Skip to main content
SpringerLink
Log in

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

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

α :

Nondimensional conductance factor

Ω:

Full domain of the two-phase problem

Ω i :

Domain defined by tetrahedron i

Ω ij :

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

Γ:

Part of Ω occupied by the solid phase

Γ i :

Domain occupied by solid particle i

Θ:

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

Θ i :

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

Θ ij :

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

S ij :

Surface of the facet ij, separating tetrahedra i and j

∂X :

Contour of domain X

∂f X :

Part of contour of X intersecting the fluid phase

∂s X :

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

γ ij :

Area of ∂Θ ij in contact with spheres

\({\gamma_{ij}^k}\) :

Area of the part of ∂Θ ij in contact with sphere k

S ij :

The common facets of tetrahedra Ω i and Ω j

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

Area of the fluid part S ij ∩ Θ of facet ij

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

Area of the intersection S ij ∩ Γ k of facet ij and sphere k

P i :

Voronoi dual (weighted center) of tetrahedra i

p′:

Microscopic (pore-scale) fluid pressure

p i :

Macroscopic fluid pressure in tetrahedra i

u′:

Microscopic fluid velocity

u :

Macroscopic fluid velocity

v :

Geometric contour velocity

q ij :

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

L ij :

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

g ij :

Hydraulic conductance of facet (throat) ij

K ij :

Hydraulic conductivity of facet (throat) ij

l 0 :

Size of the cube enclosing the flow problem

References

  • Abichou T., Benson C.H., Edil T.B.: Network model for hydraulic conductivity of sand-bentonite mixtures. Can. Geotech. J. 41(4), 698–712 (2004)

    Article  Google Scholar 

  • Aurenhammer F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)

    Article  Google Scholar 

  • Bagi K.: Analysis of microstructural strain tensors for granular assemblies. Int. J. Solids Struct. 43(10), 3166–3184 (2006)

    Article  Google Scholar 

  • Bakke S., Øren P.: 3-D pore-scale modelling of sandstones and flow simulations in the pore networks. SPE J. 2(2), 136–149 (1997)

    Google Scholar 

  • Boissonnat J.-D., Devillers O., Pion S., Teillaud M., Yvinec M.: Triangulations in CGAL. Comput. Geom. Theory Appl. 22, 5–19 (2002)

    Article  Google Scholar 

  • Bonilla, R.R.O.: Numerical simulation of undrained granular media. PhD thesis, University of Waterloo (2004)

  • Bryant S., Blunt M.: Prediction of relative permeability in simple porous media. Phys. Rev. A 46(4), 2004–2011 (1992)

    Article  Google Scholar 

  • Bryant S., Johnson A.: Wetting phase connectivity and irreducible saturation in simple granular media. J. Colloid Interface Sci. 263(2), 572–579 (2003)

    Article  Google Scholar 

  • Bryant S., King P., Mellor D.: Network model evaluation of permeability and spatial correlation in a real random sphere packing. Transp. Porous Media 11(1), 53–70 (1993)

    Article  Google Scholar 

  • Calvetti F., Combe G., Lanier J.: Experimental micromechanical analysis of a 2D granular material: relation between structure evolution and loading path. Mech. Cohes. Frict. Mater. 2(2), 121–163 (1997)

    Article  Google Scholar 

  • Catalano, E., Chareyre, B., Cortis, A., Barthélémy, E.: A pore-scale hydro-mechanical coupled model for geomaterials. In: Oñate, E., Owen, D., (eds.) Particles 2011 II International Conference on Particle-Based Methods, 2011

  • Chareyre B., Briançon L., Villard P.: Theoretical versus experimental modelling of the anchorage capacity of geotextiles in trenches. Geosynth. Int. 9(2), 97–123 (2002)

    Google Scholar 

  • Cundall P., Strack O.: A discrete numerical model for granular assemblies. Geotechnique 29, 47–65 (1979)

    Article  Google Scholar 

  • Edelsbrunner H., Shah N.R.: Incremental topological flipping works for regular triangulations. Algorithmica 15(6), 223–241 (1996)

    Article  Google Scholar 

  • Glowinski R., Pan T.W., Hesla T.I., Joseph D.D., Periaux J.: A fictious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies. J. Comput. Phys. 169(2), 363–426 (2001)

    Article  Google Scholar 

  • Hakuno, M.: Simulation of the dynamic liquefaction of sand. In: Earthquake Geotechnical Engineering, pp. 857–862. Balkema, Rotterdam (1995)

  • Hassanizadeh M., Gray W.: General conservation equations for multi-phase systems: 1. Averaging procedure. Adv. Water Resour. 2, 131–144 (1979)

    Article  Google Scholar 

  • Hilpert M., Glantz R., Miller C.T.: Calibration of a pore-network model by a pore-morphological analysis. Transp. Porous Media. 51, 267–285 (2003)

    Article  Google Scholar 

  • Jerier, J.-F., Hathong, B., Richefeu, V., Chareyre, B., Imbault, D., Donze, F.-V., Doremus, P.: Study of cold powder compaction by using the discrete element method. Powder Technol. (2010). doi:10.1016/j.powtec.2010.08.056

  • Joekar-Niasar V., Prodanovic M., Wildenschild D., Hassanizadeh S.: Network model investigation of interfacial area, capillary pressure and saturation relationships in granular porous media. Water Resour. Res. 46(6), W06526 (2010)

