a model for geotechnical analysis of flow slides and debris flows
TRANSCRIPT
A model for geotechnical analysis of flow slidesand debris flows
Xiaobo Wang, Norbert R. Morgenstern, and Dave H. Chan
Abstract: Flow slides and debris flows incorporate a broad range of sediment–fluid mixtures that are intermediate betweendry rock avalanches and hyperconcentrated flows. Following a comprehensive review of some existing analytical ap-proaches to debris flow runout analysis, a new analytical model based on energy conservation has been formulated. Thenew analytical model was developed to deepen the understanding of fundamental aspects in modeling of granular flowsand to improve the geotechnical mobility analysis of flow slides and debris flows. The Lagrangian finite difference methodwas used to solve the governing equations. The model and numerical scheme have been tested against analytical solutionsand experiments of granular flows with simplified geometries for sliding mass and basal topography. Results of granularflow simulations indicate that the model based on energy conservation performs well and is robust. The model can beused for geotechnical analysis of a wide range of dense granular flows, such as flow slides and debris flows.
Key words: flow slides, debris flows, runout analysis.
Resume : Les glissements par liquefaction et les ecoulements de debris comprennent un large eventail de melanges fluide-sediments qui sont a mi-chemin entre les avalanches de roches seches et les ecoulement hyperconcentres. Suite a une re-vue exhaustive de quelques approches existantes pour l’analyse du parcours des ecoulements de debris, un nouveau modeleanalytique base sur la conservation de l’energie a ete formule. L’objectif de ce nouveau modele est de mieux comprendreles aspects fondamentaux de la modelisation des ecoulements granulaires ainsi que d’ameliorer l’analyse de mobilite geo-technique des glissements par liquefaction et des ecoulements de debris. La methode de difference finie de Lagrange estutilisee pour resoudre les equations. Le modele et le schema numerique ont ete compares a des solutions analytiques et ades experimentations d’ecoulement granulaire ayant des geometries simplifiees de masse glissante et de topographie debase. Les resultats des simulations d’ecoulement granulaire indiquent que le modele base sur la conservation de l’energieperforme bien et est robuste. Le modele peut donc etre utilise pour des analyses geotechniques d’une grande varieted’ecoulements granulaires denses, tels que les glissements par liquefaction et les ecoulements de debris.
Mots-cles : glissement par liquefaction, ecoulement de debris, analyse de parcours.
[Traduit par la Redaction]
IntroductionFlow slides and debris flows are extremely rapid flows of
a highly concentrated mixture of water and predominantlycoarse granular material. This type of granular flow is com-posed of a poorly sorted, sediment–water mixture that com-monly contains more than 50% solids by volume. Theconstituent sediment usually varies widely in size, fromclay particles to boulders of several metres in diameter. Be-cause of high flow velocities, large impact forces, long run-
out distance, and poor temporal predictability, flow slidesand debris flows are among the most dangerous and destruc-tive natural hazards (Johnson and Rodine 1984; Jakob andHungr 2005).
Flow slides and debris flows are complex in their physicalbehavior and demand subtle theoretical descriptions and nu-merical models. However, validation of flow slide and de-bris flow simulation models against field observations isoften difficult or impossible because of the dangers involvedwith in situ experimental campaigns, uncontrollable geo-physical conditions, and unpredictable time and locations ofnatural flow events (Hutter 2005). Although laboratory testsof flow slides and debris flows of reduced size can be per-formed under well-defined and well-controlled conditions,these dense granular flows are known to be scale-dependentand runout distance is greatly influenced by the rate of porepressure dissipation (Hutchinson 1986; Iverson 1997). Thesignificance of scale effects raises questions regarding theapplication of experimental observations to predicting flowslide and debris flow behaviors in practice. Because of thesedifficulties in experimental tests and field observations, nu-merical modeling has become an important and promisingalternative in flow slide and debris flow studies.
Within the framework of dense granular flow simulations,
Received 20 November 2008. Accepted 7 April 2010. Publishedon the NRC Research Press Web site at cgj.nrc.ca on18 November 2010.
X. Wang. Thurber Engineering Limited, Edmonton, AB T6E6A5, Canada.N.R. Morgenstern. Department of Civil and EnvironmentalEngineering, University of Alberta, 3-075 Markin/CNRL NaturalResources Engineering Facility, Edmonton, AB T6G 2W2,Canada.D.H. Chan.1 Department of Civil and EnvironmentalEngineering, University of Alberta, 3-038 Markin/CNRL NaturalResources Engineering Facility, Edmonton, AB T6G 2W2,Canada.
1Corresponding author (e-mail: [email protected]).
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Can. Geotech. J. 47: 1401–1414 (2010) doi:10.1139/T10-039 Published by NRC Research Press
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depth-averaged equations have been predominantly appliedto modeling debris avalanches, debris flows, and other typesof fast-moving landslides. The depth-averaged model is es-tablished by integrating the mass and momentum conserva-tion equations in differential form over the flow depth withthe use of Leibnitz’s law. Integration of the differential formof the conservation laws over the flow depth considerablyreduces the computational time required in rapid landsliderunout analysis. Constitutive properties of the sliding massare incorporated into governing equations through the basalflow resistance in the depth-averaged model. The flow re-sistance is expressed in terms of averaged velocities whenfluid models such as the Bingham (Johnson 1970) andHerschel–Bulkley (Huang and Garcıa 1998) models areselected. The Coulomb frictional law has been widelyemployed to compute basal shear resistance in the depth-averaged model for dense granular flow simulations ingeotechnical engineering. The magnitude of resistance isdetermined by the product of overburden normal stress andcoefficient of basal friction.
In this paper, a slice-based model based on energy conser-vation was formulated on a macroscopic scale after a reviewof the existing numerical models for granular flow simula-tions. Compared with the existing depth-averaged slice-based models, our new model incorporates deformationwork and internal energy dissipation into the governingequations. The numerical method for practical implementa-tion of the governing equations is presented following themodel formulation. Model verification has been undertakenby comparing numerical predictions with analytical solu-tions and experimental observations of granular flows.
