Adsorption and Surface Diffusion of Supercritical Methane in Shale

Article pubs.acs.org/IECR

Adsorption and Surface Diffusion of Supercritical Methane in Shale Wenxi Ren, Gensheng Li,* Shouceng Tian, Mao Sheng, and Lidong Geng State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China S Supporting Information *

ABSTRACT: The primary objective of this study is to investigate the adsorption and surface diffusion of supercritical methane in shale. An adaptive Dubinin−Astakhov (ADA) model, with a term taking the adsorbed phase density, was introduced to interpret measured excess adsorption isotherms and obtain temperature-independent characteristic curves. The ADA model can also predict adsorption isotherms at different temperatures. Combining with the ADA model and the Maxwell−Stefan equation, a new model was developed to describe the surface diffusion of supercritical methane in shale. The model was successfully validated against experimental data. We also studied the effect of fugacity and temperature on the surface diffusion. With increasing temperature, the surface diffusion flux reaches a maximum and then decreases, which is the result of a trade off between the amount adsorbed and the effective Maxwell−Stefan surface diffusivity. The driving force for surface diffusion is the chemical potential gradient, which can be related to the gradient of fractional occupancy by a thermodynamic factor. The absolute adsorption isotherms of supercritical methane in shale are of Type I in the IUPAC classification scheme. At high feed fugacity, the adsorbed concentration at feed side approaches saturation. Further increasing the feed fugacity can hardly increase the driving force for the surface diffusion. But increasing temperature can increase the effective Maxwell−Stefan surface diffusivity, which leads to the increase of the surface diffusion flux.

1. INTRODUCTION Shale gas is an attractive and potential “bridge fuel” during the transition to carbon-free and renewable energy sources.1,2 Shale pore network, consisting of inorganic nanopores and organic nanopores, hosts free gas and adsorbed gas.3 A significant portion of the hydrocarbons is stored as an adsorbed phase accounting for 20−85% of the total gas-in-place (GIP).4 During shale gas production, the adsorbed molecules diffuse on pore walls, which is usually referred to as surface diffusion. For shale media, considering the great amount of adsorbed gas and large surface area, surface diffusion could be an important transport mechanism.5,6 Thus, understanding the adsorption and surface diffusion of hydrocarbons in shale is critical for shale gas resource assessment and shale gas recovery. Many industrial and scientific applications such as gas separation,7 hydrogen storage,8 and carbon dioxide sequestration9 involve adsorption and surface diffusion in nanoporous media, which is a fruitful area of research. However, the adsorption and surface diffusion of hydrocarbons in shale is not well understood. The following discussion is limited to methane, which is the main component of shale gas. Methane is in the supercritical © XXXX American Chemical Society

state under geological conditions. Previous studies show that the adsorption of supercritical methane in micropores (pore diameter < 2 nm) involves the volume filling of the adsorption space.10,11 In mesopores (2−50 nm), mainly monolayers of adsorbed gas are formed.12 For shale media, micropores are mainly housed in organic matter (also known as kerogen).13 The volume of the micropores is the key factor to control gas adsorption in shale.13,14 Moreover, supercritical methane adsorption in shale is mainly physical adsorption.15 Temperature is the main factor affecting the adsorption. For shale, the most widely used adsorption model is the Langmuir model,16 which is an ideal model for homogeneous monolayer adsorption. However, in the micropores, the dominant mechanism of adsorption is the volume filling of the adsorption space.17 Moreover, the assumption of an energetically homogeneous surface is not strictly true for shale media. Thus, the Langmuir model may be merely a curve fitting of Received: Revised: Accepted: Published: A

