Experimental and numerical analyses of apparent gas diffusion coefficient in gas shales

Fuel 258 (2019) 116123

Contents lists available at ScienceDirect

Fuel journal homepage: www.elsevier.com/locate/fuel

Full Length Article

Experimental and numerical analyses of apparent gas diffusion coefficient in gas shales

T

⁎

Ying Zhonga,b,c, Jiping Shea,b, , Hao Zhanga,b, Ergun Kurua,b,c, Bin Yanga,b, Jianchao Kuanga,b a

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology), Chengdu, Sichuan Province 610059, China College of Energy, Chengdu University of Technology, Chengdu, Sichuan Province 610059, China c Department of Civil and Environmental Engineering, University of Alberta, 16 St. and 85 Ave., Edmonton, AB T6G2R3, Canada b

A R T I C LE I N FO

A B S T R A C T

Keywords: Gas shales Knudsen diffusion Fick diffusion Surface diffusion Apparent gas diffusion coefficient

A unified model of apparent gas diffusion coefficient in gas shales has been developed by considering the combined effects of Knudsen diffusion, Fick diffusion, and surface diffusion. Results showed that the model predictions matched the measured apparent gas diffusion coefficient in Longmaxi black shale reasonably well when the correction factor for real surface diffusion was approximately equal to 0.01. Results of sensitivity analyses revealed that Knudsen diffusion and Fick diffusion control the bulk of the methane diffusion in shale nanopores. The governing mechanism of methane molecules transport through shale nanopores varied depending on the overburden pressure, pore pressure, and resultant effective stress (i.e., contributions of Fick diffusion and Knudsen diffusion to the apparent gas diffusion coefficient varied depending on the level of pore pressure and effective stress). The apparent methane diffusion coefficient also varied depending on the size of the pore space. The effective radius of pores in a formation with a relatively high effective stress is smaller than that of the one with a relatively low effective stress. Greater apparent methane diffusion coefficient is, therefore, observed in a relatively lower effective stress formation because of the increasing contributions of Fick and Knudsen diffusion. Considering the combined effects of pore pressure and effective stress, we have found that the overall impact of gas production from shale formations (i.e. the decreasing pore pressure and the corresponding increase in effective stress) was to increase apparent methane diffusion coefficient, which would result into further stimulation of the gas production from shale reservoirs.

1. Introduction Unlike the conventional reservoirs, the mechanism of gas transport in shale gas reservoirs includes viscous flow, slip flow, transition flow, as well as the desorption [1]. The gas storage and migration channels are provided by the multiscale (i.e., micro-and nano-scale) pores [2,3]. Thus, the diffusion, which is a spontaneous physical process where hydrocarbon gases migrate from a region of high concentration to one of low concentration under the action of spatial derivative (i.e., concentration gradient) [4,5], plays a critical role in the hydrocarbon gas migration, accumulation, and enrichment in gas shales [6–8]. In addition, the production of gas shales involves the desorption, diffusion, and seepage of hydrocarbon gases [9,10]. Diffusion of hydrocarbon gases through shale pores is critical for petroleum engineers to develop and produce the gas shales since it guarantees the in-situ gas mass transfer

[11,12]. Therefore, accurate evaluation and modeling of the diffusion coefficient of hydrocarbon gas in shales are essential for the shale gas production forecasting, as well as the development of shale gas reservoirs (e.g., well placement and configuration optimization) [13,14]. Many experimental studies have been conducted to investigate the gas diffusion in shales. The isobaric diffusion method (i.e., free gas molecular diffusion method) is often used to determine the gas diffusion coefficient in shales. Results from previous studies demonstrated that the gas diffusion coefficient in rocks is proportional to the porosity of the rock, and also closely related to the temperature and pressure [15,16]. Some researchers used NMR (nuclear magnetic resonance) measurement technique for estimating the hydrocarbon diffusion coefficient in organic-rich shales at equilibrium state [17–19]. The pulse-decay method has also been used to conduct the measurement of the methane diffusion coefficient in some studies [20,21]. The surface

⁎ Corresponding author at: State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology), Chengdu, Sichuan Province 610059, China. E-mail address: [email protected] (J. She).

https://doi.org/10.1016/j.fuel.2019.116123 Received 16 July 2019; Received in revised form 24 August 2019; Accepted 29 August 2019 Available online 11 September 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.

Fuel 258 (2019) 116123

Y. Zhong, et al.

ζreal κ

Nomenclature

correction factor for real surface diffusion, dimensionless molecular block coefficient (i.e., ratio of blocking velocity coefficient to forward velocity coefficient), dimensionless H(1 − κ) Heaviside function, dimensionless Kn Knudsen number, dimensionless λ mean free path length of gas molecular, m D mean diameter of the shale rock pore, m κb Boltzmann constant, 1.3805*10−23 J/K Jknudsen mass flux of molecule-pore wall collision in shale pores, kg/(m2·s) Jfick mass flux of molecule-molecule collision in shale pores, kg/(m2·s) Jbulk diffusive flux of bulk gas in shale pores, kg/(m2·s) r mean radius of the shale rock pore, m l distance of gas diffuse, m μg gas viscosity, Pa·s w1, w2 weighting coefficients of Knudsen diffusive flux and Fick diffusive flux in bulk gas diffusion, respectively tTotal average time of one collision of bulk gas molecules, s tknudsen average time consumed for one collision of molecule-pore wall, s tfick average time required for one collision of molecule-molecule, s vmolecule average velocity of gas molecule movement, m/s λTotal mean free path of bulk gas molecules, m ro initial radius of pores, m α Biot’s coefficient K bulk modulus of shale sample εL Langmuir volumetric strain constant Pconfine confining pressure, Pa

DF

methane diffusion coefficient measured by isobaric diffusion method, m2/s DTotal apparent diffusion coefficient of shale gas in pores, m2/s Dosurface theoretical surface diffusion coefficient, m2/s Dsurface surface diffusion coefficient of adsorbed gas in shale pores, m2/s Dbulk bulk gas diffusion coefficient in shale pores, m2/s C01 − C02 methane concentration difference between the two diffusion chambers at the initial time, % Ct1 − Ct2 methane concentration difference between the two diffusion chambers during a period of diffusion time t. % t time of methane diffusion, s S cross-sectional area of the thin cylindrical shale sample, m2 L length of the thin cylindrical shale sample, m V1, V2 volumes of the two diffusion chambers and their connected pipelines, respectively, 41.3 × 10−6 cm3 Cs concentration of adsorbed gas, kg/m3 θ gas coverage on the surface of pore wall, dimensionless M gas molar mass, kg/mol NA Avogadro’s constant, 6.02 × 1023/mol δ gas molecular collision diameter, m P pressure of gas, Pa PL Langmuir pressure, 8.45 × 107 Pa T absolute temperature, K ΔH isosteric heat of adsorption, J/mol R universal gas constant, 8.314 J·mol−1·K−1 ζms correction factor for surface diffusion, dimensionless ζmb correction factor for bulk diffusion, dimensionless

