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A Theoretical Approach to Model Gas Adsorption/Desorption and the Induced Coal Deformation and Permeability Change Quanshu Zeng, Zhiming Wang, Brian J. McPherson, and John D. Mclennan Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b01105 • Publication Date (Web): 03 Jul 2017 Downloaded from http://pubs.acs.org on July 13, 2017

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Energy & Fuels

1

A Theoretical Approach to Model Gas Adsorption/Desorption and the

2

Induced Coal Deformation and Permeability Change

3

Quanshu Zeng†, Zhiming Wang†,*, Brian J. McPherson‡, John D. McLennan‡

4

†

5

University of Petroleum, Beijing 102249, China

6

‡

State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China

Energy & Geoscience Institute at the University of Utah, Salt Lake City, UT 84108, United States

7 8

ABSTRACT: Recovery of coalbed methane (CBM) can trigger a series of coal-gas interactions, including methane

9

desorption, coal deformation, and associated permeability change. These processes may impact one another. A

10

primary objective of this analysis is to simultaneously quantify these interactions and their impacts during CBM

11

recovery. To achieve this and other objectives, a rigorously coupled adsorption-strain-permeability model was

12

developed. Gas adsorption, coal deformation, and cleat permeability characteristics were described using simplified

13

local density (SLD) adsorption theory, a theory-based strain model, and matchstick-based permeability models,

14

respectively. The strain model was verified against measured methane-adsorption-induced coal strain data

15

published during the past 60 years, and the coupled model was tested using well test data measured in the San Juan

16

Basin of New Mexico, USA. Results suggest that the strain model is very consistent with measured coal deformation

17

data for fluid pressure up to 80 MPa, and the average relative errors between measured and predicted results are all less

18

than 15.51%. In general, simulated volumetric strain first increases then decreases with pressure, and the maximum

19

volumetric strain occurs at approximately 20 MPa pressure, which is strikingly different from the pressure corresponding

20

to maximum Gibbs adsorption. Simulation results also suggest that methane adsorption/desorption, coal deformation,

21

permeability evolution, and their coupled impacts are quantitatively reasonable and consistent with observed data.

22

Several cleat permeability models were coupled and tested, and the improved Palmer and Mansoori model exhibited the

23

best performance among those tested. Since methane sorption-induced volumetric strain of typical San Juan Basin

24

coal is five-nine times of magnitude larger than that due to reservoir compaction, matrix shrinkage dominates and

25

leads to monotonic increase of permeability during CBM recovery.

26

KEYWORDS: coalbed methane; gas adsorption/desorption; coal deformation; permeability evolution;

27

multi-physical coupling -1-

ACS Paragon Plus Environment

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1. INTRODUCTION

29

Advances in understanding of coal-gas interactions have changed how coalbed methane (CBM) recovery

30

operations are designed and executed, and have led to reduced mining hazards to increased potential as an

31

unconventional gas resource.

32

Fundamentally, coal reservoirs are conceptualized as porous matrix blocks bounded by a well-developed cleat

33

(fracture) network and are often idealized as a collection of “matchsticks” structures1. On the one hand, matrix

34

pores with diameters from nanometers to tens of nanometer form the most significant component of porosity and

35

thus dominate gas storage. On the other hand, cleats refer to mutually orthogonal face and butt cleats and facilitate

36

fluid mobility. In general, the matrix pore contribution to fluid mobility is assumed to be small; we therefore

37

consider only cleat permeability and the general term “permeability” here refers to only cleat permeability. Cleat is

38

usually described by its aperture, length, spacing, and frequency. For typical coal reservoirs1-3, coal cleat apertures

39

usually vary from 1×10-7 to 5×10-5 m, cleat lengths vary from 0.1 to 1 m, cleat spacing vary from 0.002 to 0.07 m,

40

and a frequency of 12 to 22 per cm2. During primary recovery of coalbed methane, both reservoir compaction and

41

gas desorption will occur, and have a competing effect on matrix volumetric change. Although the volumetric

42

change is small, its impact on cleat open or closure and permeability change is still considerable. Therefore,

43

understand gas adsorption/desorption, coal deformation, permeability change, and their interactions is of significant

44

importance.

45

Several frameworks have been developed to describe the pure and mixed gas adsorption. These include the

46

Langmuir model4, the ideal adsorbed solution theory5, the real adsorbed solution theory6, the theory of volume

47

filling of micropores7, the two-dimensional equation of state model8, the One-Kondo model9, and the simplified

48

local density model10. While the former four models seem to lack of theoretical rigor for mixed gases and have

49

defects in adsorption modeling, the latter three models are found to be readily amenable to the modeling demands

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of CBM systems11.

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Similar to gas adsorption, sorption-induced strain is usually described by a Langmuir-like equation12-14. However,

52

this empirical model is confined to low pressures. Another problem is that the model parameters can only be

53

determined through history matching. To our best knowledge, only two theory-based model have been developed.

54

Bangham and Fakhoury15 believe that the surface energy of coal solid matrix decreases with adsorption, and the

55

inducing swelling increment is proportional to surface energy decrease. However, the adsorption therein is

56

calculated with Langmuir isotherm, which is inherent flawed at high pressures. Pan and Connell16 believe that the -2-


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surface energy change caused by adsorption is equal to the elastic energy change of coal rock, and apply the energy

58

balance approach to derive a theory-based strain model. However, Pan and Connell model includes the solid grain

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and pore structure geometry term, which is difficult to estimate and, therefore, has associated uncertainties.

