Adsorption-induced coal swelling and stress: Implications for methane production and acid gas sequestration into coal seams

Adsorption-induced coal swelling and stress: Implications for methane

production and acid gas sequestration into coal seams

Xiaojun Cui,1 R. Marc Bustin,1 and Laxmi Chikatamarla1

Received 14 October 2004; revised 18 June 2007; accepted 16 July 2007; published 9 October 2007.

[1] Sequestration of CO2 and H2S into deep unminable coal seams is an attractive optionto reduce their emission into atmosphere and at the same time displace preadsorbed CH4

which is a clean energy resource. High coal seam permeability is required for efficientand practical sequestration of CO2 and H2S and recovery of CH4. However, adsorption ofCO2 and H2S into coals induces strong swelling of the coal matrix (volumetric strain) andthus reduces significantly coal permeability by narrowing and even closing fractureapertures. Our experimental data on three western Canadian coals show that theadsorption-induced volumetric strain is approximately linearly proportional to the volumeof adsorbed gas, and for the same gas, different coals have very similar volumetric straincoefficient. Impacts of adsorption-induced swelling on stress and permeability aroundwellbores were analytically investigated using our developed stress and permeabilitymodels. Our model results indicate that adsorption-induced volumetric strain hassignificant controls on stress and permeability of producing and sequestrating coal seamsand consequently the potential of acid gas sequestration. Coal seams may undergo >10times enhancement of permeability around CH4-producing wellbores due to a reduction ineffective stress as a result of coal shrinking caused by methane desorption accompanying areduction in reservoir pressure. Injection of H2S and CO2 on the other hand results instrong sorption-induced swelling and a marked increase in effective stress which in turnleads to a reduction of coal seam permeability of up to several orders of magnitude.Injection of mixtures of N2 and CO2 such as found in flue gas results in weaker swelling,the amount of which varies with gas composition, and provides the greatest opportunity ofsequestering CO2 and secondary recovery of CH4 for most coals. Because of the markedswelling of coal in the presence of H2S, even minor amounts of H2S result in amarked reduction in permeability, and hence sequestration of H2S in deep coals will belikely impractical. Furthermore, high stresses resulting from sorption of acid gases willpotentially cause the coal to yield, fracture or slip, and produce fine particles, whichfurther affect permeability and thus methane production and acid gas sequestration.

Citation: Cui, X., R. M. Bustin, and L. Chikatamarla (2007), Adsorption-induced coal swelling and stress: Implications for methane

production and acid gas sequestration into coal seams, J. Geophys. Res., 112, B10202, doi:10.1029/2004JB003482.

1. Introduction

[2] The emission of greenhouse gas (e.g., CO2) or acidgas (CO2 and H2S) into the atmosphere is recognized tohave significant environmental impacts. Geological seques-tration of greenhouse gases and other pollutants is nowconsidered a viable method of reducing their emissions.Injection of CO2 into geological formations is already beingpracticed by the petroleum industry for enhanced oil recoveryand H2S is currently disposed of by its injection into depletedgas reservoirs. Because of the high internal surface area ofcoal, comparatively large volumes of CO2 and H2S can bestored in an adsorbed state as adsorbed gases have near

liquid like densities. Hence the volume of gas stored per unitvolume of coal greatly exceeds that held in pore space asfree gas (non adsorbed state) particularly at low pressurescorresponding to relatively shallow reservoirs (<1500 m).Since the gases are stored by adsorption conventionalreservoir traps are not required to retain the sequesteredgas and the opportunity for future leakage is thus greatlyreduced. Another benefit of the sequestration of CO2 andH2S in coal seams is that these gases have a higheradsorption affinity than methane. Their sequestration intocoals displaces CH4 methane from the coal and thus resultsin enhanced production of coal bed methane. Thereforesequestration of CO2 and H2S into deep, unminable coalseams is an attractive option with economic incentives.[3] Coal is a readily combustible rock composed of more

than 50% by weight and 70% by volume organic matterderived mainly from high plants and minor mineral matter,water and gases. During the progressive diagenesis of coal,

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B10202, doi:10.1029/2004JB003482, 2007

1Department of Earth and Ocean Sciences, University of BritishColumbia, Vancouver, British Columbia, Canada.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2004JB003482$09.00

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varying amounts of gas (of which CH4 is predominant) isgenerated and the organic matrix progressively becomesmicroporous. The resultant microporosity in coal has avariably large surface area and hence a high capacity tostore by adsorption the gases that are generated duringdiagenesis. The degree of microporosity and hence theadsorption capacity of coal progressively increases withcoal rank, which is a measure of the level of diagenesis ofthe coal. Adsorption varies with the type of organic matterpresent, with vitrinite rich coals having the highest adsorp-tion capacity.[4] The practical sequestration of CO2 and H2S into

unmineable coal seams through injection requires that thecoal seams have interconnected pathways for injected gasesto penetrate into coal seams efficiently. Besides the primary,mainly micropores (i.e., pores with diameters <10�6 m),coal seams invariably have secondary fractures/cleatsformed during coalification or in response to tectonic events[Laubach et al., 1998]. The cleat networks are the majorconduits for gas and water flow in coal seams, whereas gasadsorbed into microporous coal particles moves by diffu-sion. The cleat permeability contributed by the cleat net-work may vary significantly due to the hydromechanicalresponses of coal seams during gas injection. For example,injection of gas into coal seams elevates the cleat porepressure significantly, widens the cleat apertures, andenhances the permeability. In contrast, adsorption of acidgas induces strong swelling of coals, consequently narrow-ing the cleat aperture and reducing the permeability. Modelshave been formulated to account for the dynamic perme-ability changes during primary coal bed methane recovery[e.g., Palmer and Mansoori, 1996; Pekot and Reeves, 2003;Sawyer et al., 1990; Shi and Durucan, 2004]. These studieswere mainly limited to modeling CH4 recovery and fewstudies have considered the impact on permeability of CO2

or H2S injection. The possible permeability variationinduced by acid gas (CO2 or H2S) or flue gas (CO2 + N2

or H2S + N2) injection and consequent influence on acid gassequestration remain poorly investigated because of a lackof experimental data.[5] In this study, we present experimental adsorption

data of CH4, CO2, H2S, and N2 and their associatedvolumetric strains measured for three western Canadiancoals (Ardley, Wolf Mountain, and Quinsam coals). Treat-ing the adsorption-associated volumetric strain analogouslyto the thermal effects of a poroelastic medium, we developedseveral stress models suitable for coal seams in basins withdifferent stratigraphic and tectonic settings to study theeffective stress and permeability distribution around well-bores during methane recovery and acid gas injection. Withthe constraints of our experimental data from the aforemen-tioned coals, the possible variations of stress and coal seampermeability during methane production and acid gas injec-tion are investigated analytically and their implications formethane production and acid gas sequestration into thosecoals are discussed.

2. Experimental Gas Adsorption and CoalSwelling

[6] Gas is mostly adsorbed onto the internal surface ofmicroporous coal particles. The adsorbed gas volume (Vgi)

can be described by the Langmuir isotherm for pure gas i orextended Langmuir isotherm for gas component i of a gasmixture [e.g., Yang, 1987], taking the form

Vgi ¼VLicip=pLi

1þ pPni¼1

ci=pLi

; ð1Þ

where pLi and VLi are the Langmuir pressure and volume forgas i, both determined from measurements on the puregases; ci is the mole fraction of gas component i in free gasmixture; p is the pressure.[7] The Langmuir pressure ( pL) and volume (VL) are

derived from a series of isothermal adsorption experiments.The underlying principle for the adsorption experiments ismass balance. Coal samples with measured mass/volumeand porosity are put into a sample cell with a known voidvolume. A known volume of the experimental gas isinjected into the sample cell. As gas in the sample cell isadsorbed, there is a corresponding decrease in pressure thatis proportional to the volume adsorbed. The gas pressure inthe sample cell is computer monitored until equilibriumpressure is reached. The volume adsorbed is calculated asthe difference between the gas injected and the gas in thevoid volume in the sample cell. Corrections are made toaccount for the change in void volume of the sample cellsince the adsorbate occupies space. Gas of known mass isrepeatedly injected at progressively higher pressures into theequilibrated sample cell and, the cumulatively adsorbed gas(Vg) at a newly equilibrated pressure ( p) can be determined.Repeating the process, a series of adsorbed gas volumes(Vg) at different pressures ( p) are obtained and then curve-fitting equation (1) yields the Langmuir pressure ( pL) andvolume (VL) for the gas. Coal samples are normally crushed(�60 meshes) to shorten the adsorption time by reducingthe diffusion length of the gas. The experimental apparatusis also maintained at a fixed temperature (isothermal). Formeasuring the sorption-induced volumetric strain fourresistance-type strain gauges are adhered onto the coresurface (i.e., core 37 mm in diameter and about 100 mmin length were used in this study). The strain gauges arelocated 180� apart, two axial and two radial, to measurethe strain variations caused by adsorbed gas (Vg) at theequilibrated pressures (p) relative to the initial state. Theadsorption-induced volumetric strain is then determined byassuming an isotropic swelling/shrinking of the coal duringadsorption or desorption. For all calculations the Peng andRobinson [1976] equation of state was utilized. The detailedexperimental setup and procedure were described elsewhere[Chikatamarla et al., 2004]. Experiments also show that theexpansive strain can be recovered upon desorbing the pre-adsorbed gas.[8] In this study, the Langmuir constants for CH4, CO2,

