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International Journal ofRock Mechanics and Mining Sciences

journal homepage: www.elsevier.com/locate/ijrmms

Mechanistic analysis of coal permeability evolution data under stress-controlled conditions

Rui Shia,b, Jishan Liuc,⁎, Mingyao Weid, Derek Elsworthe, Xiaoming Wanga

a Key Laboratory of Tectonics and Petroleum Resources, Ministry of Education, China University of Geosciences, Wuhan 430074, Chinab Beijing Key Laboratory for Precise Mining of Intergrown Energy and Resources, China University of Mining and Technology, Beijing 100083, Chinac Department of Chemical Engineering, School of Engineering, The University of Western Australia, 35 Stirling Highway, WA 6009, Australiad State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, Chinae Department of Energy and Mineral Engineering, G3 Centre and Energy Institute, The Pennsylvania State University, University park, PA 16802, USA

A R T I C L E I N F O

Keywords:Coal permeabilityTransient effective stressConstant effective stressConstant confining pressure

A B S T R A C T

At present, two types of experiments under stress-controlled conditions were normally conducted to measurecoal permeability: constant confining pressure (CCP) tests and constant effective stress (CES) ones. The originalrationale of this situation was to assume that the impacts of effective stresses and gas sorption-induced matrixswelling/shrinking on coal permeability could be separated and investigated individually. In this study, wecollected coal permeability data measured under both conditions with a purpose to see if this original rationalewas appropriate. This goal was achieved through collection of experimental permeability data under the CCPconditions; collection of experimental permeability data under the CES conditions; and comparison of thoseexperimental data with solutions of the poroelastic theory. For CCP tests, the permeability ratios change fromreductions (less than 1.0) to enhancements (greater than 1). These changes are bounded by an upper envelopeand a lower one. The upper envelope is corresponding to the solution of free-swelling while the lower one zero-swelling. For CES tests, the permeability ratios also change within an upper envelope and a lower one. The upperenvelope is equal to 1.0 corresponding to the solution of free-swelling while the lower one zero-swelling.Through these comparisons, we found that permeability data for both types of tests are confined within theporoelastic solutions for two extreme boundary conditions: free-swelling and zero-swelling. These findingssuggest that permeability ratios for both constant confining tests and constant effective stress tests are primarilydetermined by the matrix-fracture interactions, including sorption-induced swelling/shrinking, through tran-sient effective stresses in matrixes and fractures.

1. Introduction

Coal permeability experiments can be divided into displacementcontrolled ones and stress controlled ones. For displacement-controlledexperiments, uniaxial strain experiments are normally used to study theevolution of coal permeability. Many permeability models under uni-axial strain conditions were derived,1–6 and the most widely used isproposed by Palmer and Mansoori,3 some scholars have further provedit 7,8. Although these permeability models are under uniaxial strainconditions but only a few of experiments are under these conditions.9–11

As for the stress controlled condition experiments, two types ofexperiments are normally conducted to investigate the impact of coaldeformation on the evolution of coal permeability. One is to keep thetotal stress as constant while the other is to keep the effective stress asconstant. When the total stress is constant, the effective stress decreases

as the gas pressure increases. Under this condition, the effective stress isbelieved to be the important reason for the evolution of coal perme-ability. When the effective stress is constant (this can be achievedthrough keeping the increment of total stress the same as that of gaspressure), the effective stress impact is eliminated. Under this condi-tion, the gas sorption is believed to be the important reason for theevolution of coal permeability. These two hypotheses have beenguiding the experimental research of coal permeability for decades.

The primary goal of CCP tests is to measure the influence of effectivestress and associated processes on the evolution of coal permeability.For examples, CCP tests were used to investigate the impact of effectivestress and the combined adsorption/desorption effect on the evolutionof permeability;10,12–22 to simulate the change of permeability in CO2-ECBM process;23,24 to investigate the temperature effect on perme-ability;25,26 to study the influence of fracture geometry and water-

https://doi.org/10.1016/j.ijrmms.2018.07.003Received 9 December 2017; Received in revised form 1 July 2018; Accepted 27 July 2018

⁎ Corresponding author.E-mail address: jishan.liu@uwa.edu.au (J. Liu).

International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

1365-1609/ © 2018 Elsevier Ltd. All rights reserved.

T

content on permeability;27,28 to study the influence of cleat volumecompressibility on permeability;16 to study the influence of slippageeffect29 on permeability;16–18 to study the permeability evolution ofpropped artificial fractures in coal on injection of CO2,30 and study thedynamic permeability in the process of gas injection/depletion.31–33

These examples illustrate the importance of CCP tests in a broadspectrum of applications. In all of these studies of gas injection condi-tion, coal permeability data can be classified as three categories: per-meability increases directly with the increasing of injection pres-sure;12–15,23,25,27,30 permeability decreases initially with the increasingof injection pressure, and then rebounds;14,27,30 and permeability de-creases with the increasing of injection pressure and nearly show norebound.16,17 In all of these studies of gas depletion condition, coalpermeability data can also be classified as three categories: perme-ability decreases directly with the decreasing of pore pressure;10,13,34

permeability decreases initially with the decreasing of pore pressure,and then rebounds lower than the initial permeability;10,12 permeabilitydecreases initially with the decreasing of pore pressure, and then re-bounds larger than the initial permeability.12,13,18,35

The primary goal of CES tests is to measure the influence of gasadsorption/desorption on the evolution of coal permeability and theassociated processes. For examples, the CES tests were conducted toinvestigate the impact of gas adsorption/desorption on the evolution ofpermeability.2,36–47 Some previous studies use CES tests to study theinfluence of the size of effect stress on permeability.2,38,40 The CES testswere also used to investigate the effect of slippage effect on perme-ability,36,38–40 and the sensitivity of permeability to pore pressure.48

Different from the CCP tests all of the permeability data from the CEStests decrease with the increasing of pore pressure, but the decliningrate is different.

