Depth Averaged Models for Fast Landslide Propagation

Depth Averaged Models for Fast Landslide Propagation: Mathematical, Rheological and Numerical Aspects M. Pastor, T. Blanc, B. Haddad, V. Drempetic, Mila Sanchez Morles, P. Dutto, M. Martin Stickle, P. Mira & J. A. Fernández Merodo Archives of Computational Methods in Engineering State of the Art Reviews ISSN 1134-3060 Arch Computat Methods Eng DOI 10.1007/s11831-014-9110-3

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Author's personal copy Arch Computat Methods Eng DOI 10.1007/s11831-014-9110-3

Depth Averaged Models for Fast Landslide Propagation: Mathematical, Rheological and Numerical Aspects M. Pastor · T. Blanc · B. Haddad · V. Drempetic · Mila Sanchez Morles · P. Dutto · M. Martin Stickle · P. Mira · J. A. Fernández Merodo

Received: 26 November 2013 / Accepted: 26 November 2013 © CIMNE, Barcelona, Spain 2014

Abstract This paper presents an overview of depth averaged modelling of fast catastrophic landslides where coupling of solid skeleton and pore fluid (air and water) is important. The first goal is to show how Biot–Zienkiewicz models can be applied to develop depth integrated, coupled models. The second objective of the paper is to consider a link which can be established between rheological and constitutive models. Perzyna’s viscoplasticity can be considered a general framework within which rheological models such as Bingham and cohesive frictional fluids can be derived. Among the several alternative numerical models, we will focus here on SPH which has not been widely applied by engineers to model landslide propagation. We propose an improvement, based on combining Finite Difference meshes associated to SPH nodes to describe pore pressure evolution M. Pastor (B) · T. Blanc · V. Drempetic · P. Dutto · M. M. Stickle · J. A. F. Merodo Department of Applied Mathematics, ETS Ingenieros de Caminos, UPM, Madrid, Spain e-mail: [email protected] P. Mira Centro de Estudios y Experimentación de Obras Públicas (CEDEX), Madrid, Spain P. Mira Department of Geotechnical Engineering, Escuela de Ingenieros de Obras Públicas, UPM, Madrid, Spain B. Haddad Universidad de Castilla La Mancha, Toledo, Spain J. A. F. Merodo Instituto Geológico y Minero de España, IGME, Madrid, Spain M. S. Morles Departmento de Ingenieria Vial, Universidad Centroccidental Lissandro Alvarado, Barquisimeto, Venezuela

inside the landslide mass. We devote a Section to analyze the performance of the models, considering three sets of tests and examples which allows to assess the model performance and limitations: (i) Problems having an analytical solution, (ii) Small scale laboratory tests, and (iii) Real cases for which we have had access to reliable information.

1 Introduction Landslides cause severe economic damage and a large number of casualties every year around the world. There is a wide variety of types of landslides, depending on the materials involved and triggering mechanism. The time scale involved can range from minutes to years. Landslides and failure of slopes are caused by changes in the effective stresses, variation of material properties or changes in the geometry. Changes of effective stresses can be induced either directly, as consequence of variation of the external forces (earthquakes, human action), or indirectly through pore pressures (rainfall effects). Variations in material properties can be caused by processes of degradation (weathering and chemical attack). Finally, geometry can change because of natural causes (erosion) or human action (excavation, construction, reshaping...). The main types of landslides, according to Dikau et al. [1] are the following: (i) Fall, (ii) Topple, (iii) Slides, (iv) Lateral spreading, (v) Flows, and (vi) complex movements. Slides are characterized by a mass movement over a clear failure surface, which can be idealized as a plane (translational slides) or circular (rotational slides). In the former case, the movement is assumed to be a translation along the failure plane, while in the latter; the upper mass of soil rotates as a rigid solid following the curved surface.

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Author's personal copy M. Pastor et al.

On the other hand, flows can be described as fluid like movements where individual particles travel separately. The velocity can range between some meters per day (slow earth flows) and tens of kilometres per hour (debris flows, flowslides). The soil accommodates to changes of slope, contours obstacles and fills cavities. From a mechanical point of view, it can be said that, in the case of slides, the failure mechanism consists of localization of shear deformation on a thin surface [2]. In the case of flows, failure is of another different mechanical type, which is described as diffuse. The study of diffuse mechanisms is more recent, and we should mention the work of Darve who named it [3] Details could be also found in [4]. This mechanism of failure is characteristic of soils presenting very loose or metastable structures with a strong tendency to compact under shearing, like poorly compacted deposits which experiment a sudden collapse with important build up of pore pressures and liquefaction in some cases. One paramount feature is that effective stresses can approach zero, the material behaving like a viscous fluid in which buildings can sink, as it happened during the 1966 earthquake of Niigata in Japan. When this failure mode takes place in a slope, the mass of mobilized soil can propagate downhill, evolving into flow slides or mudflows. Therefore, using a proper constitutive model is crucial. It is important to notice here the fundamental role played by the pore pressures generated during the initiation phase. Depending on the relative lengths of the time scales of propagation and consolidation, the movement, once initiated, can continue even at slopes much smaller than the effective friction angle. Indeed, the generated pore pressures can explain how submarine landslides can develop and propagate [5]. It has to be mentioned here that this mode of failure can be exhibited also by non-saturated soils such as those of volcanic origin. Indeed, collapse of the material under the loading induced by an earthquake can make the air pore pressure increase. The phenomenon is controlled by two characteristic time scales, a characteristic time for the consolidation and a characteristic time of loading. If the former is much larger than the latter, there will be not enough time for dissipation of air pore pressures, and the material will arrive to a dry liquefaction. Bishop [6] describes the failure of a fly ash tip at Jupille in Belgium and refers to the explanation provided by Calembert and Dantinne [7]. Some of the catastrophic landslides caused in El Salvador by the 13th of January 2001 earthquake can be explained by this mechanism. The term “air pore pressure” in dried soils is rather unusual in geomechanics, and the behaviour of dry materials is considered as drained under normal loading situations because of both air compressibility and small consolidation times. However, under special conditions such as those mentioned

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in the preceeding paragraph, the pressure of the interstitial air will cause the effective stress to change. Mining operations produce wastes which can be stored in several alternative ways. In the case of materials of very high water content, tailing dams can be constructed to impound the tailings. They are located close to the mines, and their height is continuously increased during mining operation as needed to impound more material. The soil is transported to the pond in pipelines or flumes, and discharged into it. Both the coarser fractions used to build the dam and the finer of the impounded material have very low relative densities and high water contents. In the case of waste dumps, the material is usually carried to the tip and then dumped from it, which results again in very low densities. Because of the combination of low densities and high water contents, failure of these poorly compacted materials is frequently induced by liquefaction. Once the structure fails, propagation of the fluidized material can be very fast, taking important tolls in human life, properties and environmental damage. We can mention here the cases of Aberfan in South Wales (Fig. 1, reported by Bishop [6], Anhui in China (Han & Wang [8]) and those of Rocky Mountains coal mine dumps described by Dawson et al. [9]. Rock avalanches are a case where rock blocks disaggregate into much smaller particles, and the final behaviour is that of a frictional fluid. Of course, both types are idealizations of reality. It is possible to The main objective of engineers and geologists is to minimize the risk. It is important therefore, to have a precise knowledge of both the triggering mechanism and the run out. In some occasions, it is possible to prevent the landslide— or to decrease the probabilities of its occurrence. In others, especially in large landslides, the run out has to be analyzed. The purpose of this paper is to present some mathematical, constitutive-rheological, and numerical models recently developed which can be applied to model the propagation of fast landslides. The paper is structured as follows. (i)

Section 2 is devoted to describe how Biot–Zienkiewicz models can be applied to derive depth integrated models accounting for pore pressure evolution inside the landslide mass. (ii) Sections 3 and 4 describe rheological models used for fluidized geomaterials, and shows how a link between rheological and constitutive models exists, based on Perzyna’s viscoplasticity. (iii) Section 5 deals with the discretization of the models described in previous Sections, which is done using SPH. (iv) Finally, Sect. 6 will show a set of validation tests and examples, including (a) problems for which an analyti-

Author's personal copy Depth Averaged Models for Fast Landslide Propagation

Fig. 1 Aberfan flowslide (1966) (National Archives, Kew, UK)

cal solution exists (ii) Small scale laboratory tests, and (iii) real events for which we have had access to reliable information.

2 Mathematical Model 2.1 Introduction Coupling between the solid skeleton and the pore fluids is of paramount importance on the behaviour of geomaterials and geostructures, and in particular on landslide triggering and propagation [10–13]. The first mathematical model describing the coupling between solid and fluid phases was proposed by Biot [14,15] for linear elastic materials. This work was followed by further development at Swansea University, where Zienkiewicz and coworkers [16–20], extended the theory to non-linear materials and large deformation problems. In fact, the first successful comparison of mathematical and physical models is due to him [19].

It is also worth mentioning the work of Lewis and Schrefler [21], Coussy [22] and de Boer [23,24]. In de Boer [24], the author presents applications to other materials such as metallic porous bodies and biological materials such as bone tissue. It can be concluded that the geotechnical community have incorporated coupled formulations to describe the behaviour of foundations and geostructures since the early 1980s. Indeed, analyses of earth dams, slope failures and landslide triggering mechanisms have been carried out using such techniques during last decades. The important coupling which exists between solid particles and pore fluids (air, water, etc.) appears also in other areas, in industrial processes and devices such as fluidized beds. The study of these problems has been done considering a mixture of several interacting phases, developing mass and linear momentum balance equations [25]. Mixture theories have been developed by Green and Adkin [26], Green [27] and Bowen [28]. This approach was also studied by Li and Zienkiewicz [29] and Schrefler [30]. The interested reader is referred to the texts by de Boer [31] and Zienkiewicz et al. [20]. Concerning the study of landslide propagation, coupled formulations arrived at a later stage. We can mention here the work of Hutchinson [32], who proposed a sliding consolidation model to predict run out of landslides, Iverson and Denlinger [33], Pastor et al. [34,35] and Quecedo et al. [36]. The effect of pore pressure has been described by Major and Iverson [37], who provided experimental data describing the evolution of basal pore water pressure in debris flows. Models including two fluids have been proposed recently by Pitman and Le [38] and Pudasaini [39]. These models, derived from mixture theory, can be applied to debris flow problems where the relative movement of soil particles and water can be important. The case of segregation has been studied by Gray and Thornton [40] and Gray and Ancey [41,42]. It is interesting to note that these modern two fluid models use the same equations presented by Zienkiewicz and Shiomi [17]. The purpose of this Section devoted to mathematical models which can be applied to landslide propagation analysis is to present a set of hierarchical models based on the Biot– Zienkiewicz equations. From the general model which allows important relative movements between fluid and solid phases, we will develop the u-pw model, and depth integrated models. 2.2 Biot–Zienkiewicz Model Fundamentals We will assume here that the landslide material consists of solid particles and a pore fluid (water or air) which fully saturates its pores, leaving out of this analysis the case of unsaturated soils.

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Concerning the solid phase, we will introduce the density of the particles ρs and characterize the structure by its porosity n (volume fraction of voids in the mixture) or the void ratio e which is related to porosity by n=

e 1+e

(1)

It is interesting to note that an alternative decomposition of the total Cauchy stress acting on the mixture is: σ = (1 − n)σs + nσw

where σ s and σ w are the stresses acting on the solid and fluid. If we define the partial stresses

Alternatively, a solid fraction can be defined as = 1 − n and represents the fraction of the volume occupied by solids. Fluid and solid densities will be denoted by ρw and ρs . We will introduce the concept of phase densities as the mass of components per unit volume of mixture, which are given by:

σ (s) = (1 − n)σs σ (w) = −npw I

ρ (s) = (1 − n) ρs ρ (w) = nρw

Moreover, the effective stress is obtained as (2)

We will define material derivatives following the solid

d (s) dt

and the fluid

d (w) dt

as:

∂ d (s) = + vsT .grad dt ∂t ∂ d (w) T = + vw .grad dt ∂t

(3)

where vs and vw are the velocities of solid and fluid particles in the mixture. Both derivatives are related by: d (s) d (w) = + (vw − vs )T .grad dt dt

(4)

In classical Soil Mechanics, the analysis is performed in a Lagrangian framework moving with the solid skeleton, and the velocity of the fluid is characterized by its Darcy’s velocity w, related to its real velocity by: vw = vs +

w n

(5)

from which we obtain: d (s) wT d (w) = + .grad dt dt n

We will further assume that the pore fluid has a viscosity which can be neglected, and the total stress tensor σ can be decomposed into an effective σ and a hydrostatic component pw I where I is the second order identity tensor and pw the pore water pressure: σ = σ − pw I where we have taken compressions as negative.

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(7)

(9)

we have: σ = σ (s) + σ (w)

σ = (1 − n) (σs + pw I)

(10)

(11)

These aspects are described in detail in the texts by Zienkiewicz et al. [20] and Lewis and Schrefler [21]. The balance of mass equations for both components are: d (s) ((1 − n) ρs ) + (1 − n) ρs div vs = 0 dt d (w) (nρw ) + nρw div vw = 0 dt which can be combined as: n d (w) ρw (1 − n) d (s) ρs + +div vs + div w = 0 ρs dt ρw dt

(12) (13)

(14)

If we introduce the volumetric stiffness for the solid grains and the pore water K s and K w , above equation can be written as: n d (w) pw (1 − n) d (s) pw + + div vs + div w = 0 Ks dt K w dt

(15)

Concerning balance of momentum equations, the form proposed by Zienkiewicz and Shiomi [17] is: d (w) vw = − n grad pw + nρw b + n Rw dt d (s) vs = div σ − (1 − n) grad pw (1 − n) ρs dt + (1 − n) ρs b + (1 − n) Rs = 0

nρw (6)

(8)

(16)

(17)

where b is the body forces vector; Rs and Rw characterize the interaction between phases: n Rw = − (1 − n) Rs = −R

(18)

For a Darcy flow, R is given by: −1 −1 R = n 2 kw w (vw − vs ) = nkw

(19)


where kw is the permeability tensor. Its component have dimensions [kw ] = L 3 T M −1 In the case of isotropy, is given by: kw =

k intr μw

(20)

where k intr is the intrinsic permeability with dimensions L 2 and μw is the viscosity of pore fluid. In geotechnical analysis, the permeability is usually defined in a slightly different way as: k¯w = kw gρw

(21)

where g is the acceleration of gravity. In this case, its dimensions are L T −1 . Other alternatives, such as that used by Pitman and Le [38] (see [25]), can be used when the relative velocity is larger: R=

n (1 − n) (ρs − ρw ) g (vw − vs ) VT n m

(22)

where VT is the terminal velocity of solid particles falling in the fluid; g the acceleration of gravity; m a constant. The equations describing the balance of mass and momentum for the mixture are, therefore (12), (13), (16) and (17), which coincide with those proposed by Pitman and Le [38] in their two-fluid model. In the case of debris flows, deformability of water and solid grains can be assumed to be very small, resulting on: div vs + div w = 0

(23)

They have to be complemented by suitable constitutive relations and kinematic equations relating velocities to rate of deformation tensors for both phases.

