International Journal of Coal Geology 113 (2013) 1–10
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International Journal of Coal Geology
j ourna l homepage: www.e lsev ie r .com/ locate / i j coa lgeo
Permeability prediction of coalbed methane reservoirs duringprimary depletion
Shimin Liu a,⁎, Satya Harpalani b
a Department of Energy and Mineral Engineering, Pennsylvania State University, University Park, PA 16802, USAb Department of Mining and Mineral Resources Engineering, Southern Illinois University, Carbondale, IL 62901, USA
Permeability increase in coalbed methane (CBM) reservoirs during primary depletion, particularly in the SanJuan Basin, is a well accepted phenomenon. It is complex since it is influenced by stress conditions and coalmatrix shrinkage associated with gas desorption. Understanding the variations in coal permeability is criticalin order to reliably project future gas production, or consider other gas migration issues in the reservoir. Sincesorption-induced strain plays a critical role in changing the permeability, typically observed, the theoreticalstrain model should be incorporated into the permeability prediction models. An effort is made in this paperto couple the recently developed Liu and Harpalani sorption-induced strain model with various permeabilitymodels. The model first calculates the theoretical coal matrix shrinkage strain and, using the calculated strain,various commonly used permeability models are applied to two sets of field data. The results of the coupledmodels show that the agreement between the predicted permeability and that observed in the field is verygood. The merit of the coupled models is that it can theoretically predict the permeability with less experi-mental work, making it a more time efficient and economical technique compared to models used in the past.
Coalbed methane (CBM) refers to gas formed as a part of thegeological processes of coal formation, and is contained in varyingquantities within all coals. With increased energy demand and signif-icant advances made in development of CBM industry during the lastthree decades in the United States, CBM has become a resource ofglobal significance, with emergence of active CBM plays in China,Canada, Australia and India. CBM reserves and production in the USare both almost 10% of the total values (Palmer, 2010), which is sig-nificant. The rate at which CBM production has increased, from nearzero in 1980 to more than 1.9 trillion cu ft (TCF) in 2009, is just asstriking.
Scientific understanding of, and production experience with, coalbedmethane hasmatured significantly in the last decade although, relativelyspeaking, CBM industry is still young and further studies will continue inorder to better understand the production behavior. Compared to con-ventional natural gas reservoirs, CBM reservoirs have several uniquecharacteristics: (1) very low effective porosity/permeability, whichoften hinders the development of CBM plays; (2) adsorption storagemechanism since most of the methane produced in coal is stored in themicropores in adsorbed form (Gray, 1987); (3) desorption-inducedmatrix shrinkage since this results in opening up of the cleat which, inturn, results in increased fracture permeability in the low pressure
rights reserved.
range; and (4) the ‘negative decline curve’, that is, production increasesat the beginning and then slowly decreases over time (Harrison andGordon, 1984), which is a distinct feature of CBM production wells.
With advances in scientific knowledge and field observations, anumber of analytical permeability models have been developed topredict the unique permeability behavior of CBM reservoirs (Cui andBustin, 2005; Gray, 1987; Liu et al., 2012; Ma et al., 2011; Palmer andMansoori, 1998; Palmer et al., 2007; Pekot and Reeves, 2003;Robertson and Christiansen, 2006; Seidle and Huitt, 1995; Shi andDurucan, 2005). In all of these models, the influence of sorption-induced volumetric strain on coal permeability has been integratedeither by empirical approaches, such as the Langmuir-type curve(Levine, 1996), or experimentally measured data. Clarkson et al.(2010) theoretically coupled the Pan and Connell (2007) and Palmeret al. (2007) permeability models to match the field permeability dataobtained using the production data analysis (PDA). Finally, a new theo-retical model was recently derived to quantify the sorption-inducedvolumetric strain based on the surface energy change theory (Liu andHarpalani, in press).
This paper is aimed at incorporating the recently developedmatrix strain model into the existing and widely used analyticalpermeability models and evaluating its performance. Following this,effort is made to compare the modified permeability model predic-tions against published field permeability increase data. It providesa basis for the different CBM simulators to theoretically incorporatethe sorption-induced strain into establishing the changes in perme-ability during history matching and projection of gas production. It
2 S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
also provides a theoretical basis to improve the performance of CBMreservoir simulators with only basic geomechanical and adsorptionparameters as input. These parameters are either available in geologicreport or easy to obtain through experiments. Hence, the couplingeffort of the strain and permeability model will avoid to carry outexpensive field tests and/or time consuming experiments, for instance,steady-state shrinkage and swelling tests.
2. Review of sorption-induced strain and permeability models
2.1. Sorption-induced strain models
The first reported study of coal matrix volumetric response tosorption of gas can be traced back to Moffat and Weale (1955).They observed 0.2% to 1.6% volumetric increase when coal blockswere subjected to methane pressure of up to ~15 MPa. However, formethane pressure between 15 and 71 MPa, the volume of coal blockeither decreased or remained constant, which was attributed to thechanges in the solid volume of coal, effectively the grain compressibil-ity effect.
After initiation of the CBM industry in the early 1980's in the US, andavailability of production data for a few years, coal matrix shrinkageresearch gained a great deal of attention due to its impact on changesin the cleat/fracture aperture, inducing a considerable increase in thecleat permeability during later part of life of CBMwells. The coal matrixshrinkage and swelling strain has been quantified in the laboratory byseveral researchers (George and Barakat, 2001; Harpalani and Chen,1997; Harpalani and Mitra, 2010; Harpalani and Schraufnagel, 1990;Levine, 1996; Robertson, 2005; Seidle and Huitt, 1995; Wang et al.,2011). In each one of these laboratory studies, matrix volumetric in-crease/decrease with methane adsorption/desorption was reported.However, no theoretical explanation to unlock the relationship be-tween volumetric strain and sorption of gas was presented, making itdifficult to integrate the actual effect of the phenomenon into CBM res-ervoir permeability prediction models.
