Transport in Porous Media

, 81:35

Polynomial Filtration Laws for Low Reynolds Number Flows Through Porous Media

  • Matthew Balhoff
  • Andro Mikelić
  • Mary F. Wheeler
Article

DOI: 10.1007/s11242-009-9388-z

Cite this article as:
Balhoff, M., Mikelić, A. & Wheeler, M.F. Transp Porous Med (2010) 81: 35. doi:10.1007/s11242-009-9388-z

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 Mech 222:647–663, 1991) and Barree and Conway (SPE Annual technical conference and exhibition, 2004), even if the quadratic model were valid at moderate Reynolds numbers in the laminar flow regime, then the permeability extrapolated on a Forchheimer plot would not 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.

Keywords

Forchheimer Inertial effects Homogenization Cubic law 

List of symbols

v

Physical velocity [L/t]

p

Pressure [F/L2]

V

Characteristic velocity [L/t]

μ

Viscosity [F/L2t]

ρ

Density [M/L3]

L

Characteristic length M

\({\mathcal{P}}\)

Characteristic pressure [F/L2]

ΔP

Pressure drop

Re

Reynolds number

Fr

Froude’s number

Ω

Reservoir

Y

Unit cell

Ys

Solid part of the unit cell

YF

Pore

Ωε

Pore space (the fluid part of Ω)

ε

Ratio between the pore size and the reservoir size L

φ

Porosity

F

Dimensionless forcing term

vε

Dimensionless physical velocity

pε

Dimensionless pressure

K

Dimensionless permeability

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Matthew Balhoff
    • 1
  • Andro Mikelić
    • 2
    • 3
  • Mary F. Wheeler
    • 4
  1. 1.Petroleum and Geosystems EngineeringThe University of Texas at AustinAustinUSA
  2. 2.Université de LyonLyonFrance
  3. 3.Institut Camille Jordan, UFR MathématiquesUniversité Lyon 1Lyon Cedex 07France
  4. 4.Institute for Computational and Engineering ScienceThe University of Texas at AustinAustinUSA

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