Existing mathematical models for granularflow simulations
Mathematical modeling of granular flows was originallyintroduced by Savage and Hutter (1989, 1991). Based onmass and momentum conservation equations for flow on arough inclined plane, and using the depth-averaging processand making scaling arguments, Savage and Hutter (1989)derived the one-dimensional, depth-averaged equations forthe shallow free-surface flow of dry granular materials. Themodel assumes that a moving granular mass behaves as acohesionless Coulomb frictional material and the relation-ship between shear and normal stresses on internal andrough bounding surfaces obeys the Coulomb friction law.Multi-dimensional extensions of the Savage–Hutter modelhave been formulated for analyzing dry granular flows.
Iverson (1997) and Iverson and Denlinger (2001) intro-duced Coulomb mixture theory and derived the governingequations for a wide spectrum of grain–fluid mixture flowsbased on a two-phase analysis. The Coulomb mixture modelassumes that solids and interstitial fluids in debris flows be-have constitutively as Coulomb frictional materials andNewtonian viscous fluids, respectively. Advection–diffusionequations are postulated to describe pore pressure changesin response to the movement of solids (Iverson and Denlin-ger 2001). Denlinger and Iverson (2004) developed a mathe-matical depth-averaged model using a rectangular Cartesiancoordinate system to simulate granular avalanches across ir-regular three-dimensional terrains. The equations are solved
numerically using a hybrid finite volume and finite elementscheme. Fluxes of mass and momentum across vertical cellboundaries are computed using the finite volume methodand stresses accompanying the deformation within the gran-ular avalanche are calculated using the finite elementmethod. The stresses are first evaluated based on constitutiverelations for an isotropic linear elastic material and then cor-rected using the Coulomb yield criterion. The model wastested against laboratory experiments of dry sand avalanchesacross irregular basal topography. Good agreement was ob-tained between theoretical prediction and experimental tests(Iverson et al. 2004).
Pitman and Le (2005) proposed a two-fluid model forgranular flows of the mixture of solid particles and fluid.Continuity and momentum equations are formulated explic-itly for both solid and fluid phases in the two-fluid model.Interactions between particles and the fluid are taken intoaccount in the model with a velocity-dependent force. Puda-saini et al. (2005a, 2005b) extended the Savage–Huttermodel for debris flow simulation by including pore pres-sures in the model. The extended model has been applied toanalyzing debris flow flume tests, and good agreement isobtained between theoretical predictions and experimentalobservations (Pudasaini et al. 2005a, 2005b). However, thepore pressures are not predicted; they are merely assumedby employing an advection–diffusion equation similar tothose proposed by Iverson and Denlinger (2001).
The Savage–Hutter model and its generalized versionshave been tested against laboratory experiments of rapidgranular flows over a wide variety of bed topographies(Hutter and Koch 1991; Greve and Hutter 1993; Wieland etal. 1999; Iverson et al. 2004). Theoretical predictions werefound to be in good agreement with experimental measure-ments. The Savage–Hutter model and its various generalizedversions have been established as the leading models in thearea of dry granular flow analysis (Pudasaini and Hutter2007). However, their broad applications are limited to mod-eling of dry granular flows over the simple topography ana-lyzed in most laboratory experiments.
Slice-based dynamic analysis has been developed andused in studies of fast-moving gravitational flows, such asrapid landslides, debris flows, and liquefaction flow slides.Hungr (1995), Tinti et al. (1997), Miao et al. (2001), andKwan and Sun (2006) are examples. In the slice-basedmodel, the flowing mass is represented by an ensemble ofcontiguous slices (two dimensions) or blocks (three dimen-sions) that are subjected to gravitational forces, basal resist-ance, and internal forces. While sliding down over aspecified sliding surface, the slices interact with each other,dissipating energy along their bases and exchanging momen-tum between adjacent slices. The shape of the slices can bechanged during the flow process, but total volume is con-served and no mass can penetrate between slices. Newton’slaws of motion are employed to define the relationship be-tween slice movement and forces acting on each slice. ALagrangian description is generally adopted to easily deter-mine locations of the slices. A general survey of themomentum-based models indicates that the methods of cal-culation of gravitational forces and basal resistance are verysimilar. It is the treatment of internal forces that distin-guishes one method from another (Wang 2008).
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Hungr (1995) presented a dynamic model (DAN) for run-out analysis of rapid landslides. DAN is based on an explicitsolution of the Saint-Venant equations with the integrationof a variety of constitutive relationships that have beenwidely applied in debris flow modeling. The displaced mate-rials of a landslide are represented by a number of boundaryelements and mass elements. Formulation of the governingequations is based on the law of conservation of momentumfor the boundary elements. Mass conservation is applied tothe mass elements to calculate changes in flow depth and in-ternal deformation within the landslide. Flow resistancealong the sliding surface is determined using the constitutivelaw specified in the analysis. The resultant longitudinal pres-sures acting on the boundary elements are determined fromthe product of the hydrostatic pressure gradient of the equiv-alent fluid and a lateral pressure coefficient (Hungr 1995).The value of the lateral pressure coefficient is dependent onthe coefficients of active and passive earth pressures, thestiffness coefficient, and the average tangential strain of themass elements. Initially, the coefficient of lateral pressure atrest is used for each element. The magnitude of the lateralcoefficient is then increased or decreased by a value equalto the product of the incremental strain times a stiffness co-efficient. The lateral coefficient can vary between the mini-mum and maximum values, which correspond to the activeand passive states, respectively. Hungr (2003) proposed thatthe values of the active and passive earth pressures can bedetermined based on internal frictional strength of the slid-ing material using the equation proposed by Savage andHutter (1989).
An improved debris mobility model (DMM) was pre-sented by Kwan and Sun (2006). DMM is based on themodel for the dynamic analysis of rapid landslides (DAN)developed by Hungr (1995). The formulation of flow resist-ance in DMM is based on the entire wetted perimeter of theflow channel. For simplicity, the cross section of the flowchannel is approximated by a trapezoid in the modifiedDMM formulation. Calculation of internal forces in DMMis based on the hydrostatic gradient and lateral pressure co-efficient, which is dependent on the local internal deforma-tion states of the landslide. The method proposed by Hungr(1995) is used in DMM to calculate the coefficient of lateralearth pressure. The total flow resistance consists of resistan-ces developed at the base of each slice and on the two sideslopes of the channel. The basal resistance is calculated us-ing the same procedure as in the original formulation ofDAN (Hungr 1995). The resistances on side slopes are de-termined from the sliding mass above the side slopes andthe constitutive law specified in the analysis. Applicationsof DMM and DAN in modeling rapid landslides in HongKong show that very similar front velocities are predictedby both models (Kwan and Sun 2006). However, the back-calculated strengths from DAN are greater than those deter-mined from the improved DMM due to incorporation of re-sistances on the side slopes in the DMM model.
