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Journal of Rock Mechanics and Geotechnical Engineering 12 (2020) 461e472

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Controlling effects of differential swelling index on evolution of coalpermeability

Chuanzhong Jiang a,b, Zhenfeng Zhao c, Xiwei Zhang a, Jishan Liu d,*, Derek Elsworth e,Guanglei Cui a

aKey Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang, 110819, ChinabBeijing Key Laboratory for Precise Mining of Intergrown Energy and Resources, China University of Mining and Technology, Beijing, 100083, ChinacOil & Gas Technology Research Institute, Changqing Oilfield Company, Xi’an, 710018, ChinadDepartment of Chemical Engineering, School of Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, AustraliaeDepartment of Energy and Mineral Engineering, G3 Centre and Energy Institute, The Pennsylvania State University, University Park, PA16802, USA

a r t i c l e i n f o

Article history:Received 23 November 2019Received in revised form27 January 2020Accepted 5 February 2020Available online 8 April 2020

Keywords:Coal permeabilityDifferential swelling behaviorGas adsorptionEquilibrium pressure

* Corresponding author.E-mail address: [email protected] (J. Liu).Peer review under responsibility of Institute of R

nese Academy of Sciences.

https://doi.org/10.1016/j.jrmge.2020.02.0011674-7755 � 2020 Institute of Rock and Soil MechanicNC-ND license (http://creativecommons.org/licenses/b

a b s t r a c t

Coal permeability measurements are normally conducted under the assumption that gas pressure in thematrix is equalized with that in fracture and that gas sorption-induced swelling/shrinking strain isuniformly distributed within the coal. However, the validity of this assumption has long been questionedand differential strain between the fracture strain and the bulk strain has long been considered as theprimary reason for the inconsistency between experimental data and poroelasticity solutions. Althoughefforts have been made to incorporate the impact into coal permeability models, the fundamental natureof those efforts to split the matrix strain between fracture and coal bulk remains questionable. In thisstudy, a new concept of differential swelling index (DSI) was derived to theoretically define the relationamong sorption-induced strains of the coal bulk, fracture, and coal matrix at the equilibrium state. DSIwas a function of the equilibrium pressure and its magnitudes were regulated by the Langmuir constantsof both the matrix and the coal bulk. Furthermore, a spectrum of DSI-based coal permeability modelswas developed to explicitly consider the effect of differential strains. These models were verified with theexperimental data under the conditions of uniaxial strain, constant confining pressure, and constanteffective stress.� 2020 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

As an “unconventional” gas resource, coalbed methane (CBM) isa natural product of coalification process (Moore, 2012). CBM haslong been considered as a source of disasters for underground coalmining, but now is recognized as a valuable energy resource (Flores,1998). Therefore, reasonable disposal of CBM released from un-derground coal mining can not only reduce gas explosion hazardbut also recover this clean energy. Depending on reservoir condi-tions, CBM can be extracted using reservoir-pressure depletionmethod or enhanced CBM (ECBM) recovery methods. For ECBM,nitrogen (N2) and/or carbon dioxide (CO2) are injected into coal

ock and Soil Mechanics, Chi-

s, Chinese Academy of Sciences. Pry-nc-nd/4.0/).

seam to desorb and displace CBM (Puri and Yee, 1990; Gunter et al.,1997).

One of the key reservoir parameters for CBM production andCO2 sequestration in coal seams is coal permeability. Gas is mainlystored in a form of adsorption in coal seams, and both CBM recoveryand CO2 injection trigger a series of desorption or adsorptionassociated processes in the coal seams (Chen et al., 2012; Pereraet al., 2013). During CBM production, the fracture pressure draw-down induces gas desorption from the coal matrix, and the des-orbed gas flows into the fracture network. During desorption, thecoal matrix shrinks. As a direct consequence of this matrixshrinkage, the fractures may dilate, and fracture permeability in-creases correspondingly (e.g. Harpalani and Schraufnagel, 1990a).Conversely, CO2 geological sequestration and CO2-ECBM mayreverse these processes.

Significant experimental efforts have been made to investigatethe characteristics of coal permeability evolution. Coal permeabilityexperiments can be generally divided into stress-controlled

oduction and hosting by Elsevier B.V. This is an open access article under the CC BY-

http://creativecommons.org/licenses/by-nc-nd/4.0/


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C. Jiang et al. / Journal of Rock Mechanics and Geotechnical Engineering 12 (2020) 461e472462

(constant confining pressure and constant effective stress) anddisplacement-controlled ones (uniaxial strain condition and con-stant volume condition) (Shi et al., 2018). Under following fourboundary conditions, permeability shows different evolutioncharacteristics:

(1) Laboratory-measured coal permeability in terms of adsorb-ing gasses such as CH4 and CO2 can change from reduction toenhancement instantaneously under constant confiningpressure condition (Robertson, 2005; Pini et al., 2009;Siriwardane et al., 2009; Wang et al., 2011, 2019; Vishalet al., 2013).

