A Model for Gas Transport in Dual-Porosity Shale Rocks with Fractal

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Thermodynamics, Transport, and Fluid Mechanics

A Gas Transport Model in Dual-Porosity Shale Rocks with Fractal Structures Jinze Xu, Keliu Wu, Zhandong Li, Yi Pan, Ran Li, Jing Li, and Zhangxin Chen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00021 • Publication Date (Web): 09 Apr 2018 Downloaded from http://pubs.acs.org on April 10, 2018

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Industrial & Engineering Chemistry Research

A Gas Transport Model in Dual-Porosity Shale Rocks with Fractal Structures 1

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Jinze Xu, 1*Keliu Wu, 2Zhandong Li, 3Yi Pan, 1Ran Li, 1Jing Li, 1Zhangxin Chen

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta,

Canada T2N 1N4 2

College of Petroleum Engineering Institute, Northeast Petroleum University, Daqing,

Heilongjiang, China 163318 3

College of Petroleum Engineering, Liaoning Shihua University, Fushun, Liaoning, China

113001

*Email: [email protected] Tel: +1-(403)966-3673

Abstract A model for shale gas flow in a fractal dual-porosity rock is built and well validated with experimental data. Results indicate that: (1) the gas transport in shale natural fractures gradually owns higher priority in the dual-porosity rock with the increase of fracture porosity; (2) the ratio of minimum fracture aperture to a fracture aperture and the fractal dimension of fracture together contribute to the fracture aperture distribution. More fractures with higher aperture exist in shale rock with higher minimum fracture aperture, which benefits in the gas apparent permeability; (3) a natural fracture-dominate region enlarges with decreasing fractal dimension of fracture aperture and increasing minimum fracture aperture, and this is beneficial to the improvement of gas apparent permeability. Keywords: dual-porosity rock; fractal; gas transport; surface diffusion; Knudsen diffusion

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ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Introduction Shale gas is one of the most significant clean energy resources for human beings in the past decade1,2. Owing to the horizontal drilling and hydraulic fracturing technologies, shale gas has been exploited rapidly in North America3,4. Study of gas flow in dual-porosity rocks is one of the numerous challenges for further studies on evaluating gas transport efficiency, optimizing well production and economically developing shale gas reservoirs5-9. In shale rocks, gas can be stored in combination of shale rock matrix and fractures10-12. These nanopores commonly exhibit shapes of bubbles-like, ellipses and rectangles in a cross section12-15 and the tapered characteristic (the pore size decreases or increases along the length of the pore) in the axial direction16-19. Pore size is as low as 1 nm and as high as 100 um20-24, which can be represented by fractal model25-26. Fractures with aperture ranging from 0.01 um to 10 mm and porosity ranging from 0.001 to 0.05 are created in shale gas reservoirs due to stress variation and gas production27-30. These fractures are generally simplified to parallel planar fractures to approximately describe the gas transport behavior27,31. Previous experimental and theoretical studies indicated that the distribution of fracture apertures also yields to a power law25-27. Pores and fractures in shale rocks have been shown to be fractal by experimental studies25-27,28,31-37. Previous theoretical studies also indicated that the fractal theory is effective to characterize the gas transport behavior in fractal dual-porosity media6-9. Free gas and adsorbed gas coexist in shale gas reservoirs, where the contribution of adsorbed gas to the total gas in place ranges from 20% to 80%38-40. Previous studies on shale gas generally summarized the gas transport mechanisms in shale matrix as viscous flow, Knudsen diffusion and surface diffusion38-42. As the adsorbed gas does not exist in fractures, 2


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the gas transport mechanisms in fractures are generally summarized as viscous flow and Knudsen diffusion27,31. Adsorbed gas is thus one of the main differences between gas transport in shale matrix and gas transport in natural fractures. Riewchotisakul and Akkutlu (2016)43 studied the adsorbed gas transport based on nonequilibrium-molecular-dynamics simulations. Their studies indicated the adsorbed gas transport cannot be ignored, especially in capillaries with the size smaller than 10 nm. Kou et al. (2017) 44 studied the natural gas transport in organic-rich shale based on molecular dynamic simulation. They reported that the velocity of the cluster diffusion of the adsorbed methane is dependent on pressure drop across the capillary. Alafnan and Akkutlu (2018) 45 built matrix-fracture gas transport model and their results showed the apparent permeability with the mobile adsorbed phase is higher than that with the immobile adsorbed phase. Many models have been proposed to explore the gas transport mechanisms in shale rocks. Javadpour (2009)46 applied the Knudsen diffusion into the study of shale gas in nanopores. Ren et al. (2016) claimed that the slippage effect can be considered into the Knudsen diffusion42. Xu et al. (2018) studied the shale gas flow in fractal matrix47. Rahmanian et al. (2012) performed pore-scale simulations for tight gas transport in fractures based on the mechanisms of viscous flow and Knudsen diffusion31. Wu et al. (2015) investigated the gas transport behavior in fractures in a shale rock based on a gas model with considering viscous flow and Knudsen diffusion27. Zheng and Yu (2012) established a permeability model for gas flow in dual-porosity media and obtained good validations with experimental data6. Miao et al. (2015) employed fractal theory to study gas behavior in dual-porosity rocks and a good application is obtained7. Li et al. (2016) obtained a good history match based on a fractal 3