A block-based model was proposed by Tinti et al. (1997)to simulate movements of flow slides over slopes. Intersliceforces in the block-based model are expressed in terms of aninteraction coefficient, which is dependent on the dynamicstates of the slices and the instantaneous distance betweenthe slice centers. The value of the interaction coefficient
varies between 0 and 1, which represent two limiting condi-tions for interslice actions. When the interaction coefficientis equal to 0, interacting slices have the same post-interactionvelocities irrespective of their pre-interaction states, andslices move as if they are adhered to each other; this isthe maximum possible interaction. When the interactioncoefficient is equal to 1, the slices retain their pre-interac-tion velocities as if no interaction had taken place. In gen-eral, the interaction coefficient can be determined from afunction in terms of interaction intensity, deformability pa-rameter, and shape parameter (Tinti et al. 1997).
Miao et al. (2001) proposed a sliding block model for therunout analysis of rapid landslides. The model starts with alimit equilibrium assessment and incorporates mass dynam-ics and soil deformation into the calculation of slidingmovement. In the sliding block model, the sliding mass isdivided into a number of slices. Forces acting on a singleslice consist of gravity, basal resistance, and interactions be-tween slices. The initial interslice forces are determinedfrom the limit equilibrium analysis using the method of un-balanced thrust, in which the resultant interslice force is as-sumed to be parallel to the base of the preceding slice andacts at the midpoint of the height of the slice. The criticalstate corresponding to a factor of safety of 1 is consideredas the initial state. Initial acceleration of the slices is calcu-lated based on the initial unbalanced forces. After the massmovement is triggered, the interslice forces are determinedfrom the deformation energy, which is calculated using themacroscopic deformation of the slices and bulk deformationmodulus. For slices in a tensile state, the deformation energyvanishes and the interslice forces are assumed to be zero.The basal shear resistance is determined based on the over-burden normal force using the Coulomb frictional law, sim-ilar to the methods used by other sliding block models.Application of the sliding block model to two rapid land-slides in China (Miao et al. 2001) indicated that the slidingblock model can result in considerable fluctuations in thevelocity field within very short periods of time.
Formulation of the slice-based model usingthe energy conservation law
The existing slice-based models incorporate the effects ofinternal deformation on the flow dynamics using either theinteraction coefficient (Tinti et al. 1997) or deformation en-ergy (Miao et al. 2001). The interslice forces proposed inthe slice-based model are primarily dependent on the magni-tude of deformation and dynamic state of the slices. How-ever, application of the slice-based models of Tinti et al.(1997) and Miao et al. (2001) to rapid landslide simulationis either difficult or impractical because of the great uncer-tainties involved in the calculation of the interslice force andthe determination of model input parameters required. Forinstance, the significant fluctuations in the velocity field pre-dicted by Miao et al.’s (2001) model are caused mainly byinappropriate computation of the interaction forces in termsof the deformation energy. A slice-based model incorporat-ing internal energy dissipation is presented in this paper.The formulation of the new model is based mainly on theconservation of energy of slices during the flow process.
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Mechanical energy equationsThe rate of change of the kinetic energy of a material vol-
ume can be written as the sum of three parts (Aris 1962):
(1) The rate at which the body forces do work, i.e., changein potential energy.
(2) The rate at which the surface stresses do work.(3) The rate at which the internal stresses do work, i.e.,
change in internal energy due to deformation.The kinetic energy equation for a unit volume mass is
given by:
½1� rd
dt
1
2uiui
� �¼ rgiui þ
@
@xj
ðuitijÞ � tijeij
where r is the density of the fluid, t is time,uiui ¼ u2
1 þ u22 þ u2
3 is kinetic energy per unit volume, ui isthe component of the velocity vector in the xi direction, rgiis the body force per unit volume in the xi direction (wheregi is the component of the acceleration vector), tij representscomponents of the stress tensor, and eij represents compo-nents of the strain rate tensor.
An integral form of the mechanical energy equation canbe derived by integrating each term in eq. [1] over a mate-rial volume or a slice of a landslide body, V. Using Gauss’theorem, eq. [1] in the Lagrangian system becomes
½2� d
dt
ZV
1
2ruiui
� �dV ¼
ZV
rgiui dV þZ
A
uitij dAj
�ZV
tijeij dV
Each term in eq. [2] is a time rate of change: the firstterm is the rate of the kinetic energy of a material volume,the second is the rate of total work done by the body forceacting on a material volume, the third represents the totalwork done by the forces acting on the surface of a materialvolume, and the fourth represents the rate of energy dissipa-tion due to deformation of a material volume. The deforma-tion work includes two parts: the work due to volumeexpansion and the work due to irreversible deformation. Foran incompressible fluid, the work due to volume expansionis zero and the deformation work represents irreversible ki-netic energy dissipation. Thus, the term
RV tijeij dV repre-
sents a rate of loss of kinetic energy and a gain of internalenergy due to deformation of the volume.
The body force can be represented as the gradient of ascalar potential. The rate of work done by body forces canbe taken to the left-hand side of eq. [2] and can be inter-preted as a change in the potential energy. For an incom-pressible fluid and debris, r is constant and the mass withina material volume can be determined from
½3� m ¼Z
V
r dV
and eq. [2] can be rewritten as
½4� d
dt
1
2m�u2
i þ mgzW
� �¼Z
A
uitij dAj �Z
V
tijeij dV
where �ui is the average velocity component of a material vo-lume, g is gravitational acceleration, zW is the elevation of
the center of gravity of the sliding mass, and Aj is the com-ponent of the surface area vector. The left-hand side ofeq. [4] thus represents the rate of change of mechanical en-ergy (the sum of kinetic energy and potential energy).
