Evaluation and Modeling of the CO2 Permeability Variation by

Gas flow behavior during the process of CBM recovery has previously been a focus of study and involves the desorption, diffusion, and seepage of gas a...
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Evaluation and Modeling of the CO2 Permeability Variation by Coupling Effective Pore Size Evolution in Anthracite Coal Junqian Li,† Dameng Liu,*,‡ Shuangfang Lu,† Yanbin Yao,‡ and Haitao Xue† †

Unconventional Oil & Gas and Renewable Energy Research Institute, China University of Petroleum, Qingdao, Shandong 266580, PR China ‡ Coal Reservoir Laboratory of National Engineering Research Center of Coalbed Methane (CBM) Development and Utilization, School of Energy Resources, China University of Geosciences, No. 29 Xueyuan Road, Haidian District, Beijing 100083, PR China ABSTRACT: A decrease in the gas pressure and coal pore size evolution resulting from the comprehensive effects of effective stress increase, coal matrix shrinkage, gas slippage, and Knudsen diffusion impacts gas permeability. This paper focuses on anthracite coal from the Qinshui basin of China to investigate pore size evolution and its impact on CO2 permeability variation. The results show that (a) the mean pore radius of the coal core varies from 7.839 to 42.946 nm under a 0.2−2.2 MPa mean gas pressure and 2.4−5.5 MPa confining stress conditions, (b) the decreases in the mean pore radius and gas pressure during pressure depletion under constant effective stress conditions cause the permeability to gradually increase due to gas slippage and Knudsen diffusion, (c) at constant gas pressures, the mean pore radius decrease induced by the effective stress increase primarily leads to a reduction in the permeability, and (d) during pressure depletion under constant confining stress (3.7 and 4.3 MPa) conditions, the permeability changes by multiple factors including the effective stress, matrix shrinkage, gas slippage, and Knudsen diffusion. It is observed that the negative effect from the effective stress is slightly stronger (obviously weaker) than the positive effects from the matrix shrinkage, gas slippage, and Knudsen diffusion at mean gas pressures greater (less) than approximately 0.8 MPa. Finally, an empirical model was established predicting the CO2 permeability change by coupling pore size evolution with the permeability change during gas pressure depletion under constant confining stress conditions. This model comprehensively considers the four effects of effective stress, matrix shrinkage, gas slippage, and Knudsen diffusion and shows consistency with the experimental data.

1. INTRODUCTION

permeability of coals is generally less than the CH4 permeability under the same experimental conditions.12−14 As an unconventional gas-bearing reservoir, the absolute permeability of coal varies during the CBM production process, which plays a significant role on both primary and enhanced CBM recoveries.15 To date, it has been well accepted that the permeability changes as the effective stress (i.e., the stress acting on the coal framework and is approximately equal to the confining stress minus fluid pressure16,17) increases and the matrix shrinkage occurs during pressure depletion. The permeability is extremely sensitive to the effective stress and exponentially decreases with the increasing effective stress.2,4,10,18 The effect from the adsorption/desorptioninduced matrix swelling/shrinkage on the permeability is much more complicated and has been investigated experimentally by many researchers.6,19−24 In recent decades, a great number of analytical models have been developed to predict the coal permeability change considering the stress/strain, matrix swelling/shrinkage behavior, and geomechanical effect.25 These models were established by means of linking one or two of the porosity, stress, or strain to the permeability.4,6−9,19,20,26−37 Moreover, some models were commonly established according to the simplification of the coal reservoir condition primarily including uniaxial strain with constant

