Fuel 94 (2012) 110–116

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Fuel

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Laboratory measurement and modeling of coal permeability with continuedmethane production: Part 1 – Laboratory results

Abhijit Mitra, Satya Harpalani ⇑, Shimin LiuSouthern Illinois University, Carbondale, IL 62901, USA

a r t i c l e i n f o

Article history:Received 16 February 2011Received in revised form 26 September 2011Accepted 20 October 2011Available online 7 November 2011

Keywords:Coalbed methanePermeabilityUniaxial strainHorizontal stress

0016-2361/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.fuel.2011.10.052

⇑ Corresponding author. Tel.: +1 6184537918; fax:E-mail address: satya@engr.siu.edu (S. Harpalani).

a b s t r a c t

This paper, first of a two-part series, discusses the results of a laboratory-scale study completed toestablish the permeability variation trend with continued production of methane from coal-gas reser-voirs. The field condition of uniaxial strain, assumed in the analytical models developed for permeabilityprediction, was replicated in the study. The results showed that the permeability of coal increases con-tinuously, the rate of increase accelerating at low pressures. The primary reason for the increase appearsto be the decrease in horizontal stress resulting from the sorption-induced volumetric strain, the so called‘‘matrix shrinkage’’ effect. In the second part, experimental data is used to validate the commonly usedpermeability prediction analytical models and present a modification for one to improve its ability topredict permeability changes.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Increase in coalbed methane (CBM) production in the US hasbeen truly significant since the early eighties, from near zero in1980 to almost two trillion cu ft (TCF) in 2009. Interest in this re-source continues to grow worldwide, with significant activity inAustralia, Canada, China and India. With increased experience withCBM production, it has become abundantly clear that the perme-ability of coal varies with continued production. The most dramaticexamples of this are several producing reservoirs in the San JuanBasin, with permeability increases of as much as 100 times.

During drawdown of a reservoir by primary production, effec-tive stress is believed to increase, resulting in permeability reduc-tion due to the closure of cleats. However, methane is stored in coalas sorbed gas and production leads to desorption of gas. This isaccompanied by ‘‘matrix shrinkage’’, which is believed to openup the cleats, thus leading to increased permeability. In order topredict the overall changes in permeability with depletion andproject long-term gas production, several models have been devel-oped taking into account these two effects. Effort to validate thesemodels using production data has only been partially successfuland has required ‘‘tweaking’’ of the input parameters, somewhatdefeating the purpose of modeling. Since the models are basedon fundamental principles of poro-elasticity and geomechanics,and their application is independent of size, laboratory deriveddata has also been used for model validation. This has had onlylimited success, primarily because the experimental conditions

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+1 6184537455.

did not replicate the underlying principles and assumptions ofthe models properly, raising serious questions about the value ofthe validation exercise.

This paper, first part of a two-part series, presents the results ofa study, where permeability variation of core of coal taken from theSan Juan basin was established as a function of decreasing pres-sure. The experimental conditions not only best replicated in situconditions but were also identical to the founding principles usedfor development of the recent theoretical models. This is the firstreported experimental study where flow measurements weremade while coal was held under uniaxial strain condition, that is,the sample was not permitted to physically shrink as a result ofdesorption, just like it does not under in situ condition due to lat-eral confinement. Instead, the horizontal stress was adjusted whenthe core started to shrink, ensuring zero horizontal strain. As a sep-arate effort, volumetric strain associated with sorption of gas wasmeasured to obtain appropriate parameters for modeling. Thesecond part of this effort includes application of two models mostcommonly used in the San Juan basin to predict permeabilitychanges and comparison of the experimental and modeled results,and concludes by providing an insight to the strengths andweaknesses of the models.

2. Background

2.1. Structure of coal

Coal is generally characterized as a dual porosity rock, contain-ing both macropore and micropore systems, as shown in Fig. 1a.

A. Mitra et al. / Fuel 94 (2012) 110–116 111

The macropore system consists of a naturally occurring network offractures called cleats, serving as the primary pathways for gastransport. The microporosity of coal is within the coal matrixblocks, surrounded and separated by cleats, consisting of largenumber of interconnected pores that serve as the storehouse formethane in adsorbed form. For the purpose of gas flow modelingin CBM reservoirs, coal structure is best described by a bundle ofmatchstick model, as shown in Fig. 1b [1], with each matchstickrepresenting a block of coal matrix, whereas the cleats are repre-sented by the void space between the matchsticks.

