{"id":61,"date":"2021-10-02T23:21:36","date_gmt":"2021-10-02T23:21:36","guid":{"rendered":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/effective-knudsen-diffusion-coefficient\/"},"modified":"2022-01-10T20:36:40","modified_gmt":"2022-01-10T20:36:40","slug":"effective-knudsen-diffusion-coefficient","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/chapter\/effective-knudsen-diffusion-coefficient\/","title":{"raw":"6.2 Effective Knudsen Diffusion Coefficient","rendered":"6.2 Effective Knudsen Diffusion Coefficient"},"content":{"raw":"
\r\n

Equation\u00a031<\/a> 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\u00a031<\/a> 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.<\/p>\r\n

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<\/a> from measurements of volume flux of air and the corresponding pressure gradient from which he computed an apparent permeability. Thus, Klinkenberg\u2019s 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<\/em>. The graph was extrapolated to [latex]1\/\\bar{p}=0[\/latex] 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<\/em>, of the porous medium. Klinkenberg\u2019s results can be expressed by Equation\u00a037.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{k_{a}\\bar{p}}{\\mu }=\\left ( \\frac{k\\bar{p}}{\\mu } \\right )\\left ( 1+\\frac{b}{\\bar{p}} \\right )[\/latex]<\/td>\r\n(37)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The left side of Equation\u00a037 contains the measured apparent value of permeability, k<\/em>a<\/em><\/sub>, and the right side contains the intrinsic value. The parameter b<\/em> is known as the Klinkenberg parameter. Equation\u00a038 follows from Equation\u00a031<\/a> and 37.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{D^{K}\\mu }{k}=b[\/latex]<\/td>\r\n(38)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Heid et al. (1950) correlated the Klinkenberg parameter with intrinsic permeability from more than 150 measurements over the permeability range 10\u2212<\/sup>17<\/sup> to 10\u2212<\/sup>12<\/sup>\u00a0m2<\/sup> to arrive at Equation\u00a039.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle b=0.11(k)^{-0.39}[\/latex]<\/td>\r\n(39)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
k<\/em><\/td>\r\n=<\/td>\r\npermeability (must be in m2<\/sup>)<\/td>\r\n<\/tr>\r\n
b<\/em><\/td>\r\n=<\/td>\r\nKlinkenberg parameter (expressed as Pascals)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

When permeability is expressed in m2<\/sup>, Equation 39 returns a Klinkenberg parameter value with units of Pascals (Thorstenson and Pollock, 1989). Equation\u00a039 was developed from measurements with air flow through dry porous media, so values of Knudsen diffusion coefficients computed from Equations\u00a038 and 39 are specific for air in dry media and are given the symbol [latex]D_{a}^{K}[\/latex]. Coefficients for other gases in dry porous media can be determined from Equation\u00a040.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle D_{i}^{K}=\\left ( \\frac{M_{a}}{M_{i}} \\right )^{0.5}D_{a}^{K}[\/latex]<\/td>\r\n(40)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The subscripts a<\/em> and i<\/em> denote air and the gas of interest, respectively. Equation\u00a040 is Equation\u00a019<\/a> rewritten specifically for the case at hand. The procedure is as follows: 1) determine the intrinsic permeability, 2) compute b<\/em> from Equation\u00a039 and use Equation\u00a038 to compute the Knudsen diffusion coefficient for air, and 3) calculate the Knudsen diffusion coefficient for the gas of interest from Equation\u00a040.<\/p>\r\n

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, k<\/em>g<\/em><\/sub>, determined at the water content of interest, in place of the intrinsic permeability k<\/em> in the above procedure.<\/p>\r\n

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\u00a041.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle D_{a}^{K}=2.69\\times 10^{6}(k_{g})^{0.764}[\/latex]<\/td>\r\n(41)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation\u00a041 yields the Knudsen diffusion coefficient for air in cm2<\/sup>\/s corresponding to an effective gas permeability expressed in cm2<\/sup>. The values for other gases follow from Equation\u00a040.<\/p>\r\n

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.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Equation\u00a031<\/a> 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\u00a031<\/a> 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.<\/p>\n

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<\/a> from measurements of volume flux of air and the corresponding pressure gradient from which he computed an apparent permeability. Thus, Klinkenberg\u2019s 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<\/em>. 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<\/em>, of the porous medium. Klinkenberg\u2019s results can be expressed by Equation\u00a037.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{k_{a}\bar{p}}{\mu }=\left ( \frac{k\bar{p}}{\mu } \right )\left ( 1+\frac{b}{\bar{p}} \right )\"<\/td>\n(37)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The left side of Equation\u00a037 contains the measured apparent value of permeability, k<\/em>a<\/em><\/sub>, and the right side contains the intrinsic value. The parameter b<\/em> is known as the Klinkenberg parameter. Equation\u00a038 follows from Equation\u00a031<\/a> and 37.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{D^{K}\mu }{k}=b\"<\/td>\n(38)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Heid et al. (1950) correlated the Klinkenberg parameter with intrinsic permeability from more than 150 measurements over the permeability range 10\u2212<\/sup>17<\/sup> to 10\u2212<\/sup>12<\/sup>\u00a0m2<\/sup> to arrive at Equation\u00a039.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle b=0.11(k)^{-0.39}\"<\/td>\n(39)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n
k<\/em><\/td>\n=<\/td>\npermeability (must be in m2<\/sup>)<\/td>\n<\/tr>\n
b<\/em><\/td>\n=<\/td>\nKlinkenberg parameter (expressed as Pascals)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

When permeability is expressed in m2<\/sup>, Equation 39 returns a Klinkenberg parameter value with units of Pascals (Thorstenson and Pollock, 1989). Equation\u00a039 was developed from measurements with air flow through dry porous media, so values of Knudsen diffusion coefficients computed from Equations\u00a038 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\u00a040.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle D_{i}^{K}=\left ( \frac{M_{a}}{M_{i}} \right )^{0.5}D_{a}^{K}\"<\/td>\n(40)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The subscripts a<\/em> and i<\/em> denote air and the gas of interest, respectively. Equation\u00a040 is Equation\u00a019<\/a> rewritten specifically for the case at hand. The procedure is as follows: 1) determine the intrinsic permeability, 2) compute b<\/em> from Equation\u00a039 and use Equation\u00a038 to compute the Knudsen diffusion coefficient for air, and 3) calculate the Knudsen diffusion coefficient for the gas of interest from Equation\u00a040.<\/p>\n

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, k<\/em>g<\/em><\/sub>, determined at the water content of interest, in place of the intrinsic permeability k<\/em> in the above procedure.<\/p>\n

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\u00a041.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle D_{a}^{K}=2.69\times 10^{6}(k_{g})^{0.764}\"<\/td>\n(41)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation\u00a041 yields the Knudsen diffusion coefficient for air in cm2<\/sup>\/s corresponding to an effective gas permeability expressed in cm2<\/sup>. The values for other gases follow from Equation\u00a040.<\/p>\n

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.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":24,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-61","chapter","type-chapter","status-publish","hentry"],"part":111,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/61","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":4,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/61\/revisions"}],"predecessor-version":[{"id":308,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/61\/revisions\/308"}],"part":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/parts\/111"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapters\/61\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/pressbooks\/v2\/chapter-type?post=61"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/contributor?post=61"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/flux-equations-for-gas-diffusion-in-porous-media\/wp-json\/wp\/v2\/license?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}