In order to quantitatively model the sorption-induced volumetricstrain, some empirical approaches have been developed to establish alink between the measured strain and reservoir pressure. For example,Levine (1996) used a Langmuir-type model to fit the experimentally-derived linear strain data for Illinois basin coals, when flooded withmethane and CO2. With the help of this empirical model, a number ofanalytical permeability models were developed to quantitatively incor-porate the shrinkage effect (Izadi et al., 2011; Ma et al., 2011; Palmerand Mansoori, 1998; Robertson and Christiansen, 2006; Seidle et al.,1992; Shi and Durucan, 2004, 2005; Wang et al., 2012). The model isgiven as:
ε ¼ εlp
pþ Pεð1Þ
where, ε is the sorption-induced volumetric strain at pressure p, εl rep-resents themaximum strain which can be achieved at infinite pressure,and Pε is the pressure at which coal attains 50% of the maximum strain.
Other researchers (Chikatamarla et al., 2004) reported that matrixswelling is directly proportional to gas content of coal. Following this,Cui and Bustin (2005) proposed a linear relationship between thevolume of adsorbate and the sorption-induced volumetric strain,given as:
ε ¼ εgVLp
pþ PLð2Þ
where, ε is the sorption-induced volumetric strain at pressure p, εgrepresents the coefficient of sorption-induced volumetric strain; VL
and PL are the usual Langmuir Constants. However, Cui and Bustin(2005) recognized that the relationship between volumetric strain
and sorbed gas content may not always be linear and it was confirmedbyWang et al. (2011), who pointed out that the fracture geometry alsoplays a role in this relationship.
In 2007, Pan and Connell (2007) derived a theoretical model,namely the P&C model, to describe the adsorption-induced coalswelling at sorption/strain equilibrium conditions. The model utilizedan energy balance approach, which assumes that the change in sur-face energy caused by adsorption is equal to the change in the elasticenergy of solid coal. The model requires the elastic modulus of coal,adsorption isotherm and other measureable parameters, includingcoal density and porosity. This model relates strain (due to bothadsorption and pressure compression) to surface potential adsorptionas follows:
ε ¼ �Φρs
Esf x;νsð Þ � P
Es1� 2νsð Þ ð3Þ
where, Φ is surface potential of sorption and is calculated based on gasadsorption isotherms, ρs is the density of the solid adsorbent, Es is theYoung's modulus of the solid and νs is the Poisson's ratio. The functionf (x, νs), represents the structural model parameter (dimensionless)and is given as:
where, c = 1.2, and x = a/l. The parameter ‘a’ is the cylindrical radiusof the selected pore structure model and l is its length. This is the firsttheoretical model developed to date that enables estimation of thesorption related strain using parameters that are typically known forcoal. Themodel prediction results showed good agreement with exper-imental observations of coal swelling. However, P&C model includesthe solid grain and pore structure geometry term, which is difficult toestimate and, therefore, has associated uncertainties. Clarkson et al.(2010) expanded this model to describe multi-component gas sorption-induced strain using the extended Langmuir equation, but did not detailthe solid and pore structure geometry information in the publication.
Liu and Harpalani (in press) proposed a theoretical technique tomodel the volumetric changes in coal matrix during gas de/ad- sorptionusing elastic properties, sorption parameters and physical properties ofcoal. The proposedmodel is based on the principles of physics and chem-istry of a surface and the interface theory. The volumetric strain of coalmatrix for sorbing gas includes two components, sorption-induced andmechanical-induced strain. The sorption-induced strain is directly pro-portional to the decrease in surface energy and mechanical-inducedstrain is calculated by the Hooke's law. In this model, these two strainswere assumed to be purely additive. It highlights the relationships be-tween sorption and coalmatrix strain. The volumetric strain is calculatedas follows:
ε ¼ 3VLρsRTEAV0
∫p
0
1PL þ p
dp� 3 1� 2νð ÞE
∫p
0dp ð5Þ
where, PL and VL are Langmuir constants, ρs is the density of the solidadsorbent (coal), EA is the modulus of the solid expansion, E is theYoung's modulus, ν is the Poisson's ratio, Vo is the gas molar volume(22.4 m3/kmol), R is the universal gas constant and T is the absolutetemperature. The authors presented validation of the proposed modelusing laboratory volumetric strain data available in open literature overthe lastfifty years and the results, showing excellent agreement betweenmodeled and experimental data. Compared to P&C model, this modelavoids the solid grain and pore structure geometry parameters, thusreducing the number of input parameters and, more importantly, theuncertainties associated with these parameters. Moreover, Liu andHarpalani model is more transparent, easy to understand, and yet, pro-vides improved accuracy with fewer input parameters. Finally, the pro-posed model can be extended to describe mixed-gas sorption behavior,
3S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
which can be applied to enhanced coalbedmethane (ECBM) and CO2 se-questration operations as well as CBM reservoirs where the in situgas content includes significant fractions of other gases, like CO2 in thenorthern San Juan Basin and nitrogen in Illinois Basin.
2.2. Literature review of coal permeability modeling with sorption-inducedstrain
Pan and Connell (2012) presented a comprehensive review ofanalytical coal permeability models associated with both experimen-tal and field testing data. Eighteen permeability models were coveredin this overarching review. Among these eighteen models, five ofthem discussed the anisotropy effect on coal permeability variations,namely, Gu and Chalaturnyk (2010), Wang et al. (2009), Liu et al.(2010), Wu et al. (2010), as well as Pan and Connell (2011). In ourcurrent work, detailed review of four analytical permeability models(Cui and Bustin, 2005; Ma et al., 2011; Palmer and Mansoori, 1998;Shi and Durucan, 2004) are carried out since they are selected to cou-ple with the new developed sorption-induced strain model (Liu andHarpalani, in press). To avoid the duplication of review effort, othermodels discussed in Pan and Connell (2012) are not included here.However, we briefly review the most recently studies on coal perme-ability models (Liu et al., 2011; Mazumder et al., 2012; Siriwardane etal., 2012; Wang et al., 2012).