and distribution [1,34]. Previous studies reported that the mechanical deformation of shale has a significant impact on the nanopore radius and the Knudsen diffusion coefficient is space-dependent and negatively correlated with effective stress [35–37]. The thermal motion of gas molecules is closely related to the temperature and gas pressure. The kinetic energy increases with increasing temperature, while the mean free path length of gas molecules decreases with the increasing gas pressure [32,38]. In addition, pore water saturation level significantly influences the tortuosity factor in shale [39]. Consequently, the gas mass transfer behavior and gas permeability vary with changing petrophysical properties of the shale as well as with the temperature and pressure of the gas in the system. The pore pressure of the shale reservoirs decreases gradually as more gas is produced. As a result, since the overburden stress is constant, the effective stress of the rock skeleton increases gradually. Both the pore pressure and effective stress are in-situ influencing factors and would alter the methane diffusion coefficient. Although the Knudsen diffusion, the Fick diffusion, and the surface diffusion are independent, they contribute to gas mass transfer in gas shales simultaneously. Review of the previous research revealed that the studies of the total gas diffusion coefficient in shale nanopores and sensitivity analysis (i.e., individual and the coupling effects of pore pressure and effective stress on methane diffusion coefficient in shales) are rather scarce. Therefore, in this paper, we present results of an experimental study and model analyses of the methane diffusion coefficient under the conditions simulating the gas production (i.e., pore pressure decreases and the effective stress increases). A new model for estimating “Apparent gas diffusion coefficient” consisting of the coefficients of the Knudsen diffusion, the Fick diffusion, and the surface diffusion is proposed and derived to describe the diffusivity of gases in shales more realistically. Finally, sensitivity analyses were conducted to reveal the in-situ parameters (i.e., the pore pressure and effective stress)

diffusion stage and coefficient can be conveniently determined using these types of measurements. The diffusion coefficients measured by these different methods represent different physical processes [15]. Fick diffusion, the type of gas collision is dominated by the collisions with the gas molecules, is one type of bulk diffusion. Knudsen diffusion, the type of gas collision is dominated by the collisions of gas molecules on the pore wall, is another type of bulk diffusion [22]. Surface diffusion, however, is used to describe the process of methane molecules jumping between the adjacent adsorption sites on pore wall [23]. Therefore, various suitable models for the prediction of gas diffusion were developed. Many diffusion models for gas transfer in shale nanopores have been commonly developed by considering one or two of the Fick diffusion, Knudsen diffusion and surface diffusion concepts [22,24]. For example, Fick diffusion and Knudsen diffusion have been used to determine the diffusive flux of bulk gas molecules through nanocapillary pores where the diffusive flux is proportional to the concentration gradient [25,26]. Simulation results demonstrated that Knudsen diffusion is one of the main gas mass transfer mechanisms in gas shales. Contribution of the Knudsen diffusion to gas production can be up to 20% [27–29]. Surface diffusion of gas in shales is very complex as it involves the motion of the adsorbed methane molecules at the inner surface of the rock multiscale pores (i.e., the process of methane molecules jumping between the adjacent adsorption sites) [30]. The surface diffusion model was derived from hopping models [31]. Experimental and simulation results indicated that the surface diffusion coefficient increases with the increasing gas pressure [20,24]. For the sensitivity analysis of factors influencing gas diffusion coefficient, results of the extensive researches have shown that the methane diffusion in shales was significantly effected by the size of nanometer scale of shale pores and throats, the in-situ pressure and temperature of shale reservoirs [32,33], as well as the water saturation 2

Fuel 258 (2019) 116123

Y. Zhong, et al.

influencing the magnitude of the methane diffusion coefficient.

experiments:

2. Experimental program

1) Place the thin cylindrical shale sample into the core holder and apply the confining pressure. 2) Fully vacuum the system (i.e., gas booster, diffusion chamber, shale pore and so on). 3) Inject methane and nitrogen into the methane and nitrogen gas boosters, respectively. 4) Heat both the methane and nitrogen to the experimental temperature 398.15 K. 5) Fill the methane and nitrogen diffusion chambers by opening the valves of #6, #10 at the same time. Then close the valves when the pressure in both chambers are increased up to the experimental pore pressure. In addition, keep the pressure difference between the methane and nitrogen diffusion chambers at about 0 pa (i.e., the differential pressure is 0 pa). 6) Determine the gas composition and methane concentration in both diffusion chambers at the initial time by using the gas chromatograph. 7) After a period of diffusion, measure the gas composition and methane concentration in both diffusion chambers. Then, calculate the methane diffusion coefficient in shale rock. 8) Change the experimental condition according to the test matrix shown in Table 2. Repeat the steps 1) to 7) and calculate the methane diffusion coefficients under various experimental conditions.

2.1. Materials used The shale sample obtained from Silurian Longmaxi black shale formation, which is located in Southeast of Sichuan Basin in China, was used to conduct the experiments. The black shale formation is mainly composed of black silty shale and grey-green pelitic siltstone. The mineralogy-based brittleness index of the black shale formation is as high as 0.66. The cylindrical shale sample was prepared in sizes of 25 mm * 18 mm (diameter * length) by drilling parallel to the shale bedding plane without creating macroscopic fractures. Then, the shale sample was used to accurately evaluate the impacts of pore pressure and effective stress on methane diffusion coefficient in shales. The petrophysical properties and geometrical characteristics of the shale samples were summarized in Table 1. The mean pore diameter of the shale sample was determined by using BJH (Barrett, Joyner and Halenda) desorption data in BET (Brunauer Emmett and Teller) measurements as 4.897 nm. The result of the pore size distribution as shown in Fig. 1. In addition, The FESEM images shown in Fig. 2 indicate that large amounts of honeycomb-like nanopores were developed in shales. 2.2. Experimental program 2.2.1. Experimental apparatus and method Isobaric diffusion method was used to conduct the measurement of the gas diffusion coefficient in shale rocks. The schematic diagram of the apparatus for the measurement of the gas diffusion coefficient in shale rocks is shown in Fig. 3. The instrument is mainly composed of a core holder, a diffusion chamber, pressure and temperature control system, a vacuum system, gas chromatograph, a data acquisition system and so on. Isobaric diffusion method was applied by measuring the changes in gas concentration in the two diffusion chambers over a period of time. The gas concentration was determined by using a gas chromatograph. The methane diffusion coefficient through the shale rock sample can be calculated by using Eq. (1).

C − C02 ⎞ ⎛ S ⎛ 1 1⎞ ⎞ + DF = ⎛ln 01 /⎜ t ⎟ × 10−4 − C C L V V t1 t2 ⎠ ⎝ ⎝ 1 2⎠ ⎠ ⎝ ⎜

⎟

⎜

3. Theory of gas diffusion in shale nanopores As shown in Fig. 4, this theory assumes that the surface diffusion of adsorbed gas on shale pore wall and bulk gas diffusion (Knudsen diffusion and Fick diffusion) in shale nanopores are independent [24]. Thus, the apparent diffusion coefficient (DTotal) of shale gas can be expressed as:

DTotal = Dsurface + Dbulk

(2)

where, Dsurface and Dbulk are the surface diffusion coefficient of adsorbed gas and bulk gas diffusion coefficient in shale pores, respectively, m2/s. 3.1. Adsorbed gas diffusion coefficient There exists a dynamic equilibrium between adsorbed gas and bulk gas [40,41]. According to the Langmuir model of monolayer adsorption, the concentration of adsorbed gas can be expressed as [42]:

⎟

(1)

In which, DF is the methane diffusion coefficient, m2/s. C01 − C02 is the methane concentration difference between the two diffusion chambers at the initial time, %. Ct1 − Ct2 is the methane concentration difference between the two diffusion chambers during a period of diffusion time t. %. t is the time of methane diffusion, s. S is the crosssectional area of the thin cylindrical shale sample, m2. L is the length of the thin cylindrical shale sample, m. V1 and V2 are the volumes of the two diffusion chambers and their connected pipelines, respectively, 41.3 × 10−6 m3.

