Compressibility ofsorptive porous media:Part 1. Background and theoryShimin Liu and Satya Harpalani
ABSTRACT
This paper, the first of a two-part series, provides a sound back-ground of the volumetric response of sorptive porous media togas depletion under in situ boundary conditions in producing res-ervoirs. As a first step, the overall rock matrix deformation is splitinto two separate components, elastic deformation caused bymechanical decompression and the nonelastic swelling or shrink-age strain induced by adsorption or desorption of gas. The shrink-age or swelling compressibility is estimated by the first derivativeof pure adsorption or desorption strain with variations of gas pres-sure. The pore volume, or fracture, compressibility is then esti-mated by application of a semi-empirical model under uniaxialstrain conditions. Based on the proposed model, both shrinkageor swelling and pore volume compressibilities show strongpressure dependence for sorbing gases and are thus variablesfor which gas production is controlled by desorption of gas. InPart 2, the experimental work under best-replicated in situ condi-tions is described in detail along with the results obtained andapplication of the theory presented in this paper.
INTRODUCTION
The decline in fluid pressure during reservoir depletion results inchanges in the volumes of both reservoir fluids and reservoir rockformations. The volumetric response of reservoir rock to pressureand/or stress variation has long been recognized as a critical factorinfluencing production behavior (Geertsma, 1957; Zimmerman,1991), reservoir analysis, and simulation (Yale et al., 1993) inthe traditional oil and gas industry. However, these relationshipsare not clearly understood for unconventional gas reservoirs, suchas coalbed methane (CBM) and shale gas. As development of
Manuscript received July 26, 2013; provisional acceptance November 15, 2013; revised manuscriptreceived February 06, 2014; final acceptance March 24, 2014.DOI: 10.1306/03241413133
AUTHORS
Shimin Liu ∼ Department of Energy andMineral Engineering, College of Earth andMineral Sciences (EMS) Energy Institute, ThePennsylvania State University, UniversityPark, Pennsylvania 16802; [email protected]
Shimin Liu is an assistant professor at ThePennsylvania State University. He receivedhis B.S. and M.S. degrees from the ChinaUniversity of Mining and Technology,Beijing, and Ph.D. in engineering sciencefrom Southern Illinois UniversityCarbondale. His research focuses on gasshale and coalbed methane development,carbon sequestration in geologicalformations and modeling of flow in coal androcks.
Satya Harpalani ∼ Department of Miningand Mineral Resources Engineering,Southern Illinois University, Carbondale,Illinois 62901; [email protected]
Satya Harpalani is a professor at SouthernIllinois University Carbondale. He receivedhis Ph.D. from the University of California,Berkeley and M.S. from Virginia Tech. Hisresearch focuses on flow characterization ofporous media, with emphasis on coal andsandstone, including modeling andsimulation of gas flow and production fromdeep 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. 1761–1772 1761
unconventional gas resources progresses, compre-hensive treatment of volumetric response of forma-tion rocks to variations in pressure or stress isessential to assess the reservoir formation and enabledeveloping the reservoir in a safe and effective man-ner. The dynamic volumetric evolutions of uncon-ventional plays are exasperated by specializeddrilling, injection of a fluid for enhanced production,and/or carbon sequestration activities. Unfortunately,the required data are typically not available becauseof the complexity of volumetric behavior of reservoirrocks in sorbing gas environments. The volum-etric behavior can be quantitatively described by useand application of different compressibilities(Zimmerman, 1991). In this paper, we provide a theo-retical approach to model these compressibilities forsorptive porous media and uncover the interrelation-ships among them. The distinguishing features of thiswork include (1) use of in situ boundary conditions,(2) variation of deviatoric stress during gas depletion,(3) incorporation of sorptive deformation during thestress-deformation analysis for sorptive porousmedia, and (4) indirect prediction of the pore volumevariations using measureable or predictable bulk andsolid matrix deformation. The theory presented in thispaper applies to the deformation and dynamic flowbehavior because of gas depletion in unconventionalreservoirs, which, in turn, results in redistribution ofstresses. The models presented in this paper also pro-vide reliable input parameters for gas-flow modelingin unconventional reservoirs, which are commonlyassumed or obtained using empirical fits to produc-tion data.
