Compressibility of sorptiveporous media: Part 2.Experimental study on coalShimin Liu and Satya Harpalani

ABSTRACT

This paper, second of a two-part series, discusses the results ofthe experimental work conducted to estimate the different com-pressibilities of coal under both unconstrained and constrainedconditions. Under unconstrained conditions, the shrinkage orswelling compressibility ðCmÞ was measured, which was foundwith certainty to be a pressure-dependent parameter. The modelproposed to estimate Cm was able to effectively predict the varia-tion trend, although the modeled values were larger than thosecalculated using experimental results. The pore-volume com-pressibility ðCpÞ under uniaxial strain conditions for heliumdepletion was found to be a constant positive value. The valueof Cp for methane depletion, however, was found negative, indi-cating that the pore volume (cleat) increases with depletion.Moreover, its absolute value increased with decreasing methanepressure. Consistent with field permeability observations, thepermeability increases with methane depletion, and the rate ofincrease at lower pressures is higher than at high pressures. Theproposed pore-volume compressibility model was well able topredict the variation.

INTRODUCTION

Coalbed methane (CBM) has become an important part of theworld’s natural gas resource. The energy industry classifies coal-beds as unconventional gas reservoirs and works continuouslytoward developing methods to economically develop gas pro-duction from them. Most recent data from the EnergyInformation Administration (EIA) suggest that proven CBMreserves of the United States accounted for about 17.5 cubic feet(TCF) in 2010. The CBM production in 2010 was 1.9 TCF,

Copyright ©2014. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received July 26, 2013; provisional acceptance November 15, 2013; revised manuscriptreceived February 06, 2014; final acceptance March 24, 2014.DOI: 10.1306/03241413134

AUTHORS

Shimin Liu ∼ Pennsylvania State University,Department of Energy and MineralEngineering, University Park, Pennsylvania16802; szl3@psu.edu

Shimin Liu is an assistant professor atPennsylvania State University. He received aB.S. and an M.S. degree from the ChinaUniversity of Mining and Technology,Beijing, and a Ph.D. in engineering sciencefrom the Southern Illinois UniversityCarbondale. His research focuses onunconventional gas development, carbonsequestration in geological formations, andmodeling of flow in coal and rocks.

Satya Harpalani ∼ Southern IllinoisUniversity, Department of Mining andMineral Resources Engineering, Carbondale,Illinois 62901; satya@engr.siu.edu

Satya Harpalani is a professor at SouthernIllinois University Carbondale. He receivedhis Ph.D. from the University of California,Berkeley and an M.S. degree from VirginiaPolytechnic Institute and State University(Virginia Tech). His research focuses on flowcharacterization of porous media, withemphasis on coal and sandstone, includingmodeling and simulation of gas flow, andproduction from deep rocks.

ACKNOWLEDGEMENTS

The AAPG Editor thanks the followingreviewers for their work on this paper:Shugang Wang and an anonymousreviewer.

AAPG Bulletin, v. 98, no. 9 (September 2014), pp. 1773–1788 1773

accounting for 8.5% of the United States natural gasproduction (EIA, 2014). Apart from the UnitedStates, interest in other countries, particularlyCanada, Australia, China, and India, continues togrow. Interest in CBM has been further acceleratedby the U.S. Environmental Protection Agency(EPA) because methane is identified as a greenhousegas, known as significantly more damaging to theenvironment than carbon dioxide.

During drawdown of a CBM reservoir by pri-mary production, the state of stress on coal formationis believed to change continuously, resulting in defor-mation of the coal bulk, matrix as well as pore (cleat)volume. Additionally, methane is mainly stored incoal as sorbed gas, and reservoir depletion leads todesorption of gas in coal that is accompanied by anon-elastic deformation, namely, the matrix-shrinkagephenomenon believed to open up the cleats andresulting in increased permeability. The evolution ofthe volumetric response of reservoir rock to pressureand/or stress variation is one of the most importantparameters for flow behavior modeling as well asgeomechanical description, characterized by differentcoal compressibilities.

The theoretical modeling, a first-of-its-kind effortto describe the volumetric behavior of sorptive mate-rials, has been carried out by integrating sorption-induced strain into poroelastic deformation, andpresented in the first part of this two-part series (Liuand Harpalani, 2014). To validate the models, anexperimental study replicating the underlying princi-ples and assumptions of the models, as well as satis-fying the boundary conditions, was conducted. Thissecond part of the two-part series presents the resultsof the study to estimate the volumetric variation ofSan Juan coal as a function of pore pressure underboth unconstrained and constrained conditions. Thesubbituminous coal sample was collected from thenorthern San Juan Basin in New Mexico. For uncon-strained conditions, the shrinkage or swelling com-pressibility ðCmÞ was estimated for methane. Underconstrained conditions, the coal bulk compressibilityðCbÞ was measured directly for different gases.Finally, the pore-volume compressibility was com-puted using the estimated and measured values ofCm and Cb. All of the symbols in the text aredescribed in Table 1.