(2) Most experimental data (Connell et al., 2010a; Lin andKovscek, 2014; Meng et al., 2015; Chen et al., 2019; Weiet al., 2019a) exhibit significant changes under constanteffective stress condition: the permeability declines initiallyand then remains stable as the gas pressure increases.

(3) An ingenious experimental work was conducted by Mitraet al. (2012). In their experiment, coal sample was held un-der uniaxial strain condition. The coal samplewas taken fromSan Juan basin. It was not permitted to shrink laterally as aresult of gas desorption by adjusting the confining pressure.The results showed that coal permeability increases contin-uously with decrease in gas pressure. These observations areconsistent with the field observations made in different partsof the San Juan basin (Gierhart et al., 2007).

(4) A permeability test for coal was conducted under constantvolume condition to investigate the coalecleat interactionsusing helium (Wang et al., 2017). The results showed thatpermeability increases as the pore pressure increases.

Various coal permeability models have been proposed toexplain the variability of coal permeability under laboratory orfield conditions. Based on applicable boundary conditions, majorcoal permeability models were classified into two groups (Liuet al., 2011a): permeability models under conditions of uniaxialstrain and that under conditions of variable stress. In thesemodels(Gray, 1987; Palmer andMansoori, 1996; Shi and Durucan, 2003b),coal permeability is defined as a function of the coal bulk strain. Itis also assumed that the coal bulk strain caused by gas adsorptionis equal to that of the fracture (Cui et al., 2007; Zhang et al., 2008).However, this assumption may not be valid for coal (Zang et al.,2015).

This discrepancy between assumption and reality has long beenrecognized. In the study (Liu et al., 2010a), the concept of “internalswelling stress”was introduced to account for the impact of matrixswelling/shrinkage on the fracture aperture changes. In this study,the matrix strain was divided into two parts: one for the fracturecomponent and the other for the bulk strain. Based on this concept,permeability models under conditions of uniaxial strain and con-stant external stress were developed.

In other studies (e.g. Connell et al., 2010a; Lu and Connell,2010), a constant ratio of the adsorption-induced fracture strainincrement to the sorption-induced coal bulk strain increment wasintroduced. By rearranging the Shi-Durucan model, they foundthat the ratio was significantly larger than (approximately 50times) the sorption-induced coal bulk strain. Liu et al. (2010b)proposed the concept of elastic modulus reduction ratio to allo-cate matrix swelling strain to the fracture and the matrix. Byassuming that only part of total swelling strain contributes tofracture aperture change and the remaining to coal bulk defor-mation, a permeability model under variable stress conditionswas developed (Chen et al., 2012).

To explain the phenomenon of coal permeability switch fromreduction to rebound under constant confining pressure condition,the concepts of critical pressure and critical time were introducedto define the switch from local swelling to macro-swelling due togas diffusion from fractures to matrices (Liu et al., 2011b). Beforethe critical time, coal permeability was determined by the internalvolumetric transformations (local swelling) between matrixswelling and fracture void compaction; after the critical time, coalpermeability was determined by the external boundaries (macro-swelling). Similarly, the concept of critical swelling area wasintroduced to define the relation between swelling transition andcoal permeability evolution under variable stress conditions (Quet al., 2014). Subsequently, numerous coal permeability modelswere proposed (e.g. Guo et al., 2014; Wang et al., 2014; Peng et al.,2014b; Lu et al., 2016; Liu et al., 2017a). In these models, a constantstrain splitting factor was used to define the ratio of the facturesorption strain to thematrix sorption strain. In a recent study (Penget al., 2017a), the heterogeneous distribution of internal swellingwas considered by extending the concept of strain splitting factor tostrain splitting function. All strain splitting-based permeabilitymodels are summarized in Table 1. For sake of simplicity, only strainsplitting-based parameters are presented in the table. Moredetailed information can be found in the original works.

As mentioned above, the impact of the differential strain be-tween the fracture strain and the bulk strain is a challenging issue.To address this issue, a concept of differential swelling index (DSI)was introduced in this study to define the relation among sorption-induced deformations of the coal bulk, fracture, and coal matrix.Furthermore, a spectrum of DSI-based coal permeability modelswas developed to explicitly consider the effect of differentialstrains.