Coals are heterogeneous porous media and have complicated internal pore-fracture structures controlling the physical properties (such as porosity, permeability, etc.) of coalbed methane (CBM) reservoirs. Coal permeability depends upon the interconnected pore fractures within coals1 and is primarily determined by natural fracture attributes, including number, size, spacing, connectedness, aperture, degree of mineral infill, and patterns of orientation.2,3 On the basis of the impact of pore fractures on the permeability, coal was idealized as a conceptual dual-porosity structure including a matrix block and cleat/fracture system and was generally modeled by the collections of sheets, cubes, and matchsticks, of which the last was commonly utilized for coal permeability modeling.4−9 In these conceptual models, abundant pores are considered to develop in the matrix block and act as the main storage space for CBM, whereas the cleat/fracture system serves as the primary pathway for fluid transport.10 The gas flow occurring in the coal matrix is mainly through diffusion from micropores to the larger pores driven by the concentration difference and is commonly modeled by Fick’s first Law of diffusion.5 Coal permeability to different fluid media are distinctly varied. Commonly, the coal permeability to liquid is less than the gas permeability due to the effect from the gas slippage.5,11 For the gas permeability, it is closely related to the adsorption capacity of gas molecule on the pore surface of coals and is relatively greater for a weaker absorbing gas. For instance, the CO2 © XXXX American Chemical Society

Received: November 29, 2014 Revised: January 27, 2015

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DOI: 10.1021/ef502674y Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels vertical stress,6,8,26,29 uniaxial strain with constant confining stress,7,30,33 constant volume,19,38 constant confining stress,39 and so on. Overall, the models proposed were improved gradually by the assumption of coal physical properties from isotropy to anisotropy and from constant cleat compressibility to varied cleat compressibility. Gas flow behavior during the process of CBM recovery has previously been a focus of study and involves the desorption, diffusion, and seepage of gas and the two-phase flow of the gas−liquid.40,41 Particularly at the stage of single gas flow, the gas permeability change of coals is significant to CBM production. To date, observations show that as gas pressure decreases under a constant confining stress (i.e., the stress around coals, as shown in Figure 3 by Li et al.10) condition, the effective stress increases, and matrix shrinkage and gas slippage simultaneously occur in coals, resulting in a gas (CO2) permeability change.10,39 Previously, porosity variation was usually used to predict the permeability change via the porosity−permeability relationship. However, there are few models established by directly coupling the cleat/fracture attributes with the permeability, such as cleat aperture.33 In this paper, the interconnected pore-fracture network serving for the fluid flow within coals was assumed to be a series of capillary tubes. Then, on the basis of the work by Javadpour,42 which presented a theoretical relationship between the mean pore (i.e., tube) radius and the gas permeability, an empirical model was innovatively proposed to predict the absorbing-gas (CO2, used to simulate the CBM flow through coals) permeability change by correlating the pore size (i.e., effective pore fractures for fluid flow) evolution with the permeability variation during the gas pressure depletion under constant confining stress conditions. Simultaneously, the pore size evolution and its impact on the CO2 permeability variation were investigated under different experimental conditions for an anthracite coal.

effective stress was utilized to establish the prediction model of the permeability change by coupling with the pore size evolution. 2.2. Calculation of the Mean Pore Radius. Gas slippage impacts the gas flow behavior in low permeable porous media, such as anthracite coals, and causes a gas permeability increment compared to liquid. Previous research has shown that gas slippage is associated with the mean free path of gas molecules and the mean radius of pores within porous media.11 Commonly, a linear relationship between the gas permeability and the reciprocal of the mean gas pressure is considered to be a response to the gas slippage and is mathematically expressed by the Klinkenberg equation. Under a constant effective stress condition, the slippage factor of an inert gas (such as helium) can be easily obtained by the Klinkenberg equation10,39 and is given as

⎛ b ⎞ kg = ko⎜1 + ⎟ P ⎝ m⎠

(1)

Where, kg is the gas permeability, μD; Pm is the mean gas pressure, KPa; ko is the equivalent liquid permeability obtained at an infinite mean gas pressure, μD; b is the gas slippage factor, KPa. However, for an adsorbing gas (such as carbon dioxide), the linear relationship (eq 1) cannot be directly used to evaluate the gas slippage resulting from the adsorption between the gas and porous media.5 In this paper, due to the CO2 adsorption to the internal surface of the coal, gas slippage and matrix shrinkage simultaneously occur in the coal core as the gas pressure decreases under a constant effective stress condition. On the basis of this, the linear relationship between the permeability and the reciprocal of the mean gas pressure was defined as a new Klinkenberg-like equation to describe the CO2 flow and is given as

