An innovative method to characterize sorption-induced kerogen swelling in organic-rich shales

Fuel 254 (2019) 115629

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Full Length Article

An innovative method to characterize sorption-induced kerogen swelling in organic-rich shales

T

Yu Panga,b, Yongming Heb, Shengnan Chena,

⁎

a b

Department of Chemical and Petroleum Engineering, University of Calgary, AB T2N 1N4, Canada State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu University of Technology, Chengdu 610059, PR China

ARTICLE INFO

ABSTRACT

Keywords: Kerogen swelling Gas sorption Absorption Bulk and pore strains

Matrix swelling due to the gas sorption has not been properly accounted in the organic-rich shale formations. In general, the adsorbed gas attaches on the pore surface in the kerogen and clay minerals and occupies the pore volume. Meanwhile, the absorbed gas dissolves or diffuses into the matrix (solid lattice) of kerogen, resulting in the swelling of kerogen. As a result, the increase of the adsorbed and absorbed gases leads to a reduction of the pore volume for the free gas. Thus, it is essential to appropriately characterize the kerogen swelling caused by gas sorption so that to accurately determine the nanopore structure and the gas transport behavior in shale kerogen. In this study, an innovative method is developed to evaluate the swelling behavior of kerogen ascribed to the methane sorption. The method aims at characterizing the regions of adsorbed gas, absorbed gas, and free gas in the slit-shaped nanopores in shale kerogen based on the density profile determined using the Simplified LocalDensity (SLD) model. Results for Barnett and Eagle Ford shale kerogens reveal that the gas adsorption prevails when the pore pressure is less than 6 MPa, whereas the gas absorption dominates the gas sorption process when the pore pressure is larger than 6 MPa. In addition, the ratio of absorption thickness to adsorption thickness increases with the pore pressure and such ratio reaches the 34.4% and 25.4% at the pore pressure around 20 MPa for the Barnett and Eagle Ford shale kerogens, respectively. Furthermore, the bulk and pore volumetric strains of the kerogen in the Barnett and Eagle Ford shale core samples are calculated based on the swelling length of kerogen caused by the methane sorption. The calculated strains of kerogen are in line with the measured strains of the shale and coal samples published in the literature, which verifies the reliability of the proposed innovative method. Finally, the impacts of methane sorption on the surface diffusion and the gas slippage in the slit nanopores of kerogen are also discussed. The proposed method provides a practical approach to characterize the sorption-induced kerogen swelling in shales, which is difficult to measure in the laboratory. The findings of this study advance the understanding of gas sorption process and give insight into the characterization of nanopore structure and gas transport mechanism in shale kerogen.

1. Introduction

proportion of gas-in-place (GIP) in shale gas reservoirs [5,9]. Currently, there are two typical methods to measure the gas sorption capacity in the laboratory, volumetric and gravimetric methods. The volumetric method is not a direct measurement of gas sorption amount. The amount of gas sorption is calculated by the real gas law. Normally, the volumetric method is the gas expansion method, which expands an amount of sorptive gas from a pressure cell into an evacuated measuring cell during an isothermal process [10–12]. On the other hand, the gravimetric method enables to expose a testing core sample to the pure sorptive gas at constant temperature. In terms of gravimetric method, the magnetic suspension balance, which has been widely used,

Shale gas reservoirs are currently regarded as the major targets of natural gas production worldwide [1–3], and the gas sorption capacity, nanoscale porosity, and extremely low permeability are the major characteristics of shale formations [4]. The storage mechanisms of the natural gas in the shale gas reservoirs is diveided into three categories: the free gas in the pores and natural fractures, the adsorbed gas on the surface of pore walls in kerogen and clays, and the dissolved (absorbed) gas into the formation water and kerogen [5–8]. The gas sorption, including the adsorbed gas and absorbed gas, accounts for a large

⁎

Corresponding author. E-mail address: [email protected] (S. Chen).

https://doi.org/10.1016/j.fuel.2019.115629 Received 19 April 2019; Received in revised form 7 June 2019; Accepted 11 June 2019 Available online 26 June 2019 0016-2361/ © 2019 Elsevier Ltd. All rights reserved.

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allows to expose the core sample to the sorptive gas while keeping the balance located under ambient condition [7,13–15]. The volumetric method allows to measure the changes in volumetric strain and permeability while determine the gas sorption amount. However, the gravimetric method appears to be more reliable due to the direct measurement of gas sorption amount rather than the volumetric method which requires the calculation of gas sorption amount by the real gas law. The gas adsorption in shales has been extensively investigated, where kerogen and clays are the two main contributors of the gas adsorption in the shale formations. Kerogen associated with a large amount of micropores is the primary control on the gas adsorption in the shale formations [16–18]. The micropores in kerogen provides the large internal surface area for the molecular interactions between the gas and the solid, which facilitate the gas adsorption [5,16,19]. Therefore, it is observed in several studies that methane sorption capacities in shale and isolated kerogen positively correlate with total organic carbon (TOC) [16,17,20,21]. In addition, clays are able to provide the extra adsorption capacity in shales because of their vast internal surface area of the micropores and fine mesopores [18,22]. However, such capacity can be greatly reduced when the formation water and moisture content are presented in shale formations [18,23–25]. Due to the hydrophilic nature of the clay minerals, water and moisture mainly occupy the adsorption sites of clay minerals and restrict the access of methane to the adsorption sites in the nanopores [18]. The gas absorption, on the other hand, is not properly accounted in the literature. The concept of gas absorption refers to the gas dissolution in water, organic solvents, polymers, and coals [6,25–30]. It is very difficult to distinguish between the gas adsorption and the gas absorption under the experimental conditions because the gas absorption occurs immediately after the gas adsorption [5,27]. To differentiate the absorption from the adsorption, various attempts were made to depict the process of absorption [5,6,25,26,31–33]. Milewska-Duda et al. proposed the dual sorption model and the multiple sorption model to account for both adsorption and absorption phenomena in subprocesses and subsystems, respectively [26,31–33]. These two models were successfully applied to describe the adsorption and absorption of CH4 and CO2 in coal and active carbon. Etminan et al. measured the gas absorption capacity and the gas diffusion coefficient of a shale core sample using the batch-pressure-decay method [6]. In addition, Jin and Firoozabadi used the solid solution model to account for gas dissolution in kerogen by molecular simulation [25]. Moreover, Pang et al. proposed an experimental and analytical combined method to estimate the gas absorption in shale kerogen [5]. All these studies indicate that the gas absorption in organic matter takes place during the process of gas sorption, which cannot be ignored. More significantly, the rock matrix swelling induced by gas sorption has been widely observed in the coal and shale kerogen [21,34–43]. The measurements of volumetric strain induced by gas sorption have been carried out using different gases in coal and shale core samples [21,36,39–41,43]. In general, it was found that the swelling strain caused by the sorption of CH4 or CO2 is linearly proportional to the absolute gas sorption amount [21,36,43]. In addition, various models have been proposed to characterize the changes in porosity and permeability for coal considering the sorption-induced swelling and shrinkage [44–46]. Furthermore, coal matrix swelling due to the dissolved CO2 can lead to the increase of the bulk volume and surface area [34,35]. CO2 dissolves into the coal and serves as a plasticizer, which results in a change in the coal structure [27]. The volumetric strains reported in the previous studies are at the bulk sample scale, however, it is very difficult to measure the volumetric strain at pore scale. As a result, the previous studies by Cui et al. and Liu et al. assumed that the sorption-induced bulk volumetric strain is identical to the sorption-induced pore volumetric strain for the permeability models of coal [44,47]. Although this assumption could be used to describe the pore