A number of permeability models were developed to explain theexperimental observations and have been reviewed.1,3,5,8,15,49–53 In ourprevious work,54 we concluded that these models can’t explain the re-sults from stress-controlled laboratory tests (CCP tests and CES tests).Both the hypotheses and permeability models are based on the theory ofsingle poroelasticity but applied to explain the experimental data for atypical dual porosity and dual permeability system. These experimentswere conducted under the triaxial conditions, while most permeabilitymodels were developed under specific conditions such as uniaxialstrains. The experimental observations cannot be explained unless theseinconsistencies are resolved.

When a dual porosity system such as coal is assumed as the singleporoelastic medium, we hypothesize that the gas pressures between thefracture and matrix has reached equilibrium. This is why we have tomeasure the coal permeability at the equilibrium state when we use thetheory of single poroelasticity. This could take from a few days36 to a fewweeks.16 When the gas sorption was included, the time from the initialstate to the final equilibrium state might take much longer from a fewmonths to years.23,34. When coal is assumed as a dual porosity system, wehypothesize that the gas pressures between the fracture and matrix reachequilibrium gradually. We assume that the matrix pressure changes as afunction of time. We do not consider the pressure gradient in the matrixfor permeability models. These assumptions were implemented in thetheoretical analysis of permeability evolution but we still measure thepermeability at the equilibrium state.55 Because we measure the perme-ability only at the equilibrium state, the impact of interactions betweenmatrix and fracture has been excluded and the permeability data can beexplained by using the theory of single poroelasticity.

In this study, we hypothesize that if the experimental observationsare the permeability at the equilibrium state, the permeability datashould be consistent with the theory of single poroelasticity. We testthis hypothesis through collecting all of permeability data at the equi-librium state available in the literature, comparing them with the the-oretical solutions of single poroelasticity, and conducting a mechanisticanalysis of these comparisons. These results and findings are reported inthe following sections.

2. Experimental permeability under constant confining pressure

2.1. Data collection

In this section, we collected the experimental permeability dataunder the condition of constant confining pressure. In these experi-ments, the confining pressure was maintained as constant (green line)while the gas pressure (black line) increased/decreased from a lower/larger value to a larger/lower one, then the effective stress (red line)decreased/increased gradually with the increasing/decreasing of porepressure, as shown in Fig. 1 for gas injection condition. The gray rec-tangular in the pore pressure line is the time needed to reach theequilibrium state, and the blue rectangular in the pore pressure line isthe measured stage for each data point (red point) at time tdata-x. Gaspermeability was calculated using the modified Darcy's law for acompressible gas.56 Permeability was measured either by the steady-state method or the pressure transient one. For the purpose to study theimpact of coal deformation on the evolution of coal permeability, thedata by using other methods57 are not included.

Steady State Method: The specific implementation mode of steadystate method is shown in Fig. 2(a). The black vertical axis is the pres-sure value, and the blue vertical axis is the flow rate which only in-dicates the blue flow rate line. In a typical steady state experiment, thesample is placed into the triaxial core holder and both confining pres-sure and axial stresses are applied at a slow rate to establish initialconditions and are then kept constant (green line). The sample is thenvacuum desaturated to evacuate air from the system. The sample is thenflushed with the fluid to be used to an equilibrium state (light grayline), as an initial condition that it is considered that the pressure dis-tribution in sample is evenly balanced (light gray rectangular,

=p pup dn). A pressure increment (Δp) is then applied to the upstreamgas reservoir (light red line and red line) and keeps constant for eachdata measurement. The downstream pressure (gray line) is consistentwith the initial condition. The flow rate of the upstream or the down-stream is measured. The flow rate first increases slowly (light blue line)and then keep constant (blue line). The measured stage is started attime ttest, and when the flow rate is stable for enough time the pressureand flow rate data are available at time tdata (red points). And it isconsidered that the pressure distribution in sample is declining linearlyfrom the upstream to the downstream (gray gradient rectangular,

>p pup dn).Permeability of the coal sample to gas was calculated according to

the compressible form of Darcy's law,

=−

kQ P μL

A P P2( )

a a

up dn2 2

(1)

where k is the permeability (mD), Qa is the volumetric rate of flow atreference pressure Pa (cm3/s), Pa is the reference pressure (Pa), μ is thefluid viscosity (cp), L is the core sample length (cm), A is the cross-

Fig. 1. Schematic diagram of the experimental process during gas injection forCCP tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

37

section area of the core sample (cm2), Pup is the upstream pressure (Pa),and Pdn is the downstream pressure (Pa).

Pressure Transient Method: The specific implementation mode ofpressure transient method is shown in Fig. 2(b). In a typical pressuretransient experiment, the sample is placed into the triaxial core holderand both confining pressure and axial stresses are applied at a slow rateto establish initial conditions and are then kept constant (green line).The sample-reservoir system is then vacuum desaturated to evacuate airfrom the system. The sample is then flushed with the fluid to be usedand, as an initial condition, reservoirs and sample are equilibrated witha fluid at the same pressure (light gray line). It is considered that thepressure distribution in sample is evenly balanced at this state (lightgray rectangular, =p pup dn). A pressure increment (Δp) is then appliedto the upstream gas reservoir and discharged through the sample to thedownstream gas reservoir. The time taken for the discharging upstreamreservoir (red line) and the recharging downstream reservoir (blue line)to reach a new equilibrium pressure (gray line) is measured. It is con-sidered that at this state the pressure distribution in sample is evenlybalanced too (gray rectangular, =p pup dn). The pressure decay rate re-corded in the upstream reservoir and the pressure increase rate in thedownstream reservoir are used to evaluate permeability. The decaycharacteristics depend on the permeability, on the dimensions of thesample and reservoirs, and on the physical characteristics of the per-meating fluid.15,27

