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AUTHORS
Shimin Liu � Department of Energy andMineral Engineering, Pennsylvania State Uni-versity, University Park, Pennsylvania;[email protected]
Shimin Liu is an assistant professor at Pennsyl-vania State University. He received his B.S. andM.S. degrees from the China University of Miningand Technology, Beijing, and his Ph.D. in engi-neering science from Southern Illinois University,
GEOLOGIC NOTE
A new theoretical approach tomodel sorption-induced coalshrinkage or swellingShimin Liu and Satya Harpalani
Carbondale. His research focuses on coalbedmethane development, carbon sequestration ingeologic formation, and modeling of flow in coaland rocks.
Satya Harpalani � Department of Miningand Mineral Resources Engineering, SouthernIllinois University, Carbondale, Illinois;[email protected]
Satya Harpalani is a professor at SouthernIllinois University, Carbondale. He received hisPh.D. from the University of California, Berkeleyand his M.S. degree from Virginia Tech.,Blacksburg, Virginia. His research focuses onflow characterization of porous media, withemphasis on coal and sandstone, includingmodeling and simulation of gas flow andproduction from deep rocks.
ACKNOWLEDGEMENTS
The AAPG Editor thanks the anonymousreviewers who reviewed this paper.
ABSTRACT
The shrinkage or swelling of coal as a result of gas desorptionor adsorption is a well-accepted phenomenon. Its impact onpermeability changes has also been recognized for two decades.Its importance has increased significantly because of the po-tential of coals that are not likely to be mined and depleted ornearly depleted coalbed methane reservoirs to serve as CO2
repositories. This article proposes a new theoretical techniqueto model the volumetric changes in the coal matrix during gasdesorption or adsorption using the elastic properties, sorptionparameters, and physical properties of coal. The proposedmodel is based on the theory of changes in surface energy as aresult of sorption. The results show that the proposed model isin excellent agreement with the laboratory volumetric straindata presented in the literature during the last 50 yr. Further-more, the proposedmodel can be extended to describemixed-gas sorption behavior, which can be applied to enhancedcoalbed methane and CO2 sequestration operations.
EDITOR ’S NOTE
A color version of Figure 2 can be seen in theonline version of this paper.
INTRODUCTION
Coalbedmethane (CBM) or coal gas has become an importantsource of energy in the United States, Canada, Australia, andChina, with interest in several other countries growing in therecent decades. Coalbed methane production in the UnitedStates increased from near zero in 1980 to almost 2 trillion ft3
in 2009, accounting for approximately 10% of the U.S. gas
Copyright ©2013. The American Association of Petroleum Geologists. All rights reserved.
Manuscript received April 14, 2012; provisional acceptance September 24, 2012; revised manuscriptreceived November 14, 2012; final acceptance December 18, 2012.DOI:10.1306/12181212061
AAPG Bulletin, v. 97, no. 7 (July 2013), pp. 1033– 1049 1033
consumption (U.S. EIA, 2009). A distinctive fea-ture of CBMwells is the negative decline or increasein gas production rates commonly observed duringproduction, particularly in the San Juan Basin ofNew Mexico and southwestern Colorado. It isnow well accepted that the permeability of coal in-creases over time with continued production as aresult of coal matrix volumetric shrinkage associ-ated with the desorption of methane (Harpalaniand Schraufnagel, 1990; Harpalani and Chen, 1995;Mavor and Vaughn, 1998; Palmer and Mansoori,1998; Shi andDurucan, 2005;Liu et al., 2012;Mitraet al., 2012). A sound knowledge of the volu-metric response of coalmatrix to pressure variationsis, therefore, critical, given its impact when inter-preting gas production data as well as projectingfuture production.
Any change in the coal matrix volume is theresult of two opposite effects: mechanical compres-sion or expansion of solid coal resulting fromchangesin pore pressure and volumetric shrinkage or swel-ling of the solid coal matrix as a result of methanedesorption or adsorption. The sorption-inducedstrain of coal has long been recognized to be of im-mense importance in CBM operations because it isbelieved to be responsible for the increased per-meability of coal as a result of opening up of thecleat, characterized by increased cleat porosity andpermeability (Harpalani and Schraufnagel, 1990;Harpalani and Chen, 1997; Palmer and Mansoori,1998; Harpalani and Mitra, 2010). The phenom-enon gained further importance when efforts weremade to inject CO2 in coal for the purpose of en-hanced CBM (ECBM) production and carbon se-questration (Day et al., 2008; Harpalani and Mitra,2010; Hol et al., 2011, 2012). Several studies duringthe last 10 yr have given results of the volumetricstrain measurement (Harpalani and Schraufnagel,1990; Seidle andHuitt, 1995;Levine, 1996;Harpalaniand Mitra, 2010). Coal generally shrinks when a sorp-tive gas is released from its matrix and swells whena gas adsorbs onto it (Pan andConnell, 2007). Also,coal shrinkage and swelling follows the pattern of atypical adsorption isotherm (Levine, 1996; Robertson,2005; Harpalani and Mitra, 2010), suggesting that adirect relationship exists between the amount of gassorbed and the associated matrix volumetric strain.