⎛ b ⎞ kg = k∞⎜1 + c ⎟ P ⎝ m⎠

(2)

Then, we can obtain

kg = k∞bc

1 + k∞ Pm

(3)

In eqs 2 and 3, kg is the measured CO2 permeability, μD; Pm is the mean gas pressure, KPa; k∞ is the equivalent liquid permeability (i.e., the gas permeability at an infinite gas pressure where CO2 has a maximum adsorption quantity); and bc is defined as a composite factor reflecting the comprehensive effects from the gas slippage and matrix shrinkage, KPa. Another mathematical model considering the effects of gas slippage and even Knudsen diffusion43 was proposed by Javadpour42 and was successfully used to evaluate the gas permeability of shale and siltstone, within which nanopores develop.42 At the stage of the slip flow (0.001 < Kn < 0.1, Kn is the Knudsen number and is defined as the ratio of the molecular mean free path to the mean pore diameter of porous media44), the model proposed by the Javadpour42 can also be used to accurately describe the gas flow process.45 Thus, in this paper, the Javadpour equation was also attempted to describe the gas flow in the Chinese anthracite coal, within which a gas slippage phenomenon exists10,39 and is given as

2. EXPERIMENTAL SECTION 2.1. Experiment Designs. An experimental program on the CO2 permeability change and pore size evolution of coal was taken using an anthracite coal sample from the Yong’an mine located in the southeastern Qinshui basin.10 The sample was cut perpendicular to the bedding surface into a 2.54 cm (in diameter) cylindrical core with a length of 3.54 cm. The coal had a maximum vitrinite reflectance of 4.20 %Ro,max, which was obtained by a Leitz MPV-3 microscope following the Chinese Standard of GB/T 6948-1998. The permeability test was performed at the Rock Mechanics Laboratory of China University of Petroleum (Beijing, China). Core preparation, the experimental setups utilized for the CO2 permeability test, schematic diagram of the stress, and gas pressure applied to the core, and the calculation method for the permeability were detailed in previous studies.10,39 The test was carried out at an ambient temperature of 26 °C. The objective of this experimental work was to investigate the pore size evolution and CO2 permeability change under different experimental conditions and the coupling relationship between the two factors for a Chinese anthracite coal. For this purpose, the mean pore size change was first observed during the gas pressure decrease under five effective stress conditions (2.2, 2.5, 3.0, 3.5, and 4.0 MPa).10 Under each constant effective stress condition, both the confining stress and gas pressure were simultaneously increased by maintaining a positive effective stress on the core, and each effective stress condition includes five confining stress and five corresponding gas pressure conditions. Then the pore size evolution and permeability change were further investigated at the same gas pressure under decreasing effective stress conditions. Last, the measured data selected under constant confining stress conditions with decreasing gas pressure and increasing

kg =

⎛ 8RT ⎞0.5 2rμM ⎜ ⎟ 3 × 103RTρ2 ⎝ πM ⎠ ⎡ ⎞⎤ r 2 ⎛ 8πRT ⎞0.5 μ ⎛ 2 ⎟ ⎜ + ⎢1 + ⎜ − 1⎟⎥ ⎝ M ⎠ Pmr ⎝ α ⎠⎦ 8ρ ⎣

(4)

where M is the molar mass, kg/kmol; Pm is the mean gas pressure, KPa; r is the mean pore radius at the mean gas pressure of Pm, m; R is the gas constant, J/mol/K; T is the temperature, K; μ is the viscosity at the mean gas pressure of Pm and the temperature of T, Pa·s; ρ is the density at the mean gas pressure of Pm and the temperature of T, kg/ m3; and α is the tangential momentum accommodation coefficient (TMAC), fraction. Further, eq 4 can be transformed into: B

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Energy & Fuels ⎡⎛ 8πRT ⎞0.5 rμ ⎛ 2 ⎞⎤ 1 1 2 ⎟ ⎜ − 1⎟⎥ + k g = ⎢⎜ r ⎝ ⎠ ⎝ ⎠⎦ Pm 8ρ α 8ρ ⎣ M +