The transient method of Brace was widely used to conduct the gasflow experiments in the low permeability samples. The Brace methodinvolves observing the decay of a differential pressure between up-stream and downstream vessels across the sample. This pressure decayis combined with the vessel volumes in the analysis to relate the flowthrough the sample and thus determine the permeability.58 The

pressure decay curve can be modeled as:

−

−= −

p t p tp t p t

e( ) ( )

( ) ( )up dn

up dn

υt

0 0 (2)

= +υ kAμβL

V V(1/ 1/ )up dn(3)

where Pup(t)−Pdn(t) is the pressure difference between the upstreamand downstream reservoirs at time t, (Pa); and (Pup(t0)−Pdn(t0)) is theinitial pressure difference between the upstream and downstream re-servoirs at time t0, (Pa). υ is the slope of the line when plotting thepressure decay Pup(t)− Pdn(t) on semi-log paper against time. L is thecore sample length (cm), A is the cross-section area of the core sample(cm2), μ is the fluid viscosity (cp), β is the compressibility of the gas,and Vup and Vdn are the volume of the upstream reservoir and down-stream reservoir respectively, (cm3).

For the case of gas injection, we use the lowest gas pressure in anexperiment as the initial pore pressure. The permeability ratio is de-fined as the ratio of permeability at the initial pressure to that at thecurrent one. All experimental permeability ratios under constant con-fining pressures are shown in Fig. 3 where the details of the datasources are shown in Table 1. Although they spread over a wide rangeof magnitudes, they are within a lower bound and an upper one. Theseexperimental data represent a wide range of permeability measure-ments with different gases such as helium, argon, nitrogen, methaneand carbon dioxide. The injection pressure varies from 0.1 to 8.0MPawhile the confining pressure from 3.0 to 40.0 MPa.

For the case of gas depletion, we use the highest gas pressure in anexperiment as the initial gas pressure. The permeability ratio is definedas the ratio of permeability at the initial pressure to that at the currentone. As shown in Fig. 4, although all experimental permeability ratiosspread over a wide range of magnitudes, they are within a lower boundand an upper one. The details of the data sources are shown in Table 2.These experimental data represent a wide range of permeability mea-surements with different gases such as helium, methane and carbondioxide. Most data points are within the zone of permeability ratio lessthan 1.0. The pore pressure varies from 0.2 to 6.8MPa while the con-fining pressure from 3.0 to 13.8MPa.

2.2. Impact of confining pressure magnitude

According to the magnitude of confining pressure, the distributionof permeability ratios can be divided into three zones from lowerconfining pressure to higher ones for the case of gas injection. When thegas pressure (from 0 to 2MPa) is lower, nearly all high confiningpressure (> 5MPa) data points are below the k/k0 = 1 line. It indicatesthat coal permeability decreases for high confining pressures (> 5MPa)when the gas pressure is lower. When the pore pressure is larger than2MPa, the permeability data are distributed both in the upper and the

Fig. 2. Schematic diagram of the steady state experimental method and pressure transient experimental method.

Fig. 3. Statistical distribution of coal permeability ratios during gas injectionfor CCP tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

38

Table1

Expe

rimen

talmeasuremen

tde

tails

during

gasinjectionforCCPtests.

Autho

rsYea

rCoa

lrank

Origin

Samplesize

Gas

Metho

dPo

repr

essu

reCon

fining

pressu

rePe

rmea

bility

Temp.

Equilibr

ium

Time

Note

Harpa

lani

and

Zhao

1989

n/a

BlackWarrior

basin,

America

Plug

s:D:3

8.1mm,

L:76

.2mm

CH4;H

eSteady

state

0.8–

6.2MPa

11.72MPa

0.47

–11.04

μDn/

an/

a/

Harpa

lani

and

Schr

aufnag

el19

90b

n/a

Picean

ceba

sin,

Colorad

o,America;

Plug

s:D:3

8.1mm,

L:76

.2mm

CH4;H

eSteady

state

1.65

–6.16MPa

10.3;11

.7MPa

0.53

–2.04μD

;n/

an/

a/

BlackWarrior

basin,

America

1.42

–6.70MPa

0.32

–1.78μ

D

Rob

ertson

and

Chr

istian

sen

2005

High-vo

latile

bituminou

sco

al;

Subb

itum

inou

s,low-

contam

inan

tco

al

Gilson

seam

,Boo

kCliff

sco

alfield,

Uinta-Piceanc

eba

sin,

Utah,

America;

Plug

s:D:5

0.8mm

N2;C

H4;

CO2

Steady

state

0.48

–5.58MPa

6.89

5MPa

57–2

92mD;

26.7

°C24

hwhe

nch

anging

gas

type

/

And

ersonseam

,Pow

der

River

basin,

Gillette,

Wyo

ming,

America

0.01

77–0

.085

34mD

Guo

etal.

2007

Sub-bituminou

sco

alMan

nvilleGroup

,Alberta,

Can

ada

Plug

s:D:3

3.75

mm,

L:85

.5mm

CO2

Steady

state

2.2–

5.6MPa

10.5

MPa

0.03

–0.05mD

23°C

n/a

/

Pini

etal.

2009

Highvo

latile

Cbituminou

sco

alMon

teSinn

ico

alminein

theSu

lcis

Coa

lProv

ince,

Sardinia,Italy

Plug

s:D:2

5.4mm,

L:36

mm

N2;H

e;CO2

Pressure

tran

sien

t0.48

–7.75MPa

6–14

MPa

0.05

–12.07

D45

°C≥

2da

ys/

Han

etal.

2010

Anthraciteco

alYan

gqua

nco

al,Q

inshui

Basin,

China

Plug

s:D:2

8.5mm,

L:21

.2mm

Ar

Steady

state

0.2–

4.2MPa

10–4

0MPa

2.1–

1102

.0nD

45°C

n/a

Samples

withan

dwitho

utcleats

Wan

get

al.

