Prediction of Adsorption Isotherms of Multicomponent Gas Mixtures in

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Prediction of Adsorption Isotherms of Multicomponent Gas Mixtures in Tight Porous Media by the Oil–GasAdsorption Three-Phase Vacancy Solution Model Yizhong Zhang, Shanshan Yao, Maolin Zhang, Xiang Zhou, Haiyan Mei, and Fanhua Zeng Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b02762 • Publication Date (Web): 16 Nov 2018 Downloaded from http://pubs.acs.org on November 17, 2018

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Prediction of Adsorption Isotherms of Multicomponent Gas Mixtures in Tight Porous Media by the Oil–Gas-Adsorption Three-Phase Vacancy Solution Model Yizhong Zhang,1,2 Shanshan Yao,2 Maolin Zhang,1 Xiang Zhou,2 Haiyan Mei,3 Fanhua Zeng2* 1

Innovation Centre of Unconventional Gas and Oil, Yangtze University, Wuhan, China, 410300

2

Petroleum System Engineering, University of Regina, Regina, SK, Canada, S4S 0A2

3

Faculty of Petroleum Engineering, Southwest Petroleum University, Chengdu, China, 610500

Keywords: Multicomponent; phase equilibrium; shale gas; oil–gas-adsorption three-phase; vacancy solution model.

Abstract For unconventional reservoirs, the effect of adsorption on phase equilibrium cannot be neglected as the process occurs in extremely tight porous media. This work focuses on the adsorption prediction of multicomponent mixture systems in tight porous media with the oil–gasadsorption three-phase equilibrium model to a gas sample in the literature. The revisited vacancy solution model of adsorption by Bhatia and Ding is introduced to study the adsorption behaviors of the mixture. The gas and adsorbed phases are assumed to be the solutions of adsorbates in a hypothetical solvent called “vacancy” and the vacancy is treated as an additional component

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engaged in the phase equilibrium in this theory. Instead of using parameters extracted from the multicomponent adsorption data, this method takes advantage as it accurately predicts gas mixture adsorption equilibrium with consideration of nonideal behaviour in the adsorbed phase from pure gas adsorption isotherms over wide ranges of conditions, which could be efficient in terms of cost and time. It can explain the competitive adsorption phenomenon, which is proved during the adsorption process of the gas mixture. The experimental data in the literature of CH4-C2H6 binary gas mixtures of different compositions with a pressure ranging from 0 to 125 bar under the temperatures of 40°C, 50°C, and 60°C is restudied in this work. The prediction results are compared with two other methods including the extended Langmuir model (ELM) and the multicomponent potential theory (MPTA). This method shows an improved precision with less than 5% mean absolute percentage error in all cases. In addition to predications of desorption for the depletion process, the vacancy solution model has the potential in future work to give simulations for other production operations, such as CO2 or N2 injection for the displacement of hydrocarbons in shales.

1. Introduction The tight gas reservoir occurs commonly in the form of free gas and adsorbed gas in the pores of a formation with a small amount of dissolved gas in reservoir oil or water.1,2,3 The research on tight unconventional reservoirs is particularly challenging due to the heterogeneity of the porefracture structure, extra-low rock permeability,4,5,6 varied rock mineralogy,7,8,9 and complex fluid phase behavior.10 In the study of adsorption phase equilibrium, it is tedious to collect the multicomponent data and a reliable method to estimate the mixture equilibrium from single component adsorption isotherm is preferred. The vacancy solution model takes advantage as it accurately predicts gas mixture adsorption equilibrium with consideration of nonideal behaviour


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in the adsorbed phase from pure gas adsorption isotherms over wide ranges of conditions. The adsorbed sites on the adsorbent surface were considered as a solution, which were engaged in the phase equilibrium with the free gas. The ideal adsorbed solution model (IAS) was proposed by Myers and Prausnitz in 1965 based on the Gibbs adsorption isotherm theory, which treated the adsorbed phase as an ideal gas solution but ignored the molecular interactions and compositional dependency.11 The heterogeneous ideal adsorbed solution (HIAS) model improved from IAS model was based on the theory that there were many adsorption patches/sites on a solid surface (which were independent of each other) and the adsorption energy distribution showed different degrees of heterogeneity.12,13 Different energy distribution functions could be applied to describe the heterogeneous distribution of adsorption energy such as the Langmuir-uniform-distribution (LUD) model and the dual-site Langmuir (DSL) model.14,15,16 One of the main problems with the HIAS model was that it is a combination of the heterogeneous adsorption energy distribution model and the ideal adsorbed solution (IAS) theory, so there were always differences between predictions and experiments, which were attributed to the non-idealities in the adsorbed phase and were accounted for by not using the adsorbed-phase activity coefficients (i.e. parameters describing the composition dependency of nonideal solute-solvent interactions during pure and multicomponent adsorption and solute-solute interactions during multicomponent interaction).17,18 Thus, the ideal solution model mentioned previously was developed into the real adsorption solution model (RASM) by changing the activity coefficient for the adsorbed phase was no longer equal to one (γ≠1).19,20 Based on the previous studies, vacancy solution model (VSM) was first proposed by Dubinin21 when he studied the single component gas adsorption on molecular sieves. He treated the single component adsorption as an equilibrium between two “vacancy solutions” with different


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compositions and derived an equation of state used for osmotic equilibrium. Lucassen-Reynders modified the model when he studied the non-ideal adsorption behaviour of a mixed surfactant system from pure component adsorption data.