1034 Geologic Note
This article includes a brief review of the ex-perimental and theoretical studies on coal matrixshrinkage and swelling associated with the sorp-tion of methane and CO2. A theoretical model isthen proposed, based on the change in the surfaceenergy of coal, to describe the sorption-inducedvolumetric changes. The model developed as-sumes the coal to be isotropic and homogeneous.Following this, the results are presented from anexperimental study of coal matrix shrinkage andswelling that was conducted using coal from thenorthern San Juan Basin. Finally, the proposedmodel is tested using experimental results andother data published in the last 50 yr.
Nomenclature and symbols are defined in theAppendix.
BACKGROUND
A significant effort has been devoted toward under-standing the behavior of coal shrinkage and swel-ling during CBM production. Coal matrix shrink-age resulting fromgas desorption has been reportedby several researchers (Moffat and Weale, 1955;Harpalani and Schraufnagel, 1990; Harpalani andChen, 1995; Seidle andHuitt, 1995; Levine, 1996;Harpalani and Chen, 1997; George and Barakat,2001;Robertson, 2005;Harpalani andMitra, 2010).Throughout this article, the shrinkage and swellingresponse to gas pressure refer only to matrix strain.
Moffat and Weale (1955) were among the firstto report a volumetric increase ranging from 0.2%to 1.6% when coal blocks, with ranks from lowvolatile bituminous to semianthracite, were sub-jected to methane pressure of as much as approx-imately 15 MPa (2176 psi). However, for meth-ane pressure between 15 and 71 MPa (2176 and10,298 psi), the volume of coal block either de-creased or remained constant, which was attrib-uted to the changes in solid volume and the graincompressibility effect.
Harpalani and Schraufnagel (1990) studied thevolumetric strain of coal in methane and heliumenvironments. For helium, coal matrix volumewasfound to decrease with an increase in gas pressureresulting from the compression of solid grains of
coal matrix. For methane, a swelling of the coalmatrix was reported with an increase in gas pres-sure, themeasured increase inmatrix volumebeingapproximately 0.5% at 6.9MPa (1000.8 psi).Withdecreasing methane pressure, the matrix volumedecreased. Harpalani and Schraufnagel (1990) con-cluded that desorption of gas causes the shrink-age of the coal matrix. They also observed that thesorption-induced strain exhibited a curvilinearform, with a steep slope at lower pressure, whichbecame gentler at higher pressures (Harpalani andSchraufnagel, 1990; Harpalani and Chen, 1995).
Seidle and Huitt (1995) summarized eightprevious studies of coal matrix shrinkage compres-sibility (Cm), defined as the change in coal matrixvolume per unit change in pore pressure of a sorb-ing gas, ranging in value from 1.25 × 10−4 MPa–1
(8.62 × 10−7 psi−1) to a high of 9.5 × 10−2 MPa–1
(6.55 × 10−4 psi−1), a difference of almost three or-ders of magnitude. Seidle and Huitt also conductedseveral tests to determine the shrinkage and swel-ling coefficients using high-volatile bituminous Ccoals (based on the classification of coal rank innumerous publications, e.g., van Krevelen, 1961)obtained from the San Juan Basin. In their study,the samples were first subjected to a methane pres-sure of up to 13.8 MPa (2001.5 psi), followed bya CO2 cycle of up to 6.2 MPa (899.2 psi). Theestimated swelling coefficient was reported to be30.4 microstrain-ton per cubic meter for methaneand 27.6 microstrain-ton per cubic meter forCO2. Also, the sorption-induced strain was foundto be proportional to the amount of gas sorbed.
Later, Levine (1996) proposed a Langmuir-typemodel to fit the volumetric strain, as follows:
e ¼ elp
p+Peð1Þ
where, e is the sorption-induced volumetric strainat pressure, p; e1 represents the maximum strain,which can be achieved at infinite pressure; and Peis the pressure at which coal attains 50% of themaximum strain. Levine’s (1996) measurementsshowed a linear swelling ratio for CO2 of 0.41%at 3.1 MPa (449.6 psi) and 0.18% for methane at5.2MPa (754.2 psi) using high-volatile bituminouscoal samples from the Illinois Basin.
Harpalani and Chen (1997) conducted an ex-perimental investigation to estimate the matrixshrinkage coefficient. Thematrix volumetric strainwas measured using six strain gauges. The strainwas measured for decreasing methane pressuresfrom 10.4 MPa (1508.3 psi) to atmospheric. Theoverall matrix shrinkage coefficient was calculatedto be 3.34 × 10−2 MPa–1.
George and Barakat (2001) measured volu-metric strains of New Zealand subbituminous Bcoals (based on the classification of coal rank innumerous publications, e.g., van Krevelen, 1961)for adsorption at 4.0 MPa (580.2 psi) for CO2,methane, nitrogen, and helium. The measured vol-umetric strain was 2.1% for CO2, 0.4% for meth-ane, and 0.2% for nitrogen.