BACKGROUND AND PREVIOUS STUDIES
A sound knowledge of the volumetric response ofcoals and gas shales to pressure variation aids theunconventional-reservoir operator’s interpretation ofthe gas production behavior and, more importantly,predicts future production. The incremental changein pore volume with respect to pore pressure is calledthe pore-volume compressibility (Cp); the incrementalchange in bulk volume with respect to pore pressuredefines the bulk volume compressibility (Cb). All ofthe symbols in the text are described in Table 1. Coal
Table 1. Nomenclature
Symbol Definition and Units
A Cross-sectional area (m2)a, b Sorption Langmuir constants (a: m3∕t; b: Pa−1)Cb Bulk-volume compressibility (Pa−1)Cbc Bulk compressibility with changing confining stress
(Zimmerman, 1991) (Pa−1)Cbp Bulk compressibility with changing pore pressure
(Zimmerman, 1991) (Pa−1)Cg Grain compressibility under hydrostatic conditions
(Zimmerman, 1991) (Pa−1)Cm Shrinkage or swelling compressibility (Pa−1)Cp Pore-volume compressibility (Pa−1)Cpc Pore-volume compressibility with changing
confining stress (Zimmerman, 1991) (Pa−1)Cpp Pore-volume compressibility with changing pore
pressure (Zimmerman, 1991) (Pa−1)Cs Solid matrix compressibility (Pa−1)D, D1, D2 Intermolecular distance (m)E Young’s Modulus (Pa)EA Modulus of solid expansion related to adsorption
and desorption (Pa)p Gas pressure (Pa)Pc Confining stress (Pa)
Pp Pore pressure (Pa)
Ps Stress experience by solid phase (Pa)R Universal gas constant (J × mol−1 × K−1)T Temperature (K)Vb Bulk volume of rock sample (m3)
Vm Matrix volume (m3)
Vp Pore volume of rock sample (m3)
Vs Solid-phase volume (m3)
V 0 Gas molar volume (m3∕mol)
Greek symbolsσa Vertical stress related to overburden (Pa)εh Horizontal or lateral strain (dimensionless)σ External stress (Pa)ϕ Porosity (dimensionless)εl , Pε Parameters of Langmuir match to volumetric strain
because of matrix shrinkage (εl : dimensionless;Pε: Pa)
εa Adsorption volumetric strain (dimensionless)ρ Solid-phase density (kg∕m3)εb Bulk-volume strain (dimensionless)εs Solid-phase strain (dimensionless)εp Pore-volume strain (dimensionless)ν Poisson’s ratio (dimensionless)
1762 Compressibility of Sorptive Material, Background and Theory
and gas shale with organic matter, both fractured andsorptive media, exhibit a dilation phenomenonbecause of changes in surface energy caused by sorp-tion or immersion into a sorbing gas. The incrementalchange in volume of rock matrix with respect to sorb-ing gas pressure caused by the dilation phenomenon isdefined as the shrinkage or swelling compressibility(Cm), which is a unique behavior exhibited by uncon-ventional formation rocks. The rock matrix deforma-tion can be split into two separate components: first,the elastic deformation related to mechanical decom-pression of the solid matrix and, second, the non-elastic swelling or shrinkage strain induced by adsorp-tion or desorption. These compressibilities areextremely important for gas production from uncon-ventional resources because they can dynamicallyinfluence the overall gas transportation system. Bothexperimental and theoretical approaches have histori-cally been conducted to understand the compressibili-ties of formation rock and these continue to attractsignificant attention as production from these sourcescontinues to grow.
Compressibility of Non-Sorptive Rocks
Rock compressibility has been studied extensively inthe oil and gas industry. One of the first experimentalstudies involving determination of rock pore com-pressibility was carried out by Carpenter andSpencer (1940). Their work showed that the valuesof pore-volume compressibility were on the order of4.4 × 10−4 MPa−1 (6.42 × 10−2 psi−1) and were nottemperature sensitive. Following this, Hall (1953)stated that neglecting pore-volume compressibilitymay lead to errors of a factor of two when calculatinghydrocarbon volumes in undersaturated reservoirs.Hall (1953) reported experimental work in which hecorrelated his data with porosity and formulated anempirical relationship showing that the pore-volumecompressibility decreased with increase in rockporosity. Hall’s empirical formula, however, has beensomewhat controversial. Li et al. (2004) claimed thatHall’s plot is logically incorrect because the rockswith higher porosity should be more compressiblethan rocks of low porosity. Geertsma (1957) wasamong the first, and perhaps the most respectedscholar, responsible for describing the behavior of
porous media subjected to pressure variations. Histheoretical contribution was rederivation of Biot’sequation (Biot, 1941), using the more physicallyobvious set of variables, including the pore and bulkvolumes and both confining and pore pressures.Fatt (1958a) carried out experimental studies onkerosene-saturated sandstones to evaluate the pore-volume compressibility. Fatt (1957, 1958b, 1959)conducted much of the initial experimental work tomeasure the bulk compressibility and concludedthat sandstone, with greater than 10% porosityand in the pressure range of 1 to 103 MPa (145 to1.5 × 10−4 psi), exhibited changes in bulk volumethat were approximately equal to the changes in porevolume for any change in confining stress.
