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Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
445
DOI: 10.4408/IJEGE.2013-06.B-43
3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED
BY THE VAJONT ROCKSLIDE
R
enato
VACONDIO
(*)
, S
eRena
PAGANI
(*)
, P
aolo
MIGNOSA
(*)
& R
inaldo
GENEVOIS
(**)
(*)
University of Parma, Department of Civil Engineering - Parco area delle Scienze 181/A, 43121 Parma (Italy)
(**)
University of Padova, Department of Geosciences - Via Gradenigo 6, 35131 Padova (Italy)
the border between Friuli Venezia-Giulia and Veneto
(Northern Italy), fell into the artificial reservoir of the
Vajont dam. The slide generated one of the most de-
structive waves ever documented in the literature.
The wave overflowed the dam, which remained
almost intact, and through the downstream narrow
gorge reached the Piave valley and the village of Lon-
garone, causing the loss of about 2000 people.
In the past 50 years the Vajont slide has been
deeply analyzed from a geological point of view (S
e
-
menza
et alii, 1965; m
ülleR
, 1987a; m
ülleR
, 1987b;
S
emenza
, 2001; G
enevoiS
& G
hiRotti
, 2005). Several
authors investigated also its kinematics and dynamics
(h
endRon
& P
atton
, 1985; S
emenza
& MELIDORO,
1992; e
RiSmann
& a
bele
, 2001) and most of them
agree that the rockslide can be schematized as a rigid
body (m
ülleR
, 1961; S
elli
& t
ReviSan
, 1964; d
atei
,
2005; S
uPeRchi
, 2011). There is also a general agree-
ment (S
elli
& t
ReviSan
, 1964; c
iabatti
, 1964; d
atei
,
1969) about the total volume of the slide, which was
estimated in about 270÷300 10
6
m
3
.
On the other hand few studies were devoted to the
simulation of the wave generated by the rockslide. The
older ones are mainly based on the empirical recon-
struction of the wave, through the data collected by
eye witnesses and by marks observed on the ground
after the disaster (S
elli
& t
ReviSan
, 1964; v
iPaRelli
& m
eRla
, 1968; S
emenza
, 2001; d
atei
, 2005).
More recently b
oSa
& P
etti
(2010) simulated
these phenomenon by means of a 2D Shallow Wa-
ABSTRACT
In this paper a 3D numerical modeling of the
wave generated by the Vajont slide is presented. In
order to completely describe the complex flow gener-
ated by the slide a Smoothed Particle Hydrodynam-
ics (SPH) technique was adopted. To the best of the
author knowledge this is the first attempt to describe
the events adopting a fully 3D numerical model which
discretizes the Navier-Stokes Equations.
The SPH adopted herein is a meshless Lagrangian
technique which is able to simulate the highly frag-
mented violent flows generated by the falling slide in
the Vajont artificial reservoir. Moreover the Compute
Unified Device Architecture (CUDA) of nVidia devic-
es parallelization technique has been adopted to obtain
the speed-up sufficient for the high resolution needed
to accurately describe the phenomenon.
The simulation results have been validated by com-
paring the maximum run-up and the water level in the
residual lake after the events against the ones reported
in literature. In addition to that, the 3D velocity field of
the flow during the event together with the discharge hy-
drograph which overflowed the dam has been obtained.
K
ey
words
: Vajont, rockslide, smoothed particle hydrodyna-
mics, Navier-Stokes, free-surface flows
INTRODUCTION
At 22:39 of October 9
th
1963 a catastrophic rock-
slide slipped from the northern slope of Mount Toc, on
email: rinaldo.genevois@unipd.it, phone: +39 049 8279119
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R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS
446
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
were reconstructed. Moreover, assuming that the slide
can be described as a rigid body which rotates around
an axis, its kinematics has been calculated starting
from the Newton’s second law.
The numerical model was validated by comparing
the results against: a) the position of maximum wave
run-up reported in S
emenza
et alii (1965); b) the in-
creased water level of the residual lake after the slide.
The discharge hydrograph that overflowed the dam
was also evaluated.
SPH NUMERICAL SCHEME
Through the SPH technique, the continuum is
represented as a set of discrete particles, character-
ized by their own physical properties (such as mass,
density, pressure).
The main feature of the SPH technique is to ap-
proximate a generic scalar function A(r) at any point
r, as follows:
where h is the so-called “smoothing length” and W(r-
r', h) is the weighting function or kernel. This notation
in discrete form becomes:
where the summation is extended to all the particles
within the domain of influence of the particle a (2h
for the kernel function herein adopted), m
b
and ρ
b
are
respectively the mass and the density of particle b.