Palmer and Mansoori (1998) presented the first of the most com-monly used analytical permeability model. It is based on the assump-tion of uniaxial strain conditions, that is, constant vertical stress dueto overburden and zero horizontal strain due to lateral confinement.The model is based on coal geometry being represented as a bundleof matchsticks. It is also the first-of-its-kind permeability modelincorporating fundamentals of rockmechanics. The underlying principleof the model is that a change in pore pressure and the desorption-associated matrix shrinkage influence the cleat porosity which, in turn,results in permeability changes. This model estimates thematrix shrink-age as a result of desorption in amanner similar to thermal contraction ofa material. The final form of the model to estimate the changes in cleatporosity with pore pressure is given as:
ϕϕ0
¼ 1þ Cm
ϕ0p� p0ð Þ þ εl
ϕ0
KM
−1� �
pPε þ p
− p0
Pε þ p0
� �ð6Þ
Cm is given as:
Cm ¼ 1M
� KM
þ f � 1� �
γ ð7Þ
where, K is the bulk modulus, M is the constrained axial modulus, γ isthe solid compressibility (also referred to grain compressibility), p isthe pressure and f is a fraction, its value varying between 0 and 1. Kand M are expressed as functions of the Young's modulus (E) andPoisson's ratio (ν) as follows:
ME
¼ 1� ν1þ νð Þ 1� 2νð Þ ð8Þ
KM
¼ 13
1þ ν1� ν
� �ð9Þ
The change in permeability is then determined using the followingcubic equation:
kk0
¼ ϕϕ0
� �3ð10Þ
Since this model was unable to successfully match the field datafrom San Juan basin reservoirs, it was subsequently modified (Palmeret al., 2007) to better match the observed permeability increases,
accounting for permeability variation due to cleat anisotropy andmodulichangeswith continued production. To quantify the changes in porosityfor sub-vertical cleats, one additional parameter ‘g’ was introduced,such that Eq. (7) became:
Cm ¼ gM
� KM
þ f � 1� �
γ ð11Þ
A value of zero for ‘g’ was recommended for vertical cleats and 1for horizontal cleats, leading to suppression of the pressure effectterm (first term on the right side of Eq. (6)). A comprehensive reviewof the evolution of the model is presented by Palmer (2009).
Shi and Durucan (2004) proposed a permeability model, alsobased on matchstick representation of coalbed (Reiss, 1980), wherecleat permeability is impacted by the prevailing effective horizontalstresses acting normal to the cleats. Following the effective stress–strain relationship for a homogenous, isotropic, thermo-elastic, porousmedium (Nowacki, 1975), the variation in effective horizontal stressunder uniaxial strain condition was expressed as a function of porepressure, which included a cleat mechanical compression term and amatrix shrinkage term that have competing effects on cleat/fracturepermeability, as shown below:
σ � σ0 ¼ � ν1� ν
p� p0ð Þ þ E3 1� νð Þ εl
ppþ Pε
� p0p0 þ Pε
� �ð12Þ
where, σ and σ0 are the effective horizontal stress at pressure p and p0,respectively. In this formulation also, the sorption-induced matrixstrain was analogous to thermal expansion/contraction (Palmer andMansoori, 1998). The variation in permeability was then computed bythe following equation, proposed by Seidle et al. (1992):
kko
¼ exp �3Cf σ � σoð Þh i
ð13Þ
The value of cleat compressibility, Cf, was assumed in the originalmodel to remain constant throughout the life of a producing reservoir.However, the modelers were compelled to vary the cleat compressibil-ity to match the observed permeability increases in the field during val-idation of themodel (Shi andDurucan, 2005, 2009). The basis of varyingthe cleat compressibility was purely to improve the match, with nojustification for the variation in its value during reservoir depletion,making the model somewhat controversial.
Cui and Bustin (2005) proposed a separate model, which assumedconstant reservoir loading, matrix shrinkage to be linearly propor-tional to the volume of sorbed gas and coal matrix to be much stifferthan the bulk coal. Additionally, the uniaxial strain condition wasassumed to hold. Starting with the constitutive equation for porousmedia (Neuzil, 2003), the modelers derived the following equationto predict the porosity variation with pressure drawdown:
ϕϕ¼ exp �Cf σ � σ0ð Þ � p� p0ð Þ½ �
n oð14Þ
The cubic relationship between permeability and porosity wasused to calculate coal cleat permeability change, as follows:
k ¼ k0exp �3Cf � 1þ νð Þ3 1� νð Þ p� p0ð Þ þ 2E
9 1� νð Þ εv � εv0ð Þ� �� �
ð15Þ
where, εv - εv0 is the volumetric strain change with pressure variationfrom p0 to p. The change in the mean normal stress is used to calculatethe permeability variation. It is different from the Shi and Durucanmodel, which applies only the changes in effective horizontal stress.
All three models discussed above are based on uniaxial straincondition. However, there are some researchers who believe thatthe volume of the entire reservoir does not change with depletion
4 S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
(Massarotto et al., 2009). Based on this, Ma et al. (2011) proposed asimplified permeability model based on the volumetric balancebetween the bulk coal, solid grain/matrix and cleat/fracture, usingthe constant volume assumption. The fundamental concept for thismodel is that the grain/matrix volume of coal changes dynamicallywith reservoir depletion as a result of the combined effects of me-chanical decompression and sorption-induced strain. The cleat aper-ture variation as a function of the reservoir pressure is calculatedusing the grain/matrix volumetric variation and the permeability pro-file is then computed based the matchstick geometry. In this model,the overall matchstick strain resulting from matrix shrinkage and de-crease in reservoir pressure is given as:
where, Δa/a is the horizontal strain in a single matchstick. The changein permeability is then calculated using the following equation, givenby Harpalani and Chen (1995):
kk0
¼1þ 2
ϕ0
Δaa
3
1� Δaa
ð17Þ
Instead of laying heavy emphasis on the pore volume/cleat com-pressibility, which is still a topic of controversy for CBM reservoirs,the model focuses on changes in the grain volume and then convertsthis to change in cleat volume, assuming that the total volume re-mains constant. Moreover, the authors pointed out that this modelis more “friendly” due to its transparency, validation with field datausing reasonable input parameters, simplicity and ability to measureevery parameter required in the model.
