Advertisement

Inertial Sensitivity of Porous Microstructures

  • Martin Pauthenet
  • Yohan Davit
  • Michel Quintard
  • Alessandro Bottaro
Article
  • 28 Downloads

Abstract

Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, \(Re_{\mathrm{C}}\), so that the inertial deviation occurs when \(Re_{\mathrm{C}}\sim 1\) for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, \(\ell \). We analyze the problem using the method of volume averaging and identify a length scale in the form \(\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}\), with \(C_\lambda \) a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for \(C_\lambda \) is given; \(C_\lambda \) is computed from a creeping flow simulation in the porous medium; and \(Re_{\mathrm{C}}\) can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.

Keywords

Volume-averaging method Filtration law Inertial flow Reynolds number 

List of Symbols

\(\bullet _\beta \)

Item related to the fluid phase (\(\beta \)-phase)

\(C_\lambda \)

Inertial sensitivity of the microstructure

\(\epsilon _\beta \)

Porosity

\(\ell \)

Length scale used in \(Re_{\mathrm{C}}\) (L)

\(\ell _\beta \)

Pore length scale (L)

\(F_\parallel ,F_\perp \)

Parallel and orthogonal Forchheimer terms

\(F_\lambda \)

Forchheimer number

\(\gamma _\beta \)

\(\beta \)-phase indicator

\({{\mathbf {\mathsf{{g}}}}}_{\beta }\)

Macroscopic pressure gradient (\({\hbox {ML}}^{-2}{\hbox {T}}^{-2}\))

\(K_\lambda \)

Scalar permeability (\({{\hbox {L}}^2}\))

\(\varvec{{\lambda }} \)

Direction of the intrinsic average velocity

\(\mathscr {A}_{\beta \sigma }\)

Domain of the fluid–solid interface

\(\mathscr {V}\)

Domain of the REV

\(\mathscr {V}_\beta \)

Domain of the \(\beta \)-phase

\(\mu _{\beta }\)

Dynamic viscosity of the fluid (\({{\hbox {ML}}^{-1}{\hbox {T}}^{-1}}\))

\(\nu _{\beta }\)

Kinematic viscosity of the fluid (\({{\hbox {L}}^{2}{\hbox {T}}^{-1}}\))

\(p_{\beta }\)

Pressure field (\({{\hbox {ML}}^{-2}{\hbox {T}}^{-2}}\))

\(\psi \)

Generic field

\(\mathbb {R}\)

Real space

\(Re_{\mathrm{C}},Re_\mathrm{k}\)

Critical- and permeability-based Reynolds number

\(\rho _{\beta }\)

Density of the fluid (\({{\hbox {ML}}^{-3}}\))

\(\ell _0\)

Dimension of the REV (L)

\({{\mathbf {\mathsf{{v}}}}}_{\beta }\)

Flow velocity field (\({{\hbox {LT}}^{-1}}\))

\({{\mathbf {\mathsf{{f}}}}}\)

Extension of Darcy’s law (\({{\hbox {L}}^{-2}}\))

\({{\mathbf {\mathsf{{K}}}}}_\mathrm{D}\)

Darcy permeability (\({{\hbox {L}}^2}\))

\({{\mathbf {\mathsf{{r}}}}}\)

General position vector (L)

\({{\mathbf {\mathsf{{s}}}}}_\beta \)

Constant source term (\({{\hbox {LT}}^{-2}}\))

\({{\mathbf {\mathsf{{y}}}}}_\beta = {{\mathbf {\mathsf{{r}}}}} - {{\mathbf {\mathsf{{x}}}}}\)

Position vector relative to the centroid of the REV (\({{\hbox {L}}}\))

\(\left\langle {{\mathbf {\mathsf{{v}}}}}_{\beta } \right\rangle ^\beta ,\left\langle p_{\beta } \right\rangle ^\beta \)

Intrinsic averages

\(\tilde{{{\mathbf {\mathsf{{v}}}}}}_{\beta },\tilde{p}_{\beta }\)

Spatial deviations

\({{\mathbf {\mathsf{{v}}}}}^*,p^*\)

Dimensionless spatial deviations

b

Nonlinear part of Ergun’s equation (\({{\hbox {L}}^{-1}{\hbox {T}}}\))

\(L_\mathrm{v}\)

Macroscopic length scale (L)

V

Measure of the REV (\({{\hbox {L}}^3}\))

v

Magnitude of the intrinsic average velocity (\({{\hbox {LT}}^{-1}}\))

\(V_\beta \)

Measure of the \(\beta \)-phase inside the REV (\({{\hbox {L}}^3}\))

Notes

Acknowledgements

The authors would like to thank the IDEX Foundation of the University of Toulouse for the financial support granted to AB under the project ”Attractivity Chairs.” This work was granted access to the HPC resources of CALMIP supercomputing center under the allocation 2016-p1540.