H2S, and N2 of the three Canadian coals were measured onboth coal cores and crushed coals at equilibrium moisture(or received) condition and at 25�C (Table 1). The volu-metric strains induced by CH4, CO2, H2S, and N2 adsorp-tion were measured on the coal cores. The experimental datasuggest that the volumetric strains induced by gas sorptionare approximately linearly proportional to the volume of theadsorbed gas at standard conditions for all the experimental

B10202 CUI ET AL.: COAL SWELLING AND ACID GAS SEQUESTRATION

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gases (Figure 1). Therefore the total volumetric strain eVcaused by adsorption of a pure gas or a gas mixture can beapproximated as

eV ¼Xi¼n

i¼1

egiVgi; ð2Þ

where subscript i represents a gas component i in a gasmixture, egi is the volumetric strain coefficient of gas i,given by the slope of solid lines that fit the experimentaldata in Figure 1. Thus the coefficient eg measures theswelling effect of different gases with respect to a unitvolume of adsorbed gas. Vgi is the volume of adsorbed gas i,and n is the total number of gas components, which rangesfrom 1 (pure gas) to 3 (multiple components) in this study.Here it was assumed that, for a gas mixture, the totalvolumetric strain caused by adsorption of each component

is simply a linear combination of the volumetric straininduced by adsorption of each individual gas as we onlyconsider small linear elastic deformation of coal seams andthe sorption-induced volumetric strain is linearly dependenton the volume of individual adsorbate.[9] The swelling or volumetric strain induced by sorption

of a specific gas is similar among the three coals studiedexcept for the adsorption of H2S into the Ardley coal(Figure 1). This is likely because that H2S reacted withthe moisture inside the coal matrix forming sulphuric acidwhich in turn reacted with the strain gauges. The highermoisture content of the Ardley sample (8.04%) relative toWolf Mountain (2.26%) and Quinsam (4.14%) might havecaused the early reaction with the strain gauges comparedto the other two samples, consequently resulting in thelower strain values for the Ardley coal. Among all gasesH2S induces the largest volumetric strain per unit volumeof adsorbed gas as indicated by an average egH2S of 2.5 �10�3 gram per cubic centimeter (g/cm3) for Wolf Moun-tain and Quinsam coals (Figure 1c). Voumetric strain due toH2S sorption is about three times that of CH4 (egCH4� 6.9 �10�4 g/cm3; Figure 1a); followed by CO2, which has anaverage egCO2 of about 9.9 � 10�4 g/cm3 (Figure 1b) or1.5 times that of CH4. Nitrogen induces the smallest volu-metric strain with an average volumetric strain coefficient of3.1 � 10�4 g/cm3 (Figure 1d) or half that of CH4.[10] Although the Langmuir constants determined with

coal cores or crushed coals are similar, only those data thatcould reproduce better the experimental volumetric strains

Figure 1. Experimental volumetric strains induced by gas sorption in coals. The symbols represent theexperimental data for different coals as given in Figure 1d, whereas the solid curves are fitted straightcurves passing through the origin and their slopes are defined as the swelling coefficients (eg).

Table 1. Experimental Langmuir Isotherms (as Received at 25�C)

Coal GasAdsorption

Ardley Wolf Mountain Quinsam

VL,cm3/g

pL,MPa

VL,cm3/g

pL,MPa

VL,cm3/g

pL,MPa

CH4 5.09 5.12 13.42 2.39 8.80 0.93CO2 25.72 3.38 27.78 1.85 38.83 2.06H2S 23.6 0.19 49.55 0.26 65.19 0.67N2 3.07 8.59 10.34 9.06 6.17 11.57


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at different pressures using equations (1) and (2) wereadopted here (Table 1 and Figure 2). The gas adsorptioncapacity (VL) increases in the order N2 < CH4 < CO2 < H2S,as does the adsorption affinity or strength (approximated tobe 1/pL) for the three coals. However, the adsorptionproperties for a specific gas are different between the coals.Because of their large adsorption capacity and high adsorp-tion strength, under similar pressures, H2S and CO2 causemuch larger volumetric strains than CH4 (Figure 2). Atintermediate to high pressures the volumetric strain ratios ofeH2S/eCH4 are about 8, 20 and 30 for Ardley, Wolf Mountainand Quinsam coals, respectively. Similarly, the ratios ofeCO2/eCH4 are about 7, 3, and 5, but the ratios of eN2/eCH4are 0.2, 0.3, and 0.3 for Ardley, Wolf Mountain, andQuinsam coals. Therefore the displacement of CH4 fromthe coal seams by H2S or CO2 injection without a significantpore pressure change will cause net coal swelling. In contrastinjecting N2 to displace CH4 from these coals will cause netcoal shrinking.

3. Coal Seam Permeability

[11] Most coal seams have a dual porosity structure(Figure 3). The macroscopic pore network consists of morecontinuous face cleats and less continuous butt cleats. Bothcleats are orthogonal and perpendicular to the coal beddingplanes. The more intact coal matrix blocks contain mostlymicropores. Coal bed gases are dominantly adsorbed ontothe internal surface of the microporous coal matrix. Themicropores in the coal matrices or coal particles negligiblycontribute to the permeability while the macroscopic cleats/fractures are the major avenues for water and gas flow. Thefractures or cleats are also likely more sensitive to stresschanges than those micropores distributed in coal matrices.Hence the macroscopic cleats/fractures may undertake mostof the deformation upon a load or stress change. Thereforethe coal seam or cleat permeability (k) can be described interms of cleat porosity (f) as [e.g., Walder and Nur, 1984]

k

k0¼ f

f0

� �3

; ð3Þ

where k0 is the initial cleat permeability. The permeabilityand porosity of a coal seam can be determined from the coalfabric, such as the cleat spacing (a) and aperture width (b),by using the match stick model [Harpalani and Chen, 1995;Sawyer et al., 1990]:

f ¼ 2b

aand k ¼ b3

12a: ð4Þ

In practice, initial permeability (k0) and cleat spacing (a) canbe determined by well testing and examining coal cores.Then the initial porosity (f0) can be determined withequation (4).[12] During coal bed methane production (desorption)

and acid gas injection (adsorption), fluid pressure willchange and the volumetric strain induced by gas desorptionor adsorption will also induce changes in the stress field.Variation in the stress field in turn causes the porosity tochange. Treating the sorption/desorption-induced volumetric

strain as analogous to thermal effects for a poroelasticmedium, then the stress dependence of coal seam permeabil-ity (k) can be approximately described as

k

k0¼ f

f0

� �3

¼ exp�3Dse

Kp

� �; ð5Þ

where Dse = (s � p) � (s0 � p0) is mean effective stresschange relative to the initial state, and Kp is pore volume

Figure 2. Predicted volumetric strains under differentequilibrium pressures using equations (1) and (2).


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modulus, the subscript 0 represents initial state values.Derivation of equation (5) is described in Appendix A.

