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Depth integrated modelling of fast landslidepropagationThomas Blanc a , Manuel Pastor a , Mila Sánchez Valentina Drempetic a & Bouchra Haddada

a Mathematical Modelling in Engineering Group (M2i) Department of Applied Mathematicsand Computer Science ETS de Ingenieros de Caminos, Canales y Puertos , UniversidadPolitécnica de Madrid, Carretera de la Ciudad Universitaria , 28011, Madrid, Spain E-mail:Published online: 17 Oct 2011.

To cite this article: Thomas Blanc , Manuel Pastor , Mila Sánchez Valentina Drempetic & Bouchra Haddad (2011) Depthintegrated modelling of fast landslide propagation, European Journal of Environmental and Civil Engineering, 15:sup1,51-72, DOI: 10.1080/19648189.2011.9695304

To link to this article: http://dx.doi.org/10.1080/19648189.2011.9695304

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EJECE. Special Issue 2011, pages 51 to 72

Depth integrated modelling of fast landslide propagation

Thomas Blanc — Manuel Pastor — Mila Sánchez

Valentina Drempetic — Bouchra Haddad

Mathematical Modelling in Engineering Group (M2i) Department of Applied Mathematics and Computer Science ETS de Ingenieros de Caminos, Canales y Puertos Universidad Politécnica de Madrid, Carretera de la Ciudad Universitaria 28011 Madrid, Spain

th.blanc@gmail.com

manuel.pastor@upm.es

msanchez@ucla.edu.ve

vali@gmx.de

bouchra.haddad@uclm.es

ABSTRACT. This paper deals with a simple yet effective model which can be used for prediction of propagation of fast catastrophic landslides. The main ingredients are: i) a hierarchical set of mathematical models describing the coupled behaviour between solid skeleton and pore fluids. Here we will arrive to a coupled depth integrated model; ii) a rheological model describing the behaviour of the fluidized soil; iii) a numerical model to discretize mathematical and rheological models. The model performance has been assessed using a set of benchmark tests, including some provided by the Hong Kong Geotechnical Engineering Office.

RÉSUMÉ. Cet article traite d’un modèle simple mais efficace qui peut être utilisé pour prédire la propagation des glissements de terrains rapides. Le modèle se base sur (i) un ensemble hiérarchique de modèles mathématiques qui décrivent l’interaction du solide avec le fluide interstitiel et qui aboutissent à un modèle intégré en profondeur.(ii) Un modèle rhéologique décrivant le comportement des sols fluidifiés, et (iii) un modèle numérique pour discrétiser les modèles mathématique et rhéologique. Les performances du modèle ont été évaluées utilisant un ensemble de benchmarks dont quelques un ont été fournis par l’organisme Hong Kong Geotechnical Engineering Office.

KEYWORDS: landslide, propagation, depth integrated equations, SPH, Bingham fluids, cohesive-frictional fluid.

MOTS CLÉS : glissement de terrain, propagation, équations intégrées en profondeur, SPH, fluide de Bingham, fluide cohésif-frictionnel.

DOI:10.3166/EJECE.15SI.51-72 © 2011 Lavoisier, Paris

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1. Introduction

The study of landslides and their consequences has become a multidisciplinary subject, where geographical, pedological and urban planning aspects are important. Here we will deal only with engineering aspects and will focus on the prediction of the initiation and propagation of landslides. The first aspect is important not only to know what has really happened but also to avoid possible landslides.

Concerning the propagation phase, once the landslide has taken place, it is important to know the velocity of the flow, how long will it reach, and what will be the path followed by them. In this way, it is possible to propose strategies based on channeling and protection structures. Another interesting example is that of landslide affecting a reservoir. Here, we will need to know both the speed and the mass of soil involved in the problem in order to feed the hydrodynamic model with proper data.

This paper deals with a simple yet effective model that can be used for prediction. The main ingredients are: i) a hierarchical set of mathematical models describing the coupled behaviour between solid skeleton and pore fluids. Here we will arrive to a coupled depth integrated model; ii) a rheological model describing the behaviour of the fluidized soil; iii) a numerical model to discretize mathematical and rheological models.