Cs =

4θM πδ 3NA

(3)

where, Cs is the concentration of adsorbed gas, kg/m . θ is the gas coverage on the surface of pore wall (calculated by Eq. (4)), dimensionless. M is the gas molar mass, kg/mol. NA is Avogadro’s constant, 6.02 × 1023/mol. δ is the gas molecular collision diameter, m. 3

θ= 2.2.2. Experimental procedure During the period of gas well production, the pore pressure decreases and the effective stress increases, consequently the methane diffusion coefficient changes. The test matrix used for determining the effects of pore pressure and effective stress on methane diffusion coefficient in shale sample is shown in Table 2. The following step-by-step experimental procedure was used in the

P P + PL

(4)

where, P is the pressure of gas, Pa. PL is the Langmuir pressure, 8.45 × 107 Pa, taken from Ref. [43]. Then the surface diffusion coefficient of adsorbed gas on the surface can be expressed as Eq. (5) under the force of a chemical-potential gradient [24]. The Arrhenius equation can be used to model the temperature variation of diffusion coefficients [44,45].

Table 1 Petrophysical properties and geometrical characteristics of the shale samples. Sample #

Porosity (%)

Permeability (mD)

Mean diameter of pore (nm)

Core description

Shape and size

LMX

1.16

0.0172

4.897

Black silty shale

Cylinder with a diameter of 25 mm and a length of 18 mm

3

Fuel 258 (2019) 116123

Y. Zhong, et al.

dV/dD

0.030

Cumulative pore volume

0.030

0.025

0.020

0.020

0.015

0.015

0.010

0.010

0.005

0.005

3

dV/dD (cm /g m)

3

0.025

Cumulative pore volume (cm /g)

LMX

0.000

1

2

3

4 5 6 7 8 910

0.000 200

30 40 50 60708090 1100 00

20

Pore diameter (nm) Fig. 1. Pore size distribution of shale rock sample obtained by using BJH technique (dV/dD is the differentiation of pore volume to pore diameter).

expressed as Eq. (7) by multiplying Dosurface with the correction factor (ζreal) for real surface diffusion.

o Dsurface

ΔH 0.8 ⎞ = 8.29 × 10−7T 0.5 exp ⎛− ⎝ RT ⎠ κ κ (1 − θ) + 2 θ (2 − θ) + [H (1 − κ )](1 − κ ) 2 θ 2 ⎜

⎟

2 κ θ 2

(1 − θ + )

Dsurface 0.8

ΔH ⎞ = 8.29 × 10−7T 0.5 exp ⎛− ⎝ RT ⎠ κ κ (1 − θ) + 2 θ (2 − θ) + [H (1 − κ )](1 − κ ) 2 θ 2

ζms

⎜

(5)

In which, Dosurface is the theoretical surface diffusion coefficient, m2/ s. T is the absolute temperature, K. ΔH is the isosteric heat of adsorption, J/mol. R is the universal gas constant, 8.314 J·mol−1·K−1. ζms is the correction factor for surface diffusion (defined by [24]), dimensionless. κ is the molecular block coefficient (i.e., ratio of blocking velocity coefficient to forward velocity coefficient), dimensionless. H (1 − κ) is Heaviside function, dimensionless.

H (1 − κ ) =

{

κ≥1 0, 1, 0 ≤ κ ≤ 1

⎟

(1 − θ + θ ) κ 2

2

ζms ζreal (7)

3.2. Bulk gas diffusion coefficient The diffusion of bulk gas molecules through a very small capillary pore can be characterized by Fick diffusion and Knudsen diffusion. In 1934, Knudsen defined the Knudsen number (Kn) to distinguish the gas diffusion type [32,47]. The Kn is a dimensionless number defined as the ratio of the gas molecular mean free path length to the mean diameter of the pore. The Kn can be determined by using Eq. (8).

(6)

However, the process of surface diffusion of adsorbed gas in shale organic pores is discontinuous (in Fig. 4) since the organic material is only a small part of shale matrix volume (3%–15%, given by [46]). As a result, the real surface diffusion coefficient is far less than the theoretical Dosurface. Therefore, the real surface diffusion coefficient can be

Kn =

λ D

(8)

where, Kn is the Knudsen number, dimensionless. λ is the mean free

Fig. 2. FESEM images of the shales with large amounts of honeycomb-like nanopores developed. 4

Fuel 258 (2019) 116123

Y. Zhong, et al.

Fig. 3. Schematic diagram of the experimental set-up used for the measurement of the gas diffusion coefficient in shale rocks.

path length of gas molecular, m, which can be determined by using Eq. (9). D is the mean diameter of the shale rock pore, m.

Table 2 Test matrix used for determining the effects of pore pressure and effective stress on methane diffusion coefficient in shale sample. Sample #

LMX

Case

Simulate gas production (i.e., effects of pore pressure and effective stress on methane diffusion coefficient)

κb T 2 πδ 2P

Experimental condition

λ=

Temperature (K)

Overburden stress (MPa)/ Pore pressure (MPa)/ Effective stress (MPa)

398.15

21 21 21 21 21

In which, κb is the Boltzmann constant (1.3805*10−23 J/K). If the mean diameter of the pore is larger than the gas molecular mean free path length (i.e., λ < D, Kn < 1), the type of gas collision is dominated by the collisions with the gas molecules. In this case, the process of gas diffusion in the porous medium can be characterized by Fick diffusion, as shown in Fig. 5(a). On the contrary, if the mean diameter of the pore is smaller than the gas molecular mean free path length (i.e., λ > D, Kn > 1), the type of gas collision is dominated by the collisions of gas molecules on the pore wall. The process of gas diffusion, in this case, can be characterized by Knudsen diffusion, as shown in Fig. 5(b). Therefore, considering the case of variable pore diameters

20 15 10 5 1

1 6 11 16 20

(9)

Fig. 4. Schematic diagram of gas diffusion process and mechanism in shale nanopore (01 Surface diffusion, the process of surface diffusion of adsorbed gas on shale pore wall is discontinuous; 02 Bulk gas diffusion, which consists of Knudsen diffusion and Fick diffusion), modified from [24]. 5

Fuel 258 (2019) 116123

Y. Zhong, et al.

Fig. 5. Schematic diagram of the diffusion of gas molecules through a very small capillary pore, modified from [15].

10

molecular collisions in a unit time period can be calculated by:

1 1 1 = + tTotal tknudsen t fick

Kn (dimensionless)

Knudsen diffusion

1

In which, tTotal is the average time of one collision of bulk gas molecules, s. tknudsen is the average time consumed for one collision of molecule-pore wall, s. tfick is the average time required for one collision of molecule-molecule, s. The numbers of bulk gas molecular collisions (i.e., bulk gas diffusion), molecule-pore wall collision (Knudsen diffusion), molecule-molecule collision (Fick diffusion) can also be calculated from the equations as follows:

Fick diffusion

0.1

(15)

0.01

1 v = molecule tTotal λTotal

(16)

Fig. 6. Curves of Knudsen number (Kn) versus pore diameter in shale rock (temperature is 398.15 K, gas pressure is 1 MPa).

