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Upscaled dynamic capillary pressure for two-phase flow in porous media

Published online by Cambridge University Press:  17 March 2023

Didier Lasseux Open the ORCID record for Didier Lasseux [Opens in a new window]  and
Francisco J. Valdés-Parada Open the ORCID record for Francisco J. Valdés-Parada [Opens in a new window]
Didier Lasseux
Affiliation:
I2M, UMR 5295, CNRS, Université de Bordeaux, 351 Cours de la Libération, 33405 Talence CEDEX, France
Francisco J. Valdés-Parada*
Affiliation:
División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Av. Ferrocarril San Rafael Atlixco, Núm. 186, col. Leyes de Reforma 1 A Sección, Alcaldía Iztapalapa, C.P. 09310, Ciudad de México, Mexico
*
†Email address for correspondence: iqfv@xanum.uam.mx
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Abstract

A closed expression for the average pressure difference (often called the macroscopic dynamic capillary pressure in the literature) is proposed for two-phase, Newtonian, incompressible, isothermal and creeping flow in homogeneous porous media. This upscaled equation complements the average equations for mass and momentum transport derived in a previous article. Consistently with this work, the expression is derived employing a simplified version of the volume-averaging method that makes use of elements of the adjoint method and Green's formula. The resulting equation for the average pressure difference is novel, as it shows that this quantity is controlled by the pressure gradient (and body forces) in each phase, as well as interfacial effects, and is applicable to situations in which the fluid–fluid interface is not necessarily at its steady position. The effective-medium quantities associated with the sources are all obtained from the solution of a single adjoint (or closure) problem to be solved on a (periodic) unit cell representative of the process. The average pressure difference predicted by the derived expression is validated through excellent comparisons with direct numerical simulations performed in a model porous structure.

JFM classification

Low-Reynolds-number flows: Porous media Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , R2
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

REFERENCES

Chavent, G. & Jaffré, J. 1986 Mathematical Models and Finite Elements for Reservoir Simulation. Elsevier.Google Scholar
Gray, W.G. 1975 A derivation of the equations for multi-phase transport. Chem. Engng Sci. 30 (2), 229–233.CrossRefGoogle Scholar
Gray, W.G., Bruning, K. & Miller, C.T. 2019 Non-hysteretic functional form of capillary pressure in porous media. J. Hydraul. Res. 57 (6), 747–759.CrossRefGoogle Scholar
Gray, W.G. & Miller, C.T. 2011 TCAT analysis of capillary pressure in non-equilibrium, two-fluid-phase, porous medium systems. Adv. Water Resour. 34 (6), 770–778.CrossRefGoogle ScholarPubMed
Hassanizadeh, S.M. & Gray, W.G. 1990 Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13 (4), 169–186.CrossRefGoogle Scholar
Hassanizadeh, S.M. & Gray, W.G. 1993 Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29 (10), 3389–3405.CrossRefGoogle Scholar
Joekar-Niasar, V. & Hassanizadeh, S.M. 2012 Uniqueness of specific interfacial area-capillary pressure-saturation relationship under non-equilibrium conditions in two-phase porous media flow. Transp. Porous Med. 94 (2), 465–486.CrossRefGoogle Scholar
Joekar-Niasar, V., Hassanizadeh, S.M. & Dahle, H.K. 2010 Non-equilibrium effects in capillarity and interfacial area in two-phase flow: dynamic pore-network modelling. J. Fluid Mech. 655, 38–71.CrossRefGoogle Scholar
Konangi, S., Palakurthi, N.K., Karadimitriou, N.K., Comer, K. & Ghia, U. 2021 Comparison of pore-scale capillary pressure to macroscale capillary pressure using direct numerical simulations of drainage under dynamic and quasi-static conditions. Adv. Water Resour. 147, 103792.CrossRefGoogle Scholar
Lasseux, D. & Valdés-Parada, F.J. 2022 A macroscopic model for immiscible two-phase flow in porous media. J. Fluid Mech. 944, A43.CrossRefGoogle Scholar
Marle, C.-M. 1981 From the pore scale to the macroscopic scale: equations governing multiphase fluid flow through porous media. In Flow and Transport in Porous Media: Proceedings of Euromech 143, Delft, 1981 (ed. A. Verruijt), pp. 57–61. CRC Press.Google Scholar
Muskat, M. & Meres, M.W. 1936 The flow of heterogeneous fluids through porous media. Physics 7, 346–363.CrossRefGoogle Scholar
Pyrak-Nolte, L.J., Nolte, D.D., Chen, D. & Giordano, N.J. 2008 Relating capillary pressure to interfacial areas. Water Resour. Res. 44 (6), W06408.CrossRefGoogle Scholar
Sánchez-Vargas, J., Valdés-Parada, F.J. & Lasseux, D. 2022 Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media. J. Non-Newtonian Fluid Mech. 306, 104840.CrossRefGoogle Scholar
Singh, K., Jung, M., Brinkmann, M. & Seemann, R. 2019 Capillary-dominated fluid displacement in porous media. Annu. Rev. Fluid Mech. 51 (1), 429–449.CrossRefGoogle Scholar
Starnoni, M. & Pokrajac, D. 2020 On the concept of macroscopic capillary pressure in two-phase porous media flow. Adv. Water Resour. 135, 103487.CrossRefGoogle Scholar
Whitaker, S. 1994 The closure problem for two-phase flow in homogeneous porous media. Chem. Engng Sci. 49 (5), 765–780.CrossRefGoogle Scholar