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

TL;DR

This study investigates the Darcy–Forchheimer transition in steady flows through porous media, emphasizing vortex formation and its impact on tortuosity across porosities. It combines pore-scale Navier–Stokes modeling in stochastic and simple cubic geometries with two tortuosity definitions, $T_\Omega$ and $T_s$, to reveal how recirculation zones and energy confinement drive inertia onset. The results show porosity as a key control parameter: lower porosity accelerates recirculation growth and energy concentration, causing larger deviations between $T_\Omega$ and $T_s$, while higher porosity delays these effects. The findings provide mechanistic insight into choosing inertia indicators for practical problems and link pore-scale dynamics to macroscopic transport properties.

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.

Figures (29)

• Figure 1: A schematic representation of the percolating volume ($\Omega_p$, red), recirculation volume ($\Omega_v$, green) and the boundary between the two ($\partial\Omega_{pv}$, yellow) in a flow through a periodic porous sample (obstacles depicted striped gray with their boundary $\partial \Omega_\text{solid}$ plotted with solid black line). The curves with arrows along their length represent the percolating and the recirculating streamlines. The dashed edges of the large square represent the periodic boundary
• Figure 2: Visualization of the geometries of the solid phase of the two stochastic porous samples considered in this work. The target porosities are $\phi=0.7$ (left) and $\phi=0.9$ (right). The black-edged cube denotes the boundaries of the periodic cell.
• Figure 3: Visualization of the geometries of the solid (gray) and fluid (blue, opaque) phases of the two simple cubic porous samples considered in this work. The porosities are $\phi=0.1, \> 0.59, \> 0.93, \> 0.999$, counting from the top-left to the bottom-right. The black-edged cube denotes the boundaries of the periodic cell. In the case of the lowest-porosity sample, only a few chosen obstacles residing in the periodic cell are shown for clarity.
• Figure 4: Visualization of the meshes of two chosen simple cubic porous samples considered in this work. The porosities are $\phi=0.1$ (two top subplots) and $\phi=0.999$ (two bottom subplots, for the uniform and refined discretization). The top-right subplot shows the magnified view of the top-left mesh. The black-edged cube denotes the boundaries of the periodic cell. Only $z<0.5$ half of each mesh is shown.
• Figure 5: Schematic representation of a streamline $s$ ( dashed line) as a curve tangent to the fluid velocity field vectors (black arrows) in each of the streamline's points (two chosen are shown here as black dots, $\boldsymbol{x}$ and $\boldsymbol{x'}$). An infinitesimal increment along a streamline, $d\boldsymbol{s}$ (small blue arrow), is parallel to the local velocity vector, here normalized for convenience. Any of the streamline's points identifies it uniquely ($s(\boldsymbol{x}) \equiv s(\boldsymbol{x'})$).