The role of porosity in the transition to inertial regime in porous media flows

Abstract

In this work, we investigate the fundamental physical mechanism of the transition from Darcy to inertial (Darcy-Forchheimer) regime in steady-state flows through porous media, with the focus on vortex formation. We investigate their influence on the tortuosity--Reynolds number relation during this transition for systems of various porosities. We do so by numerically solving the Navier-Stokes equations within the pore-scale of simple cubic systems and relating the observations made therein to stochastic systems of more complex geometry. We observe that the tortuosity defined by integrals over the whole fluid volume behaves similarly in both types of systems. At the same time, in simple cubic systems, the tortuosity based on averaging of the length of the streamlines diverges from the volume-integrated one when the inertia onset takes place. We show that the discrepancy between those two tortuosities at increasing Reynolds number carries information about the dynamics of the vortex growth in the system. We stipulate that those dynamics are directly governed by the porosity. Our results highlight the utility of various definitions of tortuosity as measures of inertia in porous media flows and explain the reasons for the differences between those definitions. This can lead to a more sensible choice of inertia indicators in more application-oriented problems.

Resubmitted, August 2025. Used h=1/195 data for low-porosity SC sample and updated the text accordingly. Included new Figures in Introduction and Methods, minor text changes. Updated abstract Resubmitted, November 2025. Added two intermediate porosity SC samples, corrected minor inconsistencies in the calculation of Re_k