The gradient of the generic scalar function A'(r)can
be approximated by means of an SPH interpolation, as:
which can be written in discrete form as:
In this work the quantic Wendland kernel (W
end
-
land
, 1995) is adopted:
where q=││r-r'││/h and α
D
is 21/(16πh
3
).
The movement of the particles is defined by in-
tegrating in time the Navier-Stokes equations written
for a weakly compressible fluid. In Lagrangian for-
malism the continuity equation can be written as
ter model. The slide was schematized as a moving
vertical wall which acted as a “piston” in moving the
water of the Vajont lake. After the halt of the slide the
wall was removed from the model and the “before”
terrain elevation was substituted with the “after”
configuration of the valley.
In the writers’ opinion, however, the wave can be
only approximately described by this type of sche-
matization and by means of a two-dimensional depth
averaged numerical scheme, which neglects the ver-
tical velocity component and assumes that the pres-
sure is hydrostatic.
On the other hand, three-dimensional Eulerian
models, already widespread in other Computational
Fluid Dynamic (CFD) fields, are difficult to apply in
this case due to the presence of a highly fragmented
free surface and due to the computational effort neces-
sary to adequately describe the water body.
In this work a 3D Smoothed Particle Hydrody-
namics (SPH) model was used in order to simulate
the wave generated by the Vajont rockslide. To the
authors’ knowledge, this is the first literature contribu-
tion which applies a fully 3D model to the slide move-
ment and to the wave simulation.
The “Smoothed Particle Hydrodynamics” (SPH)
is a Lagrangian meshless method originally intro-
duced in astrophysics (G
inGold
& m
onaGhan
, 1977)
and subsequently extended to Computational Fluid
Dynamics (m
onaGhan
, 1994).
This numerical technique enables the numeri-
cal simulation of free-surface flows with violent im-
pact, such as: breaking waves (d
alRymPle
& R
oGeRS
,
2006), dam-break phenomena (c
ReSPo
et alii, 2008),
interactions between waves and coastal structures
(Gó
mez
-
G
eSteiRa
& d
alRymPle
, 2004), etc.. The main
drawback of the SPH technique is the high computa-
tional cost which has prevented till now its applica-
tion to practical engineering problem with complex
geometries. Recently c
ReSPo
et alii (2011) developed
a parallel algorithm named “Dualsphysics” (http://
www.dual.sphysics.org/
) which overcame this limita-
tion by means of the Compute Unified Device Archi-
tecture (CUDA) available for nVidia devices. In this
way a speed-up of approximately 50-100 with respect
of CPU runtime of non-parallel codes was obtained.
The same open-source code is herein adopted.
Following the work of S
uPeRchi
(2011), the 3D
shape of the sliding surface and of the rockslide body
(1)
(2)
(3)
(4)
(5)
(6)
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3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
447
the density variation within the 1% c
0
is set equal to 20
times the maximum expected velocity.
Particle position is updated using the following
XSPH velocity correction (m
onaGhan
, 1994):
where ε =0.5 and ρ
ab
=(ρ
a
+
ρ
b
)
/
. This method moves
particle a with a velocity that is close to the average
velocity in its neighborhood.
Time-step Δ
t
is calculated according to the Cour-
ant-Friedrichs-Lewy condition, the forcing terms and
the viscous diffusion terms (m
onaGhan
, 1989), so Δt
is equal to:
Δt = CFL
·
min(Δt
f
, Δt
cv
);
with:
where |f
a
| is the magnitude of the force per unit of mass
for particle a, and Δt
cv
combines the Courant and the
viscous time-step controls and CFL is the Courant
number. In this work CFL
=
0.4 has been used.
Equations (7), (9) and (13) are updated in time
using a symplectic time integration algorithm (L
EIM-
KHULER
et alii, 1996). The values of density and par-
ticle position are calculated at the middle of the time
step n+1/2 as:
where t
=
nΔt. Pressure, p
a
n+1/2
, is calculated using the
equation of state (11). In the second stage the accelera-
tion dv
a
n+1/2
/dt gives the velocity and hence the posi-
tion at the end of the time step:
At the end of the time step dρ
a
n+1
/ dt is calculated
using the updated values of v
a
n+1
and r
a
n+1
.
