A Two-Phase Solid/Fluid Model for Dense Granular Flows including Dilatancy Effects

Transcription

1 23-26th May 2016, GdR EGRIN, Piriac-sur-Mer A Two-Phase Solid/Fluid Model for Dense Granular Flows including Dilatancy Effects Anne Mangeney1,1 F. Bouchut2, E. Fernandez-Nieto3, G. Narbona-Reina3, and E. H. Koné1 1Institut de Physique du Globe de Paris, University Paris Diderot, SPC, France 1 ANGE team, INRIA-Lab. Jacques Louis Lions-CEREMA, France 2LAMA, UMR-8050, University Paris Est Marne la Vallée, France 3DMA, University of Sevilla, Spain

2 Modeling of debris flows (grain/fluid) Solid volume fraction: 0.4 < < 0.8 At the field scale Thin layer approximation hm/lm =ε 1 Depth-averaged model h m b,,,, Mixture model L m u h m b Two-phase model fluid u Iceland solid v Canada Iverson, Denlinger; Denlinger, Iverson 2001, Iverson, George; George, Iverson 2014 Pitman and Le 2005, Pelanti et al. 2008, 2011

3 Modeling of debris flows (grain/fluid) Solid volume fraction: 0.4 < < 0.8 At the field scale Thin layer approximation hm/lm =ε 1 Depth-averaged model h m b,,,, Mixture model L m u h m b Two-phase model fluid u Iceland solid v Canada dilatancy! Iverson, Denlinger; Denlinger, Iverson 2001, Iverson, George; George, Iverson 2014 Pitman and Le 2005, Pelanti et al. 2008, 2011

4 Jackson s model Jackson, 2000 Mass conservation : : solid volume fraction, : fluid volume fraction,,,, * * Momentum conservation : * * Friction between the solid and fluid phases : 5 unknowns :,,,,, 4 equations A constitutive equation is required to close the system

5 Closure equation and energy dissipation Former thin layer depth-averaged models (Pitman and Le 2005, Pelanti et al. 2008) Additional free surface boundary condition instead of a closure equation: Hydrostatic pressure + depth-averaging boundary condition closure inside the domain Dissipation of energy is not ensured! θ FIRST STEP closure equation : Incompressibility of the solid phase Dissipative energy equation Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, 2015

6 Difference with Pitman and Le s model Initial volume fraction :, slope angle Thickness h m t =3 s Thickness h m t =10 s Distance x Distance x Solid volume fraction ϕ t =3 s Distance x t =3 s Distance x Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, 2015 Solid velocity v

7 Qualitative explanation of dilatancy effects Critical state : when deformed (loose) Granular medium contracts Fluid is expelled ( ) Fluid pore pressure increases Liquefaction (dense) Granular medium dilates Fluid is sucked (- ) Fluid pore pressure decreases Stiffening of the granular matrix - p fm p fm Excess pore pressure p e fm 0 p e fm 0 Coulomb friction: e. g. Schofield and Wroth, 1968, Wood, 1990, Pailha and Pouliquen, 2009

8 Differential motion of the fluid and solid phases A crucial element : Make it possible for the fluid to enter/escape the granular phase Solid free surface fluid free surface! u f u v Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, 2016

9 Boundary conditions At the free surface, for the fluid: * no tension: * kinematic condition: At the interface mixture/fluid: * kinematic condition for the solid: * Rankine-Hugoniot (mass conservation) condition: - * * Rankine-Hugoniot (momentum conservation) condition: * * stress transfer condition from the energy balance: * Navier friction condition for the fluid:

10 Boundary conditions Bottom boundary conditions: * * No penetration condition: * Coulomb friction for the solid: * Navier friction for the fluid: Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, 2016

11 Modeling of dilatancy effects Compression/dilatation of the solid phase : : dilatancy angle, Roux and Radjai, 1998 : shear strain rate T s xz T s zz ψ Pailha and Pouliquen, 2009 Closure equation : : compression : dilatation Impact of the dilatancy angle on the Coulomb friction force : Dilatation increases friction in addition to decrease of fluid pore pressure

12 Fluid pore pressure in thin layer depth-averaged models From the fluid momentum conservation in the direction normal to the slope with using the dilatancy closure equation Non-hydrostatic (excess) fluid pressure eq Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, JFM, 2016

13 Our model in the uniform immersed configuration Mass conservation : *, z - - * Momentum conservation : θ * * with,, Closure related to dilatancy :,,

14 Time scales involved in the model From the volume fraction equation and convergence to a critical state: Effect of solid viscous friction (from the solid momentum equation) : Relaxation of the relative velocity (from the solid momentum equation) : Relaxation of the fluid pressure in Iverson and George model :

15 Equilibrium state and parameters Critical state : z - Our model : at t = 0s θ Pailha-Pouliquen model : Parameters for the laboratory experiments of Pailha et al., 2008 : - - = 22.5,

16 Simple tests on submarine granular flows Laboratory experiments: Pailha et al., 2008 High viscosity : z - Pailha-Pouliquen PP Pailha-Pouliquen PP at t = 0s θ = 25 In our model dilatation compression compression dilatation

17 Simple tests on submarine granular flows Pailha-Pouliquen PP dilatation Pailha-Pouliquen PP > compression > 0 dilatation compression <

18 Simple tests on submarine granular flows High viscosity : Less difference between the models than between simulations and experiments compression dilatation Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, 2016

19 Simple tests on submarine granular flows Low viscosity : dilatation dilatation

20 Simple tests on submarine granular flows Low viscosity : dilatation!! dilatation dilatation!! dilatation

21 Conclusion Two-phase thin layer model taking into account accurately: - dilatancy effects - differential motion of the solid and fluid phases - along-slope gradients of the excess fluid pressure - insures dissipation of energy Different from the mixture model of Iverson and George explicit description of the two phases - different diffusion of the fluid pressure Numerical model for application SLIDEQUAKES Colorado Geological Survey

22 Equilibrium state and parameters Critical state : z - at t = 0s θ Parameters for the laboratory experiments of Pailha et al., 2008 : - -

23 Iverson and George model Dilatancy empirical law: Darcy law:

24 Closure equation and energy (Pitman and Le, 2005; Pelanti et al., 2008, 2011) Surface boundary condition instead of a closure equation: Grain thickness = fluid thickness FIRST STEP Incompressibility of the solid phase: Dissipative energy equation Bouchut, Fernandez-Nieto, Mangeney, Narbona-Reina, M2AN, 2015