22,23.

In this modified model, he treated the

monolayer adsorption of a surfactant as an unusual dividing surface, and activity coefficients are used to account for the non-ideal property.22,23 Suwanayuen and Danner17.18 believed this approach was sufficiently general that it could be applied to other kinds of surfaces, such as adsorption of gases on solids. They integrated the theories of both Dubinin and Lucassen-Reynders and deduced the association of mixed gas adsorption prediction from pure component adsorption data directly, which was named the Wilson Solution Vacancy Model (Wilson VSM) as it used two Wilson’s parameters to describe the non-ideality. Suwanayuen and Danner17.18 and Kaul24 showed that this model was simple to use, and it provided most accurate predictions with a particularly appealing aspect of its ability to predict highly nonideal equilibrium (i.e. adsorption azeotropes). Based on the concepts of Wilson Solution Vacancy Model, the Flory-Huggins Vacancy Solution Model (FHVSM) was developed by Cochran et al.25,26 as a new model to correct the deficiencies inherent of the former model and surpass it in accuracy. The vacancy solution model was criticized by Talu and Myers as being erroneous based on its inconsistent binary selectivity predictions in the Henry’s Law limit (except when saturation capacities for both components are the same).27 This model was revisited by Bhatia and Ding.28 The revisited vacancy solution model is reformulated in a thermodynamically consistent way that leads to the multisite Langmuir model of Nitta et al.,29 and it has been shown to satisfy the consistency criteria of binary adsorption.30 In this paper, the oil–gas-adsorption three-phase model with consideration of tight porous media is studied to recalculate the experimental data in the literature to compare several adsorption models. The revisited vacancy solution model of adsorption by Bhatia and Ding is introduced to


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describe the fugacity of the adsorbed phase. In this developed model, the effect of temperature, pressure, composition dependency and the nonideal behaviour are considered. To be critical, this paper attempts to study the competitive adsorption behaviour of the binary component gas mixture of methane and ethane in the literature and match the results of the original experimental data. The results reveal that, for the same adsorbent, the composition of a mixture in the adsorbed phase will be different from the total composition of the mixture. The adsorption of the multicomponent gas mixture is not only related to the pressure and temperature, but also depends on the gas mixture composition. Compared to other adsorption prediction methods in the literature, the model in this work gives precise prediction results for the experimental data. Moreover, it could estimate the adsorbed composition of every component in the binary gas mixture under a specific pressure and temperature based on the adsorption isotherms of a single pure gas component. Part of the materials contained in this paper were previously published in modified form in the author’s thesis.31

2. Methodology Revisited Vacancy Solution Model Regarding the pure component, the Langmuir model is the most widely used model, which is a function of pressure p , the limiting amount of adsorption nv , , a constant b , and adsorbed amount or coverage  c at that specific pressure.17,18

n v ,  c P b 1  c

(1)


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The Langmuir was modified by adding a term to describe the interaction of adsorbed gas molecules and vacancy, which is a function of Wilson’s parameters 13 ,  31 and coverage  c .17,18 Equation 2 shows the single component adsorption isotherm of the Wilson VSM.

P

n v ,  c f  13 ,  31 ,  c  bi 1   c

(2)

In Eq. 2, f  13 ,  31 ,  c  accounts for nonideality of the adsorbate mixture. bi is Henry’s law constant of component i .32 It is the proportional factor to describe the amount of dissolved gas to its partial pressure in the gas phase as the pressure approaches zero and it relates to the adsorbateadsorbent interaction at infinite dilution.

na  bi  lim  i  p 0  P 

(3)

The Wilson vacancy solution model of Suwanayuen and Danner17,18 has been successful as it estimated the mixture equilibrium from single component adsorption isotherm, but it is limited as opposed to corelating the data with various temperatures. The parameters extracted from isotherms at various temperatures sometimes exhibit erratic behavior with temperature.25 In the improved Flory-Huggins vacancy solution model by Cochran et al.25,26, the vacancy solution theory is developed in conjunction with the Flory-Huggins activity coefficient as the two Wilson activity coefficients ( 13 ,  31 ) used to describe non-ideality are highly correlated and could be reduced to one Flory-Huggins activity coefficient aiv .The four regression parameters in Eq. 2 are then reduced to three in Eq. 4.25,26 The Flory-Huggins vacancy solution model corrects the deficiencies inherent of the Wilson vacancy solution model and surpass it in accuracy.