Zimmerman et al. (1986) defined four differentcompressibilities that quantify the relationshipsbetween fractional volume changes and stress or pres-sure variations. Three relationships were derivedbetween four different compressibilities, using themicromechanical theory based on classical linearelasticity. The relationships were shown to be correctfor sandstones. This classical theory of compressibil-ity has its limitations when extended to unconven-tional reservoir rocks. First, these analyticalderivations only considered one type of boundarycondition, that of uniform hydrostatic pressure overthe entire outer surface of the porous body as well asthroughout the entire interior and pore surface.However, for in situ conditions, the deviatoric stressmust be considered to describe the volumetric behav-ior of porous rocks. Second, deformation discussed inthe theory must lie in the elastic deformation domain,which is not the case for both coal and gas shaleformations because of the unique feature of theshrinkage or swelling effect. Therefore, further inves-tigation is warranted given that the unconventionalsources of gas, like coal and shale, would continueto be an increasingly important factor in the world-wide energy scenario. Following this, Li et al.(2004) derived a new theoretical relationship betweenrock compressibility and porosity, incorporating twoparameters of rock, elastic modulus, and Poisson’sratio. The authors concluded that rock compressibil-ity increased with increasing porosity, which is con-trary to the empirical relationship established byHall (1953). Additionally, the authors also claimed
LIU AND HARPALANI 1763
that rock compressibility depended on the rigidity ofrock skeleton. Jalalh (2006a) presented a set of pore-volume compressibility values for sandstones andlimestones from Hungarian reservoirs and found thatthe pore-volume compressibility increased withincreasing temperature. Using laboratory-measureddata, Jalalh (2006b) concluded that the pore-volumecompressibility values are in poor agreement withthe published compressibility correlations.Additionally, Jalalh suggested that the rock com-pressibility measurement should be one of the routinecore measurements in the laboratory.
Compressibility of Sorptive Rocks–Coal andGas Shale
Coal and gas shale are organic-rich porous media,with the unique feature of desorption of gas as a criti-cal factor in gas production. The release of gas indu-ces a corresponding shrinkage volumetric strainassociated with depletion of the sorbing gases. Thisis a well-accepted phenomenon (Pan and Connell,2007; Liu and Harpalani, 2013a). The shrinkage andswelling deformation can be characterized as shrink-age or swelling compressibility. The conceptualmechanism of matrix shrinkage and swelling isbriefly described in this section for the sake ofcompleteness.
When there is no gas adsorbed on the surface ofcoal or gas shale matrix, the surface molecules ofthe matrix lose the mechanical balance because onlythe long-range force is applied to them. The long-range force is usually referred as the non-molecularcontact force (Israelachvili, 2011). The direction ofthe long-range force is opposite of the normal to thesurface. Compared to internal molecules or atoms,
the potential energy of the molecules or atoms onthe surface, known as surface free energy, is higherresulting in achievement of a new balance of the sur-face molecules or atoms. The intermolecular distance(D1) between the surface and adjacent molecules (oratoms) is less than the distance between internal mol-ecules (D), as shown in Figure 1A. The effective sur-face energy is not only determined by the integralvalue of the potential energy, but is also influencedby the characteristics of the gas or fluid inside thepores. Based on the least-energy principle, the lowerthe effective surface energy, the more stable is thesystem. Because of the high effective surface energyof the coal matrix surface molecules, they tend toattract other molecules to decrease the surface energy(Liu and Harpalani, 2013a).
