Study on the seepage characteristics of coal based on the Kozeny-Carman equation and nuclear magnetic resonance experiment

Fuel 266 (2020) 117088

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Study on the seepage characteristics of coal based on the Kozeny-Carman equation and nuclear magnetic resonance experiment

T

⁎

Zhen Liua,b, Wenyu Wanga,b, , Weimin Chenga,b, He Yanga,b, Dawei Zhaoa,b a

College of Mining and Safety Engineering, Shandong University of Science and Technology, 579 Qianwangang Rd, Huangdao District, Qingdao 266590, PR China State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, PR China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Kozeny-Carman equation Coal seam water injection Permeability Fractal dimension

It is of great significance to quantitatively describe the change in the permeability of the water-injected coal to improve the effect of the coal seam water injection technology. However, the current permeability model often assumes the pores of the porous medium are smooth, which is a large difference from the coarse coal matrix-pore interface. Therefore, a rough capillary bundle is used as the physical model to characterize the coal structure in this paper. Combined with fractal theory, a permeability model including the tortuosity fractal dimension and the specific surface area of the pores is established based on the traditional Kozeny-Carman equation, and the degree of influence of each factor on the permeability was obtained. Then, liquid permeability and structural parameters of the coal samples from the Daliuta Coal Mine and the Qincheng Coal Mine in China were obtained by nuclear magnetic resonance experiments, which verified the accuracy of the model. The research show that the tortuosity fractal dimension has the greatest influence on the theoretical permeability, and the theoretical permeability decreases rapidly when the tortuosity fractal dimension is between 1.05 and 1.20. Increasing the specific surface area of the pores will lead to an increase in the tortuosity fractal dimension and a decrease in the theoretical permeability. Under the different nuclear magnetic resonance experimental conditions, the theoretical permeability of the coal samples is consistent with the change in the liquid permeability and is closer to the measured value compared with the permeability models of Xu and Liu.

1. Introduction Coal is a porous medium material with complex variable pore structure. With the rapid development of hydraulic fracturing, coalbed methane extraction and other technologies [1–3], the study of coal permeability has become a key issue in the field of coal mining. The importance of coal permeability is reflected in all aspects of coal mining. For example, coal seam water injection technology can make water enter the fracture and pore structures of coal and increase the water content of the coal seam, thus effectively reduce the generation of coal dust during coal mining. Therefore, coal seam water injection technology has become the basic and effective dust reduction method in the fully-mechanized coal face in China [4,5,51–55]. The permeability is a key parameter that characterizes the seepage characteristics of coal. In the process of coal seam water injection, the permeability has a great influence on the migration law of the fluid in the fracture and pore structures of the coal, which in turn affects the water content of the coal

seam and the dust reduction effect of coal seam water injection [6–9]. In view of the importance of permeability in porous media materials, scholars have carried out a large number of theoretical studies on permeability. The most famous theoretical model for estimating the permeability of porous media is the Kozeny-Carman equation (KC equation), which was first proposed by Kozeny [10] in 1927 and then corrected by Carman [11]. For a long time, many scholars [12–16] have extended the permeability model for different porous media based on the KC equation and constantly revised it to improve the calculation accuracy of the model. Wong and Mettananda [17] found that the permeability reduction of porous media is only related to the number of particles deposited in the pores, and it is independent of the injection rate, concentration and particle deposition profile characteristics. Koponen [18] numerically simulated the creep flow of a Newtonian fluid in two-dimensional porous media by the lattice-gas method and found that effective porosity is the key physical parameter related to improving the accuracy of the simulated permeability. Lin and Liu [19]

⁎ Corresponding author at: Room 419, College of Mining and Safety Engineering, Shandong University of Science and Technology, 579 Qianwangang Rd, Huangdao District, Qingdao 266510, PR China. E-mail address: [email protected] (W. Wang).

https://doi.org/10.1016/j.fuel.2020.117088 Received 13 November 2019; Received in revised form 24 December 2019; Accepted 12 January 2020 0016-2361/ © 2020 Elsevier Ltd. All rights reserved.

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crack and pore distribution and the rough coal matrix-void interface have a great influence on the permeability of coal and the flow law of the fluid in coal [29]. To facilitate the theoretical study of the structural characteristics of porous media such as coal and its influence on the fluid flow laws, many scholars have separated complex porous media into various simple physical models and applied corresponding assumptions. Since the effect of coal seam water injection technology is closely related to the porosity, specific surface area and permeability of coal, the correct selection of the coal physical structure model is of great significance for studying the permeability characteristics of coal from a theoretical perspective. As shown in Fig. 2, Gates and Golf-Racht [30–32] established the matchstick model and the smooth capillary bundle model; the smooth capillary bundle model separates the pores into a number of equaldiameter circular tubes with a smooth surface and is widely used in the field of porous media seepage. The smooth capillary bundle model is generally divided into the straight capillary bundle and the curved capillary bundle, among which the curved capillary bundle accounts for the bending characteristics of the porous medium seepage channel and is more suitable for characterizing the physical structure of the coal than the straight capillary bundle. However, the contact surface between the coal matrix and the pores is very rough, and the curved capillary bundle model is essentially a smooth model; thus, this model is still not reasonable to characterize the coal structure. Fig. 3 is a schematic illustration of a rough capillary bundle model that accounts for the roughness of the pore surface inside the coal body and has a larger specific surface area of the pore than the curved capillary bundle model. Although the theoretical calculation method of the specific surface area is relatively uniform, the specific surface area of the pores calculated by the different physical structure models is quite different. The expressions of the specific surface area of the pore of the rough capillary bundle model and the curved capillary bundle model are given in formula (1) and formula (2), respectively [33].

