Application of FLATModel, a 2D finite volume code ... - Hydrate Project

Original Article Landslides (2008) 5:127–142 DOI 10.1007/s10346-007-0102-3 Received: 9 March 2007 Accepted: 27 September 2007 Published online: 27 November 2007 © Springer-Verlag 2007

Vicente Medina . Marcel Hürlimann . Allen Bateman

Application of FLATModel, a 2D finite volume code, to debris flows in the northeastern part of the Iberian Peninsula

Abstract FLATModel is a two-dimensional shallow-water approximation code with corrections and modifications that create a simulation tool adapted to debris-flows behaviour. FLATModel uses the finite volume method with the numerical implementation of the Godunov scheme and includes correction terms regarding the effect of flow over high slopes and curvature. Additionally, the stop-and-go phenomenon, the basal entrainment and a correction regarding the front inclination of the final deposit are incorporated into FLATModel. In addition, different flow resistance laws were integrated in the numerical code including Bingham, Herschel– Bulkley and Voellmy fluid model. Firstly, our numerical model was validated using analytical solutions of a dam-break scenario and published data on a laboratory experiment. Secondly, three real events, which occurred in the northeastern part of the Iberian Peninsula, were back-calculated. Although field observations of the three events are not very detailed, the back-analyses revealed interesting patterns on the flow dynamics, and the numerical results generally showed good agreement with field data. Comparing the different flow resistance laws, the Voellmy fluid model presents the best behaviour regarding both the flow behaviour and the deposit characteristics. Preliminary simulation runs incorporating the effect of basal entrainment offered satisfactory results, although the final volume is rather sensitive on the selected friction angle of channel-bed material. The outcomes regarding the correction of the calculated front inclination of the final deposit showed that this implementation strongly improves the simulation results and better represents steep fronts of final deposits. Keywords Debris flow . Finite volume . Spain . Pyrenees . Stop-and-go . Basal entrainment . Godunov Scheme Introduction One of the most complex processes in nature is the flow of mixtures of water and sediment as debris flows, mud flows or hyper-concentrated flows. The need to study debris flow lies in the high energy of the phenomenon that sometimes causes severe damages to structures. The dynamics of the flow can start from a single landslide and may evolve into more complex flow behaviour. There is still a big distance between the nature of the phenomena and the mathematical models, although improvements have been achieved during the last decades. Many different one-dimensional (1D) or two-dimensional (2D) models have been proposed and especially 2D simulations strongly improved insights in the dynamic behaviour of such flows (Laigle and Coussot 1997; Denlinger and Iverson 2001; Chen and Lee 2003; Pitman et al. 2003; Denlinger and Iverson 2004; McDougall and Hungr 2004; Zanuttigh and Lamberti 2004; Pudasaini et al. 2005; Rickenmann et al. 2006).

One of the main purposes of the present work is to introduce FLATModel, a 2D model that uses the finite volume method (FVM) with the Godunov Scheme. Several approximations of the geometry and the physical behaviour are implemented into FLATModel to generate an accurate code to study the flow dynamics and accumulation mechanisms. The model has been applied to two types of calibration cases and three real debris flows occurred in the northeastern part of the Iberian Peninsula. The objectives of the back-analyses of these events were manifold. Firstly, different flow resistance laws were compared. Secondly, the stop-and-go phenomenon implemented into FLATModel was evaluated. Thirdly, the effect of basal entrainment was studied. Fourthly, the correction mechanism for the calculation of the front inclination of the final deposit was tested. Description of numerical model Debris-flow physics imply conservation laws evaluated in fluid– solid–gas mixture. It is possible to define the mass and momentum conservation equations applied to this type of flow (e.g. Atkin and Craine 1976), but the problem description includes mainly three drawbacks. Firstly, a large amount of variables has to be considered including stress tensors and dynamic parameters for every phase. Secondly, the numerical challenge because of the mathematical character of conservation laws and strongly coupled relations between phases has to be resolved. Finally, the third and the most important problem refers to the constitutive relations of the mixture and the governing equations for the model. Serious attempts have been carried out to define a solution of these drawbacks and to solve them numerically (e.g. Syamlal et al. 1993), but the scales of these model applications are not comparable to debris-flows scenarios. Thus, it is necessary to introduce several simplifications to reduce the physical complexity of the problem. The first group of these simplifications focuses on the constitutive relations and mixture phases. Most of the recent models consider monophasic or two-phase (liquid and solid) mixtures. The second group of simplifications is related to the dimension and was introduced to diminish the computational needs of the simulations. The physical problem can be reduced in dimension assuming: (1) 1D models (Hungr 1995; Jin and Fread 1999; Imran et al. 2001) or (2) 2D ones (Hutter et al. 1993; O’Brien et al. 1993; Iverson and Denlinger 2001; Pitman et al. 2003; Denlinger and Iverson 2004). This reduction in dimension is generally incorporated using shallow water hypothesis and depth integration. Governing equations The general classification of FLATModel can be summarised as a monophasic, 2D model with the z-axis normal to the bed. As in other models, the development of the 2D-governing equations

