An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

International audienceObserved avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models

A two-layer shallow flow model with two axes of integration, well-balanced discretization and application to submarine avalanches

We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depthaveraging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kind of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to 30◦), strong relative errors on the avalanche dynamics and deposit (up to 60%) and on the generated water waves (up to 120%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Agence Nationale de la Recherche. FranceEuropean Research Council (ERC

Viscoplastic modeling of granular column collapse with pressure-dependent rheology

International audienceA mechanical and numerical model of dry granular flows is proposed that quantitatively reproduce laboratory experiments of granular column collapse over inclined planes. The rheological parameters are directly derived from the experiments.The so-called \mu(I) rheology is reformulated in the framework of Drucker-Prager plasticity with the yield stress and viscosity \eta(||D||,p) depending on both the pressure p and the norm of the strain rate tensor ||D||. The granular domain, velocities, stress deviator and pressure fields are calculated using a finite element method based on an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method. 2-D simulations using this model well reproduce the dynamics and deposits of collapsing granular columns. The flow is essentially located in a surface layer behind the front, whereas it is distributed over the whole depth near the front where basal sliding occurs. The computed runout distances and slopes of the deposits agree very well with the values found in the experiments. Using an easily calculated order of magnitude approximation of the mean viscosity during the flow (\eta = 1 Pa s here), we show that a Drucker-Prager rheology with a constant viscosity gives results very similar to the \mu(I) rheology and agrees with experimental height profiles, while significantly reducing the computational cost. Within the range of viscosities 0.1 < \eta < 1 Pa s, the dynamics and deposits are very similar. The observed slumping behavior therefore appears to be mainly due to the flow/no-flow criterion and to the associated strain-independent part of the "flowing constitutive relation" (i.e. related to plastic effects). However, the results are very different when an unrealistically large value of viscosity (10 Pa s) is used

Influence of the scar geometry on landslide dynamics and deposits: Application to Martian landslides

International audienceLandslides dynamics prediction remains difficult in spite of a considerable number of studies. The runout distance is widely used in analysis of landslide dynamics and in the calibration of the rheological parameters involved in numerical modeling. However, the unknown impact of the significant uncertainty in the shape of the initial released mass on the runout distance and on the overall shape of the deposit raises questions about the relevance of these approaches. The impact of the initial scar geometry on flow and distribution of the deposits is studied here using satellite data and numerical modeling of theoretical landslides, and Martian landslides informed by geomorphological analysis, by varying the initial scar geometry from spoon‐shaped to steep wall geometry. Our results show that the runout distance is a very robust parameter that is only slightly affected by the change in the geometry of the initial scar. On the contrary, the lateral extent of the deposit is shown to be controlled by the scar geometry, providing unique insights into the initial landsliding conditions on Mars and makes it possible to accurately recover the volume initially involved, an essential ingredient for volume balance calculation. A feedback analysis of Valles Marineris landslides can be drawn, showing good agreement between numerical results and geomorphological analysis; the geometry of the initial scar inferred from numerical modeling is strongly correlated with the regional tectonic history in Valles Marineris area

Numerical modeling of landquakes

International audienceThe Thurwieser landslide that occurred in Italy in 2004 and the seismic waves it generated have been simulated and compared to the seismic signal recorded a few tens of kilometers from the landslide source (i.e., landquake). The main features of the low frequency seismic signal are reproduced by the simulation. Topography effects on the flowing mass have a major impact on the generated seismic signal whereas they weakly affect low-frequency wave propagation. Simulation of the seismic signal makes it possible to discriminate between possible alternative scenarios for flow dynamics and to provide first estimates of the rheological parameters during the flow. As landquakes are continuously recorded by seismic networks, our results provide a new way to collect data on the dynamics and rheology of natural flows

A two-phase two-layer model for fluidized granular flows with dilatancy effects

International audienceWe propose a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects, based on the closure relation proposed by Roux and Radjai (1998). This relation implies that the occurrence of dilation or contraction of the granular material depends on whether the solid volume fraction is respectively higher or lower than a critical value. When dilation occurs, the fluid is sucked into the granular material, the pore pressure decreases and the friction force on the granular phase increases. On the contrary, in the case of contraction, the fluid is expelled from the mixture, the pore pressure increases and the friction force diminishes. To account for this transfer of fluid into and out of the mixture, a two-layer model is proposed with a fluid layer on top of the two-phase mixture layer. Mass and momentum conservation are satisfied for the two phases, and mass and momentum are transferred between the two layers. A thin-layer approximation is used to derive average equations, with accurate asymptotic expansions. Special attention is paid to the drag friction terms that are responsible for the transfer of momentum between the two phases and for the appearance of an excess pore pressure with respect to the hydrostatic pressure. For an appropriate form of dilatancy law we obtain a depth-averaged model with a dissipative energy balance in accordance with the corresponding 3D initial system