studied the effect of water injection temperature on the permeability of a formation through experiments. The experimental results show that the permeability of the porous media with low permeability decreased with decreasing temperature. Because the microstructure of coal is complex and disordered, it is difficult to accurately describe it with traditional Euclid geometry. Studies have shown that [20–22] naturally porous media such as coal and rock exhibit self-similar fractal scales at different scales, so some scholars [23–25] introduced fractal theory into the permeability model and made important contributions to the study of coal porosity and permeability. Xu and Yu [26] derived analytical expressions of the permeability and KC constant based on fractal geometry theory and found that the KC constant is determined by the microstructure of the porous media and is a function of porosity and fractal dimension. Based on fractal theory, Liu [27] introduced the hydraulic radius into the permeability model and verified the applicability of the model by lowfield NMR experiments. At present, research on the permeability of porous media is mostly based on the proper simplification and assumption of the porous materials. However, for coal with complex interlaced fractures and pore structures, the commonly used simplified models often ignore the roughness of the coal matrix-porosity interface, and there are certain differences from the actual structure of the coal body. Therefore, there is a large error between the theoretical permeability and the actual permeability. In addition, the theoretical study on the influence of the water intrusion conditions on the permeability of coal is not sufficiently deep. Therefore, the rough capillary bundle is used as the physical model to characterize the coal structure in this paper. Combined with fractal theory, a permeability model including the tortuosity fractal dimension and the specific surface area of the pores is established based on the traditional Kozeny-Carman equation, and the degree of influence of each factor on the permeability was obtained. Then, coal samples from the Daliuta coal mine and the Qincheng coal mine in China were selected to carry out nuclear magnetic resonance experiments to verify the accuracy of the model. The research content can provide reliable parameters for the on-site application of coal seam water injection technology.

Sr = πλLt [π (λ 2)2L0] = 4Lt λL0

(1)

Sc = πλLt [π (λ 2)2Lt ] = 4 λ

(2)

where: Sr is the pore-specific surface area of the rough capillary bundle; Sc is the pore-specific surface area of the curved capillary bundle; λ is the effective diameter of the capillary; L0 is the apparent length of the capillary; Lt is the actual length of the capillary. Formulas (1) and (2) show that the flexibility changes both the internal surface area and the volume of the capillary, while the roughness mainly affects the internal surface area of the capillary, and the effect

2. Physical structure model of the coal Coal is a kind of porous medium with coexisting cracks and pores. As shown in Fig. 1, a photoelectric radiation experiment, such as scanning electron microscopy [28], can visually reveal the complex physical structure characteristics of the coal body. Both the irregular

Fig. 1. Scanning electron microscopy imaging of the coal. 2

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Matchstick model

Straight capillary bundle model

Bending capillary bundle model

Fig. 2. Physical structure model of porous media.

interface inside the capillary. Currently, the complex and disordered porous structure of coal is often studied by fractal theory created by Mandelbrot BB [34]. Fractal theory is a powerful tool for studying complex systems with fractal features in nature [35]. The related literature [36] indicates that the fractal scaling law of the roughness boundary length Lt(λ) of a capillary having a diameter of λ is as shown in formula (4).

on the volume is negligible. The rough pore surface makes the seepage channel have a large specific surface area, and the influence of the specific surface area on the permeability is not negligible. Therefore, in this paper, a more accurate permeability expression is established using the rough capillary bundle model in Fig. 3. The following assumptions are made for the rough capillary bundle model: (1) the effective diameter of each capillary is the same; (2) the inner surface of each capillary is the same; (3) the connectivity between the capillary tubes is not considered; (4) the solid surface area inside the capillary beam model is equal to the pore surface area; and (5) the effect of the roughness of the pore-solid interface on the pore volume is negligible.

Lt (λ ) = λ1 − DT L0DT

where DT is the tortuosity fractal dimension that describes the degree of curvature of the streamline. When DT = 1, Lt(λ) = L0, which means that the flow lines or channels of the fluid are straight. A larger DT value means that the fluid flow lines are more curved. Therefore, in the extreme case, DT = 2 means that the capillary is so curved that the capillary can fill the entire two-dimensional plane. If DT = 3, it means that the curved capillary can fill the entire three-dimensional space. Yu [37] derived the expression of tortuosity fractal dimension as shown in formula (5).