November 14, 2016 January 30, 2017 March 2, 2017 March 2, 2017 DOI: 10.1021/acs.iecr.6b04432 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research experimental data.18 In addition, the Langmuir model involves temperature dependent parameters. To evaluate the temperature effect on adsorption or to estimate the GIP at different depths, it is necessary to perform the measurements of adsorption isotherms at different temperatures, which is tedious and time-consuming. Yu et al.19 used the BET model20 to describe supercritical methane adsorption in shale at 327.59 K and found that the BET model shows a better fitting than the Langmuir model. The BET model is derived on the assumption of multilayer adsorption on homogeneous surfaces.21 This assumption prevents the BET model from being meaningfully applied to microporous media.22 Moreover, like the Langmuir model, the BET model also includes temperature dependent parameters. Recent studies applied the supercritical Dubinin− Radushkevich (SDR) model,12,23 which is based on the Dubinin’s volume filling theory,24 to describe supercritical methane adsorption in shale. But in these studies, the obtained maximum adsorption capacity is temperature dependent, which is not strictly correct, because the number of available adsorption sites should be constant and temperature independent. The above-mentioned models have several inherent assumptions and theoretical inconsistencies, which limit their applicability for modeling the temperature dependence of supercritical methane adsorption in shale. Thus, a robust model is desired to model and predict the adsorption isotherms of supercritical methane in shale at different temperatures. Note that more sophisticated methods, such as the Grand Canonical Monte Carlo simulation method and density functional theory, are not discussed in this paper. Surface diffusion is an important mass transfer process.25 There are four approaches to study surface diffusion: experiment, molecular dynamics simulation, mathematical model, and a combination of these methods. Experiment is closest to reality. The main difficulty of this method is the accurate evaluation of the surface diffusion flux. For molecular dynamics simulation, it is not currently practical for reservoirscale simulation.26 In contrast, a practical mathematical model, with reasonable assumptions, not only provides useful physics insight but also yields general predictions. For surface diffusion in shale, there has been relatively little research on this approach. Yuan et al.27 derived a surface diffusion model based on the Langmuir model and Fick’s law. However, temperature effect was not considered in their work. Surface diffusion is an activated process.28 Temperature has a great effect on the process. Wu et al.6 first developed a nonisothermal model for surface diffusion in shale. The model contains an empirical parameter, the blockage parameter k. The main inconvenience of the model is the estimation of k. Currently, we are not aware of any measurement of k made on shale samples at reservoir conditions. Moreover, the model is derived based on the measured data of surface diffusion in activated carbon. Thus, the model should be treated with caution when it is used to describe surface diffusion in shale. The literature review shows that a practical model for surface diffusion in shale is still lacking. In this work, first an adaptive Dubinin−Astakhov (ADA) model is introduced to model and predict the adsorption isotherms of supercritical methane in shale at different temperatures. Next, based on the ADA model, we derive a new expression for the thermodynamic factor for the Maxwell− Stefan equation and develop a new model for surface diffusion in shale. Finally, the effects of temperature and fugacity on surface diffusion are discussed.

2. MODEL DEVELOPMENT 2.1. Supercritical Methane Adsorption. For typical shale gas reservoirs, the reservoir pressure ranges from 5 to 30 MPa, and the reservoir depth ranges from 500 to 3500 m.4,29 The reservoir temperature range is approximately 308−400 K. Thus, the present study is limited to the temperature range of 308− 400 K and the pressure range up to 30 MPa. Methane is in the supercritical state at reservoir conditions. For shale media, supercritical methane is mainly adsorbed in the micropores of organic matter.13,14 The adsorption of supercritical methane in micropores involves the volume filling of the adsorption space.10,11 Thus, the Dubinin−Astakhov (DA) model,24 which is based on the Dubinin’s volume filling theory,24 is used in this work. The DA model is given by24 ⎡ ⎛ A ⎞t ⎤ nabs = n0 exp⎢ −⎜ ⎟ ⎥ ⎣ ⎝E⎠ ⎦

A = RT ln

(1)

p0 p

(2) −1

where nabs is the absolute adsorption amount, mol·kg , n0 is the maximum absolute adsorption amount, mol·kg−1, R is the ideal gas constant, J·mol−1·K−1, T is the temperature, K, E is the characteristic energy of the adsorption system, J·mol−1, A is the adsorption potential, J·mol−1, p0 is the saturation pressure, Pa, and t is the heterogeneity parameter. When t = 2, eq 1 reduces to the Dubinin−Radushkevich (DR) model. For carbonaceous adsorbents, the heterogeneity parameter t usually ranges from 1 to 3.30 At high pressure, deviations from ideal gas behavior increase; thus, fugacity f is used to replace pressure p. The DA model can be written as follows31 ⎡ ⎛ A ⎞t ⎤ nabs = n0 exp⎢ −⎜ ⎟ ⎥ ⎣ ⎝E⎠ ⎦

(3)

f = φf p

(4)

A = RT ln

vads =

f0 f

(5)

⎡ ⎛ A ⎞t ⎤ M nabsM = n0 exp⎢ −⎜ ⎟ ⎥ ρads ⎣ ⎝ E ⎠ ⎦ ρads

(6)

where M is the molar mass of methane, kg·mol−1, f 0 is the saturation fugacity, Pa, vads is the adsorbed phase volume, m3· kg−1, ρads is the adsorbed phase density, kg·m−3, and φf is the fugacity coefficient, dimensionless. The fugacity coefficient φf can be calculated by an equation of state (EOS). In this work, we calculate φf using the Soave−Benedict−Webb−Rubin equation of state (SBWR-EOS) due to its accuracy at reservoir conditions.32 Above the critical temperature Tc the concept of saturation pressure no longer holds. Sakurovs et al.33 proposed that the DR model can be applied to supercritical conditions by the replacement of the term p0/p with ρads/ρ, where ρ is the bulk phase density, kg·m−3. But the method is inconvenient for numerical reservoir simulation because gas density is not as intuitive as pressure. Thus, the use of pseudo-saturation pressure34 is recommended. Following Amankwah and Schwarz,34 the pseudo-saturation fugacity f 0 is given by B

