6.2 Effective Knudsen Diffusion Coefficient

Equation 31 calculates the mole flux of pure gas subjected to a pressure gradient. The flux of a pure gas is comprised of both diffusive and viscous contributions as indicated by the two coefficients in parentheses. Equation 31 suggests that the effective Knudsen diffusion coefficient can be estimated from measured values of the quantity in parentheses, together with independently determined values for permeability (Thorstenson and Pollock, 1989; Webb, 2006) and that procedure is presented in the following paragraphs.

Klinkenberg (1941) was interested in the estimation of liquid permeability of porous media from measurements made with air. He determined the quantity in parentheses in Equation 31 from measurements of volume flux of air and the corresponding pressure gradient from which he computed an apparent permeability. Thus, Klinkenberg’s apparent permeability included a contribution from Knudsen diffusion. The apparent permeability was found to be a function of the mean pressure at which the experiments were conducted. A plot of apparent permeability versus the inverse mean pressure was approximately linear with slope b. The graph was extrapolated to 1/\bar{p}=0 to provide a value for the apparent permeability to air at a pressure sufficiently large to preclude a contribution by Knudsen diffusion. Because the experiments were conducted with dry porous media, the extrapolated value was taken to be the intrinsic permeability, k, of the porous medium. Klinkenberg’s results can be expressed by Equation 37.

\displaystyle \frac{k_{a}\bar{p}}{\mu }=\left ( \frac{k\bar{p}}{\mu } \right )\left ( 1+\frac{b}{\bar{p}} \right ) (37)

The left side of Equation 37 contains the measured apparent value of permeability, ka, and the right side contains the intrinsic value. The parameter b is known as the Klinkenberg parameter. Equation 38 follows from Equation 31 and 37.

\displaystyle \frac{D^{K}\mu }{k}=b (38)

Heid et al. (1950) correlated the Klinkenberg parameter with intrinsic permeability from more than 150 measurements over the permeability range 1017 to 1012 m2 to arrive at Equation 39.

\displaystyle b=0.11(k)^{-0.39} (39)


k = permeability (must be in m2)
b = Klinkenberg parameter (expressed as Pascals)

When permeability is expressed in m2, Equation 39 returns a Klinkenberg parameter value with units of Pascals (Thorstenson and Pollock, 1989). Equation 39 was developed from measurements with air flow through dry porous media, so values of Knudsen diffusion coefficients computed from Equations 38 and 39 are specific for air in dry media and are given the symbol D_{a}^{K}. Coefficients for other gases in dry porous media can be determined from Equation 40.

\displaystyle D_{i}^{K}=\left ( \frac{M_{a}}{M_{i}} \right )^{0.5}D_{a}^{K} (40)

The subscripts a and i denote air and the gas of interest, respectively. Equation 40 is Equation 19 rewritten specifically for the case at hand. The procedure is as follows: 1) determine the intrinsic permeability, 2) compute b from Equation 39 and use Equation 38 to compute the Knudsen diffusion coefficient for air, and 3) calculate the Knudsen diffusion coefficient for the gas of interest from Equation 40.

The porous media of interest in field applications are rarely dry. The presence of water in porous media reduces the characteristic dimension of the space available for gas diffusion and, therefore, causes Knudsen diffusion to be more significant than if the medium were dry. Thus, it may be important to estimate values for the Knudsen diffusion coefficients even in rather coarse-grained media. First-cut estimates of Knudsen diffusion coefficients, affected by the presence of water, can be made by using the effective gas permeability, kg, determined at the water content of interest, in place of the intrinsic permeability k in the above procedure.

Reinecke and Sleep (2002) found that this first-cut approximation overestimated the Knudsen diffusion coefficient for air when compared to experimental measurements. These authors propose the correlation of Equation 41.

\displaystyle D_{a}^{K}=2.69\times 10^{6}(k_{g})^{0.764} (41)

Equation 41 yields the Knudsen diffusion coefficient for air in cm2/s corresponding to an effective gas permeability expressed in cm2. The values for other gases follow from Equation 40.

The above procedure offers a practical way to estimate the difficult to measure Knudsen diffusion coefficients that are required for the application of the flux equations developed in the foregoing paragraphs. Note that the values of Knudsen diffusion coefficients calculated from the above procedure are already effective (macroscopic) values and do not require adjustments for tortuosity and open area.


Flux Equations for Gas Diffusion in Porous Media Copyright © 2021 by David B. McWhorter. All Rights Reserved.