2011

Anthraciteco

alNorthum

berlan

dBa

sin,

Mou

ntCarmel,

Penn

sylvan

ia,A

merica

Plug

s:D:2

5mm,

L:25

–50mm

He;

CH4;

CO2

Pressure

tran

sien

t1.0–

5.6MPa

6–12

MPa

0.67

nD−

1.65

mD

n/a

n/a

Fracturedco

als

withdifferen

tfracture

geom

etry

and

water-con

tent

Kum

aret

al.

2012

Subb

itum

inou

s/bituminou

sco

alUinta

basin,

Colorad

o,America

Plug

s:D:2

5mm,L

:50

mm

He;

CH4;

CO2

Pressure

tran

sien

t1.6–

5.7MPa

10MPa

0.01

–6.63mD

n/a

4hforCH4an

dCO2

Samples

with

differen

tmoistureleve

lsVisha

let

al.

2013

Bituminou

sco

alJh

aria

coalfield,

India

Plug

s:D:3

9mm

CO2

Steady

state

1.0–

5.0MPa

5–13

MPa

0.04

–31.0mD

26℃

n/a

/Gen

sterblum

etal.

2014

Sub-bituminou

sco

alWallonSu

bgroup

inSu

rat

Basin,

Que

enslan

d,Australia

Plug

s:D:38mm,

L:18

.68–

24.9

mm

He;

Ar

Steady

state

0.11

–0.57MPa

7.5–

19.2

MPa

0.59

–4.95mD

35℃

n/a

/

Niu

etal.

2014

Lign

iteco

alYua

nbao

shan

area,

Mon

golia

,China

Plug

s:D:50mm,

L:10

0mm

N2

Steady

state

0.5–

1.5MPa

8.5MPa

13.8–3

3.3mD

25,5

0℃

15min

/

Ran

athu

nga

etal.

2014

n/a

Hazelwoo

dop

encu

tmine,

Gippsland

,Australia

Plug

s:D:25mm,

L:50

mm

CO2;N

2Steady

state

5.0–

8.0MPa

10MPa

0.18

–0.35μ

D25

,40℃

n/a

/

Kum

aret

al.

2015

Bituminou

sco

al;

Anthraciteco

alBituminou

sco

alfrom

the

Uinta

Basin,

Colorad

o;Plug

s:D:2

5mm,

L:50

mm

He;

CO2

Pressure

tran

sien

t1.0–

6.8MPa

;1.0–

5.5MPa

10MPa

1.4–

38.0

mD;

20℃

n/a

Prop

pedartificial

fracturesin

coal

Anthracitefrom

Penn

sylvan

ia,A

merica

0.34

–3.3

mD

Men

get

al.

2015

n/a

Xua

ndon

gco

almine,

China

Plug

s:D:5

0mm,

L:10

0mm

He;

CH4;

CO2

Steady

state

0.3–

2.0MPa

3.5MPa

0.04

7–0.83

7μD

11.5

℃n/

a/

Qiu

etal.

2017

Bituminou

sco

alSo

uthe

astOrdos

Basin,

China

Plug

s:D:2

5mm,

L:25

–50mm

He;

CO2

Steady

state;

Pressure

tran

sien

t

1.3–

4.3MPa

4,6MPa

3.9–

13.5μD

n/a

n/a

/

Wan

get

al.

2017

bAnthraciteco

alCha

ngCun

coal

mine,

Cha

ngZh

iCity,

Shan

Xi

Prov

ince,C

hina

Plug

s:D:5

0mm,

L:10

0mm

He;

CH4;

CO2

Steady

state

0.5–

3.0MPa

4,8MPa

0.08

15–0

.342

6mD

20,4

0℃

0.06

–12.5h

/

Bottomleyet

al.

2017

n/a

Walloon

Coa

ls,S

urat

Basin

Cub

e:l=

h=w:

40mm

He

Steady

state

0.2–

1.5MPa

4MPa

1.5–

2.6mD

n/a

n/a

/

“n/a”represen

tstheda

tais

notav

ailable.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

39

lower part of the k/k0 = 1 line. When the confining pressure is between5 and 10MPa, the permeability data are distributed in a wide range.When the confining pressure is larger than 10MPa, most permeabilitydate are distributed in the lower zone.

According to the magnitude of confining pressure, the distributionof permeability ratios can also be divided into different zones for thecase of gas depletion. When the confining pressure is larger than12MPa, the permeability ratio first slightly decreases with the de-creasing of pore pressure and then increases sharply with the de-creasing of pore pressure. When the confining pressure ranges from 9 to12MPa, the data distribute in the middle part of the graph. The per-meability ratio firstly decreases with the decreasing of pore pressureand then increases slowly with the decreasing of pore pressure. Whenthe confining pressure is lower than 9MPa, the data spread nearly theentire graph. The permeability ratio firstly decreases with the de-creasing of pore pressure, then keep slowly decreases or increasesslowly with the decreasing of pore pressure.

2.3. Impact of gas characteristics

According to the characteristics of gas, permeability data distribu-tion can be divided to two distinct zones. When the injected gas is non-adsorbing (Ar & He), most permeability ratio data are distributed in theupper part of the graph. When the injected gas is adsorbing (N2, CH4,CO2), most permeability ratio data are distributed in the lower part ofthe graph.

For the case of gas depletion, permeability data distribution can alsobe divided into three zones according to gas characteristics. For per-meability data of CH4, they are distributed in the upper part of thegraph. The permeability ratio firstly decreases with the decreasing ofpore pressure, then increases slowly with the decreasing of pore pres-sure. For permeability data of He, permeability data distribute in thelower part of the graph. The permeability ratio decreases with the de-creasing of pore pressure and nearly shows no rebound. For perme-ability data of CO2, only one group of experiment data was tested withCO2. The permeability ratio first decreases quickly with the decreasingof pore pressure and then shows an obvious rebound.

