Symmetry properties of macroscopic transport coefficients in porous media
Résumé
We report on symmetry properties of tensorial effective transport coefficients characteristic of many
transport phenomena in porous systems at the macroscopic scale. The effective coefficients in the
macroscopic models (derived by upscaling (volume averaging) the governing equations at the underlying
scale) are obtained from the solution of closure problems that allow passing the information
from the lower to the upper scale. The symmetry properties of the macroscopic coefficients are
identified from a formal analysis of the closure problems and this is illustrated for several different
physical mechanisms, namely, one-phase flow in homogeneous porous media involving inertial
effects, slip flow in the creeping regime, momentum transport in a fracture relying on the Reynolds
model including slip effects, single-phase flow in heterogeneous porous media embedding a porous
matrix and a clear fluid region, two-phase momentum transport in homogeneous porous media, as
well as dispersive heat and mass transport. The results from the analysis of these study cases are
summarized as follows. For inertial single-phase flow, the apparent permeability tensor is irreducibly
decomposed into its symmetric (viscous) and skew-symmetric (inertial) parts; for creeping slip-flow,
the apparent permeability tensor is not symmetric; for one-phase slightly compressible gas flow in
the slip regime within a fracture, the effective transmissivity tensor is symmetric, a result that remains
valid in the absence of slip; for creeping one-phase flow in heterogeneous media, the permeability
tensor is symmetric; for two-phase flow, we found the dominant permeability tensors to be symmetric,
whereas the coupling tensors do not exhibit any special symmetry property; finally for dispersive heat
transfer, the thermal conductivity tensors include a symmetric and a skew-symmetric part, the latter
being a consequence of convective transport only. A similar result is achieved for mass dispersion.
Beyond the physical mechanisms under consideration in the present work, the reported technique can
be viewed as a general methodology applicable to any type of upscaled model obtained by volume
averaging.
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