BACKGROUND

Coal is a sorptive porous medium, widely accepted asa dual porosity rock (Seidle and Huitt, 1995; Levine,1996; Palmer and Mansoori, 1998; Shi and Durucan,2004, 2005; Cui and Bustin, 2005; Ma et al., 2011;Liu et al., 2012; Wang et al., 2012). It contains bothmacropore and micropore systems, as shown inFigure 1. The micropore system occurs throughoutthe coal matrix, providing the internal surface forstoring gas in sorbed form. The adsorption or desorp-tion occurs when the relative sorbing gas pressurevaries during gas production. This automaticallyresults in a change in the internal surface energy of

Table 1. Nomenclature

Symbol Definition and Units

a, b Sorption Langmuir constants (a: m3∕t; b: Pa−1)Cb Bulk-volume compressibility ðPa−1ÞCm Shrinkage or swelling compressibility ðPa−1ÞCp Pore-volume compressibility ðPa−1ÞCs Solid-matrix compressibility ðPa−1ÞE Young’s Modulus (Pa)EA Modulus of solid expansion caused by

adsorption and desorption (Pa)P Gas pressure (Pa)Ps Stress experience by solid phase (Pa)R Universal gas constant ðJ × mol−1 × K−1ÞT Temperature (K)V 0 Gas molar volume ðm3∕molÞGreek symbolsεl, Pε Parameters of Langmuir match to volumetric

strain because of matrix shrinkage (εl :dimensionless; Pε: Pa)

εs Solid-phase strain (dimensionless)εV Volumetric strain (dimensionless)εxx, εyy, εzz Linear strain in x-, y-, and z-directions,

respectively (dimensionless)ρ Solid-phase density ðkg∕m3Þεh Horizontal or lateral strain (dimensionless)ν Poisson’s ratio (dimensionless)ϕ Porosity (dimensionless)σ Volumetric stress (Pa)σ11, σ22, σ33 Three principal stresses in three principal

directions in Cartesian coordinate system (Pa)σa Vertical stress (Pa)σh Horizontal stress (Pa)

1774 Compressibility of Sorptive Material, Experimental Study

the coal matrix and induces deformation of thematrix. Liu and Harpalani (2013a) proposed anapproach to quantify the relationship betweenchanges in the internal surface energy and deforma-tion of coal matrix. In their study, the sorption-induced matrix macro-deformation was separatedfrom mechanical deformation, and pure macro-deformation of the coal matrix was thus describedas a function of sorbing gas pressure. Based on theirmodel, the shrinkage or swelling compressibility ofcoal became a calculable parameter, as shown inPart 1 of this two-part series (Liu and Harpalani,2014). The macropore system consists of a naturallyoccurring network of fractures called the cleat sys-tem, which provides the primary pathways for gasand water flow in coal. The cleat structure systemalso changes dynamically during pressure depletion,affected primarily by two distinct phenomena. First,the matrix non-elastic deformation is caused bydesorption of gas(es). Second, the stresses appliedon fractures change under certain boundary condi-tions. To develop a comprehensive description ofthe evolution on the fracture or cleat, non-elasticdeformation was integrated into the poroelasticbehavior of bulk coal for given boundary conditions.Part 1 of this two-part series (Liu and Harpalani,2014) provides a methodology to obtain thepore-volume evolution under best-replicated in situconditions, namely, the uniaxial strain, implying

that constant vertical stress ðσaÞ is related tounchanged overburden and zero lateral strain tolateral confinement.

EXPERIMENTAL SETUP AND PROCEDURE

To estimate different coal compressibilities, it wasnecessary to measure volumetric strain under differentexperimental conditions. Two types of conditionswere used for this study, unconstrained (free standing)and constrained (uniaxial strain). Under uncon-strained conditions, the volumetric behavior of coalmatrix was measured by continuously monitoring thevolumetric strain with changes in gas pressure. Forconstrained conditions, changes in the bulk volumeof the sample were measured. Using the proposedmodel, pore-volume compressibility was calculated.

Unconstrained Condition

Experimental SetupThe experimental setup consists of a vessel that iscapable of withstanding high pressures. To continu-ously monitor the volumetric behavior, linear defor-mation strain gauges were used. A data acquisitionsystem (DAS) was used to collect the pressure andstrain data. To eliminate thermal-induced effects oncoal-matrix strain, the high-pressure vessel was keptin a constant-temperature water bath, capable ofmaintaining the temperature to within 0.1°C (1.8°F)of the desired temperature. This not only ensuredconstant temperature throughout the experiment, butalso allowed carrying out the experiment at in situtemperature. A schematic of the experimental setupis shown in Figure 2. To monitor the pressure in thevessel precisely, a highly sensitive pressure trans-ducer, connected to a DAS capable of recording pres-sure data at short-time intervals, was used.

Sample PreparationThe coal samples for unconstrained experiments wereprocured from the northern part of the San Juan Basinin New Mexico. A 5.08 cm (2.0 in.) cylindrical corewas used for the experimental work. The core wasdrilled perpendicular to a visible bedding plane. Thetop and bottom parts of the core were trimmed, and

Figure 1. Conceptual physical structure of coal.

LIU AND HARPALANI 1775

the middle portion was used for the constrained-flowexperiment. Using the trimmed ends, four identicalsamples were prepared by splitting these into quad-rants, and samples with the fewest cleats wereselected. Three linear strain gauges were affixed oneach sample to monitor strains in the three orthogonaldirections, as shown in Figure 2. Prior to starting theactual test, the prepared samples were kept in an envi-ronmental chamber under controlled conditions oftemperature and humidity.