When gas is adsorbed on the coal matrix surface,the effective surface energy is reduced. On one hand,the gas molecules near the matrix surface attract coalor gas shale surface molecules and, after the matrixsurface molecules achieve a rebalance, the distancebetween the surface layer of molecules and the adja-cent molecules changes to D2 (D2 > D1). Normally,the force between solid molecules is greater than theforce between solid and gas molecules, as shown inFigure 1B as D2 < D. Hence, after methane getsadsorbed on the matrix, the surface layer thicknessof the matrix of sorptive material increases by D2 −D1 as shown in Figure 1. However, the matrix surfaceexpansion is resisted by the gas pressure within thepores. The gas pressure not only hampers the matrixsurface deformation in the outer normal direction,but also tends to increase the pore volume. Withincreasing gas pressure, the amount of adsorbed gasincreases and the attraction force increases. Withincreased intermolecular attraction force, the surface
Figure 1. Conceptual diagram ofsorptive matrix shrinkage and swell-ing mechanism with intermolecularand surface forces.
1764 Compressibility of Sorptive Material, Background and Theory
energy of the matrix decreases and the distancebetween surface matrix molecules and the adjacentlayer (D2) increases. Meanwhile, the higher the pres-sure, the more difficult is the deformation of coal sur-face matrix in the outer normal direction of the matrixbecause of the compression force of the pore pres-sure. After a certain pressure, the matrix volumeexpansion stops and starts to decrease with furtherincrease in pressure because the mechanical compres-sion of the solid rock dominates the volumetricbehavior (Liu and Harpalani, 2013a).
The volumetric response of coal matrix as a func-tion of desorption of gas has been studied both exper-imentally and theoretically. Coal matrix shrinkageresulting from gas desorption has been reported byseveral researchers (Moffat and Weale, 1955;Harpalani and Schraufnagel, 1990; Harpalani andChen, 1995; Seidle and Huitt, 1995; Levine, 1996;Harpalani and Chen, 1997; George and Barakat,2001; Robertson, 2005; Day et al., 2008; Harpalaniand Mitra, 2010; Majewska et al., 2010; Wang et al.,2012). A very comprehensive review is presented inLiu and Harpalani (2013a). To date, no matrix shrink-age or swelling data has been reported for gas shales.
Zheng et al. (1992) conducted a series of labora-tory studies to estimate the value of coal pore-volumecompressibility. The experiments were carried outunder both hydrostatic compression and uniaxialstrain conditions. The uniaxial strain condition,believed to best replicate the in situ condition,requires that the horizontal dimension remain con-stant. For the geometry of a collection of matchsticks,typically used to represent CBM reservoirs, this trans-lates to no change in the dimension of the collectionof matchsticks, although the horizontal dimension ofindividual matchsticks changes caused by stresschanges and matrix shrinkage effects. The verticalstress (σa) remains constant because of the unchangedoverburden load. In the reported study, two pore flu-ids, water and gases (helium, N2, and CO2), wereused. The results showed that Cp varied with chang-ing mean stress for the more cleated coal underhydrostatic stress conditions but remained constantunder uniaxial conditions when the pore fluid waswater. However, when helium and N2 were used, Cp
varied substantially greater than the range of appliedmean stress under hydrostatic conditions. When CO2
was used as the testing fluid, the overall variation inthe value of Cp was small because of the opening orclosure of cleats induced by matrix shrinkage orswelling.
In the same year, Seidle et al. (1992) carried outexperiments to measure Cp under hydrostatic stressusing brine water as the pore fluid and reported val-ues of Cp ranging from 0.0522 to 0.1373 MPa−1
(7.6 to 20 psi−1). The wide variance was related tothe uncertainties associated with the experimentalwork. More importantly, Seidle et al. (1992) datamay not be directly used for CBM production model-ing. The results did not take into account the matrixshrinkage effect because brine was used as the testingfluid. It was felt that, for purposes of permeabilitymodeling, a study on pore-volume compressibilityof coal should be carried out under in situ stress-and strain-controlled boundary conditions usingmethane as the testing fluid.
DEFINITIONS AND IN SITU CONDITIONS
This section is aimed at developing a methodology toestimate the different compressibilities of sorptivematerials under in situ conditions. The relationshipsbetween different compressibilities are mathemati-cally derived and establish the solid theoretical basisfor estimating the pore-volume compressibility.These relationships also provide a plausible meansto measure the different compressibility values inthe laboratory under replicated in situ conditions.