4. Stress Around Injecting/Producing Wellbores

4.1. Stress-Strain Constitutive Equation

[13] Gas adsorption or desorption causes significant vol-umetric strain. The volumetric strain is approximatelylinearly proportional to the volume of gas adsorbate(Figure 1). Thus the effects of gas sorption on the defor-mation of isothermal coal seams are treated analogously tothe effects of temperature on nonisothermal elastic porousmedium [e.g., Palmer and Mansoori, 1996]. Similar to theconstitutive equations of nonisothermal poroelastic medium[e.g., Palciauskas and Domenico, 1982; Neuzil, 2003], thestress (sij) and strain (eij) relation for deforming coal seamscan be described as

sij ¼E

1þ neij þ

n1� 2n

ebdij� �

þ zpdij þ KeV dij; ð6Þ

where E is Young’s modulus of the porous medium, n is thePoisson’s ratio, eb is the bulk volumetric strain, p is porefluid pressure, eV is the sorption-induced volumetric straingiven by equation (2), K = E/3(1 � 2n) is the frame bulkmodulus of the porous medium measured under fullydrained conditions, dij is Kronecker’s delta, and i or j is thedirectional index (i.e., r, q and z in cylindrical coordinates).The Biot constant z is defined as [e.g., Palciauskas andDomenico, 1982]

z ¼ 1� K

Ks

; ð7Þ

where Ks is the bulk modulus of the solid matrix. Generally,Ks is �K and hence z approaches a constant of one, asassumed in this study.

4.2. Stress Distribution Around Wellbores

[14] Most coal seams are just a few meters thick and areorientated horizontally or subhorizontally. Hence the macro-porous fracture networks which are invariably perpendicularto coal bedding are vertical or near vertical (Figure 3).

Therefore horizontal deformation of the fractured coalsinduced by changes in horizontal stresses is likely pre-dominant over the deformation in other directions. Con-sequently horizontal stresses are mainly considered in thisstudy. For gas production or acid gas injection, a wellborewith a radius of rw is assumed to be drilled into acylindrical domain with a radius of rb. Gas productionor injection is a dynamic process, in which fluid pressure,gas concentration, and stress change transiently and areclosely coupled, especially during early stage of produc-tion or injection. As a first-order approximation, here weonly consider a steady state, which simplifies the problemand still allows us to obtain insightful analytical solutions.[15] The simplifications and assumptions made in our

model include (1) uniform and isotropic hydromechanicalproperties with a unit thickness for coal seams; (2) a cylin-drical reservoir with radial symmetry andwith the wellbore atits center; (3) a constant wellbore pressure pw during gasproduction or injection; (4) a steady pressure profile chang-ing logarithmically from the wellbore pressure (pw) at thewellbore skin (rw) to an initial reservoir pressure p0 at theouter domain boundary (rb); (5) a consideration of only twostates of the stress and permeability, including an initialstate before drilling and a steady state reached after acertain time of gas production or injection; (6) at the initialstate, an initially intact coal seam with uniform gas compo-sition (ci0, only methane in this study), pressure (p0), andstress (s0) that is determined with a purely vertical strainmodel (equation (8)); and (7) at the final steady state, a coalseam with uniform gas composition of the injected gas or gasmixture that resulted from the complete displacement ofthe initial or preadsorbed methane by the injected gas.With these simplifications and assumptions and the stress-strain constitutive equation (equation (6)), a series ofanalytical solutions are derived under different stress modeswith different boundary conditions (refer to details inAppendix A). For all stress modes, the initial mean horizontalstress is given as

s0 ¼n

1� nszz þ

1� 2n1� n

p0 þE

3 1� nð Þ eV0: ð8Þ

4.2.1. Purely Vertical or Uniaxial Strain Model(U Model)[16] Purely vertical strain or uniaxial strain is widely used

for reservoir and basin compaction modeling, which allowsa helpful simplification for hydromechanical coupling withan intrinsic boundary condition of zero lateral or horizontalstrains everywhere in the domain. Under such conditions,the change in mean effective horizontal stress (Dse) is

Dse ¼�n1� n

pþ E

3 1� nð Þ eV � s0 � p0ð Þ: ð9Þ

4.2.2. Plane Strain Model (P Model)[17] Coal seams are often horizontal or subhorizontal and

confined by much stronger and thicker formations at theirtop and base. Thus horizontal deformation or strain ispredominant within coal seams and vertical deformation isnegligible due to the very small vertical-horizontal ratio ofthe coal seam. Under the plane strain mode (ezz = 0), the

Figure 3. Sketch of fracture/cleat networks of coal seamswith dual-porosity structure consisting of micropores withinmatrix and macrocleats or fractures.


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change in mean effective stress (Dse) relative to the initialstress state becomes

Dse ¼�1

2 1� nð Þ pþE

6 1� nð Þ eV � s0 � p0ð Þ

þ ECP1

2 1� 2nð Þ 1þ nð Þ ; ð10Þ

where CP1 is a constant determined with specified boundaryconditions.4.2.3. General Stress Model (G Model)[18] So far we have considered uniaxial and plane strain

models, both models assume some internal constrains fordeformation. Here we consider a more general case withoutintrinsic deformation constrains except for imposed externalboundary conditions. However, we still assume a constantoverloading of the coal seam reservoir (szz) as oftenencountered during reservoir modeling. Therefore the gen-eral stress model (G model) is actually a plane stress modelwith a constant vertical stress (it is so labeled in Figures 5and 6 in section 5.1). Under such conditions, the change inmean effective stress (Dse) is described as

Dse ¼� 1þ nð Þ

2pþ E

6eV � s0 � p0ð Þ þ n

1� nszz þ

ECG1

2 1� nð Þ ;

ð11Þ

where CG1 is a constant determined with imposed boundaryconditions to the studied domain.

4.3. Model Comparison

[19] The change in mean effective stress is closely cou-pled with reservoir pressure and associated gas adsorption ordesorption as shown by the derived models (equations (9)–(11)), and all models have a similar formula as

Dse ¼ CP p� p0ð Þ þ CeE eV � eV0ð Þ þ C0; ð12Þ

where Cp and Ce are coefficients reflecting relatively howpressure and adsorption strain would influence the effectivestress, and C0 is a constant associated with the initial andboundary conditions. Change in effective stress relative tothe initial stress state is obviously attributed to changes ofboth fluid pressure and sorption- or desorption-associatedvolumetric strain. Because of the nonlinear Langmuir typeof gas adsorption or desorption with respect to pressure(equation (1)), effective stress changes nonlinearly withpressure. However, if we exclude swelling/shrinking effects,the effective stress would change linearly with fluidpressure. Furthermore, the pressure and adsorption con-tribute oppositely to the stress variation because Cp isalways negative and Ce is positive. For example, duringmethane production, reservoir pressure is lowered relativelyto its initial value, the mechanical response of the reducedpressure increases effective stress or contributes a positivechange in effective stress, whereas the desorption of gas dueto lower pressure results in a negative change in volumetricstrain and thus a negative change or reduction in effectivestress (equation (12)).[20] Regardless of the constant C0 related to initial stress,

pressure, and boundary conditions, the uniaxial model(U model) has the smallest pressure effects on the effectivestress change and the general stress model (G model) hasthe largest effects (Figure 4). On the other hand, the U modelpredicts the strongest influence of sorption-associated swellingon effective stress change and the G model (Figure 4) predictsthe weakest influence. The plane strain model (P model) hasintermediate influence of pressure and adsorption on effec-tive stress change, but similar to that of the G model.Therefore, because of its relative dominance of volumetricstrain, the Umodel likely predicts the least increase or largestdecrease in effective stress during methane production withlowered reservoir pressure. However, the U model predictsthe largest increase or least decrease in the effective stressduring acid gas injection.[21] The coefficient Cp is much more sensitive to the

Possion’s ratio n than the coefficient Ce. The value of Cp

decreases markedly with increasing n (Figure 4) for allmodels. The Young’s modulus E only appears in front of theadsorption strain term. Therefore, for gas production, a coalseam with a smaller n and a larger E would be an optimalcombination for reducing the positive change in effectivestress due to the lowered reservoir pressure and enhancingthe negative change in effective stress caused by coalshrinkage. Consequently, such optimal coal seams havepotentially a lower increase or even a decrease in effectivestress and thus higher permeability during production.However, for acid gas injection with elevated reservoirpressure, a coal seam with larger Possion’s ratio n andsmaller Young’s modulus E is much more desirable as sucha coal seam likely undergoes less net effective stressincrease and thus a more limited reduction in permeability,facilitating more efficient acid gas sequestration.

