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Laboratory characterisation of coal reservoir permeability for primary and enhanced coalbed methane recovery
International Journal of Coal Geology 82 (2010) 252–261
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International Journal of Coal Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j c o a l g e o
Laboratory characterisation of coal reservoir permeability for primary and enhanced coalbed methane recovery Zhejun Pan ⁎, Luke D. Connell, Michael Camilleri CSIRO Petroleum Resources, Ian Wark Laboratory, Bayview Avenue, Clayton, Victoria 3168, Australia
a r t i c l e
i n f o
Article history: Received 7 May 2009 Received in revised form 14 August 2009 Accepted 27 October 2009 Available online 10 November 2009 Keywords: Cleat compressibility CO2 sequestration Coal swelling Adsorption Triaxial cell
a b s t r a c t Coal permeability is highly sensitive to the stress. Meanwhile, coal swells with gas adsorption, and shrinks with gas desorption. Under reservoir conditions these strain changes affect the cleat porosity and thus permeability. Coal permeability models, such as the Palmer and Mansoori and Shi and Durucan models, relate the stress and swelling/shrinkage effect to permeability using an approximate geomechanical approach. Thus in order to apply these models, stress–permeability behaviour, swelling/shrinkage behaviour and the geomechanical properties of the coal must be estimated. This paper presents a methodology for the laboratory characterization of the Palmer and Mansoori and Shi and Durucan permeability models for reservoir simulation of ECBM and CO2 sequestration in coal. In this work a triaxial cell was used to measure gas permeability, adsorption, swelling and geomechanical properties of coal cores at a series of pore pressures and for CH4, CO2 and helium with pore pressures up to 13 MPa and confining pressures up to 20 MPa. Properties for the permeability models such as cleat compressibility, Young's modulus, Poisson's ratio and adsorption-induced swelling are calculated from the experimental measurements. Measurements on an Australian coal are presented. The results show that permeability decreases significantly with confining pressure and pore pressure. The permeability decline with pore pressure is a direct result of adsorption-induced coal swelling. Coal geomechanical properties show some variation with gas pressure and gas species, but there is no direct evidence of coal softening at high CO2 pressures for the coal sample studied. The experimental results also show that cleat compressibility changes with gas species and pressure. Then the measured properties were applied in the Shi and Durucan model to investigate the permeability behaviour during CO2 sequestration in coal. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Permeability in coal is more complicated than that in conventional gas reservoirs. Coal swells with gas adsorption and shrinks with desorption, changing the coal cleat apertures and thus the permeability under reservoir conditions. In addition coal permeability is highly sensitive to the effective stress. A number of permeability models have been developed for coal seams. (Gray, 1987; Seidle and Huitt, 1995; Palmer and Mansoori, 1998; Gilman and Beckie, 2000; Pekot and Reeves, 2003; Shi and Durucan, 2004; Shi and Durucan, 2005; Cui and Bustin, 2005; Cui et al., 2007; Wang et al., 2009). The Palmer and Mansoori (P&M) and Shi and Durucan (S&D) coal permeability models, which include a consideration of stress–permeability behaviour and the geomechanical effects of gas adsorption-induced coal swelling, are widely used in reservoir simulation. As with the other coal permeability models that represent geomechanical effects, the Palmer and Mansoori model assumes uniaxial strain and constant
⁎ Corresponding author. Tel.: + 61 3 9545 8394; fax: +61 3 9545 8380. E-mail address: [email protected] (Z. Pan). 0166-5162/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coal.2009.10.019
vertical stress conditions. These assumptions allow a simple, concise relationship to be derived for cleat porosity changes due to both pore pressure and coal swelling/shrinkage. ϕ = ϕ0 ½1−cm ðP−P0 Þ + cl
K BP BP0 −1 − M 1 + BP 1 + BP0
ð1Þ
where ϕ is the porosity, ϕ0 is the porosity at reference pressure, P is pore pressure, P0 is the reference pore pressure, cl and B are fitting parameters for the Langmuir-like model to describe volumetric strain with gas adsorption, K is the bulk modulus, M is the constrained axial modulus (Palmer and Mansoori, 1998). Meantime, the other parameters are defined as:
cm =
M=
1 K − + f −1 cr M M
ð2Þ
Eð1−νÞ ð1 + νÞð1−2νÞ
ð3Þ
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
K=
E 3ð1−2νÞ
ð4Þ
where f is a fraction from 0 to 1, cr is grain compressibility, E is the Young's modulus and v is the Poisson's ratio. The equation below is used to relate the porosity with permeability. k = k0
ϕ ϕ0
3
ð5Þ
where k is the permeability, k0 is the permeability at reference pressure. Recently, Palmer et al. (2007) modified the original P&M model to account for the exponential increase of absolute permeability, with a newly defined cm function: cm =
g K − + f −1 cr M M
ð6Þ
where g is a geometric term related to the orientation of the natural cleat system. The Shi and Durucan model is another widely used coal reservoir permeability model which describes the permeability change from a stress approach instead of a porosity approach: σ−σ0 = −
ν EεV ðP−P0 Þ + 3ð1−νÞ 1−ν
ð7Þ
where σ is the effective horizontal stress, σ0 is the effective horizontal stress at the initial reservoir pressure, εV is the volumetric swelling/ shrinkage strain (Shi and Durucan, 2004). To relate the permeability with effective stress, the equation below is used: −3cf ðσ−σ0 Þ
k = k0 e
ð8Þ
where cf is referred to as the cleat volume compressibility with respect to changes in the effective horizontal stress normal to the cleats (Shi and Durucan, 2004). Connell and Detournay (2009) used a coupled model to investigate the key geomechanical assumptions of uniaxial strain and constant vertical stress used in the derivation of the coal permeability models. In their hypothetical cases studies it was found that while the assumption of uniaxial strain was not a significant source of error, stress arching in overlying formations could lead to a departure from the constant vertical stress assumption and introduce error. Palmer (2009) presents a consideration of the application of the existing analytical coal permeability models to coalbed methane production and identified some behaviour not accounted for in these models. However, the models offer a tractable approach to explaining the complex influences on coal permeability and there has been widespread adoption in reservoir simulation. In order to apply these relationships in the reservoir simulation, Young's modulus, Poisson's ratio and unconstrained swelling must be estimated. In addition, rock grain compressibility and cleat porosity for P&M model and cleat compressibility for the S&D model need to be estimated as well. In principle one approach to characterising the coal permeability models would be to fit them to laboratory permeability measurements. However, a complication is that they have been derived for uniaxial strain and constant vertical stress conditions. While it is possible to replicate these conditions in the laboratory a simpler and more routine arrangement is testing under hydrostatic conditions. The work described in this paper employed an integrated methodology to estimate the various properties required in order to apply either the P&M or S&D permeability models. A series of measurements of gas adsorption, coal permeability and core strain were conducted on an Australian core sample from the southern
253