Liu et al. (2011) developed a more general theory to characterizethe coal permeability variation under uniaxial strain, constant volumeand stress-controlled boundary conditions. Similar to the conceptof “swelling stress” introduced by Liu et al. (2010), additional forceand resulting stress was developed with the coal under either strainor stress constrained conditions. The “free expansion plus pushback” approach is developed to determine the amount of stress. Inthe approach, the coal bulk was firstly allowed to deform freely dueto sorption-induced strain, and following this, it was pushed backby applied effective stress to satisfy the certain boundary conditions.The total push-back strains were used to quantify the permeabilityevolution through the variation of cleat porosity. The results modeledby this approach consisted of the ARI model (Sawyer et al., 1990),P&M model, S&D model and C&B model.
Wang et al. (2012) proposed a model to describe permeabilityevolution in fractured sorbing media. The model is based on theunderstanding of interaction between fracture and matrix and frac-ture aperture evolution was theoretically investigated. The fractureaperture tends to open up due to reduction in effective stress and itwould be compacted due to sorption-induced swelling strain underthe constant total stress with variation in pore pressure. Then thecubic relationship was applied to derive the permeability model.Therefore, the model was developed to accommodate the effects ofporomechanical response and sorption-induced swelling responseon the permeability of coal.
Mazumder et al. (2012) reported permeability increase as a resultof desorption-induced matrix shrinkage in Moranbah Gas Project(MGP) field in Bowen basin, Australia. Regarding the permeabilitymodeling, P&M, S&D and C&B models were used to analyze the per-meability behavior obtained from field observation. These three ana-lytical models were fitted for parameters including Poisson's ratio,Young's modulus, initial cleat porosity and pore volume (cleat) com-pressibility. Both the field observation and modeled results shows thepermeability exhibited a high level of increase during primary
pressure drawdown. This work has strengthened that the applicabil-ity of three analytical models, which are initially developed for SanJuan basin.
Siriwardane et al. (2012) carried out a numerical simulation tomodel CBM production and CO2 injection by using PSU-COALCOMPsimulator. The shrinkage and swelling strain was incorporated intothe deformation behavior of porous medium (coal) for pressuredepletion. The linear relationship between the shrinkage/swelling strainand adsorption volume was applied in their investigation, which maybenot true for coal since the fracture geometry plays a role on the coal de-formation (Wang et al., 2011). Similar to P&M model, the permeabilityvariation was calculated by cubic law as shown in Eq. (10). Localizedcoal matrix swelling was found around the injection well, inducing arapid injection permeability drop. In addition, tracer data indicated a sig-nificant level of reservoir anisotropy and heterogeneity.
2.3. Brief discussion of other permeability models
Gu and Chalaturnyk (2005) and Palmer (2009) have reviewedanalytical models, and offered a distinction based on whether a modelis strain-based or stress-based. In the strain-based models, as the termsuggests, the coal/rock strain is the basic reason for any changes in theformation flow property. Basically, desorption of methane results in avolumetric strain of the coal matrix which, in turn, results in changesthe cleat porosity/aperture and thus in the permeability of coal. Onthe other hand, a stress-based model takes the coal/rock stress as theinitiating cause, and using it, the strain is calculated. Hence, the inter-pretation of the basis of this class of models is that desorption of meth-ane changes the volumetric strain which, in turn, changes the effectivemean or horizontal stress, and thus the permeability of coal. Based onthis, the Palmer & Mansoori and Ma et al. models are strain-basedwhile the Shi & Durucan and Cui & Bustin models are stress-based.
In the strain-based models, the shrinkage/swelling effect due todesorption/adsorption is directly subtracted/added to the initialfracture porosity. The advantage of these models is that they avoidthe controversial pore volume/cleat compressibility parameter. Theaccurate prediction of porosity change, however, becomes very criti-cal in these models. The required model input parameters, typicallyobtained from history matching exercise, makes the model userssomewhat uncomfortable. Moreover, there are two uncertain param-eters in the P&Mmodel, g and f, with typical values of g used between0.1 and 0.5. Although the modelers provide a means to estimate theparameter g, it is not very clear how its value can be estimated forCBM plays with limited production data, like the Illinois basin. Theauthors believe that the value of ‘g’ is selected primarily to matchthe production data rather than estimated from the technique providedby the modelers. The Ma et al. model (2011) depends entirely on thecoalbed reservoir geometry framework. In order to use this model, thecoalbed reservoir is taken to be of the matchstick geometry, where thematchsticks are near vertical. This is a major limitation of this model.
