Two numerical models for landslide dynamic analysis

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Abstract

Two microcomputer-based numerical models (Dynamic ANalysis (DAN) and three-dimensional model DAN (DAN3D)) have been developed and extensively used for analysis of landslide runout, specifically for the purposes of practical landslide hazard and risk assessment. The theoretical basis of both models is a system of depth-averaged governing equations derived from the principles of continuum mechanics. Original features developed specifically during this work include: an open rheological kernel; explicit use of tangential strain to determine the tangential stress state within the flowing sheet, which is both more realistic and beneficial to the stability of the model; orientation of principal tangential stresses parallel with the direction of motion; inclusion of the centripetal forces corresponding to the true curvature of the path in the motion direction and; the use of very simple and highly efficient free surface interpolation methods. Both models yield similar results when applied to the same sets of input data.

Both algorithms are designed to work within the semi-empirical framework of the “equivalent fluid” approach. This approach requires selection of material rheology and calibration of input parameters through back-analysis of real events. Although approximate, it facilitates simple and efficient operation while accounting for the most important characteristics of extremely rapid landslides. The two models have been verified against several controlled laboratory experiments with known physical basis. A large number of back-analyses of real landslides of various types have also been carried out. One example is presented. Calibration patterns are emerging, which give a promise of predictive capability.

Introduction

A necessary requirement of a quantitative hazard assessment concerned with highly mobile landslides such as debris flows, debris avalanches, flow slides and rock avalanches (cf., Hungr et al., 2001 for classification), is to predict the character and extent of their motion. This is a rapidly developing research field at present. The authors have constructed two microcomputer-based numerical models: a pseudo-three-dimensional Dynamic ANalysis (DAN), Hungr, 1995), based on a depth-averaged, one-dimensional form of the equations of motion and a three-dimensional model DAN (DAN3D) (McDougall and Hungr, 2004, McDougall and Hungr, 2005), based on a depth-averaged, two-dimensional form of the same equations. Both models use the simplifying concept of “equivalent fluid”, as described below, and are designed to provide for practical modelling of real landslides, deriving input parameter values from back-analysis of case histories. The objectives are to provide simple solutions that are easy to use, with a minimum number of parameters requiring empirical calibration, but at the same time to account for certain important characteristics of landslide motion, including complex and possibly variable rheology, heterogeneity, internal stiffness and the ability to entrain material from the path (e.g., McDougall and Hungr, 2003).

The purpose of this paper is to present the key features of the algorithms used by DAN and DAN3D.

Section snippets

Equivalent fluid concept

Landslides are complex phenomena that are much more difficult to simulate dynamically than fluids because the standard assumptions of hydrostatic, isotropic internal stresses and material homogeneity do not apply. At the same time, the different materials that can be involved may have non-Newtonian rheologies that are strongly influenced by complex interactions during highly unsteady and non-uniform motion across steep and irregular terrain.

Given such complexity, any single material

The governing equations

The two models are based on Lagrangian forms of the depth-integrated St. Venant equations, applied in curvilinear coordinates, similar to the SH model. Similar governing equations have been derived by a number of workers (e.g., Eulerian forms by: Iverson and Denlinger, 2001; Pastor et al., 2002; Mangeney-Castelnau et al., 2003; Denlinger and Iverson, 2004; Lagrangian forms by: Savage and Hutter, 1989; Gray et al., 1999; Chen and Lee, 2000 and others).

Detailed derivation of the equations behind

Basal shear resistance

The basal shear stress, τzx, opposes motion and, due to the chosen reference coordinate system orientation, is always negative. Consistent with the concept of equivalent fluid, τzx is governed by a basal rheology that may be different from the internal rheology. To allow the simulation of different types of rapid landslides involving different geological materials, a variety of basal rheological relationships can be implemented in DAN and DAN3D, including laminar, turbulent, plastic, Bingham,

The pressure terms

As mentioned above, the pressure terms in Eqs. (10), (11) depend on assumptions regarding the tangential (bed-parallel) stress state in the flowing sheet. In a fluid, of course, these are simply hydrostatic stresses, i.e., zero shear and a constant, hydrostatic normal stress. In Eqs. (10), (11), this condition would result in both kzx and kzy being zero, while kx=ky=1.0. In a flowing, deforming sheet of frictional material, on the other hand, the k coefficients are functions of longitudinal

Free surface interpolation

The continuity equation (3) is not directly used in either DAN or DAN3D. Instead, continuity is implicit through the use of control volumes, attached to the reference columns. In DAN (Hungr, 1995), the control volumes are trapezoidal “mass blocks” representing the material carried between each adjacent pair of reference columns. The flow thickness in the centre of each mass block and each reference column are established after each time step by linear interpolation. The model allows the mass

Material entrainment

Material entrainment depends on slope, flow velocity, strength and quantity of the basal material available for entrainment. Entrainment may occur at the front by “ploughing”, or beneath the body of the flowing mass as step or basal erosion (Sovilla et al., 2006). Some algorithms that predict the entrainment rate from descriptive parameters have been proposed in the literature for dry granular material (Mangeney et al., 2007b) and layered snow (Sovilla et al., 2006). However, a similar

Strain interpolation

As described in Section 5, both DAN and DAN3D use actual values of tangential (bed-parallel) strain to assign values to the earth pressure (k) coefficients of Eq. (21). In DAN, the strain is recorded directly as a function of increasing or decreasing tangential distance between adjacent pairs of reference columns. The k coefficients are initialized to the hydrostatic value of 1.0. Then, as the flowing sheet contracts or expands depending on the shape of the path, the k-coefficient is increased

Numerical implementation

The DAN model is coded in Visual Basic and runs on standard microcomputers, requiring input in the form of a path profile, central thickness of the source volume and the flow width function. The path profile is smoothed by fitting a spline function. An individual analysis lasts less than 1 min.