Robertson (2005) measured coal volumetricstrain as a result of the desorption of methane.He reported a linear strain of less than 1.0% forcoal and CO2 at pressures of as much as 5.3 MPa(768.7 psi) and 0.2% for methane at pressures of asmuch as 6.9 MPa (1000.8 psi). In addition, bothbituminous and subbituminous coals were used tomeasure the swelling strain resulting from theadsorption of CO2, methane, and nitrogen. Theresults showed that the linear strain caused byCO2 adsorption on the subbituminous coal was2.1%, more than twice that for bituminous coal at5.5MPa (797.7 psi); methane- and nitrogen-inducedcoal swelling for the two coal types was similar—approximately 0.5% at 6.9 MPa (1000.8 psi) andapproximately 0.2% at 6.9 MPa (1000.8 psi).
Harpalani and Mitra (2010) measured coal ma-trix volumetric strain for both the Illinois and SanJuan Basin coals. For Illinois bituminous coal, thevolume of coal matrix increased by approximately0.58% with methane at 5.5 MPa (797.7 psi). ForSan Juan subbituminous coal, the matrix volumeincreased by approximately 0.64%, with methaneat approximately 7 MPa (1015 psi).
Apart from the experimental studies describedabove, a theoretical model was recently devel-oped by Pan and Connell (2007), namely the P&Cmodel, to describe the adsorption-induced coalswelling at sorption-strain equilibrium conditions.The model applies an energy balance approach,which assumes that the change in surface energy
Liu and Harpalani 1035
caused by adsorption is equal to the change in theelastic energy of the solid phase of coal. The modelrequires the elastic modulus of coal, an adsorp-tion isotherm, and other measurable parameters,including coal density and porosity. This modelrelates strain (caused by both adsorption and pres-sure compression) to surface potential adsorptionas follows:
e ¼ �FrsEs
f ðx; nsÞ � PEs
ð1� 2nsÞ ð2Þ
where e is strain,F is the surface potential of sorp-tion and can be calculated based on gas adsorptionisotherms, rs is the density of the solid adsorbent,Es is the Young’s modulus of the solid, P is the so-lid-phase stress, and ns is the coal solid-phase Pois-son’s ratio. The function f (x,ns), representing thestructural model parameter (dimensionless), is gi-ven as
f ðx; nsÞ ¼½2ð1� nsÞ � ð1+ nsÞcx�½3� 5ns � 4ð1� 2nsÞcx�
ð3� 5nsÞð2� 3cxÞ
ð3Þ
where c = 1.2 and x = d/l. The parameter d is thecylindrical radius of the selected pore structuremodel, and the parameter l is its length. This isthe only theoretical model developed to date thatenables estimation of the sorption-related strainusing parameters that are typically known for coal.Themodel prediction results showed a good agree-ment with experimental observations of coal swel-ling provided by Pan and Connell (2007). It wasshown that the model is able to describe the dif-ferences in swelling behavior with respect to gas spe-cies and, at very high gas pressures, where the coalswelling ratio reaches amaximumand thendecreases.
The coal matrix swelling is proportional to theamount of gas adsorbed on coal (Cui and Bustin,2005). The sorption-induced strain is, therefore,directly related to the adsorption capacity. Theadsorption capacity at a given temperature is nor-mally described by the adsorption isotherm, whichis measured under unconstrained condition. Underin-situ conditions, the adsorption behavior can beinfluenced by the change of the state of stress car-
1036 Geologic Note
ried by coal. Hol et al. (2011, 2012) pointed outthat the application of positive effective stress tohigh volatile bituminous coal samples in equilib-rium with supercritical CO2 at a certain pressureleads to a reduction in adsorption capacity. Thiswould certainly impact the sorption-induced strain.
Coal matrix is a reactive, organic rock com-posed of extremely complex hydrocarbons. There-fore, the matrix shrinkage and swelling resultingfrom desorption and adsorption are influenced bythe complexity of coal composition and structure,which is a function of coal rank, formation tem-perature, and amount of moisture. The sorptioncapacity of coals generally increases with coal rank.Pore structure and size distribution varies with coalrank, and higher-rank coals have larger surfaceareas than low-rank coals, generally allowing themto hold more gas (Gan et al., 1972; Bustin andClarkson, 1999). Unfortunately, the influence ofgas composition and coal rank on matrix defor-mation has not been thoroughly investigated.Robertson and Christiansen (2006) did point outthat the variability of sorption-induced strain ismuch greater because of a change in gas type thana change in coal rank. Future work should be en-deavored to characterize and quantify the rela-tionship of the matrix shrinkage and swelling tocoal rank and purity.
PROPOSED MODEL
Coal is different from typical conventional reser-voir rocks because the reservoir gas exists primarilyin an adsorbed phase. Because sorption-inducedcoal swelling and shrinkage are well-accepted phe-nomena, an effort is made here to develop a theo-retical model that would allow the computation ofmatrix shrinkage or swelling compressibility of coal.This would further facilitate computation of theimportant parameter, pore volume compressibility,typically required in most modeling efforts.