Sydney basin. The coal sample was bituminous coal from the Bulli seam and cored to 4.5 cm in diameter and 10.55 cm in length. Three gases were used including helium, methane and CO2. All measurements were conducted at 45 °C. The value of this work for reservoir characterisation purposes is in estimation of the effects of swelling and effective stress on permeability rather than estimated absolute permeability which is highly sensitive to the scale of measurement and has been shown to be a property which can only reliably be determined from well testing and history matching (Mavor and Saulsberry, 1996). 2. Experimental methods 2.1. Experiment apparatus description Fig. 1 shows the schematic of the Triaxial Multi-Gas Rig used for this work. A triaxial permeability cell was used for the experimental measurement of gas adsorption and permeability under hydrostatic conditions. Radial and axial displacements are measured at each adsorption step to obtain swelling strain. Displacements are also measured at each confining pressure change to estimate the bulk modulus. Four displacement gauges are installed with two to measure the axial displacement and the other two to measure the radial displacement. The two radial displacement gauges was installed perpendicularly. The displacement gauges are not presented in Fig. 1 to keep this figure neat. The core sample, usually 5 cm in diameter and 10 to 15 cm in length, is wrapped with a thin lead foil then a rubber sleeve before it is installed in the cell. The thin lead foil is to prevent gas diffusion from the core to the confining fluid at high sample pressures (Mazumder et al., 2006). CO2 would diffuse through the rubber sleeve to the confining fluid if the lead foil was not in place. The rig is engineered to sustain a maximum pore pressure of 16 MPa and a maximum confining pressure of 20 MPa. The sample cell and other parts of the rig are in a temperature controlled cabinet to maintain constant temperature during the experiment. 2.2. Adsorption measurement Prior to gas adsorption, the void volume, Vvoid, in the cell is determined by injecting known quantities of helium from a calibrated gas injection pump. Since helium is not significantly adsorbed, the void volume can be determined from measured values of temperature, pressure and the helium volume injected into the cell. This helium void volume measurement was performed at different pressures to investigate the consistency of the calculated volume. The mass-balance equation, expressed in volumetric terms, is: Vvoid =
PΔV ZT pump
= ZP T − ZP T 2
2
1
1
ð9Þ cell
where ΔV is the volume injected from the gas injection pump, Z is the compressibility factor of helium, T is the temperature, P is the pressure, subscripts “cell” and “pump” refer to conditions in the cell and pump, respectively, and subscripts “1” and “2” refer to conditions in the cell before and after injection of gas from the pump, respectively. This void volume is used in subsequent measurements of adsorption. The Gibbs excess adsorption (also known as the excess adsorption) is calculated directly from experimental quantities. For pure-gas adsorption measurements, a known quantity, ninj, of gas (e.g., methane) is injected from the gas injection pump into the cell. Some of the injected gas will be adsorbed, and the remainder, nGibbs unads, will exist in the equilibrium bulk (gas) phase in the cell. A mass balance is used to calculate the amount adsorbed, nGibbs ads , as: Gibbs
nads
Gibbs
= ninj −nunads
ð10Þ
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Fig. 1. Schematic plot of the Triaxial Multi-Gas Rig.
The amount injected can be determined from pressure, temperature and volume measurements of the pump:
ninj
PΔV = ZRT pump
ð11Þ
The amount of unadsorbed gas is calculated from conditions at equilibrium in the cell:
Gibbs
nunads =
PVvoid ZRT
ð12Þ cell
In (11) and (12), Z is the compressibility factor of the pure gas at the corresponding conditions of temperature and pressure. The above steps are repeated sequentially at higher pressures to yield a complete adsorption isotherm (Pan, 2004).
Eq. (13) is used to calculated absolute adsorption from the measured Gibbs excess adsorption:
Abs nads
=
Gibbs nads
ρads ρads −ρgas
! ð13Þ
where ρads is the adsorbed phase density, ρgas is the gas phase density. Gas compressibility factors and densities for Helium, CH4 and CO2 are calculated from the NIST webbook at http://webbook.nist.gov/ chemistry/fluid/. 2.3. Permeability measurement under triaxial condition The transient method of Brace et al. (1968) was used because of the shorter test durations required compared to steady state measurements. The Brace method involves observing the decay of a differential pressure between upstream and downstream vessels across a sample of interest. This pressure decay is combined with the
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255
work, gases including helium, methane and CO2 were used to measure coal permeability. Gas adsorption-induced coal swelling would complicate the measurement of cleat compressibility using the approach of Seidle et al., i.e. constant confining pressure and varying the pore pressure. Instead, a series of permeability measurements at constant pore pressure but differing confining pressure were used to determine cleat compressibility. However, Eq. (16) derived by Seidle et al. (1992) is proved valid for the new approach used in this work, as illustrated in Appendix A. ð−3cf ðσ−σ0 ÞÞ
ð16Þ
k = k0 e
where k is the permeability, k0 is the permeability at reference hydrostatic stress, cf is the cleat compressibility, σ is the hydrostatic stress, σ0 is the reference hydrostatic stress. Cleat compressibility is defined by:
Fig. 2. Pressure decay curve for permeability calculation.
vessel volumes in the analysis to relate the flow through the sample and thus determine the permeability (Brace et al., 1968). The pressure decay curve can be modelled as: ðPu −Pd Þ −αt =e ðPu;0 −Pd;0 Þ
ð14Þ
where Pu − Pd is the pressure difference between the up and down stream cylinders, in the experimental facility used for this work, measured by a differential pressure transducer; Pu,0 − Pd,0 is the pressure difference between the up and down stream cylinders at initial stage, t is time and α is described below: α=
k 1 1 V + 2 R V Vd μβL u
ð15Þ
where k is permeability; β is the gas compressibility; L is the sample length; VR is the sample volume; Vu and Vd are the volume of the up and downstream cylinders. A typical experimental pressure decay cure is presented in Fig. 2. The measurement of permeability with this equipment was verified through a program of work as part of its commissioning. In one activity permeability measurements were carried out on a set of permeability standards (artificial material of known permeability) purchased from Core Lab Petroleum Services. The results of the testing against the Core Lab supplied standards shows excellent agreement as shown in Table 1. 2.4. Cleat compressibility
cf =
1 ∂ϕf ϕf ∂Pp
ð17Þ
where ϕf is cleat porosity and Pp is pore pressure. Hydrostatic stress is defined by (see e.g. Zimmerman et al., 1986): σ = Pc −mPp
ð18Þ
where Pc is the confining pressure, Pp is the pore pressure and m is the effective stress coefficient. Cleat compressibility can be estimated by fitting (16) to permeability measurements with respect to effective stress. 2.5. Swelling measurement Swelling displacements are measured simultaneously with gas adsorption at a constant effective stress, which is controlled by tracking the pore pressure. A typical strain gauge measurement is presented in Fig. 3, which shows displacement growth with time due to the gas adsorption-induced coal swelling. To interpret the displacement results into swelling strain, correction should be made to account for the effective stress effect on strain as well as the displacement caused by compression of the rubber membrane around the core sample. A separate measurement was carried out to determine the relationship between the rubber strains with pressure and a linear relationship was identified and used to correct the swelling strain. It should also be noted that the radial swelling strain measured in this work may be underestimated for matrix swelling because cleat along the core may take part of the displacement. The axial swelling
To determine cleat compressibility, Seidle et al. (1992) derived a relationship between permeability and stress by idealising coal as a collection of matchsticks. This relationship is by Eq. (16) and was used in combination with the S&D permeability model as presented in Eq. (8). In Seidle et al.'s work (1992), water was used to measure coal permeability and involved a series of permeability measurements at constant confining pressure but differing pore pressure. In this current
Table 1 Comparison between permeabilities measured with CSIRO permeability facility and permeability standards supplied by Core Lab.