The model described by equations (12) (13) (16) and (17), can be simplified by assuming that (24)

which allows writing (15) as: 1 dpw + div vs + div w = 0 Q dt

(25)

or 1 dpw + (1 − n) div vs + n div vw = 0 Q dt

n 1 (1 − n) = + Q Ks Kw

(26)

(27)

The resulting model is cast in terms of vs , vw and pw , hence its name. In some cases [19], the relative velocity of the pore fluid can be eliminated. The resulting model is the much celebrated u − pw or vs − pw model which is found in most geotechnical finite element codes used today. The equations are obtained as follows: (i) Balance of linear momentum equations for the solid and pore fluid can be combined using (24), which results on: dv = div σ − grad pw + ρb (28) dt where ρ is the mixture density and v the velocity of the solid skeleton. ρ

(ii) From the balance of momentum for the pore fluid, the Darcy’s relative velocity w is obtained as: dv + ρw kw b − kw grad pw (29) dt Next, it is substituted into the balance of mass of the pore fluid, which gives: w = −kw ρw

1 dpw dv +div vs +div −kw ρw + ρw kw b−kw grad pw = 0 Q dt dt

(30) from where we arrive to: 1 dpw + div vs − div (kw grad pw ) = 0 Q dt

2.3 The vs − vw − pw and vs − pw Biot-Zienkiewicz Models

d (w) d (s) d ≈ = dt dt dt

where we have introduced

(31)

where we have neglected [17] the term div(−ρw b + ρw (dv/dt))because body forces will not depend in general on space coordinates (except in centrifuge tests) and the space derivatives of accelerations are assumed to be small. In above equation, divvs is the volumetric part of the rate of deformation tensor: dε (32) div vs = dv = tr (d) = tr dt In general, deformation of soil skeleton is described by a suitable constitutive equation, but in some cases, we will have to add an extra component dv0 which is not included in it. For instance, volume changes caused by breaking of soil grains, thermal effects, or dilatancy caused by high rates of shear

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deformation. This idea was used in the past to enrich classical plasticity models with volumetric deformations caused by cyclic loading [20]. Equation (30) is written as: 1 dpw dεv + + dv0 − div (kw grad pw ) = 0 Q dt dt

(33)

where we have introduced dv0 , the volumetric component of the rate of deformation tensor describing the extra volumetric deformation and the mixed volumetric stiffness of grains and fluid. This model, proposed by Zienkiewicz and Shiomi [17] for the case of saturated soils is referred to as u − pw , and consists of equations (28) and (31), complemented by a constitutive relation for the soil and a kinematic relation linking the velocities to the rate of deformation tensor. The model has been extended to unsaturated soils in Zienkiewicz et al. [18], being worth mentioning the work of Lewis and Schrefler [21] in our book. The original model was developed by solids which were saturated with water. The authors have applied the model for dry soils, with voids filled with air, to model the flowslide of Las Colinas caused by the first of 2001 earthquakes. Collapse of a metastable soil caused the increase of air pressures, resulting on what it can be called dry liquefaction [4,34,43]. 2.4 Two-Fluid vs − vw − pw Depth Averaged Models We have described in previous Section models which describe, with different degrees of approximation, the coupled behaviour of mixtures of solid grains and one pore fluid, which can be water or air. Both vs − vw − pw and vs − pw models are used in geotechnical engineering, the former in cases where relative velocities between solid grains and fluid cannot be neglected. The more general equations of balance of mass (12) (13) and linear momentum (16) (17) can both be used in landslide triggering and propagation phases. Concerning propagation, the computational cost of solving numerically the equations is very high, especially in cases where the landslide propagates over long distances. Depth integrated models are a convenient simplification of 3D models, providing an acceptable compromise between computational cost and accuracy. They have been extensively used in the fields of coastal, harbour, oceanographic and hydraulics engineering since the work of Barré de Saint Venant in 1871 [44]. In the case of avalanche dynamics, Savage and Hutter [45, 46] proposed their much celebrated 1D lagrangian model, where a simple Mohr–Coulomb model allowed a description of the granular material behavior. This work was extended to 2D and more complex terrains in Hutter and Koch [47],

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Fig. 2 Reference system and notation used in the analysis

Hutter et al. [48], Gray et al. [49]. It has been applied by Laigle and Coussot [50], Mc Dougall and Hungr [51], Pastor et al. [34,35] and Quecedo et al. [36]. Concerning limitations of the model, the interested reader will find in Hutter et al. [52] a detailed discussion, being worth mentioning the text book by Pudasaini and Hutter [39]. In many cases there is an important coupling of pore water and air with the solid grains. As soil skeleton dilates (or contracts), pore pressures change, and so do effective stresses. In consequence, basal friction and mobility of the soil mass will be much affected. The first models addressing this issue are those of Hutchinson [32], who proposed a simple slidingconsolidation mechanism for a block, Iverson [53], Iverson and Denlinger [33]. A more general approach was proposed by Wang and Hutter [52], based on mixture theory. We will use the reference system given in Fig. 2 where we have depicted some magnitudes of interest which will be used in this section. Depth averaged models are obtained by integration along depth of the balance of mass and momentum equations described in previous Section. It is therefore possible to propose alternative depth integrated models of varying complexity from the three main types of equations above described. We will start with equations of balance of mass (12), (13) and linear momentum (16), (17), which describe two interacting phases with significant relative velocities. Their integration along depth has been done by Pitman and Le [38] in their two-fluid model. Depth integration results on the following set of equations:

(i) from balance of mass for the mixture and the solid, we obtain:

¯ dh + h div v¯ = 0 dt d¯ (s) ¯ h) + (1 − n) ¯ h div v¯s = 0 ((1 − n) dt

(34) (35)


In above equations an over bar over a magnitude indicates its depth averaged value. For instance: Z +h 1 ¯θ = θ d x3 (36) h Z

We have defined v¯ as v¯ = (1 − n) ¯ v¯s + n¯ v¯w

The general two-fluid depth integrated equations described above can be simplified if we assume, like in the vs − pw (w) (s) d model, that ddt ≈ ddt = dt and neglect the acceleration of the fluid relative to the solid. Equation (34) results on:

(37)

and the “quasi material derivative” as: d¯ ∂ ∂ = + v¯ j j = 1, 2 dt ∂t ∂x j

dh + h div v = 0 dt (38)

Notice that in (34), the velocity is v¯ while in (35) the velocity is v¯s . For convenience, from now on, we will drop the over bar, all magnitudes being depth integrated unless otherwise stated. (ii) from balance of momentum for solid and fluid phases, we obtain: d (s) 1 vs = (ρs − ρw ) grad (1 − n) h 2 b3 ρs h (1 − n) dt 2 1 + ρw (1 − n) grad h 2 b3 2 + ρs (1 − n) hb3 grad Z + τb + g (ρs − ρw )

hn(1 − n) (vw − vs ) VT n m (39)

and ρw hn

2.5 Biot–Zienkiewicz vs − pw Depth Averaged Model

d (w) 1 vw = ρw grad nh 2 b3 dt 2 + ρw n h b3 grad Z hn(1 − n) − g (ρs − ρw ) (vw − vs ) VT n m

(40)

where b3 is the component of the body forces vector along X 3 and pwb , τb are the excess pore pressure and shear stress at the bottom. We have assumed that interaction between solid particles end the fluid are given by (22). Concerning stress components for the solid, we have assumed a hydrostatic distribution, the more general case of active-passive Mohr–Coulomb stresses proposed by Hutter [48] being described in Pitman and Le [38]. Regarding the fluid, during depth integration we have assumed a hydrostatic state as done for solids. These four equations describe the evolution of height h, porosity n and velocities of solid and fluid. The pore pressure can be obtained from the balance of momentum equation for the fluid. In the case of Darcy’s flow, the relation is given in (30)

(41)

Balance of linear momentum equations yield, in a similar manner: ρh

1 dv = ρ grad h 2 b3 +ρhb3 grad Z +τb +ρbh (42) dt 2

In the case that the sliding mass erodes the basal surface, the equations are: dh + h div v = e R (43) dt dv 1 = ρ grad h 2 b3 +ρhb3 grad Z +τb − ρe R v¯ ρh dt 2 (44) where e R is the erosion rate [L T −1 ] The basal shear stress depends on the rheological law used. In the case of a pure frictional material, it is given by {(ρs − ρw ) (1 − n) hb3 − pwb } vvss tan φb where φb is the friction angle. 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 v¯ 2 /R, where v¯ is the modulus of the averaged velocity and R the main radius of curvature in the direction of the flow. Concerning the pore pressure distribution, we can use equation (33), and integrate it along depth, but a more accurate approximation will be presented in following Section. 2.6 Pore Pressure Evolution in Landslides Most simplifications done for landslide propagation come from their geometry (see [33,46,47]). We will start by defining L, the length of the landslide (larger than its width) and H, which characterizes its depth. We will assume that the ratio ε = H/L is much smaller than 1. It follows that equation (33) can be written as ∂ 2 pw dεv + dv0 − kw =0 dt ∂ x3

(45)

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In above equation, the material derivative is given by: ∂ ∂ d = + vk k = 1..3 dt ∂t ∂ xk

(46)

It can be approximated using depth integrated components of the velocity vector as: ∂ ∂ ∂ ∂ d ≈ + v¯1 + v¯2 + v3 dt ∂t ∂ x1 ∂ x2 ∂ x3

(47)

with an error given by (v1 − v¯1 ) ∂∂x1 + (v2 − v¯2 ) ∂∂x2 From now on, and for the sake of simplicity we will drop the over bar in the velocities when writing the Eulerian time derivative. v Next, we will assume that we can relate dε dt to the rate of variation of the effective confining pressure p as: dεv 1 dp =− dt K v dt

(48)

where K v is a suitable stiffness module and p is 1 p = − tr σ 3

(49)

1−ν (1 + ν) (1 − 2ν)

(51)

where we have added the term v¯ 2 /R which accounts for centrifugal accelerations—assumed to be independent of x3 — where v¯ is the modulus of the averaged velocity and R the main radius of curvature in the direction of the flow. The total stress σ3 depends on h, and varies with it as: dσ3 = ρb3 dt

dh d x3 − dt dt

= ρb3

x3 dh 1− dt h

(52)

which has been obtained using

where E is the elastic modulus and ν Poisson’s ratio. In a general case, the constitutive equation will provide its value. From here, and taking into account that p = p + pw p = − 13 tr σ , we arrive to ∂2 p

dpw dp w =− − K v dv0 + kw K v dt dt ∂ x32

(50)

which is the equation describing pore pressure changes inside the landslide. In order to solve this equation we will consider the landslide mass decomposed into differential elements of volume having, at a given time t, a height h and a cross section dA (Fig. 3). The pore pressure evolution equation will be solved in the deformed configuration, where we will use an updated Lagrangian approach—the reference configuration being that at time t. In the case of landslide propagation, the main responsible of changes in the total mean confining pressure p will be

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the height variations. For the differential volume element sketched in Fig. 3, we will have: σ3 = ρ (h − x3 ) (b3 − v¯ 2 /R)

If the skeleton is elastic, K v is the elastic volumetric stiffness ratio. Note that p is not the effective vertical stress except when the state of stress is isotropic. Another case of interest is that of an elastic material with zero lateral deformation 1+ν E m , the oedo(oedometric conditions), where K v = 3(1−ν) metric modulus E m being given by: Em = E

Fig. 3 Deformation of a soil column

h H Concerning effective stress, we can write: x3 = X 3

dσ3 x3 dpw dh = ρb3 1− + dt dt h dt

(53)

Next, we will have to relate increments of σ and p . When an increment of effective stress σ3 is applied to a soil element, its mean effective confining stress will change. The relation between both increments can be obtained by using a suitable constitutive equation describing its behavior. We will further assume that we have obtained in this way the relation dp = αdσ3 . From here, we arrive to: x3 K v dh K v ∂ 2 pw dpw 1− − (54) = −ρb3 dv0 + kw dt dt h α α ∂ x32 which is the equation describing the evolution of pore pressure along x3 .


Concerning the constitutive coefficient α, we will assume elastic behaviour and an oedometric state of deformation, where Kv = Em α The PDE has to be complemented by (i) An initial condition (ii) Boundary conditions at the surface and the bottom (for instance, zero at the surface and zero flow at the bottom). The Biot–Zienkiewicz vs − pw depth integrated model we propose, consist, therefore on: (i) The depth integrated equations dh + h div v = e R dt dv 1 ρ = ρ grad h 2 b3 + ρhb3 grad Z dt 2 vs + {(ρs − ρw ) (1 − n) hb3 − pwb } vs tan φb + ρbh − e R v (ii) The pore pressure evolution equation x3 K v dpw dh K v ∂ 2 pw = −ρb3 1− − dv0 + kw dt dt h α α ∂ x32 The former will be discretized—as shown in Sect. 4—with SPH, while the latter will be discretized using a set of finite difference meshes associated to each SPH node.