Understanding the adsorption-induced deforma-tion of microporous solids, particularly, carbons, isone of the long-standing problems in the physicalchemistry of solid surfaces, especially for adsorption
science. The first studies on charcoal swellingwere published by Bangham et al. (Bangham andFakhoury, 1931; Bangham, 1937) in the 1930s,followed by the investigations on both charcoal andcoals using Bangham’s theory performed by Maggs(1946). Bangham and Razouk (1938) pointed outthe following: (1) the expansion that unactivatedcharcoal undergoes when adsorbing gases or vaporsis directly proportional to its surface energy low-ering; and (2) the Gibbs adsorption equation, re-lating the adsorption, surface energy decrement ofthe charcoal, and pressure of the gas, is valid exceptwhen a change of surface phase is in progress.Bangham’s hypothesis is that the expansion of anadsorbent is directly proportional to the adsorbedfilm pressure, based on a comparison of Schofieldand Rideal’s two-dimensional equation of state,with the empirical equation relating expansion of acharcoal block to sorbed gas uptake. The phenom-enon was confirmed by adsorption-expansion mea-surements made using lower alcohols as adsorbates.Following this, Maggs (1946) proposed a theoreticalmethod to estimate the empirical constant of pro-portionality used to calculate the solid expansionwith a developed adsorbed film.
In the current work, the theory of Bangham andFakhoury (1931) and the work of Maggs (1946)is applied to predict the sorption-induced matrixstrain with gas adsorption and desorption on coal.In addition to the sorption-induced strain, themechanical induced strain is also determined bythe elastic stiffness of the solid matrix (Hooke’sLaw). The derivation of the proposed model is de-tailed in this section.
The surface energy of the coal solid matrixdecreases with adsorption of methane, and the re-sulting swelling of the solid is proportional to thedecrease in surface energy. Bangham and Fakhoury(1931) expressed the decrease in surface pressureof the adsorbate, p, as follows:
p ¼ �DF ¼ F 0 � F0 ð4Þ
where, F0 is the free surface energy of the solidbody in a vacuum environment and F′ is the freesurface energy of the same solid body with a fluidadsorbed on it.
According to the Gibbs adsorption equation(Adamson and Gast, 1997):
p ¼ RTZ p
0GdðlnpÞ ð5Þ
where p is the gas pressure, G is surface excess, R isthe universal gas constant, and T is the tempera-ture. “Surface excess amount, or Gibb’s adsorp-tion, of a component is defined as the excess ofthe amount of a component actually present inthe system over that present in a reference systemof the same volume as the real system, and inwhich the bulk concentrations in the two phasesremain uniform up to the Gibbs dividing surface”(Everett and Koopal, 2001, p. 23). The surfaceexcess for a single adsorbed substance (methane,in this case) is mathematically defined as
G ¼ nS
ð6Þ
where n is the total surface excess amount of ad-sorbed gas, in moles adsorbed per gram, and S isthe specific surface area of the solid (Adamsonand Gast, 1997). The specific surface area is a ma-terial property of a solid, an indicator of the totalsurface area per unit of mass.
The surface excess can then be estimated as
G ¼ nS¼ V
V0Sð7Þ
where V is the gas volume adsorbed by solid coal,V0 is the gas molar volume (22.4 mol/L), and S isthe specific surface area of the solid.
According to Bangham’s theory (Bangham andFakhoury, 1931), the relative linear deformation Dl
lof a solid body is directly proportional to the de-crease in surface energy, p, and the linear strain isproposed as
Dll¼ gp ð8Þ
where g is the deformation constant, a function ofthe solid (coal) physical properties. Using this andequation 5, the following expression is obtained:
Dll¼ gRT
Z p
0GdðlnpÞ ð9Þ
Liu and Harpalani 1037
Maggs (1946) proposed a relationship betweenthe deformation constant (g) and the physical andmechanical properties of solid as
g ¼ SrEA
ð10Þ
where r is the solid body density and EA is themodulus of solid expansion resulting from desorp-tion or adsorption. EA is not necessarily equal tothe Young’s modulus, E. For coal, E is greater thanEA, and the ratio ranges from 2 to 11 (Maggs,1946). Maggs also pointed out that EA does notvary greatly from coal to coal. This relationshiphas been shown to hold well for charcoals and isused here to calculate the coal matrix shrinkageand swelling. The shrinkage and swelling linearstrain model resulting from desorption or adsorp-tion, therefore, is developed using equations 9 and10 as
Dll¼ rRT
EAV0
Z p
0VdðlnpÞ ð11Þ
The Langmuir isotherm (Langmuir, 1916) isthe most commonly used adsorption model to cal-culate the gas volume adsorbed (V) as a function ofpressure. For a single-component gas adsorption,the Langmuir equation can be expressed as
V ¼ abp1+ bp
ð12Þ
where a and b are Langmuir constants. By substi-tuting equation 12 in equation 11, the linear straincan be stated as follows:
Dll¼ rRT
EAV0
Z p
0
abp1+ bp
dðlnpÞ ð13Þ
Rearranging the terms yields
Dll¼ arRT
EAV0
Z p
0
b1+ bp
dp ð14Þ