Experimental ProcedureTo obtain pure shrinkage or swelling strain in the lab-oratory, helium and methane were used. The speci-men was first subjected to increasing heliumpressure in a stepwise manner. For helium, knownto be non-sorbing, the measured volumetric strainwas purely related to the mechanical compression ofthe solid coal because of changes in the surroundinggas pressure. Helium was then bled out, and the sam-ple was subjected to methane flooding, in steps of∼1.38 MPa (200.2 psi), to a final pressure of∼6.9 MPa (1000.8 psi). Because the mechanicalcompression effect occurs with methane as well, themeasured strains were used to calculate the trueshrinkage or swelling strain by summing up the

helium and methane strains. Using the true shrinkagestrain, the shrinkage or swelling compressibility wasestimated.

Constrained Condition

Experimental SetupA new experimental setup was designed to imple-ment the continuous strain monitoring under con-strained conditions, which were, for this study, thoseof uniaxial strain. The experimental setup includedindependent control of stress and strain conditions,gas pressure (upstream and downstream), and meas-urement of gas-flow rate. The unique feature of thissetup was that it replicated the best in situ conditionsunder zero horizontal strain, that is, the core was notpermitted to physically shrink with reduction in gaspressure. In addition, vertical stress, representativeof overburden, was maintained constant throughoutthe experimental duration. A schematic of the con-strained-condition experimental setup is shown inFigure 3. The setup consisted of a triaxial cell, a cir-cumferential extensometer to monitor and controlthe shrinkage or swelling of core, a linear variabledifferential transducer (LVDT) to measure the verti-cal strain, and a loading system. The setup enabled

Figure 2. Schematic of experimental setup for measurement of matrix volumetric strain under unconstrained condition.

1776 Compressibility of Sorptive Material, Experimental Study

applying both confining and axial stress initially tosimulate the physical stress condition in situ. Thetemperature of the triaxial cell was kept constant byusing a heating jacket and temperature controller.

Sample PreparationThe middle part of the drilled cylindrical core wasused to carry out the experiment. The core was wellcleated and, therefore, suitable for the fracture–solidvolumetric strain test. The top and bottom ends ofthe core sample were properly polished for placementin the triaxial cell. Following this, the prepared coalcore was placed in the environmental chamber untilthe experiment started.

To replicate the in situ condition, all tests wereperformed under similar conditions. The initial reser-voir pressure was estimated as ∼7.6 MPa (1102 psi),the vertical stress remained constant at ∼14.5 MPa

(2103 psi) and the initial horizontal stress was∼9.6 MPa (1392 psi). The reservoir temperature wasmaintained constant at 35°C (95°F).

Triaxial Cell and Sample SealingThe triaxial cell used in this experiment was a criticalcomponent of the setup. It comprised a cell base withinlet for gas and hydraulic confining oil, perforatedsteel disks holding the coal core, and cell casing,which enclosed the entire arrangement. Prior to start-ing the experiment, the core was taken out of theexperimental chamber. The two perforated steel diskswere placed at the two ends of the sample. The pur-pose of the perforated steel disks was to distributethe gas throughout the entire cross section of the coresample and distribute the vertical stress equally to theentire core area. Following this, the core was sealedusing shrinkage tubing to prevent seepage of oil into

Figure 3. Schematic of experimental setup for measurement of permeability and volumetric strain under constrained condition.

LIU AND HARPALANI 1777

the core. The shrinkage tubing used for sealing isvery thin and rigid, and we believe that the deforma-tion of the tubing during depletion is negligible. Thecore, along with loading platens, was then attachedto the base of the triaxial cell. The circumferentialextensometer was placed around the core at mid-height. The lead wires of the extensometer were con-nected to feed-through connectors located at the cellbase. The sample, attached to the base of the cell,and the circumferential extensometer are shown inFigure 4. The gas flow was perpendicular to thebedding-plane direction. After the sample was placedin the triaxial cell, the heating blanket was affixed tothe outer surface of the cell, and insulated by wrap-ping it with heating tape and insulation pad.

Experimental ProcedureThe triaxial cell was placed in the load frame. Oncethe complete experimental setup was in place, thesample was stressed triaxially, using the loadingframe for axial stress and hydraulic pump for the

confining stress. The load was applied gradually in astepwise manner, allowing adequate time to attainstress–strain equilibrium at each step to avoid anymechanical shock to the sample. Application of thedesired stress level typically took two to three days.The sample was then flushed with helium at 100 psi(0.7 MPa) to get rid of any residual gas and air withinthe sample. It also provided a means to check for anyleakage in the gas circuit.

Next, the core was saturated with helium at amaximum pressure of 7.6 MPa (1102 psi). The injec-tion was performed in steps of 0.7 MPa (100 psi),with ∼5 hours between steps. After injection wascompleted, the sample was allowed to equilibrate for∼4 days. After the circumferential extensometer andLVDT readings stabilized, the sample was believedto be in mechanical equilibrium. The pore pressurewithin the sample was gradually reduced in steps,resulting in compression of the pore volume, which,in turn, resulted in a horizontal strain. To ensure uni-axial strain boundary conditions, the horizontal stresswas reduced to ensure zero circumferential strain.However, the vertical stress was kept constant. Duringthe entire helium pressure-depletion cycle, the verticalstrain was monitored continuously, and the bulk vol-ume compressibility was calculated for helium.