The paper is structured as follows. First, we introduce the mathematical model used within a general framework, showing the different alternative levels of approximation. A second section is devoted to describe the rheological models used for the benchmarks, followed by a section where we describe the discretization technique we have used. Finally, we present a selected set of benchmarks.

2. Mathematical model: a hierarchical set of models for the coupled behaviour

of fluidized geomaterials

The materials found in fast landslides are mixtures where we can find soil, rocks, water and air. The behaviour of the mixture can be described using alternative approximations, which can be described in a hierarchical manner, from the more complex based on mixture theory, to the simpler depth integrated models. Here, we will describe both the model used in our computations, a coupled depth integrated model incorporating pore pressures, and the framework within which it has been derived. The first mathematical model describing the coupling between solid and fluid phases was proposed by (Biot, 1941, 1955) for linear elastic materials. This work was followed by further development at Swansea University, where Zienkiewicz and co-workers (Zienkiewicz et al., 1980; 1984; 1990a; 1990b; 2000) extended the theory to non-linear materials and large deformation problems. It is also worth mentioning the work of Lewis and Schrefler (Lewis et al. 1998; Coussy, 1995; de Boer, 2000). It can be concluded that the geotechnical community have

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Depth integrated modelling of fast landslides propagation 53

incorporated coupled formulations to describe the behaviour of foundations and geostructures. Indeed, analyses of earth dams, slope failures and landslide triggering mechanisms have been carried out using such techniques during last decades.

This theoretical framework has not been applied to model the propagation of landslides until recently. We can mention here the work of Hutchinson (Hutchinson 1986), who proposed a sliding consolidation model to predict run out of landslides, Iverson and Denlinger (2001), Pastor et al. (2002) and Quecedo et al. (2004)

The different modelling alternatives and their mutual dependences can be seen in Figure 1.

Figure 1. Modelling alternatives for landslide problems

2.2. General model

The general model is based on the assumption that the mixture is composed of a solid phase and several fluid phases. The equations are: i) balance of mass and ii) balance of linear momentum for the constituents and the mixture, iii) constitutive or rheological laws describing the material behaviour of all constituents, and iv) kinematics relations linking velocities to rate of deformation tensors. The main problem with this approach is the computational cost, because of the number of unknowns and the difficulty of having to track all interfaces. The main advantage is its general character, as it can describe phenomena involving large relative displacements between solid and fluid phases. This model has been described by (Pastor et al., 2002).

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54 EJECE. Special Issue 2011

2.3. Biot-Zienkiewicz model

A first simplified model can be derived by assuming that the velocity of fluid phases relative to solid skeleton is small. In this case the equations can be cast in terms of the displacements or velocities of solid skeleton, the velocities of the pore water relative to the skeleton and the averaged pore pressure of the interstitial fluids. This model, proposed by Zienkiewicz et al. (1984), for the case of saturated soils is referred to as wu p w� � . Its main variables are i) the velocity of solid skeleton sv ,

ii) the Darcy velocity of the pore water, w and the pore pressure wp . Under certain

assumptions, which were analyzed for soil mechanics problems by (Zienkiewicz et al. 1980), it is possible to eliminate the Darcy velocity from the model. This is the most celebrated wu p� model of Zienkiewicz, which has been widely used in

geomechanics being the base of many computer codes. The resulting model consists of the following equations:

– balance of mass and momentum of pore water, which is obtained eliminating the Darcy’s velocity of the pore water:

1 v p 0w

w w w w

DpDvdiv div k b grad

Dt Q DtU U

­ ½§ ·� � � � � ® ¾¨ ¸

© ¹¯ ¿ [1]

where � �1 1 s wQ n K n K � �ª º¬ ¼ ;

– balance of momentum of the mixture

( ) ( )

s sD v

b divDt

U U V � [2]

2.4. The propagation-consolidation approximation

So far we have described general models which can be applied to general problems. The analysis of landslides, due to their shape and geometrical properties allows some interesting simplifications. First of all, we will arrive to “propagation-consolidation” models, where pore pressure dissipation takes place along the normal to the terrain surface, and next, we will describe depth integrated models, where the three dimensional problem is transformed into a two dimensional form. The propagation-consolidation model can be derived assuming that the velocity and pressure fields can be split into two components, i.e., propagation and consolidation as 0 1v v v � and 0 1w w wp p p � .

