Protocol for Permeability Measurement

Posted on November 15, 2007

A porous medium comprising of at least a stationary solid matrix and a fluid flowing through it is characterized usually by three properties. The first one, the volumetric porosity, is intuitive and is the ratio of the void (fluid) volume to the total (solid + fluid) volume of the porous medium. Permeability K (m²) and Form coefficient C (m^-1) are the two other hydraulic properties necessary to characterize the flow of a fluid through a porous medium. As viscosity and density are properties of a fluid, so are K and C of a porous medium. Earlier discussion on flow through porous media and permeametry are refreshers before we proceed to explain a research paper today.

Experiments determining K and C are usually influenced by undesired secondary effects. This results in higher uncertainties in their determination.

This note describes the suggestions we made in a recent Physics of Fluids paper [Ref. 1 given at the end] on how to avoid such secondary effects. We restrict the discussion only to the determination of permeability although the paper also discusses form coefficient measurement.

A constitutive equation defines a material property. It is necessary to reduce the number of unknowns in a balance equation. In the Navier Stokes momentum balance equation for fluid flow, the viscosity is found from its constitutive equation, the Newton's Law of viscosity. Similarly, Darcy, in his famous work [1], defined a hydraulic conductivity parameter, which evolved to what is now known as permeability K, m^2. It is a constitutive equation defining permeability as

$K = \frac {\mu}{(\Delta P_v)} U$ ------- (1)

and requires the measurement of the cross-section averaged fluid speed $U (ms^{-1})$ , usually obtained by dividing the volumetric flow rate by the cross-section area of the channel, and of the pressure-drop $\Delta P_v, (Pa)$ , in the flow of a Newtonian fluid, along a length L (m) of the channel occupied by a porous medium.

On the other hand, the differential form of the volume averaged momentum equation for the flow of a Newtonian fluid through a porous medium, with uniform, isotropic, and constant properties looks like this,

$\frac {\rho}{\phi} \left[ \frac {\partial \vec{V}} {\partial t} \right] + \frac {1} {\phi^2} \vec{V} \cdot \nabla \vec{V} = -\nabla p + \mu_{eff}\nabla^2\vec{V} - \frac {\mu}{K} \vec{V} - \rho C |\vec{V}|\vec{V}$ ------- (2)

Here, $\rho (kgm^{-3})$ and $\mu_{eff} (kgm^{-1}s^{-1})$ are the fluid density and effective dynamic viscosity. Volumetric porosity $\phi$ of the porous medium is defined as the ratio of fluid occupied volume to the total volume and is necessary to correct the increased speed of the fluid through the pores as compared to the clear (of porous medium) channel case. $V (ms^{-1})$ is the vectorial local fluid velocity and p (Pa) the local static pressure.

The terms to the left side of the equal sign of Eq. (2) represent the fluid acceleration (local and convective). To the right of the equal sign we have: pressure gradient, viscous diffusion, viscous-drag and form-drag, respectively. The sum of the viscous and form drags constitute the total drag effect imposed on the flow by the porous medium. Their absence signals absence of a porous medium and we should reclaim from Eq. (2), the standard Navier Stokes equation for clear (of porous medium) fluid flow.

For steady, fully-developed unidirectional flow, and negligible viscous diffusion and form-drag effects, the momentum balance equation Eq. (2) reduces to

$\frac {\Delta P}{L} = \frac {\mu}{K}U$ ------- (3)

which is almost identical to Eq. (1).

The difference between Eq. (1), the constitutive equation defining K and Eq. (3), the momentum balance equation, is on the pressure-drop. The implicit assumption when determining K from Eq. (1), its constitutive equation is that, once determined, the value of K would not change with changes in the porous medium length, or channel geometry, or flow rate, but only with changes in the internal structure of the porous medium. This requires the pressure-drop $\Delta P_v$ of Eq. (1) must measure the pressure-drop caused by the viscous-drag alone, induced by the internal structure of the solid porous matrix.

The momentum balance equation version of Darcy law, Eq. (3), need not have this restriction.

Obviously, the applicability of Eq. (1) for determining K is more restricted than that of the momentum balance equation, Eq. (3).

To guarantee that only the viscous-drag effect is accounted for in using Eq. (1) to determine K (i.e., for the experimentally measured pressure-drop $\Delta P_m$ to be identical to $\Delta P_v$ ), the testing channel containing the porous medium must be straight and of uniform cross-section. Else, $\Delta P_m$ could be affected by acceleration caused by channel curvature and/or variations in the cross-section.

The effect imposed on the fluid pressure-drop by the bounding walls of the channel (boundary viscous diffusion effects) would also have to be made negligible. Obviously, this last requirement might be difficult to satisfy in practice.

Two ways exist to resolve this. First one is to live with the secondary effects and include it in the pressure-drop, hoping that its effect would be an insignificant raise in the total pressure-drop, while determining K. This means Eq. (1) can be represented as

$K = \frac {\mu}{(\Delta P_{cv})} U$ ------- (4)

where $\Delta P_{cv}$ is the channel pressure drop that includes the pressure drop resulting from the porous medium viscous drag $(\Delta P_v)$ and by the secondary wall effects $(\Delta P_c)$ .

This means, the determined K would depend on the channel wall or geometry. Further, even in the absence of a porous medium, we would measure a non-zero K from Eq. (4), a result of non-zero pressure-drop caused by the channel wall effects.

Obviously, this method of resolving the issue leads to a maximum limit for the measured permeability - equal to that of the channel clear (of porous medium) fluid flow permeability.

The other method to resolve this confusion between constitutive relation and balance equation is to subtract the secondary effects from the measured pressure-drop, while determining K. This would result in a determined K that represents only the viscous drag of the porous medium.

$K = \left[ \frac {\Delta P_{cv}}{L\mu U} - \frac {2Po}{\phi D_h^2} \right]$ ------- (5)

The last term of Eq. (5) is the secondary effect by the bounding wall, which is subtracted from the measured pressure drop, before determining K. As defined in Eq. (5), K is no longer limited but can tend to infinity for a clear (of porous medium) channel because it can be shown that as the porosity $\phi \rightarrow 1$ (meaning there is no solid matrix), $(\Delta P_{cv}/L) \rightarrow (\mu U 2Po /D_h^2)$ .

The significance of all of this is, in an engineering situation of channel bounded porous medium flows, permeability K as defined by Eq. (4) has a restriction for its highest value dependent on the channel cross-section geometry. However, when defined by Eq. (5), the value of K has no upper bound. Many times parametric (numerical or CFD) studies involve wall-bounded channel flow through a porous medium using $Da > 1/(2Po)$ , where $Da = K/D_h^2$ . The results of the paper suggests such types of parametric studies are valid only provided K is defined by, and measured according to, Eq. (5).

To summarize, we find two alternative constitutive equations defining K, namely Eq. (4) and (5) exist. They differ on the inclusion or not of the bounding channel wall viscous diffusion effect on the measured $\Delta P_m$ . The simplicity of Eq. (4) makes it appealing than Eq. (10) for use. But the later part of the paper, when considering the simultaneous determination of K and C, presents arguments for the use of Eq. (5) for determining K.

References

Lage, J., Krueger, P., & Narasimhan, A. (2005). Protocol for measuring permeability and form coefficient of porous media Physics of Fluids, 17 (8) DOI: 10.1063/1.1979307

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