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According to the characteristics of coal reservoir and CBM recovery, a variety of analytical permeability models

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have been developed with three assumptions: (1) matchstick geometry, (2) uniaxial strain, and (3) constant

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overburden stress17-21. Gray22 first proposed a coal permeability model on the basis of stress change, accounting for

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both geomechanical effects and sorption-induced swelling/shrinkage. Sawyer et al.23 first proposed a coal

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permeability model based on the porosity change instead. Seidle and Huitt24 also developed a model based on

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porosity change, but only the porosity change induced by coal swelling/shrinkage was considered. Palmer and

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Mansoori25, 26 used the three assumptions mentioned above to derive a simple, concise relationship for porosity

67

changes. With the same three assumptions, Shi and Durucan27, 28 and Cui and Bustin13, 29 developed a permeability

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model from the constitutive equation of a linear poroelasticity media, however, effective horizontal stress is

69

assumed in the former study, while effective mean stress is assumed for the later study. Liu and Hapalani30 derived

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a rigorously coupled adsorption-strain-permeability model with Langmuir model, Bangham and Fakhoury’s theory,

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and matchstick-based permeability models. However, Langmuir model therein is inherent flawed at high pressures.

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Liu and Rutqvist31 introduced an internal swelling stress concept to account for the impact of matrix

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swelling/shrinkage on cleat aperture change. Connell et al.32 used the general linear poroelastic constitutive law to

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extend the permeability model to tri-axial conditions. Wu et al.33-35 further considered the non-equilibrium

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thermodynamic effects including slip and surface diffusion for gas transport in noanopores. Among these analytical

76

models, the Shi-Durucan (S&D) model, the Cui-Bustin (C&B) model, and the Palmer-Mansoori (P&M) model (and

77

its improved version) are widely used. However, no final conclusion is yet reached regarding which model yields

78

the best performance.

79

This paper is aimed at developing an internally-consistent coupled adsorption-strain-permeability model to

80

simultaneously quantify gas desorption, coal deformation, permeability change and associated coupled interactions

81

of these processes during CBM recovery. The remainder of this paper is organized as follows: section 2 presents the

82

model and couple details of gas adsorption, coal strain, and permeability; sections 3 presents the validation of strain

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model, the validation and sensitivity analysis of coupled model.

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2. MODELING

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In this section, the coupled adsorption-strain-permeability model is detailed. Gas adsorption, coal deformation, -3-


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and cleat permeability characteristics are described using simplified local density theory, a theory-based strain

87

model, and matchstick-based permeability models, respectively.

88

10 2.1. Simplified Local Density Theory. Rangarajan et al. proposed simplified local density (SLD) theory by

89

combining mean-field approximation theory and density functional theory, believing that gas adsorption/desorption

90

is a resultant effect of adsorbate-adsorbate and adsorbate-adsorbent interactions. This theory provides a consistent

91

framework that accounts for both adsorbate-adsorbate and adsorbate-adsorbent interactions, delineates the

92

adsorbent structural properties with adsorbent physical geometries, offers the opportunity for model generalizations

93

using molecular descriptions, and predicts the mixture-adsorption based solely on pure gas isotherms. Following

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Chen et al.36 and Fitzgerald et al.37, 38, several assumptions form the basis of SLD theory, including

95

(1) The chemical potential of any point above an adsorbent surface is equal to the bulk phase chemical potential;

96

(2) The chemical potential of any point above an adsorbent surface is the sum of adsorbate-adsorbate and

97 98 99

adsorbate-adsorbent interactions; (3) The adsorbate-adsorbent interaction of any point above an adsorbent surface is independent of adsorbate molecule number or temperature;

100

(4) Matrix pores are treated as perfect slits with uniformly distributed temperature and pressure;

101

(5) All adsorbate and adsorbent molecules are spherical, except for the adsorbate molecules touching slit walls.

102

Under these assumptions, the chemical potential of any point above an adsorbent surface will be the same, or

103

µ ( z ) = µbulk = µ ff ( z ) + µ fs ( z )

104

where, µ ( z ) is the chemical potential at position z, J/mol; z is the distance between adsorbate molecule and

105

adsorbent surface, m; µbulk is the chemical potential of the bulk phase, J/mol; µ ff ( z ) is the chemical potential

106

caused by adsorbate-adsorbate interaction at position z, J/mol; and µ fs ( z ) is the chemical potential caused by

107

adsorbate-adsorbent interaction at position z, J/mol.

(1)

108

Since a slit shape is assumed for coal pores with adsorbate molecules located between the parallel slit walls,

109

adsorbate molecules interact with both walls, as schematically depicted in Fig. 1. These interactions are expressed

110

by

111

µ fs ( z ) = N A Ψ fs ( z ) +Ψ fs ( Ls − z ) 

112

where, N A is the Avogadro constant, 6.02×1023mol-1; Ψ fs ( z ) is the potential energy caused by interaction

-4-


(2)

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between adsorbate molecule at position z and left wall, J; Ψ fs ( Ls − z ) is the potential energy caused by interaction

114

between adsorbate molecule at position z and right wall, J; and Ls is the slit width, m. Interactions between

115

adsorbate and adsorbent molecules can be represented by the Lennard-Jones potential function39. Specifically, the

116

interactions between adsorbate molecules and carbon planes after the fourth layer are ignored, or 4  1  d 10 1 4  d fs   fs −     ∑ 5 z'  2 i =1  z '+ ( i − 1) d ss     

