# ijege-13_bs-federico-cesali.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-12*

**MODELING OF RUNOUT LENGTH OF**

**HIGH-SPEED GRANULAR MASSES**

terstitial pressures at the base of the mass, that can

vary between null and higher than hydrostatic values

(i

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

Mohr-Coulomb (M-C) shear resistance criterium

doesn’t allow to obtain this result.

mass because high speed relative motion and colli-

sions between solid grains take place within the basal

shear layer, causing a fluidification effect (h

boundary that encloses a set of particles in grain-in-

ertial regime:

*a*

*i*

*Bagnold coefficient*”; B

*0.042*;

*ρ*

*s*

**ABSTRACT**

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**

sloping along mountain streams till their arrest is nec-

essary to identify hazardous areas. The runout length

*F. FEDERICO & C. CESALI*

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

*BASIC ASSUMPTIONS*

resent respectively the "

*shear zone*" (thickness

*s*

*s*

*s*

*b*

*block*”). The to-

*H = s*

*b(*

*t) + s*

*s*

*(t)*. The

*Ω, l*and

*H*) does not change; erosion

or deposition processes are neglected.

*shear zone*” is composed by particles that,

ers, induce appreciable fluctuations of their velocities

(granular temperature); the “

*block*” is dominated by

inertial forces and quasi-static stress.

*m*

*s*

*m*

*b*

*m*

*0*

*ρ*

*s*

*, ρ*

*b*

*s*

*b*

*(x(t)), s*

*s*

*(x(t))*are not a priori

*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-

*σ*

*lit*

*p*

*dis*

as follows:

*H=s*

*0*

*b*

*block*initial thickness;

**and**

*r***are defined to allow a rational**

*r̅**N*

*tot*

*between the lithostatic force*

ten as a function of the rate of the sliding masses(

*ẋ*):

(6)

*linear concentration*”, that is a function of solid

fraction

*v*

*s*

*d*

*p*

*(du/dy)*

*2*

*Φ*, the internal dynamic friction angle of granular

bulk.

dependence is obtained between the Bagnold’s defini-

tion of dispersive pressure and the rate x ̇ of the sliding

mass:

*p*

*dis*

*~x*

*2*

*granular temperature*” as

sliding and colliding particles with respect to the mean

value

estimate the runout length of debris flows.

cally modelled in the paper, by taking into account the

effects of

*granular temperature*and

*dispersive pres-*

sure, acting within the basal ‘

sure

*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 temperatureand the

lar temperature

*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*

**MODELING OF RUNOUT LENGTH OF HIGH-SPEED GRANULAR MASSES**

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

*Editrice*

on powers of energies:

*EFFECTS OF COLLISIONS*

*E*

*gt*

*E*

*coll*.

*E*

*gm*

*shear*

*η,*parameterϵ[0.005,0.5];

*v*

*crit,*

regime governed by the collisions.

By recalling the expression [4], it is obtained:

*ζ*of the slope may assume only two

*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*

*E*

*p,0,*

*E*

*p*

*E*

*k*

*E*

*fr*

*,*energy

ing surface;

*E*

*coll*

*E*

*gt,*

*block*” to the basal “

*shear zone*” to support the grain

inertial regime. Deriving the eq. [11], the Power Bal-

ance is obtained

*POTENTIAL AND KINETIC ENERGY: E*

*P*

*AND*

*E*

*K*

*b =*

*block; s = shear zone):*

*F. FEDERICO & C. CESALI*

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

*d*

*w*

*d*

*(w,min)*

*≤d*

*w*

*≤d*

*(w*,

*max)*

the minimum value

*d*

*w,*min

sliding surface:

*γ*

*s*

*THE ROLE OF BASAL FRICTION*

**:**

*Efr*

**.**

*Efr*is a function of

angle

*Φ*

*b*

*at the base of the block. Dispersive and inter-*

*φ*

*b*

tion along the sliding basal surface is:

**RESULTS OF PARAMETRICAL ANALY-**

**SES**

*β*

(ϵ[0,2]), e (restitution coefficient

(ϵ[0,2])

*ϵ[0,1]*),

*k*(in

*Ėgt*’s

expression [36]) and

*d*

*p*

*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

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

*x*),

*block*and

*shear zone*thicknesses (

*s*

*b*

*, s*

*s*

*zone*” is partly lost due to repeated grain inelastic col-

lisions

*(E*

*coll*

*E*

*gt*

*)*. Z

*granular temperature*(

*T*

*g*

*E coll~T*

*granular temperature*:

*E*

*gt*

*~T*

*g*

zone (s

*T*

*g*

*~x*

*2*

inelastic collisions (

*E*

*coll*

*E*

*gt*

*e*being the restitution coefficient (ϵ[0,1]); ω, ZHANG-

FODA coefficient;

*β*, coefficient

*ϵ[0,2]*and

*υ*

*s*

*d*

*p*

*ρ*

*s*

*INTERSTITIAL PRESSURES*

*p*

*w*

lines are assumed orthogonal to the motion direction

and

*p*

*w*

*(x) is assumed constant along the planar slid-*

*γ*

*w*

*d*

*w*

*d*

*w*

*=H*if the mass is dry;

*p*

*w*

sociated with the rapid change of pore volumes, cor-

responding growth of interstitial water pressures excess

(m

*et alii*, 2004). To simulate this effect,

*d*

*Tab. 1 - Assigned values to the parameters e, β, k*

**MODELING OF RUNOUT LENGTH OF HIGH-SPEED GRANULAR MASSES**

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

*Editrice*

shear zone thicknesses, collisional energy and energy

related to granular temperature, according to

*d*

*p*

*E*

*coll*

*E*

*gt*

*k*are shown in the figures 2, 3, 4 and 5.

*block*and the ‘

*shear zone*’

*k*increases (Fig. 2).

Collisional energy

*E*

*coll*

*(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).

*E*

*gt*

*k*decreases (Fig. 4). It is worth

*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).

*dp*. (the parameters

*e*,

*β*

*k*are:

*e = 0.3; β = 1.75; k = 0.7*).

*d*

*p*

*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*

*F. FEDERICO & C. CESALI*

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

*d*

*p*

*shear zone*’ becomes smaller; as a result, being the sum

of

*s*

*b*

*s*

*s*

*H*of the debris flow (con-

es (Fig. 6). Instead, the collisional energy, to an increase

of

*d*

*p*

*d*

*p*

*increases, the distance*

sociated with granular temperature

*E*

*gt*

**COMPARISON WITH RESULTS OBTAI-**

**NED ACCORDING TO CONVENTIONAL**

**RHEOLOGICAL MODELS**

and Voellmy-Model (V-M). The motion equations for

the V-M and C-M models are given by the ODE:

*M̅=0*describes the Coulomb Model (C-M); M̅≠0,

related to the ξ turbulence coefficient of Voellmy

through the equation (F

*γ*

*s*

in a schematic manner, the energy dissipation due to

granular collisions. Therefore, in the Voellmy Model,

the term

*M̅x*

*2*

*E*

*coll*

V-M), the coefficient M̅ has been determined on the

base of the coefficient

*M*of the General Model, re-

*s*

*s*

*(t)*with (

*s*

*s*

*max*

*⁄2*):

*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.

*run out*and velocity values great-

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

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.

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*

**MODELING OF RUNOUT LENGTH OF HIGH-SPEED GRANULAR MASSES**

*Editrice*

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

*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

x 10

*7*

*Kg/m; in C-M, M̅ =*

*0 kg/m*is assumed. The

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

*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**

of G-M, r

ish Columbia, Canada). The original unstable rock

mass volume was estimated as 30x10

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*

*F. FEDERICO & C. CESALI*

*V*, volume of the mass;

*H*, difference in eleva-

tion (see Fig. 14).

*θ = 38°,*

α = 0°; L = 1000 m;γ

α = 0°; L = 1000 m;

*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

*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*.

*V*of a

*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

*C*

*1*

*C*

*2*

back analysis on several debris flows, c

*C*

*1*

*= - 0.034*;

*C*

*2*

*= - 0.101*.

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.*

relationships proposed in literature is carried out. Af-

ter back analyses on “

*160 debris flows*”, r

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*

**MODELING OF RUNOUT LENGTH OF HIGH-SPEED GRANULAR MASSES**

*Editrice*

**CONCLUDING REMARKS**

*General Model*)

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*

*e*) and the

*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.

the General-Model, for different values of the micro-

mechanical parameters

*e*and

*d*

*p*

*f*as a func-

*V*related to the cases analyzed with

the G-M and the results obtained through the [49] are

shown (c

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*

*H*and

*Ω*, the

obtained by c

*Fig. 15 - Curves [f, V]; comparison between General Mod-*

*el and COROMINAS’s results*

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