AM  Vol.6 No.7 , June 2015
Saint-Venant Equations and Friction Law for Modelling Self-Channeling Granular Flows: From Analogue to Numerical Simulation
Abstract: Rock avalanches are catastrophic events involving important granular rock masses (>106 m3) and traveling long distances. In exceptional cases, the runout can reach up to tens of kilometers. Even if they are highly destructive and uncontrollable events, they give important insights to understand interactions between the displaced masses and landscape conditions. However, those events are not frequent. Therefore, the analogue and numerical modelling gives fundamental inputs to better understand their behavior. The objective of the research is to understand the propagation and spreading of granular mass released at the top of a simple geometry. The flow is unconfined, spreading freely along a 45° slope and deposit on a horizontal surface. The evolution of this analogue rock avalanche was measured from the initiation to its deposition with high speed camera. To simulate the analogue granular flow, a numerical model based on the continuum mechanics approach and the solving of the shallow water equations was used. In this model, the avalanche is described from a eulerian point of view within a continuum framework as single phase of incompressible granular material. The interaction of the flowing layer with the substratum follows a Mohr-Coulomb friction law. Within same initial conditions (slope, volume, basal friction, height of fall and initial velocity), results obtained with the numerical model are similar to those observed in the analogue. In both cases, the runout of the mass is comparable and the size of both deposits matches well. Moreover, both analogue and numerical modeling gave same magnitude of velocities. In this study, we highlighted the importance of the friction on a flowing mass and the influence of the numerical resolution on the propagation. The combination of the fluid dynamic equation with the frictional law enables the self-channelization and the stop of the granular mass.
Cite this paper: Longchamp, C. , Caspar, O. , Jaboyedoff, M. and Podladchikov, Y. (2015) Saint-Venant Equations and Friction Law for Modelling Self-Channeling Granular Flows: From Analogue to Numerical Simulation. Applied Mathematics, 6, 1161-1173. doi: 10.4236/am.2015.67106.

[1]   Heim, A. (1932) Der Bergsturz und Menschenleben. Fretz und Wasmuth Verlag, Zürich, 218 p.

[2]   Hungr, O., Evans, S.G., Bovis, M. and Hutchinson, J.N. (2001) Review of the Classification of Landslides of the Flow Type. Environmental & Engineering Geoscience, 7, 221-238.

[3]   Hsü, K.J. (1975) Catastrophic Debris Streams (Strurzstroms) Generated by Rockfall. Geological Society of America Bulletin, 86, 129-140.<129:CDSSGB>2.0.CO;2

[4]   Scheidegger, A.E. (1973) On the Prediction of the Reach and Velocity of Catastrophic Landslides. Rock Mechanics and Rock Engineering, 5, 231-236.

[5]   Nicoletti, P.G. and Sorriso-Valvo, M. (1991) Geomorphic Controlf the Shape and Mobility of Rock Avalanches. Geological Society of America Bulletin, 103, 1365-1373.<1365:GCOTSA>2.3.CO;2

[6]   Van Gassen, W. and Cruden, D.M., (1989) Momentum Transfer and Friction in the Debris of Rock Avalanches. Canadian Geotechnical Journal, 26, 623-628.

[7]   Davies, T.R.H. and McSaveney, M.J. (1999) Runout of Dry Granular Avalanches. Canadian Geotechnical Journal, 36, 313-320.

[8]   Davies, T.R.H. and McSaveney, M.J. (2003) Runout of Rock Avalanches and Volcanic Debris Avalanches. Proceedings of the International Conference on Fast Slope Movements, Naples, 11-13 May 2003, 2.

[9]   Okura, Y., Kitahara, H., Sammori, T. and Kawanami, A. (2000) The Effects of Rockfall Volume on Runout Distance. Engineering Geology, 58, 109-124.

[10]   Legros, F. (2002) The Mobility of Long-Runout Landslides. Engineering Geology, 63, 301-331.

[11]   Kent, P.E. (1966) The Transport Mechanism in Catastrophic Rock Falls. The Journal of Geology, 74, 79-83.

