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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Particle based modeling of granular-uid mixtures:from porous media to suspensions via solid-uid
transition
Bruno Chareyre (Grenoble INP - 3SR)Emanuele Catalano (Grenoble INP - 3SR)Anh Tuan Tong (Grenoble INP - 3SR)Eric Barthélémy (Grenoble INP - LEGI)
Granulaires Immergés et Suspensions en ECoulement - GISEC33Pessac, Nov. 21th 2011
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Introduction
Discrete elements modeling (DEM) becomes a standard tool forstudying the behavior of (dry) geomaterials at the microscale.
Modern problems involve couplings with intersticial uids, andDEM-based hydromechanical models are being developped actively.
Ecient numerical methods have been proposed to simulate surfacetension eects in two-phase problems (see e.g. Sholtès et al. 2009).
For uid ow however, the proposed methods are eitherover-simplifying or computationaly high-demanding, even for asingle uid.
We propose a new approach, that we apply to one-phaseincompressible Stokes ow as a rst step.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions4/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid - Particles coupling
Continuum Flow Models
The variables (porosity, uid velocity,...) are averaged in elements containing manyparticles.
A minimal number of solid grain per element to prevent numerical instability.
Low uid / solid DOF's ratio → good performance.
The uid model is not micro, hence the need for phenomenological laws and limitedpredictive power.
Problems when the phenomena are heterogeneous at the small scales (strainlocalization, hydrofracturing, internal erosion,...).
Figure: Continuum-Discrete coupling scheme. Zeghal and Shamy, 20045/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid - Particles coupling
Microscale uid models (LBM, FEM, FVM,...)
Mesh size particles sizes.
High uid / solid DOF's ratio
Small number of particles
In principle, only classical uid mechanics is introduced (Navier Stokes / Stokes)
Commonly restricted to 2D models, where tricks are needed to let the uid ow (hencekilling the previous advantage...)
Figure: Coupled DEM - LBM method. Han and Feng, 20076/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid - Solid coupling
Pore scale Finite Volumes Model (PFV)
Fluid / solid DoF's ratio ' 1
Preserves the discrete nature of DEM modelsOnly classical uid mechanics
Hundred thousands of particles in 3D
Figure: Pore scale Finite Volumes model.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions8/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid phase
Regular triangulation
Regular triangulation of particles (onetethraedron = one pore in 3D)
Voronoi tessellation
The dual Voronoi represents a map of thepore space, i.e. path for the uid.
Finite Volumes Formulation
Deforming mesh, following the deformationof the solid phase.One value of pressure per pore (piecewiseconstant eld).
Delaunay Triangulation.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid phase
Regular triangulation
Regular triangulation of particles (onetethraedron = one pore in 3D)
Voronoi tessellation
The dual Voronoi represents a map of thepore space, i.e. path for the uid.
Finite Volumes Formulation
Deforming mesh, following the deformationof the solid phase.One value of pressure per pore (piecewiseconstant eld).
Voronoi Tessellation.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Fluid phase
Regular triangulation
Regular triangulation of particles (onetethraedron = one pore in 3D)
Voronoi tessellation
The dual Voronoi represents a map of thepore space, i.e. path for the uid.