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P

n v ,  c f  iv ,c  bi 1   c

(4)

For a constant temperature, the adsorption isotherm of a pure component could be associated by the Eqs. 5 and 6:25,26

P(

ni v , c a 2  )exp( iv c ) bi 1  c 1  aivc

(5)

c  nia / nv,

(6)

As mentioned previously, the nonideal behaviour also includes the interactions among adsorbates. In this model, only binary adsorbate-adsorbate interaction is considered. The binary interaction coefficient ( ij ) is introduced as a function of the adsorbate-adsorbent interaction coefficients of the two components ( aiv and a jv ), which is listed as follows:25,26

 ij 

aiv  1 1 a jv  1

(7)

 ii  0

 ij 

(8)

1  1 (i, j  1, 2, ..., N , v) a ji  1

(9)

Since the vacancy solvent was treated as an abstract virtual entity, non-ideal characters were caused by interactions between the adsorbent and adsorbate molecules as well as interactions between adsorbate molecules. As concentration of the adsorbed phase increases, the adsorbate molecules would draw closer together. The interaction between adsorbate molecules would be


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improved and the interaction between adsorbate and adsorbent molecules would decrease. Equations 7–9 provide a method to estimate easily the complex binary adsorbate-adsorbate interaction in a specific adsorbent based on the pure component adsorption isotherms only (Eq. 5). The vacancy solution model was criticized by Talu and Myers26 as being erroneous based on its inconsistent binary selectivity predictions in the Henry’s Law limit (except when saturation capacities for both components are the same). Bhatia and Ding28 revisited the vacancy solution model and argued that the key error source is the incorrect disposition of vacancies during a g a g adsorption. The original model25,26 suggested the chemical potentials i  i and v  v at

phase equilibrium, which assumed that the adsorption of a molecule and the desorption of a vacancy are mutually uncorrelated and independent. However, in reality, the adsorption of a molecule would result in the removal of vacancy, which contradicts the assumption. Thus, they represented the adsorption process as:28

Ai ( g )  vi (a )  Ai (a )

(10)

The condition of equilibrium was given by

d (G )  ia  vi va  ig  0 dni

(11)

The mass-action law was obtained when expressing the chemical potential in terms of activities.28

aia

(ava )vi ig yi P

0

 e Gi / RT

(12)


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In the above equation, vi is the number of vacancies consumed on the adsorption of molecules a of species i . The standard state of the activities of vacancy av takes the value of 1, as the absorbent

is completely without adsorbate and all the vacancies are not occupied by gas molecules. The a standard state of the activities of the adsorbate of i th species ai equals 1 when it completely covers

g the adsorbent with no vacancies.  is the fugacity coefficient of gas phase. Gi0 is the standard i

excess free energy, which could be obtained by equation of state. In this study, the Peng-Robinson equation of state (PR EOS) is used.34 Equation 12 could be also written as Eq. 13 in terms of the activity coefficient of the vacancies and the adsorbed phase.28 0 xia ia  ig yi PeGi / RT a a vi ( xv  v )

(13)

a Cochran et al. 25,26 gave the activity coefficient of the adsorbed phase in terms of xv

N v x aj x aj ln( )  1  ln[  ] [ ]1 (1   ) (1   ) j 1 j 1 ij ij a i

N v

(14)

a In Eq. 14, when pressure P  0 , the mole fraction of adsorbed gas xi  0 and the mole fraction a of the vacancy (considered as a "pseudo-component") xv  1 . In this limit, Eq. 14 will reduce to:

 ia  (1   iv )eiv  va

(15)

Then Eq. 13 will reduce to: 0

y Peiv eGi / RT x  i , i  1, 2,..., N c (1   iv ) a i

(16)


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According to Henry’s Law limit:

ni  n v , xia  Ki

yi P RT

(17)

Combining Eqs. 16 and 17: 0

RTn v , eiv e Gi / RT Ki  (1  iv )

(18)

The condition of equilibrium in terms of activity coefficient (Eq. 13) could be rewritten in terms of Henry’s Law coefficient Ki 28

xia ia (1  iv )e iv Ki g  i yi P , i  1, 2,..., N c ( xva va )vi RTn v ,

(19)

Oil–Gas-Adsorption Three-Phase Equilibrium Model Fig. 1 shows a schematic diagram of oil–gas-adsorption three-phase system consists of N c components and the total number of moles is assumed to be 1. The total composition of each substance in the system is zi (i  1, 2,..., N c ) . The mole number of a liquid at equilibrium is L with compositions xi (i  1, 2,..., N c ) . The mole number of a gas at equilibrium is V with compositions

yi (i  1, 2,..., N c ) . The mole number of an adsorbed phase at equilibrium is A with compositions

xia (i  1, 2,..., N c ) .


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Free gas phase V, y1 , y2 , …, y Nc

Liquid phase L, x1 , x2 , …, xNc

Adsorbed phase

A, x1a ， x2a … xNa

c

Figure 1. Schematic diagram of an oil–gas-adsorption three-phase system consists of free gas phase, liquid phase, and adsorbed phase.

