ARTICLES classification about pore structure of high rank coal[6]. The diameters distribution of coal has a wide range with more than 6 grades. Considering that the above-mentioned classification is full of artificial factors, this paper will discuss the fractal classification and natural classification.
Chinese Science Bulletin 2005 Vol. 50 Supp. I 66ü71
Fractal classification and natural classification of coal pore structure based on migration of coal bed methane
1
FU Xuehai, QIN Yong, ZHANG Wanhong, WEI Chongtao & ZHOU Rongfu College of Mineral Resource and Earth science, China University of Mining and Technology, Xuzhou 221008, China Correspondence should be addressed to Fu Xuehai (email: )
Abstract According to the data of 146 coal samples measured by mercury penetration, coal pores are classified into two levels of <65 nm diffusion pore and >65 nm seeping pore by fractal method based on the characteristics of diffusion, seepage of coal bed methane(CBM) and on the research results of specific pore volume and pore structure. The diffusion pores are further divided into three categories: <8 nm micropore, 8̣20 nm transitional pore, and 20̣65 nm minipore based on the relationship between increment of specific surface area and diameter of pores, while seepage pores are further divided into three categories: 65̣325 nm mesopore, 325̣1000 nm transitional pore, and >1000 nm macropore based on the abrupt change in the increment of specific pore volume. Keywords: coal pore structure, fractal classification, coal bed methane, natural classification. DOI: 10.1360/
Coal is an anisotropic porous medium; its pore-structure is closely related to the absorbability and flowability of coal bed methane (CBM)[1]. The distribution of pore diameters of coal is the foundation for the study of CBM, including its occurrence, mutual interaction among gas, water and coal matrix, and desorption, diffusion, and seepage[2]. Pores are found in coals from nm grade to mm grade. We cannot describe quantitatively the special characteristic even by electron microscopy, densimetry, mercury penetration method, and adsorption method distribution. But the introduction of fractal mathematics has offered a chance for solving such a problem. Aiming at different purposes and based on different accuracies, scholars both at home and abroad have carried out a large amount of researches on classification of pore diameters of coal. In China, the most widely used classification in coal industry is the decimal system classification proposed by Hogot[3]. The classification systems proposed by Gan[4] and by the International Theory and Applied Chemistry Association[5] are commonly seen in some international journals of coal physics and coal chemistry. In addition, Qin et al. made some research on the natural 66
Flowing characteristics of CBM
Yang et al. thought that the flow of CBM in coal obeys the Fick diffusion law. With the help of digital solution of heat-conduction equation, they set up a diffusion model of CBM for coal fragment and proposed a concept of limit size in diffusion process[7,8]. Zhou et al. pointed out that the flow of CBM in coal basically obeys the Darcy law[9̣11]. According to the Darcy law and with the help of similarity theory, they advanced a seeping model of CBM and thought that it was feasible to use Darcy law in studying the flow mechanism of CBM. However, recently some scholars have found that, in many cases, the flow of CBM does not abide by this law. They noticed that the permeability coefficient of CBM in its flowing process changes with the diameters of the pore. This means that the diffusion and seeping of CBM is a continuous process where there exists a combined actionümulti-level diffusion and multi-level seeping. It is this combined action that determines the flow velocity of CBM. When the pore diameter is larger than the average free path of methane molecule, the seeping of methane will take place in pores. With an increase in diameter, the stable laminar flow, violent laminar flow, and turbulent flow may occur. When the diameter is smaller than the average free path of methane molecule, the methane will be diffused in pores. With a decrease in diameter, the Knudsen diffusion, surface diffusion, and solid diffusion may occur[12] (Fig. 1). 2 2.1
Fractal classification of pore radius Theoretical basis
Since the fractal concept was put forward by Mandelbort in 1975, the fractal geometry has been used to study some objects possessing similarity but no characteristic length, and has become a powerful tool for describing irregular objects. Many researches show that both pore geometry and particle geometry of materials, from atomic to crystal, have a fractal characteristic[13]. The sponge structure idea put forward by Menger can be used for simulating the characteristics of pores in coal[14]. Let a cube with a side length of R be the initial element and partition it into m small cubes with the same size. Take some of the small cubes out according to a certain rule. Then the number of the left small cubes is Nb1. By repeating this operation, the size of the left cubes becomes smaller and smaller while their number becomes larger and larger. After K times of repeated operation, the side length of the left cube is rk=R/mk, and the sum is Nbk=Nb1k
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Fig. 1. Flowing characteristic of methane in coal pores.
or
well-known Washburn equation Nbk
§R· ¨ ¸ © rk ¹
Db
R Db
C
rkDb
rkDb
Cr Db .
The overall volume is Vk
4 Nbk S rk3 . 3
(2)
Vkĝ rk3 Db . Then
2.2
(5)
In the experiment, the overall volume of pores under a given pressure equals the volume of mercury injected into the pores, expressed as dVk = dVP(r). Differentiating both sides of formula (5), we have Then Dr = [r/P(r)]dP(r).