After completing the helium cycle, the sample wasflushed with methane twice. The sample was then satu-rated with methane to a final pressure of 7.6 MPa(1102 psi) by stepwise injection. After achieving equi-librium, the same testing procedure was followed asfor helium. The primary difference was the long dura-tion required to achieve equilibrium because gasdesorption of coal is an extremely slow process. Thewhole methane depletion cycle took about five monthsfor the pressure decreasing from 7.6 MPa (1102 psi) to∼0.7 MPa (102 psi). Using the monitored strain, thebulk compressibility for methane was calculated toestimate the pore-volume compressibility.

EXPERIMENTAL RESULTS AND ANALYSIS

Unconstrained Condition

Helium Injection ResultsThe experimental phase involving helium injectionwas completed by dosing the coal quadrant samples

Figure 4. Sample wrapped in shrink tubing with circumferen-tial extensometer.

1778 Compressibility of Sorptive Material, Experimental Study

with increasing amounts of helium in a stepwise man-ner to maximum pressure of ∼6.9 MPa (1001 psi),while continuously monitoring each linear strain inthe three orthogonal directions. A sample was consid-ered in equilibrium if the strain remained stable forone day or more. This typically required three to fourdays for each pressure step. Three samples, preparedfrom the same core, were tested in this study. Also,the small strain assumption (ε2 ≪ ε) was believed tohold, and the volumetric strain of the sample was,therefore, calculated as

ϵV = ϵxx + ϵyy + ϵzz (1)

in which εV is volumetric strain, and εxx, εyy, and εzzare the linear strains in x-, y-, and z- directions,respectively.

Using the measured strain results, the volumetricstrain was calculated for each sample. Figure 5 showsthe average volumetric strain for the samples testedfor increasing helium pressure. As expected, the vol-ume of coal matrix decreased with increasing pres-sure caused by compression of the solid-coal grain.A linear relationship described the volumetric strainas a function of helium pressure well. Based on thedefinition described in Part 1 of this two-part-seriespaper (Liu and Harpalani, 2014), the solid-matrixcompressibility for coal was calculated as a constantat −1.97 × 10−1 MPa−1 ð−1.36 × 10−6 psi−1Þ underunconstrained conditions.

Methane Injection ResultsAfter completing the helium cycle, gas was bled outfrom the pressure vessel. This was followed by inject-ing methane into the sample gradually to a final pres-sure of ∼6.9 MPa (1001 psi). The volumetric straininduced because of methane injection was calculatedfor each pressure equilibrium step. As expected, thecoal matrix swelled with increasing methane pres-sure. Two samples were used to measure the volu-metric strain. Because both samples were preparedfrom the same block of coal, average strain valueswere calculated, as shown in Figure 6. The strainresults clearly exhibit a strong similarity to typicalsorption isotherms. The measured strain was, there-fore, fitted using an equation similar to theLangmuir equation used for sorption isotherms(Levine, 1996; Harpalani and Mitra, 2010), given as

ϵV = ϵlP

P + Pϵ(2)

in which εV is the sorption-induced volumetric strainat pressure P; εl represents the maximum strain,which can be achieved at infinite pressure; and Pε isthe pressure at which coal attains 50% of maximumstrain. The values of Pε and εl were estimated as5.88 MPa (853 psi) and 0.009885, respectively.Figure 6 shows excellent agreement between theLangmuir-fitted equation and experimental results.Liu and Harpalani (2013a) used this set of data tovalidate the theoretical volumetric strain model andconcluded that the proposed model can accurately

Figure 5. Volumetric strain with increasing helium pressure.Refer to the text and Table 1 for an explanation of the symbolsused.

Figure 6. Measured and modeled volumetric strain as a func-tion of methane pressure. Refer to the text and Table 1 for anexplanation of the symbols used.

LIU AND HARPALANI 1779

predict the volumetric strain with proper elastic prop-erties and sorption data.

True Shrinkage or Swelling ResultsWhen coal is flooded with methane, both the mechani-cal compression and shrinkage- or swelling-inducedstrain govern its total volumetric response. However,when the coal is exposed only to helium, the mechani-cal compression effect alone takes place. Therefore,the true, so-called, sorption-induced volumetric straincan be calculated by subtracting the mechanical-induced strain from the measured methane-injection-induced strain. This step is particularly critical whenusing any of the existing permeability models, becausethe two strains—mechanical and shrinkage induced—are included as two separate terms. The results areshown in Figure 7. Based on the definition of shrink-age or swelling compressibility, compressibility canthen be calculated by taking the first derivative of thetrue sorption-induced strain for methane. Using theempirical Langmuir-type fit, the true shrinkage- orswelling-induced strain is mathematically given as

εV = 0.009885P

P + 5.88+ 0.000197P (3)

Taking the first derivative of the above equationreduces it to the shrinkage or swelling compressibilityðCmÞ:

Cm = 0.009885 ·5.88

ðP + 5.88Þ2 + 0.000197 (4)

The shrinkage or swelling compressibility isapparently a pressure-dependent parameter. The Cm

was, therefore, plotted as a function of pressure asshown in Figure 8. The Cm increased with decreasingpore pressure. Consistent with the sorption isothermtrend, the sorption capacity is higher at lower pres-sures. The sorption-induced deformation at lowpressures is also apparently higher than at higherpressures for a unit-pressure change. In other words,coal solid matrix is more compressible at lower pres-sures because of the sorption effect.