2.2. Coal permeability

With continued gas production from a CBM reservoir, the effec-tive stress is believed to increase. The phenomenon of stress-dependent permeability has been studied and reported by severalresearchers [2–6]. Typically, an exponential decrease in permeabil-ity with increasing stress has been reported. Along with increasedeffective stress, the sorption-induced volumetric strain of coalmatrix is believed to result in increased permeability. Matrixshrinkage is a universal phenomenon and all coals shrink whenreleasing sorptive gas [7–10]. The associated increase in coalpermeability was first hypothesized by Gray [11], which was firstverified experimentally by Harpalani and Schraufnagel [10], andconfirmed in a subsequent study, clearly demonstrating an overallincrease in permeability [12]. It was also shown that the sorption-induced volumetric strain is a non-linear function of pressure [13],

(a)

(c)

Butt Cleat

Fac

e C

leat

Matrix Block Containing Micropores

Non-zero strain

Zero st

Fig. 1. Physical structure, model of co

later confirmed by Levine [14]. Furthermore, field measurementsof permeability variation with continued production in theSan Juan basin have shown increases of orders of magnitude insome CBM reservoirs [15–18]. The trend of this increase, however,has been reported to be either continuous or L-shaped where therate of increase is low initially, followed by a sharp increase atlower pressures. Finally, shrinkage of coal matrix with desorptionis also believed to reduce the effective horizontal stress, openingup the cleats further and thus increasing the permeabilitysignificantly [16].

2.3. Previous laboratory studies

All previous studies completed in the laboratory used stress-controlled conditions, where the sample was allowed to deformaxially as well as laterally. The first laboratory study reportingthe effect of matrix shrinkage on permeability increase was thatof Harpalani and Schraufnagel [10]. They reported an initialdecrease in methane permeability with decreasing gas pressureup to a point. However, with further reduction in gas pressure,permeability started to increase. Based on this, they concluded thatpermeability increases only after the rate of desorption becomessignificant. This was later confirmed by measuring helium perme-ability that showed a continuous decrease with decreasing gaspressure [10] since helium is non-sorptive. In a separate study,Harpalani and Chen [12] concluded that the change in permeabilityassociated with matrix shrinkage was linearly proportional to the

(b)

rain

al and concept of uniaxial strain.

112 A. Mitra et al. / Fuel 94 (2012) 110–116

volumetric strain. Since volumetric strain has been shown to belinearly proportional to the amount of gas desorbed [12], changein permeability should be a linear function of the desorbed gasvolume.

Robertson [19] reported an initial decrease in permeability, fol-lowed by an increase for pressure reduction from 5.5 to 0.7 MPa(800–100 psi). This suggested the stress effect to be dominant athigher pressure, causing a decrease in permeability. However, withfurther decrease in pressure, matrix shrinkage effect became morepronounced, resulting in an increase in permeability. It was alsoshown that, for core from Powder River basin, the effect of matrixshrinkage was small as permeability did not increase significantly,perhaps because of the very high initial permeability typicallyencountered in the basin.

2.4. Permeability models and concept of uniaxial strain

In order to predict the overall changes in permeability withdepletion, several models have been developed in the last ten years[18,20–24]. A brief summary of the various models has been pre-sented in Part II of this research effort. Of the existing models,the two most commonly used are the ones developed by Palmerand Mansoori [18] and Shi and Durucan [21]. Both models arebased on fundamental principles of geomechanics and shrinkagebehavior of coal, bundle of matchsticks geometry, and most impor-tantly, the assumption of uniaxial strain condition, that is, there iszero lateral strain with continued production.

The uniaxial deformation of a depleting reservoir was first pro-posed by Geertsma [25] who hypothesized that, with continuedproduction, an oil reservoir having a high lateral dimension com-pared to vertical dimension deforms mainly in the vertical direc-tion. In 1998, this was extended to CBM reservoirs as well sincethe two geometries are quite similar [18]. Uniaxial strain conditionwas subsequently described as a typical state of reservoirs atdepth, where changes in stress or pore pressure can induce strainonly in vertical direction since the reservoir is confined laterally[18]. This is illustrated in Fig. 1c for laboratory conditions, wherethe core would experience non-zero strain in the vertical/axialdirection and zero strain in the horizontal direction. The shrinkageof core as a result of gas desorption is not allowed when measuringthe pressure-dependent permeability in the laboratory; instead,the horizontal stress is reduced to ensure zero horizontal strain.This is a major deviation from all permeability experimentalstudies reported to date since the flow has always been measuredunder hydrostatic/triaxial state of stress, allowing sample defor-mation in lateral direction. The current study is, therefore, the firstof its kind where permeability of core of coal was measured underuniaxial strain condition although the condition has been used atleast once in the past to estimate pore volume compressibility [26].