Samples and mercury penetration
Fractal calculation
In the test process, the mercury can only be pressed into the micro-fractures. Only under a high pressure, can the mercury be pressed into pores in coal. In order to overcome the internal surface tension between mercury and solid, a pressure P(r) must be applied before the pore (with a diameter r) is filled by mercury. For a cylinder-shaped pore, the relation between P(r) and r meet the Chinese Science Bulletin
or 1/P(r) = r/7500.
P(r)dr+rdP(r)=0, (3)
Samples were collected from 146 coal mines in China; they are mainly the C-P coals and a few Jurassic ones. The vitrinite reflectance Ro,max is between 0.43% and 8.61%. Coal ranks are from lignite to anthracite, and genetic type is humic coal. The mercury penetration device of 9310 type made by Micromeritics Instrument Co. (USA) was used, whose highest measurable pressure is 206 MPa and the lower limit of measurable diameter is 7.2 nm. In order to eliminate the influence of micropores, mineral lingot pores, and corrosion pores on the test result, we concentrate on pores of 7.2̣5000 nm, which are mainly the pores evolved from maceral, such as retained structure pores of plants, intercrystalline pores, intergranular pores, gas pores, and so on. 2.3
(4)
where P(r) is the pressure applied, MPa; r is the radius of pores, nm; G is the surface tension of mercury, N/cm2; T is the contact angle of mercury with the solid surface (T = 140°). Solving formula (4), we have P(r)× r=7500
In formula (1), Db = log(Nb1)/log(m) is called the number of volumnal fractal dimension of pores. From formula (2) we have the volume of coal pores
dVk ĝ rk2 Db . dr
(2G cos T ) / r ,
P(r )
(1)
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(6)
Substituting (6) into (3), we have dVP ( r ) / dP (r ) v [r / P(r )]r 2 Db ;
(7)
substituting (5) into (7), we have dVP ( r ) / dP (r ) v r 2 r 2 Db v r 4 Db .
(8)
via a logarithm operation to the two sides of formula(8), we have log[dVP(r)/dP(r)]ĝ(4Db)logrĝ(Db4)logP(r). (9) Based on graphical construction according to log[dVP(r) / dP(r)] and logP(r) (dVP(r) is the volume increment of pores corresponding to the increment of pressure dP(r)), we can get the slope k (Fig. 2), which is Db4 = k, and so the voluminal fractal dimension is Db = 4 + k. 2.4
(10)
Fractal classification
In Fig. 2, there are obviously two straight lines for logP(r) in the interval of 0.9̣1.3 (corresponding to the radius of pores of 54̣85 nm), which means the radius in 67
ARTICLES
Fig. 2. Relations between log[dVP(r) /dP(r)] and logP(r). The diameter and average free path of molecules of CBM[16] (20ć, 0.101325 MPa) Gas molecules CH4 CO2 N2 Molecule diameter/nm 0.325 0.33 0.35 Average free path/nm 53 83.9 74.6
Table 1
this interval has discontinuity points. These discontinuity points are distributed in stages (Fig. 3). They are 54, 66, and 85 nm or so, averaged by 65 nm, which is in good accordance with the result by Zhao et al.[15]. 3
3.1
Fig. 3. Relations between discontinuity points and Ro.max%.