Ψ fs ( z ) = 4πρ atoms ε fs d 2fs  

117

118

(3)

where,

119

ε fs = ε ff × ε ss

120

d fs =

121

d ff + d ss

z' = z +

2 d cc 2

(4) (5)

(6)

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and ρ atoms is the carbon atom density, 3.82×1019/m2; ε fs is the potential energy caused by interaction between

123

adsorbate and adsorbent molecules, J; ε ff is the potential energy caused by interaction between adsorbate and

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adsorbate molecules, J; ε ss is the potential energy caused by interaction between adsorbent and adsorbent

125

molecules, J; d fs is the collision diameter between adsorbate and adsorbent molecules, m; d ff is the diameter of

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adsorbate molecule, m; d ss is the carbon plane spacing, 3.35×10-10m; z ' is the distance between adsorbate

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molecule and carbon atoms of the first layer, m; and d cc is the diameter of carbon atom, 1.4×10-10m.

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The chemistry potentials of bulk and adsorption phases may be described in fugacity forms, or  fbulk    f0 

(7)

 f ads ( z )    f0 

(8)

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µbulk = µ0 + RT ln 

130

µ ff ( z ) = µ0 + RT ln 

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where, interactions among adsorbate molecules can be described by a fluid equation of state. Since the

132

Redlich-Kwong equation of state (RK EOS) describes the thermodynamic properties of methane with reasonable

133

accuracy40, 41, it was selected and implemented to calculate the density and fugacity of bulk phase, and fugacity of

134

the adsorption phase, or

-5-


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p 1 aρ = − RT ρ 1 − b ρ RT (1 + b ρ )

135 136

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(9)

where, a=

137

9

(

3

3

138

b=

R 2Tc2.5 0.5 2 − 1 pcT

)

(10)

2 − 1 RTc 3 pc

(11)

1

139

where, µ0 is the reference chemical potential, J/mol; R is the gas constant, 8.314 J/(K·mol); T is the

140

temperature, K; fbulk is the fugacity of the bulk phase, Pa; f 0 is the reference fugacity, Pa; f ads ( z ) is the

141

fugacity of adsorption phase at z position, Pa; p is the pressure, Pa; ρ is the molar density, mol/m3; a is an

142

attraction term, J·m3·mol-2; b is a repulsion term, m3/mol; Tc is the critical temperature of methane, 190.56K;

143

and pc is the critical pressure of methane, 4.599×106Pa.

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To calculate the fugacity of the bulk phase and adsorption phase, the RK EOS can be rearranged as follows,

145

ln

146

ln

 fbulk pb  abulk p p ln (1 + bbulk ρbulk ) = − 1 − ln  − bulk  − p RT ρbulk RT  bbulk RT 1.5  RT ρbulk

f ads ( z ) p

=

 pb  a ( z ) p − 1 − ln  − ads  − ads 1.5 ln 1 + bads ρ ads ( z )  RT ρ ads ( z )  RT ρ ads ( z ) RT  bads RT p

(12)

(13)

147

where, the attraction term of the adsorbed phase varies with the relative position of adsorbate molecules. The

148

relation between aads ( z ) and Ls / d ff can be obtained by integrating the sum of the interactions between any

149

adsorbate molecule and all the adsorbed molecules around it10.

150

151

For Ls / d ff ≥ 3 ,   3 z 5 1  + − 3 8 d 16  ff  Ls − z 1  −  8   d 2    ff  aads ( z )  1 1 = 1 − − 3 3 abulk  z 1  Ls − z 1   8 8 − −      2   d ff 2   d ff  3( L − z ) 5 1 s  + − 3  8d ff 16  z 1  8 −  d ff 2    

1 z 3 ≤ ≤ 2 d ff 2 L 3 z 3 ≤ ≤ s − 2 d ff d ff 2

Ls 3 L z 1 − ≤ ≤ s − d ff 2 d ff d ff 2

-6-


(14)

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For 3 > Ls / d ff ≥ 2 ,  3z 5 1  + − 3  8d ff 16 L −z 1 8 s −    d ff 2     aads ( z )  3Ls 3 = − abulk  8d ff 8 3 L − z ) 5 1  ( s + − 3  8d ff 16  z 1  8 −  d ff 2    

153

154

L 1 z 3 ≤ ≤ s − 2 d ff d ff 2 Ls 3 z 3 − ≤ ≤ d ff 2 d ff 2

(15)

L 3 z 1 ≤ ≤ s − 2 d ff d ff 2

For 2 > Ls / d ff ≥ 1.5 , aads ( z )

155

abulk

 3 L =  s − 1  8  d ff 

(16)

156

where, abulk is the attraction term for the bulk phase, J·m3·mol-2; bbulk is the repulsion term for the bulk phase,

157

m3/mol; aads ( z ) is the attraction term for the adsorption phase, J·m3·mol-2; bads is the repulsion term for the

158

adsorption phase, m3/mol; ρ ads ( z ) is the density of the adsorption phase at position z, mol/m3; ρbulk is the molar

159

density of bulk phase, mol/m3.

160

Equilibrium criteria may be eventually obtained by substituting Eqs. (2), (7), and (8) into Eq. (1), or  Ψ fs ( z ) +Ψ fs ( Ls − z )  f ads ( z ) = fbulk exp  −  k BT  

161

162

(17)

where, k B is the Boltzmann constant, 1.38×10-23J/K.

