Upscaled permeability for gaseous microflows in porous media with arbitrary multiscale heterogeneities

Hierarchical porous media with multiple separate spatial scales are ubiquitous in substructures, unconventional strata, chemical engineering systems, energy storage, etc. The development of a highly accurate and highly efficient prediction model for gas transport in these multiscale porous media is of great importance for structural optimization design but remains an open research topic. In this study, we conduct a detailed multi-scale analysis of steady-state gas transport in a multiscale microporous matrix using the iterated asymptotic homogenization method. The upscaled permeability models and the governing equations for gas flow in the continuum and slip flow regimes in multiscale porous media with more than three separate spatial scales are derived accordingly. The accuracy and reliability of the developed models are confirmed and verified by the results of high-fidelity direct numerical simulation. The contribution of the first-order auxiliary cell functions to the upscaled permeability of porous media with different configurations is analyzed in detail, which provides a fundamental understanding of how these heterogeneities at each scale affect the macroscopic flow resistance and equivalent permeability. We also analyze the differences between the low-order and high-order multiscale models and the contrast between the gas transport processes in a steady and transient state. This work guides establishing highly efficient prediction models for gaseous microflows in complex porous media with arbitrary multiscale heterogeneities.