Constrained Condition

Throughout the experiment, the vertical linear strainðεzzÞ was monitored continuously. Based on the smallstrain assumption and equation 1, the vertical strain isalso the volumetric strain under uniaxial strain condi-tion because the two horizontal strains are zeroðεxx = εyy = 0Þ. The measured volumetric strain wasused to calculate the bulk compressibility of coalunder constrained condition.

Helium Depletion ResultsThe volumetric strain with reduction in helium pres-sure is shown in Figure 9. The vertical stress is con-stant throughout the experimental duration,satisfying the constant vertical stress requirement ofthe uniaxial strain condition. Using the measuredvolumetric strain, bulk compressibility for helium,which is the slope of the volumetric strain plot, was

Figure 7. Calculated true shrinkage or swelling strain. Refer tothe text and Table 1 for an explanation of the symbols used.

Figure 8. Shrinkage or swelling compressibility for methane.Refer to the text and Table 1 for an explanation of the symbolsused.

1780 Compressibility of Sorptive Material, Experimental Study

calculated as a constant at 1.3 × 10−4 MPa−1

ð9.0 × 10−7  psi−1Þ.Along with varying the pore pressure to replicate

depletion, the applied horizontal stress required con-tinuous adjustment to maintain the uniaxial straincondition. The horizontal stress decreased linearlywith decreasing helium pressure, and the relationshipis regressed as

σh = 0.8P + 3.14; (5)

in which σh is the external horizontal stress and P isthe pore pressure. The external horizontal stressdecreased with decreasing pore pressure. When porepressure was reduced, the sample tended to shrink inthe horizontal direction related to cleat deflation.The horizontal stress, therefore, had to be reduced torelax the coal.

Methane Depletion ResultsAfter completing the helium cycle, gas was flushedout and the sample was saturated with methane at∼7.6 MPa (1102 psi). Once methane equilibrium at7.6 MPa (1102 psi) was achieved, the uniaxial straincondition was maintained during the entire experi-mental procedure by adjusting the applied horizontalstress. The depletion was performed in a stepwisemanner, reducing the pressure to 6.2, 4.8, 3.4, 2.4,1.4, 0.7, and 0.35 MPa (899, 696, 493, 348, 203,102, and 51 psi), respectively. At each step, adequatetime was given to ensure sorption and mechanicalequilibrium. The entire procedure took approximately

four months. The volumetric strain was continuouslymonitored (shown in Figure 10). The volumetricstrain did not vary linearly with decreasing methanepressure, as was the case for helium, but rather inthe Langmuir style. Hence, a Langmuir-type functionwas regressed as

εV =0.01061PP + 8.1

(6)

Using the measured volumetric strain under con-strained conditions, the bulk compressibility formethane was calculated and plotted as a functionof methane pressure, as shown in Figure 11.Mathematically, the first derivative of the volumetricstrain of the plot in Figure 10 is the bulk compress-ibility for methane, quantified by the followingequation:

Cb =0.01061 × 8.1ðP + 8.1Þ2 (7)

The Cb increased from 3.5 × 10−4 to 11.1 × 10−4

MPa−1 (2.4 × 10−6psi−1 to 7.7 × 10−6  psi−1) forpressure reduction from 7.6 MPa (1102 psi) to∼0.7 MPa ð∼102 psiÞ. As expected, the bulk com-pressibility deviated from constancy because thesorption-induced strain is not a linear function ofpressure nor is the increase of deviatoric stress effectunder in situ conditions. The value of Cb, a directlymeasurable parameter in the laboratory, isrequired for calculation of the pore-volumecompressibility ðCpÞ.

Figure 10. Volumetric strain with methane depletion. Refer tothe text and Table 1 for an explanation of the symbols used.

Figure 9. Volumetric strain for helium depletion. Refer to thetext and Table 1 for an explanation of the symbols used.

LIU AND HARPALANI 1781

Similar to the helium-injection phase of theexperiment, the external horizontal stress was moni-tored under uniaxial strain conditions. Figure 12shows the corresponding changes in applied horizon-tal stress for methane depletion. For comparison, thecorresponding horizontal stress for helium depletionis also included in Figure 12. The external horizontalstress evidently decreased linearly with methane pres-sure, and is given by the following equation:

σh = 1.18P + 0.76 (8)

At complete depletion ðP = 0Þ, the applied hori-zontal stress would reduce to ∼0.76 MPa (110 psi),suggesting that complete loss of horizontal stressðσh = 0Þ does not occur under this condition although

the stress reduction is significant. In addition, the rateof stress loss for methane is more severe than forhelium, which is believed attributed to the matrix-shrinkage effect.

MODELS APPLICATION AND DISCUSSION

This section describes the effort carried out to esti-mate the matrix shrinkage and pore-volume com-pressibility under in situ stress- and strain-controlledconditions using theoretical models proposed earlier(in Part 1 of this series [Liu and Harpalani, 2014]).A comparison of the variation trend and values basedon experimental and modeled results is included.