volume change of cleats in coal, it might be inappropriate to assume that the sorption-induced pore volumetric strain is equal to the sorption-induced bulk volumetric strain based on the permeability measurements of coal [39]. Accordingly, the sorption-induced pore volumetric strain is believed to be different from the sorption-induced bulk volumetric strain for the nanoporous shale kerogen. The matrix swelling leads to a great reduction of the pore width/diameter and affects the determination of Knudsen number and the accuracy of gas flow regime selection. Thus, it is crucial to find a method to determine the effects of sorption-induced matrix swelling on the nanopore structure in shale kerogen. This study is an extension of our recent work on differentiating the absorbed gas from the adsorbed gas via the experimental and analytical investigations. In our previous study, the sorption capacities of methane on the Barnett and Eagle Ford shale core samples were measured experimentally via the volumetric method and analyzed theoretically using the SLD-PR model. Then, the absorbed gas was distinguished from the adsorbed gas by analyzing the gas expansion experiment using the Fick’s law of diffusion [5]. In this study, an innovative method was proposed to describe the swelling behavior of kerogen at both bulk and pore scales. The bulk and pore volumetric strains of the Barnett and Eagle Ford shale kerogens were calculated and compared with the published bulk and pore volumetric strains of the shale and coal samples. Our overall objective is to recognize the adsorbed gas, absorbed gas, and free gas regions in the organic nanopores and accurately describe the pore structure and gas transport behavior in shale kerogen. 2. Sorption-induced kerogen swelling characterization A novel method, which consists of the experimental measurements and the theoretical computation, is developed to depict the swelling behavior of shale kerogen due to the gas sorption. The vital parts of the proposed method are the determination of the absorbed gas amount via the experiments and the calculation of the Gibbs and absolute sorption amounts based on the density profile in the slit pore provided by the SLD-PR model. 2.1. Distinguish absorption from adsorption via experimental measurements To differentiate the absorbed gas from the adsorbed gas, the total gas sorption consisting of the adsorbed and absorbed gases is measured using the magnetic suspension sorption system (gravimetric method) first and followed by the estimation of the gas absorption amount through analyzing the results of the gas expansion measurements with the Fick’s second law of diffusion along with the results of the measured total gas sorption amount. The experimental facilities and the measurement procedures are presented in our previous study [5] and the data used in this study is demonstrated in Appendix A. The main steps for distinguishing the absorbed methane from the adsorbed methane are summarized as follows. 1) The sorption capacity of the methane on the shale core sample was evaluated by fitting the measured Gibbs (excess) sorption data with the SLD-PR model. This model describes the gas sorption behavior in a rectangular-shaped slit considering the gas-gas and gas-solid interactions. The details of the SLD-PR model are provided in Appendix B. 2) The absolute methane sorption amount for the shale core sample was derived from the measured Gibbs sorption amount based on the density profile obtained from the SLD-PR model. 3) The absolute sorption capacity attributed to the clay minerals in the shale core sample was excluded to determine the absolute sorption capacity contributed from the kerogen. 4) The absorbed methane was distinguished from the adsorbed methane through analyzing the relationship between absolute methane sorption amount in kerogen, testing pressure, and testing time in the 2

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specific surface area (in m2/g), and z is the position in the slit (in nm). To obtain the density profile, Eq. (1) is used to fit the measured Gibbs gas sorption capacity in the laboratory. The quality of the regression analysis using the SLD-PR model is controlled by four regression parameters: the specific surface area, A (in m2/g); fluid-solid interaction energy, εfs/k (in Kelvin); slit width, L (in nm); and covolume correction factor, Ab. In light of the density profile, the experimentally measured gas absorption amount in the kerogen of the shale core samples can be calculated by integrating the position-dependent density (the blue curve) in the slit pore. Due to the symmetric feature of the slit pore, the integration is demonstrated in half of the slit as illustrated in Fig. 2b. The insert graph in the right upper corner of the Fig. 2b is the enlarged image of the density profile for the absorbed gas. The integral calculation is provided by Eq. (2), which starts at the center of the adsorbed methane molecule attaching on the surface of the slit pore at σff/2 and ends at the cutoff point of the gas absorption in kerogen, from which the cutoff point of the gas absorption in the shale kerogen is solved numerically by the trapezoidal rule. Thus, the area under the blue curve in the range from the black solid line to the green solid line represents the absolute gas absorption capacity in shale kerogen.