The current study was thus undertaken not only to improve thecurrent understanding of the permeability trends observed in CBMoperations, but also provide experimental data for conditionsassumed by the modelers in order to validate the models. Usingthe measured data presented in this paper, validation of the exist-ing models is presented in the second part of the research effort.

3. Experimental work

3.1. Experimental principle

Replicating uniaxial strain condition when measuring flow insoils and oil-bearing rocks was first achieved by placing a sand-stone core inside a metal cylinder, followed by applying stressand/or pore pressure [27,28]. However, desorption of coal leadsto matrix shrinkage and thus a reduction in horizontal dimension,

which can not be restricted in this manner. A viable alternative wasto reduce the horizontal stress in order to keep the lateral dimen-sion constant, as practiced for sandstone cores [29]. This necessi-tated using a technique to monitor the horizontal strain causeddue to matrix shrinkage, which was achieved by employing a cir-cumferential extensometer to measure the changes in the core cir-cumference. Since the overburden depth remains constant duringproduction, vertical stress was maintained constant throughoutthe experimental duration.

3.2. Sample collection and preparation

Core of coal was obtained from a well in the northern San JuanBasin and preserved in an environmental chamber in humid condi-tion. Prior to starting the experiment, the two core ends were cutand split into three parts for measurement of sorption- inducedmatrix strain with exposure to helium and methane. The remain-ing core dimensions were then reduced to enable sealing it inthe triaxial cell for flow measurement. The finished core was againplaced in the environmental chamber to achieve moistureequilibrium.

3.3. Experimental setup and procedure

In order to replicate the conditions in situ, controlling and mon-itoring external stress conditions and gas pressure are critical. Theexperimental setup for permeability measurement, therefore, in-cluded independent control of stresses, gas pressures upstreamand downstream, and a means for measurement of gas flowrate.The setup consisted of a triaxial cell, a circumferential extensome-ter to monitor and control the shrinkage of core and a loading sys-tem. The temperature of the entire setup was kept constant. Aschematic of the experimental setup is shown in Fig. 2.

The testing procedure started with stressing the core verticallyand horizontally to replicate the mechanical conditions in situ. Thepore pressure was then applied by injecting methane. The horizon-tal strain due to mechanical loading and methane adsorption wasmonitored by the circumferential extensometer. After achievingthe desired pore pressure level (representative of reservoir pres-sure prior to commencement of production), the sample was al-lowed to equilibrate. Methane flow was then established throughthe core by applying and maintaining a small pressure gradientacross the two ends. Using the measured flowrate under equilib-rium conditions, permeability was estimated. The procedure wasrepeated for a step-wise decrease in gas pressure, maintaining zerolateral strain by reducing the horizontal stress appropriately foreach step until the pore pressure reduced to approximately0.3 MPa (50 psi).

3.4. Matrix strain experimental setup and procedure

Details of the experimental setup and procedure for measure-ment of matrix strain are given in prior publications [30,31]. Thesetup for measurement of matrix shrinkage/swelling consisted ofhigh pressure vessels, a pressure monitoring and recording systemfor each vessel, and a data acquisition system (DAS) to measureand record the strain. The algebraic sum of the three linear strains(ex, ey, ez) gave the volumetric strain for the sample for a step-wiseincrease in gas pressure, first using helium and then methane.

4. Results and discussion

Three cores were tested, one from the San Juan basin and theother two from Illinois basin. However, results obtained for the corefrom San Juan basin are presented here since the emphasis of the

Circumferential Extensometer

Triaxial Cell

Fixed Volume Cylinder Water Bath

Heat-Shrink Tubing

Gas Cylinder

Heating Blanket

Hydraulic Pump

Perforated Disc

To Atmosphere

To Atmosphere

DAS

Vertical Stress Horizontal

Hydraulic Line Gas Flow

Gas Line

2-Way Valve

3-Way Valve

Relief Valve

Pressure

Fig. 2. Permeability experimental setup.

0

5

10

15

20

0 1 2 3 4 5 6 7

Pore Pressure (P), MPa

k/k 0

Fig. 3. Laboratory results showing permeability-pore pressure relationship.