The discretization of solid boundary conditions
is still an open problem in SPH models. In this work
the “dynamic boundary particles” method is adopted
(c
ReSPo
et alii, 2007). This method guarantees that
complex domains can be easily discretized and leads
to a good compromise between accuracy and compu-
tational costs. The boundaries are discretized by a set
where v is the velocity vector and ρ is the density.
Discretizing the▼
v
by means of the SPH interpolation
Equation (6) becomes:
where▼
a
W
ab
is the gradient of the kernel function. The
summation of Equation (7) is over all the particles with-
in the region of compact support of the kernel function.
The momentum conservation equation in a con-
tinuum field is:
where p is the pressure and Θ is the dissipative term.
In SPH notation, Equation (8) can be written as:
in which p
a
and ρ
a
are respectively the pressure and
density for the particle a (same thing goes for the par-
ticle b) and π
ab
is the artificial viscosity (M
ONAGHAN
,
1994) defined as follows:
with:
In Eq. (10) r
ab
=
r
a
-
r
b
, v
ab
=
v
a
v
b
being r
k
and v
k
the position and the velocity corresponding to particle
k (a or b); c
ab
(c
a
+
c
b
)/2, η
2
=0.01h
2
, α
v
is a free param-
eter that needs to be tuned. In this work the value of
α
v
=
0.2 is adopted.
In the SPH method the fluid is simulated as weak-
ly compressible using the following equation of state
(b
atcheloR
, 1974) which connects pressure p and
density ρ:
where B can be written as:
in which ρ
0
is the reference density (1000 kg/m
3
), c
0
is the speed of sound at the reference density, p is the
pressure and γ is a dimensionless parameter taken
equal to 7. In the numerical scheme the speed of sound
c
0
is conveniently reduced with respect to its physical
value to obtain reasonable time steps (according to the
Courant-Friedrichs-Lewy condition). In order to keep
(7)
(8)
(9)
(10)
(12)
(11)
(13)
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R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS
448
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
Similarly, the shape of the slide body after the fall
has been reconstructed by intersecting the post-slide
topography (obtained from Lidar survey data) with
the surface obtained by subtracting the volume of the
slide from the pre-slide topography (Fig. 1).
Once defined the shape of the rockslide body, the
XYZ coordinates of the barycenter in the pre-slide
configuration b
pre
has been calculated discretizing the
entire volume of the slide by means of N parallelepi-
ped columns with a regular squared base Δx = Δy = 5
m and variable height d. Assuming constant density
for the entire mass, the following expression holds:
where V
tot
is the total volume of the slideand x
i
is the
vector of coordinates of the center of mass of the i-th
parallelepiped. The coordinates of the barycenter of
the rockslide mass after the event, b
post,
was obtained
in a similar way. It follows that the distance between
the two centers of gravity d
b
is equal to 326 m.
In order to define the movement of the slide from
the initial to the final configuration, we made the fol-
lowing assumptions (c
iabatti
, 1964; d
atei
,
2005):
- the entire mass is concentrated in the barycenter
of the slide;
- the barycenter moved from b
pre
to b
post
along a cir-
cular arc with radius R (which needs to be defined);
- the axis of rotation is normal to the vertical plane
containing the position of the two centers of gravity.
On the basis of the previous assumptions it fol-
lows that the equation in the 3D space of the rotation
axis (and so the entire movement of the rockslide, as-
sumed as a rigid body) depends only on the value of
the radius R. In other words, once R is defined, the
coordinate vector of the i-th point of the mass slide
of solid particles that satisfy the same equations of the
fluid ones (Equation (7) and (9)), but their position
is not calculated using Equation (13) as for the fluid
particles. The boundary particles which describe the
Vajont valley do not modify their position during the
simulation, whereas the boundary particles which dis-
cretize the landslide body are moved according to the
velocity assigned to the slide.
KINEMATICS OF THE ROCKSLIDE
Recent and current researches, carried out at
theGeosciences Department (University of Padova,
Italy), focus on the relationships between the land-
slide mass before and after the failure event. In par-
ticular, the obtained results show that the volume
of the slid mass is just a little bit higher than that of
the original in-situ mass, the difference being appar-
ently in the range of inevitable errors due to the low
accuracy of the pre-failure maps.
Besides, the shape of the sliding surface has
been generally assumed as a "chair" shape (S
uPeRchi
,
2011), but the difference between the real shape and
that of the corresponding circular arc has been consid-
ered by many Authors negligible.