The DAN3D model has been coded in C++ and runs on an IBM-compatible personal computer. The programme reads spatial data in the form of user-created grid files, which contain the following data at nodal

Verification testing

Verification of models of this type usually requires the analysis of controlled laboratory experiments, in which the rheological kernel as well as the controlling parameters are independently known, or comparison with other, independently developed models. The DAN model was verified by analysing several laboratory experiments involving dry sand flow over smooth base, a laboratory flume experiment involving laminar flow of oil of known viscosity and comparisons with other model results (Hungr,

Example result

The use of the 2D model DAN and its 3D counterpart DAN3D is illustrated by the example of a rock avalanche from the Swiss Alps. On May 30, 1946, a magnitude 4.4 earthquake triggered a rock avalanche in the Andins Valley in Valais, Switzerland (CREALP, 2001). The event initiated as a slide in limestone on the south face of an eastern arête of Six des Eaux Froides. Normal faults dipping at about 40–43° to the south formed the main failure surface between elevations 2300 and 2700 m (CREALP, 2001).

Conclusion

The two models represent versatile tools for simple, albeit somewhat approximate, DAN of rapidly moving landslides of various types. DAN and DAN3D are based on the same theory, use similar numerical solution methods and produce similar results when applied to similar data. The system of governing equations used in DAN and DAN3D has been derived from first principles both mathematically and phenomenologically using a straightforward series of explicit and justifiable assumptions and

Acknowledgements

This work was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) post-graduate scholarship. The authors would like to thank Drs. J.D. Rouiller and Raphaël Mayoraz of CREALP (Centre de Recherche sur L’Environnement Alpin) in Switzerland for providing assistance and information on the case study. Nikolai Hungr assisted with the analyses of the case history.

References (47)

  • B. Voight et al.

    Rockslides and avalanches: an introduction

  • P. Brufau et al.

    1D mathematical modelling of debris flow

    Journal of Hydraulic Research

    (2000)
  • H. Chen et al.

    Numerical simulation of debris flows

    Canadian Geotechnical Journal

    (2000)
  • H. Chen et al.

    Erosional effects on runout of fast landslides, debris flows and avalanches: a numerical investigation

    Geotechnique

    (2006)
  • V.T. Chow

    Open-Channel Hydraulics

    (1959)
  • CREALP

    Éboulement du 30 Mai, 1945 du Six des Eaux Froides près du Rawyl (Valais). (Rockslide of May 30, 1945 at Six des Eaux Froides near Rawyl, Valais)

    (2001)
  • R.P. Denlinger et al.

    Flow of variably fluidized granular masses across three-dimensional terrain, 2. Numerical predictions and experimental tests

    Journal of Geophysical Research

    (2001)
  • R.P. Denlinger et al.

    Granular avalanches across irregular three-dimensional terrain: 1. Theory and computation

    Journal of Geophysical Research

    (2004)
  • H. Erlichson

    A mass-change model for the estimation of debris-flow runout: a second discussion: conditions for the application of the rocket equation

    Journal of Geology

    (1991)
  • R. Fell et al.

    Rapid landslides, the importance of understanding mechanisms and rupture surface mechanics, 8th Glossop Lecture

    Quarterly Journal of Engineering Geology and Hydrogeology

    (2007)
  • J.M.N.T. Gray et al.

    Gravity-driven free surface flow of granular avalanches over complex basal topography

    Proceedings of the Royal Society of London, A

    (1999)
  • O. Hungr

    A model for the runout analysis of rapid flow slides, debris flows, and avalanches

    Canadian Geotechnical Journal

    (1995)
  • O. Hungr

    Dynamics of rapid landslides

  • Hungr, O., Evans, S.G., 1996. Rock avalanche runout prediction using a dynamic model. In: Proceedings of the Seventh...
  • O. Hungr et al.

    Entrainment of debris in rock avalanches; an analysis of a long run-out mechanism

    Bulletin of the Geological Society of America

    (2004)
  • O. Hungr et al.

    Review of the classification of landslides of the flow type

    Environmental and Engineering Geoscience

    (2001)
  • O. Hungr et al.

    Rapid flow slides of coal mine waste in British Columbia, Canada

  • Hungr, O., Corominas, J., Eberhardt, E., 2005. State of the Art Paper #4, Estimating landslide motion mechanism, travel...
  • R.M. Iverson

    The physics of debris flows

    Reviews of Geophysics

    (1997)
  • R.M. Iverson et al.

    Flow of variably fluidized granular masses across three-dimensional terrain, 1. Coulomb mixture theory

    Journal of Geophysical Research

    (2001)
  • R.M. Iverson et al.

    Granular avalanches across irregular three-dimensional terrain: 2. Experimental tests

    Journal of Geophysical Research

    (2004)
  • H.J. Körner

    Reichweite und Geschwindigkeit von Bergstürzen und Fließschneelawinen. (Reach and velocity of rockslides and powder snow avalanches)

    Rock Mechanics

    (1976)
  • D. Laigle et al.

    Numerical modeling of mudflows

    Journal of Hydraulic Engineering

    (1997)

    • Cited by (0)

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