Std_223A Std_223B Std_162A
Standard perm (md)
Our measurement (md)
1.70 6.21 0.185
1.74 6.33 0.147
Fig. 3. Displacement change with time.
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strain measured represents the matrix swelling in that direction because there is no cleat perpendicular to the axial direction. However, the cleat system in the core used in this work is consolidated, thus it is still reasonable to assume that the measured radial swelling strain represents the real matrix swelling strain in those directions. Volume swelling is approximately represented by: εV = εr1 + εr2 + εa
ð19Þ
where εV is the volumetric swelling, εr1 and εr2 are the two radial strains perpendicular to each other, εa is the axial strain, which is the average of the results by the two axial displacement gauge. 2.6. Modulus and Poisson's ratio Radial and axial strains are measured with respect to effective stress at constant pore pressure and corrected for the strain due to the rubber membrane. Thus: K=
ΔPc εV
ð20Þ
where K is the bulk modulus. A separate uniaxial stress test was performed on the coal core. A load was applied in the axial direction and the axial and radial displacements are monitored and used to calculate the Young's modulus and Poisson's ratio. E=
F=A Δl = l
ð21Þ
where F is the load, A is the area of the interception of the core, Δl is the displacement in the axial direction, l is the length of the core. Poisson's ratio can be calculated from: υ=−
εr εa
ð22Þ
where εr is the radial strain and εa is the axial strain. Poisson's ratio can also be calculated from the relationship between bulk modulus and Young's modulus: K=
E 3ð1−2υÞ
ð23Þ
3. Experimental results 3.1. Adsorption results Methane and CO2 adsorption results are shown in Fig. 4. Both isotherms can be modelled by the Langmuir model with reasonable
Fig. 4. Methane and CO2 adsorption results.
Table 2 Langmuir constants. Component
VL (std m3/tonne)
PL (MPa)
Methane CO2
27.0 53.8
2.96 5.20
accuracy. For this coal, the amount of CO2 adsorbed is about twice that of methane adsorbed over the measured pressure range. The Langmuir constants for the two gases are listed in Table 2. 3.2. Permeability results Permeability measured using helium with respect to pore pressure at a constant pressure difference between confining and pore pressures of 3.0 MPa is shown in Fig. 5. The pressure difference is defined in Eq. (24). These helium permeability measurements show a slight permeability decline with increasing pore pressure. Because helium has a very low adsorption capacity to coal, the matrix swelling and associated change of the cleat apertures are regarded as negligible. Thus the decline in permeability can be partly attributed to the Klinkenberg (1941) effect, especially at low pressure region and partly attribute to effective stress difference when m is less than unity. Effective stress is defined in Eq. (25). However, permeability decline with Helium is not as significant compared to the permeability results measured by methane and CO2, which will be shown below. ΔP = Pc −Pp
ð24Þ
σ = Pc −mPp
ð25Þ
Permeability measured using methane is shown in Fig. 6. It can be seen that the permeability measured by methane under a constant pressure difference decreases with increasing pore pressure. This permeability decrease attributes to three factors: (1) the Klinkenberg (1941) effect, especially low pressure region, (2) cleat volume decrease due to matrix swelling under triaxial stress conditions, and (3) effective stress increase under constant pressure difference condition. As expected the impact of the effective stress on permeability is also significant, which can be seen from the permeability results at constant pore pressure but with varying confining pressures. Since at constant pore pressure, the change in confining pressure is equal to the change in effective stress and the change in pressure difference. The permeability decreases almost 50% from pressure difference of 2.0 MPa to 6.0 MPa at pore pressure of 0.9 MPa. However, the magnitude of the decrease reduced to around 25% from pressure difference of 2.0 MPa to 6.0 MPa at pore pressure of 12.8 MPa. Permeability measured using methane is lower than that measured using helium, due to partly the Klinkenberg effect, which
Fig. 5. Permeability measured using helium.
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
Fig. 6. Permeability measured using methane.
257
Fig. 9. Relationship between permeability using methane and effective stress.
suggests that permeability measured by different gas species differs, and partly coal swelling by methane adsorption to reduce cleat opening under triaxial conditions. Permeability measured using CO2 is presented in Fig. 7. The permeability measured by CO2 is slightly lower than that measured by methane at the same pore pressure. This is also attributed to the different Klinkenberg effect and different swelling behaviour. Moreover, the permeability decreases more than 50% from pore pressure of 3.0 MPa to 13.0 MPa at constant pressure difference of 2 MPa and the permeability decreases more than 70% from pore pressure of 3.0 MPa to 13.0 MPa at pressure difference of 6 MPa. The declines with CO2 are larger than those with methane, possibly due to higher swelling effect with CO2 adsorption in coal.
Fig. 10. Relationship between permeability using CO2 and effective stress.
3.3. Cleat compressibility
Fig. 7. Permeability measured using CO2.
The relationship between pressure difference and permeability measured using helium, methane and CO2 are shown in Figs. 8–10, respectively. Cleat compressibility can be calculated using (16) and is listed in Table 3 and also plotted in Fig. 11. It can be seen from Fig. 11 that the cleat compressibility slightly decreases with increasing pore pressure for methane and increases with increasing pore pressure for CO2. The permeability measurement by helium with pore pressure of 2.1 MPa was the first series of permeability, thus the core may not be well consolidated. Hence, the compressibility determined using these permeability measurements may be over estimated. The cleat compressibility by helium with pore pressure of 10.1 MPa is about 0.05, which is comparable to that determined by methane. It should also be noted that cleat volume change due to changing effective stress may change the Klinkenberg effect on the gas permeability, although the pore pressure remains constant. However, the gas pressures in these measurements are relatively high. Thus the Klinkenberg effect should be relatively small. Hence, the dominant
Table 3 Cleat compressibility summary. Pore pressure (MPa)
Pore pressure Cf Pore pressure Cf Cf (MPa− 1) (MPa) (MPa− 1) (MPa) (MPa− 1)
Helium 2.1 10.1 Fig. 8. Relationship between permeability using helium and effective stress.