3. Establishment of the permeability model Permeability is a key parameter to characterize the seepage characteristics of coal, and it is also the main influencing factor of the coal seam water injection effects. The traditional Kozeny-Carman equation [10,11] is shown in formula (3), which is applied to various fields such as the seepage and is widely used as a starting point for many other permeability models.

K = ϕλ2 (32τav )

(4)

(5)

DT = 1 + ln τav ln(L0 λ ) where τav

(3)

is the analytical solution of the mean tortuosity. 2

(1 1 − ϕ − 1) + 1 4 ⎤ 2 Substituting τav = ⎡1 + 1 − ϕ 2 + 1 − ϕ (1 − 1 − ϕ ) ⎢ ⎥ ⎣ ⎦ formula (4) and formula (5) into formula (1), the effective diameter expression of the rough capillary bundle expressed by the pore specific surface area and the tortuosity fractal dimension is obtained as shown in formula (6). According to formula (6), the determination of the effective diameter λ of the rough capillary bundle model is mainly determined by the specific surface area of the pore and the roughness of the pore-solid interface.

where φ is the porosity of the porous medium, λ is the pore diameter, and τav is the tortuosity, which characterizes the degree of bending in the direction of the streamline. The main difference between the rough capillary bundle model and the smooth capillary bundle model is that the former considers the roughness of the pore-coal matrix interface according to the complex structure of the coal body, and the roughness mainly changes the specific surface area of the pore. Therefore, based on the pore-specific surface area of the rough capillary bundle model, a new permeability expression is established based on the traditional Kozeny-Carman equation. In formula (1), Lt is the actual length of the rough pore-tube wall

λ = exp[DT−1·ln(4L0DT − 1 Sr )]

(6)

As shown in formula (7), formula (6) is substituted into formula (3) to obtain the permeability expression based on the rough capillary

Rough capillary bundle model Fig. 3. Rough capillary beam model. 3

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bundle model. The permeability is a function of the porosity, the tortuosity fractal dimension, the average tortuosity and the specific surface area of the pore.

K = [ϕ (32τav )]·exp[2DT−1·ln(4L0DT − 1 Sr )]

Table 2 Normalized indicators.

(7)

By drawing the radar chart [38], the degree of influence of each parameter on the theoretical permeability of the rough capillary bundle can be intuitively reflected. According to the theoretical model of permeability, the evaluation index system for determining the influencing factors of the theoretical permeability includes the specific surface area of the pore, the tortuosity fractal dimension, the porosity, the effective diameter and the average tortuosity. The determination of the parameters of the rough capillary bundle model needs to be combined with the real structural features of the coal body. The Huoduote method [39] can classify the pore structure of coal into four types: micro pores (λ ≤ 10 nm), small pores (10 nm ≤ λ ≤ 100 nm), medium pores (100 nm ≤ λ ≤ 1000 nm) and large pores (λ ≥ 1000 nm), wherein the seepage pores are the laminar flow permeation interval of the coal, that is, λ ≥ 100 nm. According to this classification, the effective diameter of the rough capillary bundle is approximately 100–1000 nm, and the porosity is 0.06–0.15. As shown in Table 1, after determining the effective diameter and porosity, the average tortuosity value can be obtained according to the formula 2

Average tortuosity

Tortuosity fractal dimension

Specific surface area of pore/nm−1

100 200 300 400 500 600 700 800 900 1000

0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

8.7145 7.5251 6.6334 5.9402 5.3858 4.9326 4.5551 4.2359 3.9625 3.7258

1.1627 1.1600 1.1550 1.1495 1.1440 1.1386 1.1335 1.1286 1.1240 1.1195

0.3486 0.1505 0.0884 0.0594 0.0431 0.0329 0.0260 0.0212 0.0176 0.0149

Average tortuosity

Tortuosity fractal dimension

Specific surface area of pore

0.00 0.11 0.22 0.33 0.44 0.55 0.67 0.78 0.89 1.00

0.00 0.11 0.22 0.33 0.44 0.55 0.67 0.78 0.89 1.00

0.00 0.24 0.42 0.56 0.67 0.76 0.83 0.90 0.95 1.00

0.00 0.06 0.18 0.31 0.43 0.56 0.68 0.79 0.90 1.00

0.00 0.59 0.78 0.87 0.92 0.95 0.97 0.98 0.99 1.00

4. Discussion 4.1. Application of the permeability model 4.1.1. Nuclear magnetic resonance experiment and determination of the pore-specific surface area The nuclear magnetic resonance experiment can measure the T2 distribution, porosity and liquid permeability of the coal sample. T2 can reflect the pore distribution and specific surface area of the coal more reliably, and the relationship between T2 and the specific surface area is shown in formula (8). The T2 with the largest semaphore amplitude corresponds to the pores with the largest distribution proportion and has the greatest influence on the permeability. Therefore, when calculating the specific surface area of the pores using formula (8), the T2 value (T2max) corresponding to the maximum peak should be selected.