5. Results

[22] With constraints of experimental data of gas adsorp-tion and associated volumetric strains from three westernCanadian coals, we apply our models to study the stress andpermeability variations during gas production and acid gas

Figure 4. Influence coefficients of pressure and volu-metric strain for effective stress changes with differentmodels. UM, uniaxial strain model; GM, general stressmodel; and PM, plane strain model. Note that the CP isnegative and Ce is positive for all models and thus pressureand volumetric strain have opposite influences on effectivestress changes.


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injection and their implications for methane production andacid gas sequestration.Model parameters are listed in Table 2.As discussed in section 4.3, the uniaxial strain (U model)and the general stress model (G model) represent two end-members for the effects of fluid pressure and adsorption-associated strain on stress and permeability under the sameconditions, which is also underscored by a series of analyt-ical results. Therefore here we mainly present resultsdetermined with the U model and G model.[23] Methane production is simulated by specifying a low

wellbore pressure pw and assuming that the adsorbedmethane in the microporous matrix is in equilibrium withthe fluid pressure in cleat/fracture networks. Acid gassequestration is simulated by specifying a high wellborepressure pw and assuming that the composition of gasmixture is constant throughout the coal seam and has thesame composition as the injected gas.

5.1. Effects of Sorption-Induced Volumetric Strain

[24] To highlight the significant controls of desorptionand adsorption on stress and permeability, the experimentaldata from Wolf Mountain coal which has intermediateadsorption properties of the studied coals was investigated.During coal bed methane production, if the volumetricstrain is not considered, the U model predicts >2 MPaincrease in effective stress near the wellbore, and theG model predicts >1 MPa increase in effective stress whenthe pressure is reduced by 3MPa near thewellbore (Figure 5a).The cleat permeability is decreased to less than one half ofits initial value for the U model and about one half for theG model (Figure 5b). However, if the desorption strain isincluded, the effective stress will be reduced throughout thewhole domain as predicted by the U model. In contrast, theG model predicts a slight increase in effective stress faraway from the wellbore and slight decrease in effectivestress near the wellbore. Therefore the resultant permeabilitymay be significantly increased (U model) or nearlyunchanged (G model). The G model, with either zerodisplacement or constant stress boundary conditions at theouter boundary of the studied domain, does not predictmarked changes in the stress field on the coal seampermeability (Figure 5). The plane strain model (P model)yields intermediate values between the G and U models asexpected from the discussion in section 4.3. Therefore the

results suggest that the volumetric strain induced by meth-ane desorption during coal bed gas production significantlycounteracts the reduction in permeability caused by pressuredecrease alone, or even outweighs the pressure effects andresults in an increase in permeability.[25] To sequestrate gas (e.g., CO2), it must be pumped

into coal seams at pressures higher than the initial reservoirpressure p0 (assuming no previous depletion of methanefrom the coal seam has occurred). Therefore, if there is nosorption-induced swelling, the effective stress will bereduced due to the elevated fluid pressure during gasinjection as suggested by the models (equations (9)–(12)).This is also demonstrated by the analytical results based onthe Wolf Mountain coal with different stress models(Figure 6a) where a several fold increase in permeabilitynear the wellbore (Figure 6b) is predicted. However, due tothe strong coal swelling induced by CO2 adsorption, the

Figure 5. Impacts of volumetric strain on (a) horizontaleffective stress and (b) cleat permeability during methaneproduction from the Wolf Mountain coal. The curves withthe solid symbols represent results predicted by modelsincluding volumetric strain effects, but the curves with opensymbols are predicted by the same stress model excludingvolumetric strain effects. CSBC, constant stress specified atthe outer domain boundary. Others have the same modelparameters unless labeled.

Table 2. Model Parameters

Variables Value

Constant vertical stress szz, MPa 8.1Initial pressure p0, MPa 3Initial permeability k0, 10

�15 m2 5Cleat spacing a, cm 0.5Initial cleat porosity f0, % 0.27Young’s modulus E, GPa 3Poisson’s ratio n 0.3Producing wellbore pressure pw, MPa 0.1Injecting wellbore pressure pw, MPa 6.0Wellbore radius rw, m 0.1Studied cylindrical domain radius rb, m 2000Initial coal seam gas composition cCH4 1.0Pure acid gas injection cCO2 or cH2S 1.0Flue gas injection, cN2 or cCO2 or cH2S 0.5


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effective stress in the whole domain is markedly elevated by>15 MPa, even at the outer boundary with a constantreservoir pressure assuming a zero displacement boundarycondition (Figure 6a). Consequently the permeability issignificantly reduced by >3 orders of magnitude(Figure 6b). As expected, the U model predicts the largestincrease in effective stress and reduction in permeability incontrast to the results predicted by P model with a zerodisplacement boundary condition. Also the permeabilitynear the wellbore is most significantly reduced due to higherpressure and greater CO2 adsorption compared to zonesfarther from the wellbore.[26] It is interesting to note that the G model and the

P model with a constant mean stress imposed at the outerboundary of the domain, predict a relatively small increase(P model) or even a decrease (G model) in effective stressand thus consequently a relative small variation (several

fold) in permeability (Figure 6). These results reflect theimposed boundary conditions. With a constant stress at theouter boundary, a coal seam can swell relatively easily andnot result in a high stress build up. In contrast to modelswith zero displacement boundary conditions, the coal seamis restricted to deform to accommodate the sorption-inducedswelling, consequently resulting in a large increase in stress.Nevertheless, the sorption-induced swelling has a remark-able negative effect on the permeability during acid gasinjection due to its relative stronger swelling effects andlarger adsorption capacity than the other gases in the coalseams.

5.2. Applications for Western Canada Coals

[27] Our developed models are applied here to the threerepresentative western Canadian coals, the Ardley, WolfMountain and Quinsam coals. Analytical results based onthe experimental data from these coals with a wide range ofadsorption properties provide a complete view on howstress and permeability will change during methane produc-tion and acid gas sequestration. As shown in section 5.1.(Figures 5 and 6), the uniaxial model (U model) predicts themost significant changes in stress and permeability amongthe three models for all coals. Therefore results from theU model are presented here to highlight the possiblemaximum changes in the effective stress and permeabilityfor methane production and acid gas sequestration.[28] For comparison purposes, hydromechanical proper-

ties, initial reservoir condition, and wellbore pressure areassumed to be the same for the different cases unless notedotherwise. Furthermore, the adsorption/desorption associatedswelling coefficient for a specific gas is taken to be the samefor different coals inasmuch as it has very similar values forall the coals (Figure 1). Additional parameters user are listedin Tables 1 and 2.5.2.1. Stress and Permeability Change During MethaneProduction[29] The Ardley coal has the smallest Langmuir volume

and highest Langmuir pressure for methane among the threecoals. Under the same initial reservoir conditions, theArdley coal has the least preadsorbed in situ methane torelease. Therefore a small, nearly linear decrease in effectivestress and a slight enhancement in the permeability arepredicted for the Ardley coal (Figure 7). The Wolf Mountaincoal desorbs more methane with pressure reduction andconsequently a much larger reduction in effective stress(about 6 MPa) and a twelve fold enhancement in perme-ability are predicted near the wellbore. Because of the lowLangmuir pressure of Wolf Mountain coal, reduction ofeffective stress and increase in permeability are small in theregion away from the wellbore (e.g., r > 100 rw). ForQuinsam coal there is more significant enhanced perme-ability near the wellbore at low pressures and a muchstronger nonlinear dependence of pressure than the othercoals (Figure 7).[30] With the same change in effective stress, coal seams

with a large initial porosity (f0) undergo a more significantreduction in permeability as a result of the correspondinglylarger pore volume modulus Kp (i.e., f0K) (equation (5)).With n = 3 and E = 3 GPa, permeability is enhanced in thewhole domain for all three coals. However, the Quinsamcoal with n = 0.35 and E = 1.5 GPa has slightly reduced

Figure 6. Impacts of volumetric strain on (a) horizontaleffective stress and (b) cleat permeability during CO2

sequestration into the Wolf Mountain coal. The curves withthe solid symbols represent results predicted by modelsincluding the volumetric strain effects, but the curves withopen symbols are predicted by the same stress modelsexcluding volumetric strain effects. CSBC, constant stressspecified at the outer domain boundary. Others have thesame model parameters unless labeled.