To simplify the volumetric strain calculationbased on linear strain, a further assumption of iso-tropic elastic behavior of coal was made. The au-thors realize that this may be an oversimplification,
1038 Geologic Note
yet it serves as an ideal starting point. The adsorp-tion volumetric strain (ea), therefore, is three timesthe linear strain, given as
ea ¼ 3Dll
¼ 3arRTEAV0
Z p
0
b1+ bp
dp ð15Þ
Equation 15 shows that the coal matrix vol-ume increases with adsorption of gas caused by thepositive linear strain.However, coalmatrix volumeis also constrained by the stress or pressure actingon it. Based on rockmechanic theory (Goodman,1989), the mechanical strain (em) caused by stressor pressure alone is given as
em ¼ � 3Ps
Eð1� 2nÞ ð16Þ
where Ps is the stress experienced by the solid, E isthe Young’s modulus, and n is the Poisson’s ratio.The linear mechanical strain caused by stress orpressure is given as
elm ¼ �Ps
Eð1� 2nÞ ð17Þ
where elm is mechanical linear strain.Combining the sorption-induced and stress-
pressure compression strains, the overall linear andvolumetric strains can be given as follows:
el ¼Dll+ elm ¼ arRT
EAV0
Z p
0
b1+ bp
dp
� ð1� 2nÞE
Z p
0dp ð18Þ
e ¼ ea + em ¼ 3arRTEAV0
Z p
0
b1+ bp
dp
� 3ð1� 2nÞE
Z p
0dp ð19Þ
The assumptionmade in equations 18 and 19 isthat the two effects are purely additive. The pro-posed model separates the sorption-induced strainand the mechanical induced strain under uncon-fined conditions. The net effect of gas pressureon the volumetric change of coal matrix is, there-fore, dependent on the competition between these
opposing effects. Hence, under conditions wherethe coal seam consists of a stiff material (high bulkmodulus) exhibiting large adsorption-induced swel-ling, the volumetric response to changes in gaspressure would be dominated by swelling overcompression andwill lead to a strong reduction inpermeability. Several references on the competi-tion between swelling and elastic effects have beenrecently published in determining fracture aper-tures and permeability with variation in gas pres-sure (Liu et al., 2011; Hol et al., 2012).
The derived model differs from that of Panand Connell (2007), which is based on the energybalance approach, assuming that the surface en-ergy change caused by adsorption is equal to thechange in the elastic energy of the coal solid. Theproposed model is based on the theory of changesin surface energy, where the linear deformation ofthe solid coal is directly proportional to the changein the surface energy, as reported and confirmed inthe Bangham’s theory (Bangham and Fakhoury,1931) and the work of Maggs (1946) for coals.Compared with the Pan and Connell (2007) model,the proposedmodel avoids the solid-grain and pore-structure geometry parameters, thus reducing thenumber of input parameters and the associated un-certainties. Furthermore, the proposed model ismore transparent, fairly easy to understand, andyet, provides improved accuracy with fewer input
parameters. Moreover, for engineering purposes,the proposed model is able to isolate the sorption-induced strain for different variations of pressureand stress conditions.
EXPERIMENTAL WORK
Experimental Setup
The experimental setup for the study was designedto enable the measurement of volumetric straininduced caused by the mechanical compressionand shrinkage or swelling resulting from the de-sorption and adsorption of gas. A schematic of theexperimental setup is illustrated in Figure 1. Thesetup consisted of high-pressure vessels, pressuremonitoring and recording system for each vessel,and a data acquisition system to monitor the strains.To ensure that temperature over the entire dura-tion of the experiment remained constant, thehigh-pressure vessels were kept in a large constant-temperature water bath.
Sample Preparation
Samples for the shrinkage and swelling experimentswere prepared using core drilled from subbituminous
Figure 1. Schematic presenta-tion of the experimental setupused for the shrinkage andswelling experiment.
Liu and Harpalani 1039
coal blocks collected from the northern San JuanBasin. Two samples were prepared from coal blocksto determine the shrinkage and swelling of thecoal matrix associated with the sorption of meth-ane. The upper part of the core was split into fourquadrants, and samples with the least cleats wereselected. Figure 2 shows a typical sample sche-matically and pictorially. Three strain gauges wereaffixed on each sample to monitor strains in thethree orthogonal directions. Subsequent to prep-aration, the test specimens were kept in an envi-ronmental chamber under controlled conditionsof temperature and humidity until initiating theexperiment.
1040 Geologic Note
Experimental Procedure
The samples were subjected to increasing methanepressure in a stepwise manner, in steps of approx-imately 1.4 MPa (203.1 psi) to a final pressure at7 MPa (1015 psi). Using the measured strains, thematrix volumetric strain was calculated from thealgebraic sum of the three linear strains.
EXPERIMENTAL RESULTS
Figure 3 shows the volumetric strain as a function ofmethane pressure. The strain as a function of pressure
Figure 2. Schematic and pictorial dia-grams of the sample with strain gauges.