CO2
CH4 0.0848 0.0485
0.9 3.4 7.5 12.8
0.0507 0.0472 0.0468 0.0366
3.0 6.4 9.8 13.3
0.0606 0.0654 0.1046 0.1211
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Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261 Table 4 Model parameters for Pan and Connell swelling model. ρS (g/cc)
Es (GPa)
υs
x
1.6
1.08
0.4
0.5
Table 5 Model parameters for Langmuir-like swelling model.
Fig. 11. Cleat compressibility by helium, methane and CO2.
3.4. Swelling strain Methane and CO2 adsorption-induced coal swelling strain results are shown in Fig. 12. It can be seen from the figure that there are slight anisotropy in the measurements of swelling behaviour for both methane and CO2. The Pan and Connell (2007) swelling model was applied to the swelling measurements and the results are also shown in Fig. 12. The Pan and Connell swelling model applies an energy balance approach, which assumes that the surface energy change caused by adsorption is equal to the elastic energy change of the coal solid. Thus swelling strain can be described by adsorption isotherm and coal geomechanical properties. A simplified version of the Pan and Connell model incorporating a Langmuir adsorption isotherm can be written as: ρs P f ðx; υs Þ− ð1−2υs Þ Es Es
ð26Þ
where ε is linear strain, R is the universal gas constant, T is temperature, L and B are Langmuir adsorption constants, ρS is the coal solid density, ES and υS are the Young's modulus and Poisson's ratio for coal solid, x is the coal structure parameter, and f is a structure function. Details of this model can be found in the literature (Pan and Connell, 2007). The values of the model parameters are listed in Table 4. The model represents the swelling stain with reasonable accuracy. The swelling strain can also be fitted by Langmuir-like equation. The parameters are listed in Table 5.
Fig. 12. Methane and CO2 adsorption-induced swelling.
εV (−)
Pε (MPa)
0.030 0.052
12.0 16.0
3.5. Modulus and Poisson's ratio
effect on permeability in these measurements is the effective stress when the pore pressure is constant.
ε = RTL lnð1 + BPÞ
Component Methane CO2
The bulk modulus measurements are shown in Fig. 13 where the values under methane and CO2 adsorption are very similar. The only point which is an outlier is that measured at a CO2 pore pressure of 10 MPa which is about 10%–20% lower than the other values. As suggested by some researchers (e.g., White et al., 2005), coal softens with CO2 adsorption. However, there is no obvious coal softening for the coal sample studied. This suggests that each coal may have a unique behaviour with respect to softening with CO2 adsorption. On average, the bulk modulus for this coal is about 1620 MPa. Three load and strain tests were performed on the core and these results are presented in Table 6. Poisson's ratio was calculated in these tests from the bulk modulus and Young's modulus. In this case, the average bulk modulus of 1620 MPa and average Young's modulus of 791 MPa were used which led to a Poisson's ratio of 0.418. 4. Application of laboratory measurements to permeability models From the above measurement, significant cleat compressibility change with pressure and gas type was observed. To identify the impact of variable cleat compressibility on permeability, two simple assessments were performed using the S&D permeability model and the measured data by defining the cf as a function of the gas composition, as defined by the following relationship: cf = yCO2 cf;CO2 + yCH4 cf;CH4
ð27Þ
where yCO2 and yCH4 are the CO2 and methane molar fractions in the gas phase, cf,CO2 and cf,CH4 are the compressibilities using CO2 and
Fig. 13. Bulk modulus measured at each pore pressure.
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259
Table 6 Load and strain test results. Stress (MPa)
Axial Strain
E (MPa)
1.825 1.864 1.787
0.00232 0.00231 0.00229 Average:
786.6 807.0 780.0 791.2
methane at the same pressure as the mixture. The mixture swelling was predicted by the Pan and Connell swelling model, which can predict mixed-gas adsorption-induced swelling strain from pure-gas adsorption-induced swelling (Pan and Connell, 2007):
ε = RT
N ρs P f ðx; υs Þ ∑ Li lnð1 + Bi yi PÞ− ð1−2υs Þ Es Es i=1
ð28Þ
where i is the ith gas component, yi is the molar fraction of the ith gas component in the gas phase. The linear swelling/shrinkage strain is approximately one third of volumetric swelling/shrinkage strain. Eq. (27) is then applied to (8) and Eq. (28) is applied to Eq. (7) assuming that linear swelling/shrinkage strain is approximately one third of volumetric swelling/shrinkage strain to investigate permeability change due to swelling and cleat compressibility change. Fig. 14 presents an example using the S&D permeability model for a constant pore pressure of 9 MPa, with the gas composition changing from pure methane to pure CO2. Fig. 14 compares the permeability calculated with a variable cf using (27) and a constant cf of 0.05. The results are shown in Fig. 14, in which it can be seen that permeability declines more significantly with variable cleat compressibility. The difference is larger at higher CO2 compositions due to the larger difference in cleat compressibility. In the second example the pore pressure varies from 9 MPa to 14 MPa and at the same time the methane composition in the gas phase decreases linearly from 1 to 0. The results are presented in Fig. 15. The permeability decline using variable cleat compressibility is much larger than that of using constant cleat compressibility. Similar analyses using the P&M permeability model can also be performed to investigate permeability change due to swelling and cleat compressibility change, by assuming an initial cleat porosity, which cannot be directly measured in the laboratory. However the calculated permeability behaviour using the P&M model is very sensitive to the initial cleat porosity. An arbitrary cleat porosity could be misleading to the interpretation of the permeability behaviour.
Fig. 15. Permeability prediction with changing pressure using S&D model.
Thus analyses using the P&M permeability model were not included in this work. 5. Conclusions This work has presented a methodology of laboratory characterisation of coal core for CBM, ECBM and CO2 sequestration in coal. The parameters used in the popular P&M and S&D permeability models can be obtained through a series of permeability and strain measurements after gas adsorption equilibration at various pore and confining pressures. An example of this procedure is presented for an Australian coal sample from the southern Sydney Basin. The behaviour of permeability with respect to effective stress is well described using the accepted exponential relationship. As expected the measurements show permeability declines with increasing pore pressure at constant effective stress in response to coal swelling with gas adsorption with the magnitude of the decline depending on the gas type. Simultaneous measurements of coal strain found that the coal sample swells about 2.4% in CO2 at 13.5 MPa and about 1.6% in methane at 12.8 MPa. The swelling strains were modelled by Pan and Connell swelling model with reasonable accuracy and can be explained with empirical Langmuir-like equations too. The bulk modulus was measured with gas adsorption and no evidence was found of coal softening with CO2 adsorption for this coal. The measurements also show that the cleat compressibility changes with pore pressure and gas type, which suggests the cleat compressibility should not be regarded as constant for some coals. In addition, variable cleat compressibility may have significant impact on permeability predictions suggested by the simple analysis conducted in this work. Acknowledgement Financial support for this work was provided by CSIRO Energy Transformed Flagship. Appendix A. Permeability–stress relation in triaxial condition
Fig. 14. Permeability prediction at constant pressure using S&D model.