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permeability at moderately low pressure far from the well-bore but enhanced permeability at markedly low pressurenear the wellbore (Figure 7). For the Ardley coal, larger nand smaller E causes a significantly reduction in permeabil-ity because of less shrinkage due to lower initial methanecontent, which is consistent with the discussions in section 4.3.5.2.2. Stress and Permeability Change During CO2

Injection[31] Adsorption of CO2 into Quinsam coal causes the

largest increase in effective stress (�30 MPa) and up to6 orders of magnitude reduction in permeability of thestudied coals due to its largest Langmuir volume (Table 1and Figure 8). The Ardley and Wolf Mountain coals havesimilar CO2 Langmuir volumes, and thus similar increasesin effective stress and decreases in permeability are pre-dicted. The approximately 20 MPa increase in effectivestress predicted for Ardley and Wolf Mountain coals ismuch less than the 30 MPa increase in effective stress forQuinsam coal (Figure 8a). However, the permeability ofArdley and Wolf Mountain coals is still reduced by 4 orders

of magnitude relative to the initial values but nearly 2 ordersof magnitude higher than that of Quinsam coal (Figure 8b).[32] Nitrogen for all three coals has the least swelling

effects among the gases studied due to its small Langmuirvolume and high Langmuir pressure as suggested by its lowswelling coefficient (i.e., egN2 = 3.1 � 10�4 g/cm3, nearlyone third that for CO2, egCO2 = 9.9 � 10�4 g/cm3, and halfthat for CH4, egCH4 = 6.7 � 10�4 g/cm3). Thereforeinjection of flue gas or a gas mixture of nitrogen and acidgas (i.e., 50% N2 + 50% CO2) significantly reduces theswelling effects on stress and permeability relatively tothose caused by pure acid gas injection (Figure 8). Forexample, there are about 2 orders of magnitude improve-ment in permeability relative to the low permeability caused

Figure 7. Predicted (a) effective horizontal stress and(b) permeability variations during methane production fordifferent coals using the uniaxial strain model. Curves withopen symbols are predicted with parameters listed in Tables 1and 2, and curves with the same shape symbols represent thesame coal with different model parameters as labeled.

Figure 8. Predicted (a) effective horizontal stress and(b) permeability variations for CO2 injection with CH4

displacement for different coals using the uniaxial strainmodel. Solid curves with open symbols are predicted withparameters listed in Tables 1 and 2, and curves consisting ofthe same shape symbols (either solid or open) represent thesame coal with different model parameters as labeled. The1 represents results for flue gas or gas mixture of 50% N2 +50% CO2 injection, and 2 represents results for coal seamswith stronger pore volume modulus or apparently a largerinitial porosity.


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by pure CO2 sequestration. However, even for flue gasinjection, there is still a >10 MPa increase in effective stress(Figure 8a) and a >2 orders of magnitude permeabilityreduction (Figure 8b). With a larger initial porosity or largerpore volume modulus, there is about a 10-fold reduction inpermeability upon injecting flue gas but about a 50-foldreduction in permeability upon injecting pure CO2 (see theQuinsam coal in Figure 8b).5.2.3. Stress and Permeability Change During H2SInjection[33] Injection of H2S at elevated pressure with displace-

ment of preadsorbed CH4 causes increases in effective stressof 75 MPa for Ardley coal, 160 MPa for Wolf Mountain

coal, and �190 MPa for Quinsam coal using the uniaxialstrain model (U model, Figure 9a). Such marked increasesin stress can be attributed to the high Langmuir volume ofH2S (5 times larger than CH4) and correspondingly highvolumetric strain (>3 times greater than CH4) and very lowLangmuir pressure. A permeability reduction of >15 ordersof magnitude is predicted for the coals with H2S injection(Figure 9b). Even with injection of a mixture of 50% N2 and50% H2S with a large initial cleat porosity of 1%, perme-ability is still reduced nearly by 5 orders of magnitudealthough permeability reduction is markedly less than thatfor pure H2S injection (Figure 9). These results indicate thatit is likely impossible to sequestrate H2S into coal seamsbecause of the dramatically reduction in permeability asso-ciated with the strong coal swelling.[34] The increase in horizontal effective stress predicted

by the U model and G and P models with a zero displace-ment outer boundary condition is unrealistic high as coalseams may already yield or slip on preexisted fractures orfaults or cleats before the high stress builds up. However, ifa constant mean stress is specified at the outer boundaryconditions, the P and G models predict very limited varia-tions in stress and permeability. For example, application ofthe G model to Wolf Mountain coal predicts <5 MPaincrease in effective stress and approximately fivefold de-crease in permeability near the injection wellbore (Figure 9).

6. Discussions and Implications

6.1. Model Limitations and Implications

[35] Different models have been proposed to describe thepermeability change of coal seams during coal bed methanerecovery in this paper and in previous studies [e.g., Palmerand Mansoori, 1996; Sawyer et al., 1990; Shi and Durucan,2004]. Most of those models utilize an exponential depen-dence of porosity and permeability on stress and pressureexcept for the linear dependence of porosity in the modelproposed by Palmer and Mansoori [1996]. These modelsalso treat the sorption-induced strain analogously to thethermal strain for a poroelastic medium except for the modelof Sawyer et al. [1990] which does not consider volumetricstrains. Also most models, including ours, assume that onlyhorizontal stress affects the coal seam cleat/fracture net-works and thus the permeability. This is a reasonableassumption as most cleats/fractures are extension fracturesperpendicular to coal bedding and most exploited coal bedsare horizontal or nearly horizontal. Even if horizontalfractures occur within coal seams, horizontal fractures willbe closed and have limited contributions to coal seampermeability due to the thick overlying formations abovecoal seams (thus large vertical confining stress szz andeffective vertical stress).[36] In order to simplify the problemmost previous studies

[e.g., Shi and Durucan, 2004; Palmer and Mansoori, 1996]have assumed purely vertical strain or uniaxial strain todetermine the stress distribution around the wellbore. Themodel proposed by Shi and Durucan [2004] is similar to theuniaxial strain model (U model) in the present study exceptfor slight differences in how volumetric strain is calculated.It is interesting to note that the uniaxial strain model appearto provide better matches to the field data than other models(e.g., the model by Palmer and Mansoori [1996]) as

Figure 9. Predicted (a) effective horizontal stress and (b)permeability variations for H2S injection with CH4

displacement for different coals using the uniaxial strainmodel. Solid curves with open symbols are predicted withparameters listed in Tables 1 and 2, and curves consisting ofthe same shape symbols (either solid or open) represent thesame coal with different model parameters as labeled. The 1represents results for flue gas or gas mixture of 50% N2 +50% H2S injection and 2 represents results for coal seamswith stronger pore volume modulus or apparently a largerinitial porosity. GS represents that the curve is predicted bythe general stress model with a specified constant stress atouter domain boundary.


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suggested by a recent study [Shi and Durucan, 2004]. Theassumed purely vertical strain or uniaxial strain is not arealistic physical mode although not discussed in previousstudies. With the assumption of purely vertical strain, thereis no horizontal deformation, strain or displacement(Figure 10) and thus no horizontally narrowing or broaden-ing of the vertical fractures/cleats will occur. However, thepermeability contributed by those vertical fractures is pre-sumed to depend solely on the horizontal stress that acts onthe fractures and narrows or broadens cleat apertures hor-izontally. This implies a horizontal strain and thus is notconsistent with the assumption of purely vertical strain. Ifthe U model was correctly applied to simulate the stressdistribution in coal seams, it would not be the verticalfractures operating during gas production but the horizontalfractures that are not common in coal seams. Thus theU model has theoretical flaws in modeling stress distribu-tion and permeability in a producing or a injecting coalseam reservoir. That the U model appears to match fielddata better may simply be a coincidence or the arbitrarilyadjusting model parameters lead to a better fit. Furthermore,the U model assumes small or no lateral variations involumetric strain and fluid pressure as encountered in basincompaction studies where it is widely used. During coal bedgas production or acid gas injection, the assumption of smallor no lateral variations in fluid pressure or volumetric strainno longer holds, especially during the early stages ofproduction or injection. This further makes the U modelinappropriate for studying stress and permeability variationaround producing/injecting coal seam wellbores.[37] Besides the intrinsic flaws, the U model predicts an

unrealistically high effective stress field during acid gasinjection (Figures 7–9), which is also predicted by the

