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International Conference Vajont2013 - 8 October

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- 1. „Modeling of runout length of high-speed granular masses‟ Francesco Federico & Chiara Cesali University of Rome “Tor Vergata” - Rome, Italy Department of Civil Engineering and Information Engineering
- 2. INTRODUCTION Rapid granular flow prediction is linked to the understanding of the flow mechanisms. Several in situ observations pointed out that the runout length, kinematical characters depend on the geometry of the slope, the debris volume, the susceptibility of the involved material (Bozzano, Martino, Prestininzi, 2005). Advanced lab experiments highligthed the key role played by some micro-mechanical parameters. An analytical model to estimate the runout length of granular masses, accounting for both global and micro-mechanical parameters, is proposed. 2
- 3. Dependence of the runout length “L” on volume “V” of the debris flow. Correlations L (V) or H/L(V) are shown. RICKENMANN D. (1999), „Empirical Relationships for Debris Flows‟, Natural Hazards 19. M = material volume; He = H = difference in elevation; L = total travel distance; Lf = runup; mapp = H/L STRAUB S. (1997), „Predictability of long runout landslide motion‟, Springer. LOCAT P. et al. (2006), „Fragmentation energy in rock avalanches‟, Can. Geotech. J. 43. 3
- 4. COMPARISON WITH EMPIRICAL RELATIONSHIPS 7000 Rickenmann 1 (1999) - (R1) Rickenmann 2 (1999) - (R2) 6000 (R1) 5000 L [m] L = 30(VH)1/4 (R2) 0.8 0.7 4000 r tan 3000 Scheidegger (1973) - (S) Corominas (1996) - (C) Davies (1982) - (D) H L L = 1.9∙V0.16∙H0.83 2000 0.6 1000 0.5 log10 (H/L) = -0.15666∙log10(V) + 0.62419 0 1 2 3 4 5 6 x 10 r V [m3] Empirical Relationships (S) 0.4 log10 (H/L) = -0.085∙log10(V) + 0.047 (C) 0.3 1/(H/L) = 1/tan(fb) + 0.5∙(9.98∙V0.32)/H (D) 0.2 0.1 1 2 3 3 V [m ] The runout length L depends only on the involved “global” geometrical parameters, „V‟ and „H‟. A great variability of the results is observed. 4 5 x 10 6 4
- 5. PROPOSED MODEL Hypotheses: The granular sliding body is composed by two layers of equal basal area Ω and length l: within the "shear layer" (thickness ss) random and turbulent fluctuations of particles displacements at high rate induce a „granular temperature‟ (Zhang & Foda, 1999). The available energy is dissipated through collisions effects and frictional resistance; inertial forces act on the overlying body (thickness sb). The global geometry (Ω, l and H) does not change if erosion or deposition processes are neglected. Otherwise, the geometry may vary according to simple geometrical rules as a function of the rate of the sliding mass and of a critical value vcr,e beyond which the erosion phenomenon occurs. The sum of the masses of the shear layer (ms) and the overlying block (mb) First slope (q): equals the total sliding mass m: runout ms and mb vary along the slopes as a function of rate as well as of erosion phenomena. ms(x(t))=ρs ss(x(t)) Ω mb(x(t))=ρb sb(x(t))Ω m = mb(x(t))+ ms(x(t)) Transition zone The thicknesses sb(x(t)), ss(x(t)) along the path (x(t)) are not a priori known. Counterslope (): runup 5
- 6. PROPOSED MODEL: Energy Balance, constant mass The involved energies are: Ep, potential energy; Ek, kinetic energy; Ecoll, energy dissipated through collisions; Egt, energy related to granular temperature; Efr, energy dissipated due to the friction. The energy-balance equation is: The corresponding Power Balance is: Power of the potential energy: shear layer: shear layer: Power of the kinetic energy: shear layer: They are written as a function of the traveled distance x(t), the rate , the thicknesses , . 6