S&D and C&B models are in the stress-based category. Unlike thestrain-based models, the shrinkage/swelling effect due to desorption/adsorption is incorporated in the stress change term. In this category,the pore volume/cleat compressibility serves as the bridge betweenpermeability and stress changes. Therefore, accuracy of the pore volumecompressibility becomes the key factor during any modeling exercise.For both families of models, there is an assumption behind the formula-tion and, that is, the effective stress (Biot's) coefficient is constant andequal to unity. This assumption is not strictly correct for porousmedia, as pointed out by a few researchers in the past (Geertsma,1957; Lade and Boer, 1997; Nur and Byerlee, 1971; Skempton, 1960;Skempton and Bishop, 1954; Wang et al., 2012; Zhao et al., 2003).Several studies have shown that the effective stress coefficient isstrongly stress-dependent (Elsworth and Bai, 1992; Murdoch andGermanovich, 2006; Rice and Cleary, 1976). It can decrease to 0.1 inrock when the sample is loaded to a stress of 100 MPa (Schmitt and
5S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
Zoback, 1989). Wang et al. (2012) pointed out that the Biot coefficientis not the same for fracture and matrix due to their distinct features.But the estimation method of Biot coefficients for fracture and matrixwas not provided (Wang et al., 2012). Given that the thrust of ourpaper is to couple the strain model with existing analytical permeabil-ity models and to test for validation of field observed data, the Biotcoefficient is kept as a constant of one. The field permeability dataused in this study are collected from open literature, the pore pressureis only up to about 6.5 MPa, which is relatively low compared to100 MPa (Schmitt and Zoback, 1989) and thus it is justified to use aconstant value during the course of depletion.
A quantitative comparison of the S&D and C&B models ispresented below. According to Cui and Bustin (2005), the two modelscan be rearranged as follows:
ð18; 19Þ
There are two competing terms governing the permeability varia-tion during depletion. The first term is the increased effective stresswith reduction in gas pressure, denoted as the stress term. The secondterm is the matrix shrinkage term due to gas desorption, denoted asthe strain term. For the stress term, the coefficients are ν
1−ν and1þν
3 1−νð Þ for S&D and C&B, respectively. For coal, the Poisson's ratio isalways less than 0.5. Thus, the term 1þν
3 1−νð Þ is larger thanν
1−ν. Obviously,the permeability predicted by the C&B model decreases more thanthat by S&D model for the same pressure drawdown. For the strainterm, coefficient of C&B model is smaller than that in the S&D model.Hence, the permeability increase predicted by the C&B model is lowercompared to S&D model for the same pressure drawdown. For thesetwo reasons, the permeability predicted by C&B model always liesbelow the S&D model, as illustrated in Fig. 1. It is, therefore believedby the authors that these two models provide the upper and lowerlimiting responses of the real CBM permeability changes as a functionof reservoir pressure. The actual permeability behavior should liewithinthese two limits.
1
10
100
k/k
0
Reservoir Pressure, psi
S&D ModelC&B Model
p0
Discrepancy ZoneDiscrepancy ZoneDiscrepancy Zone
Fig. 1. Illustration of permeability behavior using S&D model and C&B model.
3. Coupling Liu and Harpalani sorption-induced strain andpermeability models
3.1. Sorption-induced strain
Liu and Harpalani (in press) presented a new theoretical modelfor the sorption-induced volumetric strain. The model includes twocompeting components, first the sorption-induced strain and secondthe mechanical compression term shown in Eq. (5). For P&M, S&Dand C&Bmodels, the desorption-inducedmatrix shrinkage is analogouswith thermal shrinkage and is quantified by an empirical method. Thetheoretical matrix shrinkage model, previously proposed by Liu andHarpalani (in press), is applied in this paper. In order to couple thismodel with different permeability models, the stress-dependent-strain component must be dropped for two reasons. First, the mechan-ical compression of coal is not analogous to thermal shrinkage sincethe two mechanisms are entirely different. Second, the permeabilitymodels already take this component into account and, by includingthis, it would effectively be counted twice. Therefore, according toEq. (5), the sorption-induced strain alone for different pressures canbe given as:
Δε ¼ 3VLρsRTEAV0
∫p2
p1
1PL þ P
dp ð20Þ
where,Δε is the sorption-induced incremental volumetric strain, p1 andp2 are reservoir pressures at conditions one and two. This is the keyequation to integrate the shrinkage term into different permeabilitymodels.
3.2. Coupling of Liu & Harpalani and P&M models
Clarkson et al. (2010) gave a general form of the P&Mmodel equa-tion as:
kk0
¼ ϕϕ0
� �3¼ 1þ Cm
ϕ0p� p0ð Þ þ 1
ϕ0
KM
� 1� �
Δε� �3
ð21Þ
Clarkson et al. (2010) first applied the Pan and Connell shrinkagestrain model, presented in Eq. (3), to replace the shrinkage term (Δε).Unfortunately, the stress dependency term in Eq. (3) was not dropped.This may very well be the reason for the authors using somewhat unre-alistic values of input parameters in order to obtain a good match withfield data. For example, the values of Poisson's ratio used were 0.484and 0.43, which are relatively high for coal (Bell and Jones, 1989).Furthermore, the initial cleat porosity was matched to be 0.03%, whichis somewhat low according to previously published values, but waslater increased to 0.125% by adjusting the sorption-induced strainparameters and changing the initial condition.
Liu and Harpalani (in press) model for estimation of matrix strainand P&M permeability model were coupled by substituting Eq. (20)for the Δε term in Eq. (21). The coupled model does not require theLangmuir-type fit of the strain data to calculate the matrix shrinkageterm.
3.3. Coupling of Liu & Harpalani and S&D models
Using Eqs. (12) and (13), S&D model can be rewritten as the fol-lowing equation using incremental matrix shrinkage strain (Δε):
kk0
¼ exp �3Cf � ν1� ν
p� p0ð Þ þ E3 1� νð ÞΔε
� �� �ð22Þ
By inserting Eq. (20) into (22), the coupled model of Liu &Harpalani and S&D is obtained.
0
5
10
15
20
0 1 2 3 4 5 6 7
k/k @
6.5
MP
a
Reservoir Pressure, MPa
Fig. 2. Permeability ratio obtained from field data (after Clarkson et al., 2008). (Basepermeability is taken to be the value at reservoir pressure of ~6.5 MPa).
