Improved apparent permeability models of gas flow in coal with Klinkenberg effect
Introduction
Convective flow, known as Darcy flow, is pressure-driven flow of gas through porous media. Darcy flow is the primary driver in micro-scale flow, even very small pressure gradients will cause larger flux than flux generated by very steep concentration gradients [1]. However, Knudsen diffusion is significant when the mean free path of gas molecules approaches the average diameter of pore throats in porous media [2]. The Knudsen number characterises the deviation from Darcy flow:where Kn is the Knudsen number, λ is the mean free path of gas molecules, and wpore is the width of pore throat in porous media. The phenomenon of Knudsen flow was first applied to petroleum engineering problems by Klinkenberg [3], known as Klinkenberg effect or gas slippage. A linear relationship between measured gas permeability and absolute permeability was proposed based on:where ka is the measured or apparent permeability, k0 is the intrinsic or absolute permeability of the porous medium, Pm is the mean gas pressure, and b is the so-called ‘Klinkenberg coefficient’. It is obvious that for a certain gas pressure, the value of b is significant for the difference between apparent permeability and absolute permeability. Various correlations for Klinkenberg coefficient are available in the literature. Basically, there are three forms for the determination of b, one is given as [4], [5].where c is a constant (typically taken as 0.9), μ is the kinetic viscosity of the fluid, M is the fluid molecular weight, R is the universal gas constant, and T is the absolute temperature. It can be seen that b is determined by both fluid properties (μ, M) and porous media property (wpore). Since the value of wpore is difficult to measured directly, the following two forms are extensively used for the investigation of gas permeability in rock, e.g. sandstone, gas sand, sedimentary rock and shale [6], [7], [8], [9], [10], [11].
andwhere α and β are constants measured by experiments, k∞ is Klinkenberg’s corrected permeability at a very large gas-phase pressure, and ϕ is fractional porosity. It should be noted that k∞ is equal to k0 in these studies, and porous media properties (wpore and ϕ) are considered independent of the gas properties. The value of k∞ can be calculated from the straight-line intercept on a Klinkenberg plot of apparent permeability versus the reciprocal mean gas pressure (as shown in Fig. 1).
Klinkenberg effect is significant in fine grained, low permeability porous media [11]. Compared with conventional low permeability gas reservoirs, coal reservoirs have high compressibility, and the unique swelling/shrinkage feature due to gas adsorption/desorption in coal matrix. Primary gas production (CBM recovery) triggers the reduction of gas pressure in the fracture network (coal cleat). Consequently, the lower pressure increases effective stress, and when the gas pressure drops below desorption point, coal seam gas diffuses from coal matrix to cleat and the coal matrix shrinks. The former induces the compaction of cleat, while the latter increases the cleat width [12]. Thus, the transient intrinsic permeability of coal depends on the net influence of these two dual competing mechanisms [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. However, both laboratory observations and in situ measurements are apparent permeability, which incorporates not only the two dual competing mechanisms, but also the influence of Klinkenberg effect. With the decrease of gas pressure in CBM extraction, Klinkenberg effect becomes more significant, however limited researches have been focused on this phenomenon.
Previous analyses illustrate the influence of Klinkenberg effect on apparent permeability of gas flow in rocks, but the basic assumption of Eqs. (4), (5), that the porosity of porous media is independent of the gas properties is not valid for coal. Furthermore, the parameter k∞ denotes the permeability at a very large gas pressure at which Klinkenberg effect can be neglected, but the sorption-induced-swelling of coal matrix determines that the porous media properties cannot remain unchanged with the alternation of gas pressure, even if the confining stress [19], [28], [30], [31], [32] or effective stress [33], [34], [35], [36] keeps constant. In other words, k0 is a transient parameter for coal, and the value of Klinkenberg coefficient b determined by k∞ from Eqs. (4), (5) is not the same as that determined by k0, because the cleat width differs as shown in Eq. (3). Therefore, explicit prediction of permeability changes in CBM recovery requires understanding of these complex mechanisms, include the ‘coupled processes’ between stress and sorptive chemist [12], as well as its influence on Klinkenberg effect, especially the change of Klinkenberg coefficient b.
This paper includes a brief review of the previous studies on gas permeability in coal with Klinkenberg effect. Then the apparent permeability models are developed under both constant effective stress condition and reservoir condition, respectively. The developed models incorporate a variable Klinkenberg coefficient and its influences are also discussed in this paper.
Section snippets
Review of previous research
Some excellent reviews of coal permeability, including not only model development but also field and laboratory permeability data, have been presented in recent years [12], [37], [38], [39], [40], however, the influence of Klinkenberg effect was rarely discussed in these reviews. Thus, a brief review and discussion of previous researches of Klinkenberg effect on coal permeability is presented in the following section.
Development of Klinkenberg coefficient model under constant effective stress
For solving the problems mentioned above, and accurately evaluating the role of Klinkenberg effect in permeability change under constant effective stress, a modified model based on the work of Harpalani and Chen [33] is presented in this section. By defining different width of coal cleat for helium and methane at the same gas pressure as, wHe and wMe, Eqs. (8), (9) can be rewritten as:
and
From Eqs. (17), (18), we obtain:
Two
Apparent permeability model with Klinkenberg effect for coal
Comparing with constant effective stress condition, the more widely applied assumption with respect to the coal reservoir conditions are uniaxial strain [63] and constant overburden stress [17]. Under these assumptions, a general model for representing apparent permeability model with variable Klinkenberg coefficient is developed in this section.
Discussion
Obviously, part of the apparent permeability measured from a tight gas reservoir such as coalbed attributes to Klinkenberg effect. Conventional permeability models do not consider Klinkenberg effect, however these models can predict most part of the permeability change successfully. It may be attributed to the misuse of the measured initial permeability, as it is the apparent permeability rather than absolute permeability. So when the gas pressure is relative high, the permeability change due
Conclusions
Klinkenberg effect of adsorptive gas flow in coal seam is different from conventional gas reservoir, the Klinkenberg coefficient cannot be treated as constant, due to the high compressibility and the matrix swelling of coal. Previous studies used the models from rock engineering and overlooked the unique feature of coalbed reservoir.
Based on the bundled matchstick conceptual model of coal, a modified model for isolating Klinkenberg effect from the permeability change under constant effective
Acknowledgments
The 1st author gratefully acknowledges scholarship provided by the China Scholarship Council. This work is also supported by the Natural Science Foundation of China (No. 51174212).
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