致密多孔介质中气体视渗透率的有效联系外文文献翻译

Effective Correlation of Apparent Gas Permeability

in Tight Porous Media

Faruk Civan

Abstract:Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accom-plished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the charac-teristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.

Keywords:Tight porous media·Apparent gas permeability·Rarefaction coefficient·Klinkenberg gas slippage factor · Tortuosity

1 Introduction

Description of various gaseous flow regimes through tight porous media has drawn con-siderable attention because the convetional Darcy’s law cannot realist ically describe the variety of the relevant flow regimes other than the viscous flow regime. For example, Javadpour et al. (2007)have determined that gas flow in shales deviates from behavior described by the conventional Fick’s and Darcy’s laws. Therefore, many attempts have been made in describing the transfer of gas through tight porous media under various regimes. Such efforts are of utmost practical importance when dealing with extraction of hydro-carbon gases from unconventional gas reservoirs, such as shale-gas and coal-bed methane reservoirs.

Skjetne and Gudmundsson (1995),and Skjetne and Auriault (1999)theoretically investigated the wall-slip gas flow phenomenon in porous media based on the Navier-Stokes equation, but did not offer any correlation for the Klinkenberg effect. Wu et al. (1998)developed analytical procedures for determination of the Klinkenberg coefficient from laboratory and well tests, but did not provide any correlation. Having reviewed the vari-ous correlations available, Sampath and Keighin (1982) proposed an improved correlation for the Klinkenberg coefficient of the N2 gas in the presence of water in porous media, expressed here in the consistent SI units as

where b k is in Pa, K∞ is in m2, and φ is in fraction. The significance of this correlation is that its exponent is very close to the ?0.50 exponent value obtained by theoretical analysis in this article.

Beskok and Karniadakis (1999)developed a unified Hagen–Poiseuille-type

equation covering the fundamental flow regimes in tight porous media, including continuum fluid flow, slip flow, transition flow, and free molecular flow conditions. Ability to describe all four flow regimes in one equation alone is an outstanding accomplishment. However, the empirical correlation of the available data of the dimensionless rarefaction coefficient is a mathe-matically complicated trigonometric function. As demonstrated in this article, much accu-rate correlation of the same data can be accomplished using a simple inverse-power-law function.

Florence et al. (2007)made an attempt at utilizing the Hagen–Poiseuille-type equation of Beskok and Karniadakis ( 1999)to derive a general expression for the apparent gas perme-ability of tight porous media and correlated some essential parameters by means of exper-imental data, including the Klinkenberg gas slippage factor and the Knudsen number by ignoring the effect of tortuosity, although it is an important factor especially in tight porous media. Therefore, although their overall methodology is reasonable, their formulation and data analysis procedure require some critical improvements as pointed out in this article when attempting to apply the Hagen–Poiseuille-type equation, originally derived for pipe flow to tight-porous media flow. Their treatment neglects a number of important issues. The Hagen–Poiseuille-type equation of Beskok and Karniadakis( 1999) has been derived for a single-pipe flow. When the bundle of tortuous tubes realization of the preferential flow paths in tight-porous media ( Carman 1956) is considered, the number and tortuosity of the pref-erential flow paths formed in porous media should be taken into account as the important parameters.

Further, the approach taken by Florence et al.( 2007) for correlation of the Klinkenberg gas slippage factor is not correct and consequently their correlation cannot represent the data over the full range of the gas molecular mass (commonly called weight). These errors are corrected in this article by a rigorous approach which leads to a very accurate correlation of their data with a coefficient of regression almost equal to 1.0. In addition, the present analysis lends itself to a practical method by which tortuosity of tight porous media can be determined using the flow data obtained by conventional gas flow tests. To the author’s knowledge, such a method does not presently exist in the literature.

Hence, the primary objectives of this article are threefold:

(1)Correlation of the rarefaction coefficient

(2)Derivation of the apparent gas permeability equation

(3)Correlation of the Klinkenberg gas slippage factor

These issues are resolved and verified in the following sections by theoretical means and rigorously analyzing experimental data.

2 Correlation of the Rarefaction Coefficient

Beskok and Karniadakis (1999)derived a unified Hagen–Poiseuille-type equation for volu-metric gas flow q h through a single pipe, given below:

where the flow condition function f(K n) is given by

where Kn is the Knudsen number given by

where R h and L h denote the hydraulic radius and length of flow tube, and λ denotes the mean-free-path of molecules given by ( Loeb 1934)

where p is the absolute gas pressure in Pa, T is the absolute temperature in K, M is the molecular mass in kg/kmol, R g = 8314 J/kmol/K is the universal gas constant, and μ is the viscosity of gas in Pa.s.