Derivation of governing equations based on theconservation of energy
The sliding mass is divided into a series of contiguous sli-ces, as shown in Fig. 1. Forces acting on a typical slice ofwidth b and height h are shown in Fig. 2. In the figures, bis the width of the slice, h is the height of the slice, N isthe normal force, subscript n is the slice number, P is theinterslice force, T is the shear force acting along the base ofthe slice, �u is the mean velocity of the slice along the baseof the slice, W is the weight of the slice, q is the inclinationof the base of the slice with respect to the horizontal, andsubscripts L and R denote properties on the left and rightsides, respectively, of a slice.
As indicated in Figs. 1 and 2, the derivation of governingequations is based on the deformation in the flow direction.Effects of side-wall deformation perpendicular to the flowdirection are not included in the formulation. With the as-sumption that the interslice shear forces in the vertical direc-tion are negligible, the rate of total work done by the surfaceforces in eq. [4] is given by
½5�Z
A
uitij dAj ¼ PL �uL cosqL � PR �uR cosqR � T �u
where qL and qR are inclinations between velocities and cor-responding interslice forces acting on the left and rightsides, respectively, of the boundary.
The rate of change of the total potential energy of a slicein eq. [4] can be expressed as the sum of the rate of poten-tial energy changes due to the displacement and deformationof the slice, as shown in Fig. 2
½6� d
dtðmgzwÞ ¼
d
dtðmgzw1Þ þ
d
dtðmgzw2Þ
½7� m ¼ rbh
where (d/dt)(mgzw1) represents the rate of change of poten-tial energy due to displacement of the center of the gravity,(d/dt)(mgzw2) represents the rate of change of the potentialenergy due to deformation of the slice, and r is the meandensity of the sliding mass. Thus
½8� d
dtðmgzW1Þ ¼ �mg�u sinq
½9� d
dtðmgzW2Þ ¼ �
1
2mgh�ezz
where �ezz is the mean vertical strain rate of a slice defined inFig. 3.
Using eqs. [5] to [9], eq. [4] becomes
½10� d
dt
1
2m�u2
� �¼ mg�u sinq þ 1
2mgh�ezz þ PL �uL cosqL
� PR �uRcosqR � T �u�ZV
tijeij dV
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Summing both sides of eq. [10] for all slices (k = 0 to nin Fig. 4) gives the rate of change of the kinetic energy ofthe overall sliding mass:
½11� d
dt
Xn
k¼0
1
2mk �u2
k
!¼Xn
k¼0
mkg �uk sinqk þ1
2ðh�ezzÞk
� �
�Xnk¼0
Tk �uk �Xnk¼0
ZVk
tijeij dVk
It is obvious that the rates of work done by interslice forces
cancel each other in the preceding derivation. Equation [11]states that the rate of the change of kinetic energy of a slidingmass is equal to the sum of the rate of potential energychange, the rate of work done by resistance force along thebase of the sliding mass, and the rate of deformation work.
Numerical scheme for the analytical modelbased on energy considerations
Depth-averaged governing equations can be solved usingeither a Lagrangian or Eulerian scheme. Two finite difference
Fig. 3. Uniform deformation assumed in the slice-based model. �exx, mean horizontal strain rate.
Fig. 2. Forces on a typical slice.
Fig. 1. Representative slice in the slice-based model.
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methods — one Lagrangian the other Eulerian — have beentested against flume experiments on dry granular flows(Savage and Hutter 1989). Simulations indicated that the Eu-lerian scheme based on the upwind flux correction methodgave poor predictions of experimental avalanches with gen-eral initial profiles. The Lagrangian finite difference approach(Savage and Hutter 1989, 1991) is simple, efficient, and reli-able for the granular rapid flow problem that involves a freesurface, dry bottom, and moving boundaries. The approach isbased on a Lagrangian, moving mesh, finite differencescheme in which the flowing material is divided into quadri-lateral cells in two dimensions or triangular prisms in threedimensions. Boundary locations are determined for each timestep. The depth of a cell is calculated from the cell volumeand boundary locations. Numerical simulations of the flumeexperiments on dry granular flows showed very good agree-ment between theoretical predictions and experimental data(Savage and Hutter 1989, 1991; Wieland et al. 1999).
Numerical methods proposed by Hungr (1995), Tinti et al.(1997), Miao et al. (2001), and Kwan and Sun (2006) forsolving the governing equations of fast-moving landslideshave many similarities. These numerical methods are basedon the Lagrangian finite difference scheme similar to thatformulated by Savage and Hutter (1989).
Following the procedures of Savage and Hutter (1989), theLagrangian finite difference scheme is presented here for solv-ing the equations of the slice-based model with internal energydissipation. In the Lagrangian scheme, the sliding mass is dis-cretized into a number of slices, as shown in Fig. 4. In Fig. 4,(hc)k denotes the average height of the center of slice k and(xc)k, (xb)k, and (xb)k+1 denote the locations of the center (sub-script c) and boundaries (subscript b) of slice k. Mass conser-vation of slice k during the landslide motion indicates
½12� d
dtðVkÞ ¼ 0
where Vk is the volume of slice k. The mean height of aslice at time t can thus be determined from
½13� ðhcÞtk ¼Vk
ðxbÞtkþ1 � ðxbÞtkSolving the governing equations of rapid landslides re-
quires determination of the positions of the boundaries ofeach slice at time t. The numerical scheme assumes that allthe variables involved in the calculation at t + Dt are knownfrom the previous time t, where Dt is the time step interval.In the framework of the Lagrangian finite differencescheme, the governing eq. [10] can be written as
½14� EtþDtk � Et
k
Dt¼ _W
t
k
½15� Ek ¼1
2mk �u2
k
½16� _Wk ¼ mkg�uk sinqk þ1
2mkghkð�ezzÞk þ ðPL �uL cosqLÞk
� ðPR �uR cosqRÞk � Tk �uk �ZVk
tijeij dV
where Ek is the kinetic energy of slice k, and _Wk is the sumof the rate of the work done by body force, surface force,and energy dissipation due to deformation of slice k. Ek and_Wk are determined from eqs. [15] and [16], respectively.
