Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

Abstract

In this study, we use the method of homogenization to develop a filtration law in porous media that includes the effects of inertia at finite Reynolds numbers. The result is much different than the empirically observed quadratic Forchheimer equation. First, the correction to Darcy's law is initially cubic (not quadratic) for isotropic media. This is consistent with several other authors (Mei and Auriault, J Fluid Mech 222:647-663, 1991; Wodié and Levy, CR Acad Sci Paris t.312: 157-161, 1991; Couland et al. J Fluid Mech 190:393-407, 1988; Rojas and Koplik, Phys Rev 58:4776-4782, 1988) who have solved the Navier-Stokes equations analytically and numerically. Second, the resulting filtration model is an infinite series polynomial in velocity, instead of a single corrective term to Darcy's law. Although the model is only valid up to the local Reynolds number, at the most, of order 1, the findings are important from a fundamental perspective because it shows that the often-used quadratic Forchheimer equation is not a universal law for laminar flow, but rather an empirical one that is useful in a limited range of velocities. Moreover, as stated by Mei and Auriault (J Fluid 123 36 M. Balhoff et al.be the intrinsic Darcy permeability. A major contribution of this study is that the coefficients of the polynomial law can be derived a priori, by solving sequential Stokes problems. In each case, the solution to the Stokes problem is used to calculate a coefficient in the polynomial, and the velocity field is an input of the forcing function, F, to subsequent problems. While numerical solutions must be utilized to compute each coefficient in the polynomial, these problems are much simpler and robust than solving the full Navier-Stokes equations.

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

“…Given that in real porous media there is a large variance in pore geometry, it is unlikely that non‐ Darcy or Forchheimer flow can be explicitly described by a single function for all porous media and for all high Re flow conditions. Therefore, it is expected that different functions may be needed for site and medium specific scenarios, even up to, for example, a 5th order polynomial [ Balhoff and Wheeler , 2009; Balhoff et al , 2010].…”

Time asymptotics of non-darcy flows controlled by total flux on the boundary

“…Different modifications were introduced since then to model and explain experimental and real life data (cf. for example, [7]- [13] and the references therein). The most comprehensive discussion of current problems is presented in the recent paper [14].…”

Geometric multigrid methods for Darcy–Forchheimer flow in fractured porous media

“…The new model, referred to as Forchheimer's law, combines the contribution of viscous and inertial effects: at low flow rates, the viscous effect is dominant and the model reduces to Darcy's law; at increasing flow rates, however, the inertial effect gains relevance and plays a significant role [12]. Remarkably, other nonlinear correction terms -e.g., cubic [13], polynomial [11] or exponential [14]have also been proposed in the literature.…”