When the fugacity of liquid phase and free gas phase equals ( f i L  f iV ) and the condition of equilibrium between adsorbed phase and free gas phase is reached (Eq. 19), the equilibrium will be realized. Detail calculation is with a combination of material balance equations, the normalization of compositions equations, and thermodynamic equilibrium equations both in the void case and the capillarity case (see detailed equations in Supplemental Information). Through the above equations, for a given temperature and pressure, the fugacity of each component in the free gas, liquid, and adsorbed phases in different cases could be obtained. The adsorbed phase is considered as a film on the capillary surface, and the radius of the capillary is amended. Then the phase equilibrium is achieved by the equality of each component in the liquid and vapour phases (Fig. s1 in Supplemental Information presents the flow chart of this calculation process).


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3. Results and Discussion The oil–gas-adsorption three-phase equilibrium model with the revisited vacancy solution model in this work is used to recalculate the binary gas mixtures of CH4 and C2H6. To make a better comparison, data from the experimental results of Wang et al. are studied.33 Both of the pure component isotherms of CH4 and C2H6 and CH4-C2H6 binary component adsorption isotherms with different compositions of shale samples extracted from the Marcellus formation in the Appalachian Basin were measured in their study. The test pressure ranged from 0 to 125 bar for pure methane adsorption and from 0 to 40 bar for pure ethane adsorption. All the isotherms were tested under three different temperatures: 40°C, 50°C, and 60°C. Their experiment tested the adsorption isotherms of the CH4-C2H6 binary component gas mixture with three groups of different compositions (i.e. C2H6 mole fraction: 4%, 7%, and 10%). Pure Component Adsorption Isotherms i In order to obtain the three parameters of nv , bi , and aiv in Eq. 5, the pure component

adsorption isotherms in Wang’s study33 are regressed. The results are shown in Table 1. The table i

indicates that the higher the temperature, the lower the limiting amount of adsorption nv and constant bi would be. The value of the gas-solid interaction aiv increases with temperature without considering that the only inconformity is the aiv of ethane at 40 °C. Table 1. Regression parameters of the pure component adsorption isotherms from the adsorption experimental data in the study of Wang et al.33 Temperature

Component

40°C

CH4

Regression Coefficient niv , （lb mol/lb）

bi

（lb mol/lb.psia）

Value 1.40×10-4 7.65×10-7


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C2H6

aiv

1.00×10-2

niv , （lb mol/lb）

1.25×10-2

bi


aiv v , i

n CH4 50°C C2H6

CH4 60°C C2H6

bi

（lb mol/lb）


1.58×10-5 19.10 1.26×10-4 6.46×10-7

aiv

7.34×10-2


8.78×10-3

bi


1.09×10-5

aiv

16.20


1.33×10-4

bi


3.41×10-7

aiv

1.04×10-1


1.17×10-2

bi


aiv

6.04×10-6 17.80

Figures 2 and 3 present the matching results of the adsorbed amount of methane and ethane, respectively, by applying the regression parameters from the experimental data in Table 1and using the pure component isotherm correlation in Eq. 5.


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2.50

Adsorption amount (mg/g)

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2.00

1.50 Cal.40 C

1.00

Cal.50 C Cal.60 C Exp.Data of Wang et al. 40 C

0.50

Exp.Data of Wang et al. 50 C Exp.Data of Wang et al. 60 C

CH4

0.00 0

20

40

60

Pressure(bar)

80

100

120

Figure 2. Comparison of calculated adsorption isotherms and the adsorption experimental data of pure methane at temperature 40°C, 50°C, and 60°C.


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6.00

Adsorption amount (mg/g)

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5.00 4.00 3.00

Cal.40 C Cal.50 C

2.00

Cal.60 C Exp.Data of Wang et al. 40 C

1.00

Exp.Data of Wang et al. 50 C Exp.Data of Wang et al. 60 C

C2H6

0.00 0

10

20

Pressure(bar)

30

40

Figure 3. Comparison of calculated adsorption isotherms and the adsorption experimental data of pure ethane at temperature 40°C, 50°C, and 60°C. Fig. 2 shows an asymptotic trend of the methane adsorption isotherms to the maximum adsorbed amount when the pressure becomes high (i.e. larger than 80bar). The adsorption ethane in Fig.3 can easily reach this maximum adsorbed amount of methane at a relative low pressure. Figures 2 and 3 suggest that the adsorption capability of ethane is larger than that of methane in tight pore structures, as under the same pressure, the adsorbed mole number of ethane is larger than the adsorbed mole number of methane.