The equations of the propagation-consolidation model are:

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Depth integrated modelling of fast landslides propagation 55

0 divDv

bDt

U U V � [3]

with 0div 0v ;

3 3

w wv

Dp pc

Dt x x

§ ·ww ¨ ¸w w© ¹

[4]

where vc is the coefficient of consolidation.

2.5. Depth integrated model with coupled pore pressures

Many flow-like catastrophic landslides have average depths which are small in comparison with their length or width. In this case, it is possible to simplify the 3D propagation-consolidation model described above by integrating its equations along the vertical axis. The resulting 2D depth integrated model presents an excellent combination of accuracy and simplicity providing important information such as velocity of propagation, time to reach a particular place, depth of the flow at a certain location, etc.

Depth integrated models have been frequently used in the past to model flow-like landslides. It is worth mentioning the pioneering work of Hutter and coworkers (Savage et al., 1991; Hutter et al., 1991), and those of (Laigle et al., 1997; Mc Dougall et al., 2005), and the authors (Pastor et al., 2002, 2008; Quecedo et al., 2004). We will use the reference system given in Figure 2 where we have depicted some magnitudes of interest which will be used in this section.

It is worth mentioning the difficulty of obtaining directly a Lagrangian form of the depth integrated equations, because the vertical integration is not performed in a material volume. Sometimes, it has been found convenient to refer to an equivalent 2D continuum having as velocities of their material points the depth integrated velocities. This cannot be considered as a Lagrangian formulation, because the moving points have no exact connection with material particles. It can be denominated either “quasi lagrangian”, or arbitrary lagrangian eulerian (ALE) formulation. To derive a quasi lagrangian formulation of the depth integrated equations, we will first introduce a “quasi material derivative” as:

jj

dv

dt t x

w w �w w

[5]

from where we obtain the “quasi lagrangian” form of the balance of mass, depth integrated equation as:

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jR

j

vdhh e

dt x

w�

w [6]

where Re is the erosion rate 1[ ]L� and h is the flow depth.

Figure 2. Reference system and notation used in the analysis

The balance of momentum equation is:

� �2 *3

1 1 1

2B B

i ij i i R ii j

dh v b h h b h N t e v

dt x xV

U U

w w§ ·� � � �¨ ¸w w© ¹

[7]

where we have introduced the decomposition:

*ij ij ijpV G V � � [8]

with 3 2p b hU and *ij ij ijpV V G � .

The term Bit is the i-th component of the normal stress acting on the basal

surface and � � � �1/22 2

1 2 1BN Z x Z xª º w w � w w �¬ ¼

,where Z is the height of the basal

surface.

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Depth integrated modelling of fast landslides propagation 57

It is important to note that we have to include the effect of centripetal accelerations, which can be done in a simple manner by integrating along the vertical the balance of momentum equation, and assuming a constant vertical

acceleration given by 2 /V R , where V is the modulus of the averaged velocity and R the main radius of curvature in the direction of the flow. Finally, after integrating the vertical consolidation equation in depth we arrive at:

� �3

Z h

ww v

Z

pdp h c

dt x

�w

w

[9]

Next, assuming an approximation of the pore pressure as:

� � � � � �1 2 3 1 2 31

, , , , ,Npw

w k kk

p x x x t P x x t N x

¦ [10]

from where, taking:

� � � �3 3

(2 1)cos 1,

2k

kN x x Z k Npw

hS

� � [11]

and keeping only the first term, we have:

� � � � � �1 2 3 1 1 2 3, , , , , cos2

wp x x x t P x x t x Zh

S � [12]

from where we obtain the depth integrated equation:

2

1124

v

dPc P

dt h

S [13]

which is the quasi-lagrangian form of the vertically integrated 1D consolidation equation. It is important to note that the results obtained above depend on the rheological model chosen, from which we will obtain the basal friction and the depth

integrated stress tensor *ijV .