⎛ 8RT ⎞0.5 2μM ⎜ ⎟ r 3 2⎝ 3 × 10 RTρ πM ⎠

(5)

According to the proximity of the two models (eqs 3 and 5) at the stage of gas slip flow, a relationship between the mean pore radius (r) and the equivalent liquid permeability (k∞) can be established under constant effective stress conditions and is expressed as

⎛ 8RT ⎞0.5 2μM 1 2 ⎜ ⎟ r = k r + ∞ 3 2⎝ 8ρ 3 × 10 RTρ πM ⎠

(6)

Then, we can obtain

16μM ⎛ 8RT ⎞ ⎜ ⎟ r = 8ρk ∞ 3 × 103RTρ ⎝ πM ⎠ 0.5

r2 +

(7)

Because of the introduction of k∞ and r, both the effects from the matrix shrinkage and gas slippage on the permeability were thus considered in eq 6. Letting A = ((16μM)/(3 × 103RTρ))((8RT)/(πM)0.5 and B = 8ρk∞, the mean pore radius of coals is finally obtained as r=

B+

A2 A − 4 2

(8)

2.3. Calculation of TMAC. In eq 4, another key parameter is the TMAC, which plays an important role in determining the gas molecule amount of wall slip at the solid surface.46 It depends upon the roughness of the pore surface and gas type, temperature, pressure, among others, and generally ranges from 0 to 1,46,47 while it is sometimes greater than one,48 reflecting the retroreflection phenomenon of gas molecules on the pore surface. In addition, some experiments also revealed that the TMAC is primarily dependent on the nature of the solid surface, such as atomic roughness and its adsorptive capacity to gas molecules.46,49 In this study, a relationship can be used for estimating the TMAC and is given as

k∞bc =

⎞ ⎛ 8πRT ⎞0.5 rμ ⎛ 2 ⎜ ⎟ ⎜ − 1⎟ ⎝ M ⎠ 8ρ ⎝ α ⎠

(9)

Then, the TMAC can be obtained as 2

α= 1+

8k∞bcρ M 0.5 rμ 8πRT

(

)

(10)

3. RESULTS 3.1. Permeability Change and Pore Size Evolution. As shown in Figure 1, the permeability linearly increases with the increase in the reciprocal of the mean gas pressure under the five effective stress conditions, which directly indicates the appearance of the gas slippage phenomenon. On the basis of the Klinkenberg-like equation (eq 2), the mean pore radius and the corresponding TMAC values were calculated, respectively, at each gas pressure (or confining stress) under the five effective stress conditions by means of eqs 8 and 10, as listed in Table 1. The permeability of the coal core is extremely low and ranges from 4.339 to 20.129 μD under mean gas pressures of 0.2−2.2 MPa and confining stresses of 2.4−5.5 MPa; the mean pore radius of the coal core varies from 7.839 to 42.946 nm under the experimental conditions. In addition, the TMAC values range from 1.926 to 1.973. This may be associated with the roughness of the pore surface within coals and the affinity of coal for CO2, but requires further verification. Both the permeability and mean pore radius of the coal core are sensitive to gas pressure. Under constant effective stress

Figure 1. Relationship between CO2 permeability and the reciprocal of the mean gas pressure.

conditions, the permeability gradually increases as the gas pressure decreases, particularly at mean gas pressures less than approximately 0.8 MPa (Figure 2a). Considering the pore size evolution, the change tendencies of the mean pore radius are characterized by a decrease with decreasing gas pressure and are similar to each other under different effective stress conditions, as shown in Figure 2b. The changes in the permeability and mean pore radius were investigated during the effective stress increase process at constant gas pressures. The results showed that the permeability gradually decreases in an approximately linear manner as the effective stress increases (Figure 3a), and the mean pore radius of the coal core is also approximately negatively related to the effective stress (Figure 3b). In C

DOI: 10.1021/ef502674y Energy Fuels XXXX, XXX, XXX−XXX

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Energy & Fuels Table 1. CO2 Permeability and Corresponding Mean Pore Radius for the Corea σe (Mpa)

P1 (Mpa)

Pm (Mpa)