A two-phase solid-fluid model for dense granular flows including dilatancy effects: comparison with submarine granular collapse experiments

We simulate here the collapse of granular columns immersed in a viscous fluid based on a simplified version of the model developed by [2]. The simulation quite well reproduces the dynamics and deposit of the granular mass as well as the excess pore fluid pressure measured in the laboratory experiments of [10] owing that dilatancy effects and pore pressure feedback are accounted for. In particular, the difference in the behaviour of initially loose and dense columns is reproduced numerically

Deducción y simulación numérica de un nuevo modelo de avalanchas submarinas

En este trabajo se estudia un nuevo modelo de tipo Savage-Hutter para avalanchas submarinas. El modelo obtenido es de tipo aguas someras bicapa en el que la capa superior representa al fluido y la capa inferior representa a un estrato de material granular. En la deducción del modelo se han tenido en cuenta diferentes leyes constitutivas en los tensores de esfuerzos, la porosidad del estrato de sedimento o roca y un término de fricción de tipo Coulomb. El modelo obtenido puede aplicarse en el estudio de problemas de avalanchas submarinas y algunos tipos de tsunamis. Finalmente se presentan la discretización del modelo obtenido mediante técnicas de vol´umenes finitos bien equilibrados y un ensayo numérico donde se modela una avalancha submarina

A new Savage-Hutter type model for submarine avalanches and generated tsunami

In this paper we present a new two-layer model of Savage-Hutter type to study submarine avalanches. A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e. g. water). The model is derived in a system of local coordinates following a non-erodible bottom and takes into account its curvature. We prove that the model verifies an entropy inequality, preserves water at rest for a sediment layer and their solutions can be seen as particular solutions of incompressible Euler equations under hydrostatic assumptions. Buoyancy effects and the centripetal acceleration of the grain movement due to the curvature of the bottom are considered in the definition of the Coulomb term. We propose a two-step Roe type solver to discretize the presented model. It exactly preserves water at rest and no movement of the sediment layer, when its angle is smaller than the angle of repose, and up to second order all stationary solutions. Finally, some numerical tests are performed by simulating submarine and sub-aerial avalanches as well as the generated tsunami

Multilayer models for shallow two-phase debris flows with dilatancy effects

We present here a multilayer model for shallow grain-fluid mixtures with dilatancy effects. It can be seen as a generalization of the depth-averaged model presented in Bouchut et al. (2016) [6], that includes dilatancy effects by considering a two-layer model, a mixture grain-fluid layer and an upper fluid layer, to allow the exchange of fluid between them. In the present work the approximation of the mixture layer is improved including normal variations of the velocities and concentrations of the two phases thanks to the multilayer approach. In the model presented here dilatancy effects induce in particular a non-hydrostatic pressure for both phases related to the excess pore fluid pressure. Contrary to the single-layer model, the computation of this excess pore pressure entrains a serious difficulty due to the multilayer approach. We identified here one of the main numerical difficulty of solving two-phase shallow debris flows models: the strongly non-linear behaviour and abrupt changes of the excess pore fluid pressure when starting from non-equilibrium conditions. We propose a simplified approach to approximate the excess pore fluid pressure in the simple case of uniform flows in the downslope direction and quantify the error made. Our method makes it possible to introduce two or three layers in the normal directions with a reasonable approximation. Analytical solutions for uniform grain-fluid flows over inclined planes, with and without side wall friction, are calculated and compared to the proposed model. The presented model preserves the total solid granular mass as in [6]. In the numerical results, we observe that the proposed model with a two layer description of the mixture accurately represents the velocity measured at the surface of the mixture in the laboratory experiments. This is obviously poorly represented by the depth-averaged velocity in single-layer models while the other quantities (solid volume fraction, basal excess pore fluid pressure) are similar to those obtained with single-layer models. Our numerical results show a significant impact of the parameters involved in dilatancy law, in particular on the calculation of the time evolution of the excess pore fluid pressure