gas expansion measurements. 5) Characterize the separated gas absorption process with the Fickian diffusion to verify the gas absorption capacity and gas diffusion coefficient in the kerogen. Following this technique, the amounts of adsorbed and absorbed gases in both kerogen and shale core sample were obtained. 2.2. Kerogen swelling due to gas absorption Matrix swelling due to the gas sorption has been intensively studied in coals [38,39,44–46], as well as the swelling behavior of the kerogen due to the dissolution of multicomponent solvents in the organic-rich shale (mudstone) [28,29]. However, the kerogen swelling attributed to the gas diffusion/dissolution is not well recognized in shale. In this study, an innovative method is proposed to characterize the kerogen swelling. The calculation method relies on the density profile in the slit pore and the amount of absorbed gas estimated using the gas expansion technique. All the pores in the testing shale core sample are assumed as the slitshaped pores on the basis of the SLD-PR model. The schematic diagram of the slit pore in shale kerogen is demonstrated in Fig. 1. The slit pore width (L) is determined through the regression analysis using the SLDPR model coupled with the pore size distribution restriction. The detailed information of the pore size distribution can be found in Appendix C. The diffusion/dissolution depth (h) denotes the distance from the surface of the kerogen to the center of the kerogen. Thus, the 2 × h indicates the original thickness of the shale kerogen. The total surface area of the slit pore is the area available for the gas adsorption in shale kerogen and we deem that the area available for the gas adsorption is identical to the area available for the gas absorption. The adsorbed gas molecules reside on the surface of the slit pore because of the Van der Waals forces, while the absorbed gas molecules diffuse/ dissolve into the interstitial space in the solid lattice of kerogen rather than remain on the pore surface. As a result, the absorbed gas leads to the matrix swelling of kerogen and thus reduces the pore width for the free gas. The kerogen swelling is illustrated as the matrix of kerogen swells from the original thickness (dashed lines) to the current thickness (solid lines) in Fig. 1. In general, the density profile in the slit pore is displayed in Fig. 2a. The blue curve represents the density profile in the slit pore and the red dashed line indicates the density of bulk phase (gas). Based on the SLDPR model, Gibbs (excess) gas sorption capacity is calculated by:

nGibbs

sorp

where n

=

A 2

Gibbs sorp

L ff

2

Abs abs nkerogen =2

( (z )

bulk ) dz

z c absorption ff

2

(z ) dz

(2)

where α is the correction factor of the surface area for the kerogen. In light of the geometry of the slit pore in Fig. 1, it is assumed that the difference between the absolute gas sorption amounts contributed from the whole shale core sample and the kerogen respectively is ascribed to the variation in surface area between the whole shale core sample and the shale kerogen. In addition, zc-absorption denotes the distance from the slit-pore wall to the cutoff point corresponding to the gas absorption in the kerogen. Consequently, the region of zc-absorption – σff/2 in Fig. 2b represents the gas absorption amount in the kerogen. It is worth noting that although the amount of gas absorption can be calculated by Eq. (2), the absorbed gas molecules are supposed to diffuse/dissolve into the interstitial space of the kerogen rather than attach on the surface of the slit pore. To involve the effect of gas diffusion/dissolution into the calculation, the process of gas diffusing/dissolving into the bulk kerogen must be clearly understood. It was concluded in our previous study that the gas adsorption and gas absorption take place in a time sequence, and the gas adsorption is the prerequisite of the gas absorption [5]. At the beginning of the gas sorption, the free gas molecules prefer adsorbing on the surface of the slit pore, which generate an adsorption film. Subsequently, until the adsorbed gas reaches the critical gas concentration to initiate the gas transfer into the kerogen, the adsorbed gas molecules diffuse/dissolve into the kerogen through the interface between the adsorbed gas and the kerogen matrix. The absorbed gas molecules attempt to take up available absorption sites (interstitial space) in the crosslinked network of kerogen and meanwhile the free gas molecules occupy the available adsorption sites on the slit pore wall. In the end, the diffusion process stops when the gas concentration at the two sides of the interface reaches equilibrium, indicating that the balance of adsorption and absorption is achieved at a given pressure. Owing to the dynamic equilibrium for the gas molecules at the interface between the absorbed-phase and the adsorbed-phase, the density or concentration of absorbed gas should be identical to the density or concentration of adsorbed gas. Thus, Eq. (3) is developed to express the dynamic equilibrium.

ff

2

A 2

(1)

is the Gibbs sorption amount (in mmol/g), A is the

mads = Ai tads

Fig. 1. Schematic of slit-shaped pores in shale kerogen. 3

ads

=

mabs = Ai tabs

abs

(3)

Abs abs mads = nkerogen W

(4)

Abs abs mads = nkerogen W

(5)

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Denstiy Proflie

Bulk Phase Density

30000

Density (mol/m3)

25000 20000 15000 10000 5000 0

0

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

3

3.3 3.6 3.9 4.2

Pore Width (nm)

a

b Fig. 2. Density profile and bulk phase density in the slit pore.

where mads and mabs are the amounts of the gas adsorption and absorption in the shale kerogen, respectively. Ai is the area of the interface between the absorbed gas and the adsorbed gas in the kerogen. The tads and tabs are the adsorption thickness and absorption thickness, respectively. In this study, the tabs is regarded as the length of sorption-induced kerogen swelling. The ρads and ρabs are the densities of the adAbs ads sorbed and absorbed gases. and nkerogen are the absolute gas absorption and adsorption capacities in the kerogen. W is the weight of the testing core sample. The ratio of mads to mabs, which can be determined in the laboratory, is equal to the ratio of tads to tabs. The equation is written as:

Gibbs ads (nkerogen 1 ) considering the kerogen swelling is calculated using the SLD-PR model with the Lnew and Ab. The main challenge of this calculation workflow is to examine whether the Gibbs adsorption calculated using the SLD-PR model with the new slit width is identical to its value calculated by Eq. (7) with the original slit width (L). Furthermore, the Abs ads absolute gas adsorption capacity in the kerogen (nkerogen 2 ) detected in the laboratory can also be computed using the SLD-PR model with the Lnew and Ab by integrating the density profile in the range starting from the center of the adsorbed methane molecule attaching on the surface of the new slit pore to the position at tads as given by Eq. (11).

mads t = ads = k mabs tabs

Abs ads nkerogen 1 = 2

(6)

Furthermore, the Gibbs (excess) gas adsorption capacity in the Gibbs ads kerogen (nkerogen 2 ) illustrated by the area in yellow in Fig. 2b can be determined following the Eqs. (7)–(9). Gibbs sorp Gibbs ads nkerogen 2 = 2 × n yellow = nkerogen

Gibbs sorp nkerogen =

ngreen =

A 2

A 2 (z c

L ff

absorption nkerogen + 2 × ngreen

( (z )

bulk ) dz

2

absorption

ff

2

)

bulk

(7) (8) (9)

+ tads

(z ) dz

2

(11)