6 S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
3.4. Coupling of Liu & Harpalani and C&B models
Similar to S&Dmodel, the permeability variation in C&Bmodel canbe rewritten as:
kk0
¼ exp �3Cf � 1þ νð Þ3 1� νð Þ p� p0ð Þ þ 2E
9 1� νð ÞΔε� �� �
ð23Þ
The coupled model of Liu & Harpalani and C&B is formulated bysubstituting Eq. (20) for the Δε term in Eq. (23).
3.5. Coupling of Liu & Harpalani and Ma et al. models
Ma et al. model can be rearranged as the following equation usingthe incremental matrix shrinkage matrix strain (Δε):
Substituting Eq. (20) for the Δε term in Eq. (24), the Liu &Harpalani and Ma et al. coupled model is proposed.
4. Validation of coupled models
4.1. Field data validation — Case 1
The San Juan basin, located in the northwestern New Mexico andsouthwestern Colorado, is one of the most mature gas producingareas in the world. Extensive investigations have been carried outto understand the flow behavior not only for primary productionalso for the CO2-enhanced CBM operations (Koperna et al., 2009;Oudinot et al., 2011; Reeves et al., 2003). For the primary gas deple-tion, the formation permeability increase is well documented in theliterature (Clarkson and McGovern, 2003; Clarkson et al., 2008;Gierhart et al., 2007). Based on the historical CO2 injectivity profilefor the two pilot sites, Allison Unit and Pump Canyon, an overall per-meability decrease was observed in the field and confirmed throughreservoir simulation study (Oudinot et al., 2011). The shrinkage and
Table 1Input parameters for sorption-induced strain model (Liu and Harpalani model).
Input parameters for Liu and Harpalani model
ρ(t/m3) R(MPa*m3/kmol*K) T(K) EA(MPa)
1.4 0.0083143 315 1600
swelling of the coal matrix plays an important role for both primaryand enhanced CBM operations. In this current study, we focused onthe primary production induced permeability prediction.
In order to validate the coupled models presented, the formationpermeability data is essential. Fortunately, several investigationshave been carried out (Clarkson and McGovern, 2003; Clarkson etal., 2008; Gierhart et al., 2007) in order to quantify and predict thepermeability variation during the primary depletion in San Juanbasin. Among these studies, Clarkson et al. (2010) is perhaps the mostrespected work, using the production data analysis (PDA) method.This method uses production rate, reservoir pressure and flowing pres-sure data to solve for the effective permeability to gas as a function ofreservoir pressure, using the pseudo-steady-state radial flow equationfrom Darcy's Law, as follows:
kg ¼qgT ln re=rwð Þ−0:75þ sþ Dqg
h i7:03x10−4h m pRð Þ−m pwf
h i ð25Þ
where, kg is the effective permeability to gas, qg the gas surface produc-tion rate, T is the temperature, re and rw are thewell drainage radius andwellbore radius respectively, s is the skin factor, D is the inertial orturbulent flow factor, h is reservoir formation thickness, m(p) is thepseudopressure, pR and pwf are volumetric average reservoir andflowing bottomhole pressures respectively. Applying this for a SanJuan basin well, a kg plot was constructed, as shown in Fig. 2, clearlyshowing an increasing permeability trend with declining reservoirpressure.
In the present work, Liu and Harpalani model, combined with theP&M permeability model, was used first to match the field data forone well in the San Juan basin. The required input parameters forLiu and Harpalani (in press) model are consistent with the valuesmeasured/estimated by the modelers, as listed in Table 1. For P&Mmodel, the required input parameters are presented in Table 2 forthe San Juan basin coal. The Young's modulus (E) and Poisson'sratio (ν) are fixed at 2200 MPa (330,000 psi) and 0.3, based on thevalues recommended by Palmer et al. (2007). The grain compressibil-ity value used is 1.88E-04/MPa (1.36E-06/psi), obtained from our lab-oratory measurements. E/EA is set to be 2.4, as confirmed in previouswork (Liu and Harpalani, in press). The value of g used by the modeler(Palmer, 2010) is 0.3. The value of f is selected as 0.7, which is withinthe range (0.5–0.8) recommended by Palmer (2010). The other vari-able was the initial cleat porosity. Based on the matching effort, asshown in Fig. 3, the initial cleat porosity is matched to be 0.3%,which lies within the reasonable range (0.1% to 0.5%) provided bythe modelers (Palmer and Mansoori, 1998). In summary, the com-bined Liu & Harpalani and P&M model can successfully predict per-meability increase in the San Juan basin using reasonable values forall input parameters.
Next, the combined Liu & Harpalani model and Ma et al. modelwas used to match the same set of field data. The matching resultsare shown in Fig. 4. There is excellent agreement between the fielddata and modeled results. The required input parameters for thiscombined model are listed in Table 3. The Young's modulus, Poisson'sratio and adsorption data are the same as those used for the P&Mmodel. The only parameter that is different is the initial cleat porosity,a value of 0.5%, which is also reasonable. The advantage of this modelis that the controversial difficult parameters, ‘f’, ‘g’ and Cf are not
V0(m3/kmol) VL(m3/t) PL(MPa) E/EA
22.4 19.1 0.25 2.4
Table 2Input parameters for P&M permeability model.
Input parameters for P&M model
ϕ0 E (MPa) ν γ (MPa−1) g f K (MPa) M (MPa)
0.3% 2200 0.3 0.000188 0.3 0.7 1746 2883
5
10
15
20
Field Data
Liu & Harpalani and Ma et al. Modeled Results
E = 2200 MPa
0 = 0.5% ν ν = 0.3 φ
k/k @
6.5
MP
a
7S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
required to get a good match. In other word, in this combined model,every parameter can be either estimated or measured easily and has aphysical meaning that is easy to relate to. Moreover, Liu & Harpalaniand Ma et al. combined model is more transparent and easy tounderstand.
00 1 2 3 4 5 6 7
Reservoir Pressure, MPa
Fig. 4. History match for field data using Liu & Harpalani and Ma et al. models.
