The role of eddies inside pores in the transition from Darcy to Forchheimer flows

Abstract

We studied the role of intra‐pore eddies, from viscous to inertial flows, in modifying continuum‐scale flow inside pores. Flow regimes spanning Reynolds NumberRe∼ 0 to 1350 are divided into three zones – one zone follows Darcy flow, and the other two zones describe non‐Darcy or Forchheimer flow. During viscous flows, i.e.,Re < 1, stationary eddies occupy about 1/5 of the pore volume. Eddies grow when Re > 1, and their growth leads to the deviation from Darcy's law and the emergence of Forchheimer flow manifested as a characteristic reduction in the apparent hydraulic conductivity Ka. The reduction in Ka is due to the narrowing of the flow channel which is a consequence of the growth in eddies. The two zones of Forchheimer flow correspond to the changes in rate of reduction in Ka, which in turn are due to the changes in eddy growth rate. Since the characteristics of Forchheimer flow are specific to pore geometry, our results partly explain why a variety of Forchheimer models are expected and needed for different porous media.

Universal Relationship Between Viscous and Inertial Permeability of Geologic Porous Media

“…• Data Set S2 by the Forchheimer equation (Bear, 1972;Chaudhary et al, 2011;Forchheimer, 1901;Javadi et al, 2014;Liu, Li, & Jiang, 2016;Qian et al, 2007;Rust & Cashman, 2004;Zeng & Grigg, 2006;Zimmerman et al, 2004). For an incompressible fluid, the Forchheimer equation reads as…”

“…Non-Darcian flow characteristics have been ascribed to the formation and growth of eddies or recirculation zones caused by significant inertial forces (Cardenas et al, 2009;Chaudhary et al, 2011;Lee et al, 2014). Eddies are persistently and distinctly separate to the bulk flow in geologic porous media, and the apparent permeability calculated from Darcy's law continuously decreases under this hydrodynamic condition (Chaudhary et al, 2013;Sivanesapillai et al, 2014;Zhou et al, 2015).…”

“…Moreover, pore geometry can be varied due to external stress (Pyrak-Nolte & Nolte, 2016) and internal fluid-pressure (Watanabe et al, 2017), and processes such as chemical erosion (Deng et al, 2015), phase change (Wang & Cardenas, 2017), and degassing , leading to various geometries and subsequently significant variation in permeability. Investigations of hydrodynamic factors, that is, inertial effects, typically assume static pore geometry (Chaudhary et al, 2011;Lee et al, 2014). Investigations of hydrodynamic factors, that is, inertial effects, typically assume static pore geometry (Chaudhary et al, 2011;Lee et al, 2014).…”

The impact of inertial effects on solute dispersion in a channel with periodically varying aperture

“…While this is very often a reasonable assumption, 32 a variety of practical situations exist where the Reynolds number can become of order unity and larger, so that inertial effects are no longer negligible. Practical examples include flow through fractures with large aperture [33][34][35][36] and flows where the viscosity can be small such as carbon sequestration where the viscosity of supercritical CO 2 can be one or two orders of magnitude smaller than that of water. 37 Increased inertial effects play an interesting role on the structure of the flow.…”

Description of non-Darcy flows in porous medium systems

“…The current state of knowledge categorizes single-fluidphase flow into two primary regimes: a creeping flow regime, corresponding to low flow velocities in which viscous forces completely dominate inertial forces, and a regime corresponding to flow velocities for which inertial forces cannot be neglected [6][7][8][9][10][11][12][13]. These two regimes are most commonly modeled by two approximations of the macroscale momentum equation for single-fluid-phase flow through a porous media: Darcy's law and Forchheimer's equation, respectively.…”