Finite Volumes Formulation
Deforming mesh, following the deformationof the solid phase.One value of pressure per pore (piecewiseconstant eld). The PFV model.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Regular vs. Delaunay triangulation
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions11/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical Formulation - Flow
Mass conservation
∀cell →∂ρf∂t
+∇ · ρf ~u = 0 (1)
Deformable networkIncompressible uid
⇒∂Vf
∂t+
∫∂Ω
(~uf − ~uΣ) · ~ndS = 0 (2)
In discrete form :
⇒dVf
dt+
∑facets
(~uf ,k ·~n−~uΣ,k ·~n) ·Sk = 0 (3)
Fluid exchanged between pores∑facets
(~uf ,k · ~n − ~uΣ,k · ~n) · Sk =∑
neighbours,i
qi,j (4)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical Formulation - Flow
Mass conservation
∀cell →∂ρf∂t
+∇ · ρf ~u = 0 (1)
Deformable networkIncompressible uid
⇒∂Vf
∂t+
∫∂Ω
(~uf − ~uΣ) · ~ndS = 0 (2)
In discrete form :
⇒dVf
dt+
∑facets
(~uf ,k ·~n−~uΣ,k ·~n) ·Sk = 0 (3)
Fluid exchanged between pores∑facets
(~uf ,k · ~n − ~uΣ,k · ~n) · Sk =∑
neighbours,i
qi,j (4)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical Formulation - Flow
Mass conservation
∀cell →∂ρf∂t
+∇ · ρf ~u = 0 (1)
Deformable networkIncompressible uid
⇒∂Vf
∂t+
∫∂Ω
(~uf − ~uΣ) · ~ndS = 0 (2)
In discrete form :
⇒dVf
dt+
∑facets
(~uf ,k ·~n−~uΣ,k ·~n) ·Sk = 0 (3)
Fluid exchanged between pores∑facets
(~uf ,k · ~n − ~uΣ,k · ~n) · Sk =∑
neighbours,i
qi,j (4)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical Formulation - Flow
Mass conservation
∀cell →∂ρf∂t
+∇ · ρf ~u = 0 (1)
Deformable networkIncompressible uid
⇒∂Vf
∂t+
∫∂Ω
(~uf − ~uΣ) · ~ndS = 0 (2)
In discrete form :
⇒dVf
dt+
∑facets
(~uf ,k ·~n−~uΣ,k ·~n) ·Sk = 0 (3)
Fluid exchanged between pores∑facets
(~uf ,k · ~n − ~uΣ,k · ~n) · Sk =∑
neighbours,i
qi,j (4)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Pores and Grains
Local geometry
The geometry of the ow path is rather complex anddoes not dene a closed pipe.
Stokes Flow
Stokes equation implies a linear relation between the pressuremicro-gradient and the ux. Then,
qi,j = gijpi − pj
Li,j(5)
where gij is the local conductance dened for the facet ij .
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Pores and grains
Local geometry
The local geometry of the ow path is rather complex and does notdene a closed pipe.
Figure: Bryant et Blunt 1991Thompson 1997
Flow
Which expression in deningconductivity between neighbouringpores ?
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Pores and Grains 2/
Hagen-Poiseuille
We use generalized Hagen-Poiseuille law to dene localconductivities,
⇒ qi,j =AijR
2hyd
2η
∆P
Li,j(6)
where Aij is the area of the uid interface and Rhyd is thehydraulic radius dened for the facet ij .
⇒ Rhyd =Vuid
Ssolid(7)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions16/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Flux predictions
Boundary conditions
Imposed pressure on top and bottomboundariesImpermeable lateral walls
The results gives Qin = Qout
Comparisons
Numerical results from COMSOLsimulationsExperimental measurements on glassbeads for dierent ne/coarse grainsizes ratio (A.T.Tong, PhD inLab3SR)
Pressure elds
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Flux predictions
Boundary conditions
Imposed pressure on top and bottomboundariesImpermeable lateral walls
The results gives Qin = Qout
Comparisons
Numerical results from COMSOLsimulationsExperimental measurements on glassbeads for dierent ne/coarse grainsizes ratio (A.T.Tong, PhD inLab3SR)
Pressure elds
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Flux predictions
Boundary conditions
Imposed pressure on top and bottomboundariesImpermeable lateral walls
The results gives Qin = Qout
Comparisons
Numerical results from COMSOLsimulationsExperimental measurements on glassbeads for dierent ne/coarse grainsizes ratio (A.T.Tong, PhD inLab3SR)
Pressure elds
Solution FEM (a) and PFV (b)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Flux predictions
Boundary conditions
Imposed pressure on top and bottomboundariesImpermeable lateral walls
The results gives Qin = Qout
Comparisons
Numerical results from COMSOLsimulationsExperimental measurements on glassbeads for dierent ne/coarse grainsizes ratio (A.T.Tong, PhD inLab3SR)
Pressure elds
Pressure/velocity (top) and pressuregradient (bottom)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions18/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical formulation - Forces
Forces on particles(Chareyre et al., Transport in Porous Media 2011, in press
Force on particle k :
F k =
∫δΓ
k
(p · n + τ · n)ds =
∫δΓ
k
(ρgz + p∗ + τ)nds (8)
Buoyancy force
Fb =
∫δΓ
k
ρgznds (9)
Pressure forces resultant, obtained by consideringpiecewise constant pressure
Fp =
∫δΓ
k
p∗nds = Akij∆pij~nij (10)
Viscous forces resultant, obtained by momentumconservation ⇒ divσ → 0
F v =
∫δΓ
k