GEOMETRY OF THE SLIDE BODY
The 3D geometry of the sliding surface was re-
cently reconstructed by S
uPeRchi
(2011) and b
iStacchi
(2013, present Conference) (see Fig. 1) by analyzing the
pre- and post- slide maps and through seismic sections
and boreholes stratigraphy interpretation. In the present
work the 3D shape of the rockslide body before the fall
(Fig. 2) has been defined by intersecting the pre-slide
topographical map of S
emenza
et alii (1965) with the
sliding surface (S
uPeRchi
, 2011). In such way the vol-
ume of the rockslide was estimated of about 310·10
6
m
3.
Fig. 1 - Sliding surface 3D view (with the sliding sur-
face in red)
Fig. 2 - Three-dimensional representation of the body
rockslide and the valley of the Vajont
(14)
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3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
449
L
2
(R) is equal to 800 m, which correspond to the
rotation axis shown in Fig. 3. The value of R herein
obtained substantially agrees with those obtained by
D
ATEI
(1969) analizing different sections of the slide
(830 - 912 m).
In Figure 3 the traces of six cross-sections of the
Vajont valley are drawn. The cross sections are then
plotted in Fig. 4 with the following information:
- the shape of the valley before the slide (S
emenza
et alii, 1965 - red line);
- the sliding surface (S
uPeRchi
, 2011 - blue line);
- the real shape of the valley after the slide
(black line);
- the virtual surface of the slide obtained accor-
ding to the procedure previously described (gre-
en line).
From Fig. 4 it can be appreciated that the green
line reasonably superimposes with the black one. This
means that the assumed rotational movement is able
to reproduce the post-slide configuration of the val-
ley starting from the pre-slide configuration. Hence
the assumption of considering the rockslide as a rigid
object which rotates around an axis is confirmed. The
differences between virtual and real topographies
are not negligible only in section 1, which is the one
closer to the dam. This suggests that in this zone the
hypothesis that the landslide is representable as a rigid
body is not fully verified.
at the end of the movement x
i
rot
is defined as follows:
where x
i
pre
and x
cr
are, respectively, the coordinate
vectors of the i-th point before the slide and of the in-
tersection point between the vertical plane containing
b
pre
and b
post
and the rotation axis of the slide. In (15)
A and B are defined as:
with:
,
.
The x
cr
vector can be calculated from the follow-
ing equation:
x
cr
=
a
-1
x
cr
+
b
pre
with x’
cr
defined as:
where
Starting from the pre- configuration of the valley
and rotating the rockslide mass from the initial to the
final position, the virtual post- configuration of the
valley is reconstructed. This can be compared against
the real topography obtained on the basis of post-slide
surveys. The L
2
norm of the differences of the eleva-
tions of the two surfaces can be calculated as follows:
where z
i
rot
is the elevation of the i-th point of the vir-
tual surface obtained by rotating the body of the slide
from the initial to the final position, and z
i
pos
the eleva-
tion of the same point in the real post-slide topogra-
phy. The value of the radius R has been defined by
minimizing the norm of Equation (19).
The value of the radius R which minimizes the
Fig. 3 - Plan traces of some cross-sections shown on the
bathymetry of the valley before 1963 . The track of
the rotational axis is also shown
(15)
(16)
(17)
(18)
(19)
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R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS
450
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
VELOCITY AND TOTAL TIME OF THE
SLIDE
For the objectives of the present study the most
important aspects of the kinematics of the slide are the
total duration T
f
, that is the time elapsed between the
beginning and end of the movementof the slide and
the time history position of the rock mass. By analyz-
ing different 2D slices of the valley and assuming a
straight movement of each center of mass C
IABATTI
(1964) and subsequently S
ELLI
& T
REVISAN
(1964) es-
timated values of T
f
ranging between 40-45 seconds.
D
ATEI
(1969) performed a critical analysis of the
seismogram recorded in Pieve di Cadore (C
ALOI,
1966)
and extended the analysis of S
ELLI
& T
REVISAN
(1964)
assuming that the center of mass of each section moved
along a circular arc. Due to this 2D approach, each slice
of the slide rotates with a slightly different radius and
reaches the final position with different total durations
T
f,
all in the range 20-25 seconds. In the present paper
the 2D Datei’s approach was extended in 3D, obtaining
a unique total duration of the slide fall and a unique
time history of the rock mass position.
As stated before, we assume that the barycenter of
the sliding mass moves in the vertical plane containing
b
pre
and b
post
and along the circular arc with radius R
and center x
cr
.
Let’s define α as the generic angle that the tangent
to the circular arc passing through the barycenter forms
with the horizontal line (the initial and final values α
0
and α
1
are shown in Tab. 1 and Fig. 5).