163

According to SLD theory, the Gibbs excess adsorption can be obtained by integrating the density difference

164

between the adsorption and bulk phases along the slit length. Since a slit pore includes two walls, half the surface

165

area may be considered, or

166

nGibbs =

As 2

∫

3 Ls − d ff 8

3 d ff 8

 ρ ads ( z ) − ρbulk dz

(18)

167

where, nGibss is the Gibbs excess adsorption amount, mol/kg; and As is the surface area per unit mass adsorbent,

168

m2/kg.

169

2.2. Coal Strain Model. Both reservoir compaction and matrix shrinkage may occur during the primary recovery

170

of CBM, as depicted schematically in Fig. 2. These have opposite effects on coal deformation and permeability

171

evolution. Coal deformation due to reservoir compaction can be described by linear elastic theory, while that due to -7-


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172

methane desorption may be predicted on the basis of Bangham and Fakhoury15. Both strains are assumed to be

173

independent but superposed.

174 175

Bangham and Fakhoury15 believed that the adsorption of adsorbate will decrease the surface free energy of adsorbent, and the swell increment induced is proportional to the surface free energy decrease, or ∆ε sl = ∆Ψ × γ

176 177

(19)

where, Ass ρ c K ss

γ =

178

(20)

179

where, ∆ε sl is the linear strain change due to gas adsorption/desorption, dimensionless; ∆Ψ is the surface free

180

energy change due to gas adsorption/desorption per unit mass adsorbent, J/kg; γ is a deformation constant，kg/J;

181

Ass is the specific surface area of adsorbent, m2/m2; ρc is the coal density, kg/m3; and K ss is the adsorbent

182

expansion/shrinkage modulus due to gas adsorption/desorption, Pa.

183

According to the Gibbs adsorption equation, the surface free energy decrease may be characterized as ∆Ψ = RT ∫

184 185 186

189

192 193 194 195

(21)

Combining Eqs. (19) and (21), the linear strain change due to gas adsorption/desorption may be expressed as ∆ε sl =

ρ c RT K ss

∫

pn p0

n Gibbs dp p

(22)

To simplify the strain calculation, isotropy is assumed for the coal, and volumetric strain is assumed to be three times as large as linear strain12, 42, 43, or

190 191

p0

nGibbs d ( ln p ) Ass

where, p0 is the initial pressure, Pa; and pn is the current pressure, Pa.

187 188

pn

∆ε s = ∆3ε sl =

3ρ c RT K ss

∫

pn p0

n Gibbs dp p

(23)

where, ∆ε s is the volumetric strain change due to gas adsorption/desorption, dimensionless. In addition, the volumetric strain change due to reservoir compaction may be calculated on the basis of linear elastic theory and expressed as ∆ε m = −

3 (1 − 2v ) E

( pn − p0 )

(24)

where, ∆ε m is the volumetric strain change due to reservoir compaction, dimensionless; v is the Poisson’s ratio,

-8-


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196 197 198

Energy & Fuels

dimensionless; and E is the Young’s modulus, Pa. Since both strains are completely independent and may be superimposed on each other, the total strain may be calculated as

199 200 201

∆ε = ε s + ε m =

3ρc Rθ K ss

∫

pn p0

3 (1 − 2v ) nGibbs dp − ( pn − p0 ) p E

(25)

Note that nGibss changes with pressure, and strains can be represented in differential forms and calculated Gibss / p , both Eqs. (23) and (25) may be rewritten as according to Simpson’s rule. Assuming that f ( p ) = n

3ρc Rθ ∆p  f ( p0 ) + 4 f ( p1 ) + 2 f ( p2 ) + L + 2 f ( pn − 2 ) + 4 f ( pn −1 ) + f ( pn )  K ss 3 

(26)

3 (1 − 2v ) 3ρ c Rθ ∆p  f ( p0 ) + 4 f ( p1 ) + 2 f ( p2 ) + L + 2 f ( pn − 2 ) + 4 f ( pn −1 ) + f ( pn ) − ( pn − p 0 ) K ss 3  E

(27)

∆ε s =

202

203

∆ε =

204

2.3. Permeability Model. Several cleat permeability models for the specific characteristics of coal reservoirs

205

amenable to CBM recovery are widely used, including the Shi-Durucan (S&D) model, the Cui-Bustin (C&B)

206

model, and the Palmer-Mansoori (P&M) model (and its improved version). These models were applied in this study

207

and share three assumptions: (1) matchstick geometry, (2) uniaxial strain, and (3) constant overburden stress17-20.

208

The following sections provide a brief review of these models.

209

27, 28, 44, 45

2.3.1. Shi-Durucan (S&D) Model. Shi and Durucan

derived a permeability model accounting for both

210

reservoir compaction and desorption-induced matrix shrinkage, believing that the cleat permeability ratio varies

211

exponentially with effective horizontal stress normal to cleats. The S&D model was derived from the constitutive

212

equation of a linear thermo-elastic porous medium, but replaced the thermal expansion term with an analogous

213

matrix shrinkage term, or

214 215 216

217

k −3C ∆σ e =e f h k0

(28)

v E ∆p + ∆ε s 1− v 3 (1 − v )

(29)

where, ∆σ he = −

Cf = −

∂φ / φ ∂σ he

(30)

218

where, k is the cleat permeability, mD; k0 is the initial cleat permeability, mD; C f is the cleat volume

219

compressibility with respect to effective horizontal stress changes, Pa-1; ∆σ he is the effective horizontal stress

-9-


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220

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change, Pa; ∆p is the pressure change, Pa; and φ is the cleat porosity, dimensionless.