Table 1 Indicators affecting theoretical permeability. Porosity

Porosity

has a larger tortuosity fractal dimension and smaller theoretical permeability. This is because, on the one hand, the volume of the capillary does not change without changing the effective diameter, and increasing the specific surface area of the pore increases the contact area of the solid phase-gas phase interface inside the porous medium; on the other hand, when the specific surface area of the pore is constant, increasing the effective diameter of the capillary bundle will lead to an increase in the capillary volume; so in this case, the contact area of the pore-solid interface should be increased to ensure that the specific surface area of the pore does not change. Both of the above cases lead to an increase in the contact area of the solid phase-gas phase interface inside the porous medium. Increasing the contact area increases the roughness of the interface; that is, increasing the degree of bending of the interface contour and the degree of bending of the fluid flow line. This degree of bending is characterized by the tortuosity fractal dimension; that is, the more curved the fluid streamline is, the larger the tortuosity fractal dimension and the greater the flow resistance of the fluid. As shown in Fig. 7, as the tortuosity fractal dimension increases, the permeability tends to be smaller. In addition, according to Fig. 7, when 1.05 < DT < 1.20, the theoretical permeability is rapidly reduced to below 0.01 mD, which indicates that when the permeability is > 0.01 mD, the tortuosity fractal dimension has a greater influence on the theoretical permeability. When DT > 1.20, the space for the decrease in permeability is already small, so the roughness of the solid–gas phase interface has a weak influence on the permeability.

(1 1 − ϕ − 1) + 1 4 ⎤ 2 , the tortuosity τav = ⎡1 + 1 − ϕ 2 + 1 − ϕ (1 − 1 − ϕ ) ⎢ ⎥ ⎣ ⎦ fractal dimension can be obtained according to formula (5), and the specific surface area of pore is obtained according to formula (6). Since the dimensions of the indicators are inconsistent, the data in Table 1 are normalized to make them comparable. The indicators are unified into a dimensionless value between 0 and 1, and then the entropy method [40] is used to determine the weight of each indicator. The normalized processing results and the weight of each indicator are shown in Tables 2 and 3, respectively. As shown in Fig. 4, a radar chart of the factors affecting the theoretical permeability of the rough capillary bundle was plotted according to Table 3. It can be seen from Fig. 4 that there is a gap in the degree of influence of each factor on the theoretical permeability. The parameter that has the greatest influence on the permeability is the tortuosity fractal dimension, and the parameter with the least influence is the specific surface area of the pore. To further analyze the specific influence of each parameter on the permeability, as shown in Figs. 5–7 and according to formula (6) and formula (7), the relationship between the tortuosity fractal dimension, the specific surface area of the pore and the theoretical permeability is plotted. Figs. 5 and 6 show that when the effective diameter λ of the rough capillary bundle is fixed, as the specific surface area of the pore increases, the tortuosity fractal dimension gradually increases and the theoretical permeability gradually decreases; the rough capillary bundles with the same specific surface area and different effective diameter have different tortuosity fractal dimensions and theoretical permeabilities; that is, the capillary bundle model with a larger pore diameter

Effective diameter /nm

Effective diameter

1 T2 = ρ2 (S V )porosity

(8)

where T2 is the relaxation time; ρ2 is the T2 surface relaxation rate and assuming that ρ2 = 10.00 µm/s in the calculation; S is the pore surface area, cm2; V is the pore volume, cm3; and (S V ) porosity is the pore specific surface area. In this paper, coal samples from the Daliuta coal mine and the Qincheng coal mine in China were selected to carry out nuclear magnetic resonance experiments. The collected raw coal seals are sent to the laboratory for the coal sample industry analysis. The analysis results 4

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Table 3 Weights of each indicator. Index

Effective diameter

Porosity

Average tortuosity

Tortuosity fractal dimension

Specific surface area of pore

Weight

0.23

0.23

0.16

0.27

0.11

Equivalent aperture

Theoretical permeability/mD

6

0.23

Specific surface area of pore

Porosity 0.11

0.23

0.16

4

=150nm =300nm =450nm =600nm =750nm =900nm

=200nm =350nm =500nm =650nm =800nm =950nm

3 2 A stage in which the permeability decreases rapidly as the DT decreases

1 0

0.27

Tortuosity fractal dimension

=100nm =250nm =400nm =550nm =700nm =850nm =1000nm

5

1.05

Average tortuosity

1.10

1.15

1.20

1.25

1.30

1.35

1.40

DT

Fig. 4. Radar chart.

Fig. 7. Relationship curves of DT and theoretical permeability.

1.40 =100nm =700nm

1.35

=300nm =900nm

Table 4 Details of the coal samples.

