Basic Geodynamics of Landslides: III. Flow-slides

Transcription

1 International School LAndslide Risk Assessment and Mitigation LARAM School 2007 (3-15 September, Ravello, Italy) Session 1: Introduction to landslides: Landslide analysis using approaches based on: Geology, Geotechnics and Geomechanics Basic Geodynamics of Landslides: III. Flow-slides Ioannis Vardoulakis N.T.U. Athens ( 1

2 The dynamic slip circle method The dynamics of landslide run-out Flow-slides (eulerian) 2

3 The Lagrangean Approach to continuum mechanics emphasizes the particulate (material) description of a motion The pathlines of car-particles flowing in a highway: Red paths belong to receding particles; white paths to approaching particles. In Fluid Mechanics these lines are called the pathlines. We notice however this snapshot corresponds to a photograph with long exposure. Thus long exposures yield the Lagrangian view, whereas short exposures the Eulerian view 3

4 The Eulerian Approach to continuum mechanics emphasizes the spatial description of a motion Ground-wind velocities meteorologic map of a given place at a given instant. In Fluid Mechanics the integral curves of this velocity field are called the streamlines v = v( x, y, t) 4

5 Natural Phenomena: Debris flows Mud flows Avalanches Granular flows 5

6 Aeolic Perlitic flow in south Melos Island Greece (mostly harmless) 6

7 Debris flow landing on a rural road of Crete after heavy rainfall (accidental) 7

8 Santa Tecla neighbourhood of San Salvador: Las Colinas landslide of January 13, 2001, near San Salvador city after an earthquake, measuring 7.6 on the Richter scale (catastrophic). 8

9 III. Flow-slides Gray, J. M. N. T., Tai, Y.-C. and Noelle, S. (2003) Shock waves, dead-zones and particle-free regionsin rapid granular free surface flows. J. Fluid Mech. 491, Quecedo, M., Pastor, M., Herreros, M.I., Merodo, J.A.F. (2004). Numerical modelling of the propagation of fast landslides using the finite element method. International Journal for Numerical Methods in Engineering 59 (12), R.J. Roberts, A One-Dimensional Introduction to Continuum Mechanics, World Scientific, Savage, S.B. and Hutter, K. (1989). The motion of a finite mass of granular mate-rial down a rough incline. J. Fluid Mech. 199,

10 Shallow-water approximation h v = h(,) s t q = = v(,) s t A v = vet a = a e + a e t t n n a t = v v v v = + v, an = t s r 2 10

11 Material time derivative v v 1 a = (( v v) + ( v v )) t t E E E v v v = v ( x+ x, t) v ( x, t) x x E E E E E v v v = v ( x+ xt, + t) v ( x+ xt, ) v ( xt, + t) v ( xt, ) t t E E v v E a = + v t x 11

12 Mass balance h t + = s ( hv) 0 12

13 Balance of lin. momentum in normal direction : σ ds ρg cosβhds σ hdθ = ρhds n h σn = ρgcos βh+ r σ + σ = Kσ t n t ( 2 ) t ρv 2 v r ( ) sin /cos if : ϕ ϕ β ϕ K = Ka if :0 β ϕ and active plastic flow K p if :0 β ϕ and pasive plastic flow 13

14 Limit equilibrium configurations 14

15 Normal reaction stress h h =, 0 h << 1 r 2 v σ n = ρh g + + O( h ) ; g = gcosβ r 15

16 Balance of linear momentum in tangential direction : ( hσt) + ρghsinβ τn = ρhv s τ = σ tanϕ n n v v v v v + Kg h 1+ = g tanβ tanϕ 1+ t s h s g r g r 16

17 h h v + v + h = t s s v v 1 1 v 1 v + v + Kg h 1+ = g tanβ tanϕ 1+ t s h s g r g r These equations constitute the set of governing p.d. equations of flow-slides in an 1D-setting and variable topography. They are usually termed as shallow-water or depth-integrated equations and constitute a system of quasi-linear hyperbolic p.d. equations. The solutions of these equations include the formation of sharp discontinuities ( shocks ), thus such a system of hyperbolic equations is treated numerically by the so-called shock capturing techniques. 17

18 A remark made by Gray et al. (2003) concerning shock-capturing ; see references therein and Tai et al. 2001): The development of these methods has a long history starting with the classic papers of Godunov (1959), Van Leer (1979), Harten (1983) and Yee (1987), and there are now a wide range of textbooks on the subject (e.g. Le Veque 1990; Godlewski & Raviart 1996; Kröner 1997; Toro 1997). Here we have opted to use the recent high-resolution shockcapturing non-oscillatory central (NOC) scheme first introduced by Nessyahu & Tadmor (1990) and extended to multi-dimensions by Arminjon & Viallon (1995, 1999); Jiang & Tadmor (1998) and Lie & Noelle (2003). Recently Professor M. Pastor and his team have developed a 2D-model for the analysis of the propagation of fast landslides, using a finite element formulation and the Navier Stokes depth-integrated equations (Quecedo et al., 2004). 18