medium with matrix and fractures from numerical simulation8. Multi-scale gas transport studies can also be achieved by simulations. Wasaki and Akkutlu (2015)48 proposed an improved apparent-permeability model based on simulation, and their research promoted the multi-scale gas transport simulation study. However, there are some limitations for previous studies: (1) previous fractal models in dual-porosity rocks were limited for ideal gas transport as the ideal gas law is applied in these models; (2) only viscous flow is considered in most of previous models; (3) contributions of matrix and natural fracture to gas transport in dual-porosity shale rocks were not discussed; (4) how the distribution of fracture aperture affects the gas apparent permeability is in need to be studied. With emphasizing pore size and fracture aperture distributions, the model in a dual-porosity shale rock is constructed by incorporating the gas EOS (equation of state) and multiple mechanisms. The proposed model bridges pore size/fracture aperture distribution, gas properties, molecular kinetics and gas flow behaviors. Gas transport efficiency in the dual-porosity is explored and a simple way to determine the fracture porosity is also provided. Model Establishment In this study, the porous shale is assumed to be comprised of a bundle of tortuous planar fractures and pores (Figure 1). Each pore and fracture are assumed to cut through the model6-8,28,49. Free gas transport exists in both fractures and pores, and adsorbed gas transport only exists in pores15 (Figure 2). In this model, we assume the parallel flow in the dual-porosity shale rock. The geomechanical effect is not considered in this study, and will be explored in our future research. The porosity and permeability in this study are based on the

4


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reservoir condition. Details of the calculation of gas physical properties (e.g. density, viscosity…) are presented as Supporting Information.

Figure 1 Sketch of the proposed model

Figure 2 View of cross sections of pores and fractures Gas Flow in Pores Shale matrix contains a lot of pores, in which both free gas and adsorbed gas contribute to the flow. Xu et al. (2018) indicated the fractal structure of shale matrix owns a significant effect on gas flow in shale matrix47. The cumulative probability of pore size is derived as Equation (1) 47,50,51:

,

0，

(1)

5



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With considering a ratio between overall cross-sectional area from minimum pore size to maximum pore size and cross-sectional area of a rock sample, matrix porosity can be expressed by Equation (2) 47:

! "#

$%&

(2)

Xu et al. (2018) gave the expression of total conductance of different flow mechanisms as Equation (3) 47: '(

'; '@

0

3 )

) )*+ ,. /0*12

, -+

. -.-

!4 5 6 "# )*+ 789:+1

(3-a)

*+ !4 5 (?:+ 1

+

+

0

) xb ) *+ y ,. /0*12

, -+

b 1 *+ -+

. -.-

3 )

!4 5 6 "# )*+ 789:+1

4 DE 1*+ ) " . F "# A B.-C ) GH I: + 1

+ (20)

Equation (20) indicates the apparent permeability is related with many parameters (e.g. gas properties, fractal dimensions and reservoir conditions). Model Validation We validate the proposed model with the data from six experiments of shale samples. The distribution of pore size is firstly examined based on experimental observation. Alnoaimi et al. (2013)’s shale sample54 and Cretaceous shale sample from Alberta, Canada, reported by Bahadur et al. (2014)35 are used to compare with the model results. Figure 3 indicates the model results fit the experimental data well. We secondly examine the distribution of fractures with the experimental data on two Marcellus shale samples from Gale et al. (2014)28. The distribution curves of fracture apertures obtained from the model and the experiment are drawn in Figure 4. The match result shows a good application of the model to describe the fracture aperture distribution. We thirdly employ the experimental data (Ren et al. (2016)44 and Zhu et al. (2016)55) from shale core plugs to validate the total gas transport conductance ('J ). Zhu et al. (2016) performed a gas transport experiment with methane by employing a shale core plug with the length of 5 cm and the diameter of 2.5 cm. In experiments by Ren et al. (2016), the methane was selected to perform the flow test in a shale core plug with the length of 8 cm and the diameter of 4 cm. From Figure 5, we can see that the match of 'J with the experimental data of shale samples is good, which shows the potential of the application of the model. This validation also indicates the existence of parallel flow in the core plug. 9