In Situ Condition–Uniaxial Strain
During unconventional resource development, theuniaxial strain condition is widely accepted to holdby both engineers and modelers. Under this condi-tion, the lateral boundaries of a reservoir are fixedand are not allowed to move, and the vertical stressremains constant because of the unchanged overbur-den load. This translates to zero lateral strain for thewhole reservoir formation with pressure depletion,although local variations may exist, for example,caused by drawdown at a well site (Palmer, 2009).Under the strain-controlled boundary, the horizontalstress has to be decreased because of zero horizontal
LIU AND HARPALANI 1765
strain when pore (reservoir) pressure declines withdrawdown. Mathematically, the uniaxial strain condi-tion is expressed by the following equation:
dσa = dεh = 0; (1)
in which σa is the vertical stress related to the over-burden and εh is the horizontal or lateral strain.Given that the uniaxial strain condition is believedto be the best representation of in situ conditionswhen evaluating or studying reservoirs, this alsobecomes the boundary condition in derivation of therelationships between different compressibilities.This is quite different from the previous studies thathad considered hydrostatic conditions. The work hasdistinct advantages over previous studies in severalways. First, the deviatoric stress applied on the sorp-tive material is taken into account when analyzingthe volumetric behavior. Second, the compressibili-ties estimated under uniaxial strain conditions can bedirectly applied to the reservoir simulation andmaterial balance calculations rather than using empir-ical correlations based on hydrostatic data.
Definition of Different CompressibilitiesUnder Uniaxial Strain Conditions
Four types of compressibility are encountered for sorp-tive materials. Among these, three common types ofcompressibility are commonly cited in the characteri-zation of conventional porous media. These includebulk compressibility (Cb), representing the relativechanges in bulk volume of the medium; grain (solid)compressibility (Cg), describing the relative changein the solid part of the medium; and pore-volumecompressibility (Cp), describing relative changes inpore volume. Mathematical definitions of these com-pressibilities are given by Zimmerman (1991) as:
Cbc =− 1Vb
�∂Vb
∂Pc
�Pp
(2)
Cbp =1Vb
�∂Vb
∂Pp
�Pc
(3)
Cpc =− 1Vp
�∂Vp
∂Pc
�Pp
(4)
Cpp =1Vp
�∂Vp
∂Pp
�Pc
(5)
Cg =�− 1Vb
�∂Vb
∂Pc
�Pp
=1Vp
�∂Vp
∂Pp
�Pc
��ΔðPc−PpÞ=0
�; (6)
in which Vb denotes the bulk volume of the sample,Vp is the pore volume, Pc is the confining stress,and Pp represents the pore pressure. For the abovecompressibility terms, the first subscript symbolizesthe type of the compressibility (b for bulk, p for porevolume, and g for grain), the second subscriptdenotes the changing pressure and/or stress (c forconfining stress, p for pore pressure), and the sub-script outside the parentheses indicates the pressureheld constant during loading of the sample.Zimmerman’s studies (Zimmerman, 1991) are trulynot applicable under uniaxial strain conditionsbecause the reservoir always behaves passively withvariations in pore pressure. In other words, the evolu-tion of applied external stresses changes passivelywith pore pressure variations. To best describe thevolumetric behavior of unconventional reservoirrock, all compressibilities should be redefined as afunction of pore pressure rather than of the confiningstress. Therefore, the bulk compressibility, pore-vol-ume compressibility, and solid matrix compressibilityare mathematically given as:
Cb =1Vb
×dVb
dp
����dσa=dεh=0
(7)
Cp =1Vp
×dVp
dp
����dσa=dεh=0
(8)
Cs =1Vs
×dVs
dp
����dσa=dεh=0
; (9)
in which the additional term Vs is the matrix volumeof sorptive media, such as coal and gas shale.Unlike conventional reservoir rocks, the compress-ibilities for coal and gas shale have to be referred toa certain gas because different gases have differentsorption capacities. If a sorbing gas, like methane orcarbon dioxide, is employed, unique matrix shrinkageor swelling compressibility must be introduced toquantify the non-elastic deformation with respect togas pressure. This is defined as
1766 Compressibility of Sorptive Material, Background and Theory
Cm =1Vm
×dVm
dp(10)
in which Vm is the matrix volume, dVm is the changein matrix volume related to the sorption effect only,and dp is the change in pressure of the sorbing gas.