3. Experimental permeability under constant effective stress

3.1. Data collection

In this section, we collected the experimental permeability dataunder the condition of constant effective stress. In these experiments,the difference between the confining pressure (green line) and the porepressure (black line) was maintained as constant (red line), as shown inFig. 5. This was achieved through a same increment/decrement wasapplied to both the confining pressure and the pore pressure. The grayrectangular in the pore pressure line is the equilibrium stage, and theblue rectangular in the pore pressure line is the measured stage for eachdata point (red point) at time tdata−x. Permeability was measured eitherby the steady-state method or the pressure transient one.

For one particular experiment, a series of pressure increments/de-crements was conducted. The gas pressure increases/decreases from thelowest/highest magnitude to a highest/lowest one. In our review ofthese experimental data, we use the permeability ratio of the perme-ability at the lowest gas pressure to that at a new pressure. The relationsbetween experimental permeability data and gas pressures under theconstant effective stress are shown in Fig. 6 where the details of thedata sources are shown in Table 3. When the CO2 is in supercriticalphase the permeability will decrease more with injection pressure.59–61

As shown in Fig. 6, nearly all experimental permeability ratiosunder constant effective stress are greater than zero but less than unity.These experimental data represent a wide range of permeability mea-surements with different gases helium, argon, nitrogen, and carbon

dioxide. The injection pressure varies from 0.1 to 13.4MPa while theeffective stress from 1.0 to 16.0 MPa.

3.2. Impact of effective stress

Permeability ratios spread over all spaces between the no changeline (k/k0 =1) and zero line under the condition of constant effectivestress for different magnitude of effective stress. This indicates thatpermeability decreases irrespective of the effective stress magnitudes.

3.3. Impact of gas characteristics

As shown in Fig. 7, most permeability ratios decrease faster for thestrongly adsorbing gas such as CO2 and CH4 (red broken circle) than theweakly adsorbing gas N2 and non-adsorbing gas Ar & He (green brokencircle). This indicates that the permeability change under constant ef-fective stress is strongly related to the absorptivity of the injected gas.The stronger the adsorption capacity of the injected gas, the faster thepermeability decreases.

4. Mechanistic analysis

According to the poroelastic solutions the permeability is a functionof effective strain only. In this section a conceptual model of fracturepermeability under the influence of matrix deformation is introduced,and applied to analyze the mechanisms of permeability ratio distribu-tions under different conditions.

4.1. Solutions of single poroelasticity

Coal is a typical dual porosity/permeability system containingporous matrix surrounded by fractures. In this study the cleat system,fractures, joints, and faults are uniformly called the fracture system. It iscommonly assumed that Darcy flow is a result of flow in the fracturesystem and that the contribution of flow in the coal matrix to Darcyflow can be neglected.62 Thus the permeability of a coalbed is a func-tion of its fracture system.63–66 The permeability of fracture system ismuch larger than the matrix system. In order to analyze the perme-ability, we treat the fracture system as pore system and the matrixsystem as the solid parts. According to our previous work,67,68 coalpermeability can be defined as

= ⎛

⎝⎜ + ⎞

⎠⎟

kk

αϕ

Δε1f

e0 0

3

(4)

Fig. 4. Statistical distribution of coal permeability ratios during gas depletionfor CCP tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

40

Table2

Expe

rimen

talmeasuremen

tde

tails

during

gasde

pletionforCCPtests.

Autho

rsYea

rCoa

lrank

Origin

Samplesize

Gas

Metho

dPo

repr

essu

reCon

fining

pressu

rePe

rmea

bility

Temp.

Equilibr

ium

Time

Note

Harpa

lani

and

Zhao

1989

n/a

BlackWarrior

basin,

America

Plug

s:D:

38.1

mm,

L:76

.2mm

CH4;H

eSteady

state

0.36

–6.6

MPa

11.7

MPa

0.21

–14.69

μDn/

an/

a/

Harpa

lani

and

Schr

aufnag

el19

89n/

aPicean

ceba

sin,

Colorad

o,America

Plug

s:D:

38.1

mm,

L:76

.2mm

CH4

Steady

state

0.25

–6.29MPa

7.8–

13.8

MPa

0.2–

10.0μD

n/a

24hwhe

nch

anging

hydrostaticstress

/

Harpa

lani

and

Schr

aufnag

el19

90a

n/a

Picean

ceba

sin,

Colorad

o,America

Plug

s:D:

38.1

mm,

L:76

.2mm

CH4

Steady

state

0.38

–6.83MPa

10.3

MPa

1.0–

6.3μ

Dn/

a8–

10h

/

Harpa

lani

and

Schr

aufnag

el19

90b

n/a

Picean

ceba

sin,

Colorad

o,America;

BlackWarrior

basin,

America

Plug

s:D:

38.1

mm,

L:76

.2mm

CH4;H

eSteady

state

0.35

–6.81MPa

;1.84

–6.70MPa

10.3;1

1.7MPa

2.82

–6.29μD

;n/

a8–

10h

/0.32

–1.78μ

D

Wan

get

al.

2015

Anthraciteco

alJedd

oco

almine,

Hazleton,

Luzerne

Cou

nty,

Penn

sylvan

ia,

America

Plug

s:D:

25.4

mm,

L:50

.8mm

He;

CO2

Pressure

tran

sien

t0.2–

6.0MPa

6.9MPa

0.02

–10.46

μD23

°Cn/

a/

Dan

eshet

al.