A. Mitra et al. / Fuel 94 (2012) 110–116 113

study was producing reservoirs in this basin. The sample was firststressed vertically and horizontally at 14.5 MPa (2100 psi) and9.7 MPa (1400 psi) respectively, representative of stresses in situ.Following this, the sample was saturated with methane at6.2 MPa (900 psi), the initial reservoir pressure. The circumferentialstrain resulting from mechanical loading, followed by methaneadsorption, was monitored using the circumferential extensometer.After methane injection, the sample was allowed to equilibrate forapproximately fifteen days. This was necessary since achievementof sorption-induced strain equilibrium for a large sample requirestime. At the end of fifteen days, the circumferential extensometerindicated strain equilibrium, with no further lateral strain. Follow-ing this, flowrate measurement was carried out and permeabilitywas estimated at 6.2 MPa (900 psi). The pore pressure within thesample was then reduced in steps, down to 0.3 MPa (50 psi). Foreach step, the horizontal stress was reduced to ensure zero horizon-tal strain since desorption resulted in shrinkage of the samplerequiring lateral relaxation. Also, the sample was allowed to attainstrain and pressure equilibrium at each step prior to flowrate mea-surement and proceeding to the next step.

Since the objective of the experimental work was to estimatethe variation in permeability rather than determining the absolutevalues, the ratio between permeability estimated at each pressurestep to that of initial permeability, that is, permeability estimatedat 6.2 MPa (900 psi), was calculated. This ratio was then plotted asa function of pressure, which is shown in Fig. 3. The results exhibitseveral features. First, there is an overall increase in permeabilitywith decrease in pore pressure. Second, the trend shown is not inagreement with the results of any previous laboratory studies car-ried out for stress-controlled conditions [12,19]. Third, there is nopermeability rebound, as predicted by various theoretical models[18,19]. Fourth, the rate of increase is not uniform, with a small in-crease between 6.2 and 2.75 MPa (900–400 psi), which becamesignificant at lower pressures. Finally, the pressure-permeabilitytrend is in agreement with field observations made in different

parts of the San Juan basin [32] although the magnitude of increasemeasured in the laboratory is smaller. This is expected due to thelarge drainage area and the large number of cleats/fractures pres-ent in the field.

The results of the matrix strain induced by desorption of methaneare shown in Fig. 4. The sample was flooded with methane inincreasing steps of 1.4 MPa (200 psi), up to a final pressure of7 MPa (1000 psi). At the end of each injection step, linear strain atthree orthogonal directions was measured and added to estimatevolumetric strain [33]. The sorption-induced strain data was thenmodeled using a model similar to the Langmuir isotherm:

e ¼ emax �P

ðP þ PeÞð1Þ

where, e is the sorption-induced volumetric strain at pressure P,emax is the maximum strain that can be achieved at infinite pressureand Pe is the pressure at which coal attains 50% of the maximum

0

0.002

0.004

0.006

0.008

0.01

0 1 2 3 4 5 6 7

Gas Pressure, MPa

Vol

umet

ric

Stra

in,

ε

Sample 1 - Lab DataSample 2 - Lab Data

ε = 0.01075P/(P+4.05) ε = 0.01075P/(P+4.26)Modeled

Fig. 4. Laboratory results showing sorption-induced volumetric strain.

y = 1.57x

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7

Pore Pressure (P), MPa

Hor

izon

tal S

tres

s ( σ

h), M

Pa

Fig. 6. Laboratory measured variation in applied horizontal stress with porepressure.

114 A. Mitra et al. / Fuel 94 (2012) 110–116

strain. The values of Pe and emax were estimated to be 4.16 MPa(603 psi) and 0.01075 respectively. These were employed tocalculate the volumetric strain at each pressure step used in thepermeability experiment. A ratio of the measured sorption-inducedstrain with respect to that at initial pore pressure was then esti-mated and plotted along with the measured variation in permeabil-ity. This is shown in Fig. 5. It is evident from the figure that thevolumetric strain as a function of pressure is a ‘‘mirror image’’ ofthe pressure-permeability trend since both plots show that thevariation is small in the high pressure range and becomes signifi-cant only below a certain pressure. A direct relationship betweenthe two is apparent, which is in agreement with the results reportedby other researchers in the past [12], and explains the permeabilityincrease at low pressure.

