# ijege-13_bs-vacondio-et-alii.pdf

*Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università*

*Editrice*

*DOI: 10.4408/IJEGE.2013-06.B-43*

**3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED**

**BY THE VAJONT ROCKSLIDE**

(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.

gorge reached the Piave valley and the village of Lon-

garone, causing the loss of about 2000 people.

*et alii,*1965; m

(h

body (m

estimated in about 270÷300 10

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

**ABSTRACT**

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.

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.

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**

*R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS*

*International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013*

can be described as a rigid body which rotates around

an axis, its kinematics has been calculated starting

from the Newton’s second law.

run-up reported in S

*et alii*(1965); b) the in-

The discharge hydrograph that overflowed the dam

was also evaluated.

**SPH NUMERICAL SCHEME**

ized by their own physical properties (such as mass,

density, pressure).

*A*(

**r**) at any point

**r**, as follows:

*h*is the so-called “smoothing length” and

*W(*

**r**

*-*

**r'**,

*h*) is the weighting function or kernel. This notation

in discrete form becomes:

within the domain of influence of the particle

*a*(2

*h*

for the kernel function herein adopted),

*m*

*b*

*ρ*

*b*

*b.*

*A*'(

**r**)can

*q*=││

**r-r'**││/

*h*and

*α*

*D*

*πh*

for a weakly compressible fluid. In Lagrangian for-

malism the continuity equation can be written as

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.

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.

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.

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.

duced in astrophysics (G

Dynamics (m

pact, such as: breaking waves (d

*et alii,*2008),

(Gó

tional cost which has prevented till now its applica-

tion to practical engineering problem with complex

geometries. Recently c

*et alii*(2011) developed

*http://*

www.dual.sphysics.org/) which overcame this limita-

www.dual.sphysics.org/

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.

**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*

*ε*=0.5 and

*ρ*

*ab*

*=(ρ*

*a*

*+*

*ρ*

*b*

*)*

*/*

*2ρ*. This method moves

*a*with a velocity that is close to the average

velocity in its neighborhood.

*Δ*

*t*

the viscous diffusion terms (m

*Δt*

*Δt*= CFL

*Δt*

*f*

*Δt*

*cv*

*f*

*a*

*a*, and

*Δt*

*cv*

*combines the Courant and the*

*CFL*is the Courant

number. In this work

*CFL*

*et alii*, 1996). The values of density and par-

step n+1/2 as:

*t*

*=*

*n*Δ

*t*. Pressure,

*p*

*a*

tion d

**v**

*t*gives the velocity and hence the posi-

*ρ*

*a*

*n*+1

*t*is calculated

**v**

*a*

*n*+1

**r**

*a*

*n*+1

the “dynamic boundary particles” method is adopted

(c

*et alii,*2007). This method guarantees that

to a good compromise between accuracy and compu-

tational costs. The boundaries are discretized by a set

**v**is the velocity vector and

*ρ*is the density.

Discretizing the▼

**v**

*a*

*W*

*ab*

in the region of compact support of the kernel function.

*p*is the pressure and Θ is the dissipative term.

*p*

*a*

*ρ*

*a*

*a*(same thing goes for the par-

ticle

*b*) and π

*ab*

**r**

*ab*

**r**

*a*

**r**

*b*

**v**

*ab*

**v**

*a*

**v**

*b*

**r**

*k*

**v**

*k*

*k*(

*a*or

*b*);

*c*

*ab*

*c*

*a*

*c*

*b*

*η*

*h*

*v*

α

*v*

(b

*p*and

*ρ*:

*B*can be written as:

*ρ*

*c*

*p*is the

pressure and

*γ*is a dimensionless parameter taken

equal to 7. In the numerical scheme the speed of sound

c

Courant-Friedrichs-Lewy condition). In order to keep

*R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS*

*International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013*

topography (obtained from Lidar survey data) with

the surface obtained by subtracting the volume of the

slide from the pre-slide topography (Fig. 1).

configuration

**b**

*pre*

*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:

*V*

*tot*

**x**

*i*

*i*-th

parallelepiped. The coordinates of the barycenter of

the rockslide mass after the event,

**b**

*post,*

the two centers of gravity

*d*

*b*

lowing assumptions (c

**b**

*pre*

**b**

*post*

*R*(which needs to be defined);

On the basis of the previous assumptions it fol-

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

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**

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.

that of the corresponding circular arc has been consid-

ered by many Authors negligible.