In this study, a simplified shale model consisting of the kerogen (organic) matrix, inorganic matrix, and three slit pores is developed as illustrated in Fig. 4. The slit pores between the kerogen and inorganic matrixes are the interparticle pores and the slit pore between the two kerogen matrixes is the intraparticle pore. The region surrounded by the red dashed line refers to the shale kerogen. This model considers that only gas absorption will cause the kerogen swelling, the pore volume occupied by the gas adsorption on the surface of the kerogen is not accounted. The swelling length (tabs) as the orange stripes shown in Fig. 4 results in the increase of the bulk kerogen volume and the decrease of the pore volume in kerogen. To keep the consistency between the surface area expressed in Eq. (2) (two-side slit surface in SLD model) and the surface area in Fig. 4 (four-side surface), each side of the kerogen matrix represents a quarter of total surface area open to gas sorption. In general, the bulk volumetric strain of shale kerogen can be defined as:

(1) The kerogen swelling associated with gas sorption is only expressed by extending the thickness of the kerogen (2 × h in Fig. 1). In other words, the surface area of the slit pore is regarded as a constant during the gas absorption process, which indicates that the specific surface area in the kerogen (Ai × α) is constant. (2) The fluid-solid interaction energy (εfs/k) of the SLD-PR model is regarded as a constant during the gas absorption process. In the workflow, the new slit width considering the sorption-induced kerogen swelling is calculated by:

2 × tabs

2

2.3. Bulk and pore volumetric strains of shale kerogen

where nyellow and ngreen are the areas in yellow and green in Fig. 2b, respectively. bulk is the density of bulk phase (red dashed line in Fig. 2b). The tabs can be determined after obtaining the Gibbs adsorption Gibbs ads capacity in kerogen (nkerogen 2 ). The calculation workflow is shown in Fig. 3. The main assumptions of the calculation workflow are:

Lnew = L

ff ff

Finally, the swelling length of kerogen due to the gas absorption (tabs) can be determined at given pressures. The corresponding new slit width, adsorption thickness, and effective width for the free gas are also calculated accordingly. It is worth pointing out that the innovative method proposed in this study, which considers the sorption-induced kerogen swelling, can successfully improve the applicability of the SLDPR model by extending the SLD-PR model to calculate the density profile and gas sorption amount using the variable slit width.

ff

2

A 2

(10)

where Lnew is the new slit width after the kerogen swelling and L is the initial slit width. The Gibbs gas adsorption capacity in the kerogen

Sb = 4

V k0 V k0

Vk

(12)

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Fig. 3. Workflow chart to determine the absorption thickness (ABS denotes the absolute value).

Fig. 4. Schematic of the shale kerogen model. 5

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where Sb denotes the bulk volumetric strain of the kerogen, Vk and V k0 are the current and initial bulk volume of the kerogen, respectively. Accordingly, the tensile strain corresponding to the swelling is presented by a negative value. Hence, the V k0 and Vk before and after the swelling can be calculated by:

V k0 =

A 4

(4 h + L)

(13)

Vk =

A 4

(4 h + 4 tabs + Lnew )

(14)

Table 1 zc-absorption and Ab for the Barnett and Eagle Ford shale core samples. Barnett

Substituting the Eqs. (13) and (14) into the Eq. (12), the volumetric strain can be expressed as:

Sb =

2 tabs 4 h+L

(15)

Pressure (MPa)

zc-absorption (nm)

Ab

Pressure (MPa)

zc-absorption (nm)

Ab

1.99 3.992 6.009 8.008 10.05 11.98 13.99 15.99 17.98 19.99

0.1969 0.2035 0.2101 0.2166 0.2233 0.2296 0.2361 0.2426 0.2492 0.2559

0.4721 0.3361 0.2152 0.2026 0.2000 0.1998 0.1960 0.2062 0.2028 0.2127

2.02 4.01 5.99 8.02 9.98 12.01 13.98 16.01 18.02 20.03

0.1957 0.2011 0.2062 0.2115 0.2165 0.2217 0.2268 0.2321 0.2372 0.2425

0.3111 0.1657 0.0700 0.1176 0.1118 0.0900 0.0827 0.0797 0.0739 0.0698

In addition, the reduction of slit width shown in Fig. 4, which is expressed by the pore volumetric strain of kerogen can be calculated as:

Swelling length (Barnett)

Swelling length of kereogen, tabs (nm)

2 tabs Sp = L

Eagle Ford

(16)

where Sp is the pore volumetric strain of kerogen. 3. Case study for kerogen swelling In this study, Barnett and Eagle Ford shale core samples were used to reveal the kerogen swelling calculated using the proposed innovative method. As an extension of our previous study, we will focus on presenting the kerogen swelling determined following the calculation workflow (Fig. 3) in this section. For the detail of the rock information and the measurements of methane adsorption and absorption in the shale kerogens and the shale core samples, please refer to our previous study [5]. Herein, a brief summary about the experimental results is provided to state the significant characteristics and factors that will be used in the kerogen swelling calculation. First, the trends of the absolute methane sorption and adsorption amounts in the kerogens and the shale core samples match the typical Langmuir form, while the amounts of methane absorption in the kerogens are linearly correlated with the pore pressure (see Appendix A). Second, the kerogen in the Barnett shale contributes 87% of the absolute gas sorption capacity of the whole Barnett shale core sample, while the kerogen in Eagle Ford shale contributes 94.8% of the absolute gas sorption capacity of the whole Eagle Ford shale core sample. Thus, the correction factors of the surface area (α) for the Barnett and Eagle Ford shales are 0.87 and 0.948, respectively. Third, the Gibbs sorption isotherms calculated by the SLD-PR model properly fit those measured via the magnetic suspension sorption system in the laboratory. The slit pore widths (L) for the Barnett and Eagle Ford shale core samples determined from the curve fitting are 3.7 nm and 4.2 nm, respectively. Fourth, in light of the slit-pore geometry and the measured TOC content, the thicknesses of kerogen (2 × h) in the Barnett and Eagle Ford shale core samples are 10.04 nm and 3.12 nm, respectively. To determine the length of sorption-induced kerogen swelling, the absorption thickness (tabs), adsorption thickness (tads), and slit-pore width for the free gas (D) at each testing pressure for the Barnett and Eagle Ford shale core samples are computed in accordance with the calculation workflow. The used zc-absorption and the determined Ab in the calculation workflow are displayed in Table 1. The absorption thickness (tabs) indicating the length of kerogen swelling associated with gas sorption is presented in Fig. 5. It is found that the swelling length increases with the pore pressure, which demonstrates that the increase of gas diffusion/dissolution amount into the kerogen results in a more obvious swelling of shale kerogen. But, the trend of tabs versus pore pressure matches the Langmuir type (convex) curve, which implies that the increase of tabs will approach a maximum value as the pore pressure increases. Furthermore, Fig. 6