Compared with the experimental work by Seidle et al. (1992), gases instead of water were used to measure the coal permeability. To avoid possible permeability change due to gas adsorption-induced coal swelling, permeability was measured at constant pore pressure with changing confining pressure to determine the permeability– effective stress relationship. In Seidle et al.'s work (1992), the relationship between permeability and stress was determined by permeability measurements under various pore pressures with a
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constant confining pressure. In this appendix, the permeability–stress relation was derived for the experimental conditions applied in this work. Meanwhile, the permeability–stress relation derived by Seidle et al. (1992) was revisited and compared. Following Seidle et al.'s work (1992) assuming the matchstick geometrical model of coal porosity, permeability can be related to the cleat porosity by: 1 2 3 a ϕf 48
k=
ðA 1Þ
Simplifying (A-10) and using (A-1), (A-5) and (A-9): ∂k 2ð1−2υÞ 3 ∂ϕf =k − E mϕf ∂Pp ∂σ
ðA 3Þ
where ε is the strain. Because the sample is in hydrostatic stress: ε=
1−2υ σ E
∂ϕf ∂Pp
1 cf = ϕf
! ðA 12Þ Pc
3 ≪m cf For coal, 2ð12υÞ E
Thus, (A-11) can be integrated to: ðA 2Þ
where σ is the hydrostatic stress. Again following Seidle et al.'s work (1992), ∂a ∂ε =a ∂σ ∂σ
ðA 11Þ
Cleat volume compressibility, an important parameter in coal reservoir permeability, is defined as:
where k is the permeability, a is the cleat spacing, ϕf is the cleat porosity. Differentiating (A-1) with respect to hydrostatic stress: ∂k 2aϕ3f ∂a 3a2 ϕ2f ∂ϕf = + 48 ∂σ 48 ∂σ ∂σ
!
ðA 4Þ
3c − f ðσ−σ0 Þ k = k0 e m
ðA 13Þ
Using Eq. (A-6) and since Pc is constant, (A-13) can be rewritten as: ð3cf ðPp −Pp;0 ÞÞ
k = k0 e
ðA 14Þ
In this work, Pp was kept constant. Differentiating (A-6) with respect to σ, it yields: ∂Pc =1 ∂σ PP
ðA 15Þ
where E is the Young's modulus and υ is the Poisson's ratio. Differentiating (A-4) with respective to hydrostatic stress and then (A-3) can be rewritten as:
Following Seidle et al.'s work, using the chain rule, the derivative of cleat porosity with respect to hydrostatic stress can be rewritten as:
∂a 1−2υ =a E ∂σ
∂ϕf ∂ϕf ∂Pc = ∂σ ∂Pc ∂σ
ðA 5Þ
Hydrostatic stress or effective stress can be defined by (see e.g. Zimmerman et al., 1986): σ = Pc −mPp
ðA 6Þ
where Pc is the confining pressure, Pp is the pore pressure and m is the effective stress coefficient. In Seidle et al. (1992)'s work, Pc was kept constant and m was assumed to be 1. Here we revisited Seidle et al's work with a more general scenario where m is less than 1. Differentiating (A-6) with respect to σ, it yields: ∂Pp ∂σ
!
Simplifying (A-10) and using (A-1), (A-5) and (A-16), it yields: ∂k 2ð1−2υÞ 3 ∂ϕf =k + E ϕf ∂Pc ∂σ
ðA 17Þ
Define cfc as: cfc = −
1 ϕf
∂ϕf ∂Pc
ðA 18Þ Pp
Thus, (A-17) can be rewritten as: =−
Pc
ðA 16Þ
1 m
ðA 7Þ
Following Seidle et al.'s work, using the chain rule, the derivative of cleat porosity with respect to hydrostatic stress can be rewritten as: ∂ϕf ∂ϕf ∂Pp = ∂σ ∂Pp ∂σ
ðA 8Þ
ðA 9Þ
For coal, 2ð1−2υÞ ≪3cf E Thus, (A-19) can be integrated to: ð−3cf ðσ−σ0 ÞÞ
k = k0 e
ðA 20Þ
ð−3cfc ðPc −Pc;0 ÞÞ
k = k0 e
ðA 21Þ
The following is to demonstrate that cf and cfc are equal. Porosity of the cleat is defined as:
Thus (A-2) can be rewritten as: ∂k 2aϕ3f að1−2υÞ 3a2 ϕ2f ∂ϕf = + 48 48 ∂σ E ∂σ
ðA 19Þ
Since in this work, Pp is kept constant, Using Eq. (A-6), (A-20) can be rewritten as:
Using (A-7), (A-8) can be rewritten as: ∂ϕf 1 ∂ϕf =− m ∂Pp ∂σ
∂k 2ð1−2υÞ =k −3cfc E ∂σ
ðA 10Þ
ϕf =
Vp Vb
ðA 22Þ
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
where Vp is the pore volume of cleat, Vb is the bulk volume. Then (A-12) can be rewritten as: 0 cf =
1 Vp Vb
1
V ∂ p @ Vb A
∂Pp
ðA 23Þ
Pc
Simplifying (A-23) yields: ∂Vp
1 cf = Vp
∂Pp
! Pc
1 − Vb
∂Vb ∂Pp
! ðA 24Þ Pc
Define compressibilities cpp = V1p Thus (A-24) can be rewritten as:
∂Vp ∂Pp P c
and cbp =
1 Vb
cf = cpp −cbp
∂Vb ∂Pp P c
ðA 25Þ
Meantime, using (A-22), (A-18) can be rewritten as: 0 V1 p 1 @∂ Vb A cfc = − V p ∂Pc Vb
ðA 26Þ Pp
Simplifying (A-26) and yields: 1 cfc = − Vp
∂Vp ∂Pc
! + Pp
1 ∂Vb Vb ∂Pc Pp
Defining compressibilities cpc = − V1p Then (A-27) can be rewritten as:
ðA 27Þ
∂Vp ∂Pc P p
and cbc = − V1b
cfc = cpc −cbc
∂Vb ∂Pc P p
ðA 28Þ
From Zimmerman et al.'s (1986) work, cpp = cpc −cr and cbp = cbc −cr
ðA 29Þ
where cr is the solid or grain compressibility. Applying (A-29), (A-28) can be rewritten as: cf = cpc −cr −ðcbc −cr Þ = cpc −cbc
ðA 30Þ
Comparing (A-28) and (A-30), the relationship below can be obtained: cfc = cf
ðA 31Þ
Thus (A-14) and (A-21) are equivalent to obtain cleat compressibility from the two different experimental procedures. However, using gases to measure permeability enables us to investigate impact of gas species on cleat compressibility.