Fp, dimensionless

Ff, dimensionless

Ff, dimensionless

Figure 3 (a) Validation of pore size distribution with Alnoaimi et al.’s experiment, c 2.10, Z[ 81.03 }p and 4.90 }p ; (b) Validation of pore size distribution with Bahadur et al.’s experiment, c 2.42, Z[ 10 p and 3.3 }p)

ct, (mol·m)/(Pa·s)

Figure 4 Validation of fracture aperture distribution with experimental data ((a): experimental data of Marcellus shale sample 1 reported by Gale et al. (2014), cM 2.5; RZ[ 3.41 pp; R 0.14 pp; (b): experimental data of Marcellus shale sample 2 reported by Gale et al. (2014), cM 2.1; RZ[ 3.57 pp; R 0.15 pp) ct, (mol·m)/(Pa·s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fp, dimensionless


Figure 5 Validation of total gas transport conductance with experimental data of shale core plug ((a): experimental data of shale core plug reported by Zhu et al. (2016), U 5 'p, 1.25 'p , n 300 w , c 2.98 , cM 2.96 , c 1.4 , cM 1.2 , 0.12 , M 0.005 , cZ[ 30 p , c 0.8 }p , RZ[ 130 p , R 3.5 p , A 1.2 , h 30° and TM 1.5; (b): experimental data of shale core plug reported by Ren et al. (2016), U 8 'p , 2 'p , n 333.15 w , c 2.96 , cM 2.93 , c 1.4 , cM 1.1 , 10


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0.11 , M 0.004 , cZ[ 45 p , c 1 }p , RZ[ 100 p , R 0.7 p, A 1.1, h 15° and TM 1.4) Results and Discussions

In this part, the transport behavior of gas in a dual-porosity shale rock is investigated and analyzed on the basis of the developed model. Modelling parameters are Table 1 Summary of modelling parameters in the results and discussions Parameter Value Approach to obtain the parameter Read by ruler U 3 'p Read the radius by ruler and calculate YJ 5 'p Read by gauge 25 q Read by thermometer n 350 w c 2.95 c Obtain from published papers6,7,50,56,57 or experiment (e.g. 1.4 cM Small-angle and ultra-small-angle neutron scattering58) 2.95 cM 1.1 0.1 Obtain from experiment (e.g. gas expansion method59) M Obtain from empirical formula (e.g. Wang et al.’s model60) 0.0025 Z[ 33 p 1 }p Obtain from experimental experiment (e.g. Small-angle R 1 p and ultra-small-angle neutron scattering58) A 1.2 h 30° TM 1.4

We firstly compare contributions of matrix and natural fracture to total gas transport in

dual-porosity shale rocks. We define the dominant transport region (matrix or fracture) as the one with the highest molar rate ratio. Two regions are thus divided based on porosities of matrix and natural fracture: (1) region 1: matrix-dominated region; (2) region 2: natural fracture-dominated region. As shown in Figure 6(a), if the matrix porosity is higher than 0.1 and the fracture porosity is lower than 0.002, the gas transport in matrix dominates the total gas transport; otherwise the gas transport in fracture dominates. Figure 6(b) further indicates that under the same matrix porosity or fracture porosity, the apparent permeability is higher in the natural fracture-dominate region; this means the natural fracture contributes more to the 11



gas transport efficiency.

Figure 6 (a) Regime of gas transport; (b) Regime of gas transport with contour lines of apparent permeability. We secondly study the effects of fracture aperture distribution on gas transport in dual-porosity shale rocks. We plot the relationship between cumulative probability (M ) and

fracture aperture under different cM and R . From Figures 7(a) and (b), we can see that: (1)

increasing cM lifts up the curve of the M , which indicates less fractures with larger aperture exist in the rock; (2) increasing R pulls down the curve of the M and pushes the curve to a region with higher R, which means more fractures with larger aperture will be observed. As

adsorbed gas does not exist in natural fractures, existence of more fractures with larger aperture leads to a higher free gas transport ratio. As increasing R and decreasing cM bring more fractures with larger aperture, the adsorbed gas transport ratio lifts up with a decrease of R and an increase of cM as shown in Figures 7(c) and (d). Xu et al. (2017, 2018)41,47 claim that the gas apparent permeability is higher with a lower adsorbed gas ratio. The gas transport efficiency thus becomes better with higher R and lower cM as shown in Figures 7(e) and (f). Figure 7 shows with the same fracture porosity, the gas transport efficiency still varies with different fracture aperture distributions. We thirdly explore the gas transport efficiencies under different fracture porosities and 12