The velocity magnitude |v| of the barycenter can
then be calculated as follows:
From the second law of dynamics, it is then pos-
sible to derive the following equation of motion for the
barycenter:
Fig. 4 - Cross-sections (a) n.1, (b) n.2, (c) n.3, (d) n.4, (e)
n.5 and (f) n.6
Tab 1 - Values of main characteristics of the kinematics
of the slide
Fig. 5 - Angles α
0
and α
1
(a)
(b)
(c)
(d)
(e)
(f)
(20)
(21)
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3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
451
ably influence the main characteristics of the flow are
the maximum velocity magnitude of the slide as well
as its total duration T
f
.
The values calculated in the previous section
(T
f
= 27.9 s and v
max
= 18.5 m/s) were adopted in
the very first simulation. However the maximum
run-up obtained by the numerical model signifi-
cantly underestimates the historical one reported
by S
EMENZA
et alii (1965). This can be explained
by considering that the Equation (22) has been ob-
tained by making significant simplifications of the
physical phenomenon. For example the total mass
of the slide was concentrated in the barycenter, and
thus the effect of the forces exerted on the sliding
surface by the valley have been schematized by a
time-constant friction coefficient. Moreover the ef-
fect of the water body on the rockslide movement
was also neglected.
For these reasons, it was decided to simulate dif-
ferent total durations of the slide, modifying the ve-
locity magnitude accordingly.
After some trials, the best results were obtained
with a total duration of the slide (T
f
) equal to 17 sec-
onds, which corresponds to a maximum velocity v
max
equal to 30 m/s.
In the following the main results of this simulation
were described. Figure 6 shows the plan snapshots of
the free surface elevation with a time interval of 10
s, starting from the beginning of the slide movement.
After 10 seconds the water already starts to overflow
the dam, mainly in the right side of the valley. After
20 seconds the slide movement has already stopped,
while the wave front continues to rise the north side
of the Vajont valley. The flow begins to divide into
four portions: I) one overtops the dam and moves into
the Vajont gorge towards the Piave Valley (west), II)
the central part continues to climb the right side of the
valley (north), III) a small part moves south retracing
the valley furrow of Massalezza, and IV) a consistent
part propagates toward Erto (east).
At t = 30 seconds the wave reaches the maximum
where f is the friction coefficient and g is the gravity
acceleration. We remark that the Equation (21) is ob-
tained assuming that the mass moves from the initial
to the final position with a positive velocity (
/
dt
<0)
and also that the force exerted by the water mass on the
rockslide body is negligible.
With simple algebraic manipulation on Equation
(21), the following differential equation can be ob-
tained (D
ATEI
, 1969):
The term -f (dα/dt)
2
is due to the centrifugal force
and is from here on neglected because of its small con-
tribution (d
atei
, 1969).
Equation (22) can be integrated imposing that
that at t=0 α=α
0
and at t=T
f
α=α
1
and also that
dα/dt=0 both at t=0 and T=f
t
. After some algebra
(d
atei
, 1969) the velocity magnitude |v| time his-
tory can be expressed as:
where α
max
is the angle at which the velocity reaches its
maximum and it can be calculated as follows:
The time T
f
at which the barycenter reaches its
final position and hence the slide mass stops can be
calculated as:
The maximum magnitude of the velocity v
max
is
obtained as:
In Table 1 the values of f, v
max
and Tf obtained by
applying the previous equations are reported.
NUMERICAL SIMULATIONS AND RE-
SULTS
The initial water elevation in the lake was set at
the historical value of the day of the event (about 700
m a.s.l.). The three-dimensional domain (valley slopes,
slide and water bodies) was then discretized into cubic
cells of side Δx equal to 5 m and with a smoothing
length h = 1.5 Δx, leading to the number of particles
reported in Table 2.
A preliminary sensitivity analysis of the numerical
model has shown that the parameters which remark-
Tab. 2 - Number of particles of the model
(22)
(23)
(24)
(25)
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R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS
452
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
Fig. 7 Experimental (S
emenza
et alii, 1965 - red line)
and numerical (green line) run up
run-up on the right side of the valley.
Figure 6 at time 40 s shows the descent of the
wave: the flow is partially diverted (by the slide body)
to the eastern and western directions. Downstream the
dam, the water spreads rapidly in the gorge toward the
village of Longarone.