DOI: 10.1021/acs.iecr.6b04432 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

⎛ T ⎞k f0 = ⎜ ⎟ pc ⎝ Tc ⎠

Γ= (7)

where pc is the critical pressure, Pa, and Tc is the critical temperature, K. The parameter k is determined by fitting the adsorption data. The measured adsorption isotherms are the excess adsorption isotherms. For the application of the DA model to excess adsorption data, the following equation is used12 ⎛ ρ ⎞ ⎟⎟ nexc = nabs − vadsρ = nabs⎜⎜1 − ρads ⎠ ⎝

B=

(8)

(9)

RTc M ,b= b 8pc

Ozawa et al.’s method38

ρads = ρb exp[−Ω(T − Tb)], Ω = 0.0025

Dubinin’s method36

ρads = ρb exp[−Ω(T − Tb)], Ω =

(T − Tb), ρc =

ln(bρb /M ) Tc − Tb

Where ρb and Tb are the liquid density and boiling temperature at ambient pressure, respectively, b is the van der Waals volume, m3· mol−1, and Ω is the thermal expansion coefficient, K−1. For methane, ρb is equal to 422.53 kg·m−3, and Tb is equal to 111.63 K.39 a

phase density. Here, we compared the three methods listed in Table 1 to determine the best option. The detailed comparison is provided in the Supporting Information. The comparison shows that Dubinin’s method36 is the best of the three methods for estimating the adsorbed phase density. Thus, Dubinin’s method36 is used in this work. 2.2. Surface Diffusion Model. For a unary system, the Maxwell−Stefan equation40 applied to surface diffusion is

θ=

nabs n0

∇θ (14)

(15)

where Đ0 is generally referred to as the limiting diffusivity and independent of temperature, m2·s−1, Ea is the activation energy of surface diffusion, J·mol−1, Est is the isosteric heat of adsorption, J·mol−1, and α is specific to the adsorption system and can be treated as a constant.25 The value of α usually ranges between 0.3 and 1.25 Here, we take α equal to 0.8 for the following discussion. The isosteric heat of adsorption Est is independent of temperature45 but dependent on loading. However, molecular dynamic simulation and experimental data show that the isosteric heat of adsorption for methane in nanoporous media is a weak function of loading.31,46 Thus, as an approximation, the average isosteric heat of adsorption is used in eq 15.47 Due to the ultralow permeability and complex pore architecture of shale, the effective MS surface diffusivity is difficult to measure from experiments. Wu et al.6 developed an empirical equation to estimate the effective MS surface diffusivity (eq 13 in Wu et al.6). The empirical equation should be used with caution for the following two reasons. First, the empirical equation is derived by fitting the experimental data for the surface diffusion of methane in activated carbon and thus implicitly includes the pore structure properties of the activated carbon. The empirical equation may not accurately describe surface diffusion in shale. Second, the empirical equation is limited to low pressure conditions.6 Because of these limitations, we abandon the empirical equation and employ a diffusivity from the work of Falk et al.,48 in which the transport of alkanes in kerogen-like porous media at reservoir temperature and pressure (T = 423 K and p ≤ 100 MPa) was investigated by molecular dynamics simulation. The pore size distribution of the kerogen-like porous media is 0.3− 1.5 nm, which agrees with the experimental results of Clarkson et al.48,49 The simulation results show that the collective diffusivity of methane is not sensitive to pressure and equal to (3.5925 ± 0.3283) × 10−8 m2·s−1. In micropores, the contribution of surface diffusion to gas transport is dominant.6 Moreover, the collective diffusivity is identical to the effective MS diffusivity.50 Thus, in this work, the effective MS surface diffusivity at temperature of 423 K is set to 3.5925 × 10−8 m2· s−1, which is in the same order of magnitude as reported by Akkutlu and Fathi.51 Based on eq 15, we obtain

adsorbed phase densitya

ρads = ρb −

tRT( −ln θ )1 − 1/ t

⎛ −E ⎞ ⎛ −αEst ⎞ ⎟ Đe = Đ0 exp⎜ a ⎟ = Đ0 exp⎜ ⎝ RT ⎠ ⎝ RT ⎠

Table 1. Different Methods for Estimating the Adsorbed Phase Density

Dubinin−Nikolaev equation37

Đe

where ρr is the rock density, kg·m , θ is the fractional occupancy, Đe is the effective Maxwell−Stefan surface diffusivity, m2·s−1, and μ is the chemical potential, J·mol−1. The effective Maxwell−Stefan surface diffusivity is a lumped parameter accounting for the porosity and tortuosity of porous media. The temperature dependence of the Đe is given by44

where nexc is the excess adsorption amount, mol·kg . The bulk phase density ρ is calculated by the SBWR-EOS. Since the direct measurement of ρads is difficult, the adsorbed phase density ρads is usually considered to be a fitting parameter or estimated by empirical approaches. In some cases, the fitted value of ρads exceeds the liquid density of methane at its boiling point,12 which may be physically unreasonable.35 Thus, we do not use the adsorbed phase density as a fitting parameter. Table 1 lists different empirical approaches to estimate the adsorbed