general stress model (G model) and plane strain model(P model) with a zero displacement boundary condition(ZDBC) at the outer domain boundary. The G model andP model with a ZDBC behave similarly to the U modelbecause limited horizontal strains can occur to accommo-date the coal swelling induced by acid gas adsorption andresults in the unrealistic high effective stress. For example,for CO2 injection, a displacement of <10 cm is predicted bythe G model with a ZDBC. In contrast, >7-m displacementis predicted with G model with a constant stress boundarycondition (CSBC) (Figure 10). The large deformation ordisplacement of G model with a CSBBC accommodates thelarge volumetric strains induced by the acid gas adsorptionand thus results in relatively much smaller effective stressvariations (Figure 7). For methane production, the G modelwith a ZDBC also predicts much lower displacements thanthe G model with a CSBC (Figure 10).[38] Such contrasting results predicted by the same model

with different boundary conditions imply that the geologicalsetting and structure of a coal seam will have significantcontrols on effective stress and coal seam permeabilityvariations. For example, if a coal seam was cut by low-angle faults, the coal seam may be better simulated byapplying a CSBC, whereas a zero horizontal displacementboundary would be more suitable if the coal seam was cutby steep faults and in contact with mechanical strong andstable geological strata. However, the intrinsic complexityof the geological setting and structure of most coal seamsmake them fall into somewhere between the two end-member conditions. Because of the lack of constrainingdata, it is difficult to select a model that can simulate moreaccurately the stress and permeability changes for gasproduction or acid gas injection. In such cases, differentboundary conditions need to be tested to obtain realisticresults against field data. Nevertheless, the influence ofsorption-induced stress and permeability changes duringgas production and acid gas injection is significant.

6.2. Dynamic Processes Versus Steady State Models

[39] Production of methane from and injection of acid gasinto coal seam reservoirs are dynamic processes especiallyduring early stages. Effective stress changes through timedue to changes in pressure and gas adsorption or desorption.However, with time an approximate steady state can bereached. It is the steady state that is used in the present studyto derive the models describing stress around a wellbore.For producing preadsorbed methane by lowering reservoirpressure, due to the dual-porosity structure of coal seams,pressure in the cleat network is first reduced, followed bymethane being desorbed out of microporous coal matrix intothe cleat networks and then methane flows to the wellbore.If coal seams have large cleat spacing (large coal matrixblocks), gas desorption out of the coal matrix may beretarded relatively to the lowered pressure in the cleatnetworks because of inefficient diffusional transfer of meth-ane through the larger microporous coal matrix. Hence thecoal seam permeability variation may be mainly controlledby fluid pressure during the very early stages of production[Cui and Bustin, 2006]. With the depletion of gas from thecoal matrix, the permeability variation associated withvolumetric strain takes into effect and abates the influenceof pressure.

Figure 10. Radial displacements around wellbores formethane production from and CO2 injection into the WolfMountain coal. All displacements were predicted using thegeneral stress model under a constant stress (s(rb) = s0) or azero displacement (u(rb) = 0) at the outer boundarycondition as labeled. Note that a negative value representsthe displacement occurring along the positive radialdirection or outward movement away from the wellboredue to the assumption of a compressive strain being positivein this study.


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[40] For early gas production, the reservoir permeabilitymay be significantly reduced as predicted by models thatexclude the volumetric strain effects associated with de-sorption. With depletion of methane from the coal matrix,the permeability approaches the results predicted by themodels that include consideration of volumetric straineffects. Similarly, the reverse will occur with acid gasinjection, the coal seam permeability is enhanced first byelevated fluid pressure, followed by a decline in permeabil-ity induced by coal swelling upon acid gas adsorption intothe coal matrix while displacing methane. Moreover, zonesof high concentration of injected acid gas will likelydevelop in the coal seam as it will be adsorbed preferentiallyinto coal matrix. These zones close to the injection well willhave increased stress and decreased permeability as pre-dicted by our models. In contrast, in advance of the acid-gas-saturated zone, elevated pressures develop with limitedadsorption of acid gas, which results in decreased effectivestress and thus enhanced permeability. Zones with saturatedacid gas are first limited to around the wellbore and slowlyexpand into the outer zones as more acid gas is injected andsequestrated until all the domain is finally saturated with theinjected acid gas with a steady state pressure. This final stateis described by our analytical models. The migration ofinjected acid gas is controlled by the coal fabrics [Cui andBustin, 2006] and thus the dynamic stress and permeabilityaround the wellbore.[41] Overall, our models with simplifications provide

insights into stress and permeability variations around thewellbore. Of course, more thorough consideration of theeffective stress and permeability change requires rigorousnumerical investigation of the complex, dynamic, andcoupled nonlinear systems.

6.3. Variable Hydromechanical Properties

[42] To simplify the analysis of stress distribution arounda wellbore during gas production and acid gas sequestration,the coal particles or coal matrix are assumed to be incom-pressible (Ks � K). This corresponds to the material havinga constant Biot’s coefficient z = 1. For heterogeneous dual-porosity coals, the porosity of macroporous cleats or frac-tures often accounts for only a very small portion (<1%) ofthe total coal seam porosity (up to 15%). The majority ofcoal porosity occurs as micropores within the coal matrix.Therefore the assumption of incompressible coal matricesmay not hold for low gas loading or pressure as a largeamount of void micropores may become available fordeformation. Experimental measurements on different coalshave suggested that the Biot’s coefficient (z = 1 � K/Ks)varies significantly from zero at relatively low pressures toone at high pressures [Zhao et al., 2003], suggesting eitheran increasing bulk modulus K with decreasing pore pressureand/or a decreasing Ks with more deformable coal matrixdue to gas depletion. Therefore, at low pressure or gasloadings, influence of shrinking coals on effective stressand thus permeability may become predominant as thepressure effects become negligible with z approaching zero(equation (6)). Consequently, a more significant enhancedpermeability would be expected than the model resultspresented in this study. This may also explain why theuniaxial strain model is not theoretically appropriate for coalseam reservoirs but matches field data from producing wells

better because the U model includes a minimum influenceof pressure and a maximum influence of volumetric strainamong all the proposed models (Figure 4).[43] For acid gas sequestration, the reservoir pressure is

elevated. Therefore, assuming a unit Biot’s coefficientseems suitable and has been done in the present study. Withconstant mechanical properties of coals, dramatic increasesin effective stress and more than 2 and 10 orders ofmagnitude of cleat permeability loss are predicted forinjecting CO2 and H2S, respectively. However, the predictedsignificant reduction in permeability may not occur inpractice. The surfaces of cleats in coal seams developasperities, including minerals and coal fines deposited oncleat surfaces. Coal fractures or cleats cannot close onasperities and become progressively stiffer or have a strongermodulus with increasing stress [Somerton et al., 1975].Therefore the swelling-induced permeability loss is likelymuch less than the predictions although significant perme-ability reduction may still occur.[44] With a constant overburden, the dramatic increase in

horizontal stress versus a decrease in vertical effective stressduring acid gas injection likely causes the coal to yield,buckle, or slip along preexisted cleats/fractures, resulting inenhancement of permeability [e.g., Vaziri et al., 1997]. Thehigh stress associated with acid gas sequestration in coalseam may even trigger the failure of overlying confiningstratum of the coal developing leakage pathways for theinjected gas and consequently causing the acid gas seques-tration to be environmentally unsafe. Overall, for a morerigorous study of the stress and permeability variationduring gas production and acid gas sequestration, variablehydromechanical properties of coal and even the ambientformations need to be considered.

