Transport in Porous Media

, Volume 125, Issue 2, pp 211–238 | Cite as

Inertial Sensitivity of Porous Microstructures

  • Martin PauthenetEmail author
  • Yohan Davit
  • Michel Quintard
  • Alessandro Bottaro


Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, \(Re_{\mathrm{C}}\), so that the inertial deviation occurs when \(Re_{\mathrm{C}}\sim 1\) for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, \(\ell \). We analyze the problem using the method of volume averaging and identify a length scale in the form \(\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}\), with \(C_\lambda \) a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for \(C_\lambda \) is given; \(C_\lambda \) is computed from a creeping flow simulation in the porous medium; and \(Re_{\mathrm{C}}\) can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.


Volume-averaging method Filtration law Inertial flow Reynolds number 

List of Symbols

\(\bullet _\beta \)

Item related to the fluid phase (\(\beta \)-phase)

\(C_\lambda \)

Inertial sensitivity of the microstructure

\(\epsilon _\beta \)


\(\ell \)

Length scale used in \(Re_{\mathrm{C}}\) (L)

\(\ell _\beta \)

Pore length scale (L)

\(F_\parallel ,F_\perp \)

Parallel and orthogonal Forchheimer terms

\(F_\lambda \)

Forchheimer number

\(\gamma _\beta \)

\(\beta \)-phase indicator

\({{\mathbf {\mathsf{{g}}}}}_{\beta }\)

Macroscopic pressure gradient (\({\hbox {ML}}^{-2}{\hbox {T}}^{-2}\))

\(K_\lambda \)

Scalar permeability (\({{\hbox {L}}^2}\))

\(\varvec{{\lambda }} \)

Direction of the intrinsic average velocity

\(\mathscr {A}_{\beta \sigma }\)

Domain of the fluid–solid interface

\(\mathscr {V}\)

Domain of the REV

\(\mathscr {V}_\beta \)

Domain of the \(\beta \)-phase

\(\mu _{\beta }\)

Dynamic viscosity of the fluid (\({{\hbox {ML}}^{-1}{\hbox {T}}^{-1}}\))

\(\nu _{\beta }\)

Kinematic viscosity of the fluid (\({{\hbox {L}}^{2}{\hbox {T}}^{-1}}\))

\(p_{\beta }\)

Pressure field (\({{\hbox {ML}}^{-2}{\hbox {T}}^{-2}}\))

\(\psi \)

Generic field

\(\mathbb {R}\)

Real space


Critical- and permeability-based Reynolds number

\(\rho _{\beta }\)

Density of the fluid (\({{\hbox {ML}}^{-3}}\))

\(\ell _0\)

Dimension of the REV (L)

\({{\mathbf {\mathsf{{v}}}}}_{\beta }\)

Flow velocity field (\({{\hbox {LT}}^{-1}}\))

\({{\mathbf {\mathsf{{f}}}}}\)

Extension of Darcy’s law (\({{\hbox {L}}^{-2}}\))

\({{\mathbf {\mathsf{{K}}}}}_\mathrm{D}\)

Darcy permeability (\({{\hbox {L}}^2}\))

\({{\mathbf {\mathsf{{r}}}}}\)

General position vector (L)

\({{\mathbf {\mathsf{{s}}}}}_\beta \)

Constant source term (\({{\hbox {LT}}^{-2}}\))

\({{\mathbf {\mathsf{{y}}}}}_\beta = {{\mathbf {\mathsf{{r}}}}} - {{\mathbf {\mathsf{{x}}}}}\)

Position vector relative to the centroid of the REV (\({{\hbox {L}}}\))

\(\left\langle {{\mathbf {\mathsf{{v}}}}}_{\beta } \right\rangle ^\beta ,\left\langle p_{\beta } \right\rangle ^\beta \)

Intrinsic averages

\(\tilde{{{\mathbf {\mathsf{{v}}}}}}_{\beta },\tilde{p}_{\beta }\)

Spatial deviations

\({{\mathbf {\mathsf{{v}}}}}^*,p^*\)

Dimensionless spatial deviations


Nonlinear part of Ergun’s equation (\({{\hbox {L}}^{-1}{\hbox {T}}}\))


Macroscopic length scale (L)


Measure of the REV (\({{\hbox {L}}^3}\))


Magnitude of the intrinsic average velocity (\({{\hbox {LT}}^{-1}}\))

\(V_\beta \)

Measure of the \(\beta \)-phase inside the REV (\({{\hbox {L}}^3}\))



The authors would like to thank the IDEX Foundation of the University of Toulouse for the financial support granted to AB under the project ”Attractivity Chairs.” This work was granted access to the HPC resources of CALMIP supercomputing center under the allocation 2016-p1540.

Supplementary material


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Authors and Affiliations

  1. 1.Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRSToulouseFrance
  2. 2.DICCA, Scuola Politecnica University of GenovaGenoaItaly

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