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Original Article starts from three-dimensional conservation laws applied to monophasic material and follows the typical depth integration process (Vreugdenhil 1994). At the end of this process, a 2D equation system can be obtained. The depth integration can be applied using the shallow water hypothesis and integrating the Navier–Stokes equations for thin 1D flows over smooth terrain. If we adapt this hypothesis to deal with steep slopes or 2D flows, it is necessary to carefully define all the geometric coordinates. This is of particular interest regarding debris flows because the terrain slopes and curvatures are critical for computations of such type of processes. The influence of curvatures (terrain and free surface) has been considered for a long time. A first theoretical approach was developed by Boussinesq (1877). His approach was restricted to weakly curved beds and developed with Cartesian coordinates. Fenton (1996) included new terms using Boussinesq equations to deal with strong curvatures, while Dressler (1978) modified the Saint Venant equations to include effects of curvature of streamlines for flows over curved beds. Dressler (1978) used an orthogonal bed-fitted curvilinear coordinate system. His approach is more complicated than the Cartesian approaches, but it provided a good approximation for non-hydrostatic situations related to curvature and steep slopes. In general, the definition of a fixed local Cartesian coordinate system is only possible in few cases. In this paper, the term “local” is used to specify that the flow depth is defined normal to the bed. For example, a fixed local Cartesian coordinate system can be assumed in a straight channel with constant slope. In curved channels, however, it is necessary to define a local curvilinear coordinate system as used in the models presented by Iverson and Denlinger (2001) or Gray et al. (1999). In such coordinate system, only the downslope curvature is used. On the other side, a more complex curvilinear coordinate system is necessary in curved and twisted channels. Such approach has been implemented into numerical models based on the thalweg (e.g. Pudasaini et al. 2005) or using principal curvatures (e.g. Quecedo et al. 2004). A completely different approach by Denlinger and Iverson (2004) uses a global Cartesian coordinate system integrating the equations along the vertical direction. In FLATModel, after applying integration in the direction normal to the bed and corrections related to slope and curvature, we obtain an equations system similar to the one proposed by Iverson and Denlinger (2001), Rickenmann et al. (2006) or McDougall and Hungr (2004). However, the model of Iverson and Denlinger (2001) includes lateral stresses and pore pressure that are neglected here. Finally, the basic equation of FLATModel can be expressed by: 1 1 0 0 1 0 hv h hu @B C @ B 2 @B 2 C C huv A @ hu A þ @ hu þ gp h2 A þ @ @t @x @y 2 hv2 þ gp h2 huv hv 1 0 (1) 0 B hg tan α  S  C ¼@ p x fx A   h gp tan αy  Sf y where t is the temporal coordinate, h is flow depth, u, v are the velocities in the x, y directions, respectively, gp is the corrected gravity, Sf is the energy gradient and α is the terrain surface angle. The sub-index x, y indicates vector component. 128

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The pressure term is affected by flow streamline characteristics, slope and curvature. Both slope and curvature are measured in the velocity flow path direction to modify the two momentum conservation equations, in which the terms appear. The normal stresses acting at the bottom or contact surface are affected by the centripetal acceleration that was included into the FLATModel by making two modifications. First, adding a centripetal acceleration in the momentum pressure term to obtain the corrected gravity, gp, gp ¼ g cos θ þ

jV j2 rc

(2)

where g is the gravity, θ is the angle defined by the gravity and pressure gradient, or in other words the angle defined by the horizontal plane and the velocity direction, which is tangent to the stream line, jV j is the velocity modulus and rc is the radial curvature evaluated in the velocity direction. Then, the pressure force per unit length at the face, Fp, can be computed by Fp ¼ gp

ρh2 2

(3)

where ρ is the flow density. The second modification refers to the bed normal stress, σb, which is computed by σb ¼ gp ρh

(4)

Because FLATModel calculates the depth in vertical direction, the real flow depth is determined by the correction angle θ in every model cell before applying the conservation equations. The same procedure also corrects the space increments dx, dy to reflect the real flow path length and to compute the real flux gradients. Finally, Eq. 1 can be expressed in a condensed form by @ @ @ ðU Þ þ ðF ðU ÞÞ þ ðGðU ÞÞ ¼ ðT ðU ÞÞ @t @x @y

(5)

in which U represents the physical magnitude, while F and G represent the corresponding fluxes of the hydraulic magnitudes. The independent vector T is the denominated source term.

Computational scheme FLATModel uses the FVM as described by Leveque (2002) because of its capacity to deal with discontinuities, large function gradients and other advantages of the mathematical model. The FVM is based on the discrete form of the integral of Eq. 1 but changes the integral fluxes using the divergence theorem. This is the aim of the numerical model of the governing system because it conducts to a physical interpretation of the different terms of the systems. The integral equivalence of Eq. 1 in the FVM form is given by: @ @t

ZZ

Z UdΩ þ

Ω

ZZ T ðU ÞdΩ

ðF; GÞndS ¼ S

(6)

Ω

in which Ω represents the selected volume with surface S and n is a unit normal surface vector. From Eq. 6 and using a square base of

the control volume of height, it is possible to get a discrete form equation applying the rule of the middle point for each of the lateral faces of the volume Uinþ1 j ¼ Uinj þ