Equation 2describes the fundamental flow regimes, namely the conditions of continuum fluid flow (Kn≤ 0.001), slip flow (0.001

The parameter α appearing in Eq. 3 is a dimensionless rarefaction coefficient which varies in the range of 0 < α < αo over 0 ≤Kn<∞. Beskok and Karniadakis (1999) provide an empirical correlation as:

where α1 = 4.0, α2 = 0.4, and αo is an asymptotic upper limit value of α as Kn→ ∞(representing free molecular flow condition), calculated by:

Here, b denotes a slip coefficient. They indicate that α = 0 and b= ?1 in the slip flow condition, and therefore Eq. 7 becomes:

The expression of Eq. 6 is mathematically complicated. In the following exercise, it is demonstrated that a simple inverse power-law expression as given below provides a much more accurate and practical alternative to Eq. 6 for the range of data analyzed by Beskok and Karniadakis (1999):

where A and B are empirical fitting constants. Note that Eq. (9) honors the limiting conditions of 0 < α < αo over 0 ≤ Kn < ∞. In fact, it can be shown that

Fig. 1 Present approach using Eq. 9 accurately correlates the data of both Loyalka and Hamoodi (1990)using the theoretically predicted upper limit value of αo = 1.358 and Tison and Tilford (1993) using anadjusted upper limit value of αo = 1.205. The present approach yields accurate fit of data with coefficients of regressions very close to 1.0

As illustrated in Fig. 1, the present approach using Eq. 9 accurately correlates the data of both Loyalka and Hamoodi (1990) using the theoretically predicted upper limit value of αo= 1.358 and Tison and Tilford (1993) using an adjusted upper limit value of αo= 1.205. Consequently, the data of Loyalka and Hamoodi (1990) is correlated as

Thus, A = 0.1780 and B = 0.4348. On the other hand, the data of Tison and Tilford(1993) is correlated as

Figure 2 shows that the data of Loyalka and Hamoodi (1990) can be correlated accurately by both the present correlation approach using Eq. 9 with a coefficient of regression of R2= 0.9871 and the empirical equation given by Beskok and Karniadakis (1999)with a coefficient of regression of R2= 0.9697 using the theoretically predicted value of αo = 1.358. However, as indicated by the comparison of the coefficients of regressions, the present approach yields a more accurate correlation than that of Beskok and Karniadakis(1999).

Fig. 2 Data of Loyalka and Hamoodi (1990) can be correlated accurately by both the present

correlation approach using Eq. 9 with a coefficient of regression of R2 = 0.9871 and the empirical equation given by Beskok and Karniadakis (1999) with a coefficient of regression of R2 = 0.9697 using the theoretically predicted value of αo = 1.358. However, as indicated by the comparison of the coefficients of regressions, thepresent approach yields a more accurate correlation than that of

Beskok and Karniadakis (1999)

Figure 3 shows that the present correlation with Eq. 9 using the adjusted value of αo =1.205 represents the data of Tison and Tilford (1993) accurately with a coefficient of regression of R2 = 0.9486, close to 1.0. In contrast, the empirical equation given by Beskok and Karniadakis (1999) using the adjusted value of αo = 1.19 leads to a lower quality correlation with a coefficient of regression of R2 = 0.7925, less than 1.0. As

indicated by the comparison of the coefficients of regressions, the present approach yields a much more accurate correlation than that of Beskok andKarniadakis(1999)

Fig. 3 Present correlation with Eq. 9 using the adjusted value of αo = 1.205 represents the data of Tison and Tilford (1993) accurately with a coefficient of regression of R2 = 0.9486, very close to 1.0. However, the empirical equation given by Beskok and Karniadakis (1999) using the adjusted value of αo= 1.19 leads to a lower quality correlation with a coefficient of regression of R2= 0.7925, less than 1.0. As indicated by the comparison of the coefficients of regressions, the present approach yields a much more accurate correlation than that of Beskok and Karniadakis (1999)

It is concluded that the present simple inverse power-law expression yields more accurate correlation of the dimensionless rarefaction coefficient with coefficients of regressions very close to 1.0 in both cases and therefore more suitable than the empirical equation given by Beskok and Karniadakis(1999).