Đe ∇μ RT

(13)

−3

−1

Ns = −ρr n0θ

RT E

Ns = −ρr n0E

⎡ ⎛ A ⎞ t ⎤⎛ ρ ⎞ ⎟⎟ nexc = n0 exp⎢ −⎜ ⎟ ⎥⎜⎜1 − ρads ⎠ ⎣ ⎝ E ⎠ ⎦⎝

ρb − ρc

(12)

When t = 2, eq 12 reduces to the thermodynamic factor derived by Linders et al.42 for the DR model. We assume that the effective MS surface diffusivity is independent of loading.43 Combining eq 10 with eq 12 gives

Combined with eq 3 and eq 8, we get the ADA model

Tc − Tb

1 tB( −ln θ )1 − 1/ t

(10)

(11)

The chemical potential gradient can be related to the gradient of fractional occupancy by a thermodynamic factor Γ.41 For the ADA model the thermodynamic factor Γ is C


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Industrial & Engineering Chemistry Research ⎡ −E ⎛ T ⎞⎤ Đe = 3.5925 × 10−8 exp⎢ a ⎜ 0 − 1⎟⎥ ⎠⎦ ⎣ RT0 ⎝ T

(16)

where T0 is the reference temperature and equal to 423 K. The effective MS surface diffusivity given by eq 16 implicitly includes the porosity and tortuosity of the kerogen-like porous media. The surface diffusion flux is calculated by integrating eq 14 Ns = −ρn0

Đe δ

∫θ

θpermeate

feed

1 dθ tB( −ln θ )1 − 1/ t

(17)

where δ is the thickness of porous media, m, and θfeed and θpermeate are the fractional occupancies at feed and permeate sides, respectively. The integral in eq 17 is evaluated using a 4point Gauss−Legendre quadrature due to its accuracy and efficiency.

3. RESULTS AND DISCUSSION 3.1. Modeling of Measured Adsorption Data. The present study is limited to the temperature range of 308−400 K. The ADA model is used to model supercritical methane adsorption in shale. The maximum absolute adsorption amount n0, the characteristic energy E, the heterogeneity parameter t, and k are used as fitting parameters. The adsorption data were obtained from Rexer et al.,52 Tian et al.,23 and Gasparik et al.14 In their work, methane adsorption isotherms were measured at different temperatures. For a specific adsorption system, the parameters (n0, E, t, and k) were obtained by a global fitting procedure. The fitting performance was assessed by the rootmean-square error (RMSE), which is defined as num

SSR =

∑ (xi − yi )2 i=1

RMSE =

SSR num

(18)

(19)

where num is the number of data points, SSR is the sum of squares of residuals, xi is the measured value for excess adsorption, and yi is the calculated value for excess adsorption. The fitting results are graphically represented in Figure 1. For brevity, we only show the results of sample 4-08, HAD7090, and HAD119 in Figure 1. The other adsorption isotherms and the ADA model fits are presented in Figures S3−S5. The figures show that the experimental data can be well fitted by the ADA model. The RMSE ranges from 1.10 × 10−3 to 2.70 × 10−3. Figure 1a and Figure S3 show that the excess isotherms have a maximum. Moreover, with increasing temperature, the fugacity at which the maximum occurs shifts to a higher fugacity.14 This phenomenon was also reported by Mosher et al.,53 who used the Grand Canonical Monte Carlo simulations to investigate methane adsorption in shale. Specifically, for sample 4-08, the maximum values are 0.083, 0.078, and 0.073 mol·kg−1 at 308.55 K (8.22 MPa), 323.55 K (8.47 MPa), and 338.55 K (8.90 MPa), respectively. Under supercritical conditions, when the micropores are fully loaded with methane, the density profiles of the adsorbed methane in the micropores are not sensitive to fugacity change while the bulk density still varies with fugacity.54 Consequently, the excess adsorption amount increases with the increase of fugacity, reaches a maximum, and then decreases with further increasing fugacity.

Figure 1. (a) Excess adsorption isotherms of sample 4-08 and the ADA model fits. Circles represent the experimental data of Tian et al.23 (b) Excess adsorption isotherms of HAD7090 and the ADA model fits. Circles represent the experimental data of Rexer et al.52 (c) Excess adsorption isotherms of HAD119 and the ADA model fits. Circles represent the experimental data of Gasparik et al.14