2. Concept of differential swelling index (DSI)

As shown in Introduction, coal permeability models can beclassified into four categories. (1) The gas sorption-induced strainresults in fracture strain only (Palmer and Mansoori, 1996; Shi andDurucan, 2003a); (2) The gas sorption-induced strain results in coalbulk strain only (Gray, 1987; Zhang et al., 2008); (3) The gassorption-induced coal bulk strain equals the fracture strain(Zimmerman et al., 1986; Cui and Bustin, 2005); and (4) The ratiobetween two of the gas adsorption-induced three strain compo-nents (matrix strain, fracture strain and bulk strain) is a constant(Lu and Connell, 2010). The assumption that gas sorption-inducedstrain results in fracture strain only is not consistent with someexperimental observations and significantly overestimates the ef-fects of matrix swelling on permeability changes (Robertson, 2005).The assumption that adsorption only causes coal bulk strain, bycontrast, would significantly underestimate the effect of adsorp-tion. The third hypothesis that the adsorption-induced coal bulkstrain is the same as the fracture strain may also underestimate thefracture strain (Connell et al., 2010b). The strain splitting approach(Zang et al., 2015; Peng et al., 2017a) is generally empirical. Infollowing section, a theoretical approach is developed.

2.1. DSI model

Coal is a typical porous medium that consists of both matricesand fractures. In this study, the butt/face cleat system, fractures,bedding plies, and joints are all called the fracture system. Micro-scopic experimental techniques can be used to investigatesorption-induced strains in coal (Mao et al., 2015). It is found thatwith adsorption of gases, thematrix adsorption deformationmakes

Table 1Summary of strain splitting-based permeability models.

Expression Parameter definition Source

Ds ¼ � DPþ EðfDεs � DεfIÞ1� v

f ¼ εSI/εS is a constant between zero and one,and εSI is the internal swelling strain Liu et al. (2010a)

k ¼ k0 exp½3ð1� gÞ~εðSÞb � ~εðSÞb is the coal bulk strain increment, and g is theratio between sorption-induced fracture strainand coal bulk strain.

Connell et al. (2010a)

k=k0 ¼ km0

kf0 þ km0

�1þ Rm

fm0

pmK

�3þ kf0

kf0 þ km0

�1þ 1� Rm

fm0ðDεV � DεsÞ

�3 Rm is the elastic modulus reduction ratio thatrepresents the partition of the total strain Liu et al. (2010b)

k=k0 ¼

8>>><>>>:

�1þ a

f0

�� Dεs þ p

Ks

��3ðp � pcÞ

�1þ a

f0

�� εLpcpc þ PL

þ pcKs

��3�1þ a

f0

�p� pcK

��3ðp > pcÞ

pc is the critical pressureLiu et al. (2011)

k=k0 ¼ ð1þ gεeÞ3 ðk � kcÞð1þ bεeÞ3 ðk > kcÞ

kc is the fracture permeability at the switchingpoint Qu et al. (2014)

k=k0 ¼ e�3cfDðs�pÞ�3bDεs b ¼ 1� Ap� po

Pc þ p� p0is the strain splitting

factor

Peng et al. (2017a)

k=k0 ¼ ½expð�CfDse � SfDεsÞ�3 Sf is the ratio of fracture strain change to theincremental volumetric swelling strain Chen et al. (2012)

k=k0 ¼1� a

f0K½ðs� s0Þ � ðP � P0Þ� �

fmf0

�εmaxPP þ pL

� εmaxP0P0 þ PL

�3fm ¼ dVmf

dVmis the effective coal matrix

deformation factor

Guo et al. (2014)

k=k0 ¼ exp� 3cf

�ðs� s0Þ� ðp� p0Þþ f

E3ð1� 2vÞ

εSmaxpεðp� p0Þ

ðpþ pεÞðp0 þ pεÞ� f defines the internal swelling partition

Lu et al. (2016)

k=k0 ¼"1� 3f εL

ff0

�pm

pL þ pm� ppL þ pm0

�� s� s0 � aðpf � pf0Þ � bðpm � pm0Þ

Kf

#3 f is the internal swelling coefficientLiu et al. (2017b)

kz=kz0 ¼(1þ 1

f0

"3ð1� 2vbÞðp� p0Þ

Eb� εL

�FIp

pL þ p� FI0p0pL þ p0

�#)3 FI is the internal swelling ratioWang et al. (2014)

C. Jiang et al. / Journal of Rock Mechanics and Geotechnical Engineering 12 (2020) 461e472 463

the internal fracture compressed and the coal bulk swells (Karacan,2003; Pone et al., 2009; Dawson et al., 2012). Three sorption-induced strains corresponding to three strain components weredefined in this study: ε

sb, ε

sm, and ε

sf , representing the sorption/

desorption-induced coal bulk strain, matrix strain, and fracturestrain, respectively. The adsorption-induced strain is the ratio ofadsorption-caused volume fluctuations against original counter-part, which can be measured by a strain gage; the sorption-inducedmatrix strain, εsm, can be measured for a small sample containingonly coal matrix and no cleats; however, εsf is more difficult tomeasure experimentally (Connell et al., 2010a). To study themicrostructure of coal, micro-computed tomography (micro-CT)was applied (Ramandi et al., 2016), and it showed that coal isformed with a complex matrix-fracture system (Fig. 1a). A repre-sentative elementary volume (REV) of the coal structure is depictedin Fig. 1b, inwhich the value of initial coal bulk volume ðVb0Þ can be