3 Derivation of the Apparent Gas Permeability Equation

The Beskok and Karniadakis (1999) unified Hagen–Poiseuille-type equation (Eq. 2) derived for flow q h through a single pipe can now be applied for the volumetric gas flow through a bundle of tortuous flow paths as:

where L h denotes the length of the tortuous flow paths and n denotes the number of preferential hydraulic flow paths formed in porous media. The latter can be approximated by rounding the value calculated by the following equation to the nearest integer (Civan 2007)

where φ is porosity and A b is the bulk surface area of porous media normal to flow direction. The symbol q denotes the total volumetric flow through porous media. It can be described macroscopically by a Darcy-type gradient-law of flow, where the flow is assumed proportional to the pressure gradient given by

where K denotes the apparent gas permeability of tight porous media and L is the length of bulk porous media.

Note that Eq. 16 is used frequently, although it is not rigorously correct. The corrections required on Eq.16, such as the effect of the threshold pressure gradient (Prada and Civan1999), are provided elsewhere by Civan (2008), but are ignored here to avoid unnecessary complications for purposes of the present discussion and derivation. Nevertheless, Civan(2008) argued that such corrections are usually negligible for gaseous flow, although the validity of such claim for tight porous media needs detailed investigation.

The tortuosity factor τhof hydraulic preferential flow paths in porous media is defined by

Hence, the following expression can be derived for the apparent gas permeability by combining Eqs. 14–17:

where K∞ denotes the liquid permeability of porous media given by

Equation 19 can be rearranged to express the hydraulic tube diameter as

Alternatively, it can be shown for a pack of porous media grains that (Civan 2007)

Where ∑g denotes the specific grain surface in porous media. Hence, equating Eqs. 20

and 21 yields the well-known Kozeny-Carman equation of permeability as (Carman 1956; Civan2007)

The function f (Kn) does not appear in Eq. 22 because the intrinsic permeability K∞of porous media is only a property of porous media and does not depend on the fluid type and flow condition.The formulation presented by Florence et al. (2007) for the apparent gas permeability of tight porous media considered flow through a single straight pipe and therefore needs corrections, according to the procedure described above.

4 Correlation of the Klinkenberg Gas Slippage Factor

Under slip flow conditions, α = 0 and b = ?1, and therefore Eq. 18 combined with Eq.

3 can be written as:

Florence et al. (2007) approximate this equation for Kn << 1, upon substitution of the Kn number expression Eq. 4 as:

whereas the equation of Klinkenberg (1941) is given by:

where b k is the slippage factor. Comparing Eqs. 24 and 25 yields an expression as

Substituting Eqs. 5 and 20 into Eq. 26 results in:

where the coefficient β is defined by

Figure 4 indicates that the present Eq. 28 yields an accurate correlation for the data of Florence et al. (2007) involving the flow of various gases (hydrogen, helium, nitrogen, air, and carbon dioxide) in a given porous medium (sandstone) under isothermal conditions (assumed as 298K) as the following:

where β is in Pa.m, μ in Pa.s, and M is in kg/kmol.

The results reported in Table 1 indicate that the values calculated by Eq. 29 obtained by correlating the β coefficient divided by the viscosity versus the square-root of the molecular mass, according to Eq. 28 accurately match the data of Florence et al. (2007). In contrast, the values calculated using the correlation developed by Florence et al. (2007), also given in Table 1, differ significantly from the same data because they correlated the β coefficient directly (i.e. without dividing by the gas viscosity) by an inverse-power-law function of the molecular mass, expressed here in the consistent SI units as:

where β is in Pa.m and M is in kg/kmol. In view of Eq. 28, their approach is not rigorously correct because the effect of viscosity was ignored in their correlation. Note that Eqs. 26–28 can be used to derive an expression for the Knudsen number as

For example, applying the correlation given by Eq. 29 to Eq. 31 yields:

Therefore

Fig. 4 Present approach yields an accurate correlation of the [(β/μ), m/s] parameter versus the molecular mass (M, kg/kmol) for the data of Florence et al. (2007) for flow of various gases (hydrogen, helium, nitrogen, air, and carbon dioxide) in sandstone under 298Kisothermal conditions. The coefficient of regression is almost equal to 1.0 indicating that the present correlation approach is rigorous

Table 1 Comparison of the β values indicate that the present correlation approach reproduces the data with high accuracy