Shrinkage or Swelling Compressibility Model

Liu and Harpalani (2013a) derived a theoreticalmodel for volumetric strain as a function of sorbinggas pressure using elastic properties, sorption param-eters, and physical properties of coal. The proposedmodel is based on the principles of physics and chem-istry of a surface and interface theory. Two deforma-tion components were included, sorption-inducedand mechanical-induced, respectively. In this study,only the sorption-induced matrix strain, directly pro-portional to the decrease in surface energy, was con-sidered and used to model the shrinkage or swellingcompressibility. The proposed strain model has beenpreviously validated using the data set presented here.Detailed discussion is available elsewhere (Liu andHarpalani, 2013a). The input parameters used in thevalidation the model are presented in Table 2. As dis-cussed in Part 1 of this two-part series (Liu andHarpalani, 2014), the shrinkage or swelling com-pressibility was theoretically calculated by the fol-lowing equation:

Cm =3aρRTEAV0

·b

1 + bP(9)

in which a and b are the sorption Langmuir constants,ρ is coal solid-phase density, R is the universal gasconstant, T is temperature, EA is modulus of solidexpansion resulting from ad- or desorption, and V0

is the gas molar volume.

Figure 12. Variation in horizontal stress for decreasing heliumand methane pressure. Refer to the text and Table 1 for anexplanation of the symbols used.

Figure 11. Variation of bulk compressibility under in situ con-ditions for methane depletion. Refer to the text and Table 1 foran explanation of the symbols used.

1782 Compressibility of Sorptive Material, Experimental Study

Figure 13 shows the modeled results along withthe results using the Langmuir-type fit. The resultscalculated through the Langmuir-type-fit strain curveare labeled as measured results, because they arepurely based on experimental data. The modeledresults predict the same increasing trend for shrinkageor swelling compressibility, although the values areslightly higher than those measured. The proposedmodel, however, only requires adsorption data andmechanical properties, which are typically availablefor most CBM plays. On the other hand, experimentalwork required to establish the trend is complicated,requiring time and laboratory facilities.

Pore-Volume Compressibility Estimation

A theoretical model was derived to estimate the pore-volume compressibility of sorptive porous media inPart 1 of this two-part series (Liu and Harpalani,2014). The proposed model is based on the strain bal-ance between bulk, pore (void), and solid phase(matrix). The pore-volume compressibility of coal isquantitatively related to bulk and solid-phase com-pressibilities through cleat porosity. Referring to the

theoretical derivation of Cp, the model is summarizedas

Cp =1φ

�Cb − ð1 − φÞ dϵs

dP

�����dσa=dϵh=0

(10)

in which Cb is bulk compressibility, ϕ is cleat poros-ity, εs is solid-phase strain, given as

ϵs =3aρRTEAV0

ZP

0

b1 + bP

dp −3ð1 − 2νÞ

E

ZPs

0dPs (11)

in which E is Young’s modulus, ν is Poisson’s ratio,and Ps is skeleton stress, calculated by

Ps =σ − φp1 − φ

(12)

in which σ is volumetric stress.The value of Cp of coal was estimated in this

paper for both helium and methane. A critical stepin the calculation of Cp is to accurately model thesolid-phase strain, which is impossible to measuredirectly in the laboratory given the complex natureof coal structure. The authors have previously vali-dated the solid strain model described by equation 11under free external stress conditions (Liu andHarpalani, 2013a). In this study, the original strainmodel (Liu and Harpalani, 2013a) was extended toapply under in situ conditions by considering boththe external stress and pore pressure to calculate themechanical-induced strain. Compared to the originalunconstrained condition, in which the mechanical-induced strain was modeled only as a function ofgas pressure (Liu and Harpalani, 2013a), a modifica-tion was made that included the mechanical-induceddeformation as being governed and predicted by thevariation of skeleton stress (Ps). This modification,therefore, gives a comprehensive description of thesolid-phase volume under uniaxial strain conditionsconsidering deviatoric stresses. The following twosubsections narrate the detailed procedure for

Figure 13. Modeled and measured shrinkage or swellingcompressibility. Refer to the text and Table 1 for an explanationof the symbols used.

Table 2. Input Parameters for Modeling

ϕ ðton∕m3ÞR  ðMPa × m3∕kmol × KÞ T (K) EA ðMPaÞ V 0 ðkmol∕m3Þ a ðm3∕tÞ b ðMPa−1Þ ϕ (%)

1.4 0.0083143 308 1600 22.4 19.1 0.4 0.1

LIU AND HARPALANI 1783

estimation of Cp for both helium and methane underreplicated in situ conditions.

Constrained Helium DepletionBecause helium is non-sorbing, only the mechanical-induced strain occurs. This is included in equation 11as the second term. The key feature of the constrainedcondition is the skeleton stress (Ps), given by equa-tion 12, in which σ is the volumetric stress (alsoknown as mean normal stress) responsible for chang-ing the volume of the stressed coal solid matrix. Thisis calculated as

σ =σkk3

=σ11 + σ22 + σ33

3=13I1 (13)

in which σ11, σ22, and σ33 are the three principalstresses in the three principal directions in theCartesian coordinate system, and σkk = σ11 + σ22 +σ33 and I1 is the first stress invariant, which has thesame value regardless of the coordinate system’s ori-entation. Under uniaxial strain conditions, the volu-metric stress for helium depletion is measured fordifferent pore pressures. The resulting stress is shownin Figure 14. The figure includes the correspondingvariation in horizontal stress for methane depletionas well. The relationship between pore pressure andvolumetric stress for helium is fitted using a linearequation as

σ = 0.55P + 6.95 (14)

Substituting equation 14 into equation 12 andsimplifying gives the following:

Ps =�0.55 − ϕ1 − ϕ

�P +

6.951 − ϕ

(15)