261
References Brace, W.F., Walsh, J.B., Frangos, W.T., 1968. Permeability of granite under high pressure. Journal of Geophysical Research 73 (6), 2225–2236. Connell, L.D., Detournay, C., 2009. Coupled flow and geomechanical processes during enhanced coal seam methane recovery through CO2 sequestration. International Journal of Coal Geology 77 (1–2), 222–233. Cui, X., Bustin, R.M., 2005. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. American Association of Petroleum Geologists Bulletin 89 (9), 1181–1202. Cui, X., Bustin, R.M., Chikatamarla, L., 2007. Adsorption-induced coal swelling and stress: implications for methane production and acid gas sequestration into coal seams. Journal of Geophysical Research 112, B10202. Gilman, A., Beckie, R., 2000. Flow of coalbed methane to a gallery. Transport in Porous Media 41, 1–16. Gray, I., 1987. Reservoir engineering in coal seams: Part 1 — the physical process of gas storage and movement in coal seams. SPE Reservoir Engineering, February 1987, 28–34. Klinkenberg, L.J., 1941. The permeability of porous media to liquids and gases. Drill. and Prod. Prac, API. 200–213. Mavor, M.J., Saulsberry, J.L., 1996. Testing coalbed methane wells. In: Saulsberry, J.L., Schafer, P.S., Schraufnagel, R.A. (Eds.), A Guide to Coalbed Methane Reservoir Engineering. Mazumder, S., Karnik, A.A., Wolf, K.-H.A.A., 2006. Swelling of coal in response to CO2 sequestration for ECBM and its effect on fracture permeability. SPE Journal 11 (3), 390–398. Palmer, I.D., 2009. Permeability changes in coal: analytical modelling. International Journal of Coal Geology 77 (1–2), 119–126. Palmer, I., Mansoori, J., 1998. How permeability depends on stress and pore pressure in coalbeds: a new model. SPEREE 1 (6), 539–544 SPE-52607-PA. Palmer, I.D., Mavor, M., Gunter, B., 2007. Permeability changes in coal seams during production and injection. International Coalbed Methane Symposium. University of Alabama, Tuscaloosa. Paper 0713. Pan, Z., 2004. Modeling of gas adsorption using two-dimensional equations of state. Ph.D Dissertation, Oklahoma State University, Stillwater, OK. Pan, Z., Connell, L.D., 2007. A theoretical model for gas adsorption-induced coal swelling. International Journal of Coal Geology 69 (4), 243–252. Pekot, L.J., Reeves, S.R., 2003. Modeling the effects of matrix shrinkage and differential swelling on coalbed methane recovery and carbon sequestration. Proceedings of the 2003 International Coalbed Methane Symposium. University of Alabama, Tuscaloosa. Paper 0328. Seidle, J.P., Huitt, L.G., 1995. Experimental measurement of coal matrix shrinkage due to gas desorption and implications for cleat permeability increase. SPE International Meeting on Petroleum Engineering, Beijing, China, 14–17 November. SPE 30010. Seidle, J.P., Jeansonne, M.W., Erickson, D.J., 1992. Application of matchstick geometry to stress dependent permeability in coals. SPE Rocky Mountain Regional Meeting, SPE 24361. Casper, Wyoming. Shi, J.Q., Durucan, S., 2004. Drawdown induced changes in permeability of coalbeds: a new interpretation of the reservoir response to primary recovery. Transport in Porous Media 56, 1–16. Shi, J.Q., Durucan, S., 2005. A model for changes in coalbed permeability during primary and enhanced methane recovery. SPEREE 8, 291–299 SPE-87230-PA. Wang, G.X., Massarotto, P., Rudolph, V., 2009. An improved permeability model of coal for coalbed methane recovery and CO2 geosequestration. International Journal of Coal Geology 77, 127–136. White, C.M., Smith, D.H., Jones, K.L., Goodman, A.L., Jikich, S.A., LaCount, R.B., DuBose, S.B., Ozdemir, E., Morsi, B.I., Schroeder, K.T., 2005. Sequestration of carbon dioxide in coal with enhanced coalbed methane recovery—a review. Energy and Fuels 19 (3), 659–724. Zimmerman, R.W., Somerton, W.H., King, S.M., 1986. Compressibility of porous rocks. Journal of Geophysical Research 91 (B12), 12,765–12,777.
Contents lists available at ScienceDirect
International Journal of Coal Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j c o a l g e o
Laboratory characterisation of coal reservoir permeability for primary and enhanced coalbed methane recovery Zhejun Pan ⁎, Luke D. Connell, Michael Camilleri CSIRO Petroleum Resources, Ian Wark Laboratory, Bayview Avenue, Clayton, Victoria 3168, Australia
a r t i c l e
i n f o
Article history: Received 7 May 2009 Received in revised form 14 August 2009 Accepted 27 October 2009 Available online 10 November 2009 Keywords: Cleat compressibility CO2 sequestration Coal swelling Adsorption Triaxial cell
a b s t r a c t Coal permeability is highly sensitive to the stress. Meanwhile, coal swells with gas adsorption, and shrinks with gas desorption. Under reservoir conditions these strain changes affect the cleat porosity and thus permeability. Coal permeability models, such as the Palmer and Mansoori and Shi and Durucan models, relate the stress and swelling/shrinkage effect to permeability using an approximate geomechanical approach. Thus in order to apply these models, stress–permeability behaviour, swelling/shrinkage behaviour and the geomechanical properties of the coal must be estimated. This paper presents a methodology for the laboratory characterization of the Palmer and Mansoori and Shi and Durucan permeability models for reservoir simulation of ECBM and CO2 sequestration in coal. In this work a triaxial cell was used to measure gas permeability, adsorption, swelling and geomechanical properties of coal cores at a series of pore pressures and for CH4, CO2 and helium with pore pressures up to 13 MPa and confining pressures up to 20 MPa. Properties for the permeability models such as cleat compressibility, Young's modulus, Poisson's ratio and adsorption-induced swelling are calculated from the experimental measurements. Measurements on an Australian coal are presented. The results show that permeability decreases significantly with confining pressure and pore pressure. The permeability decline with pore pressure is a direct result of adsorption-induced coal swelling. Coal geomechanical properties show some variation with gas pressure and gas species, but there is no direct evidence of coal softening at high CO2 pressures for the coal sample studied. The experimental results also show that cleat compressibility changes with gas species and pressure. Then the measured properties were applied in the Shi and Durucan model to investigate the permeability behaviour during CO2 sequestration in coal. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Permeability in coal is more complicated than that in conventional gas reservoirs. Coal swells with gas adsorption and shrinks with desorption, changing the coal cleat apertures and thus the permeability under reservoir conditions. In addition coal permeability is highly sensitive to the effective stress. A number of permeability models have been developed for coal seams. (Gray, 1987; Seidle and Huitt, 1995; Palmer and Mansoori, 1998; Gilman and Beckie, 2000; Pekot and Reeves, 2003; Shi and Durucan, 2004; Shi and Durucan, 2005; Cui and Bustin, 2005; Cui et al., 2007; Wang et al., 2009). The Palmer and Mansoori (P&M) and Shi and Durucan (S&D) coal permeability models, which include a consideration of stress–permeability behaviour and the geomechanical effects of gas adsorption-induced coal swelling, are widely used in reservoir simulation. As with the other coal permeability models that represent geomechanical effects, the Palmer and Mansoori model assumes uniaxial strain and constant