[12]   McDougall, S. and Hungr, O. (2004) A Model for the Analysis of Rapid Landslide Motion across Three-Dimensional Terrain. Canadian Geotechnical Journal, 41, 1084-1097.

[13]   Manzella, I. and Labiouse, V. (2008) Qualitative Analysis of Rock Avalanches Propagation by Means of Physical Modelling of Not Constrained Gravel Flows. Rock Mechanics and Rock Engineering, 41, 133-151.

[14]   Manzella, I. and Labiouse, V. (2009) Flow Experiments with Gravel and Blocks at Small Scale to Investigate Parameters and Mechanisms Involved in Rock Avalanches. Engineering Geology, 109, 146-158.

[15]   Takahashi, T. (1981) Debris Flow. Annual Review of Fluid Mechanics, 13, 57-77.

[16]   McDougall, S. (2005) A New Continuum Dynamic Model for the Analysis of Extremely Rapid Landslide Motion across Complex 3D Terrain. Ph.D. Thesis, University of British Columbia.

[17]   Mangeney-Castelnau, A., Vilotte, J.P., Bristeau, M.O., Perthame, B., Bouchut, F., Simeoni, C. and Yerneni, S. (2003) Numerical Modeling of Avalanches Based on Saint Venant Equations Using a Kinetic Scheme. Journal of Geophysical Research B, 108, EPM9.1-EPM9.18.

[18]   Mangeney-Castelnau, A., Bouchut, F., Vilotte, J.P., Lajeunesse, E., Aubertin, A. and Pirulli, M. (2005) On the Use of Saint Venant Equations to Simulate the Spreading of a Granular Mass. Journal of Geophysical Research B, 110, 1-17.

[19]   Mangeney-Castelnau, A., Bouchut, F., Thomas, N., Vilotte, J.P. and Bristeau, M.O. (2007) Numerical Modeling of Self-Channeling Granular Flows and of Their Levee-Channel Deposits. Journal of Geophysical Research, 112, Article ID: F02017.

[20]   Kelfoun, K. and Duitt, T.H. (2005) Numerical Modeling of the Emplacement of Socompa Rock Avalanche, Chile. Journal of Geophysical Research, 110.

[21]   Pirulli, M., Bristeau, M.O., Mangeney, A. and Scavia, C. (2007) The Effect of Earth Pressure Coefficient on the Runout of Granular Material. Environmental Modelling & Software, 22, 1437-1454.

[22]   Pirulli, M. (2010) On the Use of the Calibration-Based Approach for Debris-Flow Forward-Analyses. Natural Hazards and Earth System Science, 10, 1009-1019.

[23]   Pudasaini, S.P. and Hutter, K. (2007) Avalanche Dynamics. Springer, Berlin, 602 p.

[24]   Savage, S.B. and Hutter, K. (1989) The Motion of a Finite Mass of Granular Material down a Rough Incline. Journal of Fluid Mechanics, 199, 177-215.

[25]   Iverson, R.M. and Denlinger, R.P. (2001) Flow of Variably Fluidized Granular Masses across Three-Dimensional Terrain: 1. Coulomb Mixture Theory. Journal of Geophysical Research, 106, 537-552.

[26]   Andreotti, B., Forterre, Y. and Pouliquen, O. (2011) Les milieux granulaires: Entre fluide et solide. EDP Sciences, CNTS Editions, 495 p.

[27]   Pouliquen, O. (1999) Scaling Laws in Granular Flows down Rough Inclined Planes. Physics of Fluids, 11, 542-547.

[28]   Pouliquen, O. and Forterre, Y. (2001) Friction Law for Dense Granular Flows: Application to the Motion of a Mass down a Rough Inclined Plane. Journal of Fluid Mechanics, 453, 133-151.

[29]   Franz, M., Podladchikov, Y., Jaboyedoff, M. and Derron, M.-H. (2013) Landslide-Triggered Tsunami Modelling in Alpine Lakes. Italian Journal of Engineering Geology and Environment—Book Series 6, 409-416.