1 v = molecule tknudsen 2r

(17)

(according to the pore size distribution shown in Fig. 1) the values of Kn were calculated by using Eq. (8) and Eq. (9). As shown in Fig. 6, the values of Kn ranges from 0.171 to 4.284, which indicated that the bulk gas diffusion in shale multiscale pores consists of both the Knudsen diffusion and the Fick diffusion. The diffusive flux of molecule-pore wall collision (Knudsen diffusion), molecule-molecule collision (Fick diffusion) can be expressed by Eq. (9) and Eq. (11), respectively [20,48–52].

1 v = molecule t fick λ

(18)

10

Jknudsen

2r = −ζmb 3

Jfick = −ζmb

20 30 Pore diameter (nm)

8M ∂P πRT ∂l

Mκb ∂P 3Rπμg δ ∂l

40

50

where vmolecule is the average velocity of gas molecule movement, m/s. λTotal is the mean free path of bulk gas molecules, m. Substituting Eq. (16), Eq. (17) and Eq. (18) into Eq. (15) yields:

1 1 1 = + λTotal 2r λ

It is assumed that the weighting coefficients (i.e., w1 and w2) of Knudsen diffusive flux and Fick diffusive flux can be represented as the ratio of molecule-pore wall collision frequency to total collision frequency, and the ratio of molecule-molecule collision frequency to total collision frequency, respectively. Therefore, Eq. (14) can be rewritten to:

(10)

(11)

where, Jknudsen and Jfick are the mass flux of molecule-pore wall collision (Knudsen diffusion) and molecule-molecule collision (Fick diffusion) in shale pores, respectively, kg/(m2·s). r is the mean radius of the shale rock pore, m. L is the distance of gas diffuse, m. ζmb is the correction factor for bulk diffusion, dimensionless. μg is the gas viscosity, Pa·s. Therefore, the Dknudsen and Dfick can be expressed as [20,48–52]:

Dknudsen = ζmb

Dfick

2r 3

8RT πM

κ T = ζmb b 3πμg δ

Jbulk =

tTotal t Jknudsen + Total Jfick tknudsen t fick

(20)

Inserting Eq. (16), Eq. (17), Eq. (18) and Eq. (19) into Eq. (20) yields:

Jbulk =

(12)

λ 2r Jknudsen + Jfick 2r + λ 2r + λ

(21)

Inserting Eq. (10) and Eq. (11) into Eq. (21) yields: (13)

Jbulk = − 3.3. Apparent gas diffusion coefficient

2r

κb T 2 πδ2P ζmb κ T + b 2 2 πδ P

2r 3

8M ∂P 2r Mκb ∂P − ζmb κ T πRT ∂l 3Rπμg δ ∂l 2r + b 2 2 πδ P

(22)

However, the diffusive flux of bulk gas in pores is composed of Knudsen diffusive flux and Fick diffusive flux, as shown in Eq. (14) [10,15,19,24,30,47]. Determining their reasonable weighting coefficient in bulk gas diffusion is the key point to estimate the bulk gas diffusion coefficient of bulk gas in shale nanopores.

Jbulk = w1 Jknudsen + w2 Jfick

(19)

Then, according to Eq. (12) and Eq. (13), the “bulk gas diffusion coefficient”, Dbulk, m2/s, can be expressed as:

Dbulk =

(14)

where Jbulk is the diffusive flux of bulk gas in shale pores, kg/(m2·s). w1 and w2 are the weighting coefficients of Knudsen diffusive flux and Fick diffusive flux in bulk gas diffusion, respectively. The number of bulk gas

2r

κb T 2 πδ2P ζmb κ T + b 2 2 πδ P

2r 3

8RT 2r κ T + ζmb b κ T πM 3πμg δ 2r + b 2 2 πδ P

(23)

Finally, inserting Eq. (7) and Eq. (23) into Eq. (2) yields the “apparent gas diffusion coefficient”, DTotal, m2/s, which can be expressed as: 6

Fuel 258 (2019) 116123

Y. Zhong, et al.

Table 3 Summary of modeling parameters used in the calculation of gas diffusion coefficient.

DTotal ΔH 0.8 ⎞ = 8.29 × 10−7T 0.5 exp ⎛− ⎝ RT ⎠ ⎜

⎟

κ

κ

(1 − θ) + 2 θ (2 − θ) + [H (1 − κ )](1 − κ ) 2 θ 2

(1 − θ + θ ) κ 2

ζmb

2r 3

2

8RT 2r κ T + ζmb b κ T πM 3πμg δ 2r + b 2 2 πδ P

ζms ζreal +

2r

κb T 2 πδ2P κ T + b 2 2 πδ P

(24)

Porosity and permeability stress sensitivity demonstrated that any stress applied to a porous medium would tend to change the petrophysical properties (i.e., porosity and permeability) of the porous medium [53], which in turn, affects the diffusion of methane (i.e., transport properties of methane molecules) in porous medium by altering the microporosity [54]. In other words, the diffusion of methane molecules through shales is related to the connected pore volumes of the shale. In fact, there is a positive correlation between them. We also assume that the connected pore volumes of a cylindrical shale sample can be represented as a bundle of tortuous capillaries. The schematic diagram of the cross-sectional surface area of the shale is shown in Fig. 7. The effective cross-sectional area of the tortuous capillary tube bundles decreases with the increasing effective stress [53,55,56]. The stress-dependent pore radius can be expressed as [36]:

1+

α ϕK

(Pconfine − P ) −

εL P P + PL

Unit

Values

φ T P ΔH R H(1 − κ) κ ζms ζreal ζmb κb δ ro M μg α K εL PL NA

Fraction K Pa J·mol−1 J·mol−1·K−1 dimensionless dimensionless dimensionless dimensionless dimensionless J·K−1 m m kg·mol−1 Pa·s dimensionless Pa dimensionless Pa mol−1

0.0116 398.15 1 × 106–2 × 107 12,000 8.314 1 0.5 8.98 × 10−4 1 × 10−3–1 × 10−1 2.24 × 10−3 1.308 × 10−23 3.8 × 10−10 2.449 × 10−9 0.016 1.4 × 10−5 0.8 8 × 108 0.05 8.45 × 107 6.02 × 1023

methane gas. R is real gas constant, 8.314. δ is the methane gas molecular collision diameter, 3.8 × 10−10 m. The values of K, εL, and PL are taken from Ref. [36]. The value of ζreal is uncertain and initially assumed to be 1 × 10−3, 1 × 10−2 and 1 × 10−1. The more accurate value of ζreal is determined as discussed in Section 4.1. The values of the parameters taken from these references are given for gas shales. We assumed that they can also be used in the calculation of gas diffusion coefficient in Silurian Longmaxi black shale sample.

ro

r=

Parameters

(25)

where, ro is the initial radius, m. α is Biot’s coefficient, K is the bulk modulus of shale sample, εL is the Langmuir volumetric strain constant. Pconfine is the confining pressure, which is equal to the overburden stress in Table 2, Pa. 4. Results and discussions