Fig. 1. Illustration of adsorption-induced differential swelling behavior: (a) Micro-CT image oInitial equilibrium stage (initial equilibrium pressure p0); and (d) Current equilibrium stage

expressed by the sum of initial fracture volume ðVf0Þ and initialmatrix volume ðVm0Þ, i.e. Vb0¼Vf0þVm0. Coal adsorption character-istics can be generally measured under unconstrained conditions(Peng et al., 2017a). With gas adsorption, coal bulk swells andfracture is compressed. For the initial equilibrium stage as illus-trated in Fig. 1c at the initial equilibrium pressure p0, the volumesof coal bulk, matrix and fracture can be expressed as Vb0ð1þεsb

��p¼p0

Þ,Vm0ð1þ ε

sm��p¼p0

Þ, Vf0ð1 þ εsf

��p¼p0

Þ, respectively, where

εsb

��p¼p0

; εsf

��p¼p0

; εsm��p¼p0

denote the adsorption-induced strains of

coal bulk, coal fracture and coal matrix under pressure p0,respectively. Similarly, as the pressure increases, the volume for thecurrent equilibrium stage with equilibrium pressure p as shown inFig. 1d can then be written as Vb0ð1 þ ε

sb

��p¼pÞ; Vm0ð1 þ ε

sm��p¼pÞ;

Vf0ð1 þ εsf

��p¼p

Þ, respectively.

f coal structure (Ramandi et al., 2016); (b) representative elementary volume (REV); (c)(equilibrium pressure p).


Bycomparing the current stage and initial stage,we canobtain thevolumetric balance between coal matrix, coal fracture and coal bulk:

Vb0

�εsb

��p¼p � ε

sb

��p¼p0

�¼ Vm0

�εsm��p¼p � ε

sm��p¼p0

�þ Vf0

�εsf

��p¼p

� εsf

��p¼p0

�(1)

Let Dεsb ¼ εsb

��p¼p � ε

sb

��p¼p0

represent the increment of the coalbulk strain induced by adsorption, Dεsm ¼ ε

sm��p¼p � ε

sm��p¼p0

repre-

sent the increment of the coal matrix strain, and Dεsf ¼ εsf

��p¼p

�εsf

��p¼p0

represent the increment of the coal fracture strain. Using the

definition of porosity (f ¼ Vf=Vb, where Vf is the fracture volumeand Vb is the coal bulk volume), the relation between sorption-induced bulk strain increment, Dεsb, matrix strain increment, Dεsm,and fracture strain increment, Dεsf , can be expressed as

Dεsb ¼ ð1�f0ÞDεsm þ f0Dεsf (2)

where f0 is the initial porosity.Eq. (2) explicitly defines the relation among three strain compo-

nents. Experimentaldatashowthat themeasuredmatrixandcoalbulksorption strains canbefitted into Langmuir type curves (Harpalani andSchraufnagel, 1990b; Mitra et al., 2012;Wei et al., 2019b):

εis ¼

εLippþ PLi

(3)

where εis is the gas adsorption-induced strain; εL is the Langmuir

strain constant that represents the maximum adsorption strain atinfinite pressure; PL is the Langmuir pressure constant at which themeasured sorption strain is equal to 0:5εL; and the subscript i ¼b and m denotes the bulk and matrix, respectively.

Although the fracture system controls the evolution of perme-ability, it is almost impossible to measure the adsorption strain ofthe fracture in experiment. The previous work on this issue hasindicated that the value of pore strainwas around 50 times the bulkstrain for the coal under hydrostatic conditions (Connell et al.,2010b; Chen et al., 2012). As discussed above, the differentialswelling/shrinking behavior defines the relation among thesorption-induced coal fracture, coal matrix and coal bulk strains. Tofurther define the relation between Dεsb and ε

sf , the differential

swelling/shrinking index (DSI), f , was defined as f ¼ Dεsf= Dεsb.Substituting the definition of coal swelling strain Eq. (3) into Eq. (2)gives

f ¼ DεsfDεsb

¼ 1f0

�1�ð1�f0Þ

εLmPLmεLbPLb

ðpþ PLbÞðp0 þ PLbÞðpþ PLmÞðp0 þ PLmÞ

�(4)

In Eq. (4), DSI is a function of initial porosity, adsorption prop-erties of coal matrix and coal bulk, and equilibrium pressure. Allthese parameters can be measured directly in laboratory. If theLangmuir pressure constants of coal bulk and coal matrix areassumed to be equal, i.e. PLb ¼ PLm, the pressure-related terms onthe right side of Eq. (4) disappear, then this index is degeneratedinto a constant ratio:

f c ¼ DεsfDεsb

¼ 1f0

�1�ð1�f0Þ

εLm

εLb

�(5)

For this special case, the difference between Langmuir strain ofcoal matrix and coal bulk defines the differential swelling behavior.If the Langmuir strain constants are fixed, the ratios remain un-changed. This is consistent with the current assumptions (Lu and

Connell, 2010; Chen et al., 2011). For the homogeneous case, allproperties of the coal matrix and coal bulk are the same every-where, i.e. PLm ¼ PLb and εLm ¼ εLb. In this ideal case, DSI is equalto one, which is consistent with the theory of poroelasticity (Cuiand Bustin, 2005; Zhang et al., 2008).