=500nm

Coal sample

Mad

Proximate analysis (wt.%, daf)

1.30 Coal mine

Coal rank

1.25

#1- DLT

Daliuta

1.20

#2- QC

Qincheng

Long flame coal Anthracite

DT

Sample no.

Vdaf

FCd

6.29

9.7

28.98

55.03

1.83

17.35

15.84

64.98

Wt. = weight; daf = dry ash free; Mad = air-dried moisture; Ad = dry ash; FCd = dry fixed carbon.

1.15 1.10

and detailed information of the coal samples are shown in Table 4. The raw coal is processed into a cylinder with a radius of 25 mm and a height of 60 mm (L0 = 60 mm) and placed in a clamping device. The confining pressure is applied by injecting fluorine oil, and the pore water pressure is applied by injecting water via the water inlet, thereby changing the nuclear magnetic resonance experimental conditions. The specific experimental conditions and experimental results [27] are shown in Table 5. Since the seepage process of liquid in coal body mainly occurs in seepage pores, the porosity in Table 5 corresponds to the porosity of seepage pores in coal samples. According to the results of the nuclear magnetic resonance experiments, the pore specific surface areas of the #1-DLT and #2-QC coal samples as calculated by formula (8) were Spor#1 = 235.43 μm−1 and Spor#2 = 126.04 μm−1, respectively.

1.05 0.10

0.12

0.14

0.16

0.18

0.20

Specific surface area of pore/nm-1 Fig. 5. Relationship curves of pore-specific surface area and DT.

0.030

Theoretical permeability/mD

Ad

=100nm =300nm =500nm =700nm =900nm

0.025

=200nm =400nm =600nm =800nm =1000nm

0.020

4.1.2. Calculation and analysis of the tortuosity fractal dimension According to the parameters of porosity, pore-specific surface area and other parameters obtained by nuclear magnetic resonance experiment, the tortuosity fractal dimension of the coal samples was calculated under the different experimental conditions combined with formula (5) and formula (6); the relationship curve was drawn between the tortuosity fractal dimension and the experimental conditions, as shown in Fig. 8. It can be seen from Fig. 8 that with the change in the experimental conditions, variation rules of the tortuosity fractal dimension of the #1-DLT and #2-QC coal samples are the same. That is, when the pore water pressure is fixed, the tortuosity fractal dimension of the two coal samples increases with the increase in the confining

0.015

0.010

0.005 0.10

0.12

0.14

0.16

0.18

Specific surface area of pore/nm-1

0.20

Fig. 6. Relationship curves of pore-specific surface area and theoretical permeability.

5

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0.0966 0.0909 0.0905 0.0921 0.0999 0.79341 0.79341 0.79341 0.79341 0.79341

0.74742 0.46804 0.23328 0.49727 0.64621

pressure, which means that increasing the confining pressure will compress the coal body skeleton. The literature [27] indicates that the confining pressure mainly compresses the large pores and the mesopores, thereby increasing the proportion of the micropores and transition pores, which will lead to a more complex coal structure and a more curved fluid flow line. When the same confining pressure is applied, the tortuosity fractal dimension decreases with increasing pore water pressure. This is because the pore water pressure has the function of expanding the pore volume and connecting the pore space, which can reduce the complexity of the coal structure and thereby reduce the bending degree of the fluid flow line. 4.1.3. Calculation and analysis of the theoretical permeability According to the structural parameters of the coal samples obtained by the nuclear magnetic resonance experiment, the tortuosity fractal dimension and theoretical permeability of the #1-DLT and #2-QC coal samples were calculated according to formula (5) and formula (7). The results are shown in Table 6. To more intuitively compare and analyze the theoretical permeability and liquid permeability of the coal samples, the relationship between the permeability and the tortuosity fractal dimension is shown in Fig. 9, and the curve of the permeability under the different experimental conditions is plotted in Fig. 10. It can be seen from Fig. 9 that with the increase in the tortuosity fractal dimension, the liquid permeability and theoretical permeability of the #1-DLT and #2-QC coal samples generally show a decreasing trend. The tortuosity fractal dimension of the coal samples is between 1.10 and 1.14, and the variation trend of the permeability with DT is consistent with the curve of 1.10 < DT < 1.14 in Fig. 7. It can be seen from Fig. 10 that the law of variation with the experimental conditions of the theoretical permeability and the liquid permeability are consistent; that is, when the confining pressure is constant, the permeability increases with the increase in the water pressure, and conversely, when the water pressure is constant, the permeability decreases as the confining pressure increases. This is because, as shown in Fig. 8, under the experimental conditions of low water pressure and high confining pressure, the tortuosity fractal dimension of the coal samples is large. The tortuosity fractal dimension reflects the degree of bending of the fluid flow line. The more curved the flow line in, the greater the flow resistance of the fluid and the smaller the permeability of the coal. In addition, the theoretical permeability is consistent with the change in the liquid permeability, indicating that formula (7) can better predict the change in the coal permeability under the different water injection pressure and confining pressure conditions.