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Prediction of Binary Gas Mixture v ,

Figures 2 and 3 illustrate that the regression parameters of ni

, bi , and aiv in Table 1 at the

corresponding temperatures are reliable, which could precisely describe the experimental data with very small errors. These parameters could then be applied to the revisited vacancy solution model for the prediction of the adsorbed amount of methane and ethane mixtures. To be consistent, the experimental results of the CH4 and C2H6 binary mixture of Wang et al. are applied.33 The researchers tested the isotherms of the CH4-C2H6 binary gas mixture with three groups of different compositions (i.e. C2H6 mole fractions :4%, 7%, and 10%). All these three groups were tested with pressure ranged from 0 to 125 bar under three constant temperature conditions: 40°C, 50°C, and 60°C. Figures 4, 5, and 6 show the prediction results and experimental data of total weight of the adsorbed binary gas mixture per unit weight shale for the mole fraction of C2H6 of 4%, 7%, and 10% under temperature conditions 40°C, 50°C, and 60°C, respectively. The predictions of CH4C2H6 binary gas mixture isotherms are all calculated from pure adsorption isotherms in previous section. The dots on the figures are plotted from the real experimental data, which provide validation of the calculation. Comparing the prediction curves and the experimental data, the revisited vacancy solution model could provide predictions of satisfactory accuracy with the precise trend of the adsorption isotherms as well as the maximum limit of the adsorption capability, which proves its reliability.


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2.5 T= 40 °C

Adsorption Amount (mg/g)

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2.0

1.5 Exp.Data of Wang et al. 90% CH4

1.0

Cal. Data 90% CH4 Exp.Data of Wang et al. 93% CH4

0.5

Cal. Data 93% CH4 Exp.Data of Wang et al. 96% CH4 Cal. Data 96% CH4

0.0 0

20

40

60

80

Pressure (bar)

100

120

140

Figure 4. Comparison of the predicted adsorption isotherms and the experimental adsorbed amount of methane–ethane binary mixture at the temperature of 40°C with methane mole percentages of 90%, 93%, and 96%.


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2.5 T= 50 °C


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2.0

1.5

1.0

Exp.Data of Wang et al. 90% CH4 Cal. Data 90% CH4 Exp.Data of Wang et al. 93% CH4

0.5

Cal. Data 93% CH4 Exp.Data of Wang et al. 96% CH4 Cal. Data 96% CH4

0.0 0

20

40

60

80

Pressure (bar)

100

120

140



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2.5 T= 60 °C


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2.0

1.5

1.0 Exp.Data of Wang et al. 90% CH4 Cal. Data 90% CH4 Exp.Data of Wang et al. 93% CH4 Cal. Data 93% CH4 Exp.Data of Wang et al. 96% CH4 Cal. Data 96% CH4

0.5

0.0 0

20

40

60

80

Pressure (bar)

100

120

140


Figures 7, 8, and 9 show the compositions of both the adsorbed phase and the free phase of the previous predictions. With a decrease in the total mole percent of ethane, the compositions of ethane in both the adsorbed phase and the free phase reduce. However, the compositions of ethane in the adsorbed phase are always higher than the total mole fraction of ethane in the system, while the compositions of ethane in the free phase are always lower than the mole fraction of ethane in the total system. This trend becomes more significant as the pressure increases, and it proves that ethane is more competitive than methane in the adsorbed phase.


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100.0%

Composition in Adsorbed Phase

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

Page 20 of 38

80.0% CH4 in adsorbed phase - 90% CH4 in total CH4 in adsorbed phase - 93% CH4 in total

60.0%

CH4 in adsorbed phase - 96% CH4 in total C2H6 in adsorbed phase - 90% CH4 in total

40.0%

C2H6 in adsorbed phase - 93% CH4 in total C2H6 in adsorbed phase - 96% CH4 in total

20.0%

0.0% 0

20

40

60

80

100

Pressure (bar) Figure 7. The compositions in the adsorbed phase at temperature 40°C with methane mole fractions of 90%, 93%, and 96% in total.


20

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100.0%

Composition in Free Gas Phase

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

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80.0% CH4 in free gas phase - 90% CH4 in total CH4 in free gas phase - 93% CH4 in total

60.0%

CH4 in free gas phase - 96% CH4 in total C2H6 in free gas phase - 90% CH4 in total

40.0%

C2H6 in free gas phase - 93% CH4 in total C2H6 in free gas phase - 96% CH4 in total

20.0%

0.0% 0

20

40

60

80

100

Pressure (bar) Figure 8. The compositions in the free gas phase at temperature 40°C with methane mole fractions of 90%, 93%, and 96% in total.


21

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100.0%

Composition in Adsorbed Phase

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

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80.0% CH4 in adsorbed phase - at 40°C CH4 in adsorbed phase - at 50°C

60.0%

CH4 in adsorbed phase - at 60°C C2H6 in adsorbed phase - at 40°C

40.0%

C2H6 in adsorbed phase - at 50°C C2H6 in adsorbed phase - at 60°C

20.0%

0.0% 0

20

40

60

80

100

Pressure (bar) Figure 9. The compositions in the adsorbed phase with methane mole fractions of 93% in total at temperature 40°C, 50°C, and 60°C. The change of ethane mole fraction in the adsorbed phase becomes smoother as the pressure increases to the high-pressure range due to the limited pore space of the adsorbent, which is in accordance with the characteristics of the pure component isotherms of methane and ethane. The less the total mole fraction of ethane, the more competitive ethane molecules will become. For the 40°C case (Fig.7), the relative percent differences between the final fraction of ethane in the adsorbed phase and the total fraction of ethane in the system are 9.2%, 11.3% and 12.3%. The increase in the relative difference shows an increase in the competitive nature of ethane.