3 Rheology and Constitutive Models for Fluidized Soils Tout solide est un fluide qui s’ignore. (J. Lemaitre and J.-L. Chaboche, 1990) 3.1 Introduction We will discuss in this Section the behaviour of fluidized geomaterials in fast landslides, such as rock avalanches, debris flows, lahars, and flow slides. Once failure has been triggered, the behaviour of the soil mass is closer to that of fluids than to solids. It can surround obstacles and fill cavities, for instance. This is why modellers have used rheological models to describe its behaviour, using properties such as yield stress, viscosity and friction. There are many types of materials involved in fast landslides, from assemblies of rock blocks to mixtures of clay and water. Their properties will depend both on those of the solid fraction and those of the fluid. In the case of mud, the Bingham model is a reasonable choice, as it includes two material parameters, the yield stress

below which the material does not flow, and the viscosity. Bingham [54] proposed his model in 1922, and later Hohenemser and Prager [55] and Oldroyd [56] generalized Bingham model to general 3D conditions [57]. Since then, more refined models have been proposed, such as those of Coussot [58–60], Dent and Lang [61], Locat [62] and Chen [63] and Chen and Ling [64], who proposed a general 3D framework for modelling of viscoplastic cohesivefrictional fluids. Geomaterials present, at low strain rates, a series of characteristic properties: (i) Residual states after failure lie at a line in the plane p ,q

1 p = − tr (σ )q = 3J2 3

where e is the void ratio. This line is referred to as Critical State Line [65]. (ii) Failure can occur in some cases inside the Mohr Coulomb failure surface. These cases correspond, in general, to what is known as diffuse failure [3,4]. (iii) Geomaterials present a change of volume (dilation or compaction) while being sheared (dilatancy) (iv) There exists an important coupling between solid skeleton and pore fluids. At high strain rates, we could think that, once the residual state has been reached, the material will flow at states located on the Critical State Line. Consequently, at constant confining pressure, the void ratio would be constant and the fluid would flow at constant volume. However, experimental evidence which we will comment later, suggest that the Critical State Line changes, the void ratio changing with the rate of strain. As a consequence of (iii) and (iv), during both the triggering and the propagation phases of a landslide, pore pressures and, in consequence, effective stresses change. In some cases, the material tendency to compact results on an increase of pore pressure and mobility of the landslide, while in others, dilation cause pore pressures to decrease and the landslide slows down. The effect of pore pressures on landslides has been described by Okada et al. [10], Iverson [11], Pailha et al. [12] and Igwe et al. [13] among others. Dilatancy and pore pressure generation are well described by the constitutive equations used in classical geotechnical engineering. Indeed, in classical plasticity, dilatancy comes directly from the direction of plastic flow, one main ingredient of the constitutive equation. Therefore, 3D codes implementing suitable constitutive equations can model both dilatancy and pore pressure variations in a consistent way.

123


However, because of the computational cost of full 3D models, researchers have favoured the use of simpler depth integrated models described in previous Section, where the flow structure is lost and the basal friction is obtained from the depth averaged velocity. Other properties such as dilatancy are averaged too, and pore water pressure is described by simple pre-established shape functions. Most depth integrated models implementing dilatancy include an extra term accounting for it, which is used to approximate the basal pore pressure. This is the approach used by Iverson [66], Jop et al. [67], Forterre and Pouliquen [68], Pailha and Pouliquen [69], and Pastor et al. [43]. The structure of the model consists, therefore, on two parts describing the shear behaviour and the dilatancy. Bodies of granular particles, when sheared, present two other important problems: particle crushing and inverse grading. Therefore, additional compaction mechanisms have to be included in the rheological model [10,13,70,71]. Concerning inverse grading, two phase models [40,41,72] provide a suitable way to model it. The purpose of this Section is to describe classical rheological models used to study the propagation of landslides when using depth integrated models, together with the enrichment techniques which provide approximations for dilatancy. More specifically, we will show how, from general 3D rheological models, simple shear infinite landslide models provide the basal friction required, once the depth averaged velocity is known. 3.2 General 3D Rheological Models Most rheological models provide laws expressing the stress as a function of the rate of deformation tensor d = gradsym v di j =

1 2

∂v j ∂vi + ∂x j ∂ xi

Fig. 4 Simple shear flow

The invariants are defined as:

v1 = v1 (x3 ) v2 = v3 = 0

(59)

The rate of deformation tensor is:

(55)

(56)

where p¯ is a “thermodynamic” pressure, I the identity tensor, k = 0..2 scalar d the rate of deformation tensor, and k functions of the invariants of d: k = k (I1d , I2d , I3d )

(58)

In above expressions, we have chosen compressions to be negative. In order to illustrate the roles of each term, we will consider the simple shear flow illustrated in Fig. 4, where the fluid is sheared on a plane X 1 X 3 , the velocity being parallel to X 1 (Fig. 4):

⎛

0 ⎜ d = ⎝0

1 ∂v1 2 ∂ x3

The first assumption done is isotropy of the flowing material, which allows expressing the stress as σ = − p¯ I + 0 I + 1 d + 2 d 2

1 k tr d k

Ikd =

⎞ 0 21 ∂∂vx13 ⎟ 0 0 ⎠ 0 0

(60)

from where d 2 is given by: ⎛ 1 4

⎜ d2 = ⎜ ⎝0 0

∂v1 ∂ x3

2

⎞ 0 0 0 0 1 4

0

∂v1 ∂ x3

⎟ ⎟ 2 ⎠

(61)

(57) The second invariant I2d is:

Expression (56) is obtained using the representation theorems, which express a general isotropic tensor function of a tensor variable [57,73].

123

I2d

1 1 = tr d 2 = 2 4

∂v1 ∂ x3

2 (62)


The stress tensor is given by: ⎛ 0 ⎜ σ = − p¯ I + 0 I + 1 ⎝ 0 ⎛ 1 4

⎜ + 2 ⎜ ⎝0 0

∂v1 ∂ x3

2

0 21 ∂∂vx13 ⎟ 0 0 ⎠ 0 0 ⎞

1 ∂v1 2 ∂ x3

0 0 0 0 0

1 4

where γ˙ is the shear strain rate in simple shear conditions, d ∗ a representative grain diameter, p the confining pressure and ρ the bulk density. Alternatively, in the case of viscous porous fluids with viscosity μ I N is given by:

⎞

∂v1 ∂ x3

⎟ ⎟ 2 ⎠

IN = (63)

from where we obtain the components of the stress tensor immediately as: σ11 = σ33 = − p¯ + 0 + 41 2 σ22 = − p¯ + 0 σ13 = σ31 = 21 1 ∂∂vx13

∂v1 ∂ x3

2 (64)

It is important to notice that, in addition of having shear stresses which depend on the rate of shear strain ∂∂vx13 we find that the normal stresses σ11 and σ33 depend also on it. This contribution is often referred to as “dispersive stresses”. We will further assume: (i) The flow is isochoric, i.e. tr (d) = 0 (ii) As the dependence on the third invariant is difficult to obtain from experiments, it will not be taken into account. (In the case of frictional fluids, where material behaviour strongly depends on Lode’s angle, this assumption introduces a limitation) (iii) The term in d 2 provides in simple shear flows as we have seen it above, dispersive normal stresses which depend on shear strain rate which agrees with classical Bagnold [74] model. (iv) In the case 2 is neglected, it is possible to show that 1 does not depend on I3d [160], hence 1 = 1 (I2d ) and

σ = − p I + 1 (I2d ) d

(65)

μγ˙ P

(67)

Note that (I N ) provides an equilibrium volume fraction for any given I N . Once reached, the flow will be isochoric. The interpretation proposed by Pailha and Pouliquen [69] and Pastor et al. [35] is based on the idea of a transition towards a dynamical Critical State Line. George and Iverson [76] proposed a simple alternative law based on the former. Once tr(d) is accepted to be different from zero, the dispersive stresses are not anymore necessary to explain Bagnold’s experiments where dispersive stresses had been observed. So far, few models present simulations based on full 3D rheological models. In the case of Bingham fluids, there exist computational problems unless regularization of the law is introduced [77,78]. Forterre and Pouliquen [68] propose an extension to 3D of the simple shear model as: σ = − p I + τ τ = η dev (d) η =

μ (I ) p |dev (d)|

(68)

where μ (I ) is a viscosity dependent on the inertia number. Here, following Chen [63], we propose the law [35] σ = − p I + 1 dev (d)

(69)

with 1 =

m−1 s + 2μC F (4I2d ) 2 √ I2d

(70)

where s characterizes the cohesive-frictional strength of soil, μC F is a viscosity-like coefficient, and m a material parameter. In simple shear conditions, above expression particularizes to σ11 = σ22 = σ33 = − p

σ13 = σ31 = s + μC F

∂v1 ∂ x3

m

(71)

Assumption (i) is important. At first sight, we can conclude that it is logical, because soil is flowing at constant volume as it is at the Critical State Line. However, this is not exact. It is well known from experiments [75] that flowing granular materials present dilatancy, and therefore, tr(d) = 0 The observed dilatancy can be interpreted as being generated in the process of reaching a dynamic equilibrium Critical State Line. Pouliquen and coworkers [67,75] have proposed that volume fraction ϕ = ϕ (I N ) depends on the inertia number IN :

It can be seen that, if friction is zero and m = 1, we obtain Bingham model, while m = 1 leads to a Herschel Bulkley fluid. In the case of cohesionless frictional fluids, viscosity regularizes the problem, being possible to obtain velocity profiles. Finally, suitable dilatancy laws have to be added.

γ˙ d ∗ IN = √ p/ρ

1D simple shear infinite landslide models are a special case of simple shear flows where (i) flow is steady, and (ii) all mag-

(66)

3.3 Simple Shear Infinite Landslide Model

123


Fig. 5 The simple shear, infinite landslide model Fig. 6 Flow structure of a Bingham fluid in an infinite simple shear flow

nitudes are independent on the position along the landslide, which is assumed to have an infinite length (Fig. 4). In depth averaged models they are used to obtain both the velocity profile and the shear stress at the base for a given depth averaged velocity v. ¯ Indeed, when deriving depth integrated equations we have lost the flow structure along the vertical, which is needed to obtain both the basal friction. A possible solution which is widely used consist of assuming that the flow at a given point and time, 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 some necessary ingredients of depth integrated models. Referring to Fig. 5, shear stresses vary linearly with depth as: z (72) τ (z) = τb 1 − h where the basal shear stress is given by τb = ρgh sin θ and ρ is the density of the mixture. For a fluid saturated geomaterial, the component of σ3 is: σ3 = −ρd g(h − z) cos θ

(73)

where ρd = (1 − n) ρs = (1 − n) (ρs − ρw ) = ρ − ρw The effective stress is given by: σ3 = −ρd g(h − z) cos θ + pw

(74)

If we assume that the excess pore pressure in the infinite landslide can be approached by (note that, in the case of liquefaction, βw = 1): pw = βρd g(h − z) cos θ

123

we would obtain σ3 = − (1 − βw ) ρd g(h − z) cos θ

(75)

Finally, the shear strength will be given by s(z) = c + (1 − βw ) ρd g(h − z) cos θ tan ϕ

(76)

where c is the cohesion. Once we know both the strength and the shear stress along depth, we can obtain the velocity profile and the basal shear stress for a given rheological law. We will consider next some cases commonly used, which fall in the class described by (63)–(65). 3.3.1 Pure Cohesive Viscoplastic Fluid: Bingham Model The Bingham model is a particular case of (65) with s = τ y (which is the yield stress) and m = 1 τ = τy + μ

∂v ∂z

(77)

Depending on the fluid phase viscosity, mudflows, lahars and debris flows can be modelled as viscoplastic fluids, the most frequently used models being Bingham and HerschelBulkley (for a review, the reader is addressed to the classical text by Coussot [60]; see also Laigle and Coussot [50]). The flow structure, in the general case, will consist of two separate parts, which have been depicted in Fig. 6. In the upper part, from points S to P, the mobilized shear stress is smaller than τ y . Therefore ∂v ∂z = 0 and the velocity is constant in this region, which is referred to as a “plug”. From P to B, the mobilized shear stress is larger than τ y , and a shear zone


develops, with:

is zero and using

1 ∂v = τ − τy ∂z μ

(78)

It can be seen that, for a given slope, the height of the plug will depend on the yield stress τ y . There is a limit height below which the flow will freeze. Therefore, if we think of a Bingham fluid flowing in a channel, it will stop at the moment when the height is smaller than

τy ρg sin θ

. If more material

arrives later, the height will be larger than the critical value and the flow will start moving again. This phenomenon has been observed in Chinese rivers. After integration of (69), we arrive to the expression relating the averaged velocity to the basal friction for the infinite landslide problem: τB h v¯ = 6μ

τY 1− τB

2 τY 2+ τB

(79)

where τ B the shear stress on the bottom. This expression can be transformed into P3 (η) := η3 − (3 + a) η + 2 = 0

(80)

where we have introduced η = h P / h 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 6μv¯ a= hτY

(81)

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. [79] 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 3 P2 (η) = η2 − 2

65 57 +a η+ 16 32

σ13 = σ31 = s + μ F

∂v1 ∂ x3

m (83)

we obtain μF

∂v1 ∂ x3

m = τ (z) − s (z)

(84)

where τ (z) and s(z) are given by (72) and (73), μ F characterizes the viscous behaviour of the frictional fluid and m is a material parameter. From here we obtain: (i) the velocity profile as: z 1+m m v = vh 1 − 1 − h z 1+m m v = vh 1 − 1 − h

(85) (86)

where vh is the velocity at the surface (z = h), and the depth averaged velocity: v¯ = vh

1+m 1 + 2m

(87)

(ii) the basal shear stress τb = sb +

1 + 2m m

m

1 μ F v¯ m hm

(88)

where sb = s(0) = (1 − βw ) ρd gh cos θ tan ϕ Above expressions particularize for m = 2 to: v¯ = 3/5vh τb = (1 − βw ) ρd g(h − z) cos θ tan ϕ +