σc (Mpa)

kg (μD)

r (nm)

α

2.2

0.3 0.7 1.5 2.9 4.1

0.2 0.4 0.8 1.5 2.1

2.4 2.6 3.0 3.7 4.3

20.129 11.888 8.966 8.319 6.674

10.461 16.791 25.053 35.496 42.946

1.973 1.966 1.954 1.937 1.926

2.5

0.5 0.7 0.9 2.3 3.5

0.3 0.4 0.5 1.2 1.8

2.8 2.9 3.0 3.7 4.3

12.852 11.583 9.390 7.111 6.373

13.270 15.953 18.241 29.872 37.471

1.973 1.970 1.967 1.950 1.939

3.0

0.3 0.7 1.5 1.9 2.5

0.2 0.4 0.8 1.0 1.3

3.2 3.4 3.8 4.0 4.3

17.104 10.401 7.314 6.471 6.329

8.662 14.188 21.325 24.160 27.959

1.970 1.963 1.951 1.945 1.938

3.5

0.3 0.7 0.9 1.5 2.9

0.2 0.4 0.5 0.8 1.5

3.7 3.9 4.0 4.3 5.0

15.391 9.270 7.827 6.467 5.270

7.839 12.991 14.917 19.609 27.889

1.970 1.963 1.960 1.951 1.934

4.0

0.5 0.7 1.5 1.9 2.9

0.3 0.4 0.8 1.0 1.5

4.3 4.4 4.8 5.0 5.5

10.544 8.264 5.517 5.148 4.339

9.432 11.490 17.456 19.810 24.880

1.964 1.960 1.947 1.941 1.928

a σe is effective stress (MPa); σc is confining stress (MPa); P1 is inlet gas pressure (MPa); Pm is mean gas pressure (MPa), and is equal to (P1 + P2)/2, in which P2 is atmospheric pressure in our study; kg is coal permeability to gas (CO2); r is mean pore radius (nm); α is tangential momentum accommodation coefficient, fraction.

Figure 2. Relationships of the mean gas pressure with CO2 permeability (a) and the mean pore radius (b).

Figure 3. Relationships of the effective stress with CO2 permeability (a) and the mean pore radius (b).

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Energy & Fuels

Substituting eqs 11 and 12 into eq 4, an equation predicting the CO2 permeability change in the Chinese anthracite coal with a gas pressure decrease under a constant confining stress condition was obtained as

addition, the mean pore radius is much greater at a high pressure than at a low pressure for the same effective stress condition. Furthermore, the change gradients of the mean pore radius with the effective stress increase are gradually reduced as the gas pressure decreases and are 5.8943, 4.1827, and 2.9490 nm/MPa for the gas pressures of 1.5, 0.8, and 0.4 MPa, respectively. This also implies sensitivity of the permeability to effective stress decreases during gas pressure depletion. 3.2. Coupling of the Permeability Change with Pore Size Evolution. Under the constant confining stress condition used for assuming the CBM reservoir condition during the CBM production, the mean pore radius of coal also varies and is comprehensively influenced by both coal matrix shrinkage and effective stress increase associated with a decrease in gas pressure. It follows that the mean pore radius can be correlated with the gas pressure. As presented in Figure 4a, the mean pore radius was ideally considered to be positively proportional to the gas pressure. This relationship was mathematically described as

r = aPm + b

kg =

2μM(aPm + b) ⎛ 8RT ⎞0.5 ⎜ ⎟ 3 × 103RTρ2 ⎝ πM ⎠ ⎡ ⎞⎤ ⎛ 2 ⎛ 8πRT ⎞0.5 μ ⎟ + ⎢1 + ⎜ − 1⎟⎥ ⎜ ⎝ M ⎠ (aPm + b)Pm ⎝ cPm + d ⎢⎣ ⎠⎥⎦ (aPm + b)2 8ρ

(13)

The CO2 permeability change calculated by the model (eq 13) shows an excellent consistency with measured results (Figure 5).

(11)

Figure 5. Calculated results from the proposed permeability model and the experimental data.