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Comparison to Other Prediction Models To show the precision of the vacancy solution model further, the prediction results are compared to two other gas mixture adsorption prediction methods. One method is the multicomponent potential theory of the adsorption (MPTA) model, improved by Shapiro and Stenby35 based on Polanyi’s potential theory.36 This is the model used to make predictions in the original experimental data source in the study by Wang et al.33 Another method is the widely-used extended Langmuir model (ELM) for the adsorption of gas mixtures on heterogeneous surfaces first proposed by Kapoor et al..37 For this method, pure component isotherms are correlated in the formula shown in Eq. 20:

n

VmbP 1  bP

(20)

The mole numbers of gas adsorbed in the mixture could be calculated as:

ni 

Vmi bi Pyi 1   bi Pyi

(21)

Figure 10 shows the comparison of different prediction methods to predict the adsorption of the methane–ethane mixture with different compositions at temperature 40°C. To be consistent with the original experimental data, the prediction results are calculated in milligram per unit grams of adsorbent with consideration of molar weight. Comparing the experimental data and the three prediction methods, it is clear that the vacancy solution model has the highest precision with the least errors, and it could accurately predict the maximum adsorption limit of the shale sample to the gas mixtures as well.


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2.5


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

Page 24 of 38

2.0 Exp.Data of 90% CH4 (Wang et al.)

1.5

VSM 90% CH4

1.0

ELM 90% CH4 0.5 MPTA of 90% CH4 (Wang et al.) 0.0 0

20

40

60

80

100

120

140

Pressure (bar) (a)


24

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2.5


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

Energy & Fuels


1.5

VSM 93% CH4

1.0


20

40

60

80

100

120

140

Pressure (bar) (b)


25

Energy & Fuels

2.5


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

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1.5

VSM 96% CH4

1.0


20

40

60

80

100

120

140

Pressure (bar) (c) Figure 10. Comparison of different prediction methods and experimental data of methane– ethane binary mixture adsorption isotherms at a temperature 40°C with methane mole fractions of (a) 90%, (b) 93%, and (c) 96%.

Mean absolute percentage error (MAPE) is used to measure the prediction accuracy of theses prediction methods:

MAPE 

1 n Value pre  Valueexp | 100%)  (| Value n i 1 exp

(22)

Table 2 shows the mean absolute percentage error (MAPE) of the CH4-C2H6 binary mixture adsorption isotherms at temperature 40°C of the above three methods. The vacancy solution model has the highest precision with the least MAPE and could accurately predict the performance of the


26

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Energy & Fuels

adsorption of the binary gas mixture as well as the maximum adsorption capacity of the shale sample to this gas mixtures. Table 2. The mean absolute percentage error (MAPE) of the three prediction methods (VSM, MPTA and ELM) of the methane–ethane binary mixture adsorption isotherms at temperature 40°C with methane mole fractions of: (a) 90%, (b) 93%, and (c) 96%. Prediction Method VSM MPTA ELM

Mean Absolute Percentage Error (MAPE) 90% CH4 93% CH4 96% CH4 4.67% 4.40% 3.41% 12.25% 13.83% 6.90% 4.53% 6.56% 6.34%

4. Conclusions In this work, the adsorption of multicomponent mixture is studied with the oil–gas-adsorption three-phase equilibrium model. The predictions of gas mixture are compared with the experimental results under same conditions in literature together with two other methods including the extended Langmuir model (ELM) and the multicomponent potential theory (MPTA). Three conclusions can be drawn: (1) It is suggested that the oil–gas-adsorption three-phase equilibrium model with the revisited vacancy solution model in this work is reliable to predict the adsorption of a gas mixture from only the pure component isotherms. The model provides predictions of multicomponent gas mixture adsorption of satisfactory accuracy and it shows a better result than the multicomponent potential theory of adsorption methods (MPTA) and the extended Langmuir model (ELM). (2) The composition change in both the adsorbed phase and the free phase are presented in this model. The vacancy solution model takes advantage in describing the mechanism of


27

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Page 28 of 38

competitive adsorption phenomenon in which heavier hydrocarbons are easier to be adsorbed by the adsorbent than lighter hydrocarbons. (3) The model could explain the competitive adsorption phenomenon, which is proven during the adsorption process of the gas mixture. In addition to predications of desorption during the depletion process, it could give simulations for some other production operations in future application, such as CO2 or N2 injection for the displacement of hydrocarbons in shale.