3. Behaviour of fluidized soils: rheological modelling alternatives

When obtaining the depth integrated equations described in the preceding Section, we have lost the flow structure along the vertical, which is needed to obtain both the basal friction and the depth integrated stress tensor. A possible solution which is widely used consist of assuming that the flow at a given point and time,

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with known depth and depth averaged velocities has the same vertical structure than a uniform, steady state flow. In the case of flow-like landslides this model is often referred to as the infinite landslide, as it is assumed to have constant depth and move at constant velocity along a constant slope. This infinite landslide model is used to obtain necessary items in our depth integrated model. We will present next some models frequently found in landslide propagation modelling.

3.1. Bingham fluid

In the case of Bingham fluids, there exists an additional difficulty, because it is not possible to obtain directly the shear stress on the bottom as a function of the averaged velocity. The expression relating the averaged velocity to the basal friction for the infinite landslide problem is:

2

1 26

B Y Y

B B

hv

W W W

P W W

§ · § · � �¨ ¸ ¨ ¸

© ¹ © ¹ [14]

where P is the viscosity, YW the yield stress, and BW the shear stress on the bottom.

This expression can be transformed into:

� � � �33 : 3 2 0P aK K K � � � [15]

where we have introduced /Ph hK which is the ratio between the height of the

constant velocity region or plug to the total height of the flow, and the non dimensional number a defined as

6

Y

va

h

P

W [16]

It is first necessary to obtain the root of a third order polynomial. To decrease the

computational load, several simplified formulae have been proposed in the past. The authors introduced in Pastor et al. (2004) a simple method based on obtaining the second order polynomial which is the best approximation in the uniform distance sense of the third order polynomial, which is given by:

� � 22

3 57 65

2 16 32P aK K K

§ · � � �¨ ¸

© ¹ [17]

Knowing the non-dimensional number a , the root is obtained immediately.

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Depth integrated modelling of fast landslides propagation 59

3.2. Frictional fluid

One simple yet effective model is the frictional fluid, especially in the case where it is used within the framework of coupled behaviour between soil skeleton and pore fluid. Without further additional data it is not possible to obtain the velocity distribution. This is why depth integrated models using pure frictional models cannot include information concerning depth integrated stresses V .

Concerning the basal friction, it is usually approximated as � �tanb v iv vW V I � ,

where vV is the normal stress acting on the bottom. Sometimes, when there is a high

mobility of granular particles and drag forces due to the contact with the air are important it is convenient to introduce the extra term. proposed by Voellmy’s

(1955), which includes the correction term 2gvU [ , where [ is the Voellmy

turbulence parameter.

In some cases, the fluidized soil flows over a basal surface made of a different material. If the friction angle between both materials G is smaller than the friction angle of the fluidized soil, the basal shear stress is given by:

tan ib b b

vgh

vW U Ic � [18]

where the basal friction bI is:

� �min ,bI G I [19]

This simplified model can implement the effect of pore pressure at the basal

surface. In this case, the basal shear stress will be:

� �' tan b ib v b w

vp

vW V I � � [20]

We can see that the effect of the pore pressure is similar to decreasing the

friction angle.

3.3. Cohesive-frictional fluid

The cohesive-frictional model proposed in 3D by the authors (Pastor et al. 2009) particularizes for the case of a simple shear flow to:

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60 EJECE. Special Issue 2011

11 22 33

113 31

3

m

CF

p

vs

x

V V V

V V P

�

§ ·w � ¨ ¸

w© ¹

[21]

where cos 'sins c pM M � .

Two particular cases of interest are the Bingham fluid ( 0 1yc mM W )

and the Herschel-Bulkley fluid. For flow of granular materials we will use 0 2c m . The basal friction term results on (Pastor et al. 2009):

2

2

25cos tan

4b d CF

vgh

hW U T M Pc � [22]

which is a law of similar structure than Voellmy’s:

2

cos tanb d

vgh gW U T M U

9

­ ½° °c �® ¾

° °¯ ¿ [23]

where 9 is a material parameter. If we compare both expressions, we can see that

both incorporate a quadratic term depending on the averaged velocity.