*GEOMETRY OF THE SLIDE BODY*

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

*et alii*(1965) with the

*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*

**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*

*L*

*2*

*R*) is equal to 800 m, which correspond to the

*R*herein

obtained substantially agrees with those obtained by

D

plotted in Fig. 4 with the following information:

- the shape of the valley before the slide (S

*et alii,*1965 - red line);

en line).

From Fig. 4 it can be appreciated that the green

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.

**x**

*i*

*rot*

**x**

*i*

*pre*

*and*

**x**

*cr*

*i*-th point before the slide and of the in-

tersection point between the vertical plane containing

**b**

*pre*

**b**

*post*

*A*and

*B*are defined as:

**x**

*cr*

**x**

*cr*

*a*

**x**’

*cr*

**b**

*pre*

**x’**

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*

*z*

*i*

*rot*

*is the elevation of the*

*i*-th point of the vir-

from the initial to the final position, and

*z*

*i*

*pos*

*the eleva-*

phy. The value of the radius

*R*has been defined by

minimizing the norm of Equation (19).

*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*

*R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS*

*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**

total duration

*T*

*f*

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

*T*

*f*

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*,

a unique total duration of the slide fall and a unique

time history of the rock mass position.

*b**pre*

*b**post*

*R*

**x**

*cr*

*α*as the generic angle that the tangent

with the horizontal line (the initial and final values

*α*

*α*

*are shown in Tab. 1 and Fig. 5).*

*v*| of the barycenter can

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)****3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE**

*Editrice*

the maximum velocity magnitude of the slide as well

as its total duration

*T*

*f*

*.*

*T*

*f*

*v*

*max*

run-up obtained by the numerical model signifi-

cantly underestimates the historical one reported

by S

*et alii*(1965). This can be explained

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.

locity magnitude accordingly.

*T*

*f*

*v*

*max*

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).

*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 (

*dα*

*/*

*dt*

*<0)*

rockslide body is negligible.

tained (D

*-f (dα/dt)*

*2*

*is due to the centrifugal force*

tribution (d

*t=0*

*α=α*

*0*

*t=T*

*f*

*α=α*

*1*

*dα/dt=0*both at

*t*=0 and

*T=f*

*t*

*v*| time his-

*α*

*max*

*T*

*f*

*at which the barycenter reaches its*

calculated as:

*v*

*max*

*f*,

*v*

*max*

**NUMERICAL SIMULATIONS AND RE-**

**SULTS**

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.

*Tab. 2 - Number of particles of the model*

*R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS*

*Fig. 7 Experimental (S*

*emenza*

*et alii, 1965 - red line)*

*and numerical (green line) run up*

to the eastern and western directions. Downstream the

dam, the water spreads rapidly in the gorge toward the

village of Longarone.

*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).

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.

in the residual lake (towards Erto) reaches a maximum

speed of around 20 m/s.

mum, the flow is concentrated in the left hydraulic

side of the dam, with maximum water thickness of

about 50 m.

*Fig. 6 - Water elevation in the numerical model after 0,*

*10, 20, 30 and 40 s.*

**3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE**

*Editrice*

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

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

mum peak discharge of about 160·10

reported in Fig. 10) are shown in Fig. 11.

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

*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*

*R. VACONDIO, S. PAGANI, P. MIGNOSA & R. GENEVOIS*

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**

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)*

**3D SPH NUMERICAL SIMULATION OF THE WAVE GENERATED BY THE VAJONT ROCKSLIDE**

*Editrice*

downstream the dam and in the Piave valley.

**ACKNOWLEDGEMENTS**

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.

the maximum historical run-up a total duration of

the slide movement equal to 17 s is required.

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.

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