0.00367 0.00204 0.00130 0.00220 0.00226 0.1125 0.1100 0.1047 0.1154 0.1253

#2-QC

water water water water water

pressure pressure pressure pressure pressure

8 – confining pressure 12 8 – confining pressure 16 8 – confining pressure 20 10 – confining pressure 16 13 – confining pressure 16

φ Liquid permeability /mD φ

Coal sample

Test condition /MPa

T2max/ms

Liquid permeability /mD

Z. Liu, et al.

0.42476 0.42476 0.42476 0.42476 0.42476 water water water water water

pressure pressure pressure pressure pressure

8 – confining pressure 12 8 – confining pressure 16 8 – confining pressure 20 10 – confining pressure 16 13 – confining pressure 16

To further verify the accuracy of the permeability model, the data obtained from the nuclear magnetic resonance experiments of the #1DLT and #2-QC coal samples were substituted into the theoretical models of permeability as established by Xu (formula (9)) [26] and Liu (formula (10)) [27], respectively. The theoretical permeability of each model is compared with the liquid permeability, and the comparison curves are shown in Fig. 11. It can be seen from Fig. 11 that with the change in experimental conditions, the theoretical permeability of each model has the same change law as the liquid permeability, indicating that each permeability model can be used to predict the change in the coal permeability under different water injection pressures and confining pressure conditions. However, the theoretical permeability calculated by formula (7) is the closest to the liquid permeability, indicating that in the above three permeability models, the permeability model based on the rough capillary bundle (formula (7)) is the most suitable for estimating the value of the permeability under the different water injection conditions. This is because the permeability model of Xu is based on a uniform porous medium model, which is characterized by obtaining a KC constant expression without empirical constants, and thus can accurately calculate the permeability of a uniform porous

#1-DLT

T2max/ms Test condition /MPa Coal sample

Table 5 Specific experimental conditions and experimental results [27].

4.2. Comparison of the permeability models

6

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1.145

Water pressure 8MPa

1.145

Confining pressure 16MPa

1.140

1.140

1.135

#2-QC

1.130

1.130

1.125

DT

DT

1.135

1.125 1.120 1.120

1.115

#1-DLT

1.115

1.110 1.105 14

1.110 12

14

16

18

8

20

Confining pressure/MPa

10

12

Water pressure/MPa

Fig. 8. Relationship curves of the tortuosity fractal dimension and experimental conditions. Table 6 Results of the theoretical permeability. τav

DT

Spor/μm−1

k/mD

8 – confining pressure 12 8 – confining pressure 16 8 – confining pressure 20 10 – confining pressure 16 13 – confining pressure 16

4.832 4.933 5.162 4.721 4.380

1.1167 1.1184 1.1222 1.1147 1.1086

235.43 235.43 235.43 235.43 235.43

0.004968 0.004959 0.004940 0.004979 0.005016

8 – confining pressure 12 8 – confining pressure 16 8 – confining pressure 20 10 – confining pressure 16 13 – confining pressure 16

5.561 5.885 5.910 5.814 5.391

1.1347 1.1398 1.1402 1.1387 1.1320

126.04 126.04 126.04 126.04 126.04

0.017132 0.017060 0.017055 0.017075 0.017174

Coal sample

Test condition /MPa

#1-DLT

water water water water water

pressure pressure pressure pressure pressure

#2-QC

water water water water water

pressure pressure pressure pressure pressure

0.0052

1.0

#1-DLT Theoretical permeability Liquid permeability

0.0051

#2-QC Theoretical permeability Liquid permeability

0.8 0.7

0.0050

Permeability/mD

Permeability/mD

0.9

0.0049 0.0048 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 1.1075 1.1100 1.1125 1.1150 1.1175 1.1200 1.1225

0.6 0.5 0.4 0.3 0.2 0.01725 0.01720 0.01715 0.01710 0.01705 0.01700 1.1300

1.1325

1.1350

1.1375

1.1400

DT

DT

Fig. 9. Relationship curves between the permeability and the tortuosity fractal dimensions.

the actual coal structure parameters of the coal sample, a more accurate theoretical permeability can be obtained.

medium. Liu further improved the permeability model by introducing the hydraulic radius theory. However, the coal body is not a uniform porous medium, and its internal pore and fracture structure is complicated and staggered. The coal matrix-porosity interface is very rough. From Fig. 4, it can be seen that the average tortuosity, the tortuosity fractal dimension and the specific surface area of the pore all have a nonnegligible effect on the permeability. Formula (7) fully considers the roughness of the pore surface of the coal body, and the rough capillary bundle is used as the physical structure model of the coal. Combined with the nuclear magnetic resonance experiment to obtain