Figure 3. Measured and mod-eled volumetric strain as a func-tion of pressure.
follows a trend very similar to that of a sorptionisotherm. Similar to Levine’s (1996) analysis, thestrainsweremodeledusing equation1.ThemodeledLangmuir-type plot is shown in Figure 3, exhibitingan excellent agreement between the measured andmodeled strains. Because the two samples were pre-pared from coal blocks in the same area, the averagestrain values were calculated, as shown in Figure 4,along with the modeled results using the Langmuir-type equation. The difference, although only slight,is caused by the heterogeneous nature of coal.
MODEL VALIDATION
Harpalani and Mitra’s Data
Harpalani and Mitra (2010) presented two setsof shrinkage and swelling data, one each for coals
taken from the San Juan and Illinois basins. Meth-ane pressure was increased in steps of approxi-mately 1.38 MPa (200 psi) to a final pressure of5.5MPa (797.7 psi) for the Illinois coal and 7MPa(1015 psi) for the San Juan coal. For the San Juancoal, the input parameters are presented in Table 1.The solid-phase coal density is 1.4 t/m3, a reason-able value for the San Juan coal. The temperaturewas maintained during the experiment. The val-ue of Young’s modulus is given in their article as5422 MPa (786,395 psi). However, the value isfairly high for coal. The investigation of Bell andJones (1989) gave a distribution of the Young’smodulus and Poisson’s ratio for coal as shown inFigure 5. The Young’smodulus of coal is distributedmainly between 2000 and 5000 MPa (290,075 and725,189 psi), with only one value being 5500MPa(797,708 psi). From open literature, the Young’smodulus and Poisson’s ratio for San Juan Basin is
Figure 4. Average measuredand modeled (using Langmuirfit) volumetric strain as a func-tion of gas pressure.
Table 1. Input Parameters Required for Modeling of Different Coal Types*
Data Source
Coal Type T (K) a (m3/t) b (MPa–1) r (t/m3) n EA (MPa)Liu and H
E (MPa)
arpalani
Ratio (E/EA)
Harpalani and Mitra (2010)
San Juan 308 19.1 0.4 1.4 0.37 1450 3450 2.4 Harpalani and Mitra (2010) Illinois 308 13.7 0.26 1.4 0.398 780 2119 2.7 Current work San Juan 308 19.1 0.4 1.4 0.29 1600 3800 2.4 Levine (1996) Illinois 308 28 0.294 1.4 0.32 1900 4400 2.3 Moffat and Weale (1955) Coal D 308 25 0.35 1.305 0.35 1900 4500 2.4 Moffat and Weale (1955) Coal H 308 25 0.37 1.298 0.37 1900 4300 2.3*T = temperature; a and b = sorption Langmuir constants; r = coal solid-phase density; EA =modulus of solid expansion resulting from adsorption and desorption; E = Young’s modulus.
1041
presented in Table 2. Hence, Young’s modulus wasselected to be 3450 MPa (500,380 psi), which isreasonable for the San Juan Basin. The Poisson’sratio is given in our article at 0.37, which is rea-sonable. The Langmuir adsorption constants weremeasured in the laboratory and given in theHarpalaniand Mitra (2010) article. The modulus of solid ex-pansion, EA, was regressed and estimated to be1450 MPa (210,305 psi). The ratio of E and EA isestimated to be 2.38. Figure 6 shows the measuredvolumetric strain and modeled values as a function
1042 Geologic Note
of pressure. The results show an excellent agree-ment between modeled and experimental data.
For the Illinois Basin coal, the Young’s modulus,Poisson’s ratio, and Langmuir adsorption constantswere given in Harpalani and Mitra (2010) andLevine (1996) and are summarized in Table 1. Themodulus of expansion, EA, was regressed fromthe experimental data to be 780MPa (113,129 psi).The ratio of E and EA was estimated to be 2.71. Themeasured and modeled volumetric strain resultsfor the Illinois coal are presented in Figure 7. The
Figure 5. Elastic mechanicalproperties of coal (from Bell andJones, 1989).
Table 2. Elastic Properties and Reported Ranges of Values for Coal
Input Parameter
Range Reference Selected ValueYoung’s modulus (E) (MPa)
2070–4140 Levine (1996) 3800 3600 Mavor and Vaughn (1998) 855–3070 Palmer and Mansoori (1998) 3450 Seidle et al. (1995) 5422 Harpalani and Mitra (2010)Poisson’s ratio (n), dimensionless
0.23–0.4 Levine (1996) 0.29 0.39 and 0.35 Palmer et al. (2007) 0.33 Seidle et al. (1995) 0.37 Harpalani and Mitra (2010) 0.32 Gray (1987) 0.20–0.50 Shi and Durucan (2005)
results using the proposed model are, again, in ex-cellent agreement with the experimental data.
Application of Experimental Results
For the current experimental study, the max-imum gas pressure was 7 MPa (1015 psi). The
coal bulk density was measured in the laboratoryto be 1.38 t/m3, and the solid phase coal density,1.4 t/m3, a reasonable value for San Juan coals.The temperature is the value at which the exper-iment was conducted. Because the coal sampleswere obtained from San Juan Basin, the Langmuiradsorption constants stated in Harpalani andMitra’s
Figure 6. Modeled adsorption-induced swelling for San Juancoal (Harpalani and Mitra,2010).
