Experimental and Modeling Study on the Effect of Shale Composition

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Experimental and modeling study on the effect of shale composition and pressure on methane diffusivity Wei Dang, Shu Jiang, Jinchuan Zhang, Fengqin Wang, Jia Tao, Xiaoliang Wei, Xuan Tang, Chenghu Wang, and Qian Chen Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b03349 • Publication Date (Web): 24 Jan 2019 Downloaded from http://pubs.acs.org on January 29, 2019

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Experimental and modeling study on the effect of shale composition and pressure on methane diffusivity

Wei Danga,b,*, Shu Jiangb,c, Jinchuan Zhangd,e, Fengqin Wanga, Jia Taod,e, Xiaoliang Weid,e, Xuan Tangd,e, Chenghu Wangf, Qian Cheng

a

School of Earth Sciences and Engineering, Xi’an Shiyou University, Xi’an 710065, China

b

Energy & Geoscience Institute, University of Utah, Salt Lake City 84112, USA

c

State Key Laboratory of Sedimentary Basin and Energy Resources & Faculty of Earth

Resources, China University of Geosciences, Wuhan 430074, China d

School of Energy Resources, China University of Geosciences, Beijing 100083, China

e

Key Laboratory of Shale Gas Exploration and Evaluation, Ministry of Land and Resources,

China University of Geosciences, Beijing 100083, China f

Key Laboratory of Crustal Dynamics, Institute of Crustal Dynamics, China Earthquake

Administration, Beijing 100085, China g Petroleum

Exploration and Production Research Institute, SINOPEC, Beijing 100083, China

S1

ACS Paragon Plus Environment

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Abstract：In this study, we present a comprehensive experimental and modeling study on six shale samples to investigate the effect of shale composition and pressure on methane diffusivity. By correlating shale composition, pressure and methane diffusivity, it was found that both macro- and micropore diffusivity decreased with increasing pressure, while total organic carbon (TOC) content had little effect on macropore diffusivity and negatively affected micropore diffusivity. This phenomenon may be a result of nonporous organic matter (OM) in transitional shale acting as solid material instead of porous media, occupying micropore volume, prolonging gas diffusion length and increasing diffusion resistance. Clay minerals with connected microstructures positively affect micropore diffusivity while negatively affecting macropore diffusivity, which may be a result of the filling of macropores with clay particles, blocking gas diffusion pathways and increasing gas diffusion resistance. Brittle minerals have a positive effect on macropore diffusivity and a negative effect on micropore diffusivity, which is similar to their effects on macro- and micropore volume, respectively. Moreover, the effect degree of shale composition and pressure on methane diffusivity changed over methane absorbing and diffusing, which may be related to the fact that methane diffusion occurs in different pores associated with brittle minerals, clay or OM at different pressure steps.

Keywords: transitional shale; pore property; diffusion modeling; methane diffusivity; influencing factors.

1. Introduction Unlike conventional gas reservoirs, gas diffusion in organic-rich shale with low permeability S2


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and porosity is one of the critical transport mechanisms controlling gas production capacity, and its importance, with respect to simulation, production and optimization of the exploration and development of shale gas reservoir, has been largely reported.1-9 Generally, gas diffusion property is indicated by the diffusion coefficient, which is defined as the amount of gas through a unit area of a porous media in an unit time. As early as sixty years ago, Antonov 10 attempted to conduct a pioneering work to experimentally measure the diffusion coefficient of light hydrocarbons in rocks of different lithologies. Since then, most researchers try to investigate the gas diffusion property, e.g. gas diffusion coefficient, in coal or shale samples.4, 6, 7, 11-22 Among above studies, a significant amount of work have been conducted in modeling the diffusion coefficient for methane or other gas in coal or shale samples. For example, Clarkson and Bustin 11 used isotherm adsorption and diffusion modeling methods to investigate the effect of pore structure and pressure on diffusion behavior in several coal samples and found that the adsorption and diffusion behavior of coals can be explained in terms of relative proportions of micro-, meso-, and macropores. The unipore diffusion model can reasonably describe the adsorption rate data of coals with a uniform micropore structure, while the bidisperse diffusion model is more suitable for coals with a complicated pore structure. Busch, et al.