STRESS–PRESSURE RELATIONSHIPSUNDER IN SITU CONDITIONS
In view of the continuum, three stresses exist at anypoint within bulk coal and gas shale, which aretreated as porous media. The three stresses includeexternal stress, internal pore pressure, and skeletonstress (a scalar). The skeleton stress is defined as thestress in the solid matrix that exists throughout thematrix, shown in Figure 2. These three stresses aredesignated σ, p, and Ps, respectively. The skeletonstress, which is not an independent parameter,depends on the external and internal (pore pressure)stresses and can be related to the two other stresses.This is explained diagrammatically in Figure 2.Assume that the sample of cross section A is sub-jected to an external stress, σ. If the pore space isfilled with a gas at pressure p and the solid experien-ces an average stress of Ps, then the force balanceyields
σA = ϕAp + ð1 − ϕÞAPs; (11)
in which, ϕ is the porosity. Dividing the above equa-tion by A gives
σ = ϕp + ð1 − ϕÞPs (12)
This relationship is referred to as the stress and/orpressure distribution equation. It is used to obtain avalue for Ps when σ, p, and ϕ are known. The onlyassumption implied in its derivation is that the arealporosity is equal to the volumetric porosity. Thiswas confirmed and applied to porous media byZimmerman (1991) during the derivation of effectivestress.
PROPOSED SHRINKAGE OR SWELLINGCOMPRESSIBILITY MODEL
Based purely on definition, the shrinkage or swellingcompressibility is the volumetric change of the matrixcaused by sorption alone related to unit change in thesorbing gas pressure. Because of the lack of gas shaleshrinkage and swelling data, we here presentthe theoretical derivation using coal as an example.The volumetric strain of coal is typically describedby the Langmuir-type function and was first proposedby Levine (1996) as
ε = εlp
p + Pε; (13)
in which ε is the sorption-induced volumetric strain atpressure p, εl represents the maximum strain, whichcan be achieved at infinite pressure, and Pε is thepressure at which coal attains 50% of the maximumstrain. This model is widely used for coal permeabil-ity modeling (Palmer and Mansoori, 1998; Shi andDurucan, 2005; Ma et al., 2011; Liu et al., 2012;Wang et al., 2012). It should be pointed out that thisLangmuir-type strain model can only be applied fora certain pressure range (∼0 to 15 MPa [∼0 to2176 psi]), after which the volumetric strain deviatesfrom the modeled relationship (Moffat and Weale,1955; Pan and Connell, 2007; Liu and Harpalani,2013a). Moreover, this model cannot be used to cal-culate the shrinkage or swelling compressibilitybecause the strain depicted by this model combinesthe effects of sorption-induced and mechanical-induced strains. To calculate Cm, pure shrinkage or
Figure 2. Illustration of stress and/or pressure distributionrelationship.
LIU AND HARPALANI 1767
swelling strain is required. Liu and Harpalani (2013a)proposed a theoretical model to quantify the volumet-ric behavior of coal with respect to both methane andcarbon dioxide. This model separates the sorption-induced strain from the mechanical-induced strainthus providing a theoretical basis for calculationof Cm. According to the strain model, the puresorption-induced strain is given as
εa =3aρRTEAV0
Zp
0
b1 + bp
dp; (14)
in which a and b are sorption Langmuir constants, ρis coal solid-phase density, R is universal gasconstant, T is temperature, EA is modulus of solidexpansion resulting from adsorption or desorption,and V0 is gas molar volume. For the sorptionLangmuir constants, “a” is known as the Langmuirvolume (m3∕ton), and “b” is known as pressure con-stant (MPa−1) (Harpalani, et al., 2006).
By definition, the first derivative of equation 14gives Cm, shown as follows:
Cm =dεadp
=�3aρRTEAV0
Zp
0
b1 + bp
dp� 0
(15)
Simplifying the above equation gives:
Cm =3aρRTEAV0
×b
1 + bp(16)
Cm is thus a function of gas pressure, its valuedecreasing with increasing pressure.
PROPOSED PORE-VOLUMECOMPRESSIBILITY MODEL
Laboratory Measurement Technique
Theoretically, the pore-volume compressibility canbe measured using the basic definition of Cp. Fromequation 8, if the pore volume strain (dVp∕Vp) canbe obtained successfully, Cp can be estimateddirectly. However, measuring the pore volume strainis difficult because of the extremely low fractureporosity for both coal and gas shale formations.Therefore, an indirect method was developed to
estimate Cp, using parameters that are measureablein the laboratory with reasonable confidence.