2017

Ahigh

-volatile

bituminou

sco

alBo

wen

Basin,

Australia

Plug

s:D:

61mm,

L:95

mm

CH4

Steady

state

0.5–

2.5MPa

3MPa

4.14

–5.60mD

35°C

9–17

days

Perm

eabilitywas

measuredat

stag

eswhe

restrain

rate

approx

imatelyzeroed

dueto

equilib

rium

ofde

sorption

proc

ess

andeff

ective

stress

Bottomleyet

al.

2017

n/a

Walloon

Coa

ls,S

urat

Basin

Cub

e:l=

h=w:

40mm

CH4

Steady

state

0.3–

1.5MPa

4MPa

1.5–

2.3mD

n/a

n/a

/

“n/a”represen

tstheda

tais

notav

ailable.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

41

= − −Δε Δε Δε Δεe v s d (5)

or

= −−

ΔεΔσ Δp

Ke (6)

where k is the permeability of coal sample, k0 is the initial permeabilityof coal sample, α is the Biot coefficient, ϕf 0 is the initial fracture systemporosity, Δεe is the total effective volumetric strain, Δεv is total volu-metric strain increment, K is the bulk modulus of coal, σ is the meanconfining pressure, p is the injected pore pressure, Δεs is the gas sorp-tion-induced volumetric strain, Δεd is the gas diffusion-induced volu-metric strain, that caused by gas diffusion from fracture to matrix in-duced matrix swelling.

Substituting Eq. (6) into Eq. (4), we obtain

= ⎛

⎝⎜ + ⎛

⎝−

− ⎞⎠

⎞

⎠⎟

kk

αϕ

Δσ ΔpK

1f0 0

3

(7)

This model is derived based on the fundamental principles of por-oelasticity with the following assumptions [67]: coal is a homogeneous,isotropic and elastic continuum; strains are much smaller than thelength scale; gas contained within the pores is ideal, and its viscosity isconstant under isothermal conditions; the rate of gas flow through thecoal is defined by Darcy's law; conditions are isothermal, coal is satu-rated by gas.

When the mean confining pressure remains unchanged, =Δσ 0, weobtain

= ⎛

⎝⎜ + ⎛

⎝⎞⎠

⎞

⎠⎟

kk

αϕ

ΔpK

1f0 0

3

(8)

When the effective stress remains unchanged, − =Δσ Δp 0, weobtain

= ⎛

⎝⎜ + ⎛

⎝⎞⎠

⎞

⎠⎟ =k

kα

ϕ K1 0 1

f0 0

3

(9)

According to Eqs. (8) and (9), if we assume case A= =ϕ K MPa0.001 2700f 0 (high coal rank) or case B= =ϕ K MPa0.01 270f 0 (low coal rank) then we can get the theore-

tical solution of the two different types of the stress-controlled experi-ments, constant confining pressure (CCP) tests and constant effectivestress (CES) tests, as shown in Fig. 8.

4.2. Comparison with experimental data

According to the solutions of single poroelasticity as illustrated inFig. 8, coal permeability increases monotonically during the injectionfor constant confining pressure tests, remains unchanged with the gaspressure for constant effective stress tests, and decreases monotonicallyduring the depletion. These solutions are derived on the equilibriumcondition when pressures in both matrix and fracture are equalized. Allpermeability measurements were also conducted under the sameequilibrium assumption. Our hypothesis is that experimental datashould match with the analytical solutions if this equilibrium assump-tion was valid for both the analytical solutions and the experimentalmeasurements. In the following, we check this hypothesis by comparingthese solutions with experimental data as presented above.

4.2.1. CCP tests for gas injectionAs shown in Fig. 3, coal permeability ratios change within a wide

range from significant reduction (the ratio is less than 1) to enhance-ment (the ratio is larger than 1). When the equilibrium condition is met,the whole coal sample swells and so does each component. This re-presents the maximum enhancement of permeability for each gaspressure. Therefore, the analytical solution is the upper envelop of thepermeability distribution. The fact that all permeability data is belowthis line suggests that permeability measurements were conductedunder the non-equilibrium condition.

4.2.2. CCP tests for gas depletionAs shown in Fig. 4, coal permeability ratios change also within a

wide range from significant reduction (the ratio is less than 1) to en-hancement (the ratio is larger than 1). When the equilibrium conditionis met, the whole coal sample shrinks and so does each component. Thisrepresents the maximum reduction of permeability for each gas pres-sure. Therefore, the analytical solution is the lower envelop of thepermeability distribution. The fact that all permeability data is abovethis line suggests that permeability measurements were conducted alsounder the non-equilibrium condition.

4.2.3. CES testsAs shown in Figs. 6–7, coal permeability ratios change within a wide

range from significant reduction (the ratio is less than 1) to no-change(the ratio is equal to 1). When the equilibrium condition is met, thewhole coal sample remains unchanged. This represents the maximumchange of permeability for each gas pressure. Therefore, the analyticalsolution is the upper envelop of the permeability distribution. The factthat nearly all permeability data is below this line suggests that per-meability measurements were conducted under the non-equilibriumcondition.

4.3. A conceptual model of mechanistic analysis

The analysis above has proved that permeability measurementswere conducted under the non-equilibrium condition. This suggeststhat the interactions between matrixes and fractures must be taken into

Fig. 5. Schematic diagram of the experimental process for CES tests.

Fig. 6. Statistical graph of the coal permeability ratio on the impact of porepressure for CES tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

42

Table3

Expe

rimen

talmeasuremen

tde

tails

forCES

tests.

Autho

rsYea

rCoa

lrank

Origin

Samplesize

Gas

Metho

dPo

repr

essu

reEff

ective

stress

Perm

eability

Temp.

Equilibr

ium

Time

Note

Harpa

lani

and

Che

n19

97n/

aSa

nJu

anBa

sin,

America

Plug

s:D:89mm

He

Pressure

tran

sien

t0.6–

6.2MPa

5.4MPa

0.02

–0.4

mD

44.4

°C≥

3da

ys/

Al-ha

waree

1999

Bituminou

sco

alCoa

lValleyan

dCardina

lRiver

mines,Hinton,

Can

ada

Plug

s:D:25.4mm,

L:38

.1–6

3.5mm

CO2

Steady

state

3.7–

7.2MPa

6.0,

10.0,

16.0

MPa

1.1–

143.2mD

52°C

n/a

/

Linet

al.

