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 ChareyreEmail author
  • Andrea Cortis
  • Emanuele Catalano
  • Eric Barthélemy


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.


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

List of Symbols


Nondimensional conductance factor


Full domain of the two-phase problem


Domain defined by tetrahedron i


union of tetrahdra (S ij , P i ) and (S ij , P j )


Part of Ω occupied by the solid phase


Domain occupied by solid particle i


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


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


part of Ω ij occupied by the fluid phase (throat)


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


Area of ∂Θ ij in contact with spheres


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


The common facets of tetrahedra Ω i and Ω j

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

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


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


Voronoi dual (weighted center) of tetrahedra i


Microscopic (pore-scale) fluid pressure


Macroscopic fluid pressure in tetrahedra i


Microscopic fluid velocity


Macroscopic fluid velocity


Geometric contour velocity


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


Dynamic viscosity


Length of throat ij


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


Hydraulic conductance of facet (throat) ij


Hydraulic conductivity of facet (throat) ij


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
    Email author
  • 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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