speed decreases up to stop.
ENERGY AND POWER BALANCES
The energy balance of the sliding mass is ex-
pressed by the equation:
(11)
E
p,0,
initial potential energy; E
p
, potential energy;
E
k
, kinetic energy of the sliding mass; E
fr
, energy
lost due to the (Coulomb’s) friction along the slid-
ing surface; E
coll
and E
gt,
energies transferred from the
block” to the basal “shear zone” to support the grain
inertial regime. Deriving the eq. [11], the Power Bal-
ance is obtained
:
(12)
POTENTIAL AND KINETIC ENERGY: E
P
AND
E
K
The potential energy is expressed as follows (b =
block; s = shear zone):
(13)
The corresponding power is obtained deriving [13]:
(14)
First slope:
(15)

(16)
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ues must be assigned. The length d
w
varies in the range:
d
(w,min)
≤d
w
≤d
(w,max)
. If the sliding granular mass always
transfers positive normal stresses to the basal surface,
the minimum value d
w,min
can be deduced by imposing
the equilibrium along the direction perpendicular to the
sliding surface:
(41)
γ
s
is the unit weight of the sliding mass. The interstitial
pressure’s resultant is expressed as follows:
(42)
THE ROLE OF BASAL FRICTION
The role of basal friction: Efr. Efr is a function of
the weight W of the sliding mass, the dynamic friction
angle Φ
b
at the base of the block. Dispersive and inter-
stitial pressures reduce the friction energy dissipation.
By including the effect of interstitial pressures,
the basal friction resistance is expressed as follows:
(43)
φ
b
assumes a constant value along the slopes. The
power related to the energy dissipated due to the fric-
tion along the sliding basal surface is:
(44)
RESULTS OF PARAMETRICAL ANALY-
SES
The proposed model depends on few param-
eters pertaining to the micromechanical behaviour: β
(ϵ[0,2])
, e (restitution coefficient ϵ[0,1]), k (in Ėgt’s
expression [36]) and d
p
(grain diameter). First of all,
by changing arbitrarily the parameters e and β, we in-
vestigate the effect of the parameter k, for the assigned
values of remaining parameters: L=1000 m; θ=38°;
α=0°, Φ
b
=18°, l=300 m; Ω=15000 m
2
; H=35 m; d
p
=
0.1 m; interstitial pressure resultant U ≠ 0 (d
w
= 0). In
Table 1, the set of selected values is shown
Results in terms of the (runout and runup) total
length (x), block and shear zone thicknesses (s
b
, s
s
),
zone” is partly lost due to repeated grain inelastic col-
lisions (E
coll
) and partly stored as granular temperature
(E
gt
). Z
hang
& F
oda
(1997) have shown that the power
of the energy lost in granular collisions is related to the
granular temperature (T
g
) according to the relation:
E coll~T
g
(3/2)
. o
gawa
(1978) observed that the energy
stored in the grain-inertial regime is proportional to the
granular temperature: E
gt
~T
g
.
Granular temperature, in turn, is proportional to
the mean velocity of the grains composing the shear
zone (s
avage
& J
eFFrey
, 1981), according to the re-
lation: T
g
~x
2
. These relations are respectively applied
to formulate the powers of energies lost due to grain
inelastic collisions (E
coll
) as well as stored in granular
temperature (E
gt
):
(34)
(35)
being, according to previous analyses (F
ederico
&
F
avaTa
, 2011):
(36)
(37)
(38)
e being the restitution coefficient (ϵ[0,1]); ω, ZHANG-
FODA coefficient; β, coefficient ϵ[0,2] and υ
s
being
the solid franction; d
p
, the characteristic diameter of
the grains, ρ
s
, the solid phase density
(39)
INTERSTITIAL PRESSURES
The interstitial pressure p
w
(x ) at the base of the
mass affects the friction dissipated energy. Isopiezic
lines are assumed orthogonal to the motion direction
and p
w
(x) is assumed constant along the planar slid-
ing surface (i
verson
, 1997):
(40)
γ
w
is the specific weight of the water; d
w
= 0 if the mass
is saturated; d
w
=H if the mass is dry; p
w
may exceed
the hydrostatic value due to the mechanical effects as-
sociated with the rapid change of pore volumes, cor-
responding growth of interstitial water pressures excess
(m
usso
et alii, 2004). To simulate this effect, d
w
< 0 val-
Tab. 1 - Assigned values to the parameters e, β, k
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157
sults are reported in terms of runout length, block and
shear zone thicknesses, collisional energy and energy
related to granular temperature, according to d
p
.
collisional energy (E
coll
) and energy related to granular
temperature (E
gt
), for different values of the parameter
k are shown in the figures 2, 3, 4 and 5.
