thermo-poro-mechanics of catastrophic...

Thermo-poro-mechanics of catastrophic landslides: An attempt to reach a lifetime

estimate

Ioannis Vardoulakis N.T.U. Athens

(http://geolab.mechan.ntua.gr)

EU RTN DIGA

Institute for Geotechnical Engineering

Swiss Federal Institute of Technology Zurich ETHZ, Switzerland

Thursday, 26 January 2006

Landslides Lifetime I. Vardoulakis Jan. 2006

2

Acknowledgements

Manolis Veveakis, PhD cand., Geolab, NTUA

Project Pythagoras I/EPEAEK II, supported by the European Union and the Greek Ministry of National Education


3

Müller-Saltzburg (1964) in his Report “…The Rock Slide in the

Vajont Valley” summarizes his impressions as follows:

“…the interior kinematic nature of the mobile mass, after havingreached a certain limit velocity at the start of the rock slide, must

have been a kind of thixotropy. This would explain why the

mass appears to have slid down with an unprecedented velocity

which exceeded all expectations. Only a spontaneous decrease

in the interior resistance to movement would allow one to explain the fact that practically the entire potential energy of the

slide mass was transformed without internal absorption of

energy into kinetic energy…Such a behaviour of the sliding

mass was beyond any possible expectation; nobody predicted it

and the author believes that such a behavior was in no way predictable…”.


4

The Vajont landslide


5


6

Analysis of the catastrophic phase (Vardoulakis, 2002)


7

Related bibliography

Habib, P. (1967). Sur un mode de glissement des massifs rocheaux. C. R. Hebd. Seanc. Acad. Sci., Paris, 264, 151-153.

Lachenbruch, A.H. (1980). Frictional heating, fluid pressure and the resistance to fault motion. J. of Geophys. Res., 85, 6097-6112.

Voight, B. and C. Faust (1982).Frictonal heat and strength loss in some rapid landslides. Géotechnique, 32, 43-54.

Vardoulakis, I. (2002). Dynamic thermo-poro-mechanical analysis of catastrophic landslides. Géotechnique, 52, (3),157-171.

Sulem, J. Vardoulakis, I., Ouffroukh, H. and Perdikatsis, V. (2005). Thermo-poro-mechanical properties of the Aigion fault clayey gouge - application to the analysis of rapid shearing. Soils & Foundations, 45 (2): 97-108.

Rice J., (2006). Heating and weakening of faults during earthquake slip, J. Geophys. Res. In print.


8

The frictional pendulum model: Vaiont slide, sect. 5 (Vardoulakis, 2002)


9

The rapidly deforming shear-band


10

Thermo-poro-mechanical softening


11

“ Slow” ring-shear test results used for the analysis of the slide’s catastrophic phase


12

thermo-plastic collapse of clays


13

C

1106.2

o

4pc

ecc

−⋅−≈α+α=α


14

temperature and pore-pressure isochrons


15

(1) ideal plasticity

(2) displacement softening (3) displ. & velocity softening


16

Practical result from this analysis:

Thermal pressurization is the most probable mechanism, which explains the total loss of strength during the final stage of the slide ranging from 2 sec to 20 sec.


17

Analysis of the accelerated creeping phase (Veveakis & Vardoulakis 2006)


18

Related bibliography

Gruntfest I.J, 1963. Thermal Feedback in Liquid Flow; Plane Shear at Constant Stress, Trans. Soc. Rheology, v.7, p. 195-207.

Dold J.W, 1985. Analysis of the early stage of thermal runaway, Q. J. Mech. Appl. Math, 38, p. 3.

Chen, H.T.Z, Douglas A.S. and Malek-Madani, R. , 1989, An asymptotic stability condition for inhomogeneous simple shear, Quartr. Appl. Math., 47, 247-262.

Leroy Y. and Molinari, A.,1992. Stability of steady states in shear zones, J. Mech. Phys. Solids, 40 (1), 181-212.

Galaktionov V.A. and Vazquez J, 2002. The problem of Blow Up in Nonlinear Parabolic Equations, Disc. Cont. Dyn. Systems, 8, p. 399-433.

Vardoulakis, I., 2002. Steady shear and thermal run-away in clayey gouges. Int. J. Solids and Structures, 39,3831-3844.

Helmstetter, A., Sornette, D., Grasso, J.-R., Andersen, J. V., Gluzman,S., and Pisarenko, V. (2004). Slider block friction model for landslides: Application to Vaiont and La Clapiere landslides. Journal of Geophysical Research, v.109, B002160.

Veveakis E. and Vardoulakis, I., 2006, Thermo-poro-mechanics of catastrophic landslides: An attempt to reach a lifetime estimate, submitted.


19

Observed slide creep, triggered probably by a micro-quake

activity during reservoir filling


20

Critical block failure mechanism for

‘Section 5’ of the Vaiont slide (cf.