2.2. DSI-based permeability model

In this section, DSI was included in the coal permeability model.It is commonly assumed that the fracture system controls the gasflow in coal while the flow in the coal matrix can be neglected (Purlet al., 1991). Therefore, the evolution of permeability is mainlycontrolled by the behavior of fracture system (Sparks et al., 1995).Based on volumetric balance, the volumetric deformation of thecoal bulk dVb and the fracture dVf consists of two parts (Gray,1987;Connell et al., 2010a), i.e. one caused by effective stress dVE

b and dVEf

(dVb ¼ dVEb þ dVS

b ), and the other by gas adsorption dVSb and dVS

f

(dVf ¼ dVEf þ dVS

f ). Accordingly, for the strain increment, one can

obtain that dεb ¼ dεEb þ dεSb and dεf ¼ dεEf þ dεSf . Based on this,the volumetric strain of coal bulk ðdVb=VbÞ and the volumetricstrain of fractures ðdVf=Vf Þ can be expressed as follows (Detournayand Cheng, 1993; Zhang et al., 2008)

dVbVb

¼ � 1Kb

ðds�adpÞ þ dεsb (6)

dVfVf

¼ � 1Kf

ðds� bdpÞ þ dεsf (7)

where s ¼ �skk=3 is the mean compressive stress; Kb and Kf arethe bulk moduli of coal bulk and coal fracture, respectively; and a

and b are the Biot’s coefficients. According to the definition of coalporosity, we deduce the following expression:

dVbVb

¼ dVm

Vmþ df1� f

(8)

dVfVf

¼ dVm

Vmþ dfð1� fÞf (9)

Solving Eqs. (8) and (9), we obtain

dff

¼ dVfVf

� dVbVb

(10)

Substituting Eqs. (6) and (7) into Eq. (10) yields

dff

¼

1Kb

� 1Kf

!dsþ

1Kf

� 1Ks

� a

Kb

!dpþdεsf � dεsb (11)

As the bulk modulus Kb is commonly several orders of magni-tude larger than the fracture volume modulus Kf (Peng et al.,2017a), by assuming that 1=Kb � 1=Kfz� 1=Kf , the fracturecompressibility can be defined as cf ¼ 1=Kf (Lu et al., 2016):

dff

¼ � cf dðs�pÞþdεsf � dεsb (12)

Integrating Eq. (12) with time gives

f

f0¼ exp

n� cf ½ðsc �sc0Þ� ðp� p0Þ� þ

�Dεsf �Dεsb

�o(13)


By substituting the definition Eq. (4) of DSI into Eq. (13) yields

f

f0¼ exp

n� cf ½ðsc � sc0Þ� ðp�p0Þ�þ ðf �1ÞDεsb

o(14)

The relationship between permeability and fracture porosity canbe described by the cubic law (Chilingar, 1964; Chen et al., 2013; Cuiet al., 2020) as

kk0

¼�f

f0

�3(15)

where k is the permeability, and k0 denotes the initial value ofpermeability.

Solving Eqs. (14) and (15), general form of the new permeabilitymodel can be expressed as

kk0

¼ exp� 3cf ½ðsc � sc0Þ� ðp�p0Þ� þ3ðf �1Þ

$

�εLbp

pþ PLb� εLbp0p0 þ PLb

�(16)

If equal adsorption-induced fracture and bulk strains areassumed (e.g. Cui and Bustin, 2005; Zhang et al., 2008), theadsorption-induced fracture and bulk strains cancel each other out.Assuming that the material is homogeneous, and the DSI is a con-stant (f ¼ 1), then the permeability model is degenerated as

kk0

¼ expn� 3cf ½ðsc � sc0Þ� ðp�p0Þ�

o(17)

Eq. (17) is the permeability model derived from the classicaltheory of poroelasticity.

2.3. Permeability models under different boundary conditions

The general permeability model defined by Eq. (16) can beapplied to a range of boundary conditions from constant confiningpressure to constant volume, as illustrated in Fig. 2.

