Transport in Porous Media

, Volume 92, Issue 2, pp 473–493 | Cite as

Pore-Scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

  • Bruno Chareyre
  • Andrea Cortis
  • Emanuele Catalano
  • Eric Barthélemy
Article

Abstract

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element simulations of the Stokes equations for assemblies of 8–200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients.

Keywords

Viscous flow Granular material Solid fluid coupling Pore-network Finite volumes 

List of Symbols

α

Nondimensional conductance factor

Ω

Full domain of the two-phase problem

Ωi

Domain defined by tetrahedron i

Ωij

union of tetrahdra (Sij, Pi) and (Sij, Pj)

Γ

Part of Ω occupied by the solid phase

Γi

Domain occupied by solid particle i

Θ

Part of Ω occupied by the fluid phase (pore space)

Θi

Part of Ωi occupied by the fluid phase (pore)

Θij

part of Ωij occupied by the fluid phase (throat)

Sij

Surface of the facet ij, separating tetrahedra i and j

\({\partial X}\)

Contour of domain X

\({\partial^{\rm f} X}\)

Part of contour of X intersecting the fluid phase

\({\partial^{\rm s} X}\)

Part of contour of X intersecting (or in contact with) the solid phase

γij

Area of ∂Θij in contact with spheres

γijk

Area of the part of \({\partial\Theta_{ij}}\) in contact with sphere k

Sij

The common facets of tetrahedra Ωi and Ωj

\({A^{\rm f}_{ij}}\)

Area of the fluid part \({S_{ij}\cap \Theta}\) of facet ij

\({A^k_{ij}}\)

Area of the intersection \({S_{ij}\cap \Gamma_k}\) of facet ij and sphere k

Pi

Voronoi dual (weighted center) of tetrahedra i

p

Microscopic (pore-scale) fluid pressure

pi

Macroscopic fluid pressure in tetrahedra i

u′

Microscopic fluid velocity

u

Macroscopic fluid velocity

v

Geometric contour velocity

qij

Flux through facet ij

\({V^{\rm f}_i}\)

Fluid volume contained in pore i

\({R_{ij}^{\rm h}}\)

Hydraulic radius of throat ij

\({R_{ij}^{{\rm eff}}}\)

Effective radius of throat ij

μ

Dynamic viscosity

Lij

Length of throat ij

\({F_{x}^y}\)

Forces exerted by the fluid on the solid phase, x and y denote different terms in forces decomposition

gij

Hydraulic conductance of facet (throat) ij

Kij

Hydraulic conductivity of facet (throat) ij

l0

Size of the cube enclosing the flow problem

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Bruno Chareyre
    • 1
  • Andrea Cortis
    • 2
  • Emanuele Catalano
    • 1
  • Eric Barthélemy
    • 1
  1. 1.Grenoble INPGrenoble Cedex 9France
  2. 2.Earth Sciences DivisionLawrence Berkeley National LaboratoryBerkeleyUSA

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