The thicknesses of the block and the ‘shear zone
slightly change if the parameter k increases (Fig. 2).
Collisional energy E
coll
, set the coefficient of restitution
(e = 0.3), to decrease of the parameter k (and therefore
of β), increases, while holding almost unchanged, to
the decrease of k, fixed the coefficient β (Fig. 3).
The energy associated with granular tempera-
ture E
gt
decreases if k decreases (Fig. 4). It is worth
observing that if the parameters e, β provide small
values of k, the sliding mass reaches unrealistic high
speeds (more than 40 m/s), usually obtained by Cou-
lomb Model. Fixed parameter β, a decrease of k (and
thereby the increase of e) gets an increase of both
maximum speed and runout length, while fixed e, if
k decreases, the runout length decreases too (Fig. 5).
Effect of the parameter dp. (the parameters e, β
and k are: e = 0.3; β = 1.75; k = 0.7).
The following values of the parameter d
p
are as-
signed: d
p
=0.05 m; d
p
=0.10 m; d
p
= 0.15 m). The re-
F
ig. 2 - Curves [sb(x(t)), x(t)];[ss(x(t)), x(t)]; )];[H, x(t)]
for different values of parameter k
Fig. 3 - Collisional energy for different values of k
Fig. 4 - Energy E
gt
related to the granular temperature T
g
,
for different values of parameter k
Fig. 5 - Rate v of the sliding granular mass, for different
values of parameter k
Fig. 6 - Curves [s
b
(x(t)), x(t)]; [s
s
(x(t)), x(t)]; [v, x(t)] for
different values of parameter d
p
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If the diameter d
p
decreases, the thickness of the
shear zone’ becomes smaller; as a result, being the sum
of s
b
and s
s
equal to the height H of the debris flow (con-
stant value), the thickness of the sustained block increas-
es (Fig. 6). Instead, the collisional energy, to an increase
of d
p
, increases (Fig. 7). If d
p
increases, the distance
traveled, the maximum speed (Fig. 6) and the energy as-
sociated with granular temperature E
gt
(Fig. 7) decrease.
COMPARISON WITH RESULTS OBTAI-
NED ACCORDING TO CONVENTIONAL
RHEOLOGICAL MODELS
Results obtained through the General-Model (G-M)
are compared with solutions of Coulomb-Model (C-M)
and Voellmy-Model (V-M). The motion equations for
the V-M and C-M models are given by the ODE:








(45)
M̅=0 describes the Coulomb Model (C-M); M̅≠0,
the Voellmy Model (V-M). The M̅ parameter can be
related to the ξ turbulence coefficient of Voellmy
through the equation (F
ederico
& F
avaTa
, 2011):
(46)
γ
s
being the specific weight of the bulk mass. The Voe-
llmy’s turbulent component of resistance describes,
in a schematic manner, the energy dissipation due to
granular collisions. Therefore, in the Voellmy Model,
the term M̅x
2
, describes the energy dissipation in gran-
ular collisions (E
coll
). For a better comparison between
the results obtained through the models (G-M, C-M,
V-M), the coefficient M̅ has been determined on the
base of the coefficient M of the General Model, re-
placing s
s
(t) with (s
s
max
⁄2):
(47)
If the [47] applies, the energy dissipated due to col-
lisions is almost equal for both G-M and V-M models.
The following values of the parameters are as-
signed: θ = 30°, α = 0°; L = 1000 m; γ
w
=10 kN/m
3
;
ρ
b
=2105 Kg/m
3
; d
w
= 0; H = 25 m; V = 18750 m
3
; m =
4•10
7
kg; k = 0,7; φ
b
= 18°; d
p
= 0.05 m; Ω = 750 m
2
; l
= 100 m; e = 0.3; β = 1.75; in V-M, M̅=8 x 10
4
Kg/m;
in C-M, M̅ = 0 kg/m is assumed. The figures 8, 9, 10
and 11 show the G-M, V-M and C-M results.