⁎ Corresponding author. Tel.: + 61 3 9545 8394; fax: +61 3 9545 8380. E-mail address: [email protected] (Z. Pan). 0166-5162/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.coal.2009.10.019
vertical stress conditions. These assumptions allow a simple, concise relationship to be derived for cleat porosity changes due to both pore pressure and coal swelling/shrinkage. ϕ = ϕ0 ½1−cm ðP−P0 Þ + cl
K BP BP0 −1 − M 1 + BP 1 + BP0
ð1Þ
where ϕ is the porosity, ϕ0 is the porosity at reference pressure, P is pore pressure, P0 is the reference pore pressure, cl and B are fitting parameters for the Langmuir-like model to describe volumetric strain with gas adsorption, K is the bulk modulus, M is the constrained axial modulus (Palmer and Mansoori, 1998). Meantime, the other parameters are defined as:
cm =
M=
1 K − + f −1 cr M M
ð2Þ
Eð1−νÞ ð1 + νÞð1−2νÞ
ð3Þ
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
K=
E 3ð1−2νÞ
ð4Þ
where f is a fraction from 0 to 1, cr is grain compressibility, E is the Young's modulus and v is the Poisson's ratio. The equation below is used to relate the porosity with permeability. k = k0
ϕ ϕ0
3
ð5Þ
where k is the permeability, k0 is the permeability at reference pressure. Recently, Palmer et al. (2007) modified the original P&M model to account for the exponential increase of absolute permeability, with a newly defined cm function: cm =
g K − + f −1 cr M M
ð6Þ
where g is a geometric term related to the orientation of the natural cleat system. The Shi and Durucan model is another widely used coal reservoir permeability model which describes the permeability change from a stress approach instead of a porosity approach: σ−σ0 = −
ν EεV ðP−P0 Þ + 3ð1−νÞ 1−ν
ð7Þ
where σ is the effective horizontal stress, σ0 is the effective horizontal stress at the initial reservoir pressure, εV is the volumetric swelling/ shrinkage strain (Shi and Durucan, 2004). To relate the permeability with effective stress, the equation below is used: −3cf ðσ−σ0 Þ
k = k0 e
ð8Þ
where cf is referred to as the cleat volume compressibility with respect to changes in the effective horizontal stress normal to the cleats (Shi and Durucan, 2004). Connell and Detournay (2009) used a coupled model to investigate the key geomechanical assumptions of uniaxial strain and constant vertical stress used in the derivation of the coal permeability models. In their hypothetical cases studies it was found that while the assumption of uniaxial strain was not a significant source of error, stress arching in overlying formations could lead to a departure from the constant vertical stress assumption and introduce error. Palmer (2009) presents a consideration of the application of the existing analytical coal permeability models to coalbed methane production and identified some behaviour not accounted for in these models. However, the models offer a tractable approach to explaining the complex influences on coal permeability and there has been widespread adoption in reservoir simulation. In order to apply these relationships in the reservoir simulation, Young's modulus, Poisson's ratio and unconstrained swelling must be estimated. In addition, rock grain compressibility and cleat porosity for P&M model and cleat compressibility for the S&D model need to be estimated as well. In principle one approach to characterising the coal permeability models would be to fit them to laboratory permeability measurements. However, a complication is that they have been derived for uniaxial strain and constant vertical stress conditions. While it is possible to replicate these conditions in the laboratory a simpler and more routine arrangement is testing under hydrostatic conditions. The work described in this paper employed an integrated methodology to estimate the various properties required in order to apply either the P&M or S&D permeability models. A series of measurements of gas adsorption, coal permeability and core strain were conducted on an Australian core sample from the southern
253
Sydney basin. The coal sample was bituminous coal from the Bulli seam and cored to 4.5 cm in diameter and 10.55 cm in length. Three gases were used including helium, methane and CO2. All measurements were conducted at 45 °C. The value of this work for reservoir characterisation purposes is in estimation of the effects of swelling and effective stress on permeability rather than estimated absolute permeability which is highly sensitive to the scale of measurement and has been shown to be a property which can only reliably be determined from well testing and history matching (Mavor and Saulsberry, 1996). 2. Experimental methods 2.1. Experiment apparatus description Fig. 1 shows the schematic of the Triaxial Multi-Gas Rig used for this work. A triaxial permeability cell was used for the experimental measurement of gas adsorption and permeability under hydrostatic conditions. Radial and axial displacements are measured at each adsorption step to obtain swelling strain. Displacements are also measured at each confining pressure change to estimate the bulk modulus. Four displacement gauges are installed with two to measure the axial displacement and the other two to measure the radial displacement. The two radial displacement gauges was installed perpendicularly. The displacement gauges are not presented in Fig. 1 to keep this figure neat. The core sample, usually 5 cm in diameter and 10 to 15 cm in length, is wrapped with a thin lead foil then a rubber sleeve before it is installed in the cell. The thin lead foil is to prevent gas diffusion from the core to the confining fluid at high sample pressures (Mazumder et al., 2006). CO2 would diffuse through the rubber sleeve to the confining fluid if the lead foil was not in place. The rig is engineered to sustain a maximum pore pressure of 16 MPa and a maximum confining pressure of 20 MPa. The sample cell and other parts of the rig are in a temperature controlled cabinet to maintain constant temperature during the experiment. 2.2. Adsorption measurement Prior to gas adsorption, the void volume, Vvoid, in the cell is determined by injecting known quantities of helium from a calibrated gas injection pump. Since helium is not significantly adsorbed, the void volume can be determined from measured values of temperature, pressure and the helium volume injected into the cell. This helium void volume measurement was performed at different pressures to investigate the consistency of the calculated volume. The mass-balance equation, expressed in volumetric terms, is: Vvoid =
PΔV ZT pump
= ZP T − ZP T 2
2
1
1
ð9Þ cell
where ΔV is the volume injected from the gas injection pump, Z is the compressibility factor of helium, T is the temperature, P is the pressure, subscripts “cell” and “pump” refer to conditions in the cell and pump, respectively, and subscripts “1” and “2” refer to conditions in the cell before and after injection of gas from the pump, respectively. This void volume is used in subsequent measurements of adsorption. The Gibbs excess adsorption (also known as the excess adsorption) is calculated directly from experimental quantities. For pure-gas adsorption measurements, a known quantity, ninj, of gas (e.g., methane) is injected from the gas injection pump into the cell. Some of the injected gas will be adsorbed, and the remainder, nGibbs unads, will exist in the equilibrium bulk (gas) phase in the cell. A mass balance is used to calculate the amount adsorbed, nGibbs ads , as: Gibbs
nads
Gibbs
= ninj −nunads
ð10Þ
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Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
Fig. 1. Schematic plot of the Triaxial Multi-Gas Rig.