Supplementary material

References

  1. Agnaou, M., Lasseux, D., Ahmadi, A.: From steady to unsteady laminar flow in model porous structures: an investigation of the first hopf bifurcation. Comput. Fluids 136, 67–82 (2016)CrossRefGoogle Scholar
  2. Amiri, A., Vafai, K.: Analysis of dispersion effects and non-thermal equilibrium, non-Darcian, variable porosity incompressible flow through porous media. Int. J. Heat Mass Transf. 37(6), 939–954 (1994)CrossRefGoogle Scholar
  3. Andrade Jr., J.S., Street, D.A., Shinohara, T., Shibusa, Y., Arai, Y.: Percolation disorder in viscous and nonviscous flow through porous media. Phys. Rev. E 51(6), 5725 (1995)CrossRefGoogle Scholar
  4. Andrade Jr., J.S., Costa, U.M.S., Almeida, M.P., Makse, H.A., Stanley, H.E.: Inertial effects on fluid flow through disordered porous media. Phys. Rev. Lett. 82(26), 5249 (1999)CrossRefGoogle Scholar
  5. Antohe, B.V., Lage, J.L.: A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transf. 40(13), 3013–3024 (1997)CrossRefGoogle Scholar
  6. Aydın, O., Kaya, A.: Non-Darcian forced convection flow of viscous dissipating fluid over a flat plate embedded in a porous medium. Transport Porous Media 73(2), 173–186 (2008)CrossRefGoogle Scholar
  7. Beavers, G.S., Sparrow, E.M.: Non-Darcy Flow Through Fibrous Porous Media. American Society of Mechanical Engineers, New York (1969)Google Scholar
  8. Brace, W.F., Walsh, J.B., Frangos, W.T.: Permeability of granite under high pressure. J. Geophys. Res. 73(6), 2225–2236 (1968)CrossRefGoogle Scholar
  9. Carman, P.C.: Fluid flow through granular beds. Chem. Eng. Res. Design 15, 150–166 (1937)Google Scholar
  10. Chandesris, M., Serre, G., Sagaut, P.: A macroscopic turbulence model for flow in porous media suited for channel, pipe and rod bundle flows. Int. J. Heat Mass Transf. 49(15), 2739–2750 (2006)CrossRefGoogle Scholar
  11. Chauveteau, G., Thirriot, C.: Régimes d’écoulement en milieu poreux et limite de la loi de Darcy. La Houille Blanche (2), 141–148 (1967)Google Scholar
  12. Chikhi, N., Clavier, R., Laurent, J.-P., Fichot, F., Quintard, M.: Pressure drop and average void fraction measurements for two-phase flow through highly permeable porous media. Ann. Nucl. Energy 94, 422–432 (2016)CrossRefGoogle Scholar
  13. Clavier, R.: Étude expérimentale et modélisation des pertes de pression lors du renoyage d’un lit de débris. Ph.D. thesis, Institut National Polytechnique de Toulouse, France (2015)Google Scholar
  14. Clavier, R., Chikhi, N., Fichot, F., Quintard, M.: Experimental investigation on single-phase pressure losses in nuclear debris beds: identification of flow regimes and effective diameter. Nucl. Eng. Design 292, 222–236 (2015)CrossRefGoogle Scholar
  15. Clavier, R., Chikhi, N., Fichot, F., Quintard, M.: Experimental study of single-phase pressure drops in coarse particle beds. Nucl. Eng. Design 312, 184–190 (2017)CrossRefGoogle Scholar
  16. Darcy, H.: Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris (1856)Google Scholar
  17. Davit, Y., Quintard, M.: Technical notes on volume averaging in porous media I: how to choose a spatial averaging operator for periodic and quasiperiodic structures. Transport Porous Media 119, 1–30 (2017)CrossRefGoogle Scholar
  18. de Carvalho, T.P., Morvan, H.P., Hargreaves, D.M., Oun, H., Kennedy, A.: Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams. Transport in Porous Media 117, 1–26 (2017)CrossRefGoogle Scholar
  19. De Lemos, M.J.: Turbulence in Porous Media: Modeling and Applications. Elsevier, Oxford (2012)Google Scholar