4.2. Field data validation — Case 2
Shi and Durucan (2005) reported a set of field permeability data(after McGovern, 2004) for a San Juan basin well. In their study, theauthors achieved a closematch to the pressure dependent permeabilityincrease with depletion by artificially scaling down the cleat compress-ibility from 0.14 MPa−1 to 0.091 MPa−1, a reduction of 33%, as the res-ervoir pressure is reduced from 2.1 MPa to near zero pressure. Themodelers provided no justification for doing so other than to obtain agood match. They repeated the history matching exercise (2009),where the cleat compressibility was assumed to be a stress-dependentparameter by directly applying McKee et al.'s work (1988). The initialcleat compressibility was matched in the range of 0.116 MPa−1 to0.594 MPa−1.
In this study, we used the Liu and Harpalani model, combinedwith the S&D model, to match this set of field data, as well as Liuand Harpalani and C&B models. As in any history matching exercise,it is important to use reservoir properties that are representative ofthe field under study. We used the same mechanical properties,Young's modulus (E = 2900 MPa) and Poisson's ratio (ν = 0.35),used by Shi and Durucan (2004, 2005). The adsorption constantsare the same for San Juan basin, as listed in Table 1. For combinedLiu and Harpalani and S&D models, constant cleat compressibility ismatched to be 0.075 MPa−1, solely to obtain a good match, asshown in Fig. 5. For comparison, Liu and Harpalani and C&B modeledresults are also included in Fig. 5, using the exactly same input param-eters as for S&D model. It can be seen that Liu and Harpalani and S&Dmodels predict a much higher permeability increase than the coupledLiu and Harpalani and C&B models.
In order to improve the Liu and Harpalani and C&B model results,another history matching exercise was carried out using the sameYoung's modulus (E = 2900 MPa) and Poisson's ratio (ν = 0.35). The
0
5
10
15
20
0 1 2 3 4 5 6 7Reservoir Pressure, MPa
Field Data
Liu & Harpalani and P&M Modeled Results
E = 2200 MPa
g = 0.3f =0.7
0 = 0.3%
ν ν = 0.3 γ γ = 1.88E-04/MPa
φ
k/k @
6.5
MP
a
Fig. 3. History match for field data using Liu & Harpalani and P&M models.
cleat compressibility was matched to be 0.15 MPa−1, twice the valueused for the S&D model. The history matching results are illustrated inFig. 6. There is good agreement between the field data and modeledresults.
The authors would like to point out that the stress-based models,S&D model and C&B, rely heavily on the availability of an accurateestimate for the cleat compressibility. Furthermore, the two modelscan obtain fairly good match with the assumption of constant cleatcompressibility throughout the life of a producing reservoir althoughthe two values are very different. Based on the results presented, it isdifficult to conclude which of the two models is superior unless a de-pendable estimate for the value of cleat compressibility is available,along with a reasonable justification that its value remains constant.In fact, calling one better than the other would not be fair.
Finally, an effort was made to validate the Liu and Harpalanimodel and Ma et al. model with this set of field data. The results areshown in Fig. 7. The mechanical properties of the reservoir are thesame as those used for the S&D/C&B models (E = 2900 MPa andν = 0.35). The initial fracture porosity was matched to be 0.6%,which is reasonable for CBM reservoirs. The history matching resultsindicate that there is excellent agreement between the observed fielddata and modeled results.
5. Discussion
Several theoretical or empirical models have been proposed to de-scribe the complex relationship between permeability and variationsin stress, pore pressure and continued sorption of gas. Two groups ofanalytical permeability models have been discussed in this paper, thatis, strain-based and stress-based. Every model includes the effect ofmatrix shrinkage, but different mechanisms, to arrive at the variationin permeability. In the strain-based models, P&M and Ma et al.models, the matrix shrinkage directly affects the cleat/fracture poros-ity, a key factor impacting the permeability. In the stress-basedmodel, including both S&D and C&B, the matrix shrinkage effect firstinduces a change in the stress condition due to the strain–stress rela-tionship, and the change in permeability follows, controlled by thestress changes through cleat compressibility. Due to the uncertainty
Table 3Input parameters for Ma et al. permeability model — Case 1.
Input parameters for Ma et al. model
ϕ0 E (MPa) ν
0.5% 2200 0.3
0
2
4
6
8
10
0 1 2 3 4 5 6Reservoir Pressure, MPa
Field DataLiu & Harpalani and S&D Modeled ResultsLiu & Harpalani and C&B Modeled Results
E = 2900 MPa= 0.35
Cf = 0.075/MPa ν ν
k/k @
5.5
MP
a
Fig. 5. History match for field data using Liu & Harpalani and S&D and Liu & Harpalaniand C&B models.
0
2
4
6
8
10
0 1 2 3 4 5 6Reservoir Pressure, MPa
Field Data
Liu & Harpalani and Ma et al. Modeled Results
E = 2900 MPa= 0.35
0 = 0.6% ν ν φ
k/k @
5.5
MP
a
Fig. 7. History match for field data using Liu & Harpalani and Ma et al. models — Case 2.
8 S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
about the value and constancy of cleat compressibility, we preferstrain-based models because these are more transparent and easy tounderstand. However, once the cleat compressibility is estimatedaccurately, either in the laboratory or field, the stress-based modelsmay end up gaining wide application in numerical simulation.