Supplemental Information Material balance equations

L V  A  1

(s.1)

yiV  xi L  Axia  zi (i  1, 2,..., N c )

(s.2)

Normalization of compositions Nc

z i 1

i

Nc

x i 1

i

Nc

y i 1

i

1

(s.3)

1

(s.4)

1

(s.5)


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Energy & Fuels

Nc

x i 1

a i

1

(s.6)

The equilibrium coefficients between free gas and liquid phase are defined as K VL  yi / xi i

(i  1, 2,..., N c )

(s.7)

The equilibrium coefficients between adsorbed phase and free gas phase are defined as K VA  yi / xia i

(i  1, 2,..., N c )

(s.8)

Then, according to Eqs. s.1 to s.8, the following equations could be obtained

yi 

K iVL K iVA zi (i  1, 2,..., N c ) K iVA  ( K iVL  1)VK iVA  ( K iVL  K iVA ) A

(s.9)

xi 

K iVA zi (i  1, 2,..., N c ) K iVA  ( K iVL  1)VK iVA  ( K iVL  K iVA ) A

(s.10)

K iVL zi x  VA (i  1, 2,..., N c ) K i  ( K iVL  1)VK iVA  ( K iVL  K iVA ) A

(s.11)

a i

Nc

Nc

i 1

i 1

Nc

Nc

KiVL KiVA zi 1 KiVA  ( KiVL  1)VKiVA  ( KiVL  KiVA ) A

(s.12)

KiVA zi xi   VA 1  VL VA VL VA i 1 i 1 K i  ( K i  1)VK i  ( K i  K i ) A

(s.13)

 yi  

Nc

Nc

i 1

i 1

 xia  

KiVL zi 1 KiVA  ( KiVL  1)VKiVA  ( KiVL  KiVA ) A

(s.14)


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Page 30 of 38

Thermodynamic equilibrium equations The chemical potentials of each substance should be equal at thermodynamic equilibrium. For the liquid phase and vapour phase, the fugacity of each component in both phases should be equal:

fi L  iL  xi  P L  fiV  iV  yi  PV

iL  P L K  yi / xi  V V i  P VL i

(i  1, 2,..., N c )

(i  1, 2,..., N c )

(s.15)

(s.16)

f and  are the fugacity and the fugacity coefficient, respectively. For given system temperature and pressure, the fugacity coefficients could be determined by the equation of state. In this work, the Peng-Robison equation of state (PR EOS) is used34

RT ln(iV )  RT ln(

   P  f iV RT   dvV  RT ln Z V )    V   vV yi P  ni vV ,T ,n jV ( ji ) vV   

(s.17)

RT ln(iL )  RT ln(

   P  fi L RT   dvL  RT ln Z L )    L   vL xi P  n v  i vL ,T ,n jL ( ji ) L   

(s.18)

R is the universal gas constant; ni is the mole numbers of i th component in each phase; vi is the

molar volume of the i th component in each phase; Z is the Z-factor; and T is the temperature. The equilibrium coefficients between free gas and the adsorbed phase could be obtained by applying the fugacity of the adsorbed phase from the modified vacancy solution model (Eq. 19). Void (or bulk volume or PVT cell) case


30

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Energy & Fuels

In a void case, the interface could be considered a plane.

K

VL i

iL  yi / xi  V i

(i  1, 2,..., N c )

(s.19)

Capillarity Case In the reservoir formation, the hydrocarbon fluids flow in the porous media which could be considered as a combination of various capillary with different sizes. If the capillary pressure is not negligible, especially in the case that the reservoir fluids flow in the tight reservoir formation with small pore radius (i.e. less than 1 m ), then according to the Young-Laplace equation: 38

Pc  PV  P L 

2σcos r

P L  PV  Pc  PV 

(s.20)

2σcos r

(s.21)

r is the radius of capillary and θ is the contact angle. As the oil and gas system is not soluble in water, the surface tension could be obtained by the Macleod-Sugden equation:39



1/4 og

Nc

L V xi   mol yi )   Pi (  mol

(s.22)

i 1

L L where  og is the interfacial tension between the oil phase and gas phase, and  mol and  mol are

the molar density of the liquid phase and the molar density of the gas phase, respectively. For a given temperature and pressure, the chemical potential of each component in the vapour, liquid, and adsorbed phases in different cases could be obtained. The adsorbed phase is considered


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Page 32 of 38

as a film on the capillary surface and the radius of the capillary is amended. Then the phase equilibrium is achieved by the equality of chemical potential of each component in the adsorbed phase, liquid phase and vapour phase. Figure s1 depicts this calculation process as a flow chart.

f iV  f i L

Figure s1. Flow chart of the oil–gas-adsorption three-phase equilibrium calculation with consideration of capillary pressure.


32

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Energy & Fuels

Corresponding Author *Fanhua Zeng, Phone：+1 (306) 586 6366, Email: [email protected]

Acknowledgement The authors would like to acknowledge the fellowships from the University of Regina and the Innovation Center of Unconventional Oil and Gas Resources of Yangtze University. We are grateful for support from the Chinese 13th 5 Year Plan National Science and Technology Major Projects (Sub-Issue Number: 2016ZX05049005-010 and 2016ZX05060-026).