12

applied a

simplified bidisperse modeling approach for the sorption kinetics of CO2 and CH4, and a perfect fit of the experimental data were obtained. Similar experimental and modeling studies were also conducted by Zhao, 23 Shi and Durucan 20 and Cui, et al. 24 More recently, Yuan, et al. 19 and Dang, et al. 25 conducted experimental and modeling study on marine shale samples collected from South China, and methane macro- and micropore diffusivity data at different pressure steps were reported. In addition, some of these studies also discussed the effect of experimental S3


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conditions on methane diffusivity. For example, Busch, et al.

Page 4 of 39

12

parameters increased for an increased pressure, while Cui, et al.

reported that the diffusion 24

reported a clear negative

correlation of macro- and micropore diffusivities with increasing pressure over a broad pressure range for CO2, CH4, and N2, and Yuan, et al.

19

also stated that the methane macro- and

micropore diffusivity decreased with increasing pressure, and the increased moisture could reduce the pore radius and block pore throats, exhibiting a negative effect on gas diffusivities. Apart from the effect of experimental conditions, the following properties of the medium in which diffusion takes place also have a significant effect on methane diffusivity:

3

(1)

petrophysical properties; 3 (2) pore property; 11, 19 (3) physical-chemical properties of different diffusing compounds;

3, 11, 24

(4) composition including minerals and OM. Currently, a large

amount of research works have been conducted to investigate the effects of petrophysical, pore structure and physical-chemical properties on diffusivity. However, only few papers document the effect of composition on methane diffusivity. For example, in an early study conducted by Schloemer and Krooss, 5 they compared the methane diffusivity data reported by themselves and Matthews,26 and found that the clay minerals in sediment rock has negative effect on gas diffusivity, however in their paper no further explanation was reported. Recently, some researchers investigated the effect of organic materials in coal samples on gas adsorption rate and diffusion characteristics, and found that the porous organic material in coal samples increases diffusion resistance and negatively affects the gas diffusion capacity

27

and gas

adsorption rate. 28 So far, no such studies were conducted to comprehensively relate the mineral compositions, OM, pore structure, and pressure to methane diffusivity. In this paper, we carried out a comprehensive study to investigate the effect of compositions S4


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and pressure on methane diffusivity in organic-rich shale. As previous researchers did, the methane diffusion parameters including macro- and micropore diffusivity were obtained using isotherm adsorption combining bidisperse diffusion modeling. So, in order to distinguish the macropores and micropores in here, this research used a value of 50 nm to divide pores into the micro- and macropores, which corresponds to pores smaller than 50 nm and pores larger than 50 nm, respectively, instead of using the conventional pore classification proposed by Sing 29, and the reason will be elaborated later. Then, several techniques including nitrogen adsorption, mercury intrusion and scanning electronic microscopy (SEM) were used to characterize pore property in these shale samples. Finally, correlations between shale compositions, gas pressure and methane diffusivity were presented, and the varied effects of shale compositions and pressure over the entire process of methane adsorption were discussed.

2. Experimental shale samples and methods 2.1 Shale samples In this study, a set of six shale samples from the Lower Permian marine-terrestrial transitional shale, collected from the Mouye-1 well in the Southern North China Basin (SNCB) (Figure 1), were prepared for this experimental and modeling study. Among these shale samples, three were collected from the Shanxi Formation, and the other three were collected from the Taiyuan Formation. The detailed information of these shale samples, including lithology, sampling locations, sampling depth and shale properties, are shown in Figure 1 and Table 1. The Lower Permian transitional black shale intervals, which formed in coastal swamp settings between terrestrial and marine environments throughout the whole North China platform,30,

31

are

generally characterized by high TOC contents, high thermal maturity, gas-prone OM and high S5


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clay mineral contents.