25 1 μ F v¯ 2 4 h2

(89)

It is interesting to note the similarity with Voellmy’s law [86]: (82)

Knowing the non dimensional number a, the root is obtained immediately. Cohesive viscoplastic fluid models have been successfully used to model both laboratory scale tests and real events [34,50,79–85]. 3.3.2 Pure Frictional Viscoplastic Fluid Frictional viscoplastic fluids are used to model fast landslides where the friction is important. If we assume that cohesion

v¯ 2 τb = (1 − βw ) ρd g(h − z) cos θ tan ϕ + ρg ς

(90)

The difference with the proposed model consists on the Voellmy coefficient ς being dependent on h. In two dimensional applications, the basal friction along X i is given by: v¯i 25 1 2 τbi = − (1−βw ) ρd g(h −z) cos θ tan ϕ + μ F v¯ 2 |v| ¯ 4 h (91)

123


If the viscosity coefficient μ F is zero, the velocity profile cannot be obtained unless an additional assumption is done. In this sense, viscosity regularizes the problem. Basal friction is given by: τb = (1 − βw ) ρd g(h − z) cos θ tan ϕ

(92)

All the cases we have considered so far correspond to landslides which are homogeneous along depth. In some cases, there may be a saturated basal layer of width h s = αh where the shear deformation concentrates. In the case of a pure frictional fluid we will have: −σ3b = ρgh (1 − α) cos θ

cos θ

= ρgh 1 − α ρρw cos θ = ρgh Aw cos θ

(93)

√ 3 3 J3 1 θ = − arcsin 3 2 J 3/2 2

(94)

from where we obtain sb = Aw (1 − βw ) ρgh cos θ tan ϕ

(95)

sb = Aw (1 − βw ) ρgh cos θ tan ϕ

(96)

with sb = s(z = 0) If the width of the saturated shear layer is very small, α being close to zero and Aw being close to 1, it results on: (97)

In the case of a viscoplastic frictional fluid, if we assume that shear deformation concentrates on the saturated layer, we obtain: 3/α 2 v¯ 2 25 b b (98) τ = s + μF 2 4 h 5 − 2α where we have assumed that the rheological behaviour is 3/α given by (84) with m = 2. The factor 5−2α accounts for the relation between the averaged velocities in the shear layer v¯s and in the whole vertical profile: v¯s2 = v¯ 2

123

3 5 − 2α

6 sin φ 3 − sin φ sin 3θ

(101)

In above equation, the Lode’s angle θ is given by

σ3b denoting the stress at the bottom along X 3 . Concerning the increment of excess pore pressure at the b, bottom pw

s b = (1 − βw ) ρgh cos θ tan ϕ

(100)

where is the void ratio at p = 1; λ is a material parameter characterizing the slope of the CSL on (e, ln p ); M(θ ) is related to friction angle at residual conditions by: M (θ ) =

where we have introduce ρw , 0 ≤ Aw ≤ 1 Aw = 1 − α ρ

pb = βw ρgh Aw cos θ

In classical soil mechanics, residual conditions are defined as states where the soil is sheared at constant effective stresses, and it is assumed that residual conditions take place on the Critical state Line (CSL) [65], defined by: e = − λ ln p q = M (θ ) p

3.3.3 The Case of a Basal Saturated Layer

+ ρd gh s

3.4 A Note on Dilatancy

2 (99)

(102)

with J3 = 13 tr s 3 At residual conditions, where pq = M (θ ) the soil shears at constant volume, and dilatancy, defined as the ratio between p v volumetric and shear rates of plastic strain dg = dε p is zero dεs Constitutive equations provide suitable expressions for dilatancy. One simple example is obtained by assuming that dilatancy varies linearly with the distance to the CSL as: p

dg =

dεv p = (1 + α) (M (θ ) − η) dεs

(103)

with η = pq being the stress ratio. Under small rates of shear strain, soil dilatancy is not constant when the soil is sheared, as loose soils tend to compact while dense soils dilate. This fact has been considered to play an important role in triggering of landslides and early stages of propagation [66,69]. Simple plasticity models such as those using a Mohr– Coulomb yield surface will predict dilation during plastic shearing of the soil unless a non-associated flow rule is used. Indeed, at residual conditions dilatancy should be zero. Unfortunately, the implementation of dilatancy rules in depth integrated models present the difficulty of introducing a dilatancy angle which should not be constant, as it depends both on the stress state and the material history. In the case of landslides, dilatancy laws have to be modified when soil crushing occurs or when the soil flows at shear strain rates much higher than those used in classical soil mechanics tests. Concerning the latter, laboratory tests performed in rheometers have shown that:


From above ideas, an alternative could be: eC S L ,dyn = eC S L + β1 I N ε˙ v= − β2 eC S L ,dyn − e γ˙

(107)

Once we have the dilatancy law, it can be implemented in (50) taking dv0 = ε˙ v If the depth integrated model does not include a description of pore pressure variation along depth, the value to be used is either a depth integrated approximation of the dilatancy or its value at the base. In both cases, the accuracy will be smaller.

Fig. 7 Interpretation of constant p and constant volume rheometer tests

(i) If a granular material is sheared at constant confining pressure, it will dilate when shear strain rate increases [87] (ii) If shearing is done at constant volume, the pressure will increase with the shear strain rate [74]

4 Perzyna’s Viscoplasticity Based Rheological Models 4.1 Introduction There exists an interesting similitude between viscoplastic fluid rheological models of the type τ =s+μ

∂v ∂z

m (108)

More elaborated experiments performed by the GDR MiDi [75] have provided considerable insight in the behaviour of granular fluids. Experimental results on several types of rheometers [68, 69] show that volume fraction decreases linearly with the inertia number as:

from where the rate of shear strain can be written as ∂v 1 = 1/m (τ − s)1/m (109) ∂z μ

eq = max + (min − max ) I N

and Perzyna elasto-viscoplastic models, where the relation between the effective stress and the rate of deformation tensor is given by

(104)

which suggests the existence of an unique volume fraction (porosity or void ratio) for a given inertia number I N . A variant of this law has been used by George and Iverson [76] The authors (Pastor et al. [35]) have proposed the law: eC S L ,dyn = eC S L + β1 (I2d )

(105)

where β1 is a material parameter, eC S L the void ratio at CSL (at the same p’) and eC S L ,dyn the void ratio at the dynamic CSL corresponding to I2d Concerning dilatancy, Roux and Radjai [88] and Pailha and Pouliquen [69] propose laws of the type:

1 d = k3 − eq γ˙ dt

(106)

which describes the evolution towards the dynamic CSL. Alternatively, Pastor et al. [35] proposed dg = −β2 eC S L ,dyn −e where β2 is a material parameter. Figure 7 illuseC S L trates the idea, interpreting both constant pressure and constant volume rheometer tests.

σ = D e : d − d vp

(110)

Above, D e is the elastic constitutive tensor, d the rate of deformation tensor, and d vp its viscoplastic component. We have assumed an additive decomposition of the rate of deformation tensor into elastic and viscoplastic components. The viscoplastic component of the rate of deformation tensor is given by Perzyna [89,90]. d vp = γ n g φ (F)

(111)

In above equation, the symbol . . . represents the Macaulay φ = φ if φ ≥ 0 brackets: ; γ is the fluidity parameter; = 0 otherwise n g is a unit norm tensor characterizing the direction of the plastic flow; φ (F) is an arbitrary function There are several alternatives for φ (F), one of which is φ (F) =

F − F0 F0

N (112)

123


where N is a model parameter and F a function describing a convex surface in the stress space. The value F0 characterizes the stress below which no viscoplastic flow occurs. Considering the 1D shear case, if φ (F) is chosen as φ (F) = (F − F0 ) N , F = τ and F0 = s we will have:

∂v ∂z

vp

= γ (τ − s) N

(113)

which will coincide with (108) when the elastic part of the viscoplastic strain rate is negligible and γ =

1

We can ask ourselves if Perzyna’s viscoplastic models will provide similar results than those obtained with the rheological laws described in Sect. 3 when applied to infinite landslide problems. If so, it will be possible to reproduce with a single model both the triggering and propagation of the landslide in a most simple manner. To explore this possibility, we will choose three different models and study properties of the landslide such as velocity profile and stress distribution. The models will be a classical viscoplasticity model of Von Mises type, a Mohr–Coulomb and a more general Cam Clay model. In the case of Von Mises yield criterion, it depends only on the second invariant of the deviatoric stress tensor, and is written as: f =q −Y =0

(114) √ where q is related to J2 as q = 3J2 and Y is a measure of the material cohesion. F will be chosen as F = q and the initial size of the yield surface will be given by F0 = Y0 The size of the yield surface will vary according to a suitable hardening/softening law. Here we will assume that the size of the yield surface will be proportional to the increase of the equivalent deviatoric plastic strain, ε¯ vp (115)

We will also assume an associated flow

rule, which means = 0 coincides with σ that the plastic potential surface g

the yield surface F σ , internal variables = 0 and in consequence: ∂g g=q and m = ∂σ Alternatively, the Cam Clay model uses [91]:

q 2 + M 2 p p − pc = 0

(116)

(117)

where p is the hydrostatic pressure defined in equation, q is the deviatoric stress, and M is the slope of the Critical State

123

M=

6 sin φ 3 − sin φ sin 3θ

(118)

and pc is a hardening parameter characterizing the size of the ellipsoid. The hardening/softening rule is given by the relation between the size of the yield surface pc and the viscoplastic vp volumetric strain,εv : dpc vp = dεv

N = 1/m μ1/m

dY0 =H d ε¯ vp

Line d in the p - q plane, related to the friction angle ϕ and the Lode’s angle θ by

1+e λ−κ

pc

(119)

where e is the void ratio of the material, λ is the slope of the ‘Normal Consolidation line’, which is the line observed in the ln ( p) − e plot during a hydrostatic compression test, and κ is the slope of the line observed in the ln ( p) − e plot at the beginning of the unloading process. Finally, Mohr Coulomb yield surface is given by F = q − Y Y = M (θ ) p 4.2 3D Infinite Landslide Modelling In order to compare Perzyna’s to the viscoplastic fluid rheological models described in Sect. 3, we will use a 3D infinite landslide model, which differs from the Simple Shear infinite landslide model in: (i) A full 3D constitutive equation is used to describe material behaviour (ii) Transient behaviour can be studied. 3D infinite landslide models have been applied by Di Prisco and Pisanò [92], Di Prisco et al. [93] and Pisanò and Pastor [94] to study stability and failure of slopes under static and dynamic loading. We will first consider a slope 1:4 on a Von Mises material, with the following properties: Elastic modulus: E = 8.e7 Pa Poisson ratio ν = 0.3 Yield stress σ y = 0.285e5 Pa Perzyna law: γ = 0.1 s−1 N = 1 The results obtained with the Von Mises model are plotted in Figs. 8 and 9. We have used a mixed finite element formulation in effective stresses and velocities, with the TaylorGalerkin algorithm introduced by and Löhner et al. [95], Donea et al. [96] for fluid dynamics and applied to solids dynamics by Mabssout and Pastor [97] and di Prisco et al. [93].


(down slope) and Y coincide within the shear zone, differing within the plug. Figure 10 compares the velocity profiles obtained with a Bingham model—for which the analytical solution is known—with the predictions of the Von Mises Perzyna’s model. The second case we will study concerns the same slope, using this time a Perzyna Mohr Coulomb material. As before, the slope is 1:1. Soil properties selected are: Elastic modulus: E = 1.5e7 Pa Poisson ratio ν = 0.3 Friction angle φ = 22◦ Fig. 8 Velocity profile obtained using a Perzyna Von Mises model

Fig. 9 Profiles of stress (units Pa and m)

Figure 8 provides the velocity profile once steady state has been attained. The similarity with the plug found in Bingham models and given in Fig. 6 is to be noted. Concerning the stress distribution, it is possible to obtain profiles of all the components of the stress tensor. Figure 9 presents the profiles of the normal stresses along X and Y together with the shear stress. It is interesting to note that the normal stresses along X

Perzyna law: γ = 0.1 s−1 N = 1 We present in Figs. 11 and 12 the results obtained with the 3D infinite landslide model. Figure 11 provides the velocity profile once steady state has been attained. The velocity profile is linear, which can be explained by the form of Perzyna’s model selected. Concerning the stress distribution, Fig. 12 presents the profiles of the normal stresses along X and Y together with the shear stress. Now, both shear and normal stresses along X (down slope) and Y coincide (In fact the slope is 45◦ , which is the reason why the normal stress along Y and the shear stress coincide.) What it is interesting is the fact that, contrary to the assumptions usually done regarding the X stress, here we find that it is equal to the vertical stress. Finally, we will consider the case of a 1:4 slope with a Cam Clay soil. Failure has been triggered by decreasing the value of M from 1.1 to one third of it. The process has been considered drained. Soil properties selected are: Elastic modulus: E = 1.5e7 Pa

Fig. 10 Comparison between Perzyna Von Mises and Bingham models

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Fig. 11 Velocity profile obtained using a Perzyna Mohr–Coulomb model Fig. 13 Profile of velocity at steady state obtained with a Cam Clay Perzyna model

Fig. 14 Profile of components of stress tensor at steady state obtained with a Cam Clay Perzyna model Fig. 12 Profiles of stress tensor components obtained using a Perzyna Mohr–Coulomb model (units are Pa and m)

the rate of shear strain is given by: Poisson ratio ν = 0.3 Bulk densityρ = 1,500 kg/m3 Cam Clay parameters M = 1.1λ = 0.51κ = 0.09 Preconsolidation pressure pc0 = 0.285e5 Pa Perzyna law: γ = 0.1 s−1 N = 1

In above table, κ characterizes the elastic volumetric deformation dεve = κ dpp We present in Figs. 13 and 14 the results obtained with the 3D infinite landslide model. 4.3 Perzyna Based Simple Shear Infinite Landslide Models In the previous section we have obtained the velocity profiles along depth using Perzyna’s viscoplastic models. Here, we will propose a method to develop a simple rheological law for frictional materials. If we neglect elastic contributions,

123

∂v =γ ∂z

τ −s s

N (120)

where τ = ρg (h − z) sin θ and s = ρg (h − z) cos θ tan φ From where we obtain: ∂v tan θ − tan φ N =γ = ct. ∂z tan θ which results on a linear velocity profile such as the one find for Mohr–Coulomb viscoplastic model. From here, it is immediate to arrive to: τb = sb 1 +

2μv¯ h

1 N

(121)

where we have introduced μ = 1/γ , with [μ] = s, which is the inverse of the fluidity γ .