 Δt   Δt    Fi1=2 j  Fiþ1 Gi j1=2  Gi jþ1=2 þ 2 j = Δx Δy

þ Tij Δt

(7)

where a 2D grid was used, in which Uij corresponds to the average   ; Fiþ1=2j are the flows that values in the interior of the cell, Fi1=2j pass through the cell by its left and right faces and Gi j1=2 ; Gi jþ1=2 are the corresponding fluxes that enter or leave the cell through the upper and lower faces. Finally, the source term Tij is evaluated in the middle of the cell. Because of the inclusion of the centripetal forces and the pressure correction angle, θ, dependent on flow type, the system includes new non-linear terms like centripetal acceleration inside the flux terms. To reduce the complexity of the numerical solution, these elements are considered constant in the interface Riemann problem and act only as modifications of the gravity. The details of how to calculate those fluxes can be found in any document about Riemman solvers (e.g. Leveque 2002). Regarding the computations of FLATModel, we have used a Harten–Lax–Leer–Contact Riemann solver (Toro et al. 1994) in combination with the transversal solver described by Leveque (2002) and a “minmod flux limiter.” Finally, boundary conditions are solved in FLATModel as described by Medina et al. (2006). Terrain considerations FLATModel is based on Digital Elevation Model (DEM). The model equations include first and second derivatives of the terrain surface to use a coherent terrain description and to obtain a continuous surface. The creation of such a smooth surface is not simple, especially in 2D problems. FLATModel implements a Regularised Spline Tension (RST) method (Mitasova and Mitas 1993), which is a global interpolation. To apply RST, it is necessary to solve a linear equation system called “m×n”, being m the number of rows in the grid and n the columns number. The pressure correction angle depends on the flow type and should be recomputed in every time step using flow variables. Because this is computationally too time consuming, RST is only applied at the start of the computation to calculate the accurate curvatures of the terrain and pressure correction angles. Flow resistance laws applied One of the most important parts of the model is the incorporation of different rheological flow behaviours. Most of the known flow resistance laws were included into FLATModel, but only three will be discussed herein: Bingham, Herschel–Bulkley and Voellmy. All of these three models were proposed to be adequate in recent studies on debris-flow modelling (Imran et al. 2001; Naef et al. 2006; Rickenmann et al. 2006). The first two models are typically used to simulate different kind of liquids with thinning, thickening or Newtonian behaviours. The third one is used to simulate a kind of granular mixture of solids and fluid (water). The dynamic characteristic of such granular mixtures is on one hand Mohr– Coulomb behaviour, and on the other hand, it shows a turbulent

response. Recent backanalyses on debris-flow events showed that the Voellmy model reveals good characteristics to model granular debris flow (e.g. Rickenmann and Koch 1997). FLATModel incorporates the three flow resistance laws analogous as used in 1D simulation, which is a common way in 2D numerical modelling of debris flows (e.g. O’Brien 1993; Rickenmann et al. 2006). Thus, the following rheological expressions are indicated by the basal shear stress, τb. FLATModel implements Bingham rheology by the following expression (Bingham 1922):  τ b ¼ τ 0 þ μm

dV dz

 (8)

where τ0 is the threshold basal shear stress to provoke the initiation of movement, μm the Bingham viscosity of the fluid and dV/dz the velocity gradient along the depth of the flow. The Herschel-Bulkley rheology (Coussot 1994) was applied by: τb ¼ τ0 þ k

 n dV dz

(9)

where k is the consistency and n the index of the fluid behaviour. The third flow resistance law is the Voellmy model (Voellmy 1955) and was incorporated into FLATModel by: τ b ¼ gρðh cos θ tan φ þ ðV=Cz Þ2 Þ

(10)

where φ is internal friction angle of the flowing mass and Cz the Chezy coefficient. The differences that can limit the use of the Eqs. 8 to 10 are their behaviour (Fig. 1). While the frictional Voellmy model needs a slope to start the motion of the material with independence of the flow height, the Bingham and Herschel– Bulkley model only needs a flow height to start moving, even if the material is in a horizontal plane. This difference is very important, when a debris flow is modelled, and may explain why Bingham rheology did often not fit very well during back-analysis of real events. Implementation of entrainment The basal incorporation of material or entrainment is a very common characteristic of debris-flow dynamics (Takahashi et al. 1992; Rickenmann et al. 2003). A small volume of initial mass can incorporate ten times or more of its volume in some circumstances of slope and type of material (Egashira et al. 2001; Papa et al. 2004). The main condition for basal erosion is that the

Fig. 1 Stability criteria for different rheologies

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Original Article Fig. 2 Implementation of entrainment into FLATModel. a Static approach and b dynamic approach

bed shear stress of the flow is sufficiently high to incorporate part of the bed into the flow. Two different approximations were assumed to simulate the effect of entrainment: the static and the dynamic approach. Both approximations represent preliminary approaches to better understand entrainment mechanisms of debris flow and may provide general patterns on this complex topic. The static approach (Fig. 2a) can be easily described with soil mechanics concepts. It considers a static equilibrium between the flow frictional forces (bed shear forces) and the basal resistance

forces τres. This equilibrium should be achieved at each computational time step: τ b ¼ τ res

(11)

The bed shear stress represents flow friction, while the resistant shear stress can be represented by the Mohr–Coulomb failure criterion for an infinite slope τ res ¼ c þ hρg cos θ tan φbed

(12)

 Fig. 3 Test runs regarding entrainment. a Parameters of test case, b influence of bed, and c Maximum eroded depth for =3.7°, Cz ¼ 10 m1=2 s and bed =35°

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Fig. 4 Voellmy parameter related to velocity function for calculation of front inclination of final deposit

in which c is the cohesion and φbed the bulk friction angle of the bed material. Streamlines are assumed to be parallel to bed in the numerical approximation because α=θ. Because FLATModel uses a homogeneous flowing mass, it cannot distinguish the different phases of water and sediment. Two-phase models, however, should reduce the flow depth in Eq. 12 to compute the effective stresses and not the total stresses. FLATModel checks the equilibrium conditions of Eq. 11 in every time step. If there is no equilibrium, the model calculates the magnitude of entrainment necessary to achieve equilibrium by τ b þ hent ρg sin θ ¼ c þ ½h þ hent ρg cos θ tan φbed