Figure S4b shows that there is no peak in the excess adsorption amount nex. This phenomenon can be explained as follows. When increasing the fugacity the absolute adsorption amount nabs increases, and the term [1 − (ρ/ρads)] decreases. According to eq 8, when the increase of nabs dominates over the decrease of the term [1 − (ρ/ρads)], the excess adsorption amount nex increases with increasing fugacity. We further tested the ADA model using experimental data from Rexer et al.12 They measured the adsorption of supercritical methane in shale at 300−473 K. Because the present study focuses on the temperature range of 308−400 K, we selected the intermediate isotherms (308−398 K) for fitting, as shown in Figure 2. The ADA model gives a decent description of the measured data except for the 373 and 398 K isotherms. At 373 and 398 K, the ADA model results are higher than the experimental data at high fugacities (f > 9 MPa). This D


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(5.22 × 103 to 9.84 × 103 J·mol−1)31,58 and the methane−coal system (3.79 × 103 to 8.45 × 103 J·mol−1),21,59 which partially explains why the adsorption capacity of shale is lower than that of coal and activated carbon. The fitted values of t fall in the range 1−2 for heterogeneous carbonaceous materials.24 The attainment of the temperature independent characteristic curve is critical for the application of the ADA model. To obtain the temperature independent characteristic curve, we use the absolute adsorption data rather than the excess adsorption data, as suggested by Zhou and Zhou.60 Figure 3 shows the

Figure 2. Excess adsorption isotherms of Alum shale 1 and the ADA model fits. Circles represent the experimental data of Rexer et al.12 Solid lines are fits using the ADA model.

discrepancy is discussed below. As mentioned previously, with increasing temperature, higher fugacity is needed to attain the excess maximum. This phenomenon was also observed for supercritical methane adsorption in activated carbon55 and metal−organic framework56 (MOF). The ADA model captures this phenomenon, as shown in Figure 2. But the adsorption isotherms obtained by Rexer et al.12 do not show the trend. The experimental data show that the fugacities corresponding to the excess maximum are 8.81, 9.39, 8.98, and 9.22 MPa at 308, 338, 373, and 398 K, respectively. Possible reasons are as follows. First, at high temperature, the adsorption experiment of supercritical methane in shale is not a trivial task. The experimental uncertainty increases with increasing equilibrium pressure.57 Second, in Rexer et al.’s12 experiment, the equilibrium time is less than 1 h, which may not be enough for the methane−shale system to reach equilibrium status at each measuring point. The fitted parameters are listed in Table 2. The maximum absolute adsorption amounts n0 obtained from fitting are within the range (0.1−0.3 mol·kg−1) reported by Yang et al.15 Figure S6 shows a positive correlation of n0 with total organic carbon (TOC). This is because the TOC is a key control on methane adsorption.16 The parameter E characterizes the strength of the interaction of adsorbate with adsorbent. Variations in kerogen types and maturity can greatly affect the interaction between methane and shale. With increasing E, the adsorption sites become more energetic. Moreover, the fitted E is relatively low compared with that of the methane-activated carbon system

Figure 3. (a) Characteristic curves for CH4 in sample 4-08 and sample 4-64. The R2 is 0.996 for sample 4-08 and 0.996 for sample 4-64. (b) Characteristic curves for CH4 in sample 4-61 and HAR060. The R2 is 0.997 for sample 4-61 and 0.991 for HAR060. Symbols represent the experimental data of Tian et al.23 and Gasparik et al.14 Solid lines represent the characteristic curves calculated from eq 6 with the parameters listed in Table 2.

characteristic curves for CH4 in sample 4-61, sample 4-08, sample 4-64, and HAR060. The other characteristic curves are

Table 2. Fitted Parameters of the ADA Model sample 4-64a sample 4-61a sample 4-08a HAD7119b WIC7145b HAD7090b WIC149c HAD119c HAR060c Alum shale 1d

TOC (−)

n0 (mol·kg−1)

4.07% 5.44% 2.45% 7.15% 10.92% 7.41% 11.7% 7.7% 6.8% 6.35%

0.19 0.20 0.13 0.17 0.23 0.19 0.27 0.23 0.17 0.27

E (J·mol−1)

t (−)

k (−)

× × × × × × × × × ×

1.48 1.35 1.53 1.49 1.25 1.58 1.04 1.11 1.01 1.30

3.65 3.31 3.49 3.50 3.25 3.39 3.07 3.03 3.03 3.94

7.80 7.58 7.42 6.86 5.21 7.27 3.94 4.61 3.68 7.27

103 103 103 103 103 103 103 103 103 103

RMSE (−) 1.80 1.80 1.20 1.30 1.10 1.20 2.70 1.70 1.10 4.10

× × × × × × × × × ×

10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3

a

Experimental data from Tian et al.23 bExperimental data from Rexer et al.52 cExperimental data from Gasparik et al.14 dExperimental data from Rexer et al.12 E


