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Exuding porous media: deviations from Darcy's law

Published online by Cambridge University Press:  01 February 2021

Didier Lasseux Open the ORCID record for Didier Lasseux [Opens in a new window] ,
Francisco J. Valdés-Parada Open the ORCID record for Francisco J. Valdés-Parada [Opens in a new window] ,
Jean-François Thovert Open the ORCID record for Jean-François Thovert [Opens in a new window]  and
Valeri Mourzenko Open the ORCID record for Valeri Mourzenko [Opens in a new window]
Didier Lasseux*
Affiliation:
I2M, UMR 5295, CNRS, Université Bordeaux, Esplanade des Arts et Métiers, 33405Talence CEDEX, France
Francisco J. Valdés-Parada
Affiliation:
Departamento de Ingeniería de Procesos e Hidráulica, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, 09340Ciudad de México, Mexico
Jean-François Thovert
Affiliation:
Institut P′, CNRS, Université de Poitiers, ISAE-ENSMA, 11 bd Marie et Pierre Curie, TSA 41123, 86073Poitiers CEDEX 9, France
Valeri Mourzenko
Affiliation:
Institut P′, CNRS, Université de Poitiers, ISAE-ENSMA, 11 bd Marie et Pierre Curie, TSA 41123, 86073Poitiers CEDEX 9, France
*
†Email address for correspondence: didier.lasseux@u-bordeaux.fr
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Abstract

This work addresses the question of a pertinent macroscale model describing creeping, incompressible and single-phase flow of a Newtonian fluid in an exuding, rigid and homogeneous porous medium. The macroscopic model is derived by upscaling the pore-scale Stokes equations considering a normal mass flux at the solid–fluid interface. The upscaled mass equation shows that the average velocity is non-solenoidal. In addition, the macroscopic momentum equation involves a Darcy term with the classical permeability tensor accounting for macroscopic drag and a correction velocity vector which is a signature of the local fluid displacements induced by the exuding phenomenon. This correction is the sum of a term accounting for the local exuding effect and a compensation term associated with the assumption of spatial periodicity. Both the first term and the permeability tensor are obtained from the solution of the same unique and intrinsic closure problem, which corresponds to that involved in the classical Darcy's law. The upscaled model is validated by comparisons with pore-scale numerical simulations in several illustrative examples. The different configurations evidence the richness of the problem, despite the apparent simplicity of its formulation. The results of this work motivate further investigation about the influence of internal flow sources in transport phenomena in porous media.