2.3.1. Constant confining pressure conditionUnder the constant confining pressure condition as shown in

Fig. 2a, the change of the confining pressure ðsc�sc0Þ is zero. In thiscircumstance, the permeability model can be expressed as

kk0

¼ exp�3cf ðp�p0Þþ 3ðf �1Þ

�εLbp


��(18)

By substituting DSI (Eq. (4)) into Eq. (18), the permeabilitymodel in the case of constant confining pressure in an expansionform is

kk0

¼ exp3cf ðp�p0Þþ3

1f0

�1�ð1�f0Þ

εLmPLmεLbPLb


��1�

εLbppþ PLb

� εLbp0p0 þ PLb

�(19)

2.3.2. Constant effective stress conditionFor the case of constant effective stress condition as shown in

Fig. 2b, the difference between confining stress and pore pressureremains constant and the increment of effective stress is equal tozero (Cui et al., 2018):

Dse ¼ ðsc �sc0Þ � ðp�p0Þ (20)

Substituting Eq. (20) into Eq. (16), the permeability model forthe case of constant effective stress becomes

kk0

¼ exp�3ðf �1Þ

�εLbp


��(21)

By substituting Eq. (4) into Eq. (21), the permeability model inan expansion form is

kk0

¼ exp3

1f0

�1�ð1�f0Þ

εLmPLmεLbPLb

ðpþPLbÞðp0þPLbÞðpþPLmÞðp0þPLmÞ

��1

$

�εLbp

pþPLb� εLbp0p0þPLb

�(22)

2.3.3. Uniaxial strain conditionFor the case of uniaxial strain as shown in Fig. 2c, the vertical

external stress remains constant (sz � sz0 ¼ 0), the horizontaleffective stress increments can be calculated by DsEz ¼ sz � sz0 �ðp � p0Þ ¼ �ðp � p0Þ, and non-zero strain occurs in the vertical/axial direction and zero strain in the horizontal direction. The coalbulk volumetric strain increment can be represented as Dεb ¼Dεbx þ Dεby þ Dεbz. If the coal bulk strain induced by gas adsorptionis isotropic ðDεsbx ¼ Dεsby ¼ Dεsbz ¼ Dεsb=3Þ, the coal bulk strainincrement in three directions can be expressed as

Dεbx ¼ DεEbx þ DεSbx ¼ �DsEx � vDsEy � vDsEzE

þ Dεsbx (23a)

Dεby ¼ DεEby þ DεSby ¼ �DsEy � vDsEx � vDsEzE

þ Dεsby (23b)

Dεbz ¼ DεEbz þ DεSbz ¼ �DsEz � vDsEx � vDsEyE

þ Dεsbz (23c)

where v is the Poisson’s ratio, and E is the elastic modulus.Substituting Dεbx ¼ Dεby ¼ 0 and DsEz ¼ �Dp into Eq. (23a)-

(23c), we can obtain the governing equation for the effectivestress in horizontal direction:

DsEx ¼ DsEy ¼ EDεsb3ð1� vÞ �

v

1� vðp�p0Þ (24)

Then the average effective stress imposed on the coal can beexpressed as

Dse ¼ 13

�DsEx þDsEy þDsEz

�¼ 2EDεsb

9ð1� vÞ �1þ v

3ð1� vÞ ðp�p0Þ

(25)

Fig. 2. Schematic diagrams of various boundary conditions.


Substituting Eq. (25) into Eq. (16), we can obtain the governingequation for the permeability change under the uniaxial straincondition:

k = k0 ¼ expcf ð1þ vÞ1� v

ðp�p0Þþ3

1f0

�1�ð1�f0Þ

εLmPLmεLbPLb


��1� 2cfE3ð1� vÞ

�εLbp


�(27)

kk0

¼ expcf ð1þ vÞ1� v

ðp�p0Þþ�3ðf �1Þ� 2cfE

3ð1� vÞ�

$

�εLbp


�(26)

By substituting Eq. (4) into Eq. (26), the permeability model forthe case of uniaxial strain in an expansion form is

2.3.4. Constant volume conditionFor the case constant volume condition as shown in Fig. 2d,

the displacement of coal sample in each direction is zero, i.e. thetotal volume of coal remains unchanged. Then, the coal bulk

Table 3Properties of Anderson coal for the experiment conducted by Robertson andChristiansen (2005).

Symbol Description Value Unit

f0 Initial porosity 1.31 %

εbL

Langmuir strain constant of coal bulk 0.00931

PbL Langmuir pressure constant of coal bulk 6.11 MPa

cf Fracture compressibility 0.058 MPa�1

εmL Langmuir strain constant of coal matrix 0.0139PmL Langmuir pressure constant of coal matrix 2.487 MPa


strain increment is zero ðdεb ¼ dVb=Vb0¼ 0Þ, and sorption-induced matrix strain contributes to fracture deformation only.Ma et al. (2011) developed a permeability model under theconstant volume boundary condition by making a distinctionbetween mechanical compression-induced strain and matrixshrinking strain. In the experiment, the expansion of coal iscontrolled by adjusting confining pressure to achieve the goal ofconstant volume boundary condition (Espinoza et al., 2014).Based on above understanding, substituting dεb ¼ 0 into Eq. (6)yields

dεsb ¼ 1Kb

ðds�adpÞ (28)