τnds = Afij (pj − pi )nij (11)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical formulation - Forces
Forces on particles(Chareyre et al., Transport in Porous Media 2011, in press
Force on particle k :
F k =
∫δΓ
k
(p · n + τ · n)ds =
∫δΓ
k
(ρgz + p∗ + τ)nds (8)
Buoyancy force
Fb =
∫δΓ
k
ρgznds (9)
Pressure forces resultant, obtained by consideringpiecewise constant pressure
Fp =
∫δΓ
k
p∗nds = Akij∆pij~nij (10)
Viscous forces resultant, obtained by momentumconservation ⇒ divσ → 0
F v =
∫δΓ
k
τnds = Afij (pj − pi )nij (11)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Analytical formulation - Forces
Forces on particles(Chareyre et al., Transport in Porous Media 2011, in press
Force on particle k :
F k =
∫δΓ
k
(p · n + τ · n)ds =
∫δΓ
k
(ρgz + p∗ + τ)nds (8)
Buoyancy force
Fb =
∫δΓ
k
ρgznds (9)
Pressure forces resultant, obtained by consideringpiecewise constant pressure
Fp =
∫δΓ
k
p∗nds = Akij∆pij~nij (10)
Viscous forces resultant, obtained by momentumconservation ⇒ divσ → 0
F v =
∫δΓ
k
τnds = Afij (pj − pi )nij (11)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Validation of forces denition
Comparison with FEM results
See B. Chareyre et al. Pore-scale Modeling of ViscousFlow and Induced Forces in Dense Sphere Packings,Transport in Porous Media, (in press).
Imposed pressure on top and bottom boundaries.
Impermeable lateral walls.
Forces are extracted and compared for dierentsizes of the small sphere.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions21/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Computation cycle
System to solve
Linearized system to be solved :
[K ]p = ∆V + kimp .pimp (12)
Giving the uid forces :
f w = [G ][K ]−1(∆V + kimp .pimp) (13)
[K ], conductivity matrix (symmetric, sparse, positive dened)
p, pore pressures
∆V , pores' volumetric deformation rate
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions23/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Oedometric consolidation
Simulation
5000 particles
σ = 5000Pa
k = 3 · 10−7m/s
Coecient of consolidation :
Cv =kEoed
γ(14)
Consolidation time :
Tv =Cv t
H2(15)
Paraview Visualization Tool
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Oedometric consolidation
Simulation
5000 particles
σ = 5000Pa
k = 3 · 10−7m/s
Coecient of consolidation :
Cv =kEoed
γ(14)
Consolidation time :
Tv =Cv t
H2(15)
Paraview Visualization Tool
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Oedometric consolidation
Simulation
5000 particles
σ = 5000Pa
k = 3 · 10−7m/s
Analytical solution byTerzaghi (1923)
Evolution of interstitialpressure p(z, t)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions26/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Seabed sediment and waves action
Figure: LEGI instrumented channel.
Context
Role of external ow (waves) on the internal deformation (ex. liquefaction) ?
Complex phenomena involved, mobilizes elasto-plastic behaviour in cyclicloading conditions and liquefaction.
National Project C2D2 Hydrofond on seabed sediments and immersedfoundations.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Layout
1 IntroductionFluid - Particles coupling models
2 The modelSpatial partitioning : a key aspect
3 Stokes owAnalytical formulationFlux predictions
4 Fluid ForcesAnalytical Formulation
5 CouplingAlgorithmThe Terzaghi's problem
6 Seabed sediments simulationMotivationSimulated wave action
7 Conclusions28/40
Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Seabed sediment and waves action
An idealized wave action
A sinusoidal-shaped pressure prole is imposed on top boundary
A condition of impermeability is imposed on lateral and bottom boundaries
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Seabed Deformation
Inuence of the initial state
The amplitude of the wave A is increased linearly with time
The behaviour of an initially dense packing (n ' 0.38) vs. a loose packing(n ' 0.42) is observed
Signicant deformations start at A = 1500Pa in the dense packing, A = 300Pain the loose packing.
Figure: Dense packing.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Seabed Deformation
Inuence of the initial state
The amplitude of the wave A is increased linearly with time
The behaviour of an initially dense packing (n ' 0.38) vs. a loose packing(n ' 0.42) is observed
Signicant deformations start at A = 1500Pa in the dense packing, A = 300Pain the loose packing.
Figure: Loose packing.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Liquefaction
Figure: Piezometric pressure vs. (x,y) in the dense packing
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Liquefaction
Figure: Piezometric pressure vs. (x,y) in the loose packing
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Seabed Deformation
A more realistic wave action
Seabed loaded by a stationary wave
The scope is to analyze the relation between amplitude and frequency of thewaves, and the evolution of the sediment,...