6.4. Implications for Methane Production and AcidGas Sequestration

[45] Coal seams with smaller Langmuir volume likelyhave less preadsorbed gas to release and thus the contribu-tion of desorption-shrinkage of coal to permeability islimited. Less coal swelling would be favorable for acidgas sequestration with less permeability reduction. However,coals with smaller Langmuir volume have less capacity foracid gas sequestration. A larger sequestration capacity orLangmuir volume is desirable for storing more acid gas.Acid gas sequestration through an injecting well can also beapplied to displace the preadsorbed methane to a neighbor-ing methane-producing well for methane recovery, wheremethane is collected as a clean energy resource. Thereforeless difference between sorption of CH4 and acid gas wouldbe desirable for acid gas sequestration because less swellingwill occur due to relatively more methane desorbed or lessnet gas mass increase per unit acid gas sequestrated. High-rank coals have such properties and thus have greaterpotential for CO2 and H2S sequestration and simultaneousCH4 recovery.[46] For the three representative western Canadian coals,

permeability is enhanced during methane production. How-ever, pumping pure acid gas into these coals will be severelyhampered by the significant permeability reduction. Becauseof nitrogen’s very small Langmuir volume in those coals,injection of a N2 mixture versus pure acid gas (CO2 or H2S)can result in a markedly lower reduction of permeability and


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thus improve the sequestration or pumping efficiency withonly a mild reduction of the acid gas capacity (Figure 11). Forexample, injection of a gas mixture of 50% CO2 + 50% N2

into Quinsam coal enhances the sequestration rate markedlydue to its nearly 100-fold relative improvement of perme-ability (Figure 8b) while resulting in only about 31% lesssequestration of CO2 (Figure 11a). Similar trends also occurfor injection of a mixture of 50% N2 + 50% H2S versusinjection of pure H2S. Injection of a gas mixture of 50%H2S + 50% N2 into Quinsam coal will markedly promotethe sequestration rate due to the nearly >4 orders ofmagnitude of relatively improved permeability (Figure 9b)while only resulting in about 16% less H2S sequestration

(Figure 11b). However, if the coal has stronger pore volumemodulus (Kp) as indicated by the large initial porosity (1%versus the reference value of 0.47%) in this study, therelative improvement of permeability is reduced but stillsignificant (Figures 8 and 9). Even if permeability isimproved relative to pure acid gas injection, the coal seampermeability still is significantly reduced relatively to theinitial values. Thus the injection rate will likely be too lowto be economically feasible due to low intact or initial coalseam permeability on a scale of �10�15 m2.

7. Conclusions

[47] Hydromechanical behavior of coal seams during coalbed methane recovery and proposed sequestration of CO2

and H2S into coals is intrinsically complicated. Manyuncertainties in coal properties remain poorly understoodor studied. With constraints from our experimental data ofgas adsorption and volumetric strain from three westernCanadian coals, as a first-order approximation, we havedeveloped models to investigate stress and permeabilityvariation around wellbores drilled for methane recoveryand acid gas injection. Major findings from our presentstudy are summarized here.[48] Volumetric strains induced by adsorption of gases

into coal are linearly proportional to the volume of gasadsorbed. Per unit volume of adsorbed gas, hydrogensulphide causes the largest volumetric strain, nitrogencauses the least volumetric strain, and volumetric straininduced by sorption of unit volume of CO2 is slightly higherthan that of CH4. Furthermore, because of the muchstronger sorption of H2S and CO2 relative to CH4, undersimilar pressures, H2S and CO2 cause much larger volu-metric strains. N2 causes the least volumetric strain amongthe studied gases due to its very weak adsorption and coalswelling. Therefore sequestration of CO2 and H2S withsimultaneous displacement of CH4 from the coal seamscauses the net swelling of coal matrix, elevates the effectivestress, reduces cleat permeability, and makes sequestrationinefficient. Sequestration of pure CO2 and H2S into thethree Canadian coals studied here is impractical due to themore than 2 orders of magnitude of permeability reduction.Injection of mixture of CO2 and N2 mixture markedlypromotes the efficiency of CO2 sequestration and CH4

recovery.[49] Different models with different boundary conditions

predict contrasting results, suggesting that geological settingand coal seam structures have significant controls in effec-tive stress and permeability around injecting or producingcoal seam wellbores. The uniaxial strain model predicts themost significant changes in stress and permeability, whereasthe general stress model with a constant stress boundarycondition predicts the least or slight changes in stress andpermeability during methane production or acid gas injec-tion. Our model results, as a first order of approximation,provide insights into the possible ultimate variations instress and permeability during methane production and acidgas sequestration and consequently the potentials of meth-ane production and acid sequestration. Future thoroughstudies on the stress and permeability distribution aroundthe wellbore and their effects on gas production and acid gasinjection require rigorous modeling of the complex, cou-

Figure 11. Influences of adding N2 into injected acid gason sequestration capacity of (a) CO2 and (b) H2S for thethree western Canadian coals. All curves are produced withequation (1) with parameters listed in Table 1 and, for fluegas, assumed a constant free gas composition at all pressure(i.e., 50% N2 and 50% CO2 or H2S). Curves with symbolsare for pure acid gas sequestration/adsorption capacity, andthe corresponding solid curves pointed by a downwardarrow without symbols represent the flue gas sequestrationcapacity. The relatively reduced sequestration capacities aregiven by the percentage of the capacity for pure gases aslabeled beside the arrows.


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pled, nonlinear system and consideration of variable hydro-mechanical properties of coal seams.

Appendix A

A1. Stress-Dependent Porosity

[50] The bulk volumetric strain increment deb = dexx +deyy + dezz can be derived from equation (6) in text as

deb ¼ � dVb

Vb

¼ 1

Kds � zdpð Þ � deV ; ðA1Þ

where Vb is the bulk volume, and s is the confining pressureor the mean normal stress, (sxx + syy + szz)/3. The porevolume strain increment dep can be expressed as [e.g.,Zimmerman et al., 1986; Zimmerman, 2000]

dep ¼ � dVp

Vp

¼ 1

Kp

ds � 1

Kp

� 1

Ks

� �dp� deV ; ðA2Þ

where Vp is the pore volume, Kp is the pore modulus. Herewe assume that the adsorption-associated strains in the porevolume and microporous coal particle are the same.[51] The macroscopic porosity (f) of a coal seam is

defined as

f ¼ Vp

Vb

: ðA3Þ

Thus the porosity change of a deforming coal seam can bedescribed as

df ¼ dVp

Vb

� �¼ Vp

Vb

dVp

Vp

� dVb

Vb

� �: ðA4Þ

Substituting equations (A1)–(A3) into equation (A4) andmaking some reorganizations yield

dff

¼ 1

K� 1

Kp

� �ds � dpð Þ: ðA5Þ

Assuming constant K and Kp or independent of f,integrating equation (A5) with time yields

ff0

¼ exp1

K� 1

Kp

� �s � s0ð Þ � p� p0ð Þ½ �

� �; ðA6Þ

where subscript 0 represents the initial values. Furthermore,Kp may be approximated as f0K, and thus K is � Kp asgenerally f0 is �1. Then equation (A6) is simplified as

ff0

¼ exp�Dse

Kp

; ðA7Þ

where Dse = (s � s0) � (p � p0) is the change in effectivemean stress if z = 1, which is assumed in this study,although derivation of equation (A7) does not need such anassumption.

A2. Stress Distribution Around a Wellbore

A2.1. General Stress Model (G Model)[52] For a coal seam reservoir, a wellbore covering a

cylindrical domain is drilled for either methane productionor acid gas injection. For simplicity without losing gener-ality, with the wellbore at the center, a cylindrical coordinate

with radial symmetry is applied for field variables aroundthe wellbore. The isotropic elastic constitute between nor-mal stress and strain becomes

srr ¼E

1þ nerr þ

n1� 2n

eb� �

þ pþ E

3 1� 2nð Þ eV

sqq ¼E

1þ neqq þ

n1� 2n

eb� �

þ pþ E

3 1� 2nð Þ eV

szz ¼E

1þ nezz þ

n1� 2n

eb� �

þ pþ E

3 1� 2nð Þ eV ðA8aÞ

with

err ¼@u

@r; eqq ¼

u

r; ezz ¼

@uz@z

; eb ¼ err þ eqq þ ezz: ðA8bÞ

where u is the radial displacement in r direction and uz is thedisplacement in vertical (z) direction. With the assumedradial symmetry, the displacement in tangential direction (q)does not show up in above expressions.[53] For a coal seam reservoir with a constant overburden,

we can safely assume a constant szz and a uniform defor-mation in vertical direction. The stress equilibrium equationin cylindrical coordinate then can be approximated as

@srr

@rþ srr � sqq

r¼ 0: ðA9Þ

Substituting equation (A8) into equation (A9) and reorga-nizing it yield

@

@r

1

r

@ ruð Þ@r

¼ � 1þ nð Þ 1� 2nð Þ

E

@p

@r� 1þ n

3

@eV@r

: ðA10Þ

Integrating of equation (A10) yields

u ¼ � 1þ nð Þ 1� 2nð ÞE

Fp

r� 1þ n

3

Fe

rþ rCG1

2þ CG2

r; ðA11Þ

where Fp =Rprdr, Fe =

ReVrdr, CG1 and CG2 are integration

constants that can be determined with specified boundaryconditions. Substituting equation (A11) back to equation(A8) and manipulating the equations yield

srr ¼1� 2nð Þr2

Fp þE

3r2Fe þ

n1� n

szz þECG1

2 1� nð Þ �ECG2

1þ nð Þr2 ;

ðA12aÞ

sqq ¼ � 1� 2nð Þr2

Fp �E

3r2Fe þ 1� 2nð Þpþ E

3eV þ n

1� nszz

þ ECG1

2 1� nð Þ þECG2

1þ nð Þr2 ; ðA12bÞ

s ¼ 1� 2nð Þ2

pþ E

6eV þ n

1� nszz þ

E

1� nð ÞCG1; ðA12cÞ

where s is the horizontal mean stress defined as (srr + sqq)/2as we mainly consider the horizontal stress in this study.


