Computation of the Permeability Tensor of Non-Periodic Anisotropic Porous Media from 3D Images

Abstract

The direct proportionality between the flow rate and the pressure gradient of creeping flows was experimentally discovered by H. Darcy in the 19th century and theoretically justified a couple of decades ago using upscaling methods such as volume averaging or homogenization. X-ray computed micro-tomography (CMT) and pore-scale numerical simulations are increasingly used to estimate the permeability of porous media. However, the most general case of non-periodic anisotropic porous media still needs to be completely numerically defined. Pore-scale numerical methods can be split into two families. The first family is based on a direct resolution of the flow solving the Navier–Stokes equations under the assumption of creeping flow. The second one relies on the resolution of an indirect problem—such as the closure problem derived from the volume averaging theory. They are known to provide the same results in the case of periodic isotropic media or when dealing with representative element volumes. To address the most general case of non-periodic anisotropic porous media, we have identified four possible numerical approaches for the first family and two for the second. We have compared and analyzed them on three-dimensional generated geometries of increasing complexity, based on sphere and cylinder arrangements. Only one, belonging to the first family, has been proved to remain rigorously correct in the most general case. This has been successfully applied to a high-resolution 3D CMT of Carcarb, a carbon fiber preform used in the thermal protection systems of space vehicles. The study concludes with a detailed analysis of the flow behavior (streamlines and vorticity). A quantitative technique based on a vorticity criterion to determine the characteristic length of the material is proposed. Once the characterized length is known, the critical Reynolds number can be estimated and the physical limit of the creeping regime identified.