The porosity of coal or gas shale is defined as therelative volume of fracture pore space as a function ofthe bulk volume. Mathematically, it is given as
ϕ =Vp
Vb(17)
The bulk volume is the sum of fracture void vol-ume and solid matrix volume, given as
Vb = Vs + Vp (18)
in which Vs is the solid-phase volume.The differential form of equation 18 is
dVb = dVs + dVp (19)
Based on equations 7, 8, 9, 17, 18, and 19, thepore-volume compressibility is derived as:
Cp =1ϕ½Cb − ð1 − ϕÞCs� (20)
In this paper, Cb was measured. This is describedin detail in the second part of this two-part series (Liuand Harpalani, 2014), dealing with the experimentaldetails. In general, the bulk strain with pore pressuredepletion, under uniaxial strain conditions, can bedirectly measured in the laboratory. The value of Cb
can, therefore, be calculated through equation 7. Theparameter, Cs, can be estimated from experimentaldata. Using the two, Cp can be computed.
Theoretical Model Fundamentals andSelection
A theoretical model was developed in the paper thatallows computation of Cp of sorptive materials whencertain parameters are specified. The basis of themodel is strain balance. A relationship exists betweenthe bulk strain (εb), solid-phase matrix strain (εs), andvolumetric pore strain (εp) as shown:
εb = ð1 − ϕÞεs + ϕεp: (21)
For the uniaxial strain condition (boundary condi-tion), the volumetric responses of the bulk, solidmatrix, and pores are a set of binary functions of both
1768 Compressibility of Sorptive Material, Background and Theory
external stress and pore pressure. Mathematically,these are given as ( εb = f 1ðσ; pÞ
εs = f 2ðσ; pÞεp = f 3ðσ; pÞ
(22)
Differentiating equation 21 with respect to p, underuniaxial strain conditions, gives
dεbdp
= ð1 − ϕÞ dεsdp
+ ϕdεpdp
����dσa=dεh=0
(23)
Recalling the definitions of different compressibili-ties, the above equation can be rewritten as
Cp =1ϕ
�Cb − ð1 − ϕÞ dεs
dp
�����dσa=dεh=0
(24)
The solid-phase matrix strain (εs) is influenced bytwo separate effects, namely, sorption-induced strainand mechanical elastic strain. Based on the mostrecent work (Liu and Harpalani, 2013a), the sorp-tion-induced strain is given as equation 14.However, the mechanical-induced strain under uni-axial strain conditions is not the same as that pre-sented in Liu and Harpalani’s work because thestress carried by the solid phase is not only the porepressure p, but also the external stress. Based on thephilosophy followed by Liu and Harpalani (2013a),the mechanical-induced volumetric strain (εm) causedby stress and pressure is given as (Goodman, 1989)
εm =−3Ps
Eð1 − 2νÞ; (25)
in which E is Young’s modulus and ν is Poisson’sratio. Ps can be obtained by re-arranging equation 12as follows:
Ps =σ − ϕp1 − ϕ
(26)
Therefore, the overall solid-phase matrix strain underuniaxial strain conditions, given as the followingequation, can be calculated based on the assumptionthat the sorption-induced strain and mechanical-induced strain are purely additive:
εs =3aρRTEAV0
Zp
0
b1 + bp
dp −3ð1 − 2νÞ
E
ZPs
0dPs (27)
By inserting equation 27 into equation 24 andmeasuring Cb, the pore-volume compressibilitycan be estimated with variation of pore pressure.The model is semi-empirical in that some of theparameters are determined experimentally by actuallymeasuring one or two values of the compressibility.The variables required for implementation of thecomputational exercise are the bulk compressibility,initial fracture porosity, density of solid material,elastic moduli of the material, Poisson’s ratio,Langmuir sorption constants, and temperature.
DISCUSSION
For production of unconventional gases and CO2
sequestration, matrix shrinkage and swelling behav-ior has been shown to be important because it is asignificant factor in analysis of the dynamic per-meability or injectivity. To account for these relation-ships, several coal permeability models have beendeveloped (Sawyer et al., 1990; Palmer andMansoori, 1998; Shi and Durucan, 2005; Cui andBustin, 2005; Palmer et al., 2007; Ma et al., 2011;Wang et al., 2012; Liu and Harpalani, 2013b). Littleinformation is available for gas shales because mod-eling of shale permeability is still in its initial stages.The compressibilities of coal or gas shale are essentialto provide the input parameters for flow-modelingpurposes. All of the above permeability modelsinclude one or more compressibilities to predict thedynamic permeability, directly or indirectly.Therefore, a sound knowledge of compressibility forsorptive rocks would facilitate refinement of the cur-rent models and help users screen the models basedon data availability.