C-M model gets run out and velocity values great-
er than G-M and V-M models (Fig. 8); the reduction
of the sliding rate is caused by the additional shear re-
sistance due to grains dissipation. In Fig. 9 a compari-
son among the models is shown in terms of traveled
distance. In G-M model, the duration of the motion is
greater than in the others cases. In Fig. 10, the energies
concerned with the G-M model are shown; the initial
potential energy partly becomes kinetic energy, partly
is stored as granular temperature, partly is dissipated
owing to grains collisions and friction sliding. In Fig.
8, the block and ‘shear zone’ thicknesses concerned
with the General Model are shown. It is further in-
vestigated the influence of the volume V of the slid-
ing mass on the total runout length, for some assigned
values of parameters: L=1630 m; θ=30°; α=-15°,
φ
b
=18°, d
p
= 0.05 m; d
w
= 0, γ
w
=10 kN/m
3
; ρ
b
=2105
Kg/m
3
. Results of computations are shown in Fig. 11.
C-M run out doesn’t depend on the volume of the slid-
ing mass. In G-M and V-M, the solutions depend on
the granular volume; for small value of V, the run out
Fig. 7 - Collisional energy (E
coll
) and energy related to the
granular temperature (E
gt
) for different values of dp
Fig. 8 - Curves [v(t), x(t)]; [s
b
(x(t)), x(t)]; [s
s
(x(t)), x(t)]:
comparison between General Model, Coulomb
Model and Voellmy Model
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The separated rock mass has been shattered by
impacts against the side of the mountain during its
sliding, and probably long before it reached the bot-
tom, into myriads of fragments, some of which were
flung far out into the valley. Immediately after the
slide, an inspection was made by the Geological Sur-
vey of Canada: the slide occurred across rather than
along bedding planes and the primary cause for the
slide was found in the structure of the mountain. Water
action in summit cracks and severe weather conditions
also contributed to the disaster. In Table 2, the input
parameters obtained from conventional back analysis
of the event (c
ruden
& h
ungr
, 1986) are shown.
The runout length (x) and the rate of sliding
mass (v), for the assigned parameters d
p
= 0.05 m;
d
w
= 0; γ
w
=10 kN/m
3
; ρ
b
=2105 Kg/m
3
; e = 0.3; β =
1.75 are shown in figures 12, 13; in V-M, M̅=1.5
x 10
7
Kg/m; in C-M, M̅ = 0 kg/m is assumed. The
maximum speed and the runout length computed
according to the Coulomb Model appear remark-
ably greater than the corresponding values obtained
through the G-M and V-M models (Fig. 12). The
Voellmy Model gets values of runout less than the
values obtained by the General Model (Fig. 13) al-
though the corresponding rate appears too high. The
length increases significantly with the volume V; for
great values of V, G-M and V-M runout length tend
to the C-M runout and the travelled distance progres-
sively becomes almost independent on the sliding
mass volume.
BACK ANALYSES
Through the proposed model, the Frank slide is
first back analysed; the comparison among solutions
of G-M, r
ickenmann
s
empirical formula and c
orom
-
inas
s
analyses are then carried out.
Frank slide. The Frank slide occurred on the morn-
ing April 29, 1903, in the south western Alberta (Brit-
ish Columbia, Canada). The original unstable rock
mass volume was estimated as 30x10
6
m
3
. The debris
moved down from the east face of Turtle Mountain
across the entrance of the Frank mine, the Crowsnest
River, the southern end of the town of Frank, the main
road from the east, and the Canadian Pacific mainline
through the Crowsnest Pass.
Fig . 9 Curves [x(t), t]; comparison between General
Model, Coulomb Model and Voellmy Model
F
ig. 10 - Energies concerned with G-M solution
Fig. 11 - Curves: [x(t), V]; comparison between General
Model, Coulomb Model and Voellmy Model
Tab. 2 - Frank slide. Input parameters for back analysis
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mass):
(48)
being: V, volume of the mass; H, difference in eleva-
tion (see Fig. 14).