Figure 1. (a) Simplified structural map of Southern North China Basin (SNCB), showing the location of Mouye-1 well (modified from Dang et, al. 323232323231), and (b) stratigraphy column of Mouye-1 well, showing more detailed information of lithology, sampling locations and depth. Table 1. Organic geochemical characteristics and mineral compositions of measured shale samples in this study. Organic geochemistry Samples

Formation

Mineral compositions (%)

TOC

Ro

Brittle

(%)

(%)

minerals

2.52

3.26

2.37

SX3 TY1

SX1 SX2

TY2

Shanxi

Taiyuan

TY3

Quartz

Feldspar

Carbonates

Pyrite

41

39

2

0

0

3.47

35

32

3

0

2.21

3.57

36

31

5

3.12

3.59

30

30

3.74

3.12

46

3.25

3.01

41

Total

Illite

Kaolinite

Chlorite

I/S

59

39

6

6

12

0

63

33

11

7

14

0

0

61

41

3

0

20

0

0

0

64

29

14

9

18

44

0

1

1

56

26

4

7

17

37

2

2

0

57

21

11

6

21

I/S: Illite/Smectite

S6


clay

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2.2 Methods 2.2.1 Pore property characterization The pore sizes in measured shale samples were generally on the micrometer to nanometer scale, necessitating the use of multiple techniques for investigation. SEM is a common technique for characterizing pores in organic-rich shale, and the type of pores can be directly identified through SEM images. In this study, a Gatan Ilion Ⅱ milling system was used with an accelerating voltage of 6 KeV for 3 h to Ar-ion mill the rock flakes. The milling process removed approximately 70 μm of surface material and eventually provided a fan-shaped and flat surface. Then, a Hitachi Su 8010 scanning electron microscope was used to observe the pores at an accelerating voltage of 20-30 KeV. These images were amplified with magnification up to 100,000x, allowing for pores as small as 6 nm to be identified. To quantitatively describe the pore size distribution through SEM images, only images with magnification larger than 20,000x were processed during point counting. All the pores in the images were manually analyzed by using an image analysis tool called JMicroVision. Low-pressure nitrogen adsorption has been widely used to describe the pore properties of organic-rich shale by many researchers.33-36 To quantitatively determine the pore properties, the shale samples involved in this study were manually crushed and sieved into grains approximately 80-120 mesh in size, and nitrogen adsorption at -196.15C was conducted on a Quantachrome AUTOSORB gas adsorption system at the Key Laboratory of Shale Gas Exploration and Evaluation, Ministry of Land and Resources at the China University of Geosciences (Beijing), to determine the pore volume and pore size distribution by using adsorption data below a relative pressure (P/P0, where P is the gas vapor pressure in the system S7


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and P0 is the vapor pressure above the gas at the temperature of interest) between 0.005 and 0.99. In the past several decades, some methods including the Barrett-Joyner-Halenda (BJH) method,37 Dubinin – Radushkevich (D-R) method,38, 39 Dubinin – Astakhov (D-A) method,40, 41

Density Function Theory (DFT) method

42, 43

and Monte Carlo (MC) simulation method

44

were established to characterize pore size distribution in porous materials. Of these methods, the BJH method, which describes the capillary condensation phenomenon in a cylindrical pore, was the most popular method for pore size analysis. In our previous study, 45 we found that the pores developed in transitional shales in the study area mainly range from 2 nm to 100 nm. Therefore, the BJH method was appropriate for these measured shale samples and was selected in this study to characterize the pore size distribution. it can be described by the following equation: 𝛼𝛾𝑁𝑉1 𝑃 =― 𝑃0 𝑅𝑇(𝑟 ― 𝑡)

()