Author's personal copy Depth Averaged Models for Fast Landslide Propagation Velocity profile n1=0.5 1 0,9 0,8 0,7

z/h

0,6 0,5 0,4 0,3 0,2 0,1 0 0

0,2

0,4

0,6

0,8

1

V/Vmax Fig. 15 Velocity profile obtained with n1 = 0.5

As an alternative, it is possible to define ∂v (τ − s) N = γ n1 ∂z patm s n2

(122)

where N = n1 + n2 and patm is the atmospheric pressure or an alternative unity datum. The velocity profiles will depend now on n1, being given by z n1+1 v = vmax 1 − 1 − h

(123)

where vmax is the velocity at the top (z = h) Figure 15 presents the velocity profile obtained for n1 = 0.5, which is similar to that obtained with Cam-Clay viscoplasticity.

5 A Depth Integrated SPH Model Coupled with Finite Differences for Pore Pressures 5.1 Introduction In the past, most of numerical models were based on grids, which could be either structured (finite differences) or unstructured (finite elements and volumes). In the last three decades, a new group of “meshless” numerical methods has emerged. These methods are not based on meshes—even if they can be used as an auxiliary tool—but on points or nodes, where functions and derivatives are approximated. It is worth remembering here the pioneering work on the Diffuse Element Method introduced by Nayroles et al. [98], the Element Free Galerkin Method of Belytschko et al. [99], the hp-cloud method of Duarte and Oden [100], the Partition

of Unity Method of Babuska and Melenk [101], the Finite Point Method introduced by Oñate and Iddlesohn [102] (see also [103,104]), Material Point Model [105–109] and, finally, the Smoothed Particle Hydrodynamics Method, which is the technique which will be described here. Smoothed particle hydrodynamics (SPH) is a meshless method introduced independently by Lucy [110] and Gingold and Monaghan [111] and firstly applied to astrophysical modeling, a domain where SPH presents important advantages over other methods (see also the work of Monaghan and Lattanzio [112]). Good reviews can be found in Benz [113], Monaghan [114] or in the recent texts of Liu and Liu [115] or Li and Liu [116] and Liu and Liu [117]. SPH has been applied to a large variety of problems, such as hydrodynamics [111,118–124], metal forming [125], flow trough porous media [126], shallow water flows [127–132] and avalanche propagation [35,51,133]. Problems found in Solid Mechanics have also been solved using the SPH technique. Among the many contributions, it is worth mentioning those of Libersky and Petschek [134], Libersky et al. [135], Randles and Libersky [136], Bonet and Kulasegaram [125] and Gray et al. [137]. Soils are a special case of solids where the solid skeleton is filled with pore fluids, which interact with it. SPH has been applied to model coupling and failure problems of soils recently [138–141]. Adaptive techniques for inserting and removing nodes have been recently proposed by Lastiwka et al. [142], Feldman and Bonet [143] and Vacondio et al. [129].

5.2 SPH Fundamentals Smoothed particle hydrodynamics is based on the possibility of approximating a given function φ (x) and its spatial derivatives by integral approximations defined in terms of a kernel. In a second step these integral representations are approximated numerically by a class of numerical integration based on a set of discrete point or nodes, without having to define any “element”. The SPH method is based on the equality: φ (x) =

φ x δ x − x dx

(124)

with the additional requirement of “unity”. Traditionally, Dirac delta “function” is defined as: δ(x) =

∞x =0 0 |x| > 0

(125)

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with the additional requirement of “unity” δ(x)d x = 1

(126)

These results are the starting point for constructing smoothed particle hydrodynamics (SPH) approximations, where regular distributions are used to approximate the value of a function. The approximation is written as:

This definition, while enough for many engineering applications is not completely rigorous. This mathematical entity, the “Dirac delta” is a generalized function or a distribution. Distributions are a class of linear functionals, applications which transform functions into real numbers. They can be defined as Tw [φ] =

W (x )φ(x )d x

(127)

φ(x) =

φ(x )W (x − x, h)d x

(133)

The accuracy of SPH approximations depends on the properties of the kernel W(x,h). A special class of kernels is that of functions having radial symmetry, i.e., depending only on r: r = x − x

(134)

where W(x) is referred to as the kernel of the linear functional, and φ(x) is called a test function. Note that Tw [φ] is a real number. Some linear functionals cannot be derived from locally integrable kernels, as, for instance, the Dirac delta distribution, which transforms a test function into its value at the origin: δ [φ] = φ(0)

(128)

This distribution is called singular. Therefore, the Dirac delta it is not a proper function, but a singular distribution or generalized function. This singular distribution can be obtained as the limit of a sequence of regular distributions. For instance, let us consider the sequence:

(129)

2 1 1 x with h = Wk (x, h) = √ exp − 2 h k h π

(130)

where

We have parametrized the kernel by introducing a length scale h—or a integer k. It can be seen that, in the limit, h will tend to zero when k tends to infinity, and lim Twk [φ] = φ(0)

k→∞

(131)

Therefore, we can define the Dirac distribution in a weak sense as lim Twk [φ] = δ [φ]

k→∞

123

(135)

because it allows to write as in this case. We will use both notations in the following. The functions W(x,h) used as kernels in SPH approximations are required to fulfill the following conditions: (i) lim W (x − x, h) = δ(x)

h→0

(136)

(ii)

W (x − x, h)d x = 1

(137)

Wk (x , h)φ(x )d x

Twk [φ] =

It is convenient to introduce the notation: x − x r ξ= = h h

(132)

This condition, which follows also from (i) can be interpreted as well as the ability of the approximation to reproduce a constant or polynomial of degree zero (zero order consistency). (iii) The kernel W (x − x , h) is positive and has compact support: W (x − x, h) = 0 if x − x ≥ kh

(138)

where k is a positive integer which is usually taken as 2. (iv) The kernel W (x −x, h) is a monotonically decreasing function of x − x ξ= (139) h (v) The kernel W (x −x, h) is a symmetric function of (x − x)


It is possible to show that, under the conditions specified above, the approximation is second order accurate, i.e., φ(x) = φ(x) + O(h 2 )

(140)

In the framework of SPH formulations, several kernels have been proposed in the past. Among, them, it is worth mentioning: (i) The Gaussian kernel proposed by Gingold and Monaghan [111], and (ii) The cubic spline introduced by Monaghan and Gingold [144] and Monaghan and Lattanzio [112]. Concerning the integral representation of the derivatives in SPH, it is written as: (141) φ (x) = φ (x )W (x − x, h)d x

This expression is integrated by parts—in one dimensional problems—and, taking into account that the kernel has compact support, it results on: φ(x) = −

φ(x )W (x − x, h)d x

(142)

Classical differential operators of continuum mechanics can be approximated in the same way. We list below the gradient of a scalar function, the divergence of a vector function, and the divergence of a tensor function: 1 x −x grad φ (x) = − φ x W d with r = x −x h r (143)

div u (x) = − u x grad W d

1 u x . x − x W d (144) =− r h div σ (x) = − σ.grad W d

1 σ. x − x W d (145) =− r h These approximations of functions and derivatives are valid at continuum level. If the information is stored in a discrete manner, for instance, in a series of points or nodes, it is necessary to construct discrete approximations. The SPH method introduces the concept of “particles”, to which information concerning field variables and their derivatives is linked. But indeed, they are nodes, much in the same way than found in finite elements or finite differences. All operations are to be referred to nodes. We will therefore introduce the set of particles or nodes with K = 1...N. Of course, the level of approximation will depend on how the nodes are spaced and on their location. The classical finite element strategy of having more

Fig. 16 Nodes and numerical integration in a SPH mesh

nodes in those zones where larger gradients are expected is of application here. If we consider the approximation of a function given in (133), as the information concerning the function is only available at a set of N nodes the integral could be evaluated using a numerical integration technique of the type: φ (x I ) h =

N

φ (x J ) W (x J − x I , h)ω J

(146)

J =1

where we have used the sub index “h” to denote the discrete approximation, with ω J denoting the weights of the integration formula—which can be shown to be ω J = J = m J /ρ J , with J , m J and ρ J being the volume, mass and densities associated to node J . In order to simplify the notation, we will use introduce, defined as: φ I = φ (x I ) h =

N

φ (x J ) W (x J − x I , h)ω J

(147)

J =1

If we take into account that the kernel function has local support, i.e. it is zero when, the summation extends only to the set of Nh points which fulfill this condition: φ I = φ (x I ) h =

Nh

φ (x J ) W (x J − x I , h)ω J

(148)

J =1

Figure 16 illustrates the numerical integration procedure performed. which is a form commonly used in SPH. In the case we choose the function φ to represent the density, we will obtain, after substituting in (148) ρI =

Nh

ρJ WI J

J =1

ρI =

n

WI J m J

mJ ρJ (149)

J =1

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One interesting aspect of SPH is the existence of several alternative discretized forms for the differential operators. For instance, the gradient of a scalar function can be approximated as (basic form): grad φ I =

Nh mJ φ J grad W I J ρJ

(150)

J =1

and the three following symmetrized forms 1 m J (φ J − φ I )grad W I J ρI J φJ φI grad φ I = ρ I mJ + 2 grad W I J ρ 2J ρI J

grad φ I =

(151)

(152)

Sometimes it is preferred to use a variant of this form, which is grad φ I =

Nh mJ (φ I + φ J ) grad W I J ρJ

(ii) Introducing artificial stabilizing forces [137,148] Generally this force is called artificial stress and consists on repulsion when neighboring particles get closer (iii) Adding a new set of stress points between the SPH nodes. The stress is calculated on the stress points using the information of the SPH nodes and the linear momentum is calculated on the SPH nodes from the stress points. This method has been first proposed by Dyka for one-dimensional problems with the following SPH nodes arrangement: two stress points for one SPH node [149,150]. This method has been extended to two-dimensional problems by Randles and Libersky [136,151]. (iv) Blanc and Pastor [140,152] have proposed recently a novel SPH algorithm which is an extension of the Taylor Galerkin method proposed by Peraire et al. [153] and Donea et al. [96] and extended by Mabssout and Pastor [97] to solid dynamics problems. It uses a double set of SPH nodes.

(153)

J =1

5.3 Special Problems 5.3.1 Consistency The kernel approximation used in SPH is h 2 accurate for both functions and gradients [145]. However, at particle level this second order accuracy is lost because of boundaries, where the integration is carried out only over points which belongs to the domain, and not over the whole support of the kernel. This problem is often referred to as “boundary deficiency”. The problem was detected by Morris [145], and since then, several remedial techniques have been proposed, such as those of Randles and Libersky [136], Chen et al. [146], Liu and Liu [115]. In this way, accuracy is of first order, unless the particles are equally spaced where it recovers the original second order.

5.3.3 Boundary Conditions Boundary conditions require special treatment in SPH. In addition to the boundary deficiency problem and the solutions already mentioned, SPH practitioners use special virtual particles located at the boundaries. Monaghan [154] introduced a first type of virtual particles, often referred to as “type 1 virtual particles”, which apply a repulsive force on particles approaching the boundary, which prevents boundary penetration by them. Libersky et al. [135], and Randles and Libersky [151] introduced a second type of virtual particles, which are located symmetrically respect to the real particle approaching the boundary. These methods have been improved by Bonet et al. [155] for cases where rigid boundaries are present. In the case of shallow water waves, boundary conditions of absorbing and prescribed incoming waves are important, and have been studied by Lastiwka et al. [142] and Vacondio et al. [156].