The geometry of the problem is fixed and the fol-
lowing values of the parameters are assigned: θ = 38°,
α = 0°; L = 1000 m;
γ
w
=10 kN/m
3
; ρ
b
=2105 Kg/m
3
;
d
w
= 0; φ
b
= 18°; e = 0.3; β = 1.75. By applying the
(G-M) model, different values of parameters H, d
p
, e,
l, and Ω are assigned. The Fig. 14 shows the (runout
and runup) total length computed as a function of the
volume V. Following a suitable choice of the assigned
parameters, the travelled length estimated through the
General-Model is quite close to that one estimated by
the empical formula [48]. While the empirical formula
for runout length depends explicitly on the volume V,
in the G-M model, the runout length depends on the
volume through the parameters H, d
p
, e, l, and Ω.
s
cheidegger
(1973) correlated the volume V of a
sliding mass to the dimensionless variable f, this one
defined as the tangent of the angle Ψ (Fig. 15) formed
by the horizontal line and the straight line joining the
point of greatest potential energy of the system under
static conditions and the lower end of the debris flow,
after its arrest. The s
cheidegger
s relationship is ex-
pressed as follows
(49)
The constants C
1
and C
2
are determined through
interpolation of data on real landslides. After a careful
back analysis on several debris flows, c
orominas
(1994)
obtained the following values: C
1
= - 0.034; C
2
= - 0.101.
By interpolating values of f obtained from the val-
General Model gets the runout length greater than
that one computed through the Voellmy Model and
close to that one in situ observed. The time interval
of the motion is about 120 s for the General Model
and about 70 s for the Voellmy Model (Fig. 13). The
larger time interval for the G-M derives from the
stored energy as ‘granular temperature’ that sus-
tained the runup phase, allowing a more gradual re-
duction of the sliding rate, if compared to the reduc-
tion pertaining to the V-M or C-M models.
EMPIRICAL RELATIONSHIPS.
A comparison between the General-Model results
and runout lengths obtained through some empirical
relationships proposed in literature is carried out. Af-
ter back analyses on “160 debris flows”, r
ickenmann
(1999) proposed an empirical relation to estimate
the length D (horizontal distance traveled by sliding
Fig. 12 - Frank slide back analysis. Curves [v(t), x(t)] ob-
tained through the General, the Coulomb and the
Voellmy Models
Fig. 13 - Frank slide back analysis. Curves [x(t), t] ob-
tained through the General, the Coulomb and the
Voellmy Models
F
ig. 14 - Curves [x,V]; comparison between General Mod-
el, and RICKENMANN’s relationship
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CONCLUDING REMARKS
An original analytical model (General Model)
based on energy-balance equations and some simplified
assumptions, to estimate the runout length of granular
debris flow or avalanches, is proposed. Hypotheses
concern the geometry of the sliding mass (parallelepi-
pedal shape), sliding surface (planar surfaces) and en-
ergy dissipation (friction, collisions). The model takes
into account several experimental results reported in
technical literature. Specifically, in a granular material
sliding at high rate along a basal surface, a thin (‘shear’)
zone, with variable thickness, whose behaviour is
characterized by a regime dominated by the presence
of collisions, hosting the "granular temperature" phe-
nomenon, generates and develops in proximity of the
basal surface. The material composing the shear zone
exchanges energy and mass with the remaining upper
material (block), characterized by a regime dominated
by inertial forces. Through the balance of the involved
mechanical powers, the travelling of the granular mass
along the planar surfaces is describes by a system of
ODE, that have been numerically integrated. Paramet-
ric analyses allowed to identify the role of geometri-
cal and mechanical parameters, such as the diameter
of grains (d
p
), the coefficient of restitution (e) and the
parameter k. Finally, comparisons between the results
obtained through the General Model, the Coulomb and
Voellmy Models, as well as the back analysis of a case
and a critical examination of well known empirical re-
lationships are shown. The main limits of the proposed
model lie in the oversimplified geometry of the debris
body, in the assumption of constant total mass, in defi-
nition of the micro-mechanical parameters.
ues of the runout length related to cases analyzed with
the General-Model, for different values of the micro-
mechanical parameters e and d
p
, the results shown in
Fig. 15 are obtained.