  20. Dukhan, N., Bağcı, Ö., Özdemir, M.: Experimental flow in various porous media and reconciliation of Forchheimer and Ergun relations. Exp. Therm. Fluid Sci. 57, 425–433 (2014)CrossRefGoogle Scholar
  21. Dybbs, A., Edwards, R.: A new look at porous media fluid mechanics-Darcy to turbulent. In: Bear, J., Corapcioglu, M.Y. (eds.) Fundamentals of Transport Phenomena in Porous Media, pp. 199–256. Springer, Berlin (1984)CrossRefGoogle Scholar
  22. Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)Google Scholar
  23. Fand, R., Kim, B., Lam, A., Phan, R.: Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Eng. 109(3), 268–274 (1987)CrossRefGoogle Scholar
  24. Favier, J., Dauptain, A., Basso, D., Bottaro, A.: Passive separation control using a self-adaptive hairy coating. J. Fluid Mech. 627, 451–483 (2009)CrossRefGoogle Scholar
  25. Firdaouss, M., Guermond, J.-L., Le Quéré, P.: Nonlinear corrections to Darcy’s law at low Reynolds numbers. J. Fluid Mech. 343, 331–350 (1997)CrossRefGoogle Scholar
  26. Forchheimer, P.H.: Wasserbewegung durch boden. Z. Vereines Dtsch. Ing. 45, 1782–1788 (1901)Google Scholar
  27. Ghisalberti, M., Nepf, H.: Mixing layers and coherent structures in vegetated aquatic flows. J. Geophys. Res. 107(C2), 420 (2002)CrossRefGoogle Scholar
  28. Ghisalberti, M., Nepf, H.: Mass transport in vegetated shear flows. Environ. Fluid Mech. 5(6), 527–551 (2005)CrossRefGoogle Scholar
  29. Gosselin, F.: Mécanismes d’interactions fluide-structure entre écoulements et végétation. Ph.D. thesis, École Polytechnique (2009)Google Scholar
  30. Gosselin, F.P., de Langre, E.: Drag reduction by reconfiguration of a poroelastic system. J. Fluids Struct. 27, 1111–1123 (2011)CrossRefGoogle Scholar
  31. Goyeau, B., Songbe, J.-P., Gobin, D.: Numerical study of double-diffusive natural convection in a porous cavity using the Darcy-Brinkman formulation. Int. J. Heat Mass Transf. 39(7), 1363–1378 (1996)CrossRefGoogle Scholar
  32. Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transport Porous Media 2(6), 521–531 (1987)CrossRefGoogle Scholar
  33. Hlushkou, D., Tallarek, U.: Transition from creeping via viscous-inertial to turbulent flow in fixed beds. J. Chromatogr. A 1126(1), 70–85 (2006)CrossRefGoogle Scholar
  34. Hoffmann, J., Echigo, R., Yoshida, H., Tada, S.: Experimental study on combustion in porous media with a reciprocating flow system. Combust. Flame 111(1–2), 32–46 (1997)CrossRefGoogle Scholar
  35. Hong, J., Yamada, Y., Tien, C.: Effects of non-darcian and nonuniform porosity on vertical-plate natural convection in porous media. J. Heat Transf. 109(2), 356–362 (1987)CrossRefGoogle Scholar
  36. Jackson, G.W., James, D.F.: The permeability of fibrous porous media. Can. J. Chem. Eng. 64(3), 364–374 (1986)CrossRefGoogle Scholar
  37. Jin, Y., Kuznetsov, A.V.: Turbulence modeling for flows in wall bounded porous media: an analysis based on direct numerical simulations. Phys. Fluids 29(4), 045102 (2017)CrossRefGoogle Scholar
  38. Jin, Y., Uth, M., Kuznetsov, A., Herwig, H.: Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study. J. Fluid Mech. 766, 76 (2015)CrossRefGoogle Scholar
  39. Kim, S.Y., Paek, J.W., Kang, B.H.: Flow and heat transfer correlations for porous fin in a plate-fin heat exchanger. J. Heat Transf. 122(3), 572–578 (2000)CrossRefGoogle Scholar
  40. Klinkenberg, L.J.: The permeability of porous media to liquids and gases. In: Drilling and Production Practice. American Petroleum Institute, Washington (1941)Google Scholar
  41. Koch, D.L., Ladd, A.J.C.: Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech. 349, 31–66 (1997)CrossRefGoogle Scholar