Figure 7 shows the historical (red line - S
emenza
et
alii, 1965) and numerical (green line) maximum run-up
of the wave. The numerical model is able to reproduce
satisfactorily the central and eastern part of the run-up
edge. Close and downstream the dam, however, signifi-
cant differences between historical and numerical results
are appreciable. This is probably due to the assumption
of the slide as an unique rigid body, which is only ap-
proximately true close to the extreme borders (upstream
and downstream) of the slide (see also Fig. 4a).
Figure 8 shows six perspective snapshots of the
slide and of the wave. In the images the velocity mag-
nitude is mapped by colors. It can be observed that the
violent movement of the rockslide creates a complex
flow field with a very irregular and fragmented free
surface, difficult to be simulated with 3D Eulerian nu-
merical methods, and with 2D SWE codes too.
The maximum speeds on the wave front was esti-
mated of about 35 m/s, while the wave that propagates
in the residual lake (towards Erto) reaches a maximum
speed of around 20 m/s.
It can be observed that at around 30-40 seconds,
when the discharge leaving the dam reaches its maxi-
mum, the flow is concentrated in the left hydraulic
side of the dam, with maximum water thickness of
about 50 m.
This justify the major damages observed on the
dam crest on this side.
Due to the high computational cost, it was not possi-
Fig. 6 - Water elevation in the numerical model after 0,
10, 20, 30 and 40 s.
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3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
453
stage-volume relationship of the residual lake, the water
level corresponding to the total accumulated volume is
equal to 710 m a.s.l.. This is in good agreement with the
data available in literature: S
elli
& t
ReviSan
(1964) and
v
iPaRelli
& m
eRla
(1968) estimated the level of the re-
sidual lake at the end of the events equal to 712 m a.s.l.
The numerical simulations allowed also to recon-
struct the discharge hydrograph that overtopped the
dam and the neighboring slopes (see Fig. 9). The hy-
drograph shows two peaks: a small first one at around
15 seconds, with a discharge of about 50·10
3
m
3
/s and
a second one at around 30-40 seconds, with a maxi-
mum peak discharge of about 160·10
3
m
3
/s. The vol-
ume of the discharge hydrograph is about 15·10
6
m
3
.
The magnitude and direction of the velocity field
in three cross sections (for which the plan traces are
reported in Fig. 10) are shown in Fig. 11.
It can be observed that, after 10 seconds, while
ble to extend the simulation until the still water condition
in the residual lake was reached. The numerical simula-
tion has been interrupted at the physical time equal to
21 minutes (which corresponds to a runtime of 62 hours
with an Nvidia GeForce GTX 580 GPU). At this physical
time the water stopped flowing from the top of the slide
towards east and the volume accumulated in the residual
lake was estimated equal to 78·10
6
m
3
. Considering the
Fig. 8 - Time evolution of the wave and of the velocity module
Fig. 9 - Estimated discharge hydrograph overflowing the dam
Fig. 10 - Plan traces of the cross-sections A, B, C
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R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS
454
International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013
the slide is still moving, the water pushed north moves
with maximum speeds of 30-35 m/s. After 16 seconds,
while the slide is about to stop, the water continues to
move toward the north side of the valley.
CONCLUSION
In this paper the results of a 3D numerical simu-
lation of the wave generated by the Vajont rockslide
are presented.
To the authors’ knowledge this is the first con-
tribution to a complete description of the flow evo-
lution in the artificial lake, including the 3D veloc-
ity field and the discharge of the hydrograph which
overflowed the dam. The sensitivity analysis of the
numerical model has shown that the main param-
eter which influence the flow dynamics is the total
Fig. 11 - Modulus and vectors of the velocity field at the instants 10 and 16 s for cross-section A (a), B (b) and C (c)
background image
3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE
Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università
Editrice
455
late the propagation of the wave in the Vajont gorge,
downstream the dam and in the Piave valley.
ACKNOWLEDGEMENTS
We are grateful to Dr. Alejandro, C. Crespo and
Mr. Jose M. Dominguez for the suggestions about the
Dualsphysics code. We also want to thank Dr. Laura
Superchi for providing the initial data of the slide. The
High Perfomance Computing facilities where provid-
ed by the CINECA inter-university consortium.
duration of the fall. In order to correctly reproduce
the maximum historical run-up a total duration of
the slide movement equal to 17 s is required.
The comparison between the results of the nu-
merical simulation and the data available in literature
shows that the numerical scheme is able to reproduce,
together with the maximum run-up, also the level in
the residual lake after the event.
The numerical results can be adopted in future
works as an upstream boundary condition to simu-
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