In contrast, the values calculated by the correlation developed by Florence et al. (2007) differ from the data significantly

where b k is in Pa, μ in Pa.s, M is in kg/kmol, p is in Pa, K∞ is in m2, and φ is fraction. When applied for the N2 gas, Eq. 33 yields

The coefficient 0.0094 of this equation is in the same order ofmagnitude as the coefficient 0.0414of Eq. 1 of Sampath and Keighin (1982) but differs by a factor of 4.4. This may be explained due to the ?0.53 value of the exponent of Eq. 1 being different than the exponent value of ?0.50 of Eq. 34 and the N2 gas flow tests conducted in the presence of water in porous media instead of the dry porous media considered by Florence et al. (2007). As a bonus, Eq. 29 reveals that the tortuosity τh of the preferential hydraulic flow paths in the porous medium is given by, expressed in the consisted SI units

where R g is 8314 J/kmol-K and T is in K. Equation 35 may be used in determining the tortuosity of porous media. For example, the tortuosity is estimated to be τh = 1.0 for the sandstone sample used in the tests assuming a temperature of 298 K, according to Florence et al. (2007). However, Florence et al. (2007)mention that the actual temperaturewas unknown for the data involved in their studies and they simply assumed the value of 298K for their calculations.

5 Conclusions

The present approach rigorously accounts for the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor. Improved formulations presented in this study have been proven to be instrumental in accurate correlation of experimental data for effective description of gas flow in tight porous media. This has been demonstrated by correlating the available data more accurately than the previous attempts.

中文译文

致密多孔介质中气体视渗透率的有效联系

Faruk Civan

摘要：通过致密多孔介质的气体流动状态是被标准的哈根泊肃叶典型方程的严格应用所描述。合适的实现是在基于多孔介质中最好的流动路线实现的条件下作为一束弯曲的毛细管完成的。在这里呈现的改进之后的公式和方法论是被展示来提供准确而有意义的数据之间的关系，就多孔介质特有参数的影响而言，这些参数包括绝对渗透率、孔隙率、关于气体视渗透率的弯曲度、膨胀率和克林伯格气体滑动系数。

关键词：致密多孔介质、气体视渗透率、膨胀系数、克林伯格气体滑动系数、弯曲度

1引言

对通过致密多孔介质的不同流动气体状态的描述已经引起了人们高度的重视，这是因为传统的达西定律除了不能实际的描述粘性流动状态，还不能描述多种其它相关的流动状态。例如，Javadpour已经断定了页岩中的气体流动偏离了传统的菲克和达西定律所描述的行径。因此，已经尝试了许多方法来描述在不同状态下通过致密介质的气体转移。当从例如页岩气和煤层气储藏所等不常规的气体储藏所中抽取出碳氢化合物气体时，这些尝试是极具实际价值的。

Skjetne和Gudmundsson，Skjetne和Auriault曾经根据纳威尔斯托克斯方程理论性的研究过多孔介质中的壁面滑移气体流动现象，但并没有为克林伯格效应提供任何的联系。Wu也曾经从实验室为克林伯格系数开发了一些列解析程序并且测试良好，但仍然没有提供出任何的与其相关的联系。在评估过各方面可利用的关系后，Sampath和Keighin在水存在的情况下为多孔介质中的氮气克林伯格系数提供了一个改良的关系，其关系如下：

在这里b k的单位是Pa，K∞的单位是m2，Ф是分数。这个关系的重要性是因为它的指数非常靠近从本文中获得的-0.50的指数值。

Beskok和Karniadakis研究出了一个包含致密多孔介质中的基本流动状态的标准哈根泊肃叶方程，这些流动状态包括连续的液体流动、滑动流动、过渡流动和自由分子流动状态。有能力在一个独立的方程中表现出四种流动状态，这是一个非常杰出的成就。然而，这种不规格的可利用的膨胀系数数据的经验主义关系是一个复杂的数学三角函数。正如在本文中展现的一样，同一数据的许多准确关系不能用一个简单的逆幂律函数来完成。

Florence尝试用利用Beskok和Karniadakis提出的哈根泊肃叶典型方程为致密多孔介质的视渗透率得到一个通用的表达并且通过实验室数据让一些基本的参数联系起来，这些参数包括克林伯格滑动系数和不计弯曲的克努森数。因此，虽然他们所有人的方法论是合理的，但当使用哈根泊肃叶典型方程的应用时，他们的公式和数据分析程序正如本文所指出的一样仍然需要重要的改进。他们的实验忽略了许多重要问题。Beskok和Karniadakis