Swelling length (Eagle Ford)

0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

5

10

15

20

25

Pore Pressure (MPa)

Fig. 5. Swelling length of kerogen for Barnett and Eagle Ford shale core samples.

reveals that the adsorption thickness (tads) increases initially and then decreases with the pore pressure. The initially sharp increase of the tads indicates that the gas adsorption dominates the lower pore pressure range (P < 6 MPa) and the following decrease of the tads implies that the adsorbed gas begins to considerably diffuse into the kerogen, hence, gas absorption dominates the higher pore pressure range (P > 6 MPa). Excluding the absorption and adsorption thicknesses, the pore width for the free gas (D) initially decreases and then slightly increases. Thereby, the adsorbed and absorbed gases need to be well characterized in order to depict the free gas transport in nanoporous shale kerogen. Additionally, Fig. 7 presents that the ratio of absorption thickness to adsorption thickness (tabs/tads) increases with the pore pressure. The trend of tabs/tads versus pore pressure fits in with a slightly concave curve, which indicates that the swelling of shale kerogen is more obvious at the higher pressures. It can be seen that the absorption thickness accounts for approximately 34.4% of the adsorption thickness for the Barnett shale kerogen and the absorption thickness accounts for approximately 25.4% of the adsorption thickness for the Eagle Ford shale kerogen at the pressure close to 20 MPa. Accordingly, the kerogen swelling associated with the gas absorption deserves close attention. 4. Results and discussion In this section, the swelling effects on the bulk volume, pore volume, density profile, and gas transport mechanisms will be presented, and the feasibility of the proposed method will be discussed. 4.1. Sorption-induced volumetric strains of shale kerogen Typically, the bulk volumetric strain of the shale core sample is acquired during the gas sorption process to represent the matrix swelling caused by gas sorption. However, the swelling of kerogen due to 6

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Pore width for free gas (Barnett)

Pore width for free gas (Eagle Ford)

Adsorption thickness (Barnett)

Adsorption thickness (Eagle Ford)

3.60

0.42

3.50

0.40

3.40

0.38 0.36

3.30

0.34

3.20

0.32

3.10

0.30

3.00

0.28

2.90

0.26

2.80

0.24

2.70

0.22

2.60

0

5

10

15

20

25

Adsorption thickness, tads (nm)

Pore width for free gas, D (nm)

Y. Pang, et al.

0.20

Pore Pressure (MPa) Fig. 6. Pore width for free gas and adsorption thickness of Barnett and Eagle Ford shale core samples.

the gas absorption cannot be detected directly, because it is difficult to measure the changes in the bulk volume of the isolated kerogen. Fig. 8 depicts the calculated sorption-induced bulk volumetric strains at the testing pressures for the Barnett and Eagle Ford shale core samples. Based on the investigations of the coal matrix swelling and shrinkage due to gas sorption and desorption, the bulk volumetric strain of coal can be fitted to the Langmuir type curve given as follows [46,48].

Sb = a

P P+b

To further validate the calculated kerogen swelling, the bulk volumetric strains due to the methane sorption in Fig. 8 are compared with the measured bulk volumetric strains of the shale core sample, organic carbon, and illite with respect to the methane sorption [21,36]. Fig. 9 illustrates that the sorption-induced bulk volumetric strain is heavily dependent on the absolute methane sorption amount, where the absolute gas sorption amount in the kerogen for the Barnett and Eagle Ford shales is its measured value normalized to the TOC content. It is observed that the sorption-induced bulk volumetric strain reveals an approximately linear relationship to the absolute gas sorption amount. Therefore, the linear relations obtained from this study (Barnett and Eagle Ford) are identical to those measured in the laboratory on different rocks. In addition, the magnitude of the calculated bulk volumetric strains of kerogens is nearly on the same order of that measured on the organic carbon and an order of magnitude larger than those measured on shale core samples and illite. The parallel magnitude of the bulk volumetric strains of shale kerogen and organic carbon demonstrates the equivalent swelling behaviors, which verifies the accuracy and reliability of the proposed method to characterize the swelling length in shale kerogen. Moreover, the measured bulk volumetric strains of the shale core samples are in the middle of those measured

(17)

where a and b are fitting parameters of Langmuir type curve, and P is the pressure. As expected, the bulk volumetric strain representing the sorptioninduced kerogen swelling matches the Langmuir type curve in Fig. 8. As the main component of the coal and kerogen is the organic matter, the coal and shale are supposed to share a similar trend between the bulk volumetric strain and the pressure. Hence, the swelling length predicted using the proposed method could properly describe the swelling of shale kerogen. The fitting parameters and R2 values are displayed in Table 2. Barnett

0.40

Eagle Ford

0.35 0.30

tabs/tads

0.25 0.20 0.15 0.10 0.05 0.00

0

5

10

15

20

Pore Pressure (MPa) Fig. 7. Ratio of tabs to tads versus pore pressure for Barnett and Eagle Ford shale kerogens. 7

25

Fuel 254 (2019) 115629

Bulk volumetric strain of kerogen (-) (dimensionless)

Y. Pang, et al.

Kerogen in Barnett

Kerogen in Eagle Ford

Langmuir type fitting for Barnett

Lagmuir type fitting for Eagle Ford

0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000

0

5

10

15

20

25

Pore Pressure (MPa) Fig. 8. Bulk volumetric strain of kerogen in the Barnett and Eagle Ford shales.

(16), which are −6.43 and −2.49 times of the corresponding bulk volumetric strains of the shale kerogen (Sb). The negative sign implies the reverse direction of the swelling of bulk volume and the shrinkage of pore volume. This finding matches the observation that the sorptioninduced pore volumetric strain is proportional to the sorption-induced bulk volumetric strain in coal [39]. Therefore, to some extent, the linear correlation between the sorption-induced pore and bulk volumetric strains in the kerogen implies that the calculated swelling of kerogen using the proposed method is reasonable.

Table 2 Fitting parameters and R2 values of the Langmuir type curve. Core Sample

a

b (MPa)

R2

Barnett Eagle Ford

−0.01406 −0.02262

12.02 8.913

0.979 0.973

strains of organic carbon (or kerogen) and illite. Thus, the combination of the bulk volumetric strains of kerogen and clays may represent the bulk volumetric strain of shale core samples because the kerogen and clays are major components that lead to the swelling in shale, when gas sorption takes place. In addition, the pore volumetric strains of kerogen (Sp) for the Barnett and Eagle Ford shale core samples can be calculated by Eq.