221

2.3.2. Cui-Bustin (C&B) Model. The derivation of the C&B model was similar to that of S&D model. However,

222

Cui and Bustin13, 29 believe that permeability evolution is mainly controlled by effective mean stress, rather than

223

effective horizontal stress, or k −3 ∆σ e / K =e m p k0

224 225

(31)

where, ∆σ me = −

226

1+ v 2E ∆p + ∆ε s 3 (1 − v ) 9 (1 − v )

(32)

227

Kp = φK

228

where, ∆σ me is the effective mean stress change, Pa; K p is the pore volume modulus, Pa; and K is the bulk

229

modulus, Pa.

230

(33)

25, 2.3.3. Palmer-Mansoori (P&M) Model. The P&M model

26

is based on strain change of a linear

231

thermo-elastic porous medium (instead of stress change), ignores grain compressibility, and employs a cubic

232

relationship between permeability ratio and porosity ratio to calculate permeability change, or 3

k  φ   ∆φ  =  = + 1 k0  φ0   φ0 

233 234

3

(34)

where, ∆φ =

235

2 (1 − 2v ) (1 + v )(1 − 2v ) ∆p − ∆ε s E (1 − v ) 3 (1 − v )

(35)

236

However, cleat anisotropy suggests that pressure-dependent permeability may be largely suppressed, where

237

coalbeds are flat, cleats are vertical, and cleats are the dominant flow pathways. During the fitting with well test

238

data in the San Juan Basin, Palmer17 found that permeability forecasted by this model is at best only generally

239

consistent with permeability evolution trends observed in field production data, then he introduced a suppression

240

factor (g=0.3) into the compression term to improve its performance, or ∆φ =

241

g (1 + v )(1 − 2v ) E (1 − v )

∆p −

2 (1 − 2v ) 3 (1 − v )

∆ε s

(36)

242

where, φ0 is the initial cleat porosity, dimensionless; ∆φ is the cleat porosity change, dimensionless; and g is a

243

suppression factor, dimensionless.

244

2.4. Coupling of Adsorption, Deformation, and Permeability Evolution. Since adsorption in the coal strain

245

model here is calculated by simplified local density, while matrix shrinkage in the permeability model is calculated - 10 -


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Energy & Fuels

246

by the strain model, these processes are rigorously coupled. Specifically, the volumetric strain due to gas adsorption

247

and both effects may be described by substituting Eq. (18) into Eqs. (26) and (27), respectively. Gas adsorption,

248

coal deformation, and permeability evolution are rigorously coupled by further substituting Eq. (26) into Eqs. (28),

249

(31), and (34).

250

The solution procedure (algorithm) of the coupled model is depicted in Fig. 3. First, input data are read into

251

arrays, including pressure, temperature, physical properties of methane, physical and elastic properties, and

252

adsorption, strain and cleat parameters of coal. Then, the algorithm calculates density, fugacity, and chemical

253

potential of methane in both the bulk and adsorption phases, and calculates the adsorbate-adsorbent interaction and

254

induced chemical potential. Next, methane density in adsorption phase is calculated iteratively until the system

255

satisfies specified adsorption equilibrium criterion, and methane density in the adsorption phase along the slit

256

length is determined. Last but not least, adsorption, strain, and cleat parameters of coal are iteratively calculated

257

until predicted results coincide with the adsorption isotherm, strain data, and permeability data.

258

3. VALIDATION AND SENSITIVITY ANALYSIS

259

12, 46-51 published 3.1. Validation of Strain Model. All measured methane-adsorption-induced coal strain data

260

during the past 60 years and publicly available were assembled to evaluate the performance of the proposed strain

261

model.

262

Pressure, temperature, physical properties of methane, as well as the physical and elastic properties, adsorption

263

and strain parameters of coal, must be realized prior to the coal strain calculation. The physical properties of

264

methane were extracted from the NIST database52 while physical and elastic properties of coal were measured

265

directly from coal samples, and adsorption and strain parameters were obtained by fitting the adsorption isotherm

266

and strain data. In particular, three SLD adsorption parameters were obtained by fitting the adsorption isotherm,

267

wherein slit length and solid-solid interaction potential energy parameter were regressed with isotherm shape, and

268

surface area was regressed with isotherm magnitude.

269

All the input parameters required are summarized in Table 1, and the predicted results are plotted in Fig. 4. These

270

results suggest that the strain model can predict coal deformation fairly accurately at pressure up to 80 MPa; the average

271

relative errors between measured and predicted results are all within 15.51%, and the largest error occurs is associated

272

with coal H48 (Fig. 4(c)). For all plotted results (Fig. 4), positive expresses expansion, while negative expresses shrinkage.

273

Since volumetric strain is larger than zero in most cases, this means matrix shrinkage is more obvious than reservoir

274

compaction. In addition, the volumetric strain first increases then decreases with pressure, and the maximum volumetric

- 11 -


Energy & Fuels

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

275

strain occurs at around 20 MPa pressure, in contrast to the pressure corresponding to maximum Gibbs adsorption. For

276

coal reservoirs with initial pressure larger than the inversion pressure, permeability will first decrease then increase

277

during CBM recovery, and the lowest permeability occurs at the inversion pressure. Permeability will increase

278

monotonically for coal reservoirs with initial pressure lower than the inversion pressure.

279

53 3.2. Validation of Coupled Model. Well test data measured in the San Juan Basin were collected and to

280

evaluated the performance of the coupled model. The physical properties, elastic properties, adsorption and strain

281

parameters of a San Juan coal47 were used as input parameters, and only the cleat parameters were adjusted, as

282

indicated in Table 2.