4.1. Determining the approximate ζreal value of Longmaxi shale formation

The diffusive fluxes of the methane through shales during a period of time were precisely measured by the gas chromatograph. Meanwhile, according to the parameters given in Table 3, Wolfram Mathematica software was used to calculate the theoretical surface diffusion coefficient, Knudsen diffusion coefficient, Fick diffusion coefficient, bulk gas diffusion coefficient and apparent gas diffusion coefficient under various conditions by using Eq. (7), Eq. (12), Eq. (13), Eq. (23) and Eq. (24), respectively. In Table 3, the values of ro, φ, T and P have been given in the previous sections. The values of ζms, ζmb, α, ΔH, H(1 − κ) and κ are taken from Ref. [24]. The values of κb, M, μg and NA, are standard properties of

Since the volume ratio of organic material to shale matrix (usually ranges from 3% to 15%) is varied in different reservoirs, the correction factor (ζreal) for real surface diffusion is uncertain. Thus, as shown in Table 3, the correction factor (ζreal) for real surface diffusion is initially assumed to be 1 × 10−3, 1 × 10−2 and 1 × 10−1. Comparative analyses of the model predictions and experimental results were carried out to determine the reasonable ranges of correction factor (ζreal) for real surface diffusion, as well as to verify the accuracy of the model Eq. (24). Fig. 8 presents the comparisons of the model predictions and experimental results in the case of simulated shale gas development (i.e., pore pressure decreases and effective stress increases gradually). The

Fig. 7. Schematic diagram of the cross-sectional surface area of shale (a). The effective cross-sectional area of the tortuous capillary tube bundles decreases with increasing effective stress (b). 7

Fuel 258 (2019) 116123

Y. Zhong, et al.

20

Dexperimental real

15

=0.1

Diffusion coefficient (10

-10

2

m /s)

burial depth and 4–5 × 104 km2 areal extension. As shown in Fig. 9. The Longmaxi black shale formations from two adjacent wells (Well 1 and Well 2) may share the similar petrophysical properties, however, with different overburden stress and pore pressure due to their different burial depth and the presence of fault barrier (Case 1). On the other hand, The Longmaxi black shale formations from two adjacent wells (Well 1 and Well 3) may share the similar petrophysical properties and pore pressure, however, with different overburden stress due to their different burial depth (Case 2).

LMX

10 real

5

real

2

0 20

=0.01

4

18

6

16

8 10 12 14 Effective stress (MPa)

14 12 10 8 Pore pressure (MPa)

=0.001

16

6

4.2.1. Case 1 – effect of pore pressure on methane diffusion coefficient in shales In the investigation of the effect of pore pressure on methane diffusion coefficient in shales, we used the information from Well 1 and Well 2. The overburden stresses of Silurian Longmaxi black shale formation encountered in Well 1 and Well 2 are assumed to be 50 MPa and 40 MPa, respectively. The pore pressures of the Longmaxi formation encountered in Well 1 and Well 2 are assumed to be 35 MPa and 25 MPa, respectively. So, the effective stress of Longmaxi formation encountered in Well 1 and Well 2 are both 15 MPa. In the early stage of gas well production, the overburden stress, pore pressure and effective stress are assumed to be unchanged. Results of the model predictions of theoretical surface diffusion coefficient, Knudsen diffusion coefficient, Fick diffusion coefficient, the bulk diffusion coefficient, and the apparent methane diffusion coefficient during the early production stage of the Well 1 and Well 2 are presented in Table 5 (i.e. Case-I, different pore pressure and same effective stress conditions). As shown in Table 5, apparent methane diffusion coefficient (DTotal) in shale formation encountered in Well 2 with relatively low pore pressure is greater than that of the one encountered in Well 1 with relatively high pore pressure. For example, the DTotal in shale formation encountered in Well 1 is 4.577 × 10−10 m2/s, however, the DTotal in shale formation encountered in Well 2 is 4.811 × 10−10 m2/s. The difference in DTotal between two wells is as high as 5.11%. The DTotal in shales consists of the surface diffusion and the bulk diffusion (i.e. sum of the Knudsen diffusion and Fick diffusion). As shown in Table 5, contribution of the Dsurface to the apparent methane diffusion in shale nanopores is relatively lower than that of from Knudsen and Fick diffusion coefficients. The Dsurface in shale formation encountered in Well 1 with relatively high pore pressure is slightly greater than that of the one in Well 2 with relatively low pore pressure. For example, the Dsurface in shale formation encountered in Well 1 is 1.182 × 10−10 m2/s when the pore pressure is 35 MPa, however, the one in Well 2 is 1.093 × 10−10 m2/s (7.53% lower) when the pore pressure is 25 MPa. The contribution rate of Dsurface to DTotal in shale formation encountered in Well 1 and Well 2 are 25.52% and 22.72%, respectively. The Dbulk (i.e. sum of Knudsen and Fick diffusion coefficients), on the other hand, contributes much to apparent methane diffusion in shale nanopores. The Dbulk in shale formation encountered in Well 1 with relatively high pore pressure is smaller than that of the one in Well 2 with relatively low pore pressure. For instance, the Dbulk in shale formation encountered in Well 1 is 3.395 × 10−10 m2/s when the

18

4

2

Fig. 8. Determining correct value of ζreal by comparative analyses of the model predictions and experimental results. Apparent gas diffusion coefficient versus effective stress and pore pressure.

trends of the how apparent methane diffusion coefficient varies with the change of pore pressure or effective stress obtained from model predictions are the same as that of obtained from experimental measurements. The total methane diffusion coefficient increases with decreasing pore pressure and increasing effective stress. Besides, the value of correction factor (ζreal) for real surface diffusion significantly affects the apparent methane diffusion coefficient (Fig. 8). The results have shown that when ζreal is approximately equal to 0.01, the model predictions (DTotal) from Eq. (24) match the experimental results (Dexperimental) very closely, and the average error is minimal (average error is −5.49% in Table 4). This could be attributed to that the surface diffusion only occurs in organic materials, the volume ratio of organic material to shale matrix usually ranges from 3% to 15%, besides, the organic materials are discontinuous, as a result, bulk diffusion dominates the total diffusion coefficient, ζreal is equal to 0.01, and has so much influence on the surface diffusion, however, has less important on the total diffusion. Otherwise, as shown in Table 4, when ζreal is equal to 0.1, the model results are much larger than the experimental results, the average error is 116.17%. When ζreal is equal to 0.001, the model results are slightly smaller than the experimental results, the average error is −17.66%. Therefore, the results indicate that Eq. (24) is appropriate for describing the methane diffusion coefficient in Longmaxi black shale when ζreal is approximately equal to 0.01. Thus, we will use ζreal value of 0.01 for the sensitivity analyses of the insitu influencing factors, which will be presented in the following sections. 4.2. Determining individual effects of pore pressure and effective stress on the apparent methane diffusion coefficient in shales The Silurian Longmaxi black shale formation in the Southeast of Sichuan Basin in China is marine deposit with generally 200–4500 m Table 4 Summary of experimental results and model predictions. Effective stress/Pore pressure (MPa/MPa)

1/20 6/15 11/10 16/5 20/1 Average

Experimental results (10−10 m2/s)

6.43 6.82 6.99 7.34 8.09

Model predictions (10−10 m2/s)

Error (%)