The constitutive law and assumptions regarding the inter-slice forces and deformation work are required for calculat-ing the rate of work in eq. [16]. The basal shear resistancecan be determined based on the constitutive laws for materi-als. For Mohr–Coulomb materials, the basal resistance alongthe base of the slice can be expressed as
½17� Tk ¼ tAk ¼ ðcþ s tanfbÞAk ¼ cAk þ Nk tanfb
where Tk and Nk are shear resistance and normal force actingon the base of slice k, t is shear stress, Ak is basal area ofslice k, c is cohesion, s is normal stress, and fb is basal fric-tion angle.
Equation [17] is a generalized form of the Mohr–Coulombequation. In dense granular flow simulations, either apurely cohesive or a frictional model is generally used asa constitutive law in back-analyses of liquefaction flowslides or debris flows. It is evident that if the friction angleis equal to 0 (fb = 0), eq. [17] reduces to the purely cohe-sive model; if the cohesive strength is equal to 0 (c = 0),eq. [17] reduces to the frictional model. Wang (2008)back-analyzed case histories of liquefaction flow slides us-ing the cohesive and friction models. The effects of con-stitutive models on the dynamic analyses were discussedby comparing model predictions with field observations interms of runout distance, deposit distribution, and velocityprofiles.
In granular flow simulations, lateral pressure can be ap-proximated as a product of hydrostatic pressure and the co-efficient of lateral pressure, if the frictional model is used asa constitutive law. By neglecting interslice shear forces inthe vertical direction, the lateral pressure can be written as
½18� ðPLÞk ¼ðKLÞkgðhbÞ2k
2
½19� ðPRÞk ¼ðKRÞkgðhbÞ2kþ1
2
where (KL)k and (KR)k are the lateral earth pressure coeffi-cients on the left and right sides, respectively, of slice k;g = rg is the unit weight of the sliding mass; and (hb)k and(hb)k+1 are flow depths on the left and right sides of slice k(Fig. 4), respectively. The lateral stress coefficient can beactive, passive or hydrostatic (coefficient of lateral stressequal to 1) based on the local strain rate (velocity gradient)of a slice in the longitudinal direction.
Equations proposed by Savage and Hutter (1989) havebeen commonly used for calculating the coefficients of lat-eral earth pressure in the analysis of dense granular flows.The derivation of the Savage–Hutter equations for the lateralstresses assumes that Coulomb failure occurs simultaneouslyalong the bed and within the sliding mass. Therefore, thedefinition of the Savage–Hutter coefficients is often as-sumed to be more representative than those of Rankine lat-eral earth pressure coefficients used in classical soilmechanics. Examination of the Savage–Hutter equations in-dicates that the coefficient of lateral earth pressure would be
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greater than unity even as a slice is expanding, i.e., in an ac-tive state, if the value of the basal friction angle is close tothat of the internal friction angle (Fig. 5). This is physicallyinadmissible and inconsistent with the assumptions for themodel presented here. In the model developed in this paper,the active or passive state of stress on the left side of a sliceis dependent on whether the slice is expanding or contract-ing. It is also assumed that the contribution to the magnitudeof lateral stress from internal deformations is more signifi-cant than that from the materials of the thin shear zonealong the bed. The values of the lateral stress coefficients
are calculated for a frictional material using the Rankineequation
½20� ðKLÞk ¼
1� sinf
1þ sinf¼ tan 2 45� f
2
� �@u
@x
� �k
> 0
1@u
@x
� �k
¼ 0
1þ sinf
1� sinf¼ tan 2 45þ f
2
� �@u
@x
� �k
< 0
8>>>>>>>>><>>>>>>>>>:
Fig. 5. Coefficients of lateral earth pressure based on the Savage–Hutter (Savage and Hutter 1989, 1991) definition.
Fig. 4. Notations used for the slice-based model. h, height of the slice; b, width of the slice; subscript c, center; subscript b, boundary.
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where f is the internal friction angle.For a purely cohesive material, the internal friction angle
is equal to 0 and the total lateral forces calculated using theRankine theory are
½21� ðPLÞk ¼
1
2gðhbÞ2k � 2cðhbÞk
@u
@x
� �k
> 0
1
2gðhbÞ2k
@u
@x
� �k
¼ 0
1
2gðhbÞ2k þ 2cðhbÞk
@u
@x
� �k
< 0
8>>>>>>>>><>>>>>>>>>:
According to eq. [21], the calculated horizontal force isnegative when a slice is in the active state and (hb)k < (4c/g).Because tensile stresses occur rarely in soils, eq. [21] ismodified as follows:
½22� ðPLÞk ¼
1
2gðhbÞ2k
@u
@x
� �k
� 0
1
2gðhbÞ2k þ 2cðhbÞk
@u
@x
� �k
< 0
8>>><>>>:
Deformation of the slice is simplified as a pure shear de-formation, as shown in Fig. 3. The deformation work rate isapproximated by
½23�Z
Vk
tijeij dV ¼ ð�txx �exx þ �tzz �ezzÞkbkðhcÞk
where
ð�exxÞk ¼ �ð@=@xÞðu cosqÞk¼ �ðuR cosqR � uL cosqLÞk=bk
is the mean horizontal strain rate and �txx and �tzz are meanhorizontal and vertical stresses, respectively. Following theconventions of stress and strain representation usuallyadopted in geotechnical engineering, negative signs havebeen introduced so that compressive stresses and compres-sive strains are positive quantities.
For an incompressible fluid
½24� �exx þ �ezz ¼ 0
½25� ð�tzzÞk ¼gðhcÞk
2
Fig. 6. Comparison between analytical and numerical solutions of a dam break over a horizontal plane with a 08 basal friction angle.