    Article  Google Scholar 

  • Kafui K.D., Thornton C., Adams M.J.: Discrete particle-continuum fluid modeling of gas–solid fluidized beds. Chem. Eng. Sci. 57(13), 2395–2410 (2002)

    Article  Google Scholar 

  • Kawaguchi T., Tanaka T., Tsuji Y.: Numerical simulation of two-dimensional fluidised beds using the discrete element method (comparison between two- and three-dimensional models). Powder Technol. 96, 129–138 (1998)

    Article  Google Scholar 

  • Knudsen H., Aker E., Hansen A.: Bulk flow regimes and fractional flow in 2D porous media by numerical simulations. Transp. Porous Media 47(1), 99–121 (2002)

    Article  Google Scholar 

  • Liu Y., Snoeyink J.: A comparison of five implementations of 3D delaunay tessellation. In: Pach, J., Goodman, J.E., Welzl, E. (eds.) Combinatorial and Computational Geometry, pp. 439–458. MSRI Publications, Cambridge University Press, Cambridge (2005)

    Google Scholar 

  • Luchnikov V.A., Medvedev N.N., Oger L., Troadec J.-P.: Voronoi-delaunay analysis of voids in systems of nonspherical particles. Phys. Rev. E 59(6), 7205–7212 (1999)

    Article  Google Scholar 

  • McNamara S., Flekkøy E.G., Måløy K.J.: Grains and gas flow: molecular dynamics with hydrodynamic interactions. Phys. Rev. E 61(4), 4054–4059 (2000)

    Article  Google Scholar 

  • Mortensen N., Okkels F., Bruus H.: Phys. Rev. E 71(5), 057301 (2005)

    Article  Google Scholar 

  • Nakasa, H., Takeda, T., Oda, M.: A simulation study on liquefaction using dem. In: Earthquake Geotechnical Engineering, pp. 637–642. Balkema, Rotterdam (1999)

  • Niebling M.J., Flekkøy E.G., Måløy K.J., Toussaint R.: Sedimentation instabilities: impact of the fluid compressibility and viscosity. Phys. Rev. E 82(5), 051302 (2010)

    Article  Google Scholar 

  • Oostreom A.V., Strackee J.: The solid angle of a plane triangle. IEEE Trans. Biomed. Eng. BME-30(2), 125–126 (1983)

    Article  Google Scholar 

  • Øren P., Bakke S., Arntzen O.: Extending predictive capabilities to network models. SPE J. 3(4), 324–336 (1998)

    Google Scholar 

  • Patzek T., Silin T.: Shape factor and hydraulic conductance in noncircular capillaries: I. one-phase creeping flow. J. Colloid Interface Sci. 236(2), 295–304 (2001)

    Article  Google Scholar 

  • Piri M., Blunt M.J.: Three-dimensional mixed-wet random pore-scale network modeling of two- and three-phase flow in porous media. i. Model description. Phys. Rev. E 71(2), 026301 (2005)

    Article  Google Scholar 

  • Sanchez-Palencia, E., Zaoui, A. (eds.): Homogenization techniques for composite media, vol. 272. Lecture Notes in Physics. Springer Verlag, Berlin (1987)

  • Scholtes L., Hicher P.Y., Nicot F., Chareyre B., Darve F.: On the capillary stress tensor in wet granular materials. Int. J. Numer. Anal. Methods Geomech. 33(10), 1289–1313 (2009)

    Article  Google Scholar 

  • Smilauer, V., Catalano, E., Chareyre, B., Dorofeenko, S., Duriez, J., Gladky, A., Kozicki, J., Modenese, C., Scholtes, L., Sibille, L., Stransky, J., Thoeni, K.: Yade reference documentation. In: Smilauer, V. (ed.) Yade Documentation (2010). http://yade-dem.org/doc/

  • Thompson K.E., Fogler H.S.: Modeling flow in disordered packed beds from pore-scale fluid mechanics. AIChE J. 43(6), 1547–5905 (1999)

    Google Scholar 

  • Valavanides M.S., Constantinides G.N., Payatakes A.C.: Mechanistic model of steadystate two-phase flow in porous media based on ganglion dynamics. Transp. Porous Media 30(3), 267–299 (1998)

    Article  Google Scholar 

  • Whitaker S.: The method of volume averaging. Kluwer, Dordrecht (1999)

    Google Scholar 

  • Zeghal M., Shamy U.E.: A continuum-discrete hydromechanical analysis of granular deposit liquefaction. Int. J. Numer. Anal. Methods Geomech. 28(14), 1361–1383 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Grenoble INP, UMR CNRS 5519, LEGI, BP 53, 38041, Grenoble Cedex 9, France

    Bruno Chareyre, Emanuele Catalano & Eric Barthélemy

  2. Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94806, USA

    Andrea Cortis

Authors
  1. Bruno Chareyre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Cortis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emanuele Catalano
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eric Barthélemy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Chareyre.

Additional information

This paper has previously been published under doi:10.1007/s11242-011-9915-6. In order to include it in the special issue we reproduce it here.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chareyre, B., Cortis, A., Catalano, E. et al. Pore-Scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings. Transp Porous Med 94, 595–615 (2012). https://doi.org/10.1007/s11242-012-0057-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-012-0057-2

Keywords

Search

Navigation