ζreal = 0.1

ζreal = 0.01

ζreal = 0.001

ζreal = 0.1

ζreal = 0.01

ζreal = 0.001

14.839 14.725 14.825 15.449 17.118

5.412 5.719 6.249 7.313 9.342

4.469 4.819 5.392 6.500 8.564

130.78 115.91 112.08 110.47 111.60 116.17

−15.83 −16.14 −10.60 −0.37 15.47 −5.49

−30.49 −29.35 −22.87 −11.45 5.86 −17.66

8

Fuel 258 (2019) 116123

Y. Zhong, et al.

Fig. 9. Schematic diagram of the geological structure in the Southeast of Sichuan Basin. (1) Case 1. Well 1 and Well 2 share the similar petrophysical properties, however, with different overburden stress and pore pressure due to their different burial depth and the presence of fault barrier. (2) Case 2. Well 1 and Well 3 share the similar petrophysical properties and pore pressure, however, with different overburden stress due to their different burial depth.

pore pressure is 35 MPa, however, the Dbulk of shale formation in Well 2 is 3.718 × 10−10 m2/s (9.51% higher) when the pore pressure is 25 MPa. The contribution rates of Dbulk to DTotal in shale formations encountered in Well 1 and Well 2 are 74.18% and 77.28%, respectively. The bulk methane diffusion coefficient (Dbulk) in shales consists of Knudsen diffusion and Fick diffusion. The value of Fick diffusion coefficient in shale formation encountered in Well 1 is larger than that of the one in Well 2. The value of Knudsen diffusion coefficient in shale formation encountered in Well 1, however, is less than that of the one in Well 2. This can be explained as in the low pore pressure well (e.g., Well 2), the weighting coefficient of methane molecules transport through shale nanopores by Fick diffusion mode is reduced, while weighting coefficient of the same by Knudsen diffusion mode is increased. For instance, the Dfick in shale formation encountered in Well 1 is 2.204 × 10−10 m2/s when the pore pressure is 35 MPa, however, the Dfick in shale formation encountered in Well 2 is 2.117 × 10−10 m2/s when the pore pressure is 25 MPa. The contribution rate of Dfick to Dbulk in shale formation encountered in Well 1 and Well 2 are 64.92% and 56.94%, respectively. The Dknudsen in shale formation encountered in Well 1 is 1.191 × 10−10 m2/s when the pore pressure is 35 MPa. The Dknudsen in shale formation encountered in Well 2 is 1.601 × 10−10 m2/s when the pore pressure is 25 MPa. The contribution rate of Dknudsen to Dbulk in shale formation encountered in Well 1 and Well 2 are 35.08% and 43.06%, respectively. Knudsen diffusion and Fick diffusion are the dominant mechanisms controlling the methane diffusion through shale nanopores. The transport mechanism of methane molecules through shale nanopores varies depending on the pore pressure of the shale formation (i.e., weighting coefficients of Fick diffusion and Knudsen diffusion vary depending on the pore pressure). The apparent methane diffusion coefficient (DTotal) in a relatively low pore pressure formation is larger than that of the one in a relatively high pore pressure formation.

Table 5 Methane diffusion coefficients in shale formation encountered in Well 1 and Well 2 (with different pore pressure and same effective stress). Parameters

Well 1

Well 2

Change in %

Overburden stress (MPa) Pore pressure (MPa) Effective stress (MPa) Dsurface (10−10 m2/s) Dknudsen (10−10 m2/s) Dfick (10−10 m2/s) Dbulk (10−10 m2/s) Dtotal (10−10 m2/s)

50 35 15 1.182 1.191 2.204 3.395 4.577

40 25 15 1.093 1.601 2.117 3.718 4.811

– – – −7.53 34.42 −3.95 9.51 5.11

Table 6 Methane diffusion coefficients in shale formations encountered in Well 1 and Well 3 (under the same pore pressure and different effective stress conditions). Parameters

Well 1

Well 3

Change in %

Overburden stress (MPa) Pore pressure (MPa) Effective stress (MPa) Dsurface (10−10 m2/s) Dknudsen (10−10 m2/s) Dfick (10−10 m2/s) Dbulk (10−10 m2/s) Dtotal (10−10 m2/s)

50 35 15 1.182 1.191 2.204 3.395 4.577

40 35 5 1.182 1.239 2.293 3.531 4.713

– – – 0.00 4.03 4.04 4.01 2.98

10 DTotal

Dbulk

Diffusion coefficient (10

-10

2

m /s)

Dknudsen

5 Dfick

4.2.2. Case 2 – effect of effective stress on methane diffusion coefficient in shales In the investigation of the effect of the effective stress on the apparent methane diffusion coefficient in shales, the overburden stresses of Silurian Longmaxi black shale formation encountered in Well 1 and Well 3 are assumed to be 50 MPa and 40 MPa, respectively. The pore pressures of that encountered in Well 1 and Well 3 are both assumed to be 35 MPa. The effective stress of that encountered in Well 1 and Well 3 are 15 MPa and 5 MPa, respectively. In the early stage of gas well production, the overburden stress, pore pressure and effective stress are assumed to be unchanged. Results showing how the surface diffusion coefficient, Knudsen diffusion coefficient, Fick diffusion coefficients, the bulk diffusion coefficient and the apparent methane diffusion coefficient in shales vary with effective stress (under the same pore pressure of 35 MPa) are presented in Table 6.

Dsurface

0

2

-5 20

4

18

6

16

8

10

12

Effective stress (MPa)

14

14 12 10 8 Pore pressure (MPa)

16

6

18

4

2

Fig. 10. Methane diffusion coefficient versus effective stress and pore pressure in shales.

9

Fuel 258 (2019) 116123

Y. Zhong, et al.

results showed that the pore pressure and effective stress have significant influence on the methane diffusion coefficient in shales. Comparison of the experimental results and model predictions showed that the newly developed model of “apparent gas diffusion coefficient” in shales can be used to describe the methane diffusion coefficient in Longmaxi black shale when ζreal is approximately equal to 0.01. Knudsen diffusion and Fick diffusion are the dominant mechanisms controlling bulk of the methane diffusion through shale nanopores. The transport mechanism of methane molecules through shale nanopores varies depending on the pore pressure of the shale formation (i.e., weighting coefficients of Fick diffusion and Knudsen diffusion vary depending on the pore pressure). The apparent methane diffusion coefficient (DTotal) in a relatively low pore pressure formation is larger than that of the one in a relatively high pore pressure formation. Results also indicate that the values of the DTotal in shale formation is also pore space-dependent (i.e. function of the effective stress). The effective radius of pores (i.e. size of the effective diffusion channels) decreases with the increasing effective stress. Similarly, the apparent methane diffusion coefficient (DTotal) decreases as the effective stress increases Analyses considering the combined effects of pore pressure and effective stress have shown that the overall impact of gas production from shale formations (i.e. the decreasing pore pressure and the corresponding increase in effective stress) would be to increase the apparent methane diffusion coefficient and, as a result, stimulate the gas production from shale reservoirs further. The overall effect can be explained as the positive effect caused by the change of the driving force behind transporting methane molecules through shale nanopores (i.e. the change of diffusion type from Fick diffusion to Knudsen diffusion) due to change in pore pressure exceeds the negative effect caused by the change of the same due to the increasing effective stress (or decreasing pore radius).

As shown in Table 6, the apparent methane diffusion coefficient (DTotal) in shale formation encountered in Well 1 with relatively high effective stress is smaller than that of the one in Well 3 with relatively low effective stress. For instance, the DTotal in shale formation encountered in Well 1 is 4.577 × 10−10 m2/s, however, the DTotal in shale formation encountered in Well 3 is 4.713 × 10−10 m2/s. The difference in DTotal between two wells is about 2.98%. The difference in DTotal between the shale formations encountered in Well 1 and Well 3 is mainly due to the difference between Dbulk in two cases (due to different effective stress). Note that, Dsurface in shale formations encountered in Well 1 and Well 3 are the same, 1.182 × 10−10 m2/s. The bulk methane diffusion coefficient (Dbulk) in shales consists of Knudsen diffusion and Fick diffusion. The values of Fick diffusion coefficient and Knudsen diffusion coefficient in shale formation encountered in Well 1 are smaller than that of the ones in Well 3. For example, the Dfick and Dknudsen in shale formation encountered in Well 1 are 2.204 × 10−10 m2/s and 1.191 × 10−10 m2/s, respectively, when the effective stress is 15 MPa, however, the Dfick and Dknudsen in shale formation encountered in Well 3 are 2.293 × 10−10 m2/s and 1.239 × 10−10 m2/s (both are 4% greater than that of the ones in Well 1), respectively, when the effective stress is 5 MPa. Overall, the bulk methane diffusion coefficient (Dbulk) decreases from 3.531 × 10−10 m2/s to 3.395 × 10−10 m2/s (4.01% Reduction) as the effective stress increases from 5 MPa to 15 MPa. Results indicate that the values of the DTotal in shale formation is pore space-dependent (i.e. function of effective stress). The effective radius of pores (i.e. size of the effective diffusion channels) in a relatively high effective stress formation (i.e. shale formation in Well 1) is smaller than that of the one in a relatively low effective stress formation (i.e. shale formation in Well 3), which in turn leads to the smaller apparent methane diffusion coefficient as the effective stress increases in shale formation (e.g. 5 MPa in Well 3 to 15 Mpa in Well 1). This is mainly due to the reduction of the values of the Fick and the Knudsen diffusion coefficients with the increasing effective stress.

Acknowledgments 4.3. Coupling effects of pore pressure and effective stress on methane diffusion coefficient in shales

This work was supported by National Natural Science Foundation of China (No. 51874052 and No. 51704043), China, Open Fund (PLC 20180802) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology), China.

In fact, during the production from a shale gas well, the pore pressure will gradually decrease and the effective stress will gradually increase, both are dynamic parameters. Therefore, the coupling effect of pore pressure and effective stress on methane diffusion coefficient in shales were investigated by analyzing the gas diffusion coefficient varying with the simultaneous changes of pore pressure and effective stress, as shown in Fig. 10. The overall results indicate that for a given overburden stress, the methane diffusion coefficient increases with the decreasing pore pressure (and the correspondingly increasing effective stress) as a result of gas production. For example, when the pore pressure decreases from 20 MPa to 1 MPa, the DTotal in shale formation increases from 5.412 × 10−10 m2/s to 9.342 × 10−10 m2/s (72.61%, increase). We can, therefore, conclude that as shown in Fig. 10 the decrease of pore pressure and the corresponding increase of effective stress due to gas production have positive effects on methane diffusion coefficient in shales and would further stimulate the production of shale gas wells. This can be explained as the positive effect caused by the change of diffusion type (large amounts of methane molecules transfer through shale nanopores by Fick diffusion changed to by Knudsen diffusion) exceeds the negative effect caused by the decrease of effective pore radius.

References [1] Zhong Y, Zhang H, Shao Z, Li K. Gas transport mechanisms in micro-and nano-scale matrix pores in shale gas reservoirs. Chem Technol Fuels Oils 2015;51(5):545–55. [2] Yang R, He S, Hu Q, Sun M, Hu D, Yi J. Applying SANS technique to characterize nano-scale pore structure of Longmaxi shale, Sichuan Basin (China). Fuel 2017;197:91–9. [3] Zhang H, Zhong Y, She J, Li G. Characterization of shale matrix pore structure via experiment and model. Arabian J Geosci 2018;11(12):320. [4] Klimenko AA. Diffusion of gases from hydrocarbon deposits. Int Geol Rev 1976;18(6):717–22. [5] Krooss BM, Schaefer RG. Experimental measurements of the diffusion parameters of light hydrocarbons in water-saturated sedimentary rocks—I. A new experimental procedure. Org Geochem 1987;11(3):193–9. [6] Welte DH, Leythaeuser D. Geological and physicochemical conditions for primary migration of hydrocarbons. Naturwissenschaften 1983;70(3):133–7. [7] Yin S, Ding W. Evaluation indexes of coalbed methane accumulation in the strong deformed strike-slip fault zone considering tectonics and fractures: a 3D geomechanical simulation study. Geol Mag 2018;156(6):1052–68. [8] Yin S, Zhao J, Wu Z, Ding W. Strain energy density distribution of a tight gas sandstone reservoir in a low-amplitude tectonic zone and its effect on gas well productivity: a 3D FEM study. J Petrol Sci Eng 2018;170:89–104. [9] Yuan W, Pan Z, Li X, Yang Y, Zhao C, Connell LD, et al. Experimental study and modelling of methane adsorption and diffusion in shale. Fuel 2014;117:509–19. [10] Bai J, Kang Y, Chen M, Liang L, You L, Li X. Investigation of multi-gas transport behavior in shales via a pressure pulse method. Chem Eng J 2019;360:1667–77. [11] Gao J, Yu Q, Lu X. Apparent permeability and gas flow behavior in carboniferous shale from the Qaidam Basin, China: an experimental study. Transp Porous Media 2017;116(2):585–611. [12] Javadpour F, Ghanbarnezhad Moghanloo R. Contribution of methane molecular diffusion in kerogen to gas-in-place and production. SPE western regional & AAPG

5. Conclusions An experimental and model study was conducted to determine the individual and the coupling effects of pore pressure and effective stress on the apparent methane diffusion coefficient in shales. Experimental 10

Fuel 258 (2019) 116123

Y. Zhong, et al.

[13]

[14]

[15] [16] [17] [18]

[19]

[20]

[21]

[22]

[23] [24]

[25]

[26] [27]

[28] [29] [30]

[31] [32] [33] [34]

pacific section meeting 2013 joint technical conference. Monterey, California, USA: Society of Petroleum Engineers; 2013. p. 21. Civan F, Devegowda D, Sigal RF. Critical evaluation and improvement of methods for determination of matrix permeability of shale. SPE annual technical conference and exhibition. New Orleans, Louisiana, USA: Society of Petroleum Engineers; 2013. p. 10. Zhu H, Deng J, Jin X, Hu L, Luo B. Hydraulic fracture initiation and propagation from wellbore with oriented perforation. Rock Mech Rock Eng 2015;48(2):585–601. Chen M, Kang Y, Zhang T, You L, Li X, Chen Z, et al. Methane diffusion in shales with multiple pore sizes at supercritical conditions. Chem Eng J 2018;334:1455–65. Liu G, Zhao Z, Sun M, Li J, Hu G, Wang X. New insights into natural gas diffusion coefficient in rocks. Pet Explor Dev 2012;39(5):597–604. An Y, Althaus SM, Liu H-H, Chen J-H. Nuclear magnetic resonance measurement of methane diffusion in organic-rich shales. Fuel 2019;247:160–3. Yao Y, Liu J, Liu D, Chen J, Pan Z. A new application of NMR in characterization of multiphase methane and adsorption capacity of shale. Int J Coal Geol 2019;201:76–85. Wang J, Yuan Q, Dong M, Cai J, Yu L. Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores. Int J Heat Mass Transf 2017;115:566–79. Yang B, Kang Y, You L, Li X, Chen Q. Measurement of the surface diffusion coefficient for adsorbed gas in the fine mesopores and micropores of shale organic matter. Fuel 2016;181:793–804. Marcon V, Joseph C, Carter KE, Hedges SW, Lopano CL, Guthrie GD, et al. Experimental insights into geochemical changes in hydraulically fractured Marcellus Shale. Appl Geochem 2017;76:36–50. Miao Y, Lee J, Zhao C, Lin W, Li H, Chang Y, et al. A practical method for estimating stimulated reservoir volume in shale gas reservoirs: coupling Knudsen diffusion and surface diffusion. SPE Trinidad and Tobago section energy resources conference. Port of Spain, Trinidad and Tobago: Society of Petroleum Engineers; 2018. p. 10. Krishna R, Wesselingh JA. The Maxwell-Stefan approach to mass transfer. Chem Eng Sci 1997;52(6):861–911. Wu K, Li X, Guo C, Wang C, Chen Z. A unified model for gas transfer in nanopores of shale-gas reservoirs: coupling pore diffusion and surface diffusion. SPE J 2016;21(05):1583–611. Shabro V, Torres-Verdin C, Javadpour F. Numerical simulation of shale-gas production: from pore-scale modeling of slip-flow, Knudsen diffusion, and langmuir desorption to reservoir modeling of compressible fluid. North American unconventional gas conference and exhibition. The Woodlands, Texas, USA: Society of Petroleum Engineers; 2011. p. 11. Fathi E, Akkutlu IY. Multi-component gas transport and adsorption effects during CO2 injection and enhanced shale gas recovery. Int J Coal Geol 2014;123:52–61. Holt JK, Park HG, Wang Y, Stadermann M, Artyukhin AB, Grigoropoulos CP, et al. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 2006;312(5776):1034–7. Darabi H, Ettehad A, Javadpour F, Sepehrnoori K. Gas flow in ultra-tight shale strata. J Fluid Mech 2012;710:641–58. Firouzi M, Alnoaimi K, Kovscek A, Wilcox J. Klinkenberg effect on predicting and measuring helium permeability in gas shales. Int J Coal Geol 2014;123:62–8. Wu K, Li X, Wang C, Yu W, Chen Z. A model for surface diffusion of adsorbed gas in nanopores of shale gas reservoirs. Offshore technology conference. Houston, Texas, USA: Offshore Technology Conference; 2015. p. 23. Chen YD, Yang RT. Concentration dependence of surface diffusion and zeolitic diffusion. AIChE J 1991;37(10):1579–82. Kim C, Jang H, Lee Y, Lee J. Diffusion characteristics of nanoscale gas flow in shale matrix from Haenam basin, Korea. Environ Earth Sci 2016;75(4):350. Foroozesh J, Abdalla AIM, Zhang Z. Pore network modeling of shale gas reservoirs: gas desorption and slip flow effects. Transp Porous Media 2019;126(3):633–53. Zhang T, Li X, Wang X, Li J, Sun Z, Feng D, et al. A discrete model for apparent gas

[35] [36]

[37]

[38] [39] [40] [41] [42] [43]

[44] [45] [46] [47]

[48]

[49]

[50]

[51] [52]

[53]

[54] [55] [56]

11

permeability in nanoporous shale coupling initial water distribution. J Nat Gas Sci Eng 2018;59:80–96. Si L, Li Z, Yang Y. Influence of the pore geometry structure on the evolution of gas permeability. Transp Porous Media 2018;123(2):321–39. Wei M, Liu Y, Liu J, Elsworth D, Zhou F. Micro-scale investigation on coupling of gas diffusion and mechanical deformation of shale. J Petrol Sci Eng 2019;175:961–70. Lu Y, Chen X, Tang J, Li H, Zhou L, Han S, et al. Relationship between pore structure and mechanical properties of shale on supercritical carbon dioxide saturation. Energy 2019;172:270–85. Ialamov II, Iushkanov AA. Influence of thermal stresses on the motion of a simple gas. J Eng Phys Thermophys 1976;31:669–73. Nunn JA, Xiang Y, Al TA. Investigation of partial water saturation effects on diffusion in shale. Appl Geochem 2018;97:93–101. Medveď I, Černý R. Surface diffusion in porous media: a critical review. Microporous Mesoporous Mater 2011;142(2):405–22. Hwang S-T, Kammermeyer K. Surface diffusion in microporous media. Can J Chem Eng 1966;44(2):82–9. Langmuir I. The adsorption of gases on plane surfaces of glass, mica and platinum. J Am Chem Soc 1918;40(9):1361–403. Wu K, Li X, Chen Z. The mechanism and mathematical model for the adsorbed gas surface diffusion in nanopores of shale gas reservoirs. Sci. China Press 2015;45(5):16. Cordero JM, Alonso M. Modelling of the kinetics of sulphation of CaO particles under CaL reactor conditions. Fuel 2015;150:501–11. Oelkers EH. Calculation of diffusion coefficients for aqueous organic species at temperatures from 0 to 350 °C. Geochim Cosmochim Acta 1991;55(12):3515–29. Yang Y, Wang C, Yao J, Hu R, Sun H, Zhao J. A new method for microscopic pore structure analysis in shale matrix. Earth Sci 2015;41(6):7. Shi J, Zhang L, Li Y, Yu W, He X, Liu N, et al. Diffusion and flow mechanisms of shale gas through matrix pores and gas production forecasting. SPE unconventional resources conference Canada. Calgary, Alberta, Canada: Society of Petroleum Engineers; 2013. p. 19. Curtis ME, Cardott BJ, Sondergeld CH, Rai CS. Development of organic porosity in the Woodford Shale with increasing thermal maturity. Int J Coal Geol 2012;103:26–31. Loucks RG, Reed RM, Ruppel SC, Jarvie DM. Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett shale. J Sediment Res 2009;79(12):848–61. Ambrose RJ, Hartman RC, Diaz Campos M, Akkutlu IY, Sondergeld C. New porescale considerations for shale gas in place calculations. SPE unconventional gas conference. Pittsburgh, Pennsylvania, USA: Society of Petroleum Engineers; 2010. p. 17. Roy S, Raju R, Chuang HF, Cruden BA, Meyyappan M. Modeling gas flow through microchannels and nanopores. J Appl Phys 2003;93(8):4870–9. Kim C, Jang H, Lee J. Experimental investigation on the characteristics of gas diffusion in shale gas reservoir using porosity and permeability of nanopore scale. J Petrol Sci Eng 2015;133:226–37. Tan X-H, Li X-P, Liu J-Y, Zhang L-H, Fan Z. Study of the effects of stress sensitivity on the permeability and porosity of fractal porous media. Phys Lett A 2015;379(39):2458–65. Mehmani A, Prodanović M. The effect of microporosity on transport properties in porous media. Adv Water Resour 2014;63:104–19. Wyble DO. Effect of applied pressure on the conductivity, porosity and permeability of sandstones. J Petrol Technol 1958;10(11):57–9. David C, Wong T-F, Zhu W, Zhang J. Laboratory measurement of compaction-induced permeability change in porous rocks: implications for the generation and maintenance of pore pressure excess in the crust. Pure Appl Geophys 1994;143(1):425–56.