Coincidently, the radius of coal pores leading to fractal discontinuity points are approximately in accordance with the average free path of molecules of CBM (Table 1). Based on the characteristics of diffusion and seepage as well as average path of molecules of CBM, the pores are divided into seeping pore (>65 nm) and diffusion pore (<65 nm). 68
Natural classification of coal pores Diffusion pores
The average special pore volume of pores whose radius are smaller than <65 nm accounts for only 2.98 % of the total volume of tested pores. However, their special surface area accounts for 86.2% (Table 2). In the research on distribution of pore diameter, the relation between increment of special surface area and pore diameters should be investigated. The result shows that the ratio (dS or dS/dD) of increment of special surface area and diameter of all samples increases periodically with the decrease in pore diameter, indicating that the diameter structure of pores is distributed periodically. Generally, the relation of dS/dD and D of most samples has two abrupt points and there exists a certain difference between different ranks of coal (Table 3). All the samples have one thing in common; that is, the special surface area decreases for those pores whose radius is 8 and 20 nm (Fig. 4). 8 and 20 nm one seventh and one third, respectively, of the average free path of CBM. So, the diffusion pores can be classified into micropore (<8 nm), transitional pore (8̣20 nm), and mini-pore
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<8 nm V 0.86 Table 3 Ro/% >3.5
S 48.10
Table 2 Percentages of specific volume (V), specific surface area (S) of different pore radii % Diffusion pores Seeping pores 20̣65 nm 20̣65 nm 65̣325 nm 325̣1000 nm V S V S V S V S 0.46 20.20 1.66 17.76 21.05 12.80 24.74 0.99
Average radius of abrupt points of different ranks of coal Average radius of Average radius of Sample abrupt points for abrupt points for number specific surface specific volume/nm area/nm 38 8 24 306 1000
2.0̣3.5
34
7
20
312
1000
1.0̣2.0
32
7
17
350
986
<1.0
42
8
20
326
978
(20̣65 nm). The micro-pores are dominated by surface diffusion, while the mini-pores are mainly by Knudsenrt diffusion. 3.3
Seeping pores
The average special pore volume of >65 nm pores accounts for 97.02% of the total, while their special surface area accounts for only 13.9% (Table 2). These pores are one of the main channels for the seepage of CBM. In the research on diameter distribution of pores, the correlation between diameter and inrement of special pore volume should be investigated. The analysis shows that the special pore volume increment (dV or dV/dD) of all samples decreases periodically with the increasing radius, indicating
>1000 nm V S 51.23 0.15
that the radius structure of pores is distributed periodically. Generally, the relation of dS/dD and D of most samples has two abrupt points and there exists certain difference between different ranks of coal (Table 3). All the samples have one thing in common; that is, the special pore volume of pores, whose radii are 325 nm and 1000 nm, has a bench-like decrease (Fig. 5). Because 325 and 1000 nm are respectively 5 and 18 times of the average free path of CBM, the seeping pores can be divided into mesopore (65̣325 nm), transitional pore (325̣1000 nm), and macro-pore (>1000 nm) and the flowing of CBM can be divided correspondingly into stable laminar flow, violent laminar flow, and turbulent flow. 3.4
Classification of radius structure
Based on the above-mentioned fractal characteristics, the regularity of pore radius leading to abrupt points of special surface increment of diffusion pores and special pore volume increment of seeping pores, and the relation between radius and diffusion and seepage of CBM, the coal pores are divided into two levels and 6 categories (Table 4).
Fig. 4. Increment distribution curves of specific surface area of diffusion pores.
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Fig. 5. Distribution curves of special pore volume increment of seeping pores.
Table 4 shows that, for those micro-pores (<8 nm), CBM is characterized by surface diffusion, for the transitional pores (8̣20 nm) it is characterized by mixed diffusion (surface diffusion and Kundsen diffusion), and for the mini-pores (20̣65), it is dominated by Kundsen diffusion . In addition, for those mesopores (65̣325 nm) the flowing of CBM obeys the Darcy law and hence it appears as a stable laminar flow, for those transitional pores (325̣1000 nm), it appears as a violent laminar flow, and for the macro-pores (>1000 nm), it appears as a turbulent flow. Table 4 Classification of coal pores based on migration characteristics Level diffusion
seepage
4
Classification
Radius/nm
Flow characteristic
Micro-pore
˘8
Surface diffusion
Transitional pore
8~20
Mixed diffusion
Mini-pore
20~65
Kundsen diffusion
mesoporee
65~325
Stable laminar flow
Transitional pore
325~1000
Violent laminar flow
Macro-pore
˚1000
Turbulent flow
Conclusion
The mercury penetration can only measure those pores whose diameters are greater than 7.2 nm. The mm-level pores measured under a low pressure include most micro-fractures. If these fractures are simplified into cylin70
drical pores and then calculated by Washburn equation, it will lead to a big error. Pores smaller than 7.2 nm occupy a large percentage, forming a adsorption space for CBM. The diameter distribution of these pores has a certain regularity (especially the molecule structured pores). So pores smaller than 7.2 nm do not possess the fractal feature, indicating the existence of a lower limit of observation measurement in fractal researches. Based on the fractal study of pores from 7.2 to 5000 nm and on the diffusion and seeping characteristic of CBM, the pores are divided by 65 nm into diffusion pore (<65 nm) and seeping pore (>65 nm), of which the <65 nm pores form the adsorption and diffusion field of CBM while the >65nm pores form seeping passages of CBM. In addition, based on the relation of special surface area increment and each pore radius, the diffusion pores are divided into micro-pore (<8 nm), transitional pore (8̣20 nm), and mini-pore (>20 nm). Based on the relation of special pore volume increment and each pore radius, the seeping pores are divided into mesopore (65̣325 nm), transitional pore (325̣1000 nm), and macro-pore (>1000 nm). Acknowledgements The authors thank Peng Jinning, Hu Suohan, Li Jun, Fu Jianqiu, Chen Xinhua, Xiao Weiguo, Yao Pu, Zhao Muhua, and Wu Di. at China University of Mining and Technology for their participation in the research. This work was supported by the National Key Basic Developing Project of China (2002CB211704) and the National Natural Science Foundation of China (Grant No. 40372074).
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