5.3.2 Stability SPH presents, in materials with strength, the tensile instability problem, which consists of unphysical clumping of particles which from where the instability may grow [19]. Swegle et al. [147] described the problem in two dimensional arrangements of particles and studied the stability, which was found to depend on the properties of the kernel. To remediate the problem, several alternative methods have been proposed: (i) Use of special kernel functions [145]

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5.4 An SPH Model for Biot–Zienkiewicz v-pw Depth Integrated Equations Following the procedure outlined in previous Sections we will introduce a set of nodes {x K }with K = 1..N and the nodal variables h I height of the landslide at node I v¯ I depth averaged, 2D velocity tbI surface force vector at the bottom pwb I Pore pressure at the basal surface


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

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

(i) a fictitious volume m I moving with this node:

h I = h (x I )

= h J J WI J

m I = I h I

J

(154)

=

(ii) and p¯ I , an averaged pressure term, given by: 1 p¯ I = b3 h 2I 2

(155)

(161)

The height can be normalized, which allows improving the approximation close to the boundary nodes:

¯ ∂ v¯ j dh =0 +h dt ∂x j

(156)

∂ v¯ j d¯ h + h

= 0 j = 1, 2 dt ∂x j

(157)

(162)

Next, we will discretize the balance of linear momentum equation (44) ρ

1 dv = ρ grad h 2 b3 + ρhb3 grad Z + τb + ρbh (163) dt 2

or, introducing the averaged pressure defined above (164)

The left hand side results on: hI

The equation is written at node I as: ∂ v¯ j d¯ hI + hI = 0 j = 1, 2 dt ∂x j I

d¯ v¯ I dt

(165)

Depending on the symmetrized form chosen (see [114,154, 157]) to discretize the gradient of the pressure, we obtain the following discretized forms of the balance of momentum equation:

where the divergence term is given by: J v J grad W I J

m J WI J h I = J mJ J h J WI J

dv 1 = − grad p + b3 grad Z + τb + bh dt ρ

from which:

m J WI J

J

It is important to note that m I 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. The SPH approximation of the balance of mass equation is built from

div v I = −

(158)

J

pI + p J d¯ v¯ I = − mJ grad W I J + b3 grad Z I dt hI hJ J

or div v I = −

mJ J

hJ

v J grad W I J

(159)

Of course, we could have used any alternative symmetrized form. The discretized balance of mass equation is written as ¯ I dh dt ¯ I dh dt ¯ I dh dt

mJ = − hI h J v J grad W I J (Basic form) J = m J v I J grad W I J (1st form) J mJ = hI h J v I J grad W I J (3rd form) J

where we have introduced v I J vI J = vI − v J

(160)

1 + τbI + bh I ρ ¯ d pJ pI v¯ I = − mJ + 2 grad W I J dt h 2I hJ J + b3 grad Z I + τbI + bh I

(166)

(167)

The scheme is explicit, and we use a time step limit given by the CFL condition: h min

√ t S P H ≤ max gh I + |v I |

(168)

In above equations, we have used the basal excess pore pressure at node I pwb I which has to be obtained at each node and time step. One alternative is to use simple shape functions fulfilling boundary conditions at the surface and the basal surface. This has been used by Iverson and Denlinger

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[33], Pastor et al. [34,35,158] and Quecedo et al. [36]. This approach presents the limitation of not being able to model changes of boundary conditions at the bottom. For instance, when a landslide runs over a very permeable basal layer—or a rack—pore pressure becomes zero there, while in the body of the landslide is not zero. If a single shape function is used, once the basal value is set to zero, the pressure becomes zero in the whole depth. In order to improve the pore pressure variations, we propose to use a Finite Difference mesh at each SPH node. The equation to be solved is: ∂ 2 pw dp dpw =− − K v dv0 + kw K v dt dt ∂ x32

(169)

which can be written as ∂ 2 pw dpw = s + Cv dt ∂ x32

(170)

where we have introduced a source term s and the coefficient of consolidation Cv s = − ddtp − K v dv0 C v = kw K v

(171)

where h S M L I denotes the smoothing length at node I. The algorithm is explicit, and less accurate than the approach proposed by Bonet et al. [155], where both the mass conservation equation and (171) are solved using a Newton Raphson algorithm. So far, we have discretized the balance of mass (160), balance of momentum (166) and pore pressure dissipation (166). 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). In the cases we will show in the Section devoted to examples and applications, we have used a classical Runge Kutta fourth order algorithm. One important issue is the representation of the terrain over which the avalanche moves, as it greatly influences the

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6 Examples and Applications 6.1 Introduction

The term ddtp accounts for variations of the total confining pressure, caused, for instance, by changes in the height of the soil column. The Finite Difference scheme chosen is centered in space and forward in time, which is conditionally stable. Therefore, at every node and time step, critical step times of FD and SPH are compared, and, if, necessary, the pore pressure equation is solved using internal sub steps. We use the variable smoothing length formula proposed by Benz [113]: h I h SNMDLI M I = const

results. If we denote by Z I the height of the terrain at node I, we have to obtain (i) its gradient grad Z I , and (ii) the radius of curvature along the tangent to the node path, which can be obtained from the second order derivatives of Z. In the case of fast landslides, the terrain information is given on a digital terrain model (DTM), which consists on a series of values (xk , yk , Z k ) at the nodes of a structured grid. From here we can obtain both gradients and second derivatives at the grid nodes using a classical nodal recovery technique on a finite element mesh which nodes are those of the DTM grid. Concerning the neighbour search, we have used an auxiliary structured grid covering the part of the terrain where the SPH particles are. Spacing is taken as the minimum smoothing length. For a given SPH node, search is restricted to the cell it belongs and its neighbours. This temporary grid is valid only for a given time step. In cases where the flow is elongated, the grid can be oriented automatically following the main inertia axes of the set of SPH nodes on the plane.

The purpose of this Section is twofold. First, we will use (i) simple problems having an analytical solution, and (ii) laboratory small scale tests to compare their results against the proposed model predictions assessing the model accuracy. Then, we will analyze real landslides of several representative types analyzed by our group in Madrid for which detailed information was available. Some of the cases have been provided by Hong Kong Geotechnical Office, which in 2007 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 original position of the mobilized mass and the final position of the deposit. We will analyze the following cases: (i) Breaking of a dam impounding inviscid water over a wet flat plane. (ii) Small scale test representing a granular avalanche (iii) A representative rock avalanche: Thurwieser (2004). (iv) A lahar in the Popocatépetl volcano (Mexico, 2001) (v) Tsing Shan debris flow in Hong Kong (14th April 2000) (vi) The Aberfan flowslide, where pore pressures decreased the basal friction. In this way, we will cover most of types of fast catastrophic landslides.

Author's personal copy Depth Averaged Models for Fast Landslide Propagation Fig. 17 Dam-break over a wet bed

hL

hR

Fig. 19 A Wet dam break problem. Comparison between computed and theoretical elevation profiles at time 0.4 s Fig. 18 Computed (left) versus analytical (right) profiles of water depth at 0.0, 0.2 and 0.4 s

6.2 A Problem with Analytical Solution: Breaking of a Dam on a Wet bottom The mathematical model is a nonlinear system of hyperbolic equations which can present rarefaction and shock waves. The first example we will consider includes both types of waves. It consists of a reservoir of infinite length filled with water of constant depth hL separated by a vertical wall from a plane flooded with water of depth hR (Fig. 17). At time zero, the dam is suddenly removed, and the reservoir water enters the plane. As the water is inviscid, there are not rheological parameters to determine. The solution ([69,112], Toro [166]) consists on a rarefaction wave propagating leftwards, and a shock moving to the right. The simulation results are given in Figs. 17–20, where we can observe a good agreement between with the analytical solution. Both the rarefaction wave propagating leftwards and the shock moving to the right are correctly represented. 6.3 A Small Scale Laboratory Test: Granular Avalanches The second example is a small scale laboratory test performed by Dr. Irene Manzella at Ecole Polytechnique Fédérale de Lausanne [159]. The granular material is fine Hostun sand, with an estimated angle of friction of 34◦ . The

Fig. 20 Wet dam break problem. Comparison between computed and theoretical velocity profiles at time 0.4 s

base material is forex, with a basal friction angle of 32◦ .The total volume of sand is 301cm3 . We provide in Fig. 21 a perspective of the experimental device, where the initial position of the sand can be seen. In this example we have used the simple frictional fluid described in Sect. 3, with a single rheological parameter, the friction between the basal surface and the sliding sand, which is smaller than the internal friction angle of the sand. These parameters were measured in the laboratory, and no fitting was performed.

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Fig. 21 Layout of the Lausanne experiment on granular avalanches (Z in m)

Once the mass is released, the avalanche falls down the slope. We provide in Fig. 22 the position of SPH nodes at different times after the release. Figures 23 and 24 compare the model results with the experimental measurements. Finally, Figs. 23 and 24 provide a comparison between the experiments and the model predictions. 6.4 Thurwieser Avalanche (18th September 2004) 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 Fig. 22 Position of nodes at times 0, 1.12, 2.24 and 3.36 s

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Zebrú valley. Its propagation path extended from 3,500 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 [160] have provided the information concerning this avalanche, including a detailed digital terrain model. Figure 25 from Sossio and Crosta [160], 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 [51]. The rheological parameters chosen are: tan φ = 0.39, Voellmy coefficient 1,000 m/s2 , erosion coefficient 0.00025 m−1 . These parameters were obtained by trial en error, as the only values reported concerned times of propagation and runout. The results are given in Fig. 26, where we have plotted the avalanche evolution along time and the computed final extension together with the observed in the field. 6.5 Tsing Shan Debris Flow in Hong Kong (14th April 2000) This example concerns Tsing Shan debris flow which happened in Hong Kong on the 14th April 2000. The analysis is


Fig. 23 Comparison between experimental and laboratory results

Fig. 24 Final state: experimental results and model predictions

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Fig. 25 General view of Thurwieser rock avalanche [160]

based on the information found both in the package provided by Hong Kong Geotechnical Office and the report by King [161]. 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 150 to 1,600 cubic meters. Figure 27, taken from King [161] provides two general views of the debris flow. One important aspect is the bifurcation of the flow which can be observed in the pictures. In order to model it, we have used the frictional viscoplastic model described in (89), which particularizes to 25 1 μ F v¯ 2 , τb = ρgh cos θ tan ϕ + 4 h2 with tan φ = 0.18, zero cohesion and μ F = 0.00133 Pa s2 . We have chosen the Hungr’s erosion model, using an erosion coefficient of 0.0082m−1 . The results of the simulation are given in Fig. 28 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 Fig. 28 shows the branching. In Fig. 28, it can be seen that the deepest deposit was formed

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at the end of the lower south branch, with a 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.

6.6 A lahar in Popocatépetl volcano (Mexico) Lahars are a special case of debris flows of volcanic origin which can be triggered by eruptions, heavy rainfall. We will study here the case of a lahar that occurred at Popocatépetl volcano in 2001 [82]. Popocatépetl is a stratovolcano located in the Trans Mexican Volcanic Belt in central Mexico, with a glacier of 0.23 millions cubic meters at the summit. Lahars are not uncommon, and they travel downhill along gorges and channels which can reach villages (Huilouac, Tenenepanco). Indeed, Huilouac gorge crosses the village of Santiago Xalitzintla, which is 17 km away from Popocatépetl crater (Fig. 29).


Fig. 26 Thurwieser avalanche after 80 s with friction angle 26◦ : Computed results (colour isolines and deposit height) versus field measurements (black isolines and red line for the spreading). (color figure online)

Fig. 27 General view of the 2000 Tsing Shan debris flow [161]

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Author's personal copy M. Pastor et al. Fig. 28 Tsing Shan debris flow: model predictions versus field observations

Fig. 29 Study area showing Huilouac gorge

Concerning the material properties, we have used a Bingham fluid rheology, with a yield strength of 60 Pa and a viscosity of 45 Pa s. The initial mass of the lahar was assumed to be of 2 × 105 m3 . The terrain model used a structured grid with a spacing of 10 m. The results are presented in Fig. 30, where we can see the propagation of the lahar along Huilouac gorge. 6.7 The Aberfan flowslide (Wales 1966) As many other villages in Wales, Aberfan had coal mines which wastes were stored in dumps. The flowslide of 21st October 1966 developed at the Tip No.7, and propagated

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downhill into Aberfan, hitting the school and causing 144 deaths. The material was very loose, uncompacted coalmine waste deposited by end-tipping. At the time of failure, the height was 67 m from the toe, and the underlying terrain had a slope close to 12◦ . Information concerning Aberfan failure mechanism and material properties has been provided by Bishop et al. [162], Bishop [6], and Hutchinson [32]. Raw information material is available at the UK National Archives, from where some of the information used in this example has been obtained. According to the existent information, failure was caused by artesian pore pressures at the toe, which probably saturated the lower part of the tip, while the upper part remained unsaturated. Bishop reports that ‘the rescue work was complicated


Fig. 30 Propagation of the lahar along Huilouac gorge Fig. 31 Vertical profiles of Aberfan tip before and after 1965 flowslides (From UK National Archives, Kew)

by water which flowed, after the slip, from the sandstone at the base of the rotational slip, where the boulder clay and head was stripped off.’ Once failure was triggered, the flowslide propagated downhill 275 m, then divided into two lobes, which were referred to as ‘north’ and ‘south’ lobes because of their relative position. It was the larger south lobe that ran into the village, while the northern lobe stopped after reaching an embankment. The runout was approximately 600 m, with estimated velocities in the range 4.5-9 m/s. In the past, Aberfan has been modelled using a Bingham model for the material, which does not takes into account the pore pressure generation and propagation. For instance, Jeyapalan et al. [163] and Jin and Fread [164] obtained results which fitted well the observations choosing τy = 4,794 Pa,

μ = 958 Pa s and ρ = 1,760 kg/m3 . Even if these results are good, it is possible to argue that waste coal was not fully saturated, and the material was frictional. Of course, the apparent angle of friction introduced above will be much smaller than φ’, but vertical consolidation could have made it to change during the propagation phase. Hutchinson [32] proposed a simple ”sliding-consolidation” model in which it was clear that the combination of friction with basal pore pressures could provide accurate results of runout and velocities. This simplified method was used by the authors in Pastor et al. [34]. It is assumed that there exists a layer of saturated soil of height hs on the bottom of the flowing material [32]. The decrease in pore pressures is caused by vertical consolidation of this layer. Pore pressures on the top and bottom of this

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Author's personal copy M. Pastor et al. Fig. 32 Vertical profiles showing Aberfan flowslide propagation

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Author's personal copy Depth Averaged Models for Fast Landslide Propagation Fig. 33 Pore water contours at t =2s

Fig. 34 Location of the zero pore pressure zone

layer can be either estimated from the values of the vertical stresses or obtained directly from the results of finite element computations. The information provided in the literature do not provide enough data to perform a realistic analysis in two dimensions. Therefore, we have used a simple 1D model with the terrain

profiles sketched in Fig. 31, which are a better approximation than that of Jeyapalan et al. [163]. The main purpose of this example is to show that a depth integrated model using pore pressure dissipation can reproduce the basic patterns observed. Therefore, we have used the vertical profile given in Fig. 30 below, where we can see the profiles both before and after the flowslide. Density of the mixture ρ and friction angle φ’, have been taken as 1,740 kg/m3 and 36◦ respectively. we have chosen a consolidation coefficient Cv = 6.5.10−5 m2 /s. and assumed that a basal saturated layer of 0.06 times the height of the flowslide at the beginning. Initial pore pressure has been taken as 0.78 times the value corresponding to full

Fig. 35 Vertical profiles at different times (left original terrain; right with the rack). Note the larger runout distance when no rack is present

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Author's personal copy M. Pastor et al. Fig. 36 a Pore water pressure distribution at time 17 s. b Pore water pressure distribution at time 20 s

liquefaction. The rheological model is the simple frictional fluid. The results obtained in the simulation are given in Fig. 32 where sections of the free surface of the flowslide are given at times 0, 6, 10, 15, 20 and 30 s. In Fig. 33 we provide pore water pressure contours at time 2 s. Please note that in order to improve readability, we have expanded the graphic representation of the saturated layer and it occupies the whole mass (This is possible because of the assumption that the saturated layer depth is proportional to that of the flowslide).