Figure: Oscillating Waves.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Conclusions
The model gives correct prediction of permeabilities in spherepackings, refecting the role of porosity and PSD.
The forces induced on individual particles are computed correctly(still limited in terms of size ratio).
The fully coupled scheme is validated in the classical consolidationproblem.
Ecient implementation for large problems, an optimizedtime-integration scheme gave +50% cpu time compared to DEMwithout uid.Up to 500k particles have been simulated in single thread. Thecomputation will be fully parallel in the near future (only partialyparallel a.t.m).
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Computation cycle
System to solve
Linearized system to be solved :
[K ]p = ∆V + kimp .pimp (16)
Giving the uid forces :
f w = [G ][K ]−1(∆V + kimp .pimp) (17)
[K ], conductivity matrix (symmetric, sparse, positive dened)
p, pore pressures
∆V , pores' volumetric deformation rate
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Recent/current researches on the basis of the DEM-PFVcoupling
Extensions and Applications
Interactions between waves and sea-bed sediments (C2D2 ProjectHydrofond supported by MEDDTL, PhD Emanuele Catalano)
Extension to compressible uids for application to high pressureimpacts on concrete structures (PhD thesis Van Tran Tieng,Grenoble)
Extension to dense suspensions and bedload by introducing theeects of shear deformation at the microscale (PhD thesis DoniaMarzougui, Grenoble)
Application to granular lters (transport, plugging), Sari et al., inParticles (2011), Barcelona.
Stability of fractured rock slopes, Donze and Scholtès (2011).
Thermo-chemical eects : methane hydrates dissociation by heatingprocess (Msc thesis Fukuda T., 2010)
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Perspectives and challenges
Particles of complex shapes can be approximated by collections ofspheres (as in fractured rock mass models)
The very active research in the eld of pore-network models maygreatly inspire further extensions to multi-phase ow
The PFV approach in itslef is not restricting the complexity of theow equations : compressibility (already done), inertia, non-linearityat higher Reynolds could be introduced ; at the price of moreassumptions for the pore-upscaling and maybe dierent solvers.
Since the PFV model is assuming dense packings, it should becoupled to a CFD method for boundary value problems includingow outside the porous medium
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
Thank you for your attention.
Aknowledgements
Funding : Grenoble INP BQR program, Hydrofond project
supported by MEDDTL in the framework of the C2D2 program.
The PFV coupling is part of the open source project YADE
(http ://yade-dem.org). We thank all contributors to the Yade
project, and users for useful feedback.
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Introduction The model Stokes ow Fluid Forces Coupling Seabed sediments simulation Conclusions
B. Chareyre, A. Cortis, E. Catalano, E. Barthélémy. Pore-scale Modeling of Viscous Flow
and Induced Forces in Dense Sphere Packings. Transport in Porous Media (in press),DOI : 10.1007/s11242-011-9915-6) (2011)
L. Scholtes, B. Chareyre, F. Nicot, F. Darve. Discrete Modelling of Capillary Mechanisms
in Multi-Phase Granular Media. CMES-Computer Modeling in Engineering & Sciences,52(3) :297318, 2009.
H. Sari, B. Chareyre, E. Catalano, P. Philippe, E. Vincens. Investigation of internal
erosion processes using a coupled DEM-Fluid method. Particles 2011 II InternationalConference on Particle-Based Methods, E. Oñate and D.R.J. Owen (Eds), Barcelona(2011)
P.A. Cundall and O.D.L. Strack. A discrete numerical model for granular assemblies.
Geotechnique (1979) 29 :4765.
M Zeghal and U El Shamy. A continuum-discrete hydromechanical analysis of granular
deposit liquefaction Int. J. Numer. Anal. Meth. Geomech. (2004) 14 :(28)13611383.
Bryant S. and Blunt M. Prediction of relative permeability in simple porous media. Phys.
Rev. A (1992) 4 :(46)20042011.
V. Smilauer, E. Catalano, B. Chareyre, S. Dorofeenko, J. Duriez, A. Gladky, J. Kozicki,
C. Modenese, L. Scholtes, L. Sibille, J. Stransky, and K. Thoeni. Yade ReferenceDocumentation. Yade Documentation (V. Smilauer, ed.), The Yade Project, 1st ed.(2010) (http ://yade-dem.org/doc/.)
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