𝑙𝑛

(1)

where P/Po is the relative pressure; 𝛾𝑁 is the surface tension of liquid nitrogen; V1 is the molar volume of liquid nitrogen; R is the gas constant; T is the temperature; t is the statistical thickness calculated by the Harkins-Jura model for nitrogen adsorption; 46 and 𝛼 is a factor that accounts for the shape of the gas/liquid interface with a fixed value of 1. 47 The total pore volume was the volume of liquid nitrogen at a relative pressure of 0.99 using a nitrogen density of 0.808 g/mL. According to the BJH equation, pores smaller than 193.5 nm in diameter can be detected. 45

The total volume of pores ˂2 nm was calculated from the nitrogen volume at a relative

pressure of 0.16, and the total volume of pores ˂50 nm was calculated at a relative pressure of 0.96. S8


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Mercury intrusion analysis was conducted on a Quantachrome Pore Master mercury injector system. After placing the smashed small fragments into a dilatometer, mercury was first intruded into the dilatometer under a pressure of up to 20 psi to allow mercury to fill the large interspace between the rock grains. Then, mercury intrusion was conducted to a maximum pressure of approximately 29000 psi, at which the nanometer-scale pore throats could be penetrated. Then, the experimental pressure applied was converted to a pore-throat diameter using the Washburn equation.

48

Constants, including a contact angle of 140° and surface

tension of 0.485 N/m, were used to determine the pore size distributions. According to the equation, pore-throat diameters between 4 nm and ~10 um can be detected. The mercury total pore volume is the total mercury volume intruded into the dilatometer subtract the mercury volume intruded at low pressures (＜20 psi). 2.2.2 Methane adsorption rate measurement and diffusion model As previous researchers did,

11, 19

the gas diffusion property in this study was indirectly

estimated by using isotherm adsorption data and a diffusion modeling method. In this study, the methane adsorption isotherms were recorded at a temperature of 60 C with pressures from 0 to 6 MPa using a volumetric apparatus (3H-2000 PHD isothermal adsorption/desorption analyzer) that was manufactured by Beishide Instrument at the Key Laboratory of Shale Gas Exploration and Evaluation, Ministry of Land and Resources at the China University of Geosciences (Beijing). The instrument’s components in terms of the volumetric method have been documented in previous literature, 49, 50 and the main procedures of the methane adsorption rate can be summarized as follows: (1) the shale powder samples as previously used in the nitrogen adsorption measurement needed to be degassed first by evacuation for 10 h at 100℃; S9


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(2) a gas leak test was conducted using helium at 8 MPa for 2 h, and the acceptable leakage rate was less than 0.1 Pa/min; (3) the void volume of the sample cell, which was applied to calculate the isotherms, was determined by helium expansion; (4) helium within the sample cell was removed by evacuation; and (5) the methane adsorption was measured by increasing the test pressure, and pressure data and piston pump volume data versus time were recorded at 1-s intervals for all pressure steps, and the adsorption equilibrium time for each pressure steps was set as 90 mins; (6) repeating step (5) at sequentially higher pressures to measure a complete adsorption isotherm. By following and repeating the above steps, the adsorption amount of methane in the shale samples at any time during the adsorption process can be calculated through the following equations: 𝑔𝑎𝑠 𝑔𝑎𝑠 𝑚𝑔𝑎𝑠 𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 = 𝑚𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 ― 𝑚𝑢𝑛𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑

(2)

The amount of methane injected can be determined from the pressure, temperature and volume change in the piston pump: 𝑚𝑔𝑎𝑠 𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 =

(

𝑃∆𝑉𝑀 𝑍𝑅𝑇

)

(3)

𝑝𝑢𝑚𝑝

The amount of methane that is un-adsorbed can be determined from conditions at equilibrium in the sample cell: 𝑚𝑔𝑎𝑠 𝑢𝑛 𝑎𝑑𝑠𝑜𝑟𝑏𝑒𝑑 =