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fracture aperture distributions. In experiment, it is difficult to determine the fracture porosity, but easier to obtain the apparent permeability under different pressures. As shown in Figure 8, different gas apparent permeability is obtained under different fracture porosities and pressures. Thus the fracture porosity may be referred under a certain pressure and an apparent permeability. Figure 9 indicates: (1) if M 0.001, we get w% 750 } ; (2) if 0.001 M 0.005, we obtain 750 } w% 10000 } ; (3) if M > 0.005, we have w% >

10000 } . Specially, we can also determine the apparent permeability under a specific

pressure and fracture porosity. For instance, if the fracture porosity is 0.001 and the pressure is 25 MPa, the apparent permeability equals to 538 nd. Figure 9(a) shows isolines of w% 1000 } under different cM . With the increase of cM , isoline of w% 1000 }

gradually moves to high permeability region; this indicates the increase of cM leads to a

lower transport efficiency, which is due to the shrinkage of the natural fracture-dominated region as shown in Figure 9 (b); this is in agreement with the conclusions drawn from previous sections. Isolines of w% 1000 } under different R are plotted in Figure 10(a). The increase of R results in a larger apparent permeability, which is led by the

Ff, dimensionless

expansion of natural fracture-dominated region as shown in Figure 10(b).

Ff, dimensionless

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13


,%

a

KA, nd

a

KA, nd

K =10000 nd A

K =7500 nd A

K =5000 nd A

K =2500 nd A

K = A 500 nd K =7 50 nd A K =1 000 n A d

Figure 7 (a) fracture aperture distribution under different fractal dimensions, R 1 p; (b) fracture aperture distributions under different minimum fracture apertures, R 1 p; (c) curve of adsorbed gas ratio versus fractal dimension of fracture aperture, R 1 p; (d) curve of adsorbed gas ratio versus minimum fracture aperture, cM 2.9; (e) curve of apparent permeability versus fractal dimension of fracture aperture, cM 2.9; (f) curve of apparent permeability versus minimum fracture aperture, cM 2.9.

P, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

,%


Figure 7 Countour lines of apparent permeability of dual-porosity shale rocks under different pressures and fracture porosities.

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P, MPa

P, MPa

Figure 8 (a) Isolines of w% 1000 } under different fractal dimensions of fracture aperture; (b) Transport regimes under different fractal dimensions of fracture aperture.

P, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


P, MPa

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Figure 9 (a) Isolines of w% 1000 } under different minimum fracture apertures; (b) Transport regimes under different minimum fracture apertures. Conclusions Based our developed model for gas flow in a dual-porosity shale rock, we can have the following conclusions: (1) Natural fracture better benefits in gas transport efficiency of dual-porosity rocks. Fractal dimension of fracture and the ratio of minimum fracture aperture to a fracture aperture together contribute to the fracture aperture distribution. Decreasing fractal dimension of fracture aperture and increasing minimum fracture aperture bring more fractures with higher fracture aperture size and a higher gas apparent permeability. (2) The apparent permeability is significantly affected by fracture porosity and pressure. Higher pressure and fracture porosity yield to a higher apparent permeability of shale rocks. 15



The increase of minimum fracture aperture and the decrease of the fractal dimension of fracture aperture result in a larger apparent permeability, which is led by the expansion of natural fracture-dominated region Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Calculation of gas properties Acknowledgments The authors would like to acknowledge the NSERC/AIEES/Energi Simulation and Alberta Innovates - Technology Futures Chairs for providing research funding.

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Nomenclature Roman Symbols Y: cross-sectional area, p ; ': conductance in , (pf ∙ p/ q ∙ g; c: fractal dimension, dimensionless; cM : fractal dimension for fracture aperture, dimensionless; c : fractal dimension for pore size, dimensionless; cM : fractal dimension for the tortuosity of fractures, dimensionless; c : fractal dimension for the tortuosity of pores in shale matrix, dimensionless; : pore size, p; : gas molecular diameter, p; : cumulative probability, dimensionless; ∆: isosteric adsorption heat at 0, /pf; : Heaviside step function; _: geometric parameter, dimensionless; w% : apparent permeability, p ; o: Boltzmann constant, 1.38 × 10 , /w; U: straight length of the pore in shale matrix, p; UM : tortuous length of the fracture, p; U : tortuous length of the pore in shale matrix, p; : gas molecular weight, o/pf; LM : total number of fractures with the aperture larger than R, dimensionless; L : total number of pores with the hydraulic diameter larger than A in shale matrix, dimensionless; L% : Avogadro constant, 6.02 × 10 , pf ; }: molecular number density, p,; f : fitting coefficient, 7.9; f : fitting coefficient, 9 × 10^; f, : fitting coefficient, 0.28; : pressure, q; V : critical pressure, q; : : Langmuir pressure, q; : reduced pressure, dimensionless,