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[54] Assuming that the wellbore has a radius rw coveringa cylindrical domain with a radius of rb and at its outerboundary (r = rb) the fluid pressure is fixed at a constantpressure (p0), a steady pressure distribution along the radius (r)may be approximated as

p ¼ p0 þ q ln r=rbð Þ; ðA13Þ

where

q ¼ pw � p0

ln rw=rbð Þ ;

pw is wellbore pressure, and p0 is the uniform initialpressure in the coal seam. The volumetric strain associatedwith gas adsorption or desorption can be described byequation (1) or described as

eV ¼ eLbLp1þ bLp

; ðA14Þ

where

bL ¼X ci

pLi;

and

eL ¼ 1

bL

X egiVLici

pLi:

Thus we have the integration functions as

Fp ¼Z

prdr ¼ � Pw � p0

4 ln rw=rbð Þr2

4þ p

r2

2; ðA15aÞ

Fe ¼Z

eV rdr ¼eL2bL

("bLr

2 � 1

q

"2e½2ð�1�bLpwþqbL ln rwÞ�=qbL

Ei 21þ bLpw

qbLþ ln

r

rw

!!##); ðA15bÞ

where Ei(x) = �R�x1 e�t/tdt.

[55] For the wellbore, a stress boundary condition s(rw) =pw can be appropriately applied to the wellbore skin tomodel the production or injection at a fixed well bottompressure. For the outer boundary (r = rb), either a zero radialdisplacement (u(rb) = 0) or a constant mean stress (s(rb ) =s0 ) can be specified. When the zero displacementcondition (ZDBC) is applied at the outer boundary, solvingequations (A12a) and (A11) yields

CG1 ¼2 1� nð Þ 1þ nð Þr2w

E 1� nð Þr2b þ 1þ nð Þr2w� �

"pw þ 1� 2n

r2wFp rbð Þ � Fp rwð Þ� �

þ E

3r2wFe rbð Þ � Fe rwð Þ½ � � n

1� nszz

#ðA16aÞ

CG2 ¼1� 2nð Þ 1þ nð Þ

EFp rbð Þ þ 1þ n

3Fe rbð Þ � r2b

2CG1: ðA16bÞ

If a constant radial normal stress with a value of mean stressis applied at the outer boundary (CSBC) together with thespecified wellbore condition, solving equations (A12a) and(A12c) yields another pair of constants:

CG1 ¼2 1� nð Þ

Es0 �

1� nð Þ 1� 2nð ÞE

p0 �1� nð Þ3

e0 �2nEszz

ðA17aÞ

CG2 ¼1þ nð Þr2w

E

"s0 � pwð Þ þ 1� 2n

r2wFp rwð Þ

þ E

3r2wFe rwð Þ � 1� 2n

2p0 �

E

6eV0

#ðA17bÞ

where s0 is initial uniform stress before the wellbore drilledand eV0 is the initial volumetric strain determined with theinitial pressure p0 and an initial gas composition ci0.A2.2. Plane Strain Model (P Model)[56] Coal seams are often horizontal or subhorizontal and

confined by much stronger and thicker formation above orbeneath them. Thus horizontal deformation/strain may bepredominated within coal seams. Therefore the generalstress mode described above may be simplified to be atwo-dimensional plane strain model with ezz = 0. Then,solving the same stress balance equation (A9) yields theradial displacement as

u ¼ � 1þ nð Þ 1� 2nð ÞE 1� nð Þ

Fp

r� 1þ n3 1� nð Þ

Fe

rþ rCP1

2þ CP2

r: ðA18Þ

Then the radial and tangential stresses become

srr ¼1� 2n1� n

Fp

r2þ E

3 1� nð ÞFe

r2þ ECP1

2 1þ nð Þ 1� 2nð Þ �ECP2

1þ nð Þr2 ;

ðA19aÞ

sqq ¼ � 1� 2n1� n

Fp

r2� E

3 1� nð ÞFe

r2þ 1� 2n

1� np

þ E

3 1� nð Þ eV þ ECP1

2 1þ nð Þ 1� 2nð Þ þECP2

1þ nð Þr2 ; ðA19bÞ

s ¼ 1� 2n2 1� nð Þ pþ

E

6 1� nð Þ eV þ E

1þ nð Þ 1� 2nð ÞCP1: ðA19cÞ

Applying the boundary conditions of s(rw) = pw and u(rb) = 0(ZDBC) to equations (A19a) and (A18) yields

CP1 ¼1þ nð Þ 1� 2nð Þ

E2s0 �

1� 2n1� n

p0 �Ee0

3 1� nð Þ

; ðA20aÞ

CP2 ¼1þ nð Þr2w

E

"s0 � pwð Þ � 1� 2n

2 1� nð Þ p0 �E

6 1� nð Þ eV0

þ 1� 2n1� nð Þr2w

Fp rwð Þ þ E

3 1� nð Þr2wFe rwð Þ

#: ðA20bÞ


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Applying the boundary conditions of srr(rw) = pw ands(rb) = s0 (CSBC) yields another pair of constants:

CP1 ¼2 1� 2nð Þ 1þ nð Þr2wE 1� 2nð Þr2b þ r2w� �

"pw þ 1� 2n

1� nð Þr2wFp rbð Þ � Fp rwð Þ� �

þ E

3 1� nð Þr2wFe rbð Þ � Fe rwð Þ½ �

#ðA21aÞ

CP2 ¼1� 2nð Þ 1þ nð Þ

E 1� nð Þ Fp rbð Þ þ 1þ n3 1� nð ÞFe rbð Þ � r2b

2CP1:

ðA21bÞ

A2.3. Uniaxial Strain Model (U Model)[57] For reservoir or sedimentary compaction modeling, a

purely vertical strain or uniaxial strain is widely used. Withpurely vertical strain, intrinsic constrains of err = e�� = 0holds and thus equation (A8) can be further simplified.Simple manipulation of equation (A8) yields

s ¼ srr ¼ sqq ¼1� 2n1� n

pþ E

3 1� nð Þ eV þ n1� n

szz; ðA22Þ

which suggests that the uniaxial strain model does notrequire external boundary conditions.

A2.4. Stress Change[58] To investigate the stress and associated permeability/

porosity change during the gas production from and or acidgas injection into coal seams, we need to know the initialmean stress state (s0). In present study we assume that, priorto wellbore drilling, the coal seam is intact and saturatedwith methane. The constant vertical stress (szz) can beestimated with depth of the coal seams. Therefore the initialmean horizontal stress can be reasonably approximated withthe uniaixal stress model (equation (A22)) as

s0 ¼ srr0 ¼ sqq0 ¼1� 2n1� n

p0 þE

3 1� nð Þ eV0 þn

1� nszz ðA23Þ

where p0 is the initial reservoir pressure, and eV0 isdetermined with equation (2) with given gas adsorptionand swelling properties. Furthermore, we assume that srr,s��, and szz are the principle stress. Therefore thesummation of srr, sqq, and szz is the first invariants ofstress tensor and does not change under different coordi-nates. Thus we have

s0 ¼ srr þ sqqð Þ=2 ¼ sxx þ syy

� �=2: ðA24Þ

[59] Acknowledgments. We thank Robert Zimmerman and DouglasSchmitt for helpful comments. This research was made possible by grantsfrom the National Research Council of Canada and the Oil and GasCommission of British Columbia. We thank the Alberta Research Councilfor use of their facilities for the hydrogen sulphide experiments.

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��R. M. Bustin, L. Chikatamarla, and X. Cui, Department of Earth and

Ocean Sciences, University of British Columbia, 6339 Stores Road,Vancouver, BC, Canada, V6T 1Z4.


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Adsorption-induced coal swelling and stress: Implications for methane production and acid gas sequestration into coal seams