Apart from one exception (Clarkson et al., 2010), the sorption-induced shrinkage strain is empirically incorporated in the perme-ability models. In this paper, the sorption-induced strain has beenseparated from the mechanical compression strain, and only thesorption-induced strain is used in the different permeability modelssince the mechanical compression effect is considered in thesemodels independently. We have incorporated the Liu and Harpalanistrain model into the different permeability prediction models as afunction of stress/pressure and desorption, including P&M, Ma et al.,S&D and C&B models. Our present analysis indicates that the Liuand Harpalani sorption-induced strain model can be easily incorpo-rated into the various permeability models. Furthermore, for strain-based models, primarily the P&M and Ma et al., coupling the modelsis an effective approach to predict permeability increase in the SanJuan basin using the primary depletion technique. All of the inputparameters are in reasonable and realistic ranges. For the stress-basedmodels, good matches can be obtained by varying the cleat compress-ibility. Further studies are required to determine which one is actually“better”, rather works well. Finally, coupling of Liu and Harpalani andMa et al. models provides a valid tool to predict the permeability forthe field data.
0
2
4
6
8
10
0 1 2 3 4 5 6Reservoir Pressure, psi
Field DataLiu & Harpalani and C&B Modeled Results
E = 2900 MPa= 0.35
Cf = 0.15/MPa ν ν k/
k @5.
5 M
Pa
Fig. 6. History match for field data using Liu & Harpalani and C&B models.
Of additional interest is the comparison of P&M and Ma et al.models. The P&Mmodel is derived based on the uniaxial strain condi-tion whereas the Ma et al. model is based on the constant reservoirvolume assumption, both strain-based models. Ma et al. boasts the“friendly” features of the model as its transparency, good historymatching with field data, simplicity and ability to measure everyparameter required in the model. Obviously, this paper suggests thatthe Ma et al. model, combined with Liu and Harpalani strain model, ismore “friendly” for bothfield data sets. The coupledmodel only requiresbasic mechanical properties and adsorption constants to fit the fielddata. The P&M model is, however, more complex than the Ma et al.model, and it obviously has the potential of handling complicated per-meability problems. If a model user is interested primarily in obtainingan initial permeability trend, we strongly recommend the combined Liuand Harpalani andMa et al. models. On the other hand, P&Mmodel canbe used to obtain a more detailed permeability profiles since it takesreservoir geometry into account. However, it should be noted that theuncertainties associated with the model must be treated properly. Forexample, appropriate and realistic values of ‘g’, ‘f’ and initial cleat poros-ity must be obtained based on solid and justifiable data.
6. Summary
Coal formation permeability is complex in that it is influenced bythe stress and is affected by coal shrinkage with gas desorption andswelling with adsorption. Understanding coal permeability is criticalin order to reliably predict gas production or consider other reservoirgas migration issues. Since the sorption-induced strain plays animportant role in the variation of permeability, the strain theoreticalmodel should be incorporated into the permeability prediction models.An important aspect of this paper is that theoretical sorption-inducedstrain model (Liu and Harpalani) is coupled with different analyticalpermeabilitymodels and the coupledmodels are validated bymatchingtwo different sets of observed field data. This coupling process signifi-cantly reduces the experimental work necessary to estimate the volu-metric strain due to sorption. Based on the work discussed in thispaper, a few important conclusions can bemade. These are summarizedbelow:
(1) Liu and Harpalani (in press) sorption-induced strain model hasbeen successfully and easily incorporated into the differentpermeability models.
(2) P&M and Ma et al. models, coupled with Liu and Harpalanimodel, are effective methods to predict permeability increasein the San Juan basin CBM wells operated using primary
9S. Liu, S. Harpalani / International Journal of Coal Geology 113 (2013) 1–10
recovery. Moreover, all input parameters are within reasonableranges.
(3) S&D and C&Bmodels, combinedwith Liumodel, can be validatedby varying the cleat compressibility for a set of field datareported by Shi and Durucan (2005).
(4) The coupled model of Liu and Harpalani and Ma et al. model isvalid for both field cases with fewer input parameters and rea-sonable initial porosities.
(5) Themost important implication is the combining of the sorption-induced strain and different permeability models. This can be-come a powerful tool to predict the permeability in the earlystage of any new CBM plays. It only requires the basic adsorptiondata andmechanical properties, which are either available or rel-atively easy to estimate from laboratory during the early stagesof CBM development.
Nomenclatureε coal matrix strain, dimensionlessεl, Pε parameters of Langmuir match to volumetric strain as a
result of matrix shrinkagep reservoir pressure, MPaεg volumetric strain coefficient associated with gas sorption,
Es coal solid phase Young's modulus, MPaf(x,νs) structure model parameter, dimensionlessνs coal solid phase Poisson's ratio, dimensionlessP solid phase stress, MPac pore structure model constant (1.2), dimensionlessx ratio a/l, dimensionlessR universal gas constant, MPa*m3/(kmol*K)T temperature, K (Liu et al. model)EA modulus of the solid expansion, MPaV0 gas molar volume (22.4), m3/kmolE Young's modulus, MPaν Poisson's ratio, dimensionlessϕ cleat porosity, dimensionlessϕ0 initial cleat porosity, dimensionlessCm P&M model specified term, MPap0 initial reservoir pressure, MPa−1
K bulk modulus, MPaM constrained axial modulus, MPag pressure-strain suppression factor (Palmer), dimensionlessf a fraction between zero and one (Palmer), dimensionlessγ grain compressibility, MPa−1
k permeability at pressure p, mdk0 virgin permeability, mdσ effective horizontal stress at pressure p, MPaσ0 effective horizontal stress at pressure p0, MPaCf cleat compressibility, MPa−1
εv sorptioin induced strain, dimensionlessεvo initial volumetric strain value, dimensionlessΔaa horizontal strain for a single matchstick, dimensionlessΔε incremental matrix volumetric strain, dimensionlessp1 & p2 reservoir pressures at condition one and condition twoD inertial or turbulent flow factor, D/Mscfh formation thickness, ftkg effective permeability to gas, mdm(p) pseudopressure, psi2/cppR volumetric average reservoir pressure, psiapwf flowing bottomhole pressure, psiaqg gas surface flow rate, Mscf/Dre drainage radius, ft
The authors wish to thank Dr. Christopher Clarkson for providingthe field data for model validation.
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