提出的哈根泊肃叶典型方程已经由一个单头管流得到。当这束在致密多孔介质中更优流动路线的弯管被考虑进时，这个数字和在多孔介质中形成的更优流动路线应该被考虑为重要的参数。

其次，被Florence采取的为克林伯格滑动系数关系的方法是不正确的并且它们的关系不能代表所有范围内的气体分子质量（通常称为重量）。这些错误在本文中被一种严格的方法所纠正，这种方法用一个等于1.0的回归系数为他们的数据引出了一种准确的关系。另外，通过致密多孔介质的弯曲能利用从常规气体测试中得到的流动数据，现在的分析被应用到一种实际方法中。考虑到作者的知识，这种方法现在没有存在于文献中。

因此，本文最初的目标简化为三个：

·膨胀系数的相互关系

·视渗透率方程的推导

·克林伯格气体滑动系数的相互关系

这些问题将在接下来的部分通过理论方法和严格的分析实验数据得到解决和证实。

2膨胀率的相互关系

Beskok和Karniadakis通过一个单头管为测定气体流量得到了一个标准的哈根泊肃叶典型方程，方程如下：

流量条件函数f(Kn)为：

克努森数为：

这里Rh和Lh表示水力半径和流量管长度，λ表示平均自由径，如下所示：

这里p表示绝对压强，单位是Pa；T是绝对温度，单位是K；M表示分子质量，单位是kg/kmol；R g = 8314 J/kmol/K是普通气体常数；μ是气体粘度，单位是Pa.s。

根据Schaaf和Chambre的流动状态分级，方程2描述了基本的流动状态，也就是连续液体流态（Kn ≤ 0.001）、滑动流态（0.001 < Kn < 0.1)、过渡流态(0.1 < Kn < 10)和自由分子流态(Kn ≥ 10)。然而，Beskok和Karniadakis强调在分级中克努森数的极限是基于管流实验并且可能因其它容器的几何形状而不同。

出现在方程3中的参数α是一个在0 < α< α0over 0 ≤ Kn< ∞变化的无量纲膨胀系数。Beskok和Karniadakis提供了一个如下的经验主义关系：

这里α1 = 4.0, α2 = 0.4,α0是α如Kn → ∞（代表自由分子流动状态）的渐进上限值，计算如下：

这里b表示一个滑动系数。他们指出在滑移流动状态下α=0，b = ?1。因此，方程7衍化为：

图片1：现在的方法用方程9将Loyalka和Hamoodi用理论分析推理的α0=1.358上限值和Tison和Tilford使用的校正过的α0=1.205的上限值联系起来。现在的方法用了非常靠近1.0的回归系数而给出了精确符合的数据。

方程6的表达式是一个复杂的数学问题。在接下来的演练中，一个如下的为Beskok和Karniadakis分析的数据提供了一个更准确和实际得多的选择到方程6中的逆幂律得到了证明：

这里A和B是经验主义常数。值得注意的是方程9给出了0 < α< α0over 0 ≤ Kn< ∞的限制条件。实际上，它可以表示为：

正如在图片1中所说明的一样，现在的方法用方程9将Loyalka和Hamoodi 用理论分析推理的α0=1.358上限值和Tison和Tilford使用的校正过的α0=1.205的上限值联系起来。因此，Loyalka和Hamoodi的数据被联系为：

因此，A=0.1780和B=0.4348。另一方面，Tison和Tilford被联系为：

因此，A=0.199和B=0.365。

图片2：Loyalka和Hamoodi的数据能同时被使用一个回归系数R2=0.9871的方程9的关联方法和使用理论推理值α0=1.358的由Beskok和Karniadakis用一个回归系数R2=0.9697给出的经验主义方程所联系起来。然而，正如对比了回归系数所指出的，目前的方法得到了一个比Beskok和Karniadakis回归系数更准确的关系。

图片2显示了Loyalka和Hamoodi的数据能同时被使用一个回归系数R2=0.9871的方程9的关联方法和使用理论推理值α0=1.358的由Beskok和Karniadakis用一个回归系数R2=0.9697给出的经验主义方程所联系起来。然而，正如对比了回归系数所指出的，目前的方法得到了一个比Beskok和Karniadakis 回归系数更准确的关系。