Apart from pore pressure, the second parameter that varies dur-ing depletion, and influences permeability, is the horizontal stress.To date, several experimental [2–6] and numerical studies [21,23]have been carried out to enhance the understanding of the stress-permeability relationship. However, none of these studies wereaimed to investigate the in situ changes in horizontal stress as afunction of depletion. In order to do this, the variation in appliedhorizontal stress was first plotted against pore pressure, as shownin Fig. 6. The figure shows that the horizontal stress decreases lin-early with pore pressure, and is given as:

y ¼ 1:57x ð2Þ

where the y-axis intercept is 0.3 MPa, indicating that there is almosta complete loss of applied horizontal stress at zero pore pressure.The gradient of the straight line, 1.57, is the rate of change of ap-plied horizontal stress required to maintain uniaxial strain condi-tion with changes in pore pressure. The fact that the gradient isgreater than 1 indicates that there is a larger reduction in applied

0

5

10

15

20

0 1 2 3 4 5 6 7

Pore Pressure (P), MPa

k/k0

0.00

0.25

0.50

0.75

1.00

1.25

1.50

ε s/ ε

s0

PermeabilityVolumetric Strain Ratio

Fig. 5. Variation in laboratory measured permeability and volumetric strain.

horizontal stress than the corresponding reduction in pore pressure.The reason for this phenomenon is the sorbing nature of methanecausing volumetric strain of the coal matrix. During methanedesorption, in order to prevent the coal bulk from shrinking in thehorizontal direction, the applied horizontal stress had to be reducedand this reduction was larger than the pore pressure reduction.

For any porous media saturated with fluid, the total stress isborne in part by the granular skeleton of the porous media, andin part by the fluid saturating the pore space [33,34]. The term‘‘effective stress’’ relates the total stress and pore pressure as givenbelow:

re ¼ r� ap ð3Þ

where, r and re are total and effective stresses, respectively, p ispore pressure and a is a constant, termed Biot’s coefficient [35].Biot’s coefficient has been reported to be less than or equal to 1for different rocks [36,37]. For coal-methane system, Zhao et al.reported that a was near unity for well fractured coal [38]. The coretested (from San Juan coal) is well cleated and a being equal to 1appears to be a reasonable assumption. For uniaxial strain condi-tion, only the horizontal effective stress is considered since the ver-tical stress remains constant during depletion. The effectivehorizontal stress thus estimated was plotted as a function of porepressure, as shown in Fig. 7. The figure shows that the effective hor-izontal stress does not decrease linearly with gas depletion for theentire range of pressure reduction. In the high pressure range,6.2–2.75 MPa (900–400 psi), the decrease is slower than in thelow pressure range, 2.75–0.3 MPa (400–50 psi). This is probablydue to the significant desorption starting at 2.75 MPa (400 psi)and the associated matrix shrinkage. This phenomenon is in agree-ment with the observed and modeled relationships between per-meability and pressure [6,21,39]. The permeability increase issmall in the high pressure range compared to that measured inthe low pressure range.

σe = 0.23P + 1.9

σe = 0.84P

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8

Pore Pressure (P), MPa

Eff

ecti

ve H

oriz

onta

l Str

ess

( σe)

, M

Pa

Fig. 7. Effective horizontal stress with pore pressure (a = 1).

A. Mitra et al. / Fuel 94 (2012) 110–116 115

Since most of the past researchers have claimed the dependenceof permeability on stress to be log-linear, the experimental resultswere used to check its validity for uniaxial strain conditions. Thestress-permeability equation presented by Shi and Durucan [21] was:

logkk0¼ �3Cf ðr� r0Þ ð4Þ

where, k is permeability, r is effective horizontal stress [21] andCf is the cleat compressibility, which is a measure of the deforma-tion of cleat per unit change in pressure during depletion. The sub-script ‘‘0’’ denotes initial conditions. Eq. (4) can also be written as:

3Cf ¼log k

k0

ðr0 � rÞ ð5Þ

indicating that if logarithm of permeability ratio is plotted as afunction of reduction in effective horizontal stress, the slope of theplot would be equal to three times the value of cleat compressibil-ity. Hence, the logarithm of permeability ratio was plotted as afunction of reduction in horizontal effective stress. This is shownin Fig. 8. It is evident that the exponential relationship betweenstress and permeability does hold for coal since a straight linecould be fit through the data points. In Fig. 8, the slope of the linewas 0.276, giving a value for Cf of 0.092 MPa�1 (0.00063 psi�1).Hence, for laboratory scale, a constant value of cleat volume com-pressibility is a good approximation [21]. However, the modelershad to deviate from this assumption and proposed the use of a var-iable Cf although the deviation was based purely on a historymatching exercise with no scientific foundation [39]. The interest-ing observation about the matching exercise presented by themodelers was that they varied Cf from its original value of0.1392–0.092 MPa�1 (0.00096–0.00063 psi�1) to improve thematch with field observations below 2.1 MPa (300 psi). The exper-imental results presented in this paper estimate the value of Cf tobe 0.092 MPa�1 (0.00063 psi�1), equal to the value that gave thebest match for field data at pressure below 2.1 MPa (300 psi).