In above, we have defined for convenience *d d w wU U E Uc c � , where

� �1d snU Uc c � , and the pore pressure in excess to the hydrostatic is written as

w w wp pE' .

3.4. Erosion

One important aspect in the behaviour of catastrophic landslides and related phenomena is the erosion. This complex phenomena requires a rheological or constitutive behaviour of the interface, and depends on variables such as the flow structure, density, size of particles, and on how close are the effective stresses at the surface of the terrain to failure. We have used here the simple yet effective law

proposed by Hungr (1995) which gives the erosion rate as t sE E h v where sE can

be obtained directly from the initial and final volumes of the material and the

distance traveled as � �0ln / distances finalE V V| . Units of erosion coefficient are

1L� . It is also worth mentioning other erosion laws proposed by Blanc (2008), Issler and Jóhannesson (Issler et al., 2010) and Issler and Pastor (Issler et al., 2010).

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Depth integrated modelling of fast landslides propagation 61

4. Numerical model: the SPH approximation

To analyze the propagation of a fast landslide over a terrain, there are two main alternatives. The first is eulerian, and is based on a structured (Finite Differences) or unstructured grid (Finite Elements and Volumes) within which the material flows. The main problem here is the need of a very fine computational mesh for both the terrain information and for the fluidized soil. Lagrangian methods allow the separation of both meshes, with an important economy of computational effort. If we combine a Lagrangian method with a mesh based discretization technique, we will find problems as soon as the mesh deforms, making necessary to use mesh refinement. As alternative, meshless methods, which do not rely on meshes, avoids distortion problems in an elegant way. In this work, we have used a meshless method referred to as the smoothed particle hydrodynamic or SPH. As in any meshless method, information is linked to moving nodes. We will describe next the method in a very succinct way. Smoothed particle hydrodynamics (SPH) is a meshless method introduced independently (Lucy, 1977; Gingold et al., 1977) and firstly applied to astrophysical modelling, a domain where SPH presents important advantages over other methods. SPH is well suited for hydrodynamics, and researchers have applied it to a variety of problems, like those described in (Gingold et al., 1982; Monaghan et al., 1999; Bonet et al., 2000; Monaghan et al., 2003), just to mention a few. SPH has been also applied to model the propagation of catastrophic landslides (Bonet et al., 2005; McDougall, 2006; 2004) however in both cases, the analysis did not incorporate hydro mechanical coupling between the solid skeleton and the pore fluid, which has been proposed by the authors (Pastor et al., 2008).

4.1. An SPH method for depth integrated equations

We will introduce a set of nodes ^ `Kx with K = 1..N and the nodal variables:

– Ih height of the landslide at node I;

– Iv depth averaged, 2D velocity;

– bIt surface force vector at the bottom;

– *IV depth averaged modified stress tensor;

– 1IP pore pressure at the basal surface.

If the 2D area associated to node I is I: , we will introduce for convenience:

– a fictitious mass Im moving with this node I I Im h : ;

– and an averaged pressure term Ip , given by 23 2I Ip b h ;

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It is important to note that Im has no physical meaning, as when node I moves,

the material contained in a column of base I: has entered it or will leave it as the

column moves with an averaged velocity which is not the same for all particles in it.

There are several possible alternatives for the equations, according to the discretized form chosen for the differential operators results. We will show those obtained with the third symmetrized forms:

gradJII IJ IJ

J J

mdhh v W

dt h ¦ [24]

where we have introduced IJ I Jv v v � .

Alternatively, the height can be obtained once the position of the nodes is known as:

� �I I J J IJ J IJJ J

h h x h W m W : ¦ ¦ [25]

The discretized balance of linear momentum equation is:

2 2

2 2

grad

1 1grad

JII J IJ

J I J

B BJIJ IJ I

J II J

ppdv m W

dt h h

m W b N thh h

VV

U U

§ · � �¨ ¸¨ ¸

© ¹

§ ·� � � �¨ ¸¨ ¸

© ¹

¦

¦ [26]

Finally, the SPH discretized form of the basal pore pressure dissipation is:

2

1 14v

I II

cdP P

dt h

S � [27]

So far, we have discretized the equations of balance of mass, balance of

momentum and pore pressure dissipation. The resulting equations are ODEs which can be integrated in time using a scheme such as Leap Frog or Runge Kutta (2nd or 4th order).