Notes The authors declare no competing financial interest.

References 1. Chalmers, G. R. L .; Bustin, R. M. The Organic Matter Distribution and Methane Capacity of the Lower Cretaceous Strata of Northeastern British Columbia, Canada. Int. J. Coal Geol. 2007, 70(1-3), 223-239. 2. Chalmers, G. R. L.; Bustin, R. M. Lower Cretaceous Gas Shales in Northeastern British Columbia, Part II: Evaluation of Regional Potential Gas Resources. Bull. Can. Pet. Geol. 2008, 56(1), 22-61.


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3. Cao, T. T.; Song, Z. G.; Wang, S. B.; Jia, X. A Comparative Study of the Specific Surface Area and Pore Structure of Different Shales and Their Kerogens. Sci. China: Earth Sci. 2015, 58(4), 510-522. 4. Naraghi, M. E.; Javadpour, F. A Stochastic Permeability Model for the Shale-Gas Systems. Int. J. Coal Geol. 2015, 140, 111-124. 5. Pan, L.; Xiao, X.; Tian, H.; Zhou, Q.; Chen, J.; Li, T.; Wei, Q. A Preliminary Study on the Characterization and Controlling Factors of Porosity and Pore Structure of the Permian Shales in Lower Yangtze Region, Eastern China. Int. J. Coal Geol. 2015, 146, 68-78. 6. Ghanbari, E.; Dehghanpour, H. Impact of Rock Fabric on Water Imbibition and Salt Diffusion in Gas Shales. Int. J. Coal Geol. 2015, 138, 55-67. 7. Wan, Y.; Pan, Z.; Tang, S.; Connell, L. D.; Down, D. D.; Camilleri, M. An Experimental Investigation of Diffusivity and Porosity Anisotropy of a Chinese Gas Shale. J. Nat. Gas Sci. Eng. 2015, 23, 70-79. 8. Zolfaghari, A.; Dehghanpour, H.; Holyk, J. Water Sorption Behaviour of Gas Shales: I. Role of Clays. Int. J. Coal Geol. 2017, 179, 130-138. 9. Zolfaghari, A.; Dehghanpour, H.; Xu, M. Water Sorption Behaviour of Gas Shales: II. Pore Size Distribution. Int. J. Coal Geol. 2017, 179, 187-195. 10. Song, X.; Lü, X.; Shen, Y.; Guo, S.; Guan, Y. A Modified Supercritical Dubinin–Radushkevich Model for the Accurate Estimation of High Pressure Methane Adsorption on Shales. Int. J. Coal Geol. 2018, 193, 1-15.


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20. Gamba, G.; Rota, R.; Storti, G.; Carra, S.; Morbidelli, M. Adsorbed Solution Theory Models for Multicomponent Adsorption Equilibria. AIChE J. 2010, 35(6), 959-966. 21. Dubinin, M. M. New Experimental and Theoretical Investigations into Kinetics of Physical Adsorption by Microporous Adsorbents. Pure & Appl. Chem., 1977, 48(4), 407-414. 22. Lucassen-Reynders, E. H. A Surface Equation of State for Mixed Surfactant Monolayers. J. Colloid Interface Sci. 1972, 41(1), 156-167. 23. Lucassen-Reynders, E. H. Adsorption of Surfactant Monolayers at Gas/Liquid and Liquid/Liquid Interfaces. Prog. Surf. Membr. Sci. 1976, 10, 253-360. 24. Kaul, B. K. Correlation and Prediction of Adsorption Isotherm Data for Pure and Mixed Gases. Ind. Eng. Chem. Process Des. Dev. 1984, 23(4), 711-716. 25. Cochran, T. W.; Kabel, R. L.; Danner, R. P. The Vacancy Solution Model of Adsorption— Improvements and Recommendations. AIChE J. 1985, 31(12), 2075-2081. 26. Cochran, T. W.; Kabel, R. L.; Danner, R. P. Vacancy Solution Theory of Adsorption Using Flory‐Huggins Activity Coefficient Equations. AIChE J. 2010, 31(2), 268-277. 27. Talu, O.; Myers, A. L. Rigorous Thermodynamic Treatment of Gas Adsorption. AIChE J. 2010, 34(11), 1887-1893. 28. Bhatia, S. K.; Ding L. P. Vacancy Solution Theory of Adsorption Revisited. AIChE J. 2001, 47(9), 2136. 29. Nitta, T.; Kuro-Oka, M.; Katayama, T. An Adsorption Isotherm of Multi-Site Occupancy Model for Heterogeneous Surface. J. Chem. Eng. Jpn. 1984, 17(1), 45-52.


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39. Economou, I. G.; Tsonopoulos, C. Associating Models and Mixing Rules in Equations of State for Water/Hydrocarbon Mixtures. Chem. Eng. Sci. 1997, 52(4), 511-525.


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Prediction of Adsorption Isotherms of Multicomponent Gas Mixtures in

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