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For a frictional material, lateral stresses are computed from
½26� ð�txxÞk ¼ Kkð�tzzÞkwhere Kk is calculated by averaging the coefficients of lat-eral stress on left and right sides of the slice
½27� Kk ¼ðKLÞk þ ðKRÞk
2
For a purely cohesive material, the coefficient of lateralearth pressure is equal to 1 and the average lateral stressesinside slice k can be calculated by
½28� ð�txxÞk ¼
ð�tzzÞk@u
@x
� �k
� 0
ð�tzzÞk þ 2c@u
@x
� �k
< 0
8>>>>><>>>>>:
Using eqs. [18] to [28], the kinetic energy of slice k attime t + Dt can be determined from
½29� EtþDtk ¼ Et
k þ _Wt
kDt
The center velocity of a slice at time t + Dt is
½30� ðucÞtþDtk ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi2EtþDt
k
mk
s
The boundary velocity is approximated as
½31� ðubÞtþDtk ¼ ðucÞtþDt
k�1 þ ðucÞtþDtk
2
Displacement of the slice boundary is
½32� ðxbÞtþDtk ¼ ðxbÞtk þ
ðubÞtk cosqtk þ ðubÞtþDt
k cosqtþDtk
2Dt
where ðxbÞtk and ðxbÞtþDtk are x coordinates of boundaries ofslice k at time t and t + Dt, respectively.
The height of slice k at t + Dt is computed by
½33�ðhcÞtþDt
k ¼ Vk
½ðxbÞtþDtkþ1 � ðxbÞtþDt
k �
ðhbÞtþDtk ¼ ðhcÞtþDt
k�1 þ ðhcÞtþDtk
2
½34� ðhbÞtþDt0 ¼ 0; ðhbÞtþDt
n ¼ 0
Fig. 7. Comparison between analytical and numerical solutions of a dam break over a 308 slope with a 08 basal friction angle.
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where Vk is the volume of slice k, ðhbÞtþDtk is the height ofthe left boundary of slice k at t + Dt, and ðhcÞtþDtk is the cen-tral height of slice k at t + Dt.
At time t = 0, the velocities and kinetic energy of the sli-ces are equal to zero. The initial movement of slice k can bedetermined from momentum conservation
½35� mk
d�uk
dt¼ mkg sinqk � Tk þ ðPL � PRÞk cosqk
The velocities and displacements of slices during the firsttime step can be calculated as long as accelerations are ob-tained from eq. [35]. The motion of each slice can then bedetermined using equations of energy conservation with theLagrangian difference scheme presented above. The compu-tation proceeds until the maximum slice velocity is underthe velocity threshold specified. It is worthwhile to notethat the dynamic model based on energy consideration as-sumes that lateral pressure and basal resistance on individualslices are known and defined by the Rankine and Mohr–Coulomb equations. The momentum equilibrium of theoverall sliding mass is not examined during the calculationand assumptions about the forces do not satisfy statics. As aconsequence, the dynamic analysis using the new analyticalmodel cannot converge to the static case because kinematicsare not being considered in the formulation and the internalforce distribution implicit as calculated in the static case dif-fers from that assumed in the model. The threshold velocity
plays a key role in terminating the calculation process.Based on the classification of flow-like landslides (Hungr etal. 2001), a value of 0.05 m/s can be used as a velocitythreshold in the analyses of case histories of flow slides anddebris flows, which are among the extremely rapid class.
Model verification and numericalexperiments
Performance of the slice-based model and numericalscheme presented in previous sections has been tested bycomparing model predictions with analytical solutions ofone-dimensional granular flows and experimental observa-tions of granular slumping on a horizontal plane.
Comparison between model predictions and analyticalsolutions
Mangeney et al. (2000) presented an analytical solutionfor a one-dimensional granular avalanche over an inclinedplane. The analytical solution describes the motion of aflow front of the dam-break granular flow over an infinite,uniform slope with a Coulomb-type friction acting at thebase of the flow. It should be noted that the analytical solu-tion of Mangeney et al. (2000) can only be applied to anidealized flow where (i) lateral earth pressure is assumed tobe hydrostatic, (ii) the basal friction angle is not greater thanthe slope angle, and (iii) the flow never stops on the slope.
Fig. 8. Comparison between analytical and numerical solutions of a dam break over a 308 slope with 108 basal friction angle.
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Figures 6 to 9 present a comparison between the analyti-cal and numerical solutions of dam break scenarios on hori-zontal and inclined planes. Numerical analyses were carriedout using the model developed in this paper. An internalfriction angle of zero has been used to provide the hydro-static lateral pressure distribution in all the simulations. Fig-ure 6 shows the result of the dram break scenarios over ahorizontal plane with zero basal friction. Figures 7 to 9present comparisons between analytical solutions and nu-merical predictions for dam break scenarios on a 308 slopewith and without the basal frictional resistance applied. It isobserved that numerical simulations provide a good repre-sentation of the dam break–induced granular flows over hor-izontal and inclined planes.
Simulation of granular slumping on a horizontal planeGranular flows induced by the collapse of initially static
columns of sand over a horizontal surface were investigatedexperimentally by Lajeunesse et al. (2005). Effects of the in-itial column geometry on the flow runout behavior and in-ternal flow structure were explored in the experiments.Granular materials used in the experiments were glass beads1.15 or 3 mm in diameter. Granular materials were initiallycontained within a cylindrical or rectangular tank. Axisym-metric or two-dimensional granular flows were created byquickly raising the cylinder or removing the gate. Experi-mental observation demonstrated that the flow dynamics
and final deposit depends on the initial aspect ratio of thegranular column, which is defined as the ratio of the initialheight to horizontal extent of the column.
Numerical simulations of the spreading of granular col-umns on a horizontal plane have been conducted using thedynamic model formulated based on energy conservation.The value of the internal and basal friction angles used inthe analysis is 258, which is the average of values reportedby Lajeunesse et al. (2005). Figure 10 presents the normal-ized final profiles of numerical simulations and experimentalobservations of the spreading of columns with the same ini-tial aspect ratio, but witha different initial basal length.Comparison of theoretical and experimental final profiles inFig. 10 indicates that the dynamic analysis provides a rea-sonable prediction of the runout distance for flows inducedby the collapse of granular columns over a horizontal plane.It should be noted, as shown in Fig. 11, that when theSavage–Hutter equations for the coefficients of the lateralstresses are used in the governing equations based on energyconservation, the fit is not as good either in terms of runoutdistance or shape.