19 Shear flows inertia number v I = Ti z T i = D g g 19

20 Friction shear rate hardening 20

21 Friction shear rate softening 21

22 Steadily moving mud-flow down on a planar track An aerial view shows Saturday, Feb. 18, 2006, the extent of the landslide that buried the whole village of Guinsaugon, St. Bernard town in Southern Leyte province in central Philippines. Officials estimate those who perished in the landslide to be 1,800. [AP] 22

23 Governing equations: planar track s x, β = const., 1/ r = 0 h t + ( hv) = x 0 v v 1 ( 2 τ ) n + v + Kg h = g tan β t x h x σ n 23

24 Bagnold model (non-linear frictional-viscous model) n τ = σ tan ϕ + f ν γ, n = 3/ 2 n n w σ = ρ hg, ρ = ρ ρ n w f 0.26λ ρ ρ D 7/4 B w s g λ B = 1 n 1 n 1 min 1/3 1 γ 2 v h 24

25 Steadily moving mudslide v + v t v 1 ( 2 + Kg h ) = x h x ρ 3/2 f ν w v g tan β tanϕ 2 ρ ρgh h 3/2 v v v = const. v = + v = 0 t x x = x Vt h = H( x ) 25

26 Profile equation dh ρ 2K = tn a β tanϕ 2 dx ρ f ν 3/2 w 3/2 5/2 ρ g V H dh K dx = µ 2 m µ st dn 26

27 Remark: the action of seepage β 1 φ 2 27

28 Remark contnd. b ρ ρ = tan β tan ϕ, ρ ρ 1 2 Recall the static slope stability analysis: In case of an infinite slope under the action of seepage the safe slope inclination is, ρ 1 1 tan β < tanϕ tanϕ or β < ϕ ρ 2 2 The implication of this estimate are evident. A natural slope that after a heavy and protracted rainfall gets water-saturated (e.g. due to loss of its protecting cover-vegetation ). This slope will become unstable if the above inequality is true and we may have the triggering of the herein considered mud-flow. 28

29 Far-field solution H = lim H( x ) x dh dx x = 0 dh 2K dx = m µ µ st dn 0 2/3 3/2 5/2 b 5/3 H = b cv H V = c 29

30 Example β = 10 ϕ 5 c 0.5msec 3/2 5/3 V λ H, λ = 0.41 m 1 2/3 sec 30

31 Mud-flow profile * x * H b x =, h =, B = H H 2K dh dx ( 5/2 1 h ) = B x = 1 B h 0 1 dy y 5/2 31

32 Photo of a debris flow experiment, taken from R.M. Iverson, J.E. Costa, and R.G. LaHusen, 1992, Debris-Flow Flume at H.J. Andrews Experimental Forest, Oregon: U.S. Geological Survey Open-File Report The flowwave shows the formation of steep front and the development of roll-waves ( smovement/publications/ofr92-483/ofr92-483_inlined.html ) 32

33 Front geometry? Pouliquen,

34 The front of a rapid granular flow. The arrow indicates the direction of the flow Weak, long surface waves. Surface waves with long wavelength. 34

35 Roll waves Roll-waves are observed in open-channel hydraulics and are stair-like structures that move down-stream with a speed that is less then the flow velocity itself. Roll-waves are a possibility in debris flows. Dressler (1949) has shown that in open channel hydraulics roll waves cannot be described with patching of piece-wise continuous Bresse profiles. He showed also that the dynamic stability of these roll waves is explained from the fact that the front is a shock wave which moves with a constant celerity: Ahead of the shock the flow is super-critical, whereas behind the shock it is sub-critical. It can be shown that in this case the particles move through the shock front, from the region of small flow-height to the region of big height 35

36 Roller waves? (Dressler 1949) Forterre& Pouliquen,

37 The deposition bore Gray et al. (2003) 37

38 Rankine-Hugoniot campatibity conditions c D dd = dt + [[ h]] = h h + [[ v]] = v v + + [[ q] ] = v h v h D ( ) c h h = v h v h 1 1 K g ( h ) K g h ) c hv v hv v hv c ( = D ( ) ( ) + ( ) D ( hv) 38

39 The deposition bore v + = 0 st ( ) 1 R. H : c h h = v h D + nd R. H : K g ( h ) K g ( h ) = cd hv v hv 2 2 ( ) ( ) h K h K h cd = g h h h ( ) 39

40 The deposition bore is observed at slope inclinations where uniform flow is possible. For the simple of Coulomb type friction law this is the case when + β = ϕ K = K = 1+ sin cos 2 2 ϕ ϕ 1 cd = λ gh, h = h + h 2 ( + ) λ = 2 1+ sin ϕ cosϕ 2 cos ϕ h h + 40

41 Granular Solid-Fluid Transitions 41

42 Deposition layer (Ancey 2001) Flow-slide in cylindrical topography: deposition shock-wave (Bassanou 2000). 42

43 Run-out: smooth base Self-damming flow (Bassanou 2000) 43

44 Uphill moving "hydraulic jump" in granular flow model test in an incline with variable topography (Bassanou 2000) 44

45 Run-out: rough base, self damming (Bassanou 2000) 45