τ b  τ res ρg ðcos θ tan φbed  sin θÞ

(14)

Finally, entrainment conditions can be established for a cohesionless torrent bed material, if τ b > hρg cos θ tan φbed

(15)

The meaning of this condition is that the bed shear forces have to be greater than the resistance forces to provoke basal erosion. An additional condition regarding bed stability is necessary to guarantee solution. This conditions is given by cos θ tan φbed > sin θ ) tan φbed > tan θ

@z 1 ¼ ðτ b  τ res Þ @t ρV

(17)

in which ∂z/∂t is the velocity of entrainment. Equation 12 can be incorporated into Eq. 17 obtaining @z 1 ¼ ðτ b  c  hρg cos θ tan φbed Þ @t ρV

(18)

(13)

where hent is the entrainment or erosion depth (Fig. 2a). This expression can be transformed to obtain an explicit value of hent hent ¼

This means that the terrain collapses if its slope is greater than its internal friction angle. Once hent has been calculated, FLATModel recalculates the momentum of the cell because of the fact of the incorporation of new mass with a low quantity of momentum. Obviously, the resultant velocity diminishes because of entrainment. The second method to simulate entrainment applies a dynamic equilibrium approach. The hypothesis is nearly the same as described for the static approach (Fig. 2b). The entrainment occurs, if the bed forces are greater than the resistance. The difference of the dynamic approach is because the new incorporated material is accelerated to the mean velocity of the flow (Fraccarolo and Capart 2002). Thus, the quantity of new incorporated mass depends on the availability of momentum, given by

(16)

Applying this approach, some considerations should be done. First of all, if bed shear stresses are computed using a Coulomb flow resistance law, Eq. 11 changes into c  hρg cos θ tan φbed ¼ c  hρg cos θ tan φ

(19)

Therefore, no entrainment occurs, if the same friction angle for the flowing mass, φ, and the bed material, φbed, is used. The second consideration is related to the effective stresses and the incorporation of the pore pressure parameter λ proposed by Iverson and Denlinger (2001). If modelling includes λ, which varies between 0 (dry debris) and 1 (completely saturated debris), the basal resistance stresses change to τ res ¼ c þ hρg ð1  λÞ cos θ tan φbed

(20)

Finally, λ indicates which part of the pressure corresponds to the lithostatic stress and which part to the pore pressure. The factor

Fig. 5 Comparison between analytical solutions of one-dimensional dambreak scenarios and numerical results calculated by FLATModel. a Dam-break characterised by an initial flow depth of 20 m and material of zero bed friction across a horizontal surface. b Dambreak characterised by an initial flow depth of 10 m and material of 25° bed friction across a surface inclined by 30°

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Fig. 6 Comparison between the laboratory experiment of a sand avalanche (left column; Iverson et al. 2004) and numerical results calculated by FLATModel (right column). Flow depth is plotted on a topographic base with a contour interval of 1 cm

(1−λ) multiplies the value of tan φbed, which has the same effect as a modification of φbed. Finally, the dynamic approximation implemented in the FLATModel was applied in the present study. To validate this dynamic approach, some preliminary computations were made using theoretical cases. A 500-m-long channel was defined including during the first 100 m a steep inclination of 40° followed by a smooth part of 4° (Fig. 3a). The initial amount of material was 200 m3, and erosion is not limited at depth along the whole channel. The entrainment was tested using the Voellmy fluid model and different rheological parameters (φ, Cz). Then the total amount of incorporated material from the bed was calculated for different φbed values. Figure 3b shows the influence of the friction angle of the bed material on the flow volume for two different Voellmy configurations. As it can see, the Cz value is very important. However, it should be mentioned that if the static approach would have been used, the entrainment angle φbed could not have been smaller than the slope angle θ. If this occurs, an unstable situation is reached. In the presented case, however, FLATModel could be used because the dynamic approach was applied. Thus, Fig. 3b shows large material erosion volumes for φbed smaller than 40°. In addition, Fig. 3c presents the total eroded depth along the channel for the simulation using φ=3.7°, Cz = 10 m1/2s and φbed =35°. The maximum erosion depth exceeds 2.5 m at the end of steep slope, and the final volume is 250 m3. Implementation of “stop-and-go” mechanism Because of the dynamics of the flow and the effect of the selected rheology, the flowing mass can sometimes stop its movement. 132

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However, it can start the motion again because of the surrounding activity. This stop-and-go process is not simple to simulate in an explicit scheme, and it is important to apply some numerical restriction to have a good evaluation. The main problem considers the question: when does a cell reach conditions of final repose? The problem in an explicit numerical method is based on the fact that firstly, the fluxes at the faces of the cells are evaluated and “a posteriori,” the source frictional term is calculated. FLATModel operates in the following way to define a cell in total or final repose. Whether a cell with a null-velocity value will restart motion or is at final repose depends on the surrounding flow dynamics of the material. A cell is in motion, if the following three conditions are satisfied: (1) the cell was in motion in the last time step, and (2) some of the neighborhood cells of a null velocity cell were in motion in the last time step. FLATModel uses an explicit scheme controlled by the Courant number. This means that any wave or information that enters in a cell is incapable to flow throughout the front face of the same cell because of the Courant restriction. That means that any motion in a neighborhood cell can activate a null velocity cell until the next time step. This is a chain process that can activate a cell through the motion of a neighborhood cell. (3) The geometric slope of the material is greater than the internal friction angle of the material. Thus, FLATModel incorporates a recalculation of zones in which the angle of the material is greater than the internal friction angle, until all the surfaces have less or equal slope to that of the material friction angle. With this technique, it is possible to move cells that had been stopped before but receive material with sufficient momentum