Using equation 11, the coal solid-phase volumet-ric strain induced by helium depletion is thencalculated as

εs = −3ð1 − 2νÞ

E

ZP

0d��

0.55 − ϕ1 − ϕ

�P +

6.951 − ϕ

�(16)

Cp for helium depletion is, therefore, calculated usingequation 10 and the measured Cb, shown in Figure 9as 1.3 × 10−4 MPa−1 (9 × 10−7 psi−1). To obtain thevalue of Cp at any pressure, the fracture (cleat) poros-ity of coal bulk is required. Truly an unknown

parameter and, given the small porosities encounteredin typical CBM reservoirs in the San Juan Basin, itsvalue is difficult to measure reliably. In the literature,the highest porosity has been reported as 1.24%(Mavor and Saulsberry, 1996) and lowest as0.0457% (Mavor and Vaughn, 1998). In reservoirsimulation exercises, the cleat porosity of San Juancoal is often taken to be in the 0.1 to 0.4% range(Palmer, 2009). For this study, the commonly usedvalue of 0.1% (Palmer et al., 2007; Palmer, 2009)was used. To calculate Cp, ð3ð1 − 2νÞ∕EÞ must beeither estimated or measured. However, based onrock mechanics principles, this is actually the recipro-cal of the solid grain modulus (Ks) and a measurablequantity. The Ks is defined as being equal todP∕dεs. If we carefully examine the results of theunconstrained helium-injection test (shown inFigure 5), Ks can be estimated as ð1∕1.97E − 4ÞMPa. Therefore, equation 16 reduces to

εs = −ð1.97 × 10−4Þ ·Z

P

0d��

0.55 − ϕ1 − ϕ

�P +

6.951 − ϕ

�

(17)

Cp is then calculated as constant at 0.0218 MPa−1

(0.00015 psi−1).

Constrained Methane DepletionSimilar to helium depletion, the variation in volumet-ric stress with depletion is shown in Figure 14, and isgiven as

σ = 0.79P + 5.36 (18)

Figure 14. Volumetric stress during helium and methanedepletion. Refer to the text and Table 1 for an explanation ofthe symbols used.

1784 Compressibility of Sorptive Material, Experimental Study

In addition, the skeleton stress is calculated as

Ps =�0.79 − ϕ1 − ϕ

�P +

5.361 − ϕ

(19)

Based on experimental results, the total solid-phase strain was calculated as the sum of both truesorption-induced strain (given by equation 3) andmechanical-compression strain. Mathematically, thetotal strain is described as

εs = 0.009885P

P + 5.88+ 0.000197P − ð1.97E − 4Þ

·Z

P

0d��

0.79 − ϕ1 − ϕ

�P +

5.361 − ϕ

�(20)

Theoretically, the total solid-phase strain is given asfollows:

εs =3aρRTEAV0

Zp

0

b1 + bp

dp − ð1.97E − 4Þ

·Z

P

0d��

0.79 − ϕ1 − ϕ

�P +

5.361 − ϕ

�(21)

Therefore, Cp for methane can be estimated usingequation 10. The Cp was estimated by bothmeasured results (equation 20) and modeled data(equation 21). The results are shown in Figure 15,along with Cp for helium depletion. For methanedepletion, both the measured and modeled valueshave a negative sign, because the cleat volumeincreases with decreasing pore pressure. Physically,the negative value of compressibility means expan-sion with depletion.

From the results presented in Figure 15, the con-clusion is that Cp for helium depletion is a positiveand constant quantity, translating to the pore volume(cleat) decreasing linearly with decreasing pressure.This, in turn, translates to coal permeability decreas-ing with depletion. However, Cp for methanedepletion is not a constant value (measured and mod-eled results), but a pressure-dependent parameter.The negative sign translates to increasing cleat vol-ume with methane depletion, thus resulting inincreased permeability as confirmed in numerousprevious studies (e.g., Zahner, 1997; Gierhart et al.,

2006, 2007; Liu et al., 2012; Liu and Harpalani,2012; Mitra et al., 2012; Wang et al., 2012; Liu andHarpalani, 2013b). Physically, the effect of desorp-tion increases with decreasing methane pressure,making pore volume more compressible and resultingin an increase in absolute cleat compressibility. Thiswould lead to a non-uniform permeability increaseduring pressure depletion, the increase at lowerpressure being significantly stronger than that athigh pressures, again as observed in the field(Gierhart et al., 2006) as well as laboratory (Liu andHarpalani, 2012; Mitra et al., 2012). Additionally,the modeled Cp is predicted close to the resultsobtained using the measured strain. For this case, themodeled results are in excellent agreement with themeasured data. Given the experimental complexityand the associated uncertainties, application of themodel should be used as an option, although authorsrecommend using extreme care when doing so. Inaddition, time and equipment permitting, additionalexperimental studies should be conducted on differ-ent coal types to verify the proposed model andimprove its reliability.

Influence of Cleat Porosity on Cp Prediction

Although theoretical and experimental effort in thistwo-part series is devoted to estimate and predictpore-volume compressibility, two uncertainties in

Figure 15. Pore-volume compressibility with gas depletion(cleat porosity = 0.1%). Refer to the text and Table 1 for anexplanation of the symbols used.