Assuming Biot’s coefficient a ¼ 1, we can obtain Kbdεsb ¼ ds�dp. Substituting this relation and dεb ¼ 0 into Eq. (11) yields

dff

¼ dVfVf

¼ �cfKbdεsb þ dεsf (29)

Integrating Eq. (29) gives porositymodel for the case of constantvolume:

f

f0¼ exp

�� cfKbDε

sb þDεsf

�(30)

Based on the cubic law and by substituting the DSI into theporosity model, we can obtain the governing equation for thepermeability change under the constant volume condition:

kk0

¼ exp��

� cfE

1� 2vþ3f

��εLbp


��(31)

By substituting Eq. (4) into Eq. (31), the permeability model forthe case of constant volume can be defined in an expansion form as

kk0

¼ exp

� cfE

1� 2vþ3

1f0

�1�ð1�f0Þ

εLmPLmεLbPLb


��εLbp


�(32)

3. Model verifications

In this section, our models are compared with experimentalobservations under a spectrum of boundary conditions from con-stant confining pressure to uniaxial strain. Four sets of experi-mental data under these boundary conditions (Robertson andChristiansen, 2005; Pini et al., 2009; Connell et al., 2010a; Mitraet al., 2012) were used. In addition, we also compared with theperformances of other models for the same data sets (Connell et al.,2010a; Guo et al., 2014).

Table 2Properties of coal and fluid for the experiment conducted by Pini et al. (2009).



εbL





3.1. Constant confining pressure condition

We compared the model results of Eq. (19) with experimentaldata under constant confining pressure condition (Robertson andChristiansen, 2005; Pini et al., 2009). Basic parameters for the ex-periments are listed in Tables 2 and 3. The matrix Langmuiradsorption constants of εLm and PLm were obtained bymatching theexperimental data. The comparisons of our model against experi-mental data are illustrated in Figs. 3 and 4.

As shown in Fig. 3, our model results matched the permeabilitydata well for the Sulcis coal. In this case, PLm ¼ 2:631 MPa; εLm ¼0:0556, and the Langmuir pressure constant for thematrix and coalbulk were equal ðPLm ¼ PLbÞ. Under these conditions, the DSI was aconstant ðf ¼� 26:93Þ. This indicates that the ratio of theadsorption-induced fracture strain increment to the bulk strainincrement remains unchanged as the gas pressure increases. Wealso compared our model results with the results of the modelproposed by Connell et al. (2010a). In this particular case, two lineswere overlapped because of the same DSI value ðf ¼ g¼�26:93Þ. Asshown in Fig. 4, our newmodel results also match the permeabilitydata for the Anderson coal. In this case ðPLm ¼ 2:3 MPa; εLm ¼0:041Þ, the DSI value decreases as the gas pressure increases.

3.2. Constant effective stress condition

We compared our model performance of Eq. (22) with theexperimental data (Connell et al., 2010a) for the case of constanteffective stress, as shown in Fig. 5. The mechanical parameters andthe adsorption constants, as listed in Table 4, were taken directlyfrom this study (Connell et al., 2010a). The permeability data wereextracted from the original figures using a data extractor. Comparedwith the original data, there might be some imperceptible errors,but it will not affect the overall trend and the verification process. Inthis case ðPLm ¼ 15:9 MPa and εLm ¼ 0:022Þ, our new model re-sults matched experimental data very well for the coal sample fromSouthern Sydney basin. The value of the DSI increases as the gaspressure increases. If a constant DSI value is used, with g ¼ �18:7and g¼ �26:2, the matches are less satisfactory.

3.3. Uniaxial strain condition

We compared our model performance of Eq. (27) with theexperimental data (Mitra et al., 2012) for the case of uniaxial strain,as shown in Fig. 6. The parameters are listed in Table 5. It should benoted that the initial porosity, elastic modulus and Poisson’s

Fig. 3. Comparisons of our model results with experimental data for the Sulcis coal sample (Pini et al., 2009).


ratio were not provided in their study, thus we used the initialporosity from Young et al. (1991), elastic modulus andPoisson’s ratio from Shi et al. (2005). As shown in Fig. 6, ourmodel results matchedwell with the experimental data. In this caseðPLm ¼ 2:795 MPa and εLm ¼ 0:0195Þ, the DSI value f decreasedwith the pressure. As shown in Fig. 6, we also compared the per-formance of our model with others including SD model, modifiedSD model (Liu et al., 2012), and our constant DSI model.

4. Discussion

As shown in Section 3, the DSI plays a key role in comparison ofour model results with experimental data. Unlike previous studies,f is a function of gas pressure and regulated by Langmuir constants

Fig. 4. Comparisons of our model results with the experimental data f

ðPLm and PLb; εLm and εLbÞ for both matrices and coal bulk,respectively.