5. Examples and applications

During 2007, the Hong Kong Geotechnical Office organized a benchmarking exercise aiming to assess the accuracy of numerical and constitutive models. The participants were provided a detailed digital terrain elevation map, including the

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Depth integrated modelling of fast landslides propagation 63

original position of the mobilized mass and the final position of the deposit. We will show some of the 12 benchmark tests, which were modelled using the SPH model described in Pastor (Pastor et al., 2008).

5.1. The Tsing Shan debris flow (Hong Kong, 2007)

One of these benchmarks is the Tsing Shan debris flow which happened in Hong Kong on the 14th April 2000. As in the previous case, the analysis based on the information found both in the package provided by Hong Kong Geotechnical Office and the report by (King, 2001). This debris flow took place following rains which triggered more than 50 landslides in the area. The accumulated rainfall was 160 mm. The terrain was vegetated, and consisted of colluvial boulders. One important feature of this event is the strong erosion which made the initial mass to increase from 50 to 1 600 cubic meters. Figure 3, taken from (King, 2001) provides two general views of the debris flow. One important aspect is the bifurcation of the flow which can be observed in the pictures.

Figure 3. General view of the 2000 Tsing Shan debris flow (King, 2001)

In order to model it, we have used a frictional fluid model, with tan 0.18I ,

zero cohesion and 20.00133 Pa.sCFP . We have chosen the Hungr’s erosion

model, using an erosion coefficient of 0.0082 -1m . The results of the simulation are given in Figure 4 which provides information regarding both the position and depths of the final deposit and the track. One peculiarity of this debris flow is the bifurcation in two branches, which is a feature difficult to capture in simulations. The computed path depicted in Figure 4 shows the branching. In Figure 4, it can be seen that the deepest deposit was formed at the end of the lower south branch, with a

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64 EJECE. Special Issue 2011

maximum depth of 1.8 m. Regarding velocities, there is no available information. The model predicts a time of propagation close to 120 s. Considering that the runout was 900 m in the lowest branch, the average velocity is close to 30 km/h. The report provides a total mass deposited in the south branch of 500 cubic meters, while the computation provides a value of 525 cubic meters. Concerning the total volume of eroded soil, the report estimates it as 1 600 cubic meters, while the computations provide a value of 1 550.

Figure 4. Tsing Shan debris flow: Model predictions vs. field observations

5.2. Fei Tsui Road landslide

This landslide, which is described in Knill and Geotechnical Engineering Office 2006, occurred in August 2005 in a slope of weathered volcanic rock, grading from moderately to completely decomposed tuff. It involved 14 000 cubic meters of material. The groundwater conditions consisted on two groundwater regimes, the regional groundwater table, and a perched water table. The causes are described in the report, and are a combination of a weaker material together with an increase in groundwater pressure following a prolonged heavy rainfall. The slope has an inclination of 60º and was densely vegetated. The maximum width of the mobilized mass was 90 m, and the distance travelled 70 m. The landslide piled up against a corner of the Baptist Church some 6 m. Figure 5, taken from (Knill et al., 2006), shows a general view of the landslide.

The landslide has been modelled using a frictional fluid having an internal friction angle of 26º, following the information provided in the report. This apparent friction angle is smaller than the effective friction angle due to the existence of induced pore pressures. Taking into account the time of propagation (10 s) and the mass of soil involved, we have assumed that the time of propagation is much smaller

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Depth integrated modelling of fast landslides propagation 65

than that of pore pressure dissipation, that’s to say undrained behaviour. Figure 6 depicts the position of the landslide at times 1 s, 3s, 6 s, and 9 s, and Figure 7 provides a comparison between the real and the computed landslide extensions.