4. DISCUSSION 4.1. Impact of Pore Size Evolution on Permeability Change. 4.1.1. Under Constant Effective Stress Conditions. The impact of gas pressure on the mean pore radius of coals is primarily induced by the matrix shrinkage and increase in effective stress during the decrease in gas pressure. Thus, under a constant effective stress condition, the change in the mean pore radius is closely related to the matrix shrinkage. It is generally considered that coal matrix shrinkage occurs in coals as the gas desorbs and results in the expansion of the seepage space for fluids.7,22,23,28,30 However, as shown in Figure 2b, the mean pore radius reduces with the decrease in gas pressure under different effective stress conditions. This phenomenon is contrary to the general understanding of the effect of matrix shrinkage on pore fractures of coal. A probable interpretation is that the matrix shrinkage does not impact the pore fractures to increase the permeability under the constant effective stress conditions. Research by Liu et al.13 proposed that gas permeability does not change during pressure depletion under constant effective stress (3 and 5 MPa) conditions by numerical modeling and concluded that the coal matrix shrinkage/swelling has no effect on coal permeability. Nevertheless, because the model in Liu et al.13 did not consider gas slippage and Knudsen diffusion, the gas permeability presented was a constant during the gas pressure decrease process. In this study, CO2 permeability gradually increases when the gas pressure decreases under constant effective stress conditions (Figure 2a). This is because the effects of gas

Figure 4. Relationships of the mean gas pressure with the mean pore radius (a) and the tangential momentum accommodation coefficient (TMAC) (b).

As mentioned above, the TMAC is impacted by multiple factors. For a given gas type, porous media and experimental temperature, the TMAC is only related to the gas pressure. As presented in Figure 4b, the TMAC is negatively related to gas pressure and can be estimated by the following equation:

α = cPm + d

(12)

In eqs 11 and 12, a, b, c, and d are, respectively, 0.0215 KPa−1 (0.0185 KPa−1), 3.6523 (4.2169 KPa−1), −2.4 × 10−5 KPa−1 (−1.9 × 10−5 KPa−1), and 1.9756 (1.9672 KPa−1) under a 3.7 MPa (4.3 MPa) confining stress condition for the coal core studied in this paper. E

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Energy & Fuels slippage and Knudsen diffusion (as Kn is of greater than 0.001, the Knudsen diffusion can not be neglected50) play significant roles in the permeability change and are closely associated with the mean pore radius of coals and the gas pressure. Usually, for the same gas type and experimental temperature, both effects on the permeability gradually become more significant as the mean pore radius of porous media and the gas pressure are reduced.11,42 Javadpour42 showed that the gas permeability compared to the Darcy permeability obviously increases due to gas slippage and Knudsen diffusion, especially when the gas pressure decreases from 10 to 0 MPa and the mean pore radius decreases from 100 to 0 nm. In this study, during pressure depletion under constant effective stress conditions, the permeability change is also comprehensively induced by the gas slippage and Knudsen diffusion and is remarkable at a mean gas pressure less than about 0.8 MPa. This indicates that these two effects on the permeability are relatively slight at gas pressures greater than about 0.8 MPa. 4.1.2. Under Constant Gas Pressure Conditions. The natural cleat/fracture network is extremely sensitive to the effective stress and will gradually close up as the effective stress increases, which results in a reduction of coal permeability.2,4,10,18 With the decrease in the mean pore radius of coals, the impact of gas slippage and the Knudsen diffusion appears in response.42 As shown in Figure 3, the changes in the mean pore radius and permeability suggest that the negative effect of the effective stress is obviously larger than that of gas slippage and Knudsen diffusion, both of which cause an increase in permeability. Because of the potential offset of gas slippage and Knudsen diffusion to the reduction in permeability, it also can be observed that the fitted relationship between the effective stress and permeability is linear10,39 even though it is still expressed as an exponential function.51,52 4.1.3. Under a Constant Confining Stress Condition. According to the mean pore radius change during the gas pressure decrease under 3.7 and 4.3 MPa constant confining stress conditions (Figure 4a), the impact of the effective stress is much stronger than that of the matrix shrinkage for the coal core. In addition, during gas pressure depletion under these two constant stress conditions, the comprehensively positive effects from matrix shrinkage, gas slippage, and Knudsen diffusion are slightly weaker than the negative effect from the effective stress increase at gas pressures larger than approximately 0.8 MPa, while the positive effects are obviously stronger than the negative effect at gas pressure less than approximately 0.8 MPa (Figure 5). It also can be seen that the negative effect from effective stress is relatively stronger under 3.7 MPa confining stress condition than that under 4.3 MPa confining stress condition (Figure 5) because a greater variation in effective stress occurs as the gas pressure decrease under a lower confining stress condition (Figure 4a). 4.2. Improved Model. Previous researches commonly considered the impacts from the effective stress and matrix shrinkage on coal permeability. In a study by Li et al.,10,39 the three effects of the effective stress, gas slippage, and matrix shrinkage on CO2 permeability were evaluated for different rank coals. Furthermore, a theoretical model was established to predict CO2 permeability under a constant confining stress condition.39 However, the Knudsen diffusion and the impact of the coal pore size evolution on gas slippage and Knudsen diffusion were ignored in that model. Moreover, the three permeability increments in this model were obtained by fitted equations, which do not have physical meanings. The model