- 7. PROPOSED MODEL (constant mass) Egt , energy stored as granular temperature Granular temperature, Tg, measures the degree of agitation of solid grains, which influences the mixture bulk density and the ability of grains to avoid interlocking and to collide. Tg is determined by the average grains‟ velocity fluctuations, v‟, respect to their mean velocities (Campbell ,1990): Tg ~ < v‟ 2> Granular temperature, Tg , can be generated and maintained only by continuous conversion of bulk volume translational energy, supplied by the sliding of the moving masses, to grain fluctuation energy; this conversion occurs as grains shear, rotate, impact along irregular surfaces, collide each on others. 7
- 8. PROPOSED MODEL (constant mass) Egt, , energy stored as granular temperature OGAWA (1978) observed that the energy stored in the grain-inertial regime (Egt) is proportional to the granular temperature (Tg): Experimental observations, (Straub, 1997): thickness of shear zone ss ~ 10 dp Experimental analyses (Capart et al., 2000; Larcher, 2002; Armanini et al. 2003) allow to define a local relation between the granular temperature and the measured velocity and concentration profiles. e being the restitution coefficient of the granular phase ϵ [0,1] (as e → 1 (elastic collisions), Tg → ∞); u(z), the grain velocity; D, the average grain diameter; Fs , the solid fraction; g0 = g0(Fs ) = (1 - 0.5∙ Fs)/(1- Fs)3. Therefore, Tg depends on the square of the shear rate (Savage & Jeffrey, 1981) through a multiplier coefficient that depends, in turn, on the restitution coefficient e and the solid fraction Fs . 8
- 9. PROPOSED MODEL (constant mass) By taking into account the previous relationships, the Power of energy stored in granular temperature may be simply written as: being: , and ∙ the ratio between the powers lost (Ecoll) due to collisions and stored ∙ (Egt) through the granular temperature (Federico & Favata, 2011). By assuming binary collisions and constant average mass of the grains composing the sliding mass, the parameter b (ϵ [0,2], ) represents the ratio between Dv, the relative velocity between two colliding grains and dv, average value of the modulus of the velocity fluctuation vector. If collisions are neglected, the relative velocities Dv of all grains are null (b = 0); if, for each collision, two colliding grains, moving along the same direction, assume opposite velocity vectors, their relative velocity Dv doubles the absolute velocity of each grains (b = 2). 9
- 10. : PROPOSED MODEL (constant mass) Ecoll, energy lost due to repeated grain inelastic collisions The power of the energy lost in granular inelastic collisions is related to the granular temperature (Tg) (Jenkins & Savage, 1983): Recalling that relation holds, it is possible to define: . G(ns) describes the dependence of the power Ecoll on the solid fraction ns; w is the ZhangFoda coefficient (suggested value 0.8). 10