In Fig. 15, the values of the parameter f as a func-
tion of the volume V related to the cases analyzed with
the G-M and the results obtained through the [49] are
shown (c
esali
, 2013). An appreciable agreement be-
tween these results is observed. At the same time, the
role of micromechanical parameters is highlighted;
their influence on the definition of empirical relation-
ship cannot be neglected, although it is not easily
knowable, since the values of the runout length, (and
of the parameter f), were obtained by varying other
parameters such as H and Ω (height and basal area of
the debris flow). In particular, by decreasing e and in-
creasing d
p
, thanks to a suitable choice of H and Ω, the
trends obtained by the G-M approximates the trend
obtained by c
orominas
(1994).
Fig. 15 - Curves [f, V]; comparison between General Mod-
el and COROMINAS’s results
REFERENCES
B
agnold
r.a. (1954) - Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc.
Roy. Soc. London 225: 49-63.
B
lanc
T., P
asTor
m., d
remPeTic
m.s.v. & h
addad
B. (2011) - Depth integrated modeling of fast landslide propagation,
European Journal of Environmental and Civil Engineering.
c
esali
c., (2013) - Modelli di simulazione dello scorrimento di colate detritiche ad alta velocità. Thesis in Civil and
Environmental Engineering, University of Rome Tor Vergata, Rome.
c
orominas
J. (1994) - The angle of reach as a mobility index for small and large landslides. Canadian Geotechnical Journal,
33: 260-271.
c
ruden
d.m. & h
ungr
o. (1986) - The debris of the Frank Slide and theories of rockslide-avalanche mobility. Canadian Journal
of Earth Sciences, 23: 425-432.
F
ederico
F. & F
avaTa
g. (2011) - Coupled effects of energy dissipation and travelling velocity in the run-outsimulation of high-
speed granular masses. International Journal of Geosciences, 2 (3): 274-285
h
ungr
o. (1995) - Model for the runout analysis of rapid flow slides, debris flows, and avalanches. Canadian Geotechnical
background image
F. FEDERICO & C. CESALI
162
International Conference on Vajont - 1963-2013 - Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
Journal, 32: 610-623.
h
ungr
o. & e
vans
s.g., (1996) - Rock avalanche run out prediction using a dynamic model. Proceeding 7
th
International
Symposium on Landslides, 1: 233-238, Trondheim, Norway.
I
verson
r.m. (1997) - The physics of debris flows. American Geophysical Union.
l
aBiouse
v. & m
anZella
i. (2013) - Empirical and analytical analyses of laboratory granular flows to investigate rock avalanche
propagation. Landslides.
m
ollon
g., r
icheFeu
v., v
illard
P. & d
audon
d., (2012) - Numerical simulation of rock avalanches: influence of a local
dissipative contact model on the collective behavior of granular flows. Journal of Geophysical Research: Earth Surface.
m
usso
a., F
ederico
F. & T
roiano
g., (2004) - A mechanism of pore pressure accumulation in rapidly sliding submerged porous
blocks. Computers and Geotechnics, 31: 209-226.
o
gawa
s. (1978) - Multitemperature theory of granular materials. Proceedings of the US Japan Seminar on Continuum
Mechanical and Statistical Approaches in the Mechanics of Granular Materials, Tokyo, Gakajutsu Bunken Fukyu-Kai.
r
ickenmann
d. (1999) - Empirical relationships for debris flows. Natural Hazards, 19: 47-77.
s
avage
s.B. & J
eFFrey
d.J, (1981) - The stress tensor in a granular flow at high shear rates. J. Fluid Mech., 110: 255-272.
s
cheidegger
a.e. (1973) - On the prediction of the reach and velocity of catastrophic landslides. Rock Mechanics, 5: 231-236.
T
akahashi
T. (1981) - Estimation of potential debris flows and their hazardous zones; soft countermeasures for a disaster. Journal
of Natural Disaster Science, 3 (1).
w
ang
X., m
orgensTern
n.r. & c
han
d.h., (2010) - A model for geotechnical analysis of flow slides and debris flows. Canadian
Geotechnical Journal.
Z
hang
d. & F
oda
m. a., (1997) - An instability mechanism for the sliding motion of finite depth of bulk granular materials. Acta
Mechanica, 121: 1-19.
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