Fig. 7 Topographic maps of the three catchment indicating the flow path of the events simulated. a Location of the three drainage basins. b Cardemeller torrent at Pal, c Font de la Llum torrent at Montserrat and d Jou Torrent at La Guingueta

from the neighborhood cells. In a complex process, a cell can be moved and stopped more than once, similar to reality. Calculation of the front slope of the final deposit If the simulations use the Voellmy rheology for numerical modelling, the front inclination of the final deposit is normally similar to the Voellmy parameter φ. However, ϕ values are far from the static dry internal friction angle, φstat, which governs the slopes in the deposit boundaries of debris. To simulate this behaviour the Voellmy parameter φ has to be changed dynamically during the simulation. This ϕ adaptation is of crucial relevance during the accumulation phase, when the flow is near to stop. FLATModel

implements a correction of this feature increasing the ϕ value, if flow velocity is lower than a selected limit V1 (Fig. 4). The magnitude of φ changes linearly from the initial friction angle to the static dry internal friction angle. Model verification FLATModel was tested by two types of verification cases. Firstly, the model was checked against the 1D analytical solution of dambreak scenarios, and secondly, numerical simulation was compared to data obtained from sand avalanche experiments across an irregular laboratory flume. Additional model validation cases regarding clear-water flow are described in Bateman et al. (2006). Landslides 5 • (2008)

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3 Fig. 8 Backanalysis of the Font de la Llum event at Montserrat. a Depth of final deposit computed by best-fit Voellmy fluid model (μ=tan=0.13 and  Cz ¼ 10 m1=2 s). b Depth of final deposit computed by best-fit Bingham model (τ0 =0.75 kPa, μm =0.450 Pa s). c Maximum velocity and d maximum flow depth applying the same Voellmy parameters as in a

Analytical dam-break scenarios Mangeney et al. (2000) published a dam-break scenario, which represents a 1D movement of debris on an infinite uniform channel. The resistance to the flow generated on the channel is characterised by an angle of bed friction. The internal friction is assumed to be zero. The analytical solution of the debris motion provides flow profiles at any given time after the removal of the dam. In our first example, flow movement was simulated for an initial depth of debris of 20 m, a horizontal channel and zero friction at the bed (Fig. 5a). The flow profile simulated by FLATModel at 10 s indicates that numerical results are identically as the analytical ones. In our second example, the initial depth of debris was taken as 10 m, the channel inclination as 30° and the basal friction angle as 25°. Figure 5b illustrates flow profiles at 10 and 20 s indicating some minor differences between numerical and analytical results. These differences may affect maximum front velocity of a granular flow but seem to be acceptable for our debris-flow simulations. Differences could be reduced by applying a Runge–Kutta temporal discretisation. Laboratory experiment Sophisticated laboratory flume experiments were performed for sand avalanches to obtain insights of dry granular flow behaviour across irregular terrain (Iverson et al. 2004). In this experiment, we back-calculated experiment A, which was carried out for angular sand particles with an internal friction angle of ~44° and a variable angle of bed friction (see Iverson et al. 2004 for more detailed information on experiment setup and properties). Coulomb flow resistance law was implemented into FLATModel for this back-analysis. The two different friction angles 20 and 23.47°, which depend on the material of the flume, were used for bed friction, while internal friction was not incorporated. Comparisons between simulation results and data from the flume experiment are given by isopach maps of flow depth at different times (Fig. 6). Model predictions match many of the dynamic characteristics of the sand avalanche and errors of front velocity, final extension and depth of material are generally of minor importance. Back-analyses of real debris flows Because of the particular characteristics of the Mediterranean climate that affect the northeastern part of the Iberian Peninsula,

debris flows do not occur as frequent as in other mountainous areas. Generally, debris flows can be triggered by two different types of rainfall events: (1) short and locally restricted rainstorms of high to very high intensity that happen during summer and (2) rainfall episodes of moderate to high intensity that affect a large area during one or few days in late summer and autumn (Corominas et al. 2002; Hürlimann et al. 2003). In the present study, three events have been selected for numerical simulation by FLATModel. They took place at different locations in the Eastern Axial Pyrenees and the Catalan Pre-litoral Mountain range (Fig. 7a). These events probably represent the three most important debris flows that occurred in our study area during the last 30 years, but unfortunately, field data are limited. Thus, our simulations only provide general patterns on flow dynamics of these events. Font de la Llum torrent, Montserrat Field data The Font de la Llum catchment covers a small area of 0.46 km2 at the eastern flank of Montserrat massif. The massif is situated about 30 km northwest of Barcelona in the Catalan Pre-litoral Mountain range (Fig. 7a), and maximum altitudes reach about 1,250 m a.s.l. From a geological point of view, the bedrock of the drainage basin involves sequences of conglomerates, sandstones and lutites, which are locally covered by a colluvium layer. The higher part of the basin is characterised by very steep slopes with inclinations of up to 60°, while the lower part includes a smooth morphology with small slope angles (Fig. 7c). The selected event was triggered on June 10, 2000, by an extremely intense thunderstorm, which provoked a failure of the colluvium deposit in the highest part of the massif. The convective rainstorm lasted about 4 h including a maximum intensity of ~80 mm/h and accumulated about 160 mm rainfall. Field observations and interpretation of aerial photographs indicate that the initially failed mass of the terrain was rapidly transformed into a flow. Another characteristic was that almost the entire superficial deposit was incorporated into the flow along the trajectory. A total volume of about 10,000 m3 was estimated in the elongated accumulation zone (Fig. 7c). The morphologic and sedimentologic characteristics observed in the field suggest that the event descended the torrent in various surges of hyperconcentrated flows and/or debris flows. Simulation results FLATModel simulated the event at Font de la Llum as one surge of 10,000 m3. The high-quality DEM available was transformed into a grid of 0.3-m cell size, and runs were carried out at this precision.