I ; I

a: total transport molar rate, pf/g; ]: transport molar rate in a single pore or fracture, pf/g; n: temperature, w; nV : critical temperature, w; ?

n : reduced temperature, dimensionless, ? ; : molar volume, p, /pf; SM : fracture width, m; l: gas compressibility factor, dimensionless.

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Greek Symbols : function of gas properties and temperature in the SRK EOS, dimensionless; h: fracture dip, dimensionless; T: aspect ratio, dimensionless; : repulsion parameter in the SRK EOS, q ∙ p, /pf; : ratio between the blockage rate constant and the forward migration rate constant, dimensionless; : attraction parameter in the SRK EOS, p, /pf; : acentric factor, dimensionless; : porosity, dimensionless; Eg : surface diffusion coefficient, p /g; : gas coverage, dimensionless; R: fracture aperture, p; : gas density, o/p , ; : gas viscosity, q ∙ g; : gas viscosity at P = 1.01325×105 Pa and T = 423 K, 2.31 × 10 q ∙ g; : ratio between minimum equivalent pore diameter and maximum equivalent pore diameter in shale matrix, dimensionless, / Z[ ; M is the ratio between the minimum fracture aperture and maximum fracture aperture, dimensionless, R /RZ[ . Subscripts q: adsorbed gas; : free gas; : fracture; p: matrix; p}: minimum; pqr: maximum; o: Knudsen diffusion; : pore; g: surface diffusion; \: total; d: viscous flow.

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List of Figure Captions Figure 1 Sketch of the proposed model Figure 2 View of cross sections of pores and fractures Figure 10 (a) Validation of pore size distribution with Alnoaimi et al.’s experiment, c 2.10, Z[ 81.03 }p and 4.90 }p ; (b) Validation of pore size distribution with Bahadur et al.’s experiment, c 2.42, Z[ 10 p and 3.3 }p) Figure 4 Validation of fracture aperture distribution with experimental data ((a): experimental data of Marcellus shale sample 1 reported by Gale et al. (2014), cM 2.5; RZ[ 3.41 pp; R 0.14 pp; (b): experimental data of Marcellus shale sample 2 reported by Gale et al. (2014), cM 2.1; RZ[ 3.57 pp; R 0.15 pp) Figure 5 Validation of total gas transport conductance with experimental data of shale core plug ((a): experimental data of shale core plug reported by Zhu et al. (2016), U 5 'p, 1.25 'p , n 300 w , c 2.98 , cM 2.96 , c 1.4 , cM 1.2 , 0.12 , M 0.005 , cZ[ 30 p , c 0.8 }p , RZ[ 130 p , R 3.5 p , A 1.2 , h 30° and TM 1.5; (b): experimental data of shale core plug reported by Ren et al. (2016), U 8 'p , 2 'p , n 333.15 w , c 2.96 , cM 2.93 , c 1.4 , cM 1.1 , 0.11 , M 0.004 , cZ[ 45 p , c 1 }p , RZ[ 100 p , R 0.7 p, A 1.1, h 15° and TM 1.4) Figure 6 (a) Regime of gas transport; (b) Regime of gas transport with contour lines of apparent permeability. Figure 7 (a) fracture aperture distribution under different fractal dimensions, R 1 p; (b) fracture aperture distributions under different minimum fracture apertures, R 1 p; (c) curve of adsorbed gas ratio versus fractal dimension of fracture aperture, R 1 p; (d) curve of adsorbed gas ratio versus minimum fracture aperture, cM 2.9; (e) curve of apparent permeability versus fractal dimension of fracture aperture, cM 2.9; (f) curve of apparent permeability versus minimum fracture aperture, cM 2.9. Figure 8 Countour lines of apparent permeability of dual-porosity shale rocks under different pressures and fracture porosities. Figure 9 (a) Isolines of w% 1000 } under different fractal dimensions of fracture aperture; (b) Transport regimes under different fractal dimensions of fracture aperture. Figure 10 (a) Isolines of w% 1000 } under different minimum fracture apertures; (b) Transport regimes under different minimum fracture apertures.

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A Model for Gas Transport in Dual-Porosity Shale Rocks with Fractal

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