LIU AND HARPALANI 1785

the calculation of Cp remain unresolved. First, the ini-tial pore-volume (cleat) porosity is an unknownparameter. Second, different experimental conditions,such as uniaxial strain, constant volume, and hydro-static conditions, would give different values ofCp because the skeleton stresses would be different.In these two papers, only the uniaxial straincondition has been emphasized. The other boundaryconditions will be evaluated and addressed insubsequent work.

To quantitatively evaluate the impact of cleatporosity on pore-volume compressibility, a single-factor sensitivity analysis was performed. Fourtypical coal-cleat porosity values, based on valuespublished in open literature (Mavor and Saulsberry,1996; Mavor and Vaughn, 1998; Palmer et al.,2007; Palmer, 2009), were selected. These were0.05, 0.1, 0.5, and 1%, respectively. The other model-ing input parameters remained the same. The mod-eled results of pore-volume compressibility areshown in Figure 16. All values are apparently pres-sure dependent and negative in value for the selectedcleat porosity values. This suggests that the cleatporosity increases with depletion and the permeabil-ity likewise increases. Additionally, Cp is not sensi-tive to pore pressure for high cleat porosity values(0.5 and 1%). It tends to remain constant above4 MPa (580 psi) for cleat porosity values of 0.5 and1%. This does not imply that the cleat volume doesnot increase at high pore pressures, rather that itschange is relatively constant for equal-pressure incre-ments. However, Cp depends strongly on reservoirpressure for low cleat porosities (0.05 and 0.1%),indicating that the sorption-induced deformationdominates the porosity variation for low-cleat-porosity coals (Liu and Harpalani, 2013a). Anextrapolation of this would suggest that the pore-volume compressibility for shale rock would be evenmore sensitive to reservoir pressure given that theshale formations are much tighter (extremely lowfracture porosity).

Despite these uncertainties, the proposed modelthrows light on the variation of Cp under in situ con-ditions. More importantly, the results of the proposedmodel clearly exhibit the trend of Cp variationbecause of the sorption-induced strain and mechani-cal decompression of solid coal.

Broader Impact of this Study

The matrix shrinkage or swelling compressibility andpore-volume compressibility of sorptive porousmedia were theoretically analyzed in the first part ofthis two-part series (Liu and Harpalani, 2014).The experimental study on coal under in situ condi-tions is presented in the current paper. Both theoreti-cal model and experimental results show that theshrinkage or swelling compressibility is a pressure-dependent parameter, with higher values at lowpressures, which is expected given the strongerdesorption at low pressures. The pore-volume com-pressibility was also found to be pressure dependentduring drawdown, which is usually assumed constant(Sawyer et al., 1990; Pekot and Reeves, 2003; Shiand Durucan, 2004, 2005; Cui and Bustin, 2005).This improved understanding of the different com-pressibilities enhances flow-modeling capability,because some of the fuzzy parameters can be rigor-ously estimated and modeled based on our studies.For example, consider matrix-shrinkage compress-ibility in the ARI model (Sawyer et al., 1990; Pekotand Reeves, 2003), pore-volume compressibility inthe Shi and Durucan model (Shi and Durucan, 2004,2005), and the Cui and Bustin model (Cui andBustin, 2005). Apart from enhancement of flow mod-eling, a sound knowledge of different compressibili-ties of sorptive porous media would also improvethe understanding of unique production behavior ofunconventional sorptive reservoirs (CBM and shales)

Figure 16. Variation of pore-volume compressibility for differ-ent cleat porosities. Refer to the text and Table 1 for an explana-tion of the symbols used.

1786 Compressibility of Sorptive Material, Experimental Study

exhibiting negative decline (Seidle, 2011).Additionally, different compressibilities can providereliable input parameters for unconventional gas-wellproduction data analysis (Clarkson, 2013). To sum-marize, the theory and experimental work on differentcompressibilities presented in these two papers willnot only aid flow-behavior modeling, but also facili-tate the production performance analysis for uncon-ventional gas reservoirs.

CONCLUSIONS

This paper presents experimental work and modelapplication for shrinkage or swelling compressibilityand pore-volume compressibility of San Juan coalunder uniaxial strain conditions. The coal bulk com-pressibility was directly measured in the laboratoryunder in situ conditions. Using the experimentalresults of unconstrained and constrained conditions,the shrinkage or swelling and pore-volume compress-ibilities were calculated. Based on the work com-pleted, several important conclusions can be made,summarized as follows.

1. Cp for helium depletion is constant and positive,estimated as 0.0218 MPa−1 ð1.5 × 10−4 psi−1Þ.

2. Cp for methane depletion is not constant.Moreover, Cp in the negative domain for the entirepressure range. This translates to increasing porevolume with depletion under uniaxial strainconditions.

3. The pore-volume compressibility model is validfor the coal tested and provides a powerful tool toestimate Cp for San Juan coal under in situconditions.

4. Shrinkage or swelling compressibility is a functionof methane pressure, its value increasing withdecreasing pressure. This is consistent with thesorption trend, which suggests higher sorptioncapacity at low pressures.

5. The modeled Cp gives a reasonable prediction inthe experimental pressure range.

6. The applied volumetric stress on the coal solidgrain decreases linearly with pore-pressuredecrease for both helium and methane under uni-axial strain conditions. However, the rate ofdecrease for methane depletion is more intense

than for helium for similar gas-pressure depletionsteps.

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1788 Compressibility of Sorptive Material, Experimental Study