Fig. 9 Hydrographs of Montserrat event illustrating the “stop -and-go” phenomena. a Sections 5, 6 and 7 (see Fig. 8 for location). b Evolution of the flow area and the discharge at section 7 indicating remobilisation of material

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3 Fig. 10 Backanalysis of the debris flow occurred in the Cardemeller torrent at Pal 

applying Voellmy fluid (μ=tan=0.22 and Cz ¼ 12 m1=2 s). a Maximum flow velocity, b maximum flow depth, c depth of final deposit. d Erosion depth calculated along flow trajectory using the dynamic approach and bed =31°

First, the three flow resistance laws were compared, and a sensitivity analysis of the rheological parameters was carried out. Many different simulation runs were executed, and both the extension and the depth of the final deposit were compared with the data observed in the field. The results of this comparison indicate that the most realistic simulations were obtained by the Voellmy fluid model and material properties of μ=tanϕ=0.13 and Cz ¼ 10  m1=2 s (Fig. 8a). In contrast, no good agreement with the real deposit shape was obtained applying the Bingham model. Figure 8b illustrates outcomes calculated for τ0 =0.75 kPa, μm =0.450 Pa s and ρ=1,500 kg/m3. First approximations of τ0 could be calculated using the equilibrium equation for Bingham flows (Fig. 1). The depth of the final deposit is about 1.5 m, and the slope in this area is 5.7°. Therefore, an initial approximation of a value for τ0/ρ should be about 1.4 m2/s2. Previous studies applying viscous rheology proposed values of 0.4 (Chen and Lee 2002), 0.1 to 0.5 (Rickenmann and Koch 1997) or 0.1 to 1 m2/s2 (Pastor et al. 2004). The same references also propose values for viscosity (μm/ρ), which should be around 0.05 m2/s (in our case, the best-fit value was 0.3 m2/s). Finally, Herschel–Bulkley rheology was tested using parameter values proposed by previous works (Coussot et al. 1998). Once more, very poor results were computed using many different   parameter combinations (τ 0 ρ ¼ 0:1  1 m2 =s , k ρ ¼ 0:05  1 Pa s1=3 and n=0.33). Although this back-analysis does not include additional information on other important data such as flow velocity or flow area observed at some locations along the trajectory, the outcome indicates that the event may be characterised by a granular flow behaviour. Additionally, some detailed results of the Voellmy simulation are presented illustrating only the final part of the flow trajectory. This part includes the end of the steep section and the smooth section of the elongated accumulation area. Figure 8c illustrates maximum flow velocity, and Fig. 8d shows maximum flow depth. Simulation results reveal that the debris flow or hyper-concentrated flow reached locally a maximum velocity of about 16 m/s at the end of the steep section, while velocity strongly decreases in the lower smooth section, where accumulation starts. This local maximum velocity slightly surpasses the estimates obtained by the application of the super-elevation approach (Johnson and Rodine 1984), which gave about 12 m/s for the average front velocity at the same position. Regarding simulated flow depth, maximum values of more than 8 m were computed at the end of the steep section, where the flow is strongly confined and flow width is very small. Maximum flow depth rapidly decreases in the accumulation and values generally range from 0.2 to 2 m. The runs of the Font de la Llum event also provided illustrative data on how the stop-and-go mechanism could be simulated by FLATModel. Figure 9a shows different hydrographs at the three successive sections 5, 6 and 7. All these sections are located in the accumulation zone of the torrent, where the stop-and-go effect was clearly visible during numerical modelling. It can be seen that flow