  42. Kuwahara, F., Kameyama, Y., Yamashita, S., Nakayama, A.: Numerical modeling of turbulent flow in porous media using a spatially periodic array. J. Porous Media 1(1), 47–55 (1998)CrossRefGoogle Scholar
  43. Lage, J.L.: The fundamental theory of flow through permeable media from Darcy to turbulence. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media, pp. 1–30. Elsevier, Oxford (1998)Google Scholar
  44. Lage, J.L., Antohe, B.V.: Darcy’s experiments and the deviation to nonlinear flow regime. J. Fluids Eng. 122(3), 619–625 (2000)CrossRefGoogle Scholar
  45. Lasseux, D., Valdés-Parada, F.J.: On the developments of Darcy’s law to include inertial and slip effects. Compt. Rendus Méc. 345(9), 660–669 (2017)CrossRefGoogle Scholar
  46. Lasseux, D., Abbasian Arani, A.A., Ahmadi, A.: On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media. Phys. Fluids 23(7), 073103 (2011)CrossRefGoogle Scholar
  47. Li, L., Ma, W.: Experimental study on the effective particle diameter of a packed bed with non-spherical particles. Transport Porous Media 89(1), 35–48 (2011a)CrossRefGoogle Scholar
  48. Li, L., Ma, W.: Experimental characterization of the effective particle diameter of a particulate bed packed with multi-diameter spheres. Nucl. Eng. Design 241(5), 1736–1745 (2011b)CrossRefGoogle Scholar
  49. Lucas, Y., Panfilov, M., Buès, M.: High velocity flow through fractured and porous media: the role of flow non-periodicity. Eur. J. Mech. B/Fluids 26(2), 295–303 (2007)CrossRefGoogle Scholar
  50. Luminari, N., Airiau, C., Bottaro, A.: Effects of porosity and inertia on the apparent permeability tensor in fibrous media. Int. J. Multiph. Flow 106, 60–74 (2018)CrossRefGoogle Scholar
  51. Ma, H., Ruth, D.: Physical explanations of non-Darcy effects for fluid flow in porous media. SPE Form. Eval. 12(01), 13–18 (1997)CrossRefGoogle Scholar
  52. Masuoka, T., Takatsu, Y.: Turbulence model for flow through porous media. Int. J. Heat Mass Transf. 39(13), 2803–2809 (1996)CrossRefGoogle Scholar
  53. Mcdonald, I.F., El-Sayed, M.S., Mow, K., Dullien, F.A.L.: Flow through porous media-the Ergun equation revisited. Ind. Eng. Chem. Fundam. 18(3), 199–208 (1979)CrossRefGoogle Scholar
  54. Mei, C.C., Auriault, J.L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991)CrossRefGoogle Scholar
  55. Muljadi, B.P., Blunt, M.J., Raeini, A.Q., Bijeljic, B.: The impact of porous media heterogeneity on non-Darcy flow behaviour from pore-scale simulation. Adv. Water Resour. 95, 329–340 (2016)CrossRefGoogle Scholar
  56. Nakayama, A., Kuwahara, F.: A macroscopic turbulence model for flow in a porous medium. J. Fluids Eng. 121, 427–433 (1999)CrossRefGoogle Scholar
  57. Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, New York (1999)CrossRefGoogle Scholar
  58. Panfilov, M., Oltean, C., Panfilova, I., Buès, M.: Singular nature of nonlinear macroscale effects in high-rate flow through porous media. Compt. Rendus Méc. 331(1), 41–48 (2003)CrossRefGoogle Scholar
  59. Papathanasiou, T.D., Markicevic, B., Dendy, E.D.: A computational evaluation of the Ergun and Forchheimer equations for fibrous porous media. Phys. Fluids 13(10), 2795–2804 (2001)CrossRefGoogle Scholar
  60. Philipse, A.P., Schram, H.L.: Non-Darcian airflow through ceramic foams. J. Am. Ceram. Soc. 74(4), 728–732 (1991)CrossRefGoogle Scholar
  61. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media II: generalized volume averaging. Chem. Eng. Sci. 14, 179–206 (1994a)Google Scholar