- 11. PROPOSED MODEL EQUILIBRIUM ORTHOGONALLY TO THE SLIDING PLANAR SURFACES (SLOPE (q) AND COUNTERSLOPE ()) (constant mass) To simulate the coupled and unknown effects of the contact/collision as well as of the shear resistance, acting along the block-base profile, a splitting function r has been introduced in the equilibrium equation along the direction normal to the sliding surface. Wbcosz is balanced by the resultants of the effective and dispersive pressures acting along the base surface. The relative role of these pressures depends on the rate v and parameters h, vcrit, which modulate the „splitting rule‟: cos z • Wbcosζ (ζ = θ or α); Shear layer • , effective stresses; z • pdisp, dispersive pressures, (BAGNOLD, 1954): pdisp = 0.042ρ (λγdp)2 cosϕ (being g the velocity gradient), stress acting normally to the boundary of the particles moving at high rate in the grain-inertial regime. If a linear change of velocity, orthogonally to the sliding surfaces, is assumed, the following expression is obtained: The equilibrium equation thus gets: 11
- 12. THE ROLE OF THE FUNCTIONS r AND The splitting rule is defined as follows: η, being a parameter ϵ [0.005,0.5]; vcrit, critical speed for which the grain displacement realm dominated by the inertial forces turns towards a condition governed by the collisions. Some experimental results 1 (e.g. Savage, 1981) get: 0.9 0.8 (Nsav = 0.1, b = thickness within which the rate changes from 0 to the average value of the „shear layer‟ ). 1- r (v) r(v) 0.6 r(v), 1-r(v) The parameter η allows to model the shape of the function r. It modulates the local transition between the inertial and collisional regimes. 0.7 h h h h 0.5 0.4 0.3 vcr = 20 m/s 0.2 0.1 0 0 10 20 30 40 50 v [m/s] 60 70 80 90 100 12
- 13. THE ROLE OF THE FUNCTIONS r AND r 1 0.9 0.8 1- r (v) r(v) 0.7 r(v), 1-r(v) 0.6 vcr = 10 m/s vcr = 20 m/s 0.5 vcr = 30 m/s vcr = 40 m/s 0.4 0.3 0.2 h 0.1 0 0 10 20 30 40 50 60 70 80 90 100 v [m/s] If the critical rate (vcrit) increases, the transition between the two regimes occurs at higher values of the speed. 13
- 14. PROPOSED MODEL (constant mass) Energy dissipated due to the frictional resistance (Efr): Efr depends on the weight W of the sliding mass, the dynamic friction angle fb at the base of the block. Dispersive and interstitial pressures reduce the energy dissipated due to the friction. The basal frictional resistance is expressed as follows: Tfr = W cos z - U – (F (x2)) being It is simply assumed that the interstitial pressures pw assume a constant value along the sliding surface (s.s) (Iverson & LaHusen, 1997) and that the isopiezic lines are orthogonal to the s.s.: dw = 0, if the mass is saturated; dw = h, if the mass is dry; pw(x) may exceed the hydrostatic value (pore pressure excesses) if rapid pore volume changes following with the continuos rearrangement of grains involved in the flow motion occur. To simulate this effect, dw < 0, must be assigned (Musso, Federico, Troiano, 2004). The energy dissipated due to the friction along the s.s. is obtained by integration along the path x . The corresponding power is therefore obtained through derivation vs time t. 14
- 15. PROPOSED MODEL: VARIABLE MASS The mass of a debris flow may change due to erosion processes. Assumptions: • Erosion of the bed and of the walls of the crossed channel may occur; (i) • The erosion acts if the speed v ≥ vcr,e (critical speed of erosion); • The reduction of the volume of the d.f. (detachment, splitting) is not considered. • The geometry of the debris flow varies (constant width B = B0 of the flow or channel) according to the following laws: (ii) (iii) 15
- 16. PROPOSED MODEL: VARIABLE MASS The mass of the debris flow varies according to the law: The thicknesses of the shear layer (ss1(t)) and of the overlying granular mass (sb1(t)), by recalling the equilibrium equation, are rewritten as follows: The Power Balance (for v > vcr,e) becomes: Power due to the change of the inertial mass following the increase of the granular material. 16