hydrographs of the first two sections exhibit only one peak; nevertheless, the hydrograph of section 7 shows two peaks. These two peaks are illustrated in Fig. 9b together with the curve indicating the evolution of the flow area. Both peaks present discharges of ~12 m3/s, but the second peak is the result of an important deposit failure upstream. The red circle in Fig. 9b points out a small local failure, which can be clearly noted as a negative evolution of the flow area previous to the larger peak. Cardemeller torrent, Pal Field data The Cardemeller catchment near Pal is situated in the Principality of Andorra and covers an area of 1.7 km2. From a geological point of view, the drainage basin is settled in the Eastern Axial Pyrenees, and the basement principally consists of Devonian and CambroOrdovician rocks including meta-limestones, phyllites and slates. Bedrock is generally covered by superficial deposits of glacial origin and colluvium. The initiation of the debris flow took place in a road cut excavated in colluvium and mobilised several hundred cubic meters (Fig. 7b). The debris flow was provoked during the widespread floods of November 7 and 8, 1982. A total rainfall amount of about 180 mm in 48 h was estimated for the Pal area using nearby rain gauges that unfortunately only recorded the total daily precipitation. The initial slope failure rapidly transformed into a debris flow and strongly increased volume by important entrainment of material along the flow trajectory. A final volume of about 5,000 m3 was calculated by field observations just after the event. The accumulation zone of the debris flow covered the higher part of the fan and destroyed the road at the fan apex (Fig. 7b). Simulation results The main purpose of the back-analysis of the Cardemeller debris flow was the validation of the implementation of basal entrainment into FLATModel. In each simulation, a start volume of 500 m3 was released at the road cut, where initial failure occurred in 1982 (Fig. 7b). The topographic grid used at this catchment included rectangular cells of 1 m size. Preliminary simulations showed that the Voellmy fluid model provided again most reasonable results (Bateman et al. 2007). In this case, the rheological parameters of the final runs were μ=tanϕ=0.22  and Cz ¼ 12 m1=2 s . Regarding the analysis on basal erosion, the parameter ϕbed, which governs the entrainment condition, was assumed to be ϕbed = 31°. This value would correspond to a pore pressure parameter of 0.3 and an internal friction angle of 40° (Eq. 20), which coincide with the ones proposed by Iverson and Denlinger (2001) and Pudasaini et al. (2005). General results on the flow dynamics of the Cardemeller debris flow are presented first, while the sensitivity analysis on the basal entrainment will be described later. Figure 10a shows maximum velocity and Fig. 10b illustrates maximum flow depth. Maximum flow velocity reaches about 7 m/s in the higher confined section, whereas maximum flow depth greater than 2 m can be observed at several points along the flow trajectory. The extension of the final deposit (Fig. 10c) coincides more or less with the area observed just after the event by field data (see Fig. 7b). Most differences can be explained by the topographic information used for the simulation. Because an actual DEM was incorporated, significant variation can Landslides 5 • (2008)

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Fig. 11 Effect of entrainment during the Cardemeller debris flow applying the dynamic approach and different values of bed. a Evolution of debris-flow volume with time. b Relation between bed and final volume of debris flow

appear between this DEM and the pre-event relief. This discrepancy especially affects the depth of the final deposit, which is strongly related to morphologic anomalies. All these uncertainties, however, do not alter the runs analysing the effect of entrainment because erosion was limited to the higher part of the flow trajectory. Figure 10d shows the erosion depth computed in the higher section of the trajectory. Simulation results indicate that maximum entrainment occurs at steep and confined points, where erosion depths of up to 2 m were calculated. A curious feature is the step-and-pool morphology, which is associated with the areas of erosion. This fact seems to be correlated with both the torrent width and some kind of retrogressive scour. Apart from the erosion depth along the flow path, the evolution of the volume with time was computed (Fig. 11a). Look at the blue line that indicates the general value of ϕbed =31°. The initial volume of 500 m3 strongly increases in the first phase of the event because the debris flow passes through the steep section. The principal part of the total computed volume of about 4,100 m3 was reached after only 400 s, while very small entrainment occurs afterwards. The final volume under-estimates the total volume after the event, but this error can be fixed by a small reduction of ϕbed value, as we will see in the following. A sensitivity analysis of the parameter ϕbed was carried out, to obtain an idea of its influence on the erosion rate. Figure 11a compares the previously commented results using ϕbed =31° with five other ϕbed value, and Fig. 11b summaries the results correlating the final volume with the ϕbed value applied. Both figures indicate the strong influence of ϕbed on the erosion rate, especially for values smaller than about 30°. A similar non-linear behaviour was already observed in the theoretical runs illustrated in Fig. 3b. As a concluding remark, we can state that a total final volume of about 5,000 m3, which was estimated for the Cardemeller debris flow, would have been computed applying a ϕbed value of about 30.5°. Such a friction angle of the channel bed material may be rather small, but detailed laboratory or in situ tests would be necessary to exactly validate this parameter. Jou torrent, La Guingueta Field data The Jou torrent near La Guingueta village drains the largest of the three selected catchments covering 4.4 km2. The drainage basin is 138

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situated in the Eastern Axial Pyrenees (Fig. 7d), and bedrocks consist of shales. However, the basement rocks only crop out in the lower section, while the rest is covered by till deposits or periglacial materials. The Jou debris flow was also provoked by the 1982 floods and a total rainfall amount of 154 mm in 24 h, which was measured at a nearby rain gauge. In the night from November 7 to 8, three surges transported a total volume of about 30,000 m3 to the fan (Fig. 7d). Field data indicated that two major failures in the higher port of the catchment transformed into debris flows, but most of the accumulated material was eroded in the glacial or periglacial deposits (Bru et al. 1984). The accumulation zone of the debris flow covered a large area of the fan and damaged several houses and roads of La Guingueta village. Simulation results Field data indicate that two principal initiation zones caused the debris flow (Fig. 7d). During the numerical modelling, each initial volume was released at the same moment and assumed to be 15,000 m3, which gives a total mobilised mass of 30,000 m3 (Fig. 12a). The simulations were carried out over a topographic grid with a cell size of 2 m. Voellmy fluid model was applied again and  the best-fit flow properties were μ=tanϕ=0.1 and Cz ¼ 10 m1=2 s . Apart from the extent and the depth of the initially mobilised mass, Fig. 12a also indicates the calculated depths of the final deposit. While the extension of the inundated area obtained by the simulation coincides rather well with the one observed in the field, accumulated depths of material seem to be affected again by the uncertainties related to the DEM. As in the previous case, these errors can be explained by the fact that a DEM representing actual conditions was used for the input relief. Such a topographic error