283

The adsorption isotherm of the coal is plotted in Fig. 5. Sorption increases with pressure monotonically.

284

Sorption-induced strain, mechanical strain, and total strain with pressure are illustrated in Fig. 6. Two competing effects

285

may occur during primary recovery of CBM, including reservoir compaction and matrix shrinkage. Effective stress

286

will increase as pore pressure decreases, resulting in closure of cleats and reduction of cleat permeability. Matrix

287

shrinkage induces an opposite effect on cleat and permeability evolution, as pore pressure decreases and gas

288

desorbs. Since the volumetric strain due to methane adsorption of a typical San Juan coal is five-nine times as large

289

as that due to reservoir compaction, matrix shrinkage dominates and will lead to a monotonic increase of

290

permeability, as illustrated by Fig. 7. All the permeability models predict lower permeability at higher pressure, and

291

higher permeability at lower pressure. Among the cleat permeability models coupled, the improved P&M model

292

exhibited the best performance with respect to fitting well test data, followed by the P&M model, then the S&D model,

293

and finally the C&B model (worst).

294

3.3. Sensitivity Analysis of Coupled Model. Sensitivity analysis is further conducted on the coupled model to

295

highlight and calibrate its performance. Only the improved P&M model was coupled and analyzed. The parameters

296

studied are summarized in Table 3, including coal reservoir environment and property, elastic properties, and

297

adsorption characteristic parameters. For the typical coal seams54, 55 within 2000 m, its initial pore pressure is

298

usually between 5 MPa and 20 MPa, temperature is located between 293.15 K and 333.15 K, and initial cleat

299

porosity range from 0.1% to 1%. It should be noted that a maximum pressure of 80 MPa is selected for its potential

300

enhanced coalbed methane recovery application. For a typical coal1, 22, 56-60, its density is usually within 800-1300

301

kg/m3, Young’s modulus is between 1000 MPa and 5000 MPa, and Poisson’s ratio ranges from 0.2 to 0.4. For

302

methane adsorption on coals11, 30, 61-63, slit length is between 1 nm to 2 nm, solid-solid interaction potential energy

303

parameter ranges from 20 K to 100 K, surface area is within 20-120 m2/g, and adsorbent expansion/shrinkage

- 12 -


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304

Energy & Fuels

modulus ranges from 400 MPa to 1200 MPa.

305

The influence of the parameters mentioned above on volumetric strain and permeability changes are plotted in

306

Figs. 8 and 9. As can be observed, both volumetric strain and permeability ratio decrease with temperature, slit

307

length, and adsorbent expansion/shrinkage modulus, but increase with coal density, Young’s modulus, Poisson’s

308

ratio, solid-solid interaction potential energy parameter, and surface area. In addition, the volumetric strain will first

309

increase then decrease with pore pressure, while the permeability ratio trend is just the opposite. Both the maximum

310

volumetric strain and minimal permeability ratio occur at approximately 20 MPa pressure.

311

Fig. 8 also indicates that the influence on volumetric strain may be sorted by initial porosity, slit length, Poisson’s

312

ratio, temperature, Young’s modulus, coal density, solid-solid interaction potential energy parameter, surface area,

313

and adsorbent expansion/shrinkage modulus. The impacts of the former four parameters are assumed to be small,

314

while Young’s modulus has little effect on volumetric strain change once it reaches 3000 MPa. Within the coal

315

parameter range studied, volumetric strain is always larger than zero, and its decrease with pressure only occurs at

316

extremely high pressure and low Young’s modulus situations.

317

Fig. 9 illustrates that the influence of these parameters on volumetric strain may be in the order of slit length,

318

temperature, Young’s modulus, Poisson’s ratio, coal density, solid-solid interaction potential energy parameter,

319

surface area, adsorbent expansion/shrinkage modulus, and initial porosity. Similarly, the former two parameters has

320

little impact on permeability ratio. The influence of Young’s modulus on permeability ratio is assumed to be small

321

if it is larger than 3000 MPa. Fig. 9 also indicates that permeability ratio is larger than 1 for the majority of coal

322

parameters studied. Permeability decline at the initial production may be only observed at extremely high pressure

323

situation.

324

4. CONCLUSIONS

325

The purpose of the study detailed in this paper was to develop and demonstrate a rigorously coupled

326

adsorption-strain-permeability model, with gas adsorption, coal deformation, and cleat permeability characteristics

327

described by SLD adsorption theory, a theory-based strain model, and matchstick-based permeability models,

328

respectively. The following conclusions and recommendations are suggested, based on the results of this study.

329

(1) The proposed strain model predicts coal deformation consistent with observed data for pressure up to 80 MPa, and

330

the average relative errors between observed and predicted results are all within 15.51%. Volumetric strain first increases

331

then decreases with pressure, and the maximum volumetric strain occurs at approximately 20 MPa pressure, in contrast

332

to the pressure corresponding to maximum Gibbs adsorption.

- 13 -


Energy & Fuels

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

333

(2) The coupled adsorption-strain-permeability model forecasts realistic behaviors with respect to methane

334

adsorption/desorption, coal deformation, permeability evolution, and coupled impacts these processes may induce on one

335

another.

336

(3) For a typical coal reservoir, matrix shrinkage dominates and will lead to the monotonic increases of

337

permeability. Among the cleat permeability models coupled and tested, the improved P&M model exhibited the best

338

performance in fitting (comparison to) well test data, followed by the P&M model second, the S&D model third, and the

339

C&B model was least consistent with well test data.