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Industrial & Engineering Chemistry Research shown in Figures S7−S9. The figures show that, for each sample, the data points collapse on the characteristic curve with small scatter (R2 > 0.98). 3.2. Prediction of Isotherms at Different Temperatures. Absolute adsorption isotherm represents the actual adsorbed layer. Thus, the assessment and development of shale gas resources requires the knowledge of the absolute adsorption isotherm. The characteristic curves can be used to predict absolute adsorption isotherms at different temperatures. Figure 4 shows the predicted absolute adsorption isotherms of CH4 in

investigated conditions (T = 303 K and p < 0.5 MPa). Figure S10 shows that at low pressure (p < 2.7 MPa) the fugacity of CH4 and CO2 is close to the pressure. Moreover, at low pressure (p < 0.5 MPa), the difference between absolute adsorption amount and excess adsorption amount is negligible.12 Figure S11 shows the adsorption data and the fitted curves. The adsorption data can be well described by the ADA model. Table 4 lists the adsorption parameters. The fractional occupancy at the feed side θfeed is calculated by eq 11, and the fractional occupancy at the permeate side θpermeate is zero. The only unknown parameter is the effective MS surface diffusivity Đe, which is used as a fitting parameter. Figure 5 shows that the model predictions are in good agreement with the experimental data. The fitted Đe for CH4 is 8.75 × 10−10 m2·s−1, which is close to the value (10.4 × 10−10 m2·s−1) estimated by van de Graaf et al.43 The fitted Đe for CO2 is 1.14 × 10−10 m2·s−1, which is close to the value (1.232 × 10−10 m2· s−1) estimated by Lito et al.65 The fitted Đe implicitly includes the porosity and tortuosity of the zeolite membrane. Figure 5 also shows that the results calculated by the 4-point Gauss− Legendre quadrature and high-precision numerical integration method (Num = 1000 with the Simpson’s rule) match each other very closely, where Num is the number of subintervals of [θfeed, θpermeate]. Thus, in this scenario, the 4-point GaussLegendre quadrature offers good accuracy with computational efficiency. In addition, the model predictions are compared with experimental data from Zamirian et al.67 The experiment was conducted on a Marcellus shale core sample. Under the investigated conditions (T = 300 K and p < 2.7 MPa), the fugacity of test fluid (helium and carbon dioxide) is approximately equal to its pressure, as shown in Figure S12. Thus, the pressure is used as an approximation for the fugacity. Gas flow in shale is a complex process involving viscous flow, Knudsen diffusion, and surface diffusion.6,68,69 The apparent permeability of shale is given by

Figure 4. Predicted absolute adsorption isotherms at different temperatures. Circles represent the experimental data of Rexer et al.52 The measured data are converted to absolute adsorption values using eq 8. Solid and dashed lines are from eq 3. The parameter set is presented in Table 2.

HAD7119. The predicted absolute adsorption isotherms agree with the measured ones, and, thus, the predictions are reasonably reliable. Predictions in the measured temperature range are expected to be accurate. One should be prudent in extrapolating to lower or higher temperatures. For temperatures over 400 K, the ADA model needs further testing, which is not covered here. Thus, it is not recommended to use the ADA model at temperatures above 400 K. For natural gas adsorption in shale, the absolute adsorption isotherms usually can be classified as type I,61 according to the IUPAC classification system.62 Figure 4 also shows that the amount adsorbed decreases with increasing temperature. Specifically, the absolute adsorption amounts are 0.13 mol·kg−1 (318.15 K), 0.12 mol·kg−1 (338.15 K), and 0.09 mol·kg−1 (378.15 K) at 10 MPa. 3.3. Modeling of Surface Diffusion. There are limited experimental data on surface diffusion in shale. First, the model predictions are compared with the experimental data of gas permeation through zeolite membranes, where surface diffusion dominates gas transport.43 Table 3 gives the values of the membrane parameters. The parameters (n0, E, t, and k) were obtained by fitting the ADA model to the adsorption data of Zhu et al.63,64 Note that in this section we assume that the adsorbate behaves as an ideal gas (i.e., φf = 1) at the

Đe μ DK p ⎛ 1 μ 1 dZ ⎞ ⎜ − ⎟ + ρr n0 c RT Z ⎝ p Z dp ⎠ c p( −ln θ )1 − 1/ t ⎛ A ⎞t − 1 ⎡ ⎛ A ⎞t ⎤ ⎜ ⎟ exp⎢ −⎜ ⎟ ⎥ ⎝E⎠ ⎣ ⎝E⎠ ⎦ (20)

ka = k +

where ka is the apparent permeability, m2, μ is the viscosity, Pa· s, c is the molar density, mol·m−3, k is the intrinsic permeability, m2, DK is the Knudsen diffusion coefficient, m2·s−1, and Z is the compressibility factor. The first term in the right-hand side of eq 20 characterizes viscous flow. The second term represents Knudsen diffusion, and the third term characterizes surface diffusion. The detailed derivation of eq 20 is provided in the Supporting Information. Since helium is a nonadsorbing gas, the contribution of surface diffusion to the total flux is zero. Figure 6 demonstrates a good consistency between eq 20 and the experimental data. The parameter set is presented in Table 5. The fitted Đe for CO2 is 4.1 × 10−9 m2·s−1, which is in the same order of magnitude as the results of Prasetyo and Do,71 who measured the surface diffusivity of CO2 in activated carbon. The fitted value of τ falls within the range of 2.3−11.9 reported by Katsube et al.72 3.4. Effects of Temperature and Pressure on Surface Diffusion. In this section, we model the surface diffusion of supercritical methane in shale with the adsorption parameters