Figure 5. General view of Fei Tsui Road landslide (Knill et al., 2006)

Figure 6. Propagation of Fei Tsui Road landslide

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Figure 7. Comparison for the extent of the landslide between field measurements and computed results

5.3. Thurwieser rock avalanche

This case is a rock avalanche that occurred in the Central Italian Alps the 18th September 2004. The location was the south slope of Punta Thurwieser, and it propagated through Zebrú valley. Its propagation path extended from 3 500 m to 2 300 m of altitude, with a travel distance of 2.9 Km. The rock avalanche involved 2.2 million cubic meters. Sossio and Crosta (2007) have provided the information concerning this avalanche, including a detailed digital terrain model. Figure 8, from Sossio and Crosta (2007), provides a general view of the avalanche and its location.

This avalanche presents several modelling difficulties, such as crossing of terrains of different materials, such as the Zebrú glacier. There, the basal friction is very small, and erosion of ice and snow is possible. This entrained material can melt due to the heat generated by basal friction, providing extra water, and probably originating basal pore pressures. We have used here a simple frictional model including Voellmy turbulence. Concerning erosion, we have used the law proposed by Hungr (1995). The rheological parameters chosen are: tanI = 0.39, Voellmy coefficient 1 000 2m/s , erosion coefficient 0.00025 -1m

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Figure 8. General view of Thurwieser rock avalanche (Sossio et al., 2007)

The results are given in Figure 9, where we have plotted the avalanche evolution along time and the computed final extension together with the observed in the field.

Figure 9. Thurwieser avalanche after 80 seconds with friction angle 26º: computed results (colour isolines and deposit height) versus field measurements (black isolines and red line for the spreading)

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5.4. Modelling of Cougar Hill flowslide

We will present here the case of a flowslide where the coupling between pore pressures and the solid skeleton was crucial. Dawson (Dawson et al., 1998) have reported three cases of flowslides of coal mine waste dumps in the Western Canadian Rocky Mountains which they selected among some 50 flowslides which occurred there in the years 1972-1997.

The flowslides propagated distances up to 3 500 m, with a modal value of 500 to 1 000 m. in the histogram representing number of flowslide events against runout distances.

The case we have selected is Fording Greenhills, where in May 1992 the Cougar 7 dump failed. The mobilized volume of debris consisted in approximately 200,000 3m which slid off the 100 m high dump. It is important to remark that Wet fine grained layers were found at the foundation contact, near the crest, and in the debris.

According to Dawson, these fine grained layers played a crucial role both in the initiation and the propagation phases. Indeed, the flowslide is thought to have been triggered by liquefaction of the fine grained layers.

Dawson performed laboratory tests, from which we have obtained a density of 1 900 3kg/m an effective friction angle 0.37I c and a characteristic consolidation time of 68 s.

We have analysed this case using the depth averaged coupled, SPH model described in the preceding Sections. The results of the simulations are given in Figures 10 and 11 where we give perspectives of the flowslide extension and isolines of the debris depth and pore pressures.

Figure 10. Position of SPH nodes at 5, 10, 20 and 30 s

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Figure 11. Contours of normalized pore water pressure

6. Conclusions

Rock avalanches, flowslides, debris flows, lahars and other similar events are most complex phenomena involving complex physical mechanisms such as segregation, comminution, basal erosion, coupling with pore water, evolution of fluid properties, and thermal effects, just to mention some of them. Complete 3D models based on mixture theory and incorporating sub models for above mentioned phenomena are still very expensive from a computational point of view. Depth integrated models provide a good combination of simplification and accuracy and can provide useful results for scientists and engineers. There exist suitable discretization techniques for depth integrated models, such as finite differences, finite elements, finite volumes or the more recent meshless methods such as the SPH model used by the authors. All of them provide accurate numerical approximations of the depth integrated equations. In the authors experience, the SPH model allows to separate the computational mesh consisting of moving nodes or particles, from the topographical mesh which can have an structured nature, simplifying very much

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computations on it. In the author’s experience, the computational time can be reduced up to 30 times, as compared with unstructured finite element meshes.

Acknowledgements

The authors gratefully acknowledge the support of the Geotechnical Engineering Office, Civil Engineering and Development Department of the Government of the Hong Kong SAR in the provision of the digital terrain models for the Hong Kong landslide cases. The financial support of the Spanish MCINN (Project GeoDyn), and the EC (Project Safeland) are gratefully acknowledged.

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