It has been observed that granular slumping involves thefollowing two processes: (i) collapse and fall of the columnand (ii) spreading of the granular mass on a horizontal planeuntil it comes to rest. During these processes, the initial po-tential energy in the tall column is converted into kinetic en-ergy and is also dissipated because of internal deformation
Fig. 9. Comparison between analytical and numerical solutions of a dam break over a 308 slope with 208 basal friction angle.
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and basal friction. The complexity of the collective dynam-ics of momentum transfer and energy dissipation involved ingranular slumping highlights the difficulties in modeling thisclass of problems within the framework of classical shallowwater equations. To simulate the slumping of the granularmass, it appears that a model should be capable of account-ing for vertical momentum transfer associated with the fallof the column and also capturing the features of the subse-quent horizontal spreading. It is possible to apply a depth-averaged model to modeling sideways flow of a granularmass. Unfortunately, the initial vertical column collapse andmomentum transfer intrinsically violate the shallow waterassumptions and cannot be accounted for by shallow waterapproaches. It is evident that the theoretical predictions arefar from describing the whole process of granular slumpingand there are still many issues to be resolved. However, ithas been observed surprisingly that the simulations carriedout by using the new analytical model are able to reproducemany features of the spreading of a granular mass. Themodel based on energy considerations provides new insightsinto the approaches used for investigating granular flows.The model has been used in modeling a number of debrisflow cases (Wang 2008).
DiscussionFlow slides and debris flows generally exhibit pervasive,
fluid-like deformation. Morgenstern (1978) investigated mo-bile flows in a wide variety of geological and geomorpho-logical settings and concluded with the observations thatcharacterization of mobile soil and rock flows and the de-sign of protective structures should proceed using principlesof fluid mechanics. The fundamental equations of flow dy-namics are based on three universal laws of conservation:conservation of mass, conservation of momentum, and con-servation of energy. During the last three decades, conserva-tion laws of mass and momentum have been widely appliedin the formulation of numerical models for granular flowsimulations (Savage and Hutter 1989; Hungr 1995; Iverson1997; Denlinger and Iverson 2001, 2004; Iverson and Den-linger 2001; Iverson et al. 2004). However, energy conserva-tion has not been explicitly accounted for in the granularflow simulations.
Morgenstern (1978) pointed out that large volumes of soilor rock can become fluidized by virtue of energy transfermechanisms following instability. Iverson et al. (1997) pro-vided an extended discussion on mobilization of debris
Fig. 10. Nondimensional final deposit of granular slumps with the same initial aspect ratio of 3.2 (simulation using Rankine coefficient oflateral earth pressure).
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flows from landslides and noted that three processes are in-volved in the mobilization:
(1) Widespread Coulomb failure within the sediment mass.(2) Partial or complete soil liquefaction by high pore-fluid
pressure.(3) Conversion of landslide translational energy to internal
vibrational energy.These three processes operate simultaneously and synerg-
istically in many circumstances and the change of pore-fluidpressures and conversion of energy are crucial componentsin flow slide and debris flow mobilization and evolution.Formulation of the governing equation (eq. [10]) is basedon energy considerations and idealization of complex energytransfer mechanisms involved in mobilization, motion, anddeposition of flow slides and debris flows. Within theframework of the universal energy conservation law, energydissipation due to internal deformation is integrated intogeotechnical analysis of granular flows explicitly. Based onconservation laws of mass and energy, the analytical modeldeveloped in this paper provides new insights into quantita-tive analysis of granular flows in geotechnical settings.
Generally speaking, the momentum conservation and en-ergy conservation principles provide identical results whenapplied to certain flow problems. However, it has long beenrealized that particle interactions, such as intergrain collision
and sliding, dissipate energy in granular flows (Jaeger andNagel 1992; de Gennes 1999). The internal forces in granu-lar flows are much more complicated than the assumptionsin the existing depth-averaged continuum models. Formula-tion of governing equations for granular flows within theframework of energy conservation offers a simpler andclearer explanation based on granular flow physics and lendsitself to the incorporation of other phenomena, such as thoserelated to particle breakdown.
Conclusions
A slice-based model is proposed to simulate debris flows,flow slides, and other types of rapid landslides from a geo-technical perspective. The model is formulated based onuniversal conservation laws of mass and energy. Effects ofdeformation work and internal energy dissipation on debrisflow dynamics are accounted for explicitly in the proposedmodel.
A Lagrangian finite difference method has been presentedto solve the governing equations for the granular flow simu-lations. The terms due to deformation work in granularflows are incorporated into the solution to account for theeffects of internal energy dissipation. Simulations of simplegranular flows have been undertaken to examine the plausi-bility of the model and applicability of the numerical
Fig. 11. Nondimensional final deposit of granular slumps with the same initial aspect ratio of 3.2 (simulation using Savage–Hutter coeffi-cient of lateral earth pressure).
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scheme. The results of granular flow simulations provideevidence that the model based on the energy conservationlaw is robust.
AcknowledgementsThe authors would like to thank Professor P. Steffler at
the University of Alberta for contributions to the formula-tion of the analytical model and numerical solution pre-sented in the paper. The authors also benefitted fromdiscussions with Professor O. Hungr at The University ofBritish Columbia and Professor S.M. Olson at the Universityof Illinois at Urbana–Champaign.
This research has been sponsored by the Natural Sciencesand Engineering Research Council of Canada.
ReferencesAris, R. 1962. Vectors, tensors, and the basic equations of fluid
mechanics. Dover Publications, Inc., New York.de Gennes, B.G. 1999. Granular matter: a tentative view. Reviews
of Modern Physics, 71(2): S374–S382. doi:10.1103/RevModPhys.71.S374.
Denlinger, R.P., and Iverson, R.M. 2001. Flow of variably fluidizedgranular masses across three-dimensional terrain 2. Numericalpredictions and experimental tests. Journal of Geophysical Re-search. Solid Earth, 106(B1): 553–566. doi:10.1029/2000JB900330.
Denlinger, R.P., and Iverson, R.M., 2004. Granular avalanchesacross irregular three-dimensional terrain: 1. Theory and compu-tation. Journal of Geophysical Research - Earth Surface 109(F1):F01014.1–F01014.14. doi:10.1029/2003JF000085.