Fig. 12 Simulation results of the debris flow occurred in the Jou torrent atLa " Guingueta using the Voellmy fluid model (μ=tan=0.1 and Cz ¼ 10 m1=2 s). a Depth of initiation zone and of final deposit. b Maximum flow velocity. c Extensions of final deposits computed by different Voellmy parameters (case 1 applies the same parameters as in a); case 2: μ=tan=0.14 and Cz ¼ 10 m1=2 s) d Zone affected by simulations using different Herschel–Bulkley parameters (case 1: τ0 =1 kPa, k ¼ 1 kPa s1=3 ; case 2: τ0 =1 kPa, k ¼ 0:1 kPa s1=3 ; case 3: τ0 =0.5 kPa, k ¼ 0:1 kPa s1=3 ; case 4: τ0 =0.3 kPa, k ¼ 0:1 kPa s1=3 )

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Fig. 13 Correction of front inclination of final deposit modifying the  value (see text for detailed explanations). a Comparison of extensions of final deposit for normal Voellmy model and corrected Voellmy model using in both cases μ=tan=

provoked that the maximum depth computed for the final deposit was 3.4 m. However, this maximum value was only limited to one point located inside the artificial channel, which was constructed after 1982. Thus, the legend used for Figs. 12a and 13a ranges from 0 to 2 m. This range more or less coincides with the depth of the deposit observed after the debris-flow event. In Fig. 12b, the maximum calculated flow velocity is illustrated. Results indicate that maximum velocity is limited to the steep and strongly confined section just upstream of the fan apex. There, maximum velocity of more than 9 m/s was calculated. The sensitivity analysis of the different flow resistance laws and theirs rheological parameters generally confirmed previous results. Figure 12c compares the extension of the final deposit obtained on one side by the best-fit Voellmy parameters (μ=tanϕ=0.1 and Cz ¼ 10  m1=2 s ) and on the other side by a higher μ value of 0.14. As it could be supposed, the final deposit is shorter using a higher friction angle. Therefore, the value of this parameter is rather delicate and strongly affects the total extension of the debris-flow deposit. Additionally, simulation runs applying Herschel–Bulkley rheology were carried out (Fig. 12d). As already observed in the Font de la Llum event, a viscous flow resistance law is not able to simulate such a type of debris flow. Grain sizes of the till and periglacial materials include gravels to 140

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 0.1 and Cz ¼ 10 m1=2 s. Static friction angle was 35°. b Computed depths of final deposit along the profile illustrated in Fig. 13a (green line)

boulders in a silty matrix (Bru et al. 1984), which support the hypothesis of a granular debris flow. A large range of parameters were tested, but no good agreement was received, and especially, the flow cross-section area and lateral expansion angle of the deposit on the fan have a behaviour far from the one observed. Finally, the simulation results obtained on the fan help to visualise the effect of the correction referring to the front slope of the final deposit. Figure 13a compares the extension of the final deposit calculated using the Voellmy fluid model without the correction (called “Voellmy” in Fig. 13a) and the corrected Voellmy fluid model (called “Corrected Voellmy”). The corrected model changed the initial friction angle ϕ=5.7° into the static frictional angle φstat =35° for the critical velocities of V1 =0.3 and V2 =0.1 m/s (see Fig. 4). The improvement of this implementation is clearly visible in Fig. 13b, where longitudinal profiles at the front of the final deposit profile are illustrated. Only the corrected Voellmy model can simulate the typical steep fronts that are generally visible at debris-flow deposits. Conclusions Detailed simulation of debris-flow dynamics is a complex task. The 2D finite-volume numerical code FLATModel was applied to two types of

calibration cases and three real cases. Several general closing notes can be stated to contribute to the investigation on debris-flow modelling. Regarding the rheology, the Voellmy fluid model seems to be the best-fit flow resistance law in all the three cases modelled. However, the back-analyses carried out in the present study include many uncertainties. The principal drawback was the fact that an exact preevent topography was lacking and no rheological data on the material were available. Moreover, no exact measurements on flow depth or flow velocity were available. In spite of all these problems, the results obtained by the numerical modelling seem to be correct and coincide rather well with the general post-event data observed in the field. The extension and height of the final deposit area as well as the transversal extension of the flow agree rather well with the field observations. A real benefit of FLATModel is the implementation of the “stopand-go” mechanism. Its effects on the dynamic behaviour could be visualised at some real events, and an improvement of the outcomes was obtained especially in the accumulation zones of the flows. Another advantage of the model is the possibility to calculate basal entrainment. The implementation of this mechanism could represent rather well the effect of debris-flow erosion along the flow trajectory. In spite of the uncertainties regarding material properties, the simulated scour rates and the final volume coincide more or less with the data observed in the field just after the event. Simulation results indicated, however, that the final volume is rather sensitive on the selected friction angle of bed material. Acknowledgements This research was supported by the Spanish Research and Technology Ministry, contract BTE2002-0375, the EC-project HYDRATE, contract GOCE 037024 and the Generalitat de Catalunya, research group 2005SGR00770.

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V. Medina . A. Bateman ()) Sediment Transport Research Group, Hydraulic, Marine and Environmental Engineering Department, Technical University of Catalonia (UPC), Jordi Girona 1-3 (D1), 08034 Barcelona, Spain e-mail: [email protected] M. Hürlimann Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia (UPC), Jordi Girona 1-3 (D2), 08034 Barcelona, Spain