(

𝑃𝑉𝑣𝑜𝑖𝑑𝑀 𝑍𝑅𝑇

)

(4)

𝑠𝑎𝑚𝑝𝑙𝑒 𝑐𝑒𝑙𝑙

where m denotes the mass of methane in g, P is pressure in MPa, T is the temperature in K, M is the molar mass of methane in g/mol, Z is the compressibility coefficient of methane, R is the universal gas constant, ∆𝑉 is the volume change in the piston pump in m3, and 𝑉𝑣𝑜𝑖𝑑 is the S10


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volume of free gas in the sample cell in m3. Based on this, the methane adsorption amount at any time can be obtained, and the methane adsorption rate was generally plotted with the square root of time on the X-axis and the fractional adsorbed Mt/M∞ on the Y-axis. By fitting a bidisperse diffusion model 18, 48 described by the following equation, the gas diffusion parameters in measured shale samples can be obtained.

𝑀𝑡 𝑀∞

{

[

]} {

[

]}

𝐷′𝑎𝑡 𝐷′𝑖𝑡 1 𝛽 6 ∞ 1 ∞ 2 2 2 2 ∑ ∑ 1 ― 2 𝑛 = 1 2𝑒𝑥𝑝 ― 𝑛 𝜋 2 + 1 ― 2 𝑛 = 1 2𝑒𝑥𝑝 ― 𝑛 𝜋 𝛼 2 3𝛼 𝜋 𝑛 𝑅𝑎 𝜋 𝑛 𝑅𝑖 6

=

(5)

𝛽 1+ 3𝛼

where 𝑀𝑡 is the methane adsorption mass at time t in g; 𝑀∞ is the methane adsorption amount at infinite time or at equilibrium in g; 𝐷′𝑎 is the effective macropore diffusivity in cm2/sec; 𝐷′𝑖 is the effective micropore diffusivity in cm2/sec; t is the time in sec; 𝑅𝑎 and 𝑅𝑖 are the macrosphere and microsphere radii in cm, respectively; α is a dimensionless rate parameter, which is defined as 𝛼 = 𝐷𝑖𝑅2𝑎/𝐷𝑎𝑅2𝑖 ; β is a dimensionless parameter, which is defined as 𝛽 = [3(1 ― 𝜀𝑎)𝜖𝑖/𝜀𝑎](𝐷𝑖𝑅2𝑎/𝐷𝑎𝑅2𝑖 ); 𝐷𝑎 is the macropore diffusivity in cm2/sec; 𝐷𝑖 is the micropore diffusivity in cm2/sec; 𝜀𝑎 and 𝜀𝑖 are the macropore and micropore void fractions, respectively; and n is the number of increments, which was taken to be 100 in this study to get more accurate results, although previous researchers have stated that values of n as low as 10 can produce reasonably accurate solutions.51, 52

3. Experimental Results 3.1 Pore property 3.1.1 SEM observations S11


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Figure 2 shows the SEM observation results of the pores that developed in measured shale samples. In this study, a wide pore size distribution ranging from tens of nanometers to several microns was observed. The pores associated with clay minerals were generally intraparticle pores with pore sizes ranging from tens to hundreds of nanometers (Figure 2a-c), while pores associated with brittle minerals were either interparticle pores (Figure 2c, 2e and 2f) or intraparticle pores (Figure 2d), with pore sizes ranging from tens of nanometers to several microns. Compared to the large amounts of OM pores that were observed in marine shale, 35, 53 the OM pores in these transitional shale samples were not well formed and only several isolated pores ranging from tens to hundreds of nanometers were observed (Figure 2g-h). In most OM particles, the OM pores did not develop at all (Figure 2i-l), although the thermal maturity is extremely high (Table 1). Previous literature has reported that, compared with oil-prone OM, gas-prone OM is unfavorable to the formation of OM pores due to its limited hydrocarbongenerating ability and internal structure, especially for OM whose maceral composition is dominated by inertinite.32, 45, 54