In the following, all of the input parameters are taken fromMitraet al. (2012) and are listed in Table 3, and an initial gas pressureðp0 ¼ 0:5 MPaÞ is assumed. The DSI was calculated by use of Eq. (4).Note that the Langmuir pressure constant is PLb ¼ 4:16 MPa for theexperiment data. By changing the matrix Langmuir pressure con-stant PLm, we can obtain the evolution trends of f , as shown inFig. 7a. The corresponding evolutions of permeability under con-stant confining pressure, constant effective stress and uniaxialstrain conditions are shown in Fig. 7bed, respectively.

As shown in Fig. 7a, f decreases as pressure increases whenPLm < PLb, f increases as pressure increases when PLm > PLb, and fremains as a constant when PLm ¼ PLb. As shown in Fig. 7bed,

or the Anderson coal sample (Robertson and Christiansen, 2005).

Table 4Parameters used in matching the permeability for coal core sample from theSouthern Sydney basin of Australia.



εbL





Table 5Parameters for data matching under uniaxial strain condition (Mitra et al., 2012).



εbL




E Elastic modulus 2902 MPav Poisson’s ratio 0.35εmL Langmuir strain constant of coal matrix 0.0195PmL Langmuir pressure constant of coal matrix 2.795 MPap0 Initial equilibrium pressure 6.25 MPa

Fig. 5. Comparisons between different model results and the experimental data under constant effective stress condition for Southern Sydney basin coal sample (Connell et al.,2010a).


the magnitudes of f have a significant impact on the evolutions ofcoal permeability under different boundary conditions. Forexample, coal permeability decreases initially and then recoversas pressure increases when PLm< PLb, and coal permeability in-creases continuously as pressure increases when PLm > PLb. Thesemodel results illustrate that matrix Langmuir constants controlthe evolution of coal permeability.

Fig. 6. Comparisons between different model results and the experimental data und

As shown in Eq. (4), f is a function of pressure only and inde-pendent of boundary conditions. Under the same pressures, wecalculated all permeability evolutions for a range of boundaryconditions from constant confining stress to constant volumeconditions, as shown in Fig. 8. The upper envelope is the

er uniaxial strain condition for San Juan basin coal sample (Mitra et al., 2012).

Fig. 7. Permeability evolution with different differential swelling index.

Fig. 8. Permeability model evolution under different boundary conditions.


conventional poroelasticity solution for the case of constantconfining stress, while the lower envelope is the conventionalporoelasticity solution for the case of constant volume condition.Solutions of all other cases are within these envelopes.

5. Conclusions

In this study, a concept of DSI is proposed to theoretically definethe relation among sorption-induced strains of the coal bulk,


fracture and matrix at the equilibrium state. DSI is a function of theequilibrium pressure and its magnitudes are regulated by theLangmuir constants of both the coal matrix and the coal bulk.Furthermore, a spectrum of DSI-based coal permeability models isdeveloped to explicitly consider the effect of differential strains.These models are verified with the experimental data under theconditions of uniaxial strain, constant confining pressure andconstant effective stress. Based on the model results and verifica-tions, the following conclusions can be drawn:

(1) When the gas sorption-induced fracture strain is assumedthe same as the bulk strain, DSI is equal to unity; when theLangmuir constants for matrix are the same as the ones forcoal bulk, DSI is a constant. However, DSI is a function of theequilibrium pressure when the Langmuir constants for ma-trix are different from the ones for coal bulk. The theoreticaldevelopment of DSI concept essentially removes the basicassumptions in previous studies.

(2) An equilibrium state is typically assumed when interpretingpermeability measurements e representing the assumptionthat equilibration has been reached and that both sorption-induced changes in deformation and their impacts on theevolution of permeability have ceased. At the equilibriumstate, coal permeability is a function of equilibrium pressureand its magnitudes are regulated by the magnitudes of DSI.

Declaration of Competing Interest

The authors wish to confirm that there are no known conflicts ofinterest associated with this publication and there has been nosignificant financial support for this work that could have influ-enced its outcome.

Acknowledgments

This work was supported by National Key R&D Program of China(Grant No. 2018YFC0407006), the 111 Project (Grant No. B17009),and the Australian Research Council (Grant No. DP200101293).These supports are gratefully acknowledged.

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Jishan Liu is professor of the Department of ChemicalEngineering, School of Engineering, The University ofWestern Australia. Liu received his PhD in Mining Engi-neering from Penn State University, USA, in 1996. Afterthat, he has been working primarily at the University ofWestern Australia. His research interests cover Uncon-ventional Reservoir Multiphysics (URM) with an applica-tion to coal seam gas extraction, shale gas extraction,stacked deposits extraction, CO2 sequestration in coal, coalmine safety, and caprock sealing safety.





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