4.2. Sorption effects on gas flow behavior in shale kerogen The kerogen swelling caused by gas sorption not only affects the bulk and pore structures of the kerogen but also changes the density profile in the slit pore. Fig. 10 demonstrates the density profiles of the Kerogen in Barnett

Sorption-induced bulk volumetric strain (-)

0.1

Kerogen in Eagle Ford Shale 1 (Chen et al. 2015) Shale 2 (Chen et al. 2015)

0.01

Organic Carbon (Heller and Zoback 2014) Illite (Heller and Zoback 2014) Linear (Kerogen in Barnett)

0.001

Linear (Kerogen in Eagle Ford) Linear (Shale 1 (Chen et al. 2015)) Linear (Shale 2 (Chen et al. 2015))

0.0001

Linear (Organic Carbon (Heller and Zoback 2014)) Linear (Illite (Heller and Zoback 2014)) 0.00001 0.01

0.1

1

10

Absolute gas sorption amount (mmol/g) Fig. 9. The relationship between sorption-induced bulk volumetric strain and absolute gas sorption amount.

8

Fuel 254 (2019) 115629

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P=8.008, L=3.5718

P=2.02, L=4.1444

P=8.02, L=4.0928

P=13.99, L=3.5308

P=19.99, L=3.4962

P=13.98, L=4.059

P=20.03, L=4.029

P=1.99, L=3.7

P=8.008, L=3.7

P=2.02, L=4.2

P=8.02, L=4.2

P=13.99, L=3.7

P=19.99, L=3.7

P=13.98, L=4.2

P=20.03, L=4.2

30000

25000

25000

20000

20000 Density (mol/m3)

Density (mol/m3)

30000

P=1.99, L=3.6464

15000

15000

10000

10000

5000

5000

0

-1

-0.9

-0.8

-0.7

0

-0.6

-1

-0.9

-0.8

-0.7

Relative position in the slit pore

Relative position in the slit pore

Barnett

Eagle Ford Fig. 10. Density profile near the slit wall.

adsorbed gas in the slit pore of kerogen with the variable slit width (Lnew) for the Barnett and Eagle Ford shale kerogens. As a comparison, the density profiles of the total gas sorption in the slit pore with the constant slit width (L = 3.7 nm for Barnett shale and L = 4.2 nm for Eagle Ford shale) are also displayed in Fig. 10. To better display the results, the Fig. 10 only presents the density profiles near the slit wall at the left side and the relative location in the slit is the location (distance) to the slit center normalized to half of the total slit width. The solid lines denote the density profiles of adsorbed gas calculated with the variable slit width (Lnew) and the dashed lines depict the density profiles of the total gas sorption calculated with the constant slit width (L). It can be seen that the density calculated with the Lnew is less than its value calculated with the L at a given pressure, which indicates that the sorption-induced kerogen swelling results in the reduction of the adsorbed gas density. Consequently, the density profile in the slit pore will be overestimated when ignoring the sorption-induced kerogen swelling and failing to distinguish the absorbed gas from the adsorbed gas, and this overestimation is more obvious at the lower pressures. In addition, the reduction of the adsorbed gas density due to the sorption-induced kerogen swelling may affect the gas transport in the slit-shaped nanopores. Previous studies have suggested that the velocity of the surface diffusion is dependent on the density of adsorbed gas [49,50]. The velocity of the surface diffusion is given as:

usurf

Ds = D¯ ads

g ug

partition coefficient. For the Langmuir sorption, the surface coverage is presented by:

=

(19)

where θ is the surface sorption coverage, na is the amount of adsorbedphase per mass of adsorbent, n0 is the maximum amount of adsorbedphase corresponding to adsorbent capacity, and Cµ is the mass of gas adsorbed per solid volume. Substituting the Eq. (19) into the Eq. (18), the velocity of surface diffusion is written as:

Ds D¯ ads

usurf =

g ug

Cµ (20)

C (1 + KC )

To clearly demonstrate the impact of density profile reduction on the surface diffusion, the ratio of the velocity of surface diffusion calculated with variable slit width to the velocity of surface diffusion calculated with constant slit width

(

usurf Lnew usurf L

) is calculated. It is noted

that Ds, D¯ , K, C, g and ug in Eq. (20) are constants at a given pore pressure. Thus, the surface diffusion velocity ratio is calculated by:

usurf usurf

KCµs (1 + KC ) 2

Cµ na KC = = n0 Cµs 1 + KC

Lnew L

=

Cµ

Lnew

Cµ

L

ads L

(21)

ads Lnew

In this study, the density of adsorbed gas is calculated by averaging the density profile in the range from σff/2 to tads.

(18)

where Cµs is the maximum mass of gas adsorbed per solid volume, Ds and D¯ are the surface diffusion coefficient and the equivalent diffusivity, C is free gas concentration, ads is the density of adsorbed gas, g is the density of free gas, ug is the gas velocity, and K is the Langmuir

ads Lnew

9

= ads ¯ =

tads ff 2

tads

(z ) Lnew dz ff

2

(22)

Fuel 254 (2019) 115629

Y. Pang, et al.

Surface Diffusion Velocity Ratio

Barnett

to the slit pore width for the free gas (D) shown in Fig. 6. The Knudsen number calculated with the slit pore width for the free gas by excluding the adsorption thickness and absorption thickness (tads and tabs) is compared with its value calculated with the constant slit pore width (L = 3.7 and 4.2 for Barnett and Eagle Ford, respectively), and the results are shown in Fig. 12. It is found that the Knudsen numbers are approximately in the range from 0.1 to 1.2 in this study, indicating that the gas flow regime for the Barnett and Eagle Ford shale core samples belongs to the transition flow regime [54,55]. The Knudsen number calculated with the variable D is larger than such value calculated with the constant L. This difference is more obvious at the lower pressure range (P < 6 MPa). Although the Knudsen number calculated with the constant L does not drop to below 0.1 at pressures from 0 to 20 MPa in this study, there is a possibility that the Knudsen number may drop to the value less than 0.1 with a larger pore width or at a higher pore pressure. If it happens, the flow regime changes from the transition flow to the slip flow and this change will lead to a significant influence on gas flow mechanism. For the Knudsen number located from 0.1 to 1, second-order gas slippage may able to describe the gas flow behavior in this flow regime [56–59]. Herein, the slip coefficient function (S) is used to evaluate the effects of Knudsen number on the gas slippage. For the slit pore geometry, the S is given as:

Eagle Ford

1.30 1.20 1.10 1.00 0.90 0.80 0.70

0

5

10

15

Pore Pressure (MPa)

20

25

Fig. 11. Surface diffusion velocity ratio for Barnett and Eagle Ford shale kerogens.

ads L

= ads ¯ =

tads ff 2

(z )L dz

tads

ff

(23)

2

where the (z ) Lnew and (z )L are the density profiles from the Eqs. (11) and (2), respectively. Moreover, the ratio of Cµ Lnew to Cµ L is equal to Abs ads the ratio of nkerogen 2 shown in the Appendix A to from the Eq. (2). Herein, we calculate the surface diffusion velocity ratio for the density profiles provided in Fig. 10 and the results are presented in Fig. 11. It is found that the surface diffusion velocity ratio is larger than 1 at pressures lower than 6 MPa, which indicates that the surface diffusion will be enhanced due to the sorption-induced kerogen swelling. However, such ratio is smaller than 1 at pressures larger than 6 MPa, which implies that the surface diffusion will be attenuated because of the swelling. Since the surface diffusion prevails in the gas transport in nanopores, especially in the micropores (pore diameter < 2 nm) and fine mesopores (pore diameter 2–5 nm) at the lower pressures (P < 5 MPa) [51–53], it is worth noting that the reduction of the adsorbed gas density, which may further exaggerate the contribution from the surface diffusion to the gas transport in organic nanopores at low pressures. Moreover, neglecting the gas sorption effects on the pore width will likely lead to an inaccurate estimation of the gas flow mechanism. In general, the gas flow regimes are characterized by Knudsen number (Kn) which is defined as the ratio of mean free path ( ) to characteristic length (s). Herein, we assigned the characteristic length (s) equivalent 1.4

Kn for Barnett with D

Kn for Eagle Ford with D

Kn for Barnett with L

Kn for Eagle Ford with L

where A1 and A2 are the first-order and the second-order slip coefficients. As the second-order gas slippage (S) is a function of Knudsen number, the increase of the Knudsen number attributed to the gas sorption effect indicates that the gas slippage effect will be fortified. To demonstrate the gas slippage enhancement, Knudsen numbers from 0 to 1 in Fig. 12 are used to calculate the slip coefficient function and the results are shown in Fig. 13. SD denotes the slip coefficient function determined with the Knudsen number calculated using the D (dots in Fig. 12) and SL denotes the slip coefficient function determined with the Knudsen number calculated using the L (dashed line in Fig. 12). A1 and A2 are 1.2455 and 0.4361, which are determined based on the measurements of gas flow in nanoporous media presented in previous study [60]. The ratio of SD to SL ranges from 1.34 to 1.15 for the Barnett shale and such ratio ranges from 1.27 to 1.12 for the Eagle Ford shale. This reveals that the gas slippage effect is enhanced when the Knudsen number is calculated with the variable D. Additionally, the slippage enhancement decreases as the pore pressure increases for the Knudsen number from 0.1 to 1. In sum, the kerogen swelling resulting from the gas absorption coupled wth the pore volume occupation ascribed to the gas adsorption should be considered for the gas flow regime determination and the gas slippage calculation in shale kerogen. Barnett

Eagle Ford

1.40

1.2 1.0

1.30

0.8

SD/SL

Knudsen number

(24)

S = 1 + 6A1 Kn + 12A2 Kn2

0.6 0.4

1.20

1.10

0.2 1.00 0.0

0

5

10

15

20

25

0

5

10

15

20

Pore Pressure (MPa)

Pressure (MPa)

Fig. 13. Slippage enhancement for Knudsen number from 0.1 to 1.

Fig. 12. Knudsen number for Barnett and Eagle Ford shale core sample.

10

25

Fuel 254 (2019) 115629

Y. Pang, et al.

5. Conclusions

with the correlation between the pore and bulk volumetric strains of coal obtained from the laboratory measurements. 5. The density profile calculated with a variable slit pore width resulting from the sorption-induced kerogen swelling is less than its value calculated with a constant slit pore width, which indicates that the density profile in the slit pore will be overestimated when failing to distinguish the absorbed gas from the adsorbed gas and neglecting the sorption-induced kerogen swelling. The reduction of adsorbed gas density caused by the sorption-induced kerogen swelling is prone to exaggerate the velocity of surface diffusion in organic nanopores at low pressures. 6. The Knudsen number calculated using the variable slit pore width attributed to the gas sorption effect is larger than its value calculated using the constant slit pore width, and the difference is more obvious at the lower pressure range (P < 6 MPa). When the Knudsen number ranges from 0.1 to 1, the kerogen swelling enhances the gas slippage effect in the slit nanopores and such enhancement decreases as the pore pressure increases.

This paper presents an innovative method to describe the sorptioninduced swelling behavior of shale kerogen at both bulk and pore scales. This method involves the experimental measurements and the analytical models to investigate the influences of gas sorption on the nanopore structure and the gas transport in shale kerogen. The main conclusions from this study are drawn as follows: 1. The proposed method employs the detected amounts of the absorbed and adsorbed gases in the laboratory coupled with the density profile calculated using the SLD-PR model to characterize the regions of adsorbed gas, absorbed gas, and free gas in shale kerogen. It further improves the applicability of the SLD-PR model to calculate the density profile with the variable slit pore width. 2. The sorption-induced kerogen swelling increases with the pore pressure, which results in a reduction of the pore width. In this study, the gas adsorption governs the total gas sorption process at the lower pore pressure range (P < 6 MPa), whereas the gas absorption dominates the total gas sorption process at the higher pore pressure range (P > 6 MPa). 3. The sorption-induced bulk volumetric strain of kerogen versus the pore pressure can be well fitted with the Langmuir type of curve and a linear correlation is observed between the sorption-induced bulk volumetric strain and the absolute gas sorption amount in shale kerogen. 4. The calculated pore volumetric strain associated with the gas sorption is linearly proportional to the calculated bulk volumetric strain associated with the gas sorption in shale kerogen, which is in line

Acknowledgement This research was undertaken thanks in part to funding from Open Fund (PLC20190801) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology) and the Canada First Research Excellence Fund entitled “Global Research Initiative in Sustainable Low Carbon Unconventional Resources”. The authors gratefully acknowledge a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) to S. Chen.

