Evolution of gas transport pattern with the variation of coal particle size: Kinetic model and experiments
Graphical abstract
Introduction
Gas transport through coal or shale is presently an extensively investigated issue worldwide. Appropriate mathematic description to gas transport is beneficial to solving problems like the estimation of gas production [1,2], the prevention of gas-involved disasters [[3], [4], [5]] and the enhancement of greenhouse gas storage [6]. Generally, common gas transport models can be divided into two categories: mathematical models and empirical models [[7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]] (as listed in Table 1). The derived models are usually in the form of spherical diffusion equations or first-order relaxation models which are termed the unipore Fickian diffusion model (UFDM), the bidisperse Fickian diffusion model (BFDM), the first-order relaxation model (FRM), the dual first-order relaxation model (DFRM) and the Fickian diffusion and first-order relaxation model (FDFRM).
Crank [21] was the first to solve the spherical Fickian diffusion equation which was widely known as the UFDM (Eq. 1). In this model, the porous media are assumed to be homogeneous and the concentration on the surface and in the center stays constant. Then, Nandi [19] used the model to describe gas desorption from coal and obtain the diffusion coefficient. The simplified form, termed the model, is also widely applied in the field of gas engineering. However, because of heterogeneity of media, there is a common issue that the unipore model can hardly fit well with experimental data over the entire desorption period.where Mt and M∞ are the accumulated desorption volume at time t and the total desorption volume, respectively. D is the diffusion coefficient; t is the time; a is the diffusion path length.
Then, Ruckenstein et al. [10] proposed a more complicated model, the BFDM (Eq. 2), which assumed two constant diffusion coefficients in a dual-porosity medium: One is for macropores and the other is for micropores. This model can successfully fit desorption data for a long time. In 1999, Clarkson and Bustin [13] developed this model by replacing the linear Henry law (only appropriate for the low gas pressures) with the nonlinear Langmuir isotherm law (also appropriate for the high gas pressures). However, it is impossible to fit the model manually due to the complicated forms. Numerical computer calculation has to be adopted.where λ is the contribution of desorption from macropores or in the fast stage; D1 and D2 are diffusion coefficients of gas in macropores and micropores, respectively; t is the time; a1 and a2 are the diffusion path lengths for macropores and micropores, respectively.
The FRM is a classic sorption kinetic model that can be used to describe the chemical reaction process, the surface sorption process [22] and the dynamic particle penetration process [23]. It assumes that the overall sorption is dominated by surface interaction, chemical or physical interaction. The dynamic coefficient K is positively correlated to the diffusion coefficient D.where K, the first-order dynamic coefficient, depends upon the diffusion coefficient and is inversely proportional to the square of diffusion length.
The DFRM, which shared a similar combination form with the BFDM, was used by Busch et al. [14] to describe the gas adsorption rate in a space with a limited volume. Its simple form allows it to be easily applied to fitting, compared with the infinite series form of the diffusion model.where k1 and k2 are the first-order dynamic coefficients of gas in macropores and micropores, respectively.
The FDFRM was used by Staib et al. [17] to describe Fickian diffusion and pure relaxation:
However, for empirical models, there are various equation forms. Models such as the power function, the exponential function, the logarithmic function, the Langmuir function are all simple and easy to be applied in engineering. Among them, some models have similar forms with the mathematically derived models. For example, the Airey [9] model and the Bolt [8] model are similar with the FRM, whereas the Yang [24] model and the Nie [25] model are the approximate fitting of the UFDM. These models are usually used to infer the gas volume at certain time point, which is beneficial for the assessment of gas content and disaster risk in engineering.
Here, only the mathematically derived models are discussed. The unipore model and the bidisperse model share a mathematical transformation relation which iswhere ζ is the ratio of final desorption volume from macropores and micropores, ζ = M1∞/M2∞.
Therefore, for a dual-porosity medium, the gas transport model can be expressed as the following form:where M = M1/M1∞ and N = M2/M2∞.
This equation means that the total amount of gas desorption is determined by the contribution of two pore systems. The forms of M and N, or the flow type in each system, will not change the additive relationship or the parallel relationship in Eq. 7. M and N are totally independent.
As the first type of bidisperse model proposed, BFDM only considers the influence of Fickian diffusion. Afterwards, several other types of bidisperse models in which sorption or surface diffusion began to be considered were proposed in recent decades. The comparison between Eq. 6 and Eqs. (1), (2), (3), (4) indicates that the abovementioned mathematic models basically comprise either or both of the following parts: One is the diffusion part or the unipore model, and the other is the sorption part or the first-order relaxation model, as shown in Fig. 1. Different combinations represent different processes. If the basic unipore model is assumed to be A1 (macropores) and A2 (micropores), while similarly, the basic first-order relaxation model is taken as B1 (macropores) and B2 (micropores), then five combinations of the above four items can be found in literature:
- (1)
UFDM: A1 or A2. Under this situation, the overall process is dominated by diffusion, and only one kind of pore exists in coal.
