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A homogenised model for flow, transport and sorption in a heterogeneous porous medium

Published online by Cambridge University Press:  09 December 2021

L.C. Auton Open the ORCID record for L.C. Auton [Opens in a new window] ,
S. Pramanik Open the ORCID record for S. Pramanik [Opens in a new window] ,
M.P. Dalwadi Open the ORCID record for M.P. Dalwadi [Opens in a new window] ,
C.W. MacMinn Open the ORCID record for C.W. MacMinn [Opens in a new window]  and
I.M. Griffiths Open the ORCID record for I.M. Griffiths [Opens in a new window]
L.C. Auton
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
S. Pramanik
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, Gujarat, India
M.P. Dalwadi
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Department of Mathematics, University College London, London WC1H 0AY, UK
C.W. MacMinn
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
I.M. Griffiths*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
*
†Email address for correspondence: Ian.Griffiths@maths.ox.ac.uk
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Abstract

A major challenge in flow through porous media is to better understand the link between microstructure and macroscale flow and transport. For idealised microstructures, the mathematical framework of homogenisation theory can be used for this purpose. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow and transport. The equations are characterised by the local porosity, a local anisotropic flow permeability, an effective local anisotropic solute diffusivity and an effective local adsorption rate. These macroscale properties depend non-trivially on the two degrees of microstructural geometric freedom in our problem: obstacle size and obstacle spacing. We exploit this dependence to construct and compare scenarios where the same porosity profile results from different combinations of obstacle size and spacing. We focus on a simple example geometry comprising circular obstacles on a rectangular lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We investigate scenarios where the porosity is spatially uniform but the permeability and diffusivity are not. Our results may be useful in the design of filters or for studying the impact of deformation on transport in soft porous media.

JFM classification

Low-Reynolds-number flows: Porous media Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A34
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

‡

This author contributed significantly to this work.

References

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