Figure 7. Modeled adsorption-induced coal swelling for Illinoiscoal (Harpalani and Mitra,2010).
Liu and Harpalani 1043
(2010) work, also conducted for the same coal,were used for the current work. The Young’s mod-ulus, modulus of expansion, and Poisson’s ratiowere regressed using laboratory data. The ratio ofE and EA was estimated to be 2.38, based on re-gression of the experimental results. The esti-mated value for Young’s modulus, based on thisexercise, was 3800 MPa (551,143 psi), which isreasonable for the San Juan coal (Bell and Jones,1989). The value of the Poisson’s ratio was re-gressed simultaneously with the Young’s modulusto be 0.29, which is, again, reasonable for the SanJuan coal according to Table 2. Figure 8 shows theobserved and modeled swelling strain caused bymethane injection. Themodeled and experimentaldata are in excellent agreement.
Levine’s Data
Levine (1996) measured methane adsorption andswelling on two high volatile bituminous coal sam-ples taken from the Illinois Basin. Measurementswere made up to a maximum pressure of 5 MPa(725 psi) for methane. The Langmuir adsorptionconstants were estimated from the isothermspresented in Levine’s publication. The other re-
1044 Geologic Note
quired parameters are summarized in Table 1. TheYoung’s modulus and Poisson’s ratio were esti-mated from his experimental data to be 30 MPa(4400 psi) and 0.32, respectively, which are reason-able values for Illinois coals. The E and EA ratio wascalculated to be 2.32. Figure 9 presents the observedand modeled volumetric strain for Levine’s data.The results show that the proposed model is ca-pable of predicting the volumetric strain perfectlyas a function of pressure using reasonable adsorp-tion and rock mass properties.
Moffat and Weale’s Data
Moffat and Weale (1955) conducted an investiga-tion on the coal bulk volumetric response inducedbymethane injection at high pressures. The densityof the coal sampleswas provided in their article andis included in Table 1. Unfortunately, both therock mechanic properties and Langmuir adsorp-tion constants were not measured or included intheir article. To validate the proposed model, a re-gression was conducted using the experimental re-sults for D and H coals (as labeled by the authors,Moffat and Weale, 1955), which were low volatilebituminous coals. All parameters used in the model
Figure 8. Modeling of adsorp-tion-induced coal swelling forSan Juan coal (current work).
are listed in Table 1 and are reasonable and compa-rable with the values reported recently. Figures 10and 11 show themeasured andmodeled volumetricstrains for the two coal types. Based on the resultspresented, the proposed model is also able to suc-cessfully replicate the high-pressure adsorption vol-umetric strain behavior. In addition, swelling of
coal bulk can be offset by the solid-phase stress,which is a combination of in-situ stress and porepressure. From Figures 10 and 11, it is easy to seethat the loss of cleat volume because of adsorptionand swelling is reversed for pressures more than10 MPa (1450 psi). This phenomenon, therefore,provides breakthrough information regarding CO2
Figure 9. Modeled adsorption-induced coal swelling for Illinoiscoal (Levine, 1996).
Figure 10. Modeled methaneadsorption-induced coal swel-ling (D coal) (Moffat and Weale,1955).
Liu and Harpalani 1045
injection at high pressures because it has the po-tential to mitigate the loss of injectivity resultingfrommatrix swelling. The loss of injectivity is easilyrecoverable because of the matrix compressioneffect, which is purely mechanical. From Figures 10and 11, the pressure cutoff value for the volu-metric strain decrease is estimated to be between10 and 20 MPa (1450 and 2901 psi). This phe-nomenon was also confirmed in laboratory in-vestigations by Day et al. (2008), who reportedthat the maximum volumetric strain for Austra-lian high volatile bituminous coals was at ap-proximately 10 MPa (1450 psi), which decreasedslightly above this pressure. Moreover, it has beennoticed that the injectivity increased in the twoenhanced-CBM field studies—the U.S. AllisonUnitCO2–ECBM Pilot and the Japan Yubari test site(Ohga et al., 2005; White et al., 2005; Yamazakiet al., 2006).
DISCUSSION
The model proposed in this article is based on thetheory of change in surface energy during sorptionand that the sorption-induced volumetric strain in
1046 Geologic Note
coal matrix is proportional to these changes. Theproposed model clearly shows that the volumetricstrain of coal matrix is a combination of sorption-induced strain and the mechanical compression ofcoal. For both CBM production and CO2 seques-tration, thematrix shrinkage and swelling behaviorhas been shown to be important because it plays asignificant function in the analysis of the dynamicpermeability. To account for these relationships,several permeability models have been developed(Palmer andMansoori, 1998; Cui and Bustin, 2005;Shi and Durucan, 2005; Palmer et al., 2007; Maet al., 2011). However, most analytical or numer-ical permeability models deal with the adsorption-and desorption-induced strain using the empiricalLangmuir-fit technique. This not only increasesthe modeling effort but also results in additionalmodeling uncertainties and errors because theempirical method does not separate the stress andsorption effects. The proposed model has the abil-ity to separate the two effects and then apply thetwo to model the permeability history profile,thus improving the accuracy of a permeabilitymodel. Moreover, the proposed model can be ex-tended to analyze the in-situ CBM reservoir con-ditions, characterized by sorption- and mechan-ical induced volumetric strains associated with
Figure 11. Modeled methaneadsorption-induced coal swell-ing (H coal) (Moffat and Weale,1955).