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6.7.1 Effect of a Terrain Zone with Very High Permeability One of the advantages of incorporating a set of finite difference meshes at each SPH node is the ability to improve the quality of the predictions in cases where basal pore pressures go to zero as a consequence of the landside crossing a terrain with very high permeability—or a rack. The procedure to simulate this effect in the proposed model is based on changing the boundary condition associated to the finite difference nodes located at the contact with the terrain.


As a thought experiment, we will repeat the case of Aberfan flowslide, including now such a zone. We have selected the zone sketched in Fig. 34, where we have set the boundary condition of zero pore pressure at the bottom of all finite difference meshes associated to SPH nodes on it. Figure 35 provides a comparison of the vertical profiles at different time stations. Note that the depth has been amplified by a factor of 4. Concerning the distribution of pore pressure in the landslide, Fig. 36a, b provide the results obtained for times 17 and 20 s. The location of the rack is shown in Fig. 36b. 7 Conclusions This paper has presented mathematical, rheological and numerical models which can be applied to reproduce the propagation of fast landslides such as rock avalanches, debris flows, lahars and flowslides, taking into account coupling between solid skeleton and pore fluid. Concerning mathematical modelling, we have shown how Biot–Zienkiewicz models, widely used in geomechanics to describe the coupled behaviour of soil, can be applied in a straightforward manner to derive the equations describing the propagation of fast catastrophic landslides. We have studied the similarity which exists between constitutive models of Perzyna’s type and some widely used rheological models. This will surely provide a tool to model in a consistent manner both initiation and propagation phases. We have proposed that classical viscoplasticity models used in geomechanics are able to reproduce well the vertical profiles of velocities which simple shear depth integrated models using simple rheological laws provide. From here: (i) The use of viscoplasticity models in mesh free models will provide a suitable framework to study both the initiation and propagation of landslides. (ii) From viscoplasticity, simple rheological models can be developed and applied to landslide propagation problems. There exist several alternative numerical models which allow accurate discretization of the mathematical and constitutive model. We have focused here on SPH models, which present important advantages. Finally, we have chosen three sets of tests and examples which allows to assess the model performance and limitations: (i) Problems having an analytical solution, (ii) Small scale laboratory tests, and (iii) Real cases for which we have had access to reliable information. Acknowledgments The authors gratefully acknowledge the economic support provided by the Spanish Ministry MINECO (Projects GEODYN and GEOFLOW). The first author would like to express

his gratitude to Dra. Ma. D. Elizalde for the help provided with the documentation at the National Archives (Kew, UK), which allowed the retrieval of information concerning Aberfan flowslide. 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. Thanks are given to Dr Manzella for the experimental data concerning the granular avalanche experiments.

References 1. Dikau R, Commission E (1996) Landslide recognition: identification, movement, and clauses. John Wiley & Sons Ltd, 2. Sulem J, Vardoulakis IG (1995) Bifurcation Analysis in Geomechanics. Taylor & Francis, 3. Laouafa F, Darve F (2002) Modelling of slope failure by a material instability mechanism. Comput. Geotech. 29:301–325 4. Fernandez Merodo JA, Pastor M, Mira P, Tonni L, Herreros MI, Gonzalez E, Tamagnini R (2004) Modelling of diffuse failure mechanisms of catastrophic landslides. Comput. Methods Appl. Mech. Eng. 193:2911–2939 5. Prisco C, Nova R, Matiotti R (1995) Theoretical investigation of the undrained stability of shallow submerged slopes. Géotechnique. 45:479–496 6. Bishop AW (1973) The stability of tips and spoil heaps. Q. J. Eng. Geol. Hydrogeol. 6:335–376 7. Calembert L, Dantinne R (1964) The avalanche of ash at Jupille (Liege) on February 3rd, 1961. The commemorative volume dedicated to Professeur F.Campus. pp. 41–57. 8. Han G, Wang D (1996) Numerical Modeling of Anhui Debris Flow. J. Hydraul. Eng. 122:262–265 9. Dawson RF, Morgenstern NR, Stokes AW (1998) Liquefaction flowslides in Rocky Mountain coal mine waste dumps. Can. Geotech. J. 35:328–343 10. Okada Y, Sassa K, Fukuoka H (2004) Excess pore pressure and grain crushing of sands by means of undrained and naturally drained ring-shear tests. Eng. Geol. 75:325–343 11. Iverson RI (2000) Landslide triggering by rain infiltration. Water Resour. Res. 36:1897–1910 12. Pailha M, Nicolas M, Pouliquen O (2008) Initiation of underwater granular avalanches: Influence of the initial volume fraction. Phys. Fluids. 20:111701 13. Igwe O, Sassa K, Fukuoka H (2006) Excess pore water pressure: a major factor for catastrophic landslides. The 10th IAEG International Congress., Nottingham, UK 14. Biot MA (1955) Theory of Elasticity and Consolidation for a Porous Anisotropic Solid. J. Appl. Phys. 26:182 15. Biot MA (1941) General Theory of Three-Dimensional Consolidation. J. Appl. Phys. 12:155 16. Zienkiewicz OC, Chang CT, Bettess P (1980) Drained, undrained, consolidating and dynamic behaviour assumptions in soils. Géotechnique. 30:385–395 17. Zienkiewicz OC, Shiomi T (1984) Dynamic behaviour of saturated porous media; The generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Methods Geomech. 8:71– 96 18. Zienkiewicz OC, Xie YM, Schrefler BA, Ledesma A, Bicanic N (1990) Static and Dynamic Behaviour of Soils: A Rational Approach to Quantitative Solutions. II. Semi-Saturated Problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 429:311–321 19. Zienkiewicz OC, Chan AHC, Pastor M, Paul DK, Shiomi T (1990) Static and Dynamic Behaviour of Soils: A Rational Approach to Quantitative Solutions. I. Fully Saturated Problems. Proc. R. Soc. A Math. Phys. Eng. Sci. 429:285–309

123

Author's personal copy M. Pastor et al. 20. Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T (1999) Computational Geomechanics. John Wiley & Sons Ltd, 21. Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media. 22. Coussy O (1995) Mechanics of Porous Continua. 23. De Boer R (2000) Theory of Porous Media: Highlights in Historical Development and Current State. 24. De Boer R (2005) Trends in Continuum Mechanics of Porous Media. Springer Verlang, Berlin 25. Anderson TB, Jackson R (1967) Fluid Mechanical Description of Fluidized Beds. Equations of Motion. Ind. Eng. Chem. Fundam. 6:527–539 26. Green AE, Adkins JE (1960) Large Elastic Deformations and Nonlinear Continuum Mechanics. 27. Green AE, Naghdi PM (1969) On basic equations for mixtures. Q. J. Mech. Appl. Math. 22:427–438 28. Bowen RM (1976) Theory of Mixtures, Part I. In: Eringen AC (ed) Continuum physics III. 29. Li X, Zienkiewicz OC, Xie YM (1990) A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution. Int. J. Numer. Methods Eng. 30:1195–1212 30. Schrefler BA (1995) F.E. in environmental engineering: Coupled thermo-hydro-mechanical processes in porous media including pollutant transport. Arch. Comput. Methods Eng. 2:1–54 31. De Boer R (1996) Highlights in the Historical Development of the Porous Media Theory: Toward a Consistent Macroscopic Theory. Appl. Mech. Rev. 49:201 32. Hutchinson JN (1986) A sliding-consolidation model for flow slides. Can. Geotech. J. 23:115–126 33. Iverson RM, Denlinger RP (2001) Flow of variably fluidized granular masses across three-dimensional terrain: 1. Coulomb mixture theory. J. Geophys. Res. 106:537 34. Modelling tailings dams and mine waste dumps failures (2002) Pastor, M., Quecedo, M., Fernández Merodo, J.A., Herreros, M.I., Gonzalez, E., Mira, P. Géotechnique. 52:579–591 35. Pastor M, Haddad B, Sorbino G, Cuomo S, Drempetic V (2009) A depth-integrated, coupled SPH model for flow-like landslides and related phenomena. Int. J. Numer. Anal. Methods Geomech. 33:143–172 36. Quecedo M, Pastor M, Herreros MI, Fernández Merodo JA (2004) Numerical modelling of the propagation of fast landslides using the finite element method. Int. J. Numer. Methods Eng. 59:755– 794 37. Major JJ, Iverson RM (1999) Debris-flow deposition: Effects of pore-fluid pressure and friction concentrated at flow margins. Geol. Soc. Am. Bull. 111:1424–1434 38. Pitman EB, Le L (2005) A two-fluid model for avalanche and debris flows. Philos. Trans. A. Math. Phys. Eng. Sci. 363:1573– 601 39. Pudasaini SP, Hutter K (2007) Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches 40. Gray JMNT, Thornton AR (2005) A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. A Math. Phys. Eng. Sci. 461:1447–1473 41. Gray JMNT, Ancey C (2009) Segregation, recirculation and deposition of coarse particles near two-dimensional avalanche fronts. J. Fluid Mech. 629:387 42. Gray JMNT, Ancey C (2011) Multi-component particle-size segregation in shallow granular avalanches. J. Fluid Mech. 678:535– 588 43. Pastor M, Blanc T, Pastor MJ (2009) A depth-integrated viscoplastic model for dilatant saturated cohesive-frictional fluidized mixtures: Application to fast catastrophic landslides. J. Nonnewton. Fluid Mech. 158:142–153

123

44. Saint-Venant A (1871) Theorie du mouvement non permanent des eaux, avec application aux crues des rivieres et a l’introduction de marees dans leurs lits. Comptes-Rendus l’Académie des Sci. 73:147–273 45. Savage SB, Hutter K (1991) The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis. Acta Mech. 86:201–223 46. Savage SB, Hutter K (1989) The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199:177 47. Hutter K, Koch T (1991) Motion of a Granular Avalanche in an Exponentially Curved Chute: Experiments and Theoretical Predictions. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 334:93– 138 48. Hutter K, Siegel M, Savage SB, Nohguchi Y (1993) Twodimensional spreading of a granular avalanche down an inclined plane Part I. theory. Acta Mech. 100:37–68 49. Gray JMNT, Wieland M, Hutter K (1999) Gravity-driven free surface flow of granular avalanches over complex basal topography. Proc. R. Soc. A Math. Phys. Eng. Sci. 455:1841–1874 50. Laigle D, Coussot P (1997) Numerical Modeling of Mudflows. J. Hydraul. Eng. 123:617–623 51. McDougall S, Hungr O (2004) A model for the analysis of rapid landslide motion across three-dimensional terrain. Can. Geotech. J. 41:1084–1097 52. Wang Y, Hutter K (1999) A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol. Acta. 38:214–223 53. Iverson RI (1997) The physics of debris flows. Rev. Geophys. 35:245–296 54. Bingham EC (1922) Fluidity and Plasticity. McGraw-Hill, 55. Hohenemser K, Prager W (1932) Über die Ansätze der Mechanik isotroper Kontinua. ZAMM - Zeitschrift für Angew. Math. und Mech. 12:216–226 56. Oldroyd JG, Wilson AH (2008) A rational formulation of the equations of plastic flow for a Bingham solid. Math. Proc. Cambridge Philos. Soc. 43:100 57. Malvern LE (1969) Introduction to the Mechanics of a continuous medium. Prentice Hall, 58. Coussot P (1994) Steady, laminar, flow of concentrated mud suspensions in open channel. J. Hydraul. Res. 32:535–559 59. Coussot P (1997) Mudflow Rheology and Dynamics. Balkema, 60. Coussot P (2005) Rheometry of pastes, suspensions and granular matter. John Wiley and Sons Inc, 61. Dent JD, Lang TE (1983) A biviscous modified bingham model of snow avalanche in motion. Ann. Glaciol. 4:42–46 62. Locat J (1997) Normalized rheological behaviour of fine muds and their flow properties in a pseudoplastic regime. International Conference on Debris-Flow Hazard Mitigation: Prediction and Assessment. pp. 260–269, New York. 63. Chen C (1988) Generalized viscoplastic modeling of debris flow. J. Hydraul. Eng. 114:237–258 64. Chen C, Ling C-H (1996) Granular-Flow Rheology: Role of Shear-Rate Number in Transition Regime. J. Eng. Mech. 122:469–480 65. Parry RHG (1960) Triaxial Compression and Extension Tests on Remoulded Saturated Clay. Géotechnique. 10:166–180 66. Iverson RM (2005) Regulation of landslide motion by dilatancy and pore pressure feedback. J. Geophys. Res. 110:F02015 67. Jop P, Forterre Y, Pouliquen O (2006) A constitutive law for dense granular flows. Nature. 441:727–30 68. Forterre Y, Pouliquen O (2008) Flows of Dense Granular Media. Annu. Rev. Fluid Mech. 40:1–24 69. Pailha M, Pouliquen O (2009) A two-phase flow description of the initiation of underwater granular avalanches. J. Fluid Mech. 633:115