Greve, R., and Hutter, K. 1993. Motion of a granular avalanche ina convex and concave curved chute: experiments and theoreticalpredictions. Philosophical Transactions of the Royal Society A:Physical Sciences and Engineering, 342(1666): 573–600. doi:10.1098/rsta.1993.0033.
Huang, X., and Garcıa, M.H. 1998. A Herschel–Bulkley model formud flow down a slope. Journal of Fluid Mechanics, 374: 305–333. doi:10.1017/S0022112098002845.
Hungr, O. 1995. A model for the runout analysis of rapid flowslides, debris flows, and avalanches. Canadian GeotechnicalJournal, 32(4): 610–623. doi:10.1139/t95-063.
Hungr, O. 2003. User’s manual of DAN-W: Dynamic analysis oflandslides. Release 4 [computer program]. Hungr GeotechnicalResearch Inc., Vancouver, B.C.
Hungr, O., Evans, S.G., Bovis, M., and Hutchinson, J.N. 2001. Re-view of the classification of landslides of the flow type. Envir-onmental & Engineering Geoscience, VII: 221–238.
Hutchinson, J.N. 1986. A sliding–consolidation model for flowslides. Canadian Geotechnical Journal, 23(2): 115–126. doi:10.1139/t86-021.
Hutter, K. 2005. Geophysical granular and particle-laden flows: re-view of the field. Philosophical Transactions of the Royal So-ciety A: Mathematical, Physical and Engineering Sciences,363(1832): 1497–1505. doi:10.1098/rsta.2005.1591.
Hutter, K., and Koch, T. 1991. Motion of a granular avalanche inan exponentially curved chute: experiments and theoretical pre-dictions. Philosophical Transactions of the Royal Society A:Physical Sciences and Engineering, 334(1633): 93–138.
Iverson, R.M. 1997. The physics of debris flows. Reviews of Geo-physics, 35(3): 245–296. doi:10.1029/97RG00426.
Iverson, R.M., and Denlinger, R.P. 2001. Flow of variably fluidizedgranular masses across three-dimensional terrain 1. Coulomb
mixture theory. Journal of Geophysical Research. Solid Earth,106(B1): 537–552. doi:10.1029/2000JB900329.
Iverson, R.M., Reid, M.E., and LaHusen, R.G. 1997. Debris-flowmobilization from landslides. Annual Review of Earth and Plane-tary Sciences, 25(1): 85–138. doi:10.1146/annurev.earth.25.1.85.
Iverson, R.M., Logan, M., and Denlinger, R.P. 2004. Granular ava-lanches across irregular three-dimensional terrain: 2. Experimen-tal tests. Journal of Geophysical Research. Earth Surface109(F1): F01015.1–F01015.16. doi:10.1029/2003JF000084.
Jaeger, H.M., and Nagel, S.R. 1992. Physics of the granular state.Science, 255(5051): 1523–1531. doi:10.1126/science.255.5051.1523. PMID:17820163.
Jakob, M., and Hungr, O. 2005. Debris-flow hazards and relatedphenomena. Praxis Publishing Ltd., Chichester, UK.
Johnson, A.M. 1970. Physical processes in geology. Freeman-Cooper, San Francisco, Calif.
Johnson, A.M., and Rodine, J.R. 1984. Debris flow. In Slope in-stability. John Wiley & Sons, Chichester, UK. pp. 257–361.
Kwan, J.S.H., and Sun, H.W. 2006. An improved landslide mobi-lity model. Canadian Geotechnical Journal, 43(5): 531–539.doi:10.1139/T06-010.
Lajeunesse, E., Monnier, J.B., and Homsy, G.M. 2005. Granularslumping on a horizontal surface. Physics of Fluids, 17(10):103302. doi:10.1063/1.2087687.
Mangeney, A., Heinrich, P., and Roche, R. 2000. Analytical solu-tion for testing debris avalanche numerical models. Pure andApplied Geophysics, 157(6–8): 1081–1096. doi:10.1007/s000240050018.
Miao, T., Liu, Z., Niu, Y., and Ma, C. 2001. A sliding block modelfor the runout prediction of high-speed landslides. CanadianGeotechnical Journal, 38(2): 217–226. doi:10.1139/cgj-38-2-217.
Morgenstern, N.R. 1978. Mobile soil and rock flows. GeotechnicalEngineering, 9: 123–141.
Pitman, E.B., and Le, L.2005. A two-fluid model for avalanche anddebris flows. Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences, 363(1832):1573–1601.
Pudasaini, S.P., and Hutter, K. 2007. Avalanche dynamics: Dy-namics of rapid flows of dense granular avalanches. Springer,Berlin, Germany.
Pudasaini, S.P., Wang, Y., and Hutter, K. 2005a. Rapid motions offree-surface avalanches down curved and twisted channels andtheir numerical simulation. Philosophical Transactions of theRoyal Society A: Mathematical, Physical and EngineeringSciences, 363(1832): 1551–1571. PMID:16011933.
Pudasaini, S.P., Wang, Y., and Hutter, K. 2005b. Modelling debrisflows down general channels. Natural Hazards and Earth SystemSciences, 5(6): 799–819. doi:10.5194/nhess-5-799-2005.
Savage, S.B., and Hutter, K. 1989. The motion of a finite mass ofgranular material down a rough incline. Journal of Fluid Me-chanics, 199: 177–215. doi:10.1017/S0022112089000340.
Savage, S.B., and Hutter, K. 1991. The dynamics of avalanches ofgranular materials from initiation to runout. Part I: Analysis.Acta Mechanica, 86(1–4): 201–223. doi:10.1007/BF01175958.
Tinti, S., Bortolucci, E., and Vannini, C. 1997. A block-based the-oretical model suited to gravitational sliding. Natural Hazards,16(1): 1–28. doi:10.1023/A:1007934804464.
Wang, X. 2008. Geotechnical analysis of flow slides, debris flows,and related phenomena. Ph.D. thesis, University of Alberta, Ed-monton, Alta.
Wieland, M., Gray, J.M.N.T., and Hutter, K. 1999. Channelizedfree-surface flow of cohesionless granular avalanches in a chutewith shallow lateral curvature. Journal of Fluid Mechanics, 392:73–100. doi:10.1017/S0022112099005467.
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