340

AUTHOR INFORMATION

341

Corresponding Author

342

*E-mail: [email protected]. Phone: +86 1089734958.

343

Notes

344

The authors declare no competing financial interest.

345

ACKNOWLEDGMENTS

346

Financial support from the National Science and Technology Major Project of China (2016ZX05044005-001), High

347

School Subject Innovation Engineering Plan of China (B12033), and China Scholarships Council Program

348

(201606440099) are gratefully acknowledged.

349

REFERENCES

350

(1) Seidle, J. Fundamentals of coalbed methane reservoir engineering; PennWell Corp: Tulsa, OK USA, 2011

351 352 353 354 355 356 357

(ISBN No. 9781593700010). (2) Romeo, M. F. Coal and coalbed gas: Fueling the future; Elsevier: Waltham, MA USA, 2014 (ISBN No. 9780123969729). (3) Laubach, S. E.; Marrett, R. A.; Olson, J. E.; Scott, A. R. Characteristics and origins of coal cleat: A review. Int. J. Coal Geol. 1998, 35, 175-207. (4) Langmuir, I. The adsorption of gases on plane surfaces of glass, mica and platinum. J. Am. Chem. Soc. 1918, 40(9), 1361-1403.

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(5) Myers, A. L.; Prausnitz, J. M., Thermodynamics of mixed-gas adsorption. AIChE J. 1965, 11(1), 121-127.

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(6) Stevenson, M. D.; Pinczewski, W. V.; Somers, M. L.; Bagio, S. E. Adsorption/desorption of multicomponent

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gas mixtures at in-seam conditions. In Society of Petroleum Engineers - SPE Asia-Pacific conference; Perth,

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critical review. Int. J. Coal Geol. 2011, 87(3-4), 175-189. (19) Wang, G. G. X.; Zhang, X. D.; Wei, X. R.; Fu, X. H.; Jiang, B.; Qin, Y. A review on transport of coal seam gas and its impact on coalbed methane recovery. Front. Chem. Sci. Eng. 2011, 5(2), 139-161. (20) Pan, Z. J.; Connell, L. D. Modelling permeability for coal reservoirs: A review of analytical models and testing data. Int. J. Coal Geol. 2012, 92(1), 1-44. (21) Wu, K. L.; Li, X. F.; Guo, C. H.; Wang, C. C.; Chen, Z. X. A unified model for gas transfer in nanopores of shale-gas reservoirs: coupling pore diffusion and surface diffusion. SPE J. 2016, 21(5), 1583-1611. - 15 -


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(25) Palmer, I.; Mansoori, J. How permeability depends on stress and pore pressure in coalbeds: A new model. In Proceedings - SPE Annual Technical Conference and Exhibition; Denver, CO USA, October 6-9, 1996. (26) Palmer, I.; Mansoori, J. How permeability depends on stress and pore pressure in coalbeds: A new model. SPE Reserv. Eng. 1998, 1(6), 539-543. (27) Shi, J. Q.; Durucan, S. Drawdown induced changes in permeability of coalbeds: A new interpretation of the reservoir response to primary recovery. Transport Porous Med. 2004, 56(1), 1-16. (28) Shi, J. Q.; Durucan, S. A model for changes in coalbed permeability during primary and enhanced methane recovery. SPE Reserv. Eval. Eng. 2005, 8(4), 291-299.

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methane production and acid gas sequestration into coal seams. J. Geophys. Res. 2007, 112(B10202), 1-16.

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J. Coal Geol. 2013, 113, 1-10. (31) Liu, H. H.; Rutqvist, J. A new coal-permeability model, internal swelling stress and fracture-matrix interaction. Transport Porous Med. 2010, 82(1), 157-171. (32) Connell, L. D.; Lu, M.; Pan, Z. An analytical coal permeability model for tri-axial strain and stress conditions. Int. J. Coal Geol. 2010, 84(2), 103–114. (33) Wu, K. L.; Li, X. F.; Wang, C. C.; Chen, Z. X.; Yu, W. A model for gas transport in microfractures of shale and tight gas reservoirs. AIChE J. 2015, 61(6), 2079-2088. (34) Wu, K. L.; Chen, Z. X.; Li, X. F. Real gas transport through nanopores of varying cross-section type and shape in shale gas reservoirs. Chem. Eng. J. 2015, 281, 813-825. (35) Wu, K. L.; Li, X. F.; Wang, C. C.; Yu, W.; Chen, Z. X. Model for surface diffusion of adsorbed gas in - 16 -


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485

FIGURES

486

487

Fig. 1 Slit Pore Characterization in SLD Theory

488

489

490

Fig. 2 Reservoir Compaction and Matrix Shrinkage on Coal Deformation and Permeability Change

491

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492

493

Fig. 3 Solution Procedure for Coupled Adsorption-Strain-Permeability Model

494 - 21 -


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Page 22 of 30

495

496

(a) D Coal

(b) G Coal

(c) H Coal

(d) J Coal

(e) Charcoal Coal

(f) Fuitland Coal

(g) Ardley Coal

(h) Wolf Mountain Coal

497

498

499

500

501

502 503

Fig. 4 Volumetric Strain versus Pressure

- 22 -


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Energy & Fuels

504

505

506

(i) Anderson Coal

(j) Gilson Coal

(k) Sulcis Coal

(l) San Juan Coal

(m) Illinois Coal

(n) Hunter Valley Coal

507

508

509

510 511

Fig. 4 Volumetric Strain versus Pressure (Continued)