Hendron & Patton, 1985 and Müller,

1967)

Sub-structuring


21

The block model

0

2

0

1 dV1

g dt tan

(g gsin 1.9m / s )

1 dVtan 1

g dt

−

τ= −

′ σ δ

′ = δ ≈ ⇒

τ = σ δ − ′


22

The critical phase of accelerated creep


23

For the most part of the creeping slide the shear stress at the interface is practically constant; it

decreases only in the last minute


24

The sliding lasted less then 1 min and produced shocks which were recorded throughout Europe. The sliding mass moved some 400m at an estimated 20-30 m/s

(Tika & Hutchinson, Géotechnique 1999)


25

The heat equation (m) 2p

m ij ij2

k m

2

m loc2

m

d

p p

loc xz xz zx zx d

D 1D

Dt x j( C)

1D

t z j( C)

0 const.z

vD D D ,

z

θ ∂ θ′= κ + σ

∂ ρ

∂θ ∂ θ= κ +

∂ ∂ ρ

∂τ= ⇒ τ = τ =

∂

∂= σ +σ ≈ τ γ γ =

∂& &


26

Frictional strain-rate hardening and thermal softening of clay

Hicher (1974)

1

1

d d

NM( )

refref

m( )loc 0

( , ) const.

e

D D e

− θ−θ

θ−θ

′τ = σ µ γ θ ≈

γµ = µ ⋅

γ ⇒

=

&

&

&


27

The “solid-fuel model” or “Frank-Kamenetskii equation”

** 2 *

* *2

2

0 Bd

m

Gr et z

dGr m (Gruntfest 1963)

jk 2

θ∂θ ∂ θ= +

∂ ∂

γ = τ

&


28

Steady fault shearing

( )tanh 1 1.199679α α = ⇒ α ≈

Critical condition (loss of existence):

( )( )

*

max

*

*

* * 2max d

Gre 0z

Gr2ln cosh , e

2

∗

θ

θ

∂θ+ =

∂

θ = θ + α α =


29

LSA of the steady solution

( )

( )

* 2 * 2*

* *2 2 *

* *1 2

* 2

* 2

2

t z cosh z

F cosh C LP( , f , z ) C LQ( , f , z )

3LP( , f , z ) LegendreP u, , 1 cosh

2

3LQ( , f , z ) LegendreQ u, , 1 cosh

2

1 2 su 1

2

∂θ ∂ θ α= + θ

∂ ∂ α

= ξ α + α

α = − ξ

α = − ξ

= − α

% %%


30

The unsteady problem:

“solid-fuel model” or “Frank-Kamenetskii equation”Galaktionov & Vazquez (1997, 2002 )

[ ]*

* 2 ** *

* *2Gr e z 1,1 t 0

t z

θ∂θ ∂ θ= + ∈ − >

∂ ∂


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The Frank-Kamenetskii equation is important in combustion theory, since it presents a finite time singularity in its solution, known as the blow-up time. The mathematical theory of blow-up in parabolic equations has been actively investigated by researchers in the 60’s, mainly after general approaches to blow-up by Kaplan, Fujita, Friedman and others. However there is yet no complete theory developed, but detailed studies have been performed on a quite significant number of models of varying complexity, and there is nowadays an extensive literature on this subject.

Numerical treatment of the F-K equation turns out to be a formidable task. Linear stability analysis yields that maximum growth corresponds to infinite Ljapunow exponent. Thus, infinity is an accumulation point of the zeroes of the dispersion function, and for large times the problem becomes ill-posed. This anomalous behavior is linked to finite time blow-up. This makes the numerical approximation using the standard techniques of Galerkin method, finite differences etc, inapplicable.


32

In order to seek asymptotic solutions for this problem, we follow here Dold’s suggestion for the most robust variable grouping:

( )

* *

I

*

* * * *

I I

*2* * * *

0 I * *

I

ln(t t )

z

(t t ) a ln(t t )

zln (t t )

4 a ln(t t )

ξ = − −

η = − − −

θ = θ − β − + − −

*I

1t

Gr= dimensionless blow-up time


33

For * *It t→ the solution tends to the one of the adiabatic limit

d

m B

1 V,

t j( C) d

∂θ≈ τ γ γ ≈

∂ ρ& &

Sornette et al. (2004) estimate

I

1V U

t t=

−

( )* * * *

0 Iln Gr (t t )θ ≈ θ − −


34

asymptotic solution of the F.-K. equation:

Following Dold (1985) the natural way of approaching the considered

problem analytically is to develop a coordinate-perturbation analysis,

using a suitable variable grouping to represent the spatial concentration of the effects of the exponential generation term in the

governing equation

* *

I

*

* * * *

I I

ln(t t )

z

(t t ) a ln(t t )

ξ = − −

η = − − −

( )

* *

0 *2* *

I * *

I

1ln

zGr(t t )

