Reconciling asymmetry observations in the permeability tensor of digital rocks with symmetry expectations

Abstract

Modelling anisotropic flow in rocks requires their full permeability tensor. While theories derived from the upscaling of Stokes flow to Darcy’s law may justify the tensor symmetry, homogenisations from micro-scale rock samples often return a non-zero level of asymmetry. Since most studies dismiss these controversial observations as numerical errors, this contribution looks more closely at the physical possibility of such behaviour. Asymmetry of the permeability tensor, which induces a rotatory flow, is manifested at the micro-scale by tortuous streamlines. Conversely, when considering a larger scale – above the Representative Elementary Volume for permeability – these tortuous paths do not statistically affect the flow direction any longer. At this point, the homogenisation of Stokes flow to Darcy’s law reaches its domain of validity. We show that the asymmetry in the permeability tensor vanishes for this scale separation, regardless of the choice of boundary conditions unlike previously thought, if the boundary layer effect is disregarded.