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Macroscopic model for generalised Newtonian inertial two-phase flow in porous media

Published online by Cambridge University Press:  30 August 2023

Jessica Sánchez-Vargas Open the ORCID record for Jessica Sánchez-Vargas [Opens in a new window] ,
Francisco J. Valdés-Parada Open the ORCID record for Francisco J. Valdés-Parada [Opens in a new window] ,
Mauricio A. Trujillo-Roldán Open the ORCID record for Mauricio A. Trujillo-Roldán [Opens in a new window]  and
Didier Lasseux Open the ORCID record for Didier Lasseux [Opens in a new window]
Jessica Sánchez-Vargas
Affiliation:
Departamento de Biología Molecular y Biotecnología, Instituto de Investigaciones Biomédicas, Universidad Nacional Autónoma de México, CDMX 04510, Mexico Posgrado en Ciencias Bioquímicas, Universidad Nacional Autónoma de México, CDMX 04510, México
Francisco J. Valdés-Parada*
Affiliation:
División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, col. Vicentina 09340, Mexico
Mauricio A. Trujillo-Roldán
Affiliation:
Departamento de Biología Molecular y Biotecnología, Instituto de Investigaciones Biomédicas, Universidad Nacional Autónoma de México, CDMX 04510, Mexico Departamento de Bionanotecnología, Centro de Nanociencias y Nanotecnología, Universidad Nacional Autónoma de México. Km 107 Carretera Tijuana-Ensenada, Ensenada, Baja California, México
Didier Lasseux
Affiliation:
University of Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295, F-33400, Talence, France
*
†Email address for correspondence: iqfv@xanum.uam.mx
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Abstract

A closed macroscopic model for quasi-steady, inertial, incompressible, two-phase generalised Newtonian flow in rigid and homogeneous porous media is formally derived. The model consists of macroscopic equations for mass and momentum balance as well as an expression for the macroscopic pressure difference between the two fluid phases. The model is obtained by upscaling the pore-scale equations, employing a methodology based on volume averaging, the adjoint method and Green's formulation, only assuming the existence of a representative elementary volume and the separation of scales between the microscale and the macroscale. The average mass equations coincide with those for Newtonian flow. The macroscopic momentum balance equation in each phase expresses the seepage velocity in terms of a dominant and a coupling Darcy-like term, a contribution from interfacial tension effects and another one from interfacial inertia. Finally, the expression of the macroscopic pressure difference is obtained in terms of the macroscopic pressure gradient and body force in each phase, and interfacial terms that account for capillary effects and inertia, if present when the interface is not stationary. All terms involved in the macroscale equations are predicted from the solution of adjoint closure problems in periodic representative domains. Numerical predictions from the upscaled models are compared with direct numerical simulations for two-dimensional configurations, considering flow of a Newtonian non-wetting fluid and a Carreau wetting fluid. Excellent agreement between the two approaches confirms the pertinence of the macroscopic models derived here.

JFM classification

Low-Reynolds-number flows: Porous media Multiphase and Particle-laden Flows: Multiphase flow Non-Newtonian Flows: Rheology
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 970 , 10 September 2023 , A19
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Copyright
© Universidad Autónoma Metropolitana, Universidad Nacional Autónoma de México, and CNRS, 2023. Published by Cambridge University Press

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