- 17. PROPOSED MODEL PARAMETRICAL ANALYSES Results depend on the parameters describing the micromechanical behaviour : b (є [0,2]), e (restitution coefficient є [0,1]), k ( in Egt‟s expression) and dp (grain diameter). Collisional Energy Ecoll; Energy Egt related to the granular temperature Tg; Rate v and total runout length (L = x) for different values of parameter k (= ¼ (1-e2)b2) 40 k k k k 35 v [m/s] Parameters: q= 38 ; = 0 ; dp = 15 cm; fb = 18 ; e = 0.2, 0.5, 0.7; dw = 0 (U≠0); m = m0; h = 0.005; b = 0.5, 1, 1.5 ss(t) ~ 10dp x 10 k k k k 3 = = = = 0.54 (e=0.2, 0.42 (e=0.5, 0.13 (e=0.7, 0.05 (e=0.5, Ecoll Egt 2 gt E ,E coll [J] Ecoll 1.5 1 Egt 0 20 15 10 5 200 400 600 800 0 200 400 600 800 1000 1200 1400 1600 1800 - The thicknesses of the block (min) and the „shear layer‟ (max) slightly change if the parameter k increases, except for b < 1: the thickness of shear layer for some set of parameters, assumes a value close to 10∙dp, (experimental observations, Straub, 1997). - Collisional energy Ecoll, set the coefficient of restitution (e = 0.5), increases if the parameter k (and therefore β) increases, while Egt decreases. Fixed the coefficient β (= 1.5), if the k decreases (or e increases), Ecoll decreases and Egt increases. - Set the coefficient e (= 0.5), if k (and β) increases, the runout length decreases, while the maximun rate increases. 2.5 0 25 x [m] b b b b 0.5 0.54 (e=0.2, b 0.42 (e=0.5, b 0.13 (e=0.7, b 0.05 (e=0.5, b 30 0 11 3.5 = = = = 1000 x [m] 1200 1400 1600 1800 2000 17 2000
- 18. PROPOSED MODEL: PARAMETRICAL ANALYSES [sb(x(t)), x(t)];[ss(x(t)), x(t)]; )];[H, x(t)]; Collisional Energy Ecoll; Energy Egt related to the granular temperature Tg; Rate v and total runout length (L = x) for different values of grain diameter d p 70 dp = 0.06 m dp = 0.10 m dp = 0.14 m v(x(t)) 60 40 30 30 block 20 v(x(t)) 20 10 10 shear zone Ecoll 4 3.5 Ecoll 3 2 Egt 1.5 Egt 1 Ecoll Egt 0.5 0 0 500 1000 1500 0 500 1000 1500 2000 2500 3000 0 2500 3000 3500 3 x [m] dp = 0.06 m dp = 0.10 m dp = 0.14 m Egt 2.5 Egt e = 0.8 Ecoll Egt gt , E [J] 2 coll 1.5 E -If the average diameter dp increases, the thickness of the „shear layer‟ becomes smaller; the distance traveled and the maximum speed decrease. -dp, coupled to the restitution coefficient e, plays a key role in Egt end Ecoll; if dp increases (e is fixed), the collisional energy Ecoll increases, while the energy associated with granular temperature Egt decreases; - fixed dp, to an increase of e, Egt increases because of the reduction of the energy lost due to collisions; the runout length increases, if e increases. 2000 x [m] 11 0 e = 0.2 2.5 coll d.f. dp = 0.06 m dp = 0.10 m dp = 0.14 m gt [J] Parameters: q = 38 ; = 0; k = 0.54; fb = 22 ; e = 0.2; dw = 0 (U≠0); m = m0; h = 0.005; b = 1.5 x 10 ,E 40 x 10 4.5 60 50 v(x(t)) 5 E thicknesses [m] 50 12 70 v [m/s] Ecoll 1 Ecoll 0.5 0 0 500 1000 1500 2000 x [m] 2500 3000 3500 18
- 19. COMPARISON WITH Coulomb and Voellmy MODELS Parameters: q= 30 ; = 0; k = 0.28; fb = 18 ; e = 0.3; (dw/h = 0); m = m0; h = 25 m ; b = 1.1; dp = 0.05 m; M = 8x104 The motion laws: Kg/m M = 0, Coulomb‟s Model M ≠ 0, Voellmy‟s Model, - C-M model gets runout and rate values greater than G-M and V-M models; the runout length doesn‟t depend on the volume of the sling mass; - In G-M model, the motion lasts more than the other cases; - In G-M and V-M, the solutions depend on the granular volume; - For great values of V, G-M and V-M runout length values tend to the C-M runout length; - G-M gets greater runout lengths than V-M model, even if the 19 maximum speed is almost equal.