2 K = (πDf )(1 − DT ) 2 [4(2 − Df ) ϕ (1 − ϕ)](1 + DT ) 2 λ max [128(3 + DT − Df )]

(9) 2 K = (πDf )(1 − DT ) 2 [4(2 − Df ) ϕ (1 − ϕ)](1 + DT ) 2 rmax [128(3 + DT − Df )]

(10)

7

Fuel 266 (2020) 117088

Water pressure 8MPa

0.01725

1.2

0.01710

Theoretical permeability

0.8 0.6

0.01705

0.4

Liquid permeability

0.2

0.01700 0.004975 Theoretical permeability

0.002

0.004950

Theoretical permeability/mD

1.0

Liquid permeability/mD

Theoretical permeability/mD

1.4

#2-QC

0.01715

Liquid permeability

14

16

18

0.7

0.01715

0.6

Liquid permeability

0.01710 Theoretical permeability

0.5

0.01705 0.4 Theoretical permeability

0.00500

0.0026 0.0024 0.0022

0.00495

Liquid permeability

0.0020

#1-DLT

0.000 12

Confining pressure 8MPa

0.01720

#1-DLT 0.004925

#2-QC

Liquid permeability/mD

Z. Liu, et al.

0.0018

0.00490

20

8

9

10

11

Confining pressure/MPa

Water pressure/MPa

(a)

(b)

12

13

Fig. 10. Curves of the permeability under different experimental conditions.

5. Conclusions

contact area of the solid–gas phase interface. Thereby increasing the roughness of the interface; that is, increasing the bending degree of the fluid flow line. The bending degree increases the tortuosity fractal dimension and reduces the theoretical permeability. (2) The tortuosity fractal dimensions of the #1-DLT and #2-QC coal samples are between 1.1086–1.1222 and 1.1320–1.1402, respectively. The tortuosity fractal dimension of the coal samples increases with increasing confining pressure and decreases with increasing pore water pressure. This shows that increasing the confining pressure will compress the coal body skeleton, resulting in a more complicated coal structure and a more curved fluid flow line. The pore water pressure has the function of expanding the pore space and connecting the pores, which can reduce the complexity of the coal samples. (3) Under the different nuclear magnetic resonance experimental conditions, the theoretical permeability of the #1-DLT and #2-QC coal samples is consistent with the change in the liquid permeability, so the permeability model in this paper can be used to predict the coal permeability under different water injection conditions. In addition, compared with the permeability models of Xu and Liu, the model built in this paper uses the rough capillary bundle as the physical structure model of the coal. It fully considers the roughness of the pore surface of the coal body and obtains a more accurate

(1) The radar chart shows that among the structural parameters of the coal body, the tortuosity fractal dimension has the greatest influence on the theoretical permeability; when 1.05 < DT < 1.20, the theoretical permeability rapidly decreases to below 0.01 mD; when DT > 1.20, the space for the decrease in permeability is very small; so at this time, the tortuosity fractal dimension has a weak influence on the permeability. In addition, the specific surface area of the pore also has a nonnegligible influence on the theoretical permeability. Increasing the specific surface area of the pore increases the

300

Water pressure 8MPa

Confining pressure 16MPa

250

140

350

120

300

100

250

200

200 150

Literature 27 (Liu et al.,2018)

100

100 50 0.005

50 0.005 0.004

Equation (7)

0.004

Liquid permeability

0.003

0.003

0.001

0.001

0.000 12

14

16

18

20

Confining pressure/MPa

8

10

100 Literature 26 (Xu et al.,2008)

80

Literature 27 (Liu et al.,2018)

60

40

40

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4 Liquid permeability

0.3

0.002

0.002

120

60

Permeability/mD

Literature 26 (Xu et al.,2008)

150

140

Confining pressure 16MPa

Water pressure 8MPa

80

Permeability/mD

Permeability/mD

400

0.2

0.3 0.2

Equation (7)

0.1

0.1

0.0

12

0.0 12

Water pressure/MPa

14

16

18

20

Confining pressure/MPa

(a) #1-DLT

8

12

14

Water pressure/MPa

(b) #2-QC

Fig. 11. Comparison curves between the liquid permeability and the theoretical permeability. 8

10

Permeability/mD

A rough capillary bundle is used as the physical model to characterize the coal structure in this paper. Combined with fractal theory, a permeability model including the tortuosity fractal dimension and the specific surface area of the pores is established based on the traditional Kozeny-Carman equation, and the degree of influence of each factor on the permeability was obtained. Then, the permeability model was applied to the coal samples of the Daliuta coal mine and Qincheng coal mine in China. The theoretical permeability of the coal samples was calculated by the structural parameters obtained from the nuclear magnetic resonance experiment and was compared with the permeability model established by Xu and Liu. The conclusions are as follows:

Fuel 266 (2020) 117088

Z. Liu, et al.

theoretical permeability under different water injection conditions.