Author's personal copy Depth Averaged Models for Fast Landslide Propagation 70. Crosta GB, Imposimato S, Roddeman D (2009) Numerical modelling of entrainment/deposition in rock and debris-avalanches. Eng. Geol. 109:135–145 71. Crosta GB, Frattini P, Fusi N (2007) Fragmentation in the Val Pola rock avalanche. Italian Alps. J. Geophys. Res. 112:F01006 72. Johnson CG, Kokelaar BP, Iverson RM, Logan M, LaHusen RG, Gray JMNT (2012) Grain-size segregation and levee formation in geophysical mass flows. J. Geophys. Res. 117:F01032 73. Astarita G, Marucci G (1974) Principles of non-newtonian fluid mechanics. McGraw-Hill, 74. Bagnold RA (1954) Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear. Proc. R. Soc. A Math. Phys. Eng. Sci. 225:49–63 75. Midi GR (2004) On dense granular flows. Eur. Phys. J. E. Soft Matter. 14:341–65 76. George DL, Iverson RI (2011) A two phase debris flow model thatincludes coupled evolution of volume fractions, granular dilatancy and pore fluid pressure. Italian Journal of Engineering Geology and Environment. pp. 415–424. 77. Frigaard IA, Nouar C (2005) On the usage of viscosity regularisation methods for visco-plastic fluid flow computation. J. Nonnewton. Fluid Mech. 127:1–26 78. Muravleva L, Muravleva E, Georgiou GC, Mitsoulis E (2010) Numerical simulations of cessation flows of a Bingham plastic with the augmented Lagrangian method. J. Nonnewton. Fluid Mech. 165:544–550 79. Pastor M, Quecedo M, Gonzalez E, Herreros MI, Fernández Merodo J, Mira P (2004) Modelling of landslides (II) Propagation. Degradations and Instabilities in Geomaterials. Springer Wien, New York 80. Fraccarollo L, Papa M (2000) Numerical simulation of real debrisflow events. Phys. Chem. Earth, Part B Hydrol. Ocean. Atmos. 25:757–763 81. Naef D, Rickenmann D, Rutschmann P, McArdell BW (2006) Comparison of flow resistance relations for debris flows using a one-dimensional finite element simulation model. Nat. Hazards Earth Syst. Sci. 6:155–165 82. Haddad B, Pastor M, Palacios D, Muñoz-Salinas E (2010) A SPH depth integrated model for Popocatépetl 2001 lahar (Mexico): Sensitivity analysis and runout simulation. Eng. Geol. 114:312– 329 83. Cola S, Calabro N, Pastor M (2008) Prediction of the flow-like movements of Tessina landslide by SPH model. 10th International Symposium on landslides and engineered slopes. pp. 647–654. 84. Malet J-P, Maquaire O, Locat J, Remaitre A (2004) Assessing debris flow hazards associated with slow moving landslides: methodology and numerical analyses. Landslides. 1:83–90 85. Sanchez ME, Pastor M, Romana MG (2013) Modelling of short runout propagation landslides and debris flows. Georisk Assess. Manag. Risk Eng. Syst. Geohazards. 1–17. 86. Voellmy A (1955) Über die Zerstörungskraft von Lawinen. 87. Hanes DM, Inman DL (2006) Observations of rapidly flowing granular-fluid materials. J. Fluid Mech. 150:357 88. Roux S, Radjai F (1998) Texture-Dependent Rigid-Plastic Behavior. 229–236. 89. Perzyna P (1963) The constitutive equations for rate sensitive plastic materials. Q. Appl. Math. 20:321–332 90. Perzyna P (1966) Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9:243–377 91. Burland JR (1965) The yielding and dilation of clay (Correspondence). Geotechnique. 15:211–214 92. Di Prisco C, Pisanò F (2011) An exercise on slope stability and perfect elastoplasticity. Géotechnique. 61:923–934 93. Di Prisco C, Pastor M, Pisanò F (2012) Shear wave propagation along infinite slopes: A theoretically based numerical study. Int. J. Numer. Anal. Methods Geomech. 36:619–642

94. Pisanò F, Pastor M (2011) 1D wave propagation in saturated viscous geomaterials: Improvement and validation of a fractional step Taylor-Galerkin finite element algorithm. Comput. Methods Appl. Mech. Eng. 200:3341–3357 95. Löhner R, Morgan K, Zienkiewicz OC (1984) The solution of nonlinear hyperbolic equation systems by the finite element method. Int. J. Numer. Methods Fluids. 4:1043–1063 96. Donea J, Giuliani S, Laval H, Quartapelle L (1984) Time-accurate solution of advection-diffusion problems by finite elements. Comput. Methods Appl. Mech. Eng. 45:123–145 97. Mabssout M, Pastor M (2003) A Taylor-Galerkin algorithm for shock wave propagation and strain localization failure of viscoplastic continua. Comput. Methods Appl. Mech. Eng. 192:955– 971 98. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: Diffuse approximation and diffuse elements. Comput. Mech. 10:307–318 99. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37:229–256 100. Duarte CA, Oden JT (1996) An h-p adaptive method using clouds. Comput. Methods Appl. Mech. Eng. 139:237–262 101. Melenk JM, Babuška I (1996) The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Eng. 139:289–314 102. Oñate E, Idelsohn S (1998) A mesh-free finite point method for advective-diffusive transport and fluid flow problems. Comput. Mech. 21:283–292 103. Oñate E, Idelsohn SR, Celigueta MA, Rossi R, Marti J, Carbonell JM, Ryzhakov P, Suárez B (2011) Advances in the Particle Finite Element Method (PFEM) for Solving Coupled Problems in Engineering. Particle-Based Methods. pp. 1–49 104. Salazar F, Oñate E, Morán R (2011) Numerical modeling of landslides in reservoirs using the Particle Finite Element Method (PFEM). Risk Analysis, Dam Safety, Dam Security and Critical Infrastructure Management, 105. Wie˛ckowski Z (2004) The material point method in large strain engineering problems. Comput. Methods Appl. Mech. Eng. 193:4417–4438 106. Coetzee CJ, Vermeer PA, Basson AH (2005) The modelling of anchors using the material point method. Int. J. Numer. Anal. Methods Geomech. 29:879–895 107. Andersen S, Andersen L (2009) Modelling of landslides with the material-point method. Comput. Geosci. 14:137–147 108. Alonso EE, Zabala F (2011) Progressive failure of Aznalcóllar dam using the material point method. Géotechnique. 61:795–808 109. Jassim I, Stolle D, Vermeer P (2013) Two-phase dynamic analysis by material point method. Int. J. Numer. Anal. Methods Geomech. 37:2502–2522 110. Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron. J. 82:1013 111. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics—theory and application to non-spherical stars. Mon Not R Astron Soc 375–389 112. Monaghan JJ, Lattanzio JC (1985) A Refined Particle Method for Astrophysical Problems. Astron. Astrophys. 149:135–143 113. Benz W (1990) Smooth Particle Hydrodynamics - a Review. Proceedings of the NATO Advanced Research Workshop on The Numerical Modelling of Nonlinear Stellar Pulsations Problems and Prospects. p. 269 114. Monaghan JJ (1992) Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys. 30:543–574 115. Liu G-R, Liu MB (2003) Smoothed Particle Hydrodynamics: A Meshfree Particle Method 116. Li S, Liu WK (2004) Meshfree particle method. 117. Liu MB, Liu GR (2006) Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56:19–36

123

Author's personal copy M. Pastor et al. 118. Monaghan JJ, Kos A (1999) Solitary Waves on a Cretan Beach. J. Waterw. Port, Coastal. Ocean Eng. 125:145–155 119. Monaghan JJ, Kos A, Issa N (2003) Fluid Motion Generated by Impact. J. Waterw. Port, Coastal. Ocean Eng. 129:250–259 120. Takeda H, Miyama SM, Sekiya M (1994) Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics. Prog. Theor. Phys. 92:939–960 121. Monaghan JJ (1994) Simulating Free Surface Flows with SPH. J. Comput. Phys. 110:399–406 122. Monaghan JJ, Kocharyan A (1995) SPH simulation of multi-phase flow. Comput. Phys. Commun. 87:225–235 123. Antoci C, Gallati M, Sibilla S (2007) Numerical simulation of fluid-structure interaction by SPH. Comput. Struct. 85:879–890 124. Vila JP (1999) On particle weighted methods and smooth particle hydrodynamics. Math. Model. Methods Appl. Sci. 09:161–209 125. Bonet J, Kulasegaram S (2000) Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int. J. Numer. Methods Eng. 47:1189– 1214 126. Zhu Y, Fox PJ (2001) Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media. Transp. Porous Media. 43:441–471 127. Ata R, Soulaïmani A (2005) A stabilized SPH method for inviscid shallow water flows. Int. J. Numer. Methods Fluids. 47:139–159 128. De Leffe M, Le Touze D, Alessandrini B (2008) Coastal flow simulations using an sph formulation modelling the non-linear shallow water equations. 3RD ERCOFTAC SPHERIC workshop on SPH applications. pp. 48–54, Lausanne. 129. Vacondio R, Rogers BD, Stansby PK, Mignosa P, Feldman J (2013) Variable resolution for SPH: A dynamic particle coalescing and splitting scheme. Comput. Methods Appl. Mech. Eng. 256:132–148 130. Vacondio R, Mignosa P, Pagani S (2013) 3D SPH numerical simulation of the wave generated by the Vajont rockslide. Adv. Water Resour. 59:146–156 131. Vacondio R, Rogers BD, Stansby PK, Mignosa P (2013) Shallow water SPH for flooding with dynamic particle coalescing and splitting. Adv. Water Resour. 58:10–23 132. Xia X, Liang Q, Pastor M, Zou W, Zhuang Y-F (2013) Balancing the source terms in a SPH model for solving the shallow water equations. Adv. Water Resour. 59:25–38 133. Rodriguez-Paz M, Bonet J (2005) A corrected smooth particle hydrodynamics formulation of the shallow-water equations. Comput. Struct. 83:1396–1410 134. Libersky LD, Petschek AG (1991) Smooth particle hydrodynamics with strength of materials. Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method. pp. 248–257 135. Libersky LD, Petschek AG, Carney TC, Hipp JR, Allahdadi FA (1993) High Strain Lagrangian Hydrodynamics. J. Comput. Phys. 109:67–75 136. Randles PW, Libersky LD (2000) Normalized SPH with stress points. Int. J. Numer. Methods Eng. 48:1445–1462 137. Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput. Methods Appl. Mech. Eng. 190:6641–6662 138. Bui HH, Sako K, Fukagawa R (2007) Numerical simulation of soil-water interaction using smoothed particle hydrodynamics (SPH) method. J. Terramechanics. 44:339–346 139. Bui HH, Fukagawa R, Sako K, Ohno S (2008) Lagrangian meshfree particles method (SPH) for large deformation and failure flows of geomaterial using elastic-plastic soil constitutive model. Int. J. Numer. Anal. Methods Geomech. 32:1537–1570 140. Blanc T, Pastor M (2012) A stabilized Fractional Step, RungeKutta Taylor SPH algorithm for coupled problems in geomechanics. Comput. Methods Appl. Mech. Eng. 221–222:41–53

123

141. Blanc T, Pastor M (2012) A stabilized Runge-Kutta, Taylor smoothed particle hydrodynamics algorithm for large deformation problems in dynamics. Int. J. Numer. Methods Eng. 91:1427–1458 142. Lastiwka M, Quinlan N, Basa M (2005) Adaptive particle distribution for smoothed particle hydrodynamics. Int. J. Numer. Methods Fluids. 47:1403–1409 143. Feldman J, Bonet J (2007) Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems. Int. J. Numer. Methods Eng. 72:295–324 144. Monaghan J, Gingold R (1983) Shock simulation by the particle method SPH. J. Comput. Phys. 52:374–389 145. Morris JP (1996) Analysis of smoothed particle hydrodynamics with applications. Monash University, Australia 146. Chen JK, Beraun JE, Carney TC (1999) A corrective smoothed particle method for boundary value problems in heat conduction. Int. J. Numer. Methods Eng. 46:231–252 147. Swegle JW, Hicks DL, Attaway SW (1995) Smoothed Particle Hydrodynamics Stability Analysis. J. Comput. Phys. 116:123– 134 148. Monaghan JJ (2000) SPH without a Tensile Instability. J. Comput. Phys. 159:290–311 149. Dyka CT, Randles PW, Ingel RP (1997) Stress points for tension instability in SPH. Int. J. Numer. Methods Eng. 40:2325–2341 150. Dyka CT, Ingel RP (1995) An approach for tension instability in smoothed particle hydrodynamics (SPH). Comput. Struct. 57:573–580 151. Randles PW, Libersky LD (1996) Smoothed Particle Hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Eng. 139:375–408 152. Blanc T, Pastor M (2013) A stabilized Smoothed Particle Hydrodynamics, Taylor-Galerkin algorithm for soil dynamics problems. Int. J. Numer. Anal. Methods Geomech. 37:1–30 153. Peraire J, Zienkiewicz OC, Morgan K (1986) Shallow water problems: A general explicit formulation. Int. J. Numer. Methods Eng. 22:547–574 154. Monaghan JJ (1982) Why Particle Methods Work. SIAM J. Sci. Stat. Comput. 3:422–433 155. Bonet J, Kulasegaram S, Rodriguez-Paz MX, Profit M (2004) Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems. Comput. Methods Appl. Mech. Eng. 193:1245–1256 156. Vacondio R, Rogers BD, Stansby PK, Mignosa P (2012) SPH Modeling of Shallow Flow with Open Boundaries for Practical Flood Simulation. J. Hydraul. Eng. 138:530–541 157. Monaghan JJ (1985) Particle methods for hydrodynamics. Comput. Phys. Reports. 3:71–124 158. Pastor M, Quecedo M, González E, Herreros MI, Merodo JAF, Mira P (2004) Simple Approximation to Bottom Friction for Bingham Fluid Depth Integrated Models. J. Hydraul. Eng. 130:149– 155 159. Manzella I (2008) Dry rock avalanche propagation: unconstrained flow experiments with granular materials and blocks at small scale. 160. Sossio R, Crosta G (2007) Thurwieser Rock avalanche. 161. King JP (2001) The 2000 Tsing Shan Debris Flow. 162. Bishop AW, Hutchinson JN, Penman ADM, Evans HE (1969) Geotechnical investigations into the causes and circumstances of the disaster of 21st October 1966. In Aselection of technical reports submitted ot the Aberfan Tribunal. 163. Jeyapalan JK, Duncan JM, Seed HB (1983) Investigation of Flow Failures of Tailings Dams. J. Geotech. Eng. 109:172–189 164. Jin M, Fread DL. A.S. of C.E. (1997) One-Dimensional Routing of Mud/Debris Flows Using NWS FLDWAV Mode. Debris-flow hazards mitigation: mechanics, prediction, and assessment. pp. 687–696.

Depth Averaged Models for Fast Landslide Propagation

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