JFM classification

Low-Reynolds-number flows: Porous media Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A48
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Adler, P. 1992 Porous media: geometry and transports. Butterworth-Heinemann Series in Chemical Engineering. Butterworth-Heinemann.Google Scholar
Barrère, J. 1990 Modélisation des écoulement de Stokes et de Navier–Stokes en milieu poreux. PhD thesis, University of Bordeaux.Google Scholar
Bottaro, A. 2019 Flow over natural or engineered surfaces: an adjoint homogenization perspective. J. Fluid Mech. 877, 1–91.CrossRefGoogle Scholar
Bryden, K.M., Ragland, K.W. & Rutland, C.J. 2002 Modeling thermally thick pyrolysis of wood. Biomass Bioenergy 22, 41–53.CrossRefGoogle Scholar
Dadic, R., Light, B. & Warren, S.G. 2010 Migration of air bubbles in ice under a temperature gradient, with application to “snowball earth”. J. Geophys. Res. 115, D18125.CrossRefGoogle Scholar
Davit, Y., et al. . 2013 Homogenization via formal multiscale asymptotics and volume averaging: how do the two techniques compare? Adv. Water Resour. 62, 178–206.CrossRefGoogle Scholar
Di Blasi, C. 1994 Numerical simulation of cellulose pyrolysis. Biomass Bioenergy 7, 87–98.CrossRefGoogle Scholar
Elayeb, M., Debenest, G., Mourzenko, V. & Thovert, J.-F. 2017 Smoldering combustion in oil shales: influence of calcination and pyrolytic reactions. Transp. Porous Med. 116, 889–921.CrossRefGoogle Scholar
Erriguible, A., Bernada, P., Couture, F. & Roques, M. 2006 Simulation of convective drying of a porous medium with boundary conditions provided by CFD. Chem. Engng Res. Des. 84, 113–123.CrossRefGoogle Scholar
Flin, F., Brzoska, J.-B., Lesaffre, B., Coléou, C. & Pieritz, R.A. 2003 Full three-dimensional modelling of curvature-dependent snow metamorphism: first results and comparison with experimental tomographic data. J. Phys. D: Appl. Phys. 36, A49–A54.CrossRefGoogle Scholar
Gray, W.G. 1975 A derivation of the equations for multi-phase transport. Chem. Engng Sci. 30 (2), 229–233.CrossRefGoogle Scholar
Jiang, J. & Younis, R.M. 2015 Numerical study of complex fracture geometries for unconventional gas reservoirs using a discrete fracture-matrix model. J. Nat. Gas Sci. Engng 26, 1174–1186.CrossRefGoogle Scholar
Lachaud, J., Scoggins, J.B., Magin, T.E., Meyer, M.G. & Mansour, N.N. 2017 A generic local thermal equilibrium model for porous reactive materials submitted to high temperatures. Intl J. Heat Mass Transfer 108, 1406–1417.CrossRefGoogle Scholar
Larfeldt, J., Leckner, B. & Melaaen, M.C. 2000 Modelling and measurements of the pyrolysis of large wood particles. Fuel 79, 1637–1643.CrossRefGoogle Scholar
Lasseux, D. & Valdés-Parada, F.J. 2017 Symmetry properties of macroscopic transport coefficients in porous media. Phys. Fluids 29 (4), 043303.CrossRefGoogle Scholar
Lasseux, D., Valdés-Parada, F.J. & Porter, M.L. 2016 An improved macroscale model for gas slip flow in porous media. J. Fluid Mech. 805, 118–146.CrossRefGoogle Scholar
Mahmoudi, A.H., Hoffmann, F. & Peters, B. 2014 Detailed numerical modeling of pyrolysis in a heterogeneous packed bed using XDEM. J. Anal. Appl. Pyrolysis 106, 9–20.CrossRefGoogle Scholar
Mei, C.C., Auriault, J.-L. & Ng, C.-O. 1996 Some applications of the homogenization theory. In Advances in Applied Mechanics, pp. 277–348. World Scientific Publishing.CrossRefGoogle Scholar
Mei, C.C. & Vernescu, B. 2010 Homogenization Methods For Multiscale Mechanics. World Scientific Publishing Company.CrossRefGoogle Scholar
Mellor, M. 1960 Temperature gradients in the Antarctic ice sheet. J. Glaciol. 28 (3), 773–782.CrossRefGoogle Scholar
Mourzenko, V., Thovert, J.-F. & Adler, P.M. 2018 Conductivity and transmissivity of a single fracture. Transp. Porous Med. 123, 235–256.CrossRefGoogle Scholar
Olorode, O.M., Freeman, C.M., Moridis, G.J. & Blasingame, T. 2012 High-resolution numerical modeling of complex and irregular fracture patterns in shale gas and tight gas reservoirs. SPE Res. Eval. Engng 6, SPE–152482–MS.Google Scholar
Pozrikidis, C. 2003 Computation of the pressure inside bubbles and pores in Stokes flow. J. Fluid Mech. 474, 319–337.CrossRefGoogle Scholar
Rasoulzadeh, M., Yekta, A., Deng, H. & Ghahfarokhi, R.B. 2020 Semi-analytical models of mineral dissolution in rough fractures with permeable walls. Phys. Fluids 32 (5), 052003.CrossRefGoogle Scholar
Rönsch, S., Schneider, J., Matthischke, S., Schlüter, M., Götz, M., Lefebvre, J., Prabhakaran, P. & Bajohr, S. 2016 Review on methanation – from fundamentals to current projects. Fuel 166, 276–296.CrossRefGoogle Scholar
Sanchez-Palencia, E. 1980 Non-homogeneous media and vibration theory. Lecture Notes in Physics, vol. 127. Springer.Google Scholar
Shepel, S.V., Wakili, K.G. & Hugi, E. 2010 Vapor convection in gypsum plasterboard exposed to fire: numerical simulation and validation. Numer. Heat Transfer A 57, 911–935.CrossRefGoogle Scholar
Shreve, R.L. 1967 Migration of air bubbles, vapour figures, and brine pockets in ice under a temperature gradient. J. Geophys. Res. 72, 4093–4100.CrossRefGoogle Scholar
Slattery, J.C. 1967 General balance equation for a phase interface. Ind. Engng Chem. Fundam. 6 (1), 108–115.CrossRefGoogle Scholar
Spagnolie, S.E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105–147.CrossRefGoogle Scholar
Vetter, R., Sigg, S., Singer, H.M., Kadau, D., Herrmann, H.J. & Schneebeli, M. 2010 Simulating isothermal aging of snow. Europhys. Lett. 89, 26001.CrossRefGoogle Scholar
Vu, H.T. & Tsotsas, E. 2018 Mass and heat transport models for analysis of the drying process in porous media: a review and numerical implementation. Intl J. Chem. Engng 2018, 9456418.Google Scholar
Warning, A.D., Arquiza, J.M.R. & Datta, A.K. 2015 A multiphase porous medium transport model with distributed sublimation front to simulate vacuum freeze drying. Food Bioprod. Process. 94, 637–648.CrossRefGoogle Scholar
Whitaker, S. 1986 Flow in porous media I: a theoretical derivation of Darcy's law. Transp. Porous Med. 1 (1), 3–25.CrossRefGoogle Scholar
Whitaker, S. 1999 The Method of Volume Averaging. Springer.CrossRefGoogle Scholar