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CRediT authorship contribution statement Zhen Liu: Methodology, Resources. Wenyu Wang: Software, Writing - original draft. Weimin Cheng: Supervision. He Yang: Conceptualization. Dawei Zhao: Writing - review & editing. 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. Acknowledgments Funding: This work was supported by National Natural Science Foundation of China [grant number 51604168, 51934004]; the Key Research and Development Plan of Shandong Province, China [grant number 2019GSF111033]; and Major Program of Shandong Province Natural Science Foundation [grant number ZR2018ZA0602]. References [1] Dong J, Cheng YP, Jin K, Zhang H, Liu QQ, Jiang JY, et al. Effects of diffusion and suction negative pressure on coalbed methane extraction and a new measure to increase the methane utilization rate. Fuel 2017;197:70–81. https://doi.org/10. 1016/j.fuel.2017.02.006. [2] Wang G, Wang R, Wu MM, Fan C, Song X. Strength criterion effect of the translator and destabilization model of gas-bearing coal seam. Int J Min Sci Technol 2019;29:327–33. https://doi.org/10.1016/j.ijmst.2018.04.006. [3] Hu XM, Xie J, Xin L, Cheng WM, Liu WT, Wang ZG. Technical application of safety and cleaner production technology by underground coal gasification in China. J Cleaner Prod 2019:119487. https://doi.org/10.1016/j.jclepro.2019.119487. [4] Han WB, Zhou G, Gao DH, Zhang ZX, Wei ZY, Wang HT, et al. Experimental analysis of the pore structure and fractal characteristics of different metamorphic coal based on mercury intrusion-nitrogen adsorption porosimetry. Powder Technol 2020;362:386–98. https://doi.org/10.1016/j.powtec.2019.11.092. [5] Si LL, Li ZH, Yang YL, Zhou J, Zhou YB, Liu Z, et al. Modeling of gas migration in water-intrusion coal seam and its inducing factors. Fuel 2017;210:398–409. https:// doi.org/10.1016/j.fuel.2017.08.100. [6] Feng RM, Chen SN, Bryant S. Investigation of anisotropic deformation and stressdependent directional permeability of coalbed methane reservoirs. Rock Mech Rock Eng 2019:1–15. https://doi.org/10.1007/s00603-019-01932-3. [7] Feng RM, Chen SN, Bryant S, Liu J. Stress-dependent permeability measurement techniques for unconventional gas reservoirs: Review, evaluation, and application. Fuel 2019;256:115987https://doi.org/10.1016/j.fuel.2019.115987. [8] Erol S, Fowler SJ, Harcouët-Menou V, Laenen B. An analytical model of porositypermeability for porous and fractured media. Transp Porous Media 2017;120(2):327–58. https://doi.org/10.1007/s11242-017-0923-z. [9] Danesh NN, Chen ZW, Aminossadati SM, Kizil MS, Pan ZJ, Connell LD. Impact of creep on the evolution of coal permeability and gas drainage performance. J Nat Gas Sci Eng 2016;33:469–82. https://doi.org/10.1016/j.jngse.2016.05.033. [10] Kozeny J. Über kapillare leitung des wassers im boden Sitzungsber. Wien, Akad. Wiss 1927;136(2a):271–306. [11] Carman PC. Flow of gases through porous media. London: Butterworths; 1956. [12] Soldi M, Guarracino L, Jougnot D. A simple hysteretic constitutive model for unsaturated flow. Transp Porous Media 2017;120(2):271–85. https://doi.org/10. 1007/s11242-017-0920-2. [13] Li HT, Wang K, Xie J, Li Y, Zhu SY. A new mathematical model to calculate sandpacked fracture conductivity. J Nat Gas Sci Eng 2016;35:567–82. https://doi.org/ 10.1016/j.jngse.2016.09.003. [14] Chen D, Pan ZJ, Ye ZH, Hou B, Wang D, Yuan L. A unified permeability and effective stress relationship for porous and fractured reservoir rocks. J Nat Gas Sci Eng 2016;29:401–12. https://doi.org/10.1016/j.jngse.2016.01.034. [15] Liu T, Lin BQ, Yang W, Liu T, Kong J, Huang ZB, et al. Dynamic diffusion-based multifield coupling model for gas drainage. J Nat Gas Sci Eng 2017;44:233–49. https://doi.org/10.1016/j.jngse.2017.04.026. [16] Shafahi M, Vafai K. Biofilm affected characteristics of porous structures. Int J Heat Mass Transf 2009;52(3–4):574–81. https://doi.org/10.1016/j.ijheatmasstransfer. 2008.07.013. [17] Wong RCK, Mettananda DCA. Permeability reduction in qishn sandstone specimens due to particle suspension injection. Transp Porous Media 2010;81(1):105–22. https://doi.org/10.1016/j.ijheatmasstransfer.2008.07.013. [18] Koponen A, Kataja M, Timonen J. Permeability and effective porosity of porous media. Phys Rev 1997;56(56):3319–25. https://doi.org/10.1103/PhysRevE.56.

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