S12


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Figure 2. SEM images of the shale samples in this study, which show pores with different sizes. (a) Intra-particle pores associated with clay minerals, with pore size ranging from tens to hundreds nanometer; (b) Intra-particle pores associated with clay minerals, with pore size ranging from tens to hundreds nanometer; (c) Inter-particle pores associated with quartz and intra-particle pores associated with clay, showing that the pore size of inter-particle pores associated with quartz are much larger than S13


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intra-particle pores associated with clay; (d) Intra-particle pores associated with carbonate, with pore size larger than 2 um; (e) Inter-particle pores associated with pyrite, with pore size ranging from tens to hundreds nanometer; (f) Inter-particle pores associated with quartz, with pore size around tens nanometer; (g) Isolated pores developed in OM with circle shape; (h) OM pores with size ranging from tens to hundreds nanometer; (i) OM without pores; (j) OM without pores; (k) Energy spectrum of Fig. 2j, showing that the matter in Fig. 1g is OM; (l) OM without pores.

Figure 3 shows the image-based statistical results of different kinds of pores, indicating that the different kinds of pores have their own relative pore size ranges. For example, the pores associated with clay minerals and OM pores share a similar pore size range and were generally smaller than ~100 nm, with an average value of 48 nm and 26 nm respectively. While the pores associated with brittle minerals (e.g., quartz, pyrite and carbonate) have a wide pore size ranging from tens of nanometers to a micrometer, and mainly ranged from 50 nm to 300 nm, with an average value of 102 nm. This observation is consistent with the results reported by Chen, et al. 53.

S14


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Figure 3. Number distribution of different kinds of pores recognized through SEM images.

3.1.2 Low-pressure nitrogen adsorption and mercury intrusion Previous studies

35, 36

have shown that the pore size distribution derived from a desorption

branch is much more affected by pore networks, and only the adsorption branches where this phenomenon is absent are considered for pore size distribution. Additionally, for natural materials, such as seal rocks or shale, the interpretation of hysteresis loops has to be considered with caution since it is subject to error. For example, Clarkson et al.55 studied tight gas sandstones using USANS/SANS and gas adsorption analysis and found that the interpretation of slit-shaped pores inferred from hysteresis loops was not consistent with the SANS scattering results. Thus, only the nitrogen adsorption isotherm results are shown in Figure 4a. According to the classification of Brunauer, et al.

56

the low-pressure nitrogen adsorption isotherms for

Lower Permian transitional shales are type Ⅱ, suggesting that a wide pore size range from nanometers to microns existed in these shale samples. For all six shale samples, nitrogen volumes of 6.22-20.84 cm3/g were adsorbed at a relative pressure P/P0 value of 0.99, corresponding to a total pore volume ranging from 14.5×10-3 to 24.1×10-3 cm3/g, and the volume of pores smaller than 50 nm ranging from 10.4×10-3 to 17.4×10-3 cm3/g account for 57%~96% of the total pore volume (Table S1). As illustrated in Figure 4b, the negative relationship between the cumulative mercury volume and pore diameter shows that the amount of intruded mercury increased with decreasing pore diameter, which indicates that a majority of the mercury intruded into pores below 1 μm.

S15


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Figure 4. (a) Low pressure nitrogen adsorption isotherms and (b) plots of mercury volume versus pore diameter for the shale samples in this study.