2008

n/a

Wyo

dak-And

ersonco

alzo

ne,P

owde

rRiver

basin,

America

Plug

s:D:28mm,

L:70

mm

N2;

CH4;

CO2

Steady

state

0.5–

7.0MPa

2.76

MPa

0.19

–19.78

mD

22°C

>2h

Coa

lpa

ck

Panet

al.

2010

Bituminou

sco

alBu

lliseam

,sou

thernSy

dney

basin,

Australian

Plug

s:D:45mm,L

:105

mm

He;

CH4;

CO2

Pressure

tran

sien

t0.9–

13.4

MPa

2.0–

6.0MPa

0.05

–0.97mD

45°C

n/a

/

Che

net

al.

2011

Bituminou

sco

alTh

eBu

lliseam

insouthe

rnSy

dney

basin,

Australian

Plug

s:D:45–

45.5

mm,

L:10

1–10

5.5mm

He;

CH4;

CO2

Pressure

tran

sien

t0.9–

13.3

MPa

2.0–

6.0MPa

0.01

–0.97mD

35,4

5°C

Afew

days

toa

few

weeks

/

Liet

al.

2013

Anthraciteco

alTa

ng’a

n,Yon

g’an

and

Gushu

yuan

coal

mines,

southe

rnQinshui

basin,

China

Plug

s:D:25.3mm;L

:25

.41–

42.18mm

CO2

Steady

state

0.2–

2.1MPa

2.2–

4.0MPa

0.07

–13.59

μD26

°Cn/

a/

Xuet

al.

2013

Anthraciteco

alZh

aozh

uang

coal

minein

Jinc

heng

,So

uthe

rnQinshui

basin,

China

Plug

s:D:50mm,

L:10

0mm

CH4;

CO2

Steady

state

0.3–

2.0MPa

1.0–

5.0MPa

0.06

–0.99mD

n/a

n/a

/

Linan

dKov

scek

2014

n/a

Wyo

dakAnd

ersonco

alzo

ne,P

owde

rRiver

Basin,

Mon

tana

Plug

s:D:25.4mm,

L:25

–75mm

N2;H

e;CO2

Steady

state;

Pressure

tran

sien

t

0.1–

6.2MPa

3.0MPa

0.6–

18.0

mD

22°C

n/a

Gas

pressure

decrease

−2

grou

pof

data,7

intotal

Seom

oon

etal.

2015

Bituminou

sco

alEa

stKalim

antan,

Indo

nesia

Plug

s:D:38.1mm,

L:10

8.3mm

CH4;

CO2

Steady

state

0.6–

3.0MPa

2.07

MPa

1.6–

5.3mD

15°C

n/a

/

Liet

al.

2015

Anthraciteco

alYon

g’an

mine,

southe

astern

Qinshui

basin,

China

Plug

s:D:25.4mm,

L:35

.4mm

CO2

Steady

state

0.2–

2.1MPa

2.2–

4.0MPa

4.34

–20.13

μD26

°Cn/

aGas

pressure

decrease

Ang

gara

etal.

2016

Low

rank

rang

ing

from

lignite

tosub-

bituminou

sco

al

Kushiro

coal

mine,

Hok

kaido,

Japa

nPlug

s:D:50mm,

L:10

0mm

He;

CH4;

CO2

Steady

state

0.5–

2.5MPa

2.0–

4.0MPa

0.01

5–0.22

9mD

n/a

48h

/

Men

gan

dLi

2017

Anthraciteco

alPe

rmianSh

anxi

Form

ation,

southe

rnQinshui

Basin

Plug

s:D:25mm,

L:46

mm

N2;

CH4;

CO2

Pressure

tran

sien

t1.0–

7.0MPa

3.5MPa

1.0–

23.1

mD

20°C

12–2

4h

/

Feng

etal.

2017

n/a

SanJu

anco

alD:51mm,L

:89mm

CH4

Pressure

tran

sien

t0.34

–8.5

MPa

5.5MPa

n/a

n/a

5da

ysGas

pressure

decrease

“n/a”represen

tstheda

tais

notav

ailable.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

43

consideration to fully understand the distribution of permeability ra-tios. We use the injection of an adsorbing gas (such as CO2) as an ex-ample to illustrate how the mass transfer and the stress transfer be-tween the matrix and the fracture are coupled under the constantconfining pressure conditions, as shown in Fig. 9. The sample is placedinto the triaxial core holder and both confining pressure (σc) and axialstresses (σh) are applied to establish initial condition and kept constant( =Δσ 0). The injected pore pressure is kept constant ( =Δp 0). We takethe A-A` cross-section to analyze the deformation of the sample. We usethe ellipse to represent the fracture and the bubble to represent thesmallest component of the matrix system around the fracture. Ac-cording to our previous work,68,69 evolution of coal permeability can bedivided as three distinct stages. Prior to injection, the coal is under anequilibrium state (pressure, stress and mass contents) and no interac-tions between the matrix and the fracture occur. Post-injection, a seriesof processes initiate. First, gas instantly invades the fracture due to itsrelatively high permeability. We use this condition as a starting point(t= ts), to explain how does the permeability changes over time. Asresult of this process, a pressure difference between the matrix and thefracture is created – resulting in the diffusion of gas from the fractureinto the matrix, as shown in Fig. 9(a). As the gas molecules attach to thefracture surface and diffuse into the matrix, local strain evolves in thematrix due to both the gas adsorption and the increased gas pressure.Under this condition, the matrix swells (dark gray bubble) while thefracture narrows as shown in Fig. 9(b). Because this also occurs locallyin the vicinity of the fracture, the decrease in volume of the fracturemust be equal to the swelling volume of the matrix. As a result of thewidening of the swelling zone, the fracture permeability recovers at

time tm, which is the turning time of the permeability ratio. As the gasdiffuses further into the whole matrix of the sample, the gas pressurepropagates throughout the matrix until a new equilibrium state be-tween the fracture and the matrix is reached at time tf, as shown inFig. 9(c). In this condition, the entire matrix swells, so does the fractureas shown in Fig. 9(c).