4 a ln(t t )

θ = θ +

− + − −


35

( )

*0*

ref ref B 0

* *I

V(t ) V F(1,s) , V d e

a ln s 1 1F(1,s) 2 arctan , s t t

Gr s 2 Gr s a ln s

θ= = γ

− = = − −

&

Thermal creep model(Veveakis & Vardoulakis 2006)


36


37


38


39

Müller-Saltzburg in his 1964 report pointed that

“… The peak velocities increased progressively during the

early days of October. According to the report of the “Commissione di Inchiesta” the velocity had reached 20 cm

per day by October 9. Compared with the final velocity of

the sliding mass (about 25m/sec), all movement, even the last phase, must be considered a creeping movement up to

the very instant of the slide itself.”


40

The sudden, “last-minute” acceleration of the slide can

be explained with an almost instantaneous rise of the pore-pressure due to shear-heating. In our previous

study (Vardoulakis, 2002a), we showed that it takes no more than 10 to 20 sec for full pressurization to establish

itself.


41

If we assume that during the creeping phase the clay is nearly at critical state, then we get that the critical pressurization temperature should be between 32o 40o. This state is reached in o core of the shear zone, during the very last phase of the predicted thermally driven, drifting-creep motion.


42

The undrained-adiabatic approximation

m

2

3m

B

v

z t

(1/ 2)g 1.m / sec

2.4gr / cm

d 1.4mm

1.5Pa 1.MPa

∂τ ∂= ρ

∂ ∂

′ ≈

ρ ≈

≈

⇒ ∆τ = <<

d 0( p)′τ = σ − ∆ µ


43

**

*

** *

*

dV1 (1 p )

dT

dp(1 p )V

dT

= −α −

= β −

* *2p 1 exp T2

β ≈ − −


44

Open questions:

Triggering (dilatant hardening, Garagash & Rudnicki, 2003)

Vortex formation (Leroy & Molinari, 1992)

Transition from pressurization to fluidization


45


46


47

Mudflows

[Experiments on debris- and mudflows held at the model slider of the Laboratory of Geomaterials (Geolab) of NTUA.]


48

Red Cross officials estimated 200 people were dead

and 1,500 others missing.

"It sounded like the mountain exploded, and the whole

thing crumbled," survivor Dario Libatan told Manila

radio DZMM. "I could not see any house standing

anymore."

Massive Landslide and mudflow over the village of Guinsaugon, St. Bernard town in Southern Leyte province in central Philippines on Saturday, Feb. 18, 2006


49

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]


50

What experts did agree on was the probable impact of heavy rain in the area for up to two weeks before the landslide. "All these extreme disasters are multicausal but there's usually some single trigger at the last minute," said Hazel Faulkner, senior research fellow at the Flood Hazard Research Centre at Middlesex University, London. Edward Alan O'Lenic, National Oceanic and Atmospheric Administration Climate Prediction Center, Feb. 2, 2006. La Nina is thought to have taken hold in 2006The area received about 200cm of rain in the last 10 days, officials said. Heavy rain storms are frequent in the Philippines, and was also thought to be the trigger for the December 2003 landslide.But MrSpeechly said he was surprised by such weather in February.He said that severe storms normally ran between June and December.Prof Dave Petley, professor at the International Landslide Centre, Durham University, agreed. "This sort of rainfall and landslide action in the Philippines at this time of year is quite unusual," he said.


51

The open problem of the transition from pressurization to fluidization


52

z

v

)C(j

p

zt

tz

pc

t

p

)p(,z

1

t

v

m

0n

2

2

m

m2

2

v

0nm

∂∂

ρ

−σ′µ+

∂

θ∂κ=

∂θ∂

∂θ∂

λ+∂

∂=

∂∂

µ−σ′=τ∂τ∂

ρ=

∂∂

Momentum balance - ill-posedeness(Vardoulakis 2002, Rice 2006)


53

( )

∂

∂

γ∂µ∂

−σ′+∂∂

γµρ

−=∂∂

γµ=µ

2

2

0nm z

v)p(

z

p1

t

v

)(

&&

&

viscous regularization of momentum balance for frictional rate-hardening material


54

limited time viscous regularization(Vardoulakis, 2001)


55

Frictional visco-plastic softening of the Vaiont clay


56

( )

∂

∂−

∂

∂

γ∂µ∂

−σ′+∂∂

γµρ

−=∂∂

><γ∂µ∂

γ∇−γµ=µ

4

4

2

2

0nm

2

z

vr

z

v)p(

z

p1

t

v

0r,0,r)(:softeningrate

&&

&&&

2nd gradient, viscous regularization(R. Lattés and J.-L Lions (1967). Méthode de Quasi-

Réversibilité et Applications, Dunod, Paris)


57

2nd gradient-viscous regularization

thermo-poro-mechanics of catastrophic...

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