The combined pore size distribution curves (Figure 5) of dV/dlog(D) versus pore diameter were plotted in this study to show the pore size distribution features of measured shale samples. 34, 35, 53 As shown in Figure 5, except for shale sample SX3, a good consistency between nitrogen

adsorption and mercury intrusion pore volume distribution was observed, indicating that the combined method of nitrogen adsorption and mercury intrusion in this study could characterize pores in a wide range from nanometer to micrometer scale. In these pore size distribution curves, two general peaks were identified. The first peak appears at pore sizes of around 2-7 nm, and the second peak generally appears at pores size of around 70-100 nm. Therefore, the shale samples in this study are characterized by a bimodal pore size distribution, which is the basis for applying a bidisperse diffusion model to reasonably describe the gas diffusion process. In order to distinguish the macropores and micropores in a bidisperse diffusion model, a value of 50 nm was selected to divide the micro- and macropores, which corresponds to pores smaller than 50 nm and pores larger than 50 nm, respectively, instead of the conventional pore classification proposed by Sing 29 There are two reasons for this classification. One reason was that the value of 50 nm can roughly distinguish the two main peaks in combined pore size S16


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distribution curves, and another reason was that 50 nm is a conventional value to distinguish mesopores and macropores in conventional pore classification method, and if we introduced a new value to distinguish different pore size ranges in here, it would lead to new controversy.

Figure 5. Plots of the pore size distribution and pore volume curves from nitrogen adsorption and mercury intrusion in the measured shale samples.

3.2 Methane isothermal adsorption rate and diffusion modeling As a basis for this experimental and modeling study, the methane isothermal adsorption was conducted under six pressure steps, methane adsorption amount versus time for each shale sample was obtained (Figure 6), and only two shale samples, which were from the Shanxi and Taiyuan formations, were selected to show the experimental results here. There are two reasons for this. One reason was that the curve of methane isotherm adsorption amount versus time is not the end product that can be directly modeled by a bidisperse diffusion model and was only used to show the whole adsorption process through all pressure steps in shale samples here.

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Page 18 of 39

Another reason was that all six curves are almost the same in shape, and there is no point in listing all curve figures. Based on this, the methane isothermal adsorption rate data can be easily obtained and were plotted with the square root of time on the X-axis and the fractional adsorbed Mt/M∞ on the Yaxis (Figures S1 and S2). In this study, methane adsorption rate curves under all six pressure steps for each shale sample were established and were fitted using a bidisperse diffusion model. To avoid repeating the description of similar adsorption rate curves, however, only three pressure steps, ~0.2 MPa, ~3.3 MPa and ~5.6 MPa, were selected for plotting, and the plotting and diffusion modeling results can be found in the supplementary materials (Figures S1 and S2).

Figure 6. Methane isotherm adsorption amount versus time under six pressure steps for shale samples (a) SX1 and (b) TY3.

It is clear from Figures S1 and S2 that the bidisperse diffusion model is generally good at predicting the adsorption rate data over the entire time range at different pressure steps, even though there are slight discrepancies at very short time ranges (t＜5-10 s0.5) for the 3.3 MPa and 5.6 MPa pressure steps. As noted by Clarkson and Bustin S18


11

this slight difference can be

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due to (1) the temperature variation needed to attain thermal equilibrium in a sample cell after connecting the reference and sample cell and (2) the varied external (to shale particles) concentration during methane adsorption. The fitted diffusion parameters, including Da′/Ra2, Di′/Ri2, α and β/α, are summarized in Table 2. Here, we use the effective diffusion coefficient (D´/R2, s-1) instead of the diffusion coefficient (D, m2/s) for gas diffusion. One reason was that the effective diffusion coefficient, with a unit of s−1, could reflect the diffusion rate directly. 57 Moreover, it is not clear whether the value of R, the diffusion path length, should be set as equal to the particle radius, although it is assumed to be in the determination of the diffusion coefficient.

11

Some studies19, 28 took the sample

radius as the value of R, while Nandi and Walker 4 indicated that the value of R should be much less than the sample radius. Additionally, obtaining a representative R has always been a challenge and involved complex calculation processes. Thus, in order to avoid the uncertainty of R, the effective diffusion coefficient of D/R2 was used in this study as previous researchers did. 11, 57 As presented in Table 2, the diffusivities are varied between different shale samples at the same pressure step. This may be related to the different matrix composition characteristics between different shale samples, which will be discussed later.