Table 3. Membrane Parameters a

Silicatile-1 Silicatile-1b

zeolite layer thickness (m)

zeolite density (kg·m−3)

1 × 10−5 2 × 10−5

1.8 × 103 1.76 × 103

a

The parameters are taken from van de Graaf et al.43 bThe parameters are taken from Zhu et al.64 F


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Industrial & Engineering Chemistry Research Table 4. Adsorption Parameters adsorbate

Tc (K)

n0 (mol·kg−1)

E (J·mol−1)

t (−)

p0 (MPa)

K (−)

CH4 CO2

190.6 304.25

2.36 2.96

8.23 × 103 12.53 × 103

2 3.07

7.53c 7.19d

1.06 -

a

Silicatile-1 Silicatile-1b

a Experimental data from Zhu et al.63 bExperimental data from Zhu et al.64 cThe value is calculated from eq 7. dThe value is taken from the Dortmund Data Bank.66

Figure 7. Surface diffusion flux as a function of feed fugacity at different temperatures. The effective MS surface diffusivity is estimated by eq 16. Lines are from eq 17. The parameter set is given in Table 2.

Figure 5. Comparison of model predictions and experimental data at 303 K. Circles represent the experimental data of Zhu et al.64 and van de Graaf et al.43 GS refers to the 4-point Gauss-Legendre quadrature, and SIM refers to the high-precision numerical integration method (Num = 1000 with the Simpson’s rule). Solid and dashed lines are from eq 17. The parameter set is presented in Table 3 and Table 4.

phenomenon can be explained as follows. With increasing temperature, the adsorbed concentration at the feed side (n0·θfeed) decreases, which results in the decline of the surface concentration gradient. At low feed fugacity, the negative effect of temperature on adsorption is pronounced, which leads to that the surface diffusion flux decreases with increasing temperature. At high feed fugacity, the adsorbed concentration at the feed side approaches saturation. Thus, further increasing the feed fugacity can hardly increase the surface concentration gradient. But increasing temperature can increase the surface diffusivity, giving rise to the increase of the surface diffusion flux. This leads to the crossover of the flux curves. Specifically, when the feed fugacity is 18 MPa, the surface diffusion flux increases with temperature from 1.67 mol·m−2·s−1 (330 K) to 1.80 mol·m−2·s−1 (370 K). This finding indicates that the thermal stimulation of shale gas reservoirs has the potential to enhance the recovery of adsorbed gas by enhancing the surface diffusion and releasing the adsorbed gas. Figure 8 plots the surface diffusion flux as a function of temperature. The feed fugacity is set to 12, 15, and 18 MPa. The permeate fugacity is set to 0 MPa. All the curves in Figure 8 exhibit a maximum at a certain temperature. This observation is consistent with the experiment of Bakker et al.73 Moreover, as the feed fugacity increases, the temperature corresponding to the maximum flux increases. The maximum fluxes are 1.44, 1.63, and 1.80 mol·m−2·s−1 at 363 K (12 MPa), 371 K (15 MPa) and 377 K (18 MPa), respectively. The maximum can be explained by the trade off between surface diffusion and adsorption. Surface diffusion is considered to be an activated process. The effective MS surface diffusivity increases with

Figure 6. Comparison of the model predictions with the experimental data of Zamirian et al.67 The solid and dashed lines are calculated from eq 20. The parameter set is presented in Table 5.

for sample 4-61. The parameter set is presented in Table 2. Moreover, the thickness δ is set to 10 × 10−6 m, and the rock density is assumed to be 2.5 × 103 kg·m−3. The average isosteric heat of adsorption is 12.9 × 103 J·mol−1.23 The effective MS surface diffusivity is estimated by eq 16. Figure 7 plots the surface diffusion flux as a function of feed fugacity. The feed fugacity is varied from 0.1 to 20 MPa, and the permeate fugacity is kept constant at 0 MPa. Figure 7 shows a crossover of the flux curves at around 8.5 MPa. This Table 5. Parameter Set for Apparent Permeability

a

test fluid

φ (−)

τ (−)

re (nm)

ρr (kg·m−3)

t (−)

p0 (MPa)

n0 (mol·kg−1)

E (J·mol−1)

He CO2

0.048a

3.8b

2.6b

2.5 × 103

1.15b

6.72c

0.38b

8.51 × 103b

The value is taken from Wang et al.70 bFitting parameter. cThe value is taken from the Dortmund Data Bank.66 G


Article


high fugacity, adsorption approaches saturation, which means that there is almost no concentration difference between the feed and permeate sides. Thus, the permeance decreases as the fugacity increases on both the feed and permeate sides. Specifically, at 350 K, the surface concentration difference (n0·θfeed − n0·θpermeate) decreases from 0.01 to 0.003 mol·kg−1 when the feed fugacity increases from 5 to 20 MPa. As a consequence, the permeance decreases from 1.07 × 10−7 to 4.44 × 10−8 mol·m−2·s−1·Pa−1.