- 20. COMPARISON WITH EMPIRICAL RELATIONSHIPS Influence of micromechanical parameters INFLUENCE OF INTERSTITIAL PRESSURE Rick. 1 r tan H L Rick. 2 Parameters: q = 38 ; = 0 ; k = 0.91; fb = 15 ; e = 0.3; dp = 0.1 m; b = 2; H = 616 m The interstitial pressure greatly affects the traveled distance: the results obtained by imposing U = 0 (dw = h) better approximate the relationships by Corominas, while the ones obtained by imposing U ≠ 0 (dw = -h/3 and dw = 0) approximate the relationships proposed by Rickenmann ((1) and (2)). 20
- 21. COMPARISON WITH EMPIRICAL RELATIONSHIPS INFLUENCE OF RESTITUTION COEFFICIENT e 0.8 4500 0.7 4000 Rickenmann 1 0.6 3500 e = 0.5 (k = 0.42) 0.5 3000 Scheidegger e = 0.3 (k = 0.51) L [m] r = H/L e = 0.8 (k = 0.2) 0.4 Rickenmann 2 2500 e = 0.3 (k = 0.51) Corominas 0.3 2000 Davies 0.2 e = 0.8 (k = 0.2) 1500 e = 0.5 (k = 0.42) 0.1 0 0.5 1 1.5 2 2.5 3 V [m ] 3 3.5 4 6 x 10 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 V [m3] 5 x 10 fb = 18°, U ≠ 0 (dw = 0, hydr. cond), dp = 5 cm, h = 0.005, b = 1.5, q = 38°, = 0°; H = 616 m An agreement between G-M and empirical relationships may be obtained by varying parameters such as e and dp. As expected, if the restitution coefficient e increases, the energy lost due to collisions decreases, the energy dissipated due to the collisional effects becomes less significant, the granular temperature Tg increases and, for a prefixed volume V, the runout length increases. 21
- 22. COMPARISON WITH EMPIRICAL RELATIONSHIPS INFLUENCE OF GRAIN DIAMETER dp fb = 15°, U ≠ 0 (dw = 0, hydr. cond), e = 0.5, h = 0.005, b = 1.5 (k = 0.42), q = 30°, = 0°; H = 616 m The role of the average grain diameter dp becomes less important if the volume V increases. Relationships proposed by Scheidegger, Corominas and Davies provide smaller runout values than G-M and Rickenmann‟s criteria . 22
- 23. BACK ANALYSES The event of 12 June 1997 in the „Acquabona Creek‟ Berti, Genevois, Simoni and Tecca („Field observations of a debris flow event in the Dolomites‟ (1999)) studied the dynamics of d.f. through in situ analyses: three measurement stations, equipped with pressure sensors, geophones and cameras, provided data on pore pressures, global thickness of the d.f., flow velocity at the surface. + + No counterslope: the slope q= +18 reduces to = +7 23
- 24. ACQUABONA CREEK: FIELD OBSERVATIONS 12 Maximum Rate v (observed): 9 m/s 10 Pos. x [m] 500 700 750 875 1000 1125 1175 1250 1320 1380 1440 1630 v [m/s] 8 6 4 2 12 0 0 200 400 11 600 10 8 9 800 1000 x [m] 7 6 1200 3 2 5 4 1400 1 1600 1800 Runout (observed): 1630 m From data measured in situ, the following input parameters for a back analysis through the G-M model (variable mass) are chosen: L = 1630 m; CW = 1.5 m2/m; Ch = 0 (being the average depth of d.f. h = H0 = 2 m, case (ii)); B0 = 4 m; l0 = 75 m; W0 = 300 m2. 24
- 25. ACQUABONA CREEK (variable mass) 25 12 e = 0.6; dp = 1 cm e = 0.8; dp = 1.5 cm dw = -h/4 dw = 0b = 1.5; fb = 21 ; b = 1.5; h= 0.04; G-M: 20.2 m/s fb = 26 ; e = 0.3; dp = 1 cm h= 0.03 dw = 0 b = 0.7; Pore pressure fb = 19 ; G-M: 16.5 m/s excess at the base h= 0.035; U≠0 15 Maximum Rate v (observed): 9 m/s 10 10 G-M: 12 m/s (G-M): 10.7 m/s (G-M): 10 m/s e = 0.2; dp = 1 cm dw = h; b = 0.6; fb = 13 ; h= 0.035; Maximum Rate v (observed): 9 m/s 8 rate v [m/s] rate v [m/s] 20 e = 0.3; dp = 1 cm dw = h b = 0.6; fb = 12 ; h= 0.03; 6 (G-M): 8.5 m/s G-M: 10.5 m /s 4 Runout (G-M): 1643 m 5 2 Hydrostatic condition 0 0 200 400 e = 0.5; dp = 1 cm dw = h/2 b = 0.7; fb = 16 ; h= 0.04; 600 800 1772 m 1691 m 1680 m 1660 m 1000 x [m] 1200 1400 1600 1800 0 0 Runout (observed): 1630 m partially saturated observed 200 400 600 1760 m Runout (G-M): 1636 m U=0 800 1000 1200 1400 1600 1800 x [m] Runout (observed): 1630 m e = 0.1; dp = 1 cm dw = h; b = 0.5; fb = 13 ; h= 0.04; - Long runout (~1650-1750 m) are measured, although the initial slope assumes small value (q = +18 ), due to the positive value (+7 ) of the second slope angle. -To fit the observed values (max rate = 9 m/s; runout length = 1630 m) through the G-M model, if interstitial pressures are neglected, very small shear resistance angle fb ϵ [12-13 ] at the base of the d.f., coupled to small values of restitution coefficient e, must be assigned. - If U ≠ 0 and high values of e are imposed, the rate v increases (respect the condition U = 0), although appreciable values of fb are assigned. 25