The solid-phase matrix compressibility, shrink-age or swelling compressibility, and pore-volumecompressibility are the most commonly used parame-ters in flow-modeling exercises. The models for thesethree are proposed and quantified as equations 9,16, and 24, respectively. The proposed work inthe preceding sections has several advantages com-pared with previous studies (Geertsma, 1957; Fatt,1958a, b; Zimmerman, 1991; Li et al., 2004). First,it integrates non-elastic volumetric strain occurringin unconventional resources with the elastic
LIU AND HARPALANI 1769
deformation to describe the volumetric strain evolu-tion of sorptive rocks. Second, the boundary condi-tion of the proposed work is based on realistic insitu conditions, namely, the uniaxial strain. Becauseof the application of deviatoric stress on the forma-tion during production, the compressibilities obtainedunder uniaxial strain conditions are more representa-tive for reservoir characterization. Compared to con-ventional hydrostatic stress boundary conditions(Geertsma, 1957; Zimmerman et al., 1986), theincreasing deviatoric stress under uniaxial strain con-ditions during production can cause pore collapse andmicrocrack generation, and propagation may result inincreased pore-volume compressibility, the effect forwhich cannot be extracted when using hydrostaticconditions. Third, the proposed compressibility mod-els are independent of effective stress. The effectivestress law for sorptive material is complicatedbecause of the sorption-induced non-elastic deforma-tion encountered. Liu and Harpalani (2013c) pointedout that the effective stress coefficient (Biot coeffi-cient, α) is a pressure-dependent parameter for sorp-tive materials, such as coal and shales. Even forconventional formations without the sorption effect,the Biot coefficient (α) varies from case to case andmay result in complications when carrying out anycompressibility calculations. To avoid this compli-cated term, application of the skeleton stress Ps
describes the elastic deformation behavior. Ps is ascalar and depends on the external stress and porepressure, as shown in equation 26. Last, the parame-ters required in these models are adsorption iso-thermal constants, coal-solid density, Young’smodulus, Poisson’s ratio, and stress evolution duringproduction. Each one of these parameters has a physi-cal meaning and can either be easily measured or con-veniently estimated, which makes the model fairlytransparent and easy to use. Hence, this model pro-vides a new and simple technique to estimate thevolumetric response of sorptive porous media withvariation of external stress and pore pressure (gas)under stress- and strain-controlled boundaryconditions.
As with any model, the proposed models haveshortcomings, somewhat limiting their application.For the shrinkage or swelling compressibility model,coal was assumed isotropic and homogeneous when
estimating the strain (Liu and Harpalani, 2013a).This assumption may be a limitation, based on theresearch results presented recently (Day et al., 2008;Pone et al., 2009). However, this model provides asound basis for estimation of the coal matrix shrink-age-or-swelling compressibility by separating theshrinkage-or-swelling effect from the mechanicalcompression effect. For pore-volume compressibility,the model shown in equation 24 is semi-empiricalbecause the bulk compressibility is measured directlyin the laboratory. Additionally, pore-volume com-pressibility model can only treat the solid macrofrac-ture system. If there are any joints in the formation,which is commonly the case, the model will not beapplicable.
CONCLUSIONS
This paper presents a theoretical model for shrinkageor swelling compressibility for sorbing gases and onefor semi-empirical pore-volume compressibility ofsorptive rocks under uniaxial strain conditions. Thepore-volume compressibility model can theoreticallyestimate the compressibility with variations in bothexternal stress and pore pressure during gasdepletion. The proposed models can also provide theinput parameters for various permeability models.Based on the work completed, several important con-clusions can be made. These are summarized asfollows.
1. Shrinkage or swelling compressibility is a functionof sorbing gas pressure, its value decreasing withincreasing gas pressure.
2. The skeleton stress (a scalar) Ps dominates themechanical-induced volumetric behavior of therock. The value of Ps is a dependent variable andcan be estimated by a stress- and/or pressure-dis-tribution equation.
3. The relationship between Cb, Cs, and Cp is derivedby the strain balance equation given as equa-tion 24. It can be used to estimate pore-volumecompressibility under in situ conditions.
4. The pore-volume compressibility is a pressure-de-pendent parameter for sorbing gases, as indicatedby the first term of equation 27.
1770 Compressibility of Sorptive Material, Background and Theory
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1772 Compressibility of Sorptive Material, Background and Theory