512

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Energy & Fuels

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513

514

Fig. 5 Adsorption Isotherm of a Typical Coal Sample in the San Juan Basin

515

516

517

Fig. 6 Volumetric Strain Change of a Typical Coal Sample in the San Juan Basin

518

519

520

Fig. 7 Permeability Change with Reservoir Pressure in the San Juan Basin

521

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Page 24 of 30

Page 25 of 30

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Energy & Fuels

522

523

(a) Initial Pore Pressure

(b) Temperature

(c) Coal Density

(d) Young’s Modulus

(e) Poisson’s Ratio

(f) Slit Length

(g) Solid-Solid Interaction Potential Energy Parameter

(h) Surface Area

524

525

526

527

528

529 530

Fig. 8 Sensitivity Analysis of Model Parameters on Volumetric Strain

- 25 -


Energy & Fuels

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Page 26 of 30

531

532 533

(i) Adsorbent Expansion/Shrinkage Modulus

(j) Initial Porosity

Fig. 8 Sensitivity Analysis of Model Parameters on Volumetric Strain (Continued)

534

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Page 27 of 30

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Energy & Fuels

535

536

(a) Initial Pore Pressure

(b) Temperature

(c) Coal Density

(d) Young’s Modulus

(e) Poisson’s Ratio

(f) Slit Length

(g) Solid-Solid Interaction Potential Energy Parameter

(h) Surface Area

537

538

539

540

541

542 543

Fig. 9 Sensitivity Analysis of Model Parameters on Permeability Ratio

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Energy & Fuels

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Page 28 of 30

544

545 546

(i) Adsorbent Expansion/Shrinkage Modulus

(j) Initial Porosity

Fig. 9 Sensitivity Analysis of Model Parameters on Permeability Ratio (Continued)

547

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Page 29 of 30

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

548

Energy & Fuels

TABLES

549

Table 1 Model Fitting Results for Coal Strain Induced by Methane Adsorption Database

Moffat

Levine

Coal

T

Ls

εss/kB

As 2

E

ρc

v

3

Kss

Error

K

nm

K

m /g

MPa

Dimensionless

m /kg

MPa

%

D

308.15

1.76

22.3

91.6

4500

0.35

1305

1684

12.10

G

308.15

1.50

222.3

52.8

2890

0.40

1300

3605

13.86

H

308.15

1.55

22.9

91.4

4300

0.37

1298

2515

15.51

J

308.15

2.91

16.1

220.6

1160

0.45

1321

1883

6.73

Charcoal

308.15

1.82

21.7

340.4

2630

0.34

1522

3552

13.31

Fruitland

308.15

1.50

19.5

104.2

4400

0.32

1400

1874

5.41

Ardley

298.15

2.59

12.6

19.8

3000

0.30

1500

439

3.78

WM

298.15

1.19

21.5

49.3

3000

0.30

1340

929

2.07

Anderson

299.82

1.63

15.3

129.7

2713

0.34

1400

887

11.77

Gilson

299.82

1.72

22.3

68.1

2713

0.34

1400

719

3.14

Sulcis

318.15

1.80

66.7

83.9

2410

0.48

1400

1902

8.31

San Juan

308.15

1.14

22.6

71.6

3450

0.37

1400

1361

3.37

Illinois

308.15

2.08

18.1

51.2

2119

0.40

1400

774

7.07

HV

308.15

1.22

30.5

86.5

1150

0.44

1340

2256

1.71

Bustin

Robertson Ottiger Harpalani Pan

550 551

Table 2 Input Parameters of the Coupled Adsorption-Strain-Permeability Model Input Parameters

Value

Unit

p

6.5

MPa

T

308.15

ρc

1400

m3/kg

E

3450

MPa

v

0.37

Dimensionless

Ls

1.14

nm

ess/kB

22.6

K

As

71.6

m2/g

Kss

1361

MPa

Cf

0.10

MPa-1

Kp

4.10

MPa

φi

0.07

%

φi

0.13

%

Reservoir Environment Physical Property

K

Elastic Property

Adsporption Parameters

Strain Parameters

Cleat Parameters

552 553 554 555 - 29 -


Energy & Fuels

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556

Page 30 of 30

Table 3 Sensitivity Research Projects of the Coupled Model p0

θ

ρc

E

v

3

MPa

Dimensionless

ess/kB

As

Kss

φi

nm

K

2

m /g

MPa

%

Ls

MPa

K

kg/m

5-80

313.15

1300

3000

0.3

1.5

60

70

1200

0.5

10

293.15-333.15

1300

3000

0.3

1.5

60

70

1200

0.5

10

313.15

800-1800

3000

0.3

1.5

60

70

1200

0.5

10

313.15

1300

1000-5000

0.3

1.5

60

70

1200

0.5

10

313.15

1300

3000

0.2-0.4

1.5

60

70

1200

0.5

10

313.15

1300

3000

0.3

1-2

60

70

1200

0.5

10

313.15

1300

3000

0.3

1.5

20-100

70

1200

0.5

10

313.15

1300

3000

0.3

1.5

60

20-120

1200

0.5

10

313.15

1300

3000

0.3

1.5

60

70

400-2000

0.5

10

313.15

1300

3000

0.3

1.5

60

70

1200

0.1-1

557

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Theoretical Approach To Model Gas Adsorption ... - ACS Publications

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