  62. Quintard, M., Whitaker, S.: Transport in ordered and disordered porous media III: closure and comparison between theory and experiment. Chem. Eng. Sci. 15, 31–49 (1994b)Google Scholar
  63. Quintard, M., Kaviany, M., Whitaker, S.: Two-medium treatment of heat transfer in porous media: numerical results for effective properties. Adv. Water Resour. 20(2), 77–94 (1997)CrossRefGoogle Scholar
  64. Rojas, S., Koplik, J.: Nonlinear flow in porous media. Phys. Rev. E 58, 4776–4782 (1998)CrossRefGoogle Scholar
  65. Ruth, D., Ma, H.: On the derivation of the Forchheimer equation by means of the averaging theorem. Transport Porous Media 7(3), 255–264 (1992)CrossRefGoogle Scholar
  66. Schneebeli, G.: Expériences sur la limite de validité de la loi de Darcy et l’apparition de la turbulence dans un écoulement de filtration. La Houille Blanche (2), 141–149 (1955)Google Scholar
  67. Seguin, D., Montillet, A., Comiti, J.: Experimental characterisation of flow regimes in various porous media. I: limit of laminar flow regime. Chem. Eng. Sci. 53(21), 3751–3761 (1998a)CrossRefGoogle Scholar
  68. Seguin, D., Montillet, A., Comiti, J., Huet, F.: Experimental characterization of flow regimes in various porous media. II: transition to turbulent regime. Chem. Eng. Sci. 53(22), 3897–3909 (1998b)CrossRefGoogle Scholar
  69. Sharp, K.V., Adrian, R.J.: Transition from laminar to turbulent flow in liquid filled microtubes. Exp. Fluids 36(5), 741–747 (2004)CrossRefGoogle Scholar
  70. Skjetne, E., Auriault, J.L.: New insights on steady, nonlinear flow in porous media. Eur. J. Mech. B/Fluids 18(1), 131–145 (1999)CrossRefGoogle Scholar
  71. Souto, H.P.A., Moyne, C.: Dispersion in two-dimensional periodic porous media. Part I: hydrodynamics. Phys. Fluids 9(8), 2243–2252 (1997)CrossRefGoogle Scholar
  72. Vafai, K., Tien, C.L.: Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf. 24(2), 195–203 (1981)CrossRefGoogle Scholar
  73. Vafai, K., Tien, C.: Boundary and inertia effects on convective mass transfer in porous media. Int. J. Heat Mass Transf. 25(8), 1183–1190 (1982)CrossRefGoogle Scholar
  74. Venkataraman, P., Rao, P.R.M.: Darcian, transitional, and turbulent flow through porous media. J. Hydraul. Eng. 124(8), 840–846 (1998)CrossRefGoogle Scholar
  75. Whitaker, S.: Diffusion and dispersion in porous media. Am. Inst. Chem. Eng. 13(3), 420–427 (1967)CrossRefGoogle Scholar
  76. Whitaker, S.: The Forchheimer equation: a theoretical development. Transport Porous Media 25(1), 27–61 (1996)CrossRefGoogle Scholar
  77. Whitaker, S.: The Method of Volume Averaging. Kluwer Academic, Dordrecht (1999)CrossRefGoogle Scholar
  78. Wodie, J.-C., Levy, T.: Correction non linéaire de la loi de Darcy. Comptes Rendus de l’Académie des Sciences. Série 2, Mécanique, Physique, Chimie, Sciences de l’Univers Sciences de la Terre 312(3), 157–161 (1991)Google Scholar
  79. Wong, T.-F., David, C., Zhu, W.: The transition from brittle faulting to cataclastic flow in porous sandstones: mechanical deformation. J. Geophys. Res. Solid Earth 102(B2), 3009–3025 (1997)CrossRefGoogle Scholar
  80. Wood, B.D.: Inertial effects in dispersion in porous media. Water Resour. Res. 43(12), W12S16 (2007)CrossRefGoogle Scholar
  81. Zeng, Z., Grigg, R.: A criterion for non-Darcy flow in porous media. Transport Porous Media 63(1), 57–69 (2006)CrossRefGoogle Scholar
  82. Zimmerman, R.W., Al-Yaarubi, A., Pain, C.C., Grattoni, C.A.: Nonlinear regimes of fluid flow in rock fractures. Int. J. Rock Mech. Min. Sci. 41, 163–169 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRSToulouseFrance
  2. 2.DICCA, Scuola Politecnica University of GenovaGenoaItaly

Personalised recommendations