4. CONCLUSIONS An adaptive Dubinin−Astakhov model was used to model supercritical methane adsorption in shale. The absolute adsorption isotherms of supercritical methane in shale can be classified as type I in the IUPAC classification system. When the ADA model was applied to analyze the measured adsorption data, successful reduction to an individual characteristic curve was achieved. With the characteristic curve, the ADA model is capable of describing supercritical methane adsorption in shale at multiple temperatures. On the basis of the ADA model and the MS equation, a new surface diffusion model was introduced to describe the surface diffusion of supercritical methane in shale. The surface diffusion model was validated with the experimental data. On the basis of the model, we investigated the effect of temperature and fugacity on the surface diffusion. The major findings are summarized as follows. The driving force for surface diffusion is the chemical potential gradient, which can be related to the gradient of fractional occupancy by a thermodynamic factor.41 For type I isotherms, the driving force for surface diffusion is not a strong function of feed fugacity. At high fugacity, the adsorption approaches saturation. Thus, further increasing the feed fugacity can hardly increase the driving force for surface diffusion. But increasing temperature can enhance surface diffusion due to the increased MS surface diffusivity. With increasing temperature, the trade off between the amount adsorbed and the MS surface diffusivity results in a maximum of the surface diffusion flux. These findings provide a good starting point for the development of the thermal stimulation of shale gas reservoirs.

Figure 8. Surface diffusion flux as a function of temperature. The effective MS surface diffusivity is estimated by eq 16. Lines are from eq 17. The parameter set is shown in Table 2.

temperature while the amount adsorbed decreases. At low temperature, at which the amount adsorbed is high, the increase of the effective MS surface diffusivity is dominant. In this temperature range, the surface diffusion flux increases with increasing temperature. With further increasing temperature, the decrease of the amount adsorbed gradually dominates over the increase of the effective MS surface diffusivity, which results in the surface diffusion flux reaching a maximum and decreasing subsequently. Specifically, at the feed fugacity of 15 MPa, the surface diffusion flux decreases from the maximum value of 1.63 mol·m−2·s−1 (T = 371 K) to 1.61 mol·m−2·s−1 (T = 390 K). At higher temperature (above 500 K), the amount adsorbed is negligible, and the dominant transport mechanism turns to activated gaseous diffusion.73 We reserve the investigation of the activated gaseous diffusion for future work. Next, we show the prediction of methane permeance. The permeance is defined as

Π=

NS Δf

(21)

where Δf is the fugacity difference between the feed and permeate sides, Pa. The Δf is kept constant at 1 MPa. Figure 9 depicts the permeance versus feed fugacity at different temperatures. The permeance decreases monotonically with increasing feed fugacity, which is related to the adsorption isotherm used. For supercritical methane adsorption in shale, the absolute adsorption isotherms exhibit type I behavior. At

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.6b04432. Comparison of Dubinin’s method, the Dubinin− Nikolaev equation, and Ozawa et al.’s method for estimating the adsorbed phase density; detailed derivation of the apparent permeability equation; excess adsorption isotherms of sample 4-64, sample 4-61, HAD7119, WIC7145, WIC149, and HAR060; the maximum absolute adsorption amount n0 versus TOC relationship; the characteristic curves for CH4 in HAD7119, WIC7145, HAD7090, WIC149, HAD119, and Alum shale 1; the fugacity versus pressure relationship for CH4 and CO2 at 303 K; the adsorption isotherms of CH4 and CO2 on zeolites; and the fugacity versus pressure relationship for He and CO2 at 300 K (PDF)

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Figure 9. Methane permeance as a function of feed fugacity. The effective MS surface diffusivity is estimated by eq 16. Lines are from eq 21, using the parameter set shown in Table 2.

AUTHOR INFORMATION

Corresponding Author

*(G.L.) E-mail: [email protected]. H


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Industrial & Engineering Chemistry Research ORCID

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Wenxi Ren: 0000-0002-8904-5744 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support by the National Natural Science Foundation of China (No. 51210006, No. 51234006, and No. U1562212). We also thank Prof. Rajamani Krishna (University of Amsterdam), Dr. Xu Tang (Virginia Polytechnic Institute and State University), Prof. Freek Kapteijn (Delft University of Technology), and Dr. Jian Xiong (University of Regina) for helpful discussions.

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Adsorption and Surface Diffusion of Supercritical Methane in Shale

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