- (2)
BFDM: A1A2. Under this situation, the overall process is dominated first by macropore diffusion and then by micropore diffusion, and coal is considered to be a dual-porosity medium.
- (3)
FRM: B1 or B2. Under this situation, the overall process is dominated by sorption rate, and only one kind of pore exists in coal.
- (4)
DFRM: B1B2. Under this situation, the overall process is dominated first by macropore sorption rate and then by micropore sorption rate, and coal is considered to be a dual-porosity medium.
- (5)
FDFRM: A1B2. Under this situation, the overall process is dominated first by diffusion and then by sorption, and coal is considered to be a dual-porosity medium.
Coal is commonly considered to be a dual-porosity medium with a fracture system and a matrix system [29,30]. Differing from the abovementioned common dual-porosity structures, the structure the BFDM describes is hypothetically spherical macropore and micropore systems, as exhibited in Fig. 2. The macropore system controls the stage of rapid diffusion, whereas the micropore system controls the stage of slow diffusion. In permeability modeling, the generally used classical cubic model can be changed into other matrix shapes by considering the shape factor [[31], [32], [33]]. However, the shape of BFDM is unchangeable because of the original assumption of homogeneous spheres. If the diffusion equation and the permeability equation are linked with apparent diffusion coefficient or apparent permeability [34], it could be inferred that the BFDM is a specific case for the common dual-porosity permeability model. More shapes should be considered in the diffusion model.
Besides, the BFDM may not describe the laminar flow in fractures. Linear desorption curves are similar to the shape of Darcy's flow, but they fail to fit well with the spherical diffusion model. It may induce errors especially for the initial stage of desorption because linear facture flow generally releases first. In addition, the DFRM and the FDFRM cannot represent any coal pore structures since they are based on a surface relaxation process. They are more suitable for describing the surface sorption phenomenon. Thus, the exploration of a universal form that can describe the processes of fracture laminar flow, matrix diffusion flow and sorption in dual-porosity media is worthy of effort.
Section snippets
Basic terms for gas transport model
Commonly, there are three gas flow patterns in pore and fracture systems. In factures, gas will transport in laminar flow driven by pressure difference, whereas in the matrix, gas will transport in diffusion flow driven by concentration difference. When the transport ends, gas molecules will finally be adsorbed on the pore surface [16]. Based on the parallel relationship used in the bidisperse model, the equations for gas transport in the three patterns can be first written, and their
Experiments and results
In this section, actual experimental desorption data were fitted by the general gas transport model. Based on the standards of ISO 17246:2010 (Coal—Proximate Analysis) and MT/T752–1997 (Determine Method of Methane Adsorption Capacity in Coal), a proximate analysis and isothermal adsorption experiments were conducted to obtain compositions and adsorption ability of coal, as listed in Table 3. Then, the pore shape of sample was analyzed with the method of Liquid N2 adsorption in light of ISO
Application of SGGTM in gas transport pattern identifications
With the N2 adsorption results that the pore shape is mainly slit-shaped pores, the specific expression of Eq. 17 for the coal sample can be written as:
The fitting results of Eq. 26 are given in Table 4. All the correlation coefficients are found to be larger than 0.999, which validates correctness of the model. Besides, the contributions made by the matrix diffusion (β) and the surface sorption (1 − β − γ) increase with the decrease of
Conclusion
In previous studies, several mathematic models have been established to describe the gas transport through porous media by considering different transport mechanisms. This paper tries to summarize their internal links and obtain a general combination form that can describe different flow types in different shaped systems. The conclusions are drawn as follows:
- •
Previous used gas transport models are generally different combinations of the original unipore Fickian diffusion model and the basic
Nomenclature
- a
diffusion path length
- a1
diffusion path length of gas in macropores
- a2
diffusion path length of gas in micropores
- ac
characteristic length for different geometries
- aL, bL
Langmuir constants
- A, B, ω,ς,k
fitting coefficients
- cx
gas concentration in the region that gas is not desorbed
- D
diffusion coefficient
- D1
diffusion coefficients of gas in macropores
- D2
diffusion coefficients of gas in micropores
- Dt
dynamic diffusion coefficient
- f(t),g(t),φ(t)
functions
- K
first-order dynamic coefficient
- k1
first-order dynamic
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors are grateful to the financial support from the Beijing Municipal Natural Science Foundation (8194072), the National Science Foundation of China (Nos. 51904311, 51874314), the Fundamental Research Funds for the Central Universities (2019QY02), the State Key Laboratory Cultivation Base for Gas Geology and Gas Control (Henan Polytechnic University), China (WS2019A04). Comments by all anonymous reviewers are highly appreciated.
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