The issue of the constancy of cleat compressibility remains con-troversial. To date, two detailed discussions have been published,both by Palmer [40,41]. Based on the published experimental data[6,26], Palmer [40] pointed out that the cleat compressibility is notconstant with depletion in CBM wells. However, the laboratorydata [6,26] was obtained using either water or helium as the test-ing fluid, which can not illustrate the behavior of sorbing gases,methane and CO2. Moreover, the cleat compressibility was illus-trated by Palmer [40,41] to be a pressure-dependent parameter.Just for the record, the modeler used history matching techniqueto obtain the porosity strain/change, assuming the Palmer andMansoori model to be absolutely valid. In fact, there are still someuncertainties in the Palmer and Mansoori permeability model, forexample, f- and g-factor.

log(k/k0) = 0.276(σ0-σ)

0

0.5

1

1.5

2

2.5

0 1 2 3 4

σ0-σ, MPa

log(

k/k 0

)

Fig. 8. Variation of measured permeability with change in horizontal effectivestress.

5. Conclusions

A unique experimental technique was developed to estimatethe variation of permeability with pore pressure under stress-/strain- controlled conditions, best replicating the in situ behaviorof coal. Based on the results obtained to date, the following conclu-sions are made:

1. The improvement in laboratory results when strain-controlledconditions are applied instead of stress control is significant.The laboratory established permeability trend is closer to thatobserved in the field, at least for San Juan coal. Hence, allfuture laboratory permeability work should focus on usingstrain-controlled conditions.

2. The permeability was found to increase continuously withdecrease in pore pressure. This explains the field observationsin the San Juan basin where no permeability decline has beenobserved with continued production in several areas. Further-more, the rate of increase is not constant. At high pore pressure,the change in permeability is small, accelerating only after thepore pressure falls below a certain value. This is in agreementwith the existing theory that matrix shrinkage resulting fromdesorption enhances the cleat permeability significantly andthe shrinkage is significant only when substantial desorptionoccurs. This is also in agreement with field observations wheregas production rates increase substantially after several years ofproduction. This is in agreement with the exponential increasein permeability with depletion reported by previous research-ers [20].

3. Reduction in methane pore pressure causes significant reduc-tion in horizontal stress. However, the magnitude of stressreduction is different, �150% higher than the reduction in porepressure. This is due to the shrinkage effect as a result of meth-ane desorption and the condition of uniaxial strain.

4. There is no direct evidence in the literature about the value ofthe Biot’s coefficient for uniaxial strain condition. Hence, theconcept of effective stress, based on a value of 1 is somewhatdiscomforting. Palmer [40] pointed out that coal is a weak rockwith large grain compressibility compared to typical reservoirrocks. He also suggested that the unity value of Biot’s coefficientassumption is no longer valid. Finally, he suggested that thevalue of Biot’s coefficient is given as Ec/3(1-m). However, thereis no evidence or scientific derivation presented by Palmer tosupport this releationship. In the work presented in this paper,if a value of unity is used for a, the logarithm of permeabilityratio varies linearly with reduction in horizontal effective stress,confirming the existing belief of some modelers [21] that cleatcompressibility remains constant during production period, atleast in the laboratory scale. Given the sensitivity/importanceof this parameter, its behavior over the life of a producing res-ervoir needs further researching.

5. The assumptions of the analytical models developed to daterequire refining. This is discussed in detail in the second partof the paper dealing with use of the experimental data pre-sented here for validation of the models developed.

Acknowledgements

This work was carried out with funding from the IllinoisDepartment of Commerce and Economic Opportunity, throughthe Office of Coal Development and the Illinois Clean Coal Institute(ICCI), and BP America. The authors wish to thank these two orga-nizations for the support provided. They also wish to thank BPAmerica personnel for providing constant guidance and assistance.

116 A. Mitra et al. / Fuel 94 (2012) 110–116

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