changes in reservoir pressure and external stress.We believe that, under in-situ conditions, the ap-plied stress and constrained or confined swellingcan strongly reduce the sorption, compared tounconfined condition (Hol et al., 2011, 2012).Consequently, the adsorption parameters (a and bin Langmuir equation 12) might vary with thestate of stress, and the sorption-induced strain eawill be calculated appropriately. Meanwhile, thePs term can be expressed to represent the combi-nation of external stress and pore pressure usingporoelasticity and used to calculate themechanicalinduced strain. For simplicity in the current work,only the unconfined condition has been considered.Finally, the model emphasizes that there is no needto conduct time-consuming laboratory work to ob-tain estimates of shrinkage and swelling parameters.Isotherm data are typically available for a CBMoperation. The geomechanical parameters can bemeasured or estimated because Young’s modulusand Poisson’s ratio are fairly consistent for any coaltype.
In addition, the proposed model can be readilyextended to describe the shrinkage or swellingbehavior resulting from sorption of mixed gasesand, therefore, for the application of the existingpermeabilitymodels toCO2 sequestration or ECBMoperations. Finally, Figures 10 and 11 show thatthe coal matrix volume can decrease, thus increas-ing the cleat volume that would, in turn, increasethe injectivity in coal seams.
CONCLUSIONS
The proposed model shows how the coal matrixvolume varies with changes in gas pressure. It isbased on the change of the surface energy duringthe sorption process. Every required input param-eter in the proposed model has an associated phys-ical meaning. Based on the model validation com-pleted, several important conclusions can be made.These are summarized below:
1. The proposed model can accurately predict thevolumetric strain with proper elastic properties,
adsorption or desorption knowledge, and phys-ical properties of coal.
2. The proposed model can also successfully sim-ulate the coal volumetric behavior for high-pressure ranges, approximately beyond 15 MPa(2176 psi). It has the potential for extension toinclude modeling of mixed gases, which pro-vides a sound basis for high-pressure CO2 in-jection for sequestration and ECBM operations.
3. The estimated ratio of Young’s modulus (E)and modulus of expansion (EA) ranged from2.3 to 2.7 for different coal samples. This is inaccordance with the values provided by Maggs(1946). Furthermore, it provides a techniqueto estimate EA for different coals.
4. The proposed model provides the theoreticalbasis to develop expressions for the flow be-havior of CBM reservoirs resulting from thematrix shrinkage effect. It can be immenselyvaluable in understanding and modifying thepermeability models dealing with complex re-lationships between permeability and reservoirpressure.
5. Finally, it provides a sound knowledge of theresponse of matrix volume to variations in pres-sure and stress. Given its simplicity, this can be avaluable tool for engineering applications.
FUTURE WORK
Based on the validation presented and the promiseshown by the proposed model, we recommendthat the current model be coupled with the per-meability models to dynamically analyze perme-ability profiles of CBM production. During thiscoupling process, the knowledge of fracture me-chanics and poroeslaticity will be applied to con-sider the complex fracture system (cleats) of coal(Laubach et al., 1998) and the variation of thestate of stresses (Liu et al., 2011) during pressuredepletion. For CO2 sequestration in coals, furthereffort should be devoted to interpret the in-jectivity loss (Liu et al., 2009) and establish a CO2
injection strategy to either maintain or increaseinjectivity for the target coal seams.
Liu and Harpalani 1047
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APPENDIX:. NOMENCLATURE
e
Coal matrix strain el, Pe Parameters of Langmuir match to volumetricstrain as a result of matrix shrinkage
p Gas pressure F Surface potential r Coal solid-phase density Es Coal solid-phase Young’s modulus f(x,ns) Structure model parameter ns Coal solid phase Poisson’s ratio P Solid phase stress c Pore structure model constant (1.2) x Ratio d/l d, l Pore structure model parameter (defined byPan and Connell, 2007)
p Decrease in surface pressure of the adsorbate F Free surface energy of the solid body R Universal gas constant T Temperature G Surface excess n Total surface excess amount of adsorbed gas S Specific surface area of the solid V Gas volume adsorbed by solid coal V0 Gas molar volume Dl/l Relative linear deformation of a solid body g Deformation constant EA Modulus of solid expansion resulting fromadsorption and desorption
a, b Sorption Langmuir constants ea Adsorption volumetric strain em Mechanical induced volumetric strain elm Mechanical induced linear strain Ps Stress experience by solid phase E Young’s modulus n Poisson’s ratio
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