As illustrated in Fig. 9, coal permeability is a function of time for aconstant gas injection pressure. The magnitude of permeability for theinjection pressure varies over a wide range of magnitudes from reduc-tion (the ratio is less than 1) to enhancement (the ratio is larger than 1).Each point corresponds a state (initial state, transient state, or finalequilibrium state). In this study, we define the permeability at the finalequilibrium state as equilibrium permeability and that at the transientstate as non-equilibrium permeability. With these definitions, we canestablish the relation between the analytical solutions and experimentalmeasurements. All these solutions are for the equilibrium permeabilitywhile experimental measurements are a mixture of equilibrium per-meability and non-equilibrium permeability.

When the matrix permeability is very high (micro-crack developedor matrix permeability high), gas can diffuse from fractures into ma-trixes. Under this condition, the time from the initial equilibrium to thefinal one is short, and can be neglected. This represents the upperbound of the permeability change. When the matrix permeability isextremely low, the time from the initial equilibrium to the final one islong and cannot be neglected. If this time is extremely long, gas dif-fusion-induced swelling may take place only in the vicinity of fracturewalls. Under this condition, coal permeability is controlled primarily bythe local deformation. 100% of coal swelling/shrinkage would con-tribute to the reduction of coal permeability provided that the fracturesare much more compliant than the coal matrix.54,70,71 The analyticalsolution of this situation represents the lower bound of the permeabilitychange. In this case, the total volumetric strain is defined as

=Δε 0v (10)

From Eq. (5)

= − −Δε Δε Δεe s d (11)

Substituting Eq. (11) into Eq. (4) gives

= ⎡

⎣⎢ − + ⎤

⎦⎥

kk

αϕ

Δε Δε1 ( )f

s d0 0

3

(12)

The permeability ratio is controlled by the gas sorption-inducedvolumetric strain (Δεs) and the gas diffusion-induced volumetric strain(Δεd). The relations are

=+

ε εp

P ps LL (13)

= −ε f p p( )d 0 (14)

where the Langmuir volumetric strain, εL, is a constant representingthe volumetric strain at infinite pore pressure and the Langmuir pres-sure constant, PL, representing the pore pressure at which the measuredvolumetric strain is equal to 0.5εL. The free swelling model (Eq. (7)) andconstant volume model (Eq. (12)) are the upper bound and the lowerbound of the permeability change. For a particular experimental mea-surement, coal permeability would be in-between, as shown by grayareas in Figs. 10–12 for the cases of gas injection in CCP tests, gas de-pletion in CCP tests, and all CES tests. In this study, the area of per-meability change bounded by the analytical solution of free swellingand by that of constant volume is defined as a permeability map.

Comparing Fig. 7 with Fig. 12, we can find that the distribution ofpermeability ratios for the cases of absorbing gas is more closely to theconstant volume behavior, while the distribution of permeability ratiosfor the cases of non-absorbing gas is more closely to the free swellingbehavior.

Permeability is a function of effective stress (effective strain). In CES

Fig. 7. Gas composition analysis of CES tests.

Fig. 8. Example analytical solutions of CCP and CES tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

44

tests, the effective stress in the fracture was maintained as constantwhile the effective stress in the matrix evolves as the interaction be-tween matrix and fracture progresses. This interaction determines thepermeability map. In CCP tests, we can find that the distribution of

permeability ratios for high confining pressures is more closely to theconstant volume behavior, while the distribution of that for low con-fining pressures is more closely to the free swelling behavior.

5. Conclusions

Through comparing the experimental data of coal permeabilityevolutions under both constant confining pressures and constant ef-fective stresses, the following conclusions can be drawn:

• Experimental permeability data were obtained under the commonassumption of the equilibrium between coal matrix and fracturepressures, but not consistent with the analytical solutions under thesame assumption. For a constant fracture pressure, coal perme-ability still changes due to the gas diffusion from fractures intomatrixes. The permeability stabilizes when the fracture pressure isequalized with the matrix pressure. This process may take a verylong time because of low matrix permeability, and the equilibriumcondition may never be met in all of these laboratory tests.

• Permeability data for both constant confining tests and constanteffective stress tests are confined within the poroelastic solutions for

Fig. 9. Illustration of relations between gas diffusion in the matrix and fracture opening under the stress-controlled conditions.

Fig. 10. Map of permeability change during gas injection for CCP tests.

Fig. 11. Map of permeability change during gas depletion for CCP tests.

Fig. 12. Map of permeability change for CES tests.

R. Shi et al. International Journal of Rock Mechanics and Mining Sciences 110 (2018) 36–47

45

two extreme boundary conditions: free-swelling and zero-swelling.Evolutions of coal permeability between these poroelastic solutionsare primarily determined by the matrix-fracture interactions, in-cluding sorption-induced swelling/shrinking, through transient ef-fective stresses in matrixes and fractures.

Acknowledgements

This work is a partial result of funding by the National Key R&DProgram of China (Grant No. 2017YFC0804203), Open Research Fundof State Key Laboratory of Geomechanics and GeotechnicalEngineering, Institute of Rock and Soil Mechanics, Chinese Academy ofSciences (Z016010) and the Natural Science Foundation of China(51504235; 51474204). These sources of support are gratefully ac-knowledged.

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