- 26. ACQUABONA CREEK (variable mass) 18 observed d.f. depth 2 e = 0.3; dp = 1 cm; b = 0.6; fb = 12 . 16 U≠0, h=0.1 14 U=0, h=0.03 1.5 block U=0 (dw = h) U≠0 (dw = -h/4) rate v [m/s] thickness [m] 12 U≠0 (dw = 0) U≠0 (dw = h/2) 1 10 8 e = 0.8; dp = 1.5 cm b = 1.5; fb = 21 ; h= 0.04; dw = 0 (U ≠ 0) 6 shear layer 0.5 Rate v (observed): 9 m/s 4 2 Runout (observed): 1630 m 0 0 200 400 600 800 1000 x [m] 1200 1400 1600 1800 0 0 500 1000 1500 2000 2500 3000 3500 4000 x [m] - The thickness of the shear layer, for the different sets of assigned micromechanical parameters, falls within the range [0.4-0.7 m] and it assumes higher values if U≠0. - G-M model gets the observed values if h ϵ [0.03-0.04]; higher values of h get runout length greater than 1630 m. - High values of the parameter b must be assumed if the interstitial pressures are not neglected, coupled to high values of e. 26
- 27. ACQUABONA CREEK (constant mass): comparison between G-M, Voellmy and Coulomb models (V-M, C-M) 60 observed C-M: fb = 13 ; U = 0 (dw = h) C-M: fb = 23 ; U ≠ 0 (dw = 0) 54 m /s 50 42 m /s V-M: fb = 23 ; U ≠ 0 (dw = 0); x= 21 m/m2 C-M e = 0.8; dp = 1.5 cm b = 1.2; fb = 23 ; h= 0.04; dw = 0 (U ≠ 0) e = 0.3; dp = 1 cm b = 0.5; fb = 13 ; h= 0.025; dw = h (U = 0) rate v [m/s] 40 V-M 30 27.3 m /s V-M: fb = 13 ; U = 0 (dw = h); x=8 m/m2 20 m /s 20 Rate v (observed): 9 m/s 10 9.8 m /s 6.2 m /s 0 1550 m 0 500 1000 1500 x [m] 1767 m 1715 m 1900 m 1760 m 2000 2114 m 2500 Runout (observed): 1630 m C-M provides higher values of runout and rate v than observed values; V-M runout values approximate the measured values, although the rate v is too high. A better fitting is obtained through the proposed model if parameters e = 0.8; dp = 1.5 cm; b = 1.2; fb = 23 ; h = 0.04; dw = 0 (U ≠ 0) are assigned. 27
- 28. CONCLUDING REMARKS - An analytical (two-layers) model describing the characteristics of high speed granular masses (rate, runout length), based on energy-balance equations and several simplified assumptions, is proposed. - The model is based on several experimental results reported in technical literature („granular temperature‟, dispersive pressure, excess pore water pressure, collisions effects, …) regarding the micro-mechanical behaviour of granular masses. - The equations describing the sliding of the granular mass along two planar surfaces (slope, counterslope) have been numerically solved. - Parametric analyses and the comparison with results obtained through empirical relationships put into evidence the role played by the considered geometrical and micro-mechanical parameters. - Comparisons between the results obtained through the G-M, the C-M and V-M models, as well as some preliminary back analyses of real cases have been shown. The results obtained through the proposed model fit the observed values for some specific set of values of the considered micromechanical parameters. - The limits of the proposed model lie in the oversimplified geometry of the debris body as well as in the assumptions regarding the laws that correlate the micro-mechanical parameters and the „coupling‟ of dispersive and effective pressures acting within the shear layer that, during the rapid sliding, generates between the overlying block and the base surface. 28

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