proposed in this paper comprehensively considered the four effects from the effective stress, coal matrix shrinkage, gas slippage, and Knudsen diffusion by coupling the permeability change with the pore size evolution. First, the mean pore radius variation of coal reflects the effects of the effective stress and matrix shrinkage. Second, the changes in both the mean pore radius and the gas pressure also reflect the effects of the gas slippage and Knudsen diffusion, respectively. Furthermore, the mean pore radius of coals can be easily obtained by a function of the gas pressure. Thus, the model proposed in this paper will be practical to the permeability prediction during pressure depletion. Additionally, the model in this paper also considered the effect of temperature on gas flow. In a study by Ge et al.,45 the gas permeability slightly increased with increasing temperature. As shown in Figure 6, it also can be seen that with respect to

Figure 6. Gas (CO2) permeability changes at different experimental temperatures.

the same gas pressure, the increment of CO2 permeability increases along with the temperature. This trend is relatively greater when the mean gas pressures are larger than 0.5 MPa than at mean gas pressures less than 0.5 MPa. Overall, the impact of the temperature on gas permeability is obviously weaker than from the gas pressure, particularly at low pressures.42,45

5. CONCLUSIONS (1) This study has shown that under constant effective stress conditions, the permeability increased slightly at gas pressures greater than 0.8 MPa but significantly increased at mean gas pressures less than 0.8 MPa during pressure depletion. The mean pore radius of the coal core gradually decreased, which caused a change in permeability from the effects of gas slippage and Knudsen diffusion. (2) Under constant gas pressure conditions, the mean pore radius of the coal core decreased in an approximately linear manner. The decrease in the permeability was primarily caused by the effective stress increase and was slightly offset by the positive effects of gas slippage and Knudsen diffusion. (3) With the gas pressure decrease under constant confining stress (3.7 and 4.3 MPa) conditions, the permeability initially decreased slightly, as the gas pressures are greater than approximately 0.8 MPa but increased at lower gas pressures. During this process, the permeability was altered as a result of the comprehensive effects of the effective stress, coal matrix shrinkage, gas slippage, and Knudsen diffusion. (4) An empirical model was established predicting the permeability change by linking the mean pore radius with the permeability and is consistent with the experimental data. This model comprehensively considers the F

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Energy & Fuels

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four effects of the effective stress, coal matrix shrinkage, gas slippage, and Knudsen diffusion.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +86-10-82322322. Fax: +86-10-82326850. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was supported by the National Natural Science Foundation of China (grant nos. U1262104 and 41172134), the National Major Research Program for Science and Technology of China (grant nos. 2011ZX05034-001 and 2011ZX05062-006), the Natural Science Foundation of Shandong Province (grant no. ZR2014DP007), and the Key Program of National Natural Science Foundation (grant no. 41330313).

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DOI: 10.1021/ef502674y Energy Fuels XXXX, XXX, XXX−XXX