Appendix A The amounts of methane adsorption and absorption in the shale kerogen and the shale core sample are measured for the Barnett and Eagle Ford shales, respectively. Fig. A-1 demonstrates the absolute and Gibbs gas sorption capacities of the Barnett shale core sample and the absolute gas sorption, adsorption, and absorption capacities of the kerogen in the shale core sample. Similarly, Fig. A-2 presents those capacities of the kerogen and the shale core sample for the Eagle Ford shale. Absolute gas sorption in shale

Absolute gas sorption in kerogen

Absolute gas adsorption in kerogen

Absolute gas absorption in kerogen

Gibbs gas sorption in shale

SLD-PR

0.20

Gas sorption amount (mmol/g)

0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

5

10

15

Pore Pressure (MPa) Fig. A-1. Gas sorption amounts in kerogen and shale core sample for Barnett shale.

11

20

Fuel 254 (2019) 115629

Y. Pang, et al. Absolute gas sorption in shale

Absolute gas sorption in kerogen

Absolute gas adsorption in kerogen

Absolute gas absorption in kerogen

Gibbs gas sorption in shale

SLD-PR

0.24 0.22

Gas sorption amount (mmol/g)

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

0

5

10

15

20

Pore Pressure (MPa)

Fig. A-2. Gas sorption amounts in kerogen and shale core sample for Eagle Ford shale.

Appendix B For the SLD-PR model, the equilibrium chemical potential is the sum of the potentials from fluid-fluid and fluid-solid interactions, which is equal to the bulk fluid potential.

µ (z ) = µff (z ) + µfs (z ) = µbulk

(B-1)

where µ(z) is the chemical potential of fluid at position z. The subscripts “bulk”, “ff” and “fs” denote the bulk fluid, fluid-fluid, and fluid-solid interactions, respectively. The chemical potential of the bulk fluid can be written as a function of fugacity:

fbulk

µbulk = µ 0 (T ) + RTln

(B-2)

f0

where the subscript “0” refers to an arbitrary reference state and fbulk refers to the fugacity of the bulk fluid. Similarly, the chemical potential of a fluid-fluid interaction can be calculated using:

µ ff (z ) = µ 0 (T ) + RTln

f ff (z ) f0

(B-3)

where fff(z) is the fugacity of the fluid-fluid interaction at position z. The chemical potential of a fluid-solid interaction is given as:

µ fs (z ) = NA [

fs (z )

+

fs (L

z )]

(B-4)

where NA is Avogadro’s number and ψfs(z) and ψfs(L-z) are potential energy functions that account for a fluid molecule at position z interacting with both slit walls. According to Lee’s partially integrated 10–4 Lennard-Jones potential [61], the fluid-solid interaction can be calculated using the following equation: fs (z )

=4

atoms fs

2 fs

10 fs ' 10

5(z )

1 2

4 fs

4 i=1

'

(z + (i + 1)

ss )

4

(B-5) 2

In Eq. (B-5), ρatoms is the solid atom density, which is equal to 38.2 atoms/nm [62]. εfs is the fluid-solid interaction energy parameter. σfs is the average of σff and σss, which is expressed as σfs = (σff + σss)/2. The σff and σss represent the molecular diameter of the adsorbate and the carbon interplanar distance, respectively. The value of the carbon interplanar distance is defined as that for graphite, i.e., 0.335 nm (σss = 0.335 nm). z’ is the dummy coordinate, which is defined as z’ = z + σss/2. By substituting Eqs. (B-2)–(B-4) into Eq. (B-1), the adsorption equilibrium may be expressed as:

f ff (z ) = fbulk exp

fs (z )

fs (L

+

z) (B-6)

kT

where k is Boltzmann’s constant (k = 1.38 × 10–23 J/K). As mentioned above, the PR-EOS is applied to describe the fluid-fluid interaction. The PR-EOS can be written in terms of density as follows:

p 1 = RT (1 b)

RT [1 + (1

a (T ) 2 ) b][1 + (1 +

(B-7)

2 ) b]

where 12

Fuel 254 (2019) 115629

Y. Pang, et al.

a (T ) = b=

0.457535 (T ) R2Tc2 pc

(B-8)

0.077796RTc pc

(B-9)

In Eq. (B-8), the term α(T) is expressed as follows [62]:

(T ) = exp [(A + BTr )(1 TrC + D

+E 2

(B-10)

]

where Tr = T/Tc and A, B, C, D, and E are correlation parameters with values of 2.0, 0.8145, 0.134, 0.508 and −0.0467, respectively. The value of the acentric factor ω is set as 0.0113 in this study. Methane is the only sorptive gas used in this study. Accordingly, the values of the critical pressure (Pc), critical temperature (Tc) and diameter of the methane molecule are 4.6 MPa, 190.56 K, and 0.3758 nm, respectively. Based on Eq. (B-7), the fugacity of the bulk fluid can be expressed as:

ln

fbulk p

=

b 1

a (T ) RT (1 + 2b

b

b2 2 )

ln

p RT

pb RT

1 + (1 + a (T ) ln 2 2 bRT 1 + (1

2) b 2) b

(B-11)

Analogously, the fugacity of the sorbed gas, which accounts for the fluid-fluid interactions of the sorbed gas, can be given as:

ln

f ff (z ) p

=

1

bads (z ) bads (z )

aads (z ) (z ) 2 RT (1 + 2bads (z ) bads

2 (z ))

p RT (z )

ln

pbads RT

1 + (1 + aads (z ) ln 2 2 bads RT 1 + (1

2 ) (z ) bads 2 ) (z ) bads

(B-12)

ρ(z) in Eq. (B-12) is the density profile in the slit pore. Appendix C The pore size distributions from N2 adsorption/desorption measurements are analyzed with the DFT and BJH methods, and the results are presented in Fig. C-1. For the Barnett shale core sample, the mode of the pore width is 3.969 nm and 3.788 nm interpreted using the DFT and BJH methods, respectively. Similarly, for the Eagle Ford shale core sample, the mode of the pore width is 4.152 nm and 3.721 nm interpreted using the DFT and BJH methods, respectively. 0.010

0.0025

dV/dD (cm3/g/nm)

dV/dD (cm3/g/nm)

DFT

0.008

BJH

0.006 0.004 0.002 0.000

DFT

0.0020

BJH

0.0015 0.0010 0.0005

0

4

8

12

16

20

24

0.0000

28

Pore Width (nm) Barnett

0

4

8 12 Pore Width (nm)

16

20

Eagle Ford

Fig. C-1. Pore size distribution determined using the DFT and BJH methods [5].

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