The amount injected can be determined from pressure, temperature and volume measurements of the pump:
ninj
PΔV = ZRT pump
ð11Þ
The amount of unadsorbed gas is calculated from conditions at equilibrium in the cell:
Gibbs
nunads =
PVvoid ZRT
ð12Þ cell
In (11) and (12), Z is the compressibility factor of the pure gas at the corresponding conditions of temperature and pressure. The above steps are repeated sequentially at higher pressures to yield a complete adsorption isotherm (Pan, 2004).
Eq. (13) is used to calculated absolute adsorption from the measured Gibbs excess adsorption:
Abs nads
=
Gibbs nads
ρads ρads −ρgas
! ð13Þ
where ρads is the adsorbed phase density, ρgas is the gas phase density. Gas compressibility factors and densities for Helium, CH4 and CO2 are calculated from the NIST webbook at http://webbook.nist.gov/ chemistry/fluid/. 2.3. Permeability measurement under triaxial condition The transient method of Brace et al. (1968) was used because of the shorter test durations required compared to steady state measurements. The Brace method involves observing the decay of a differential pressure between upstream and downstream vessels across a sample of interest. This pressure decay is combined with the
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
255
work, gases including helium, methane and CO2 were used to measure coal permeability. Gas adsorption-induced coal swelling would complicate the measurement of cleat compressibility using the approach of Seidle et al., i.e. constant confining pressure and varying the pore pressure. Instead, a series of permeability measurements at constant pore pressure but differing confining pressure were used to determine cleat compressibility. However, Eq. (16) derived by Seidle et al. (1992) is proved valid for the new approach used in this work, as illustrated in Appendix A. ð−3cf ðσ−σ0 ÞÞ
ð16Þ
k = k0 e
where k is the permeability, k0 is the permeability at reference hydrostatic stress, cf is the cleat compressibility, σ is the hydrostatic stress, σ0 is the reference hydrostatic stress. Cleat compressibility is defined by:
Fig. 2. Pressure decay curve for permeability calculation.
vessel volumes in the analysis to relate the flow through the sample and thus determine the permeability (Brace et al., 1968). The pressure decay curve can be modelled as: ðPu −Pd Þ −αt =e ðPu;0 −Pd;0 Þ
ð14Þ
where Pu − Pd is the pressure difference between the up and down stream cylinders, in the experimental facility used for this work, measured by a differential pressure transducer; Pu,0 − Pd,0 is the pressure difference between the up and down stream cylinders at initial stage, t is time and α is described below: α=
k 1 1 V + 2 R V Vd μβL u
ð15Þ
where k is permeability; β is the gas compressibility; L is the sample length; VR is the sample volume; Vu and Vd are the volume of the up and downstream cylinders. A typical experimental pressure decay cure is presented in Fig. 2. The measurement of permeability with this equipment was verified through a program of work as part of its commissioning. In one activity permeability measurements were carried out on a set of permeability standards (artificial material of known permeability) purchased from Core Lab Petroleum Services. The results of the testing against the Core Lab supplied standards shows excellent agreement as shown in Table 1. 2.4. Cleat compressibility
cf =
1 ∂ϕf ϕf ∂Pp
ð17Þ
where ϕf is cleat porosity and Pp is pore pressure. Hydrostatic stress is defined by (see e.g. Zimmerman et al., 1986): σ = Pc −mPp
ð18Þ
where Pc is the confining pressure, Pp is the pore pressure and m is the effective stress coefficient. Cleat compressibility can be estimated by fitting (16) to permeability measurements with respect to effective stress. 2.5. Swelling measurement Swelling displacements are measured simultaneously with gas adsorption at a constant effective stress, which is controlled by tracking the pore pressure. A typical strain gauge measurement is presented in Fig. 3, which shows displacement growth with time due to the gas adsorption-induced coal swelling. To interpret the displacement results into swelling strain, correction should be made to account for the effective stress effect on strain as well as the displacement caused by compression of the rubber membrane around the core sample. A separate measurement was carried out to determine the relationship between the rubber strains with pressure and a linear relationship was identified and used to correct the swelling strain. It should also be noted that the radial swelling strain measured in this work may be underestimated for matrix swelling because cleat along the core may take part of the displacement. The axial swelling
To determine cleat compressibility, Seidle et al. (1992) derived a relationship between permeability and stress by idealising coal as a collection of matchsticks. This relationship is by Eq. (16) and was used in combination with the S&D permeability model as presented in Eq. (8). In Seidle et al.'s work (1992), water was used to measure coal permeability and involved a series of permeability measurements at constant confining pressure but differing pore pressure. In this current
Table 1 Comparison between permeabilities measured with CSIRO permeability facility and permeability standards supplied by Core Lab.
Std_223A Std_223B Std_162A
Standard perm (md)
Our measurement (md)
1.70 6.21 0.185
1.74 6.33 0.147
Fig. 3. Displacement change with time.
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Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
strain measured represents the matrix swelling in that direction because there is no cleat perpendicular to the axial direction. However, the cleat system in the core used in this work is consolidated, thus it is still reasonable to assume that the measured radial swelling strain represents the real matrix swelling strain in those directions. Volume swelling is approximately represented by: εV = εr1 + εr2 + εa
ð19Þ
where εV is the volumetric swelling, εr1 and εr2 are the two radial strains perpendicular to each other, εa is the axial strain, which is the average of the results by the two axial displacement gauge. 2.6. Modulus and Poisson's ratio Radial and axial strains are measured with respect to effective stress at constant pore pressure and corrected for the strain due to the rubber membrane. Thus: K=
ΔPc εV
ð20Þ
where K is the bulk modulus. A separate uniaxial stress test was performed on the coal core. A load was applied in the axial direction and the axial and radial displacements are monitored and used to calculate the Young's modulus and Poisson's ratio. E=
F=A Δl = l
ð21Þ
where F is the load, A is the area of the interception of the core, Δl is the displacement in the axial direction, l is the length of the core. Poisson's ratio can be calculated from: υ=−
εr εa
ð22Þ
where εr is the radial strain and εa is the axial strain. Poisson's ratio can also be calculated from the relationship between bulk modulus and Young's modulus: K=
E 3ð1−2υÞ
ð23Þ
3. Experimental results 3.1. Adsorption results Methane and CO2 adsorption results are shown in Fig. 