图片3显示了目前使用调整值α0=1.205的方程9能用一个R2= 0.9486, 接近1.0的回归系数准确的代表Tison and Tilford的数据。相反，Beskok和Karniadakis 给出的使用调整值α0=1.19的经验主义方程用一个R2=0.7925，小于1的回归系数导出了更低的质量关系。通过对比回归系数发现，目前的方法得到了一个比Beskok 和Karniadakis回归系数更准确的关系。

总结得出：目前的逆幂律表达式通过在两种情况下靠近1.0的回归系数得到了更准确的无量纲膨胀系数关系，因此比Beskok和Karniadakis经典主义方程更合适。

3视渗透率方程的推导

Beskok和Karniadakis统一了可以得到通过单管流量q h的哈根泊肃叶典型方程，通过一束弯曲的流动路径这种单管能被用来测定气体流量：

图片3：目前使用调整值α0 =1.205的方程9能用一个R2 = 0.9486, 接近1.0的回归系数准确的代表Tison and Tilford的数据。相反，Beskok和Karniadakis给出的使用调整值α0=1.19的经验主义方程用一个R2=0.7925，小于1的回归系数导出了更低的质量关系。通过对比回归系数发现，目前的方法得到了一个比Beskok和Karniadakis回归系数更准确的关系。

这里L h表示弯曲流动路径的长度，n表示在致密多孔介质中形成的更优水力流动路径数。后者能被接下来的方程近似的计算成最近的整数值：

这里Ф表示孔隙度，A b表示正常流动方向下多孔介质的体表面积。Q表示

通过多孔介质的总体积流量。它能被一个达西典型的流量梯度定律宏观的描述，在这里流量是假定与压力梯度成正比：

这里K表示致密多孔介质的视渗透率，L是体积多孔介质的长度。

值得注意的是虽然方程16不是完全正确，但它仍然被经常应用。Civan在别处提供了方程16所需要的修改，例如压力梯度的影响，但是为了避免对目前的讨论和推导产生不必要的复杂性，在这里这些修改被忽略了。然而，虽然Civan 辩论道这样的修改通常在气态流动中是可以被忽略的，但是这种声明的有效性仍然需要详细的调查研究。

多孔介质中的更优水力流动路径的弯曲系数τh被Hence定义为：

通过联合方程14-17，接下来的表达式能被用来求气体视渗透率：

K∞表示多孔介质的液体渗透率：

方程19能被重新整理来表达水力直径：

除此之外，它还能被解释为一包多孔介质的谷粒：

这里Σg表示致密多孔介质中谷粒的具体表面积。Hence连同方程20和21得到了著名的卡曼科泽尼渗透率方程：

函数f (Kn)并没有出现在方程22中是因为致密多孔介质的绝对渗透率K∞仅仅是致密多孔介质的一种属性并且不依赖于流体类型和流动条件。

Flofence提出的致密多孔介质的视渗透率的构想考虑了通过单头直管的流动，因此根据上述程序的描述这种构想需要修正。

4克林伯格气体滑动系数的相互关系

在滑移流动条件下，α = 0 和b = ?1，因此方程18联合方程3可以被写为：

Florence在Kn << 1的情况下，根据方程4替代了Kn后近似的求解了这个方程：

然而克林伯格的方程是被这样给出的：

b k在这里是滑动系数。比较方程24和25得到了一个这样的表达式：

把方程5和20代入到方程26后结果为：

系数β被定义为：

图片4表明：在一个给定多孔介质中且恒温条件下（假定298K）目前的方程28为Florence涉及不同气体（氢气、氦气、氮气、空气和二氧化碳）流动的数据提供了准确的关系。这个关系如下：

表一中的结果表明根据方程28，通过方程29所得的计算值准确的与Florence 的数据相匹配。相反，使用Florence研究的关系所得的计算值，同样在表一中给

出的，严重的不同于同样的数据，这是因为他们通过一个分子质量的逆幂定律函数直接得出系数β的关系，而没有根据气体粘度分级。这个关系用国际单位表示如下：

β的单位是Pa.m，M的单位是kg/kmol。观察方程28，他们的方法不是完全正确的是因为在他们的关系中粘度效应被忽略了。

方程26-28能被用来得到一个克努森数的表达式：

例如，应用方程29-31给出的关系可以得到：

因此，

图片4：目前的方法在298K恒温条件的砂岩中，为不同气体（氢气、氦气、氮气、空气和二氧化碳）流动的Flofence的数据得到了一个[(β/μ), m/s]参数对分子质量(M, kg/kmol)的准确关系。这个几乎等于1.0的回归系数表明目前的关系方法是严谨的。

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