Table 2. Bidisperse diffusion model fitting diffusion parameters for the methane adsorption rate at six pressure steps. SX1

SX2 P (MPa)

2

Da´/Ra （s-1）

Di´/Ri2 （s-1）

α

β/α

P （MPa）

Da´/Ra2 （s-1）

Di´/Ri2 （s-1）

α

β/α

0.2

1.0929

0.0689

0.117

0.4632

0.2

0.7038

0.0876

0.195

0.4875

0.8

0.7658

0.0989

0.174 9

0.0489

0.8

0.5812

0.1051

0.112 5

1.0581

2.2

0.4285

0.1008

0.235 9

0.3249

2.2

0.4101

0.0943

0.126 1

1.5801

3.3

0.4739

0.0852

0.179 2

2.2674

3.3

0.2126

0.0771

0.138 0

9.3195

4.8

0.2845

0.0654

0.321 7

5.6115

4.8

0.0089

0.0052

0.374 4

2.9589

5.6

0.0245

0.0245

1.000 7

9.1365

5.6

0.0075

0.0032

0.432 0

10.3503

0

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SX3

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TY1 P (MPa)

Da´/Ra2 （s-1）

Di´/Ri2 （s-1）

α

β/α

P （MPa）

Da´/Ra2 （s-1）

Di´/Ri2 （s-1）

α

β/α

0.2

0.7696

0.0952

0.019

0.0426

0.2

0.9152

0.0900

0.163

0.1533

0.8

0.6811

0.1179

0.090 7

0.6969

0.8

0.6694

0.1156

0.178 9

0.8379

2.2

0.5012

0.1336

0.266 6

1.5963

2.2

0.3572

0.1377

0.301 6

1.2564

3.3

0.3776

0.0815

0.170 5

9.5058

3.3

0.2221

0.1196

0.229 3

2.1636

4.8

0.2209

0.0410

0.185 6

6.2571

4.8

0.1581

0.0609

0.233 1

8.7309

5.7

0.1607

0.0269

0.267 5

12.037

5.6

0.0955

0.0283

0.244 0

1.3494

3

2

Da´/Ra2 （s-1）

Di´/Ri2 （s-1）

α

β/α

TY2

TY3

8

P (MPa)

Da´/Ra2 （s-1）

Di´/Ri2 （s-1）

α

β/α

P （MPa）

0.2

1.0542

0.0341

0.032

0.0993

0.2

1.0913

0.0543

0.049

0.1326

0.8

0.6701

0.0798

0.119 3

0.5541

0.8

0.7519

0.0697

0.126 7

0.2781

2.2

0.5401

0.1053

0.250 1

3.2922

2.2

0.3733

0.0737

0.136 3

0.0243

3.3

0.4476

0.0551

0.212 5

5.5059

3.3

0.3467

0.1193

0.184 6

8.0757

4.8

0.0227

0.0202

0.339 4

4.9387

4.8

0.2886

0.0794

0.234 4

11.7174

5.5

0.0223

0.0050

0.222 2

0.2595

5.5

0.1531

0.0303

0.353 8

14.8362

1

7

4. Discussion 4.1 Compositional control of the shale matrix over the pore volume Based on the image-based statistical results of different kinds of pores shown in Figure 3, we found that different kinds of pores have their own size ranges. For example, the pores associated with brittle minerals have a wider pore size distribution ranging from tens of nanometers to a micrometer, with pore size mainly ranged from 50 – 300 nm, while pores associated with clays and OM have a relative narrow pore size distribution, with pore size were generally smaller than 100 nm. From this statistical results of pore size, we think that the shale composition possess a significant effect on pore volume. Seeking relationships between shale compositions and pore volumes is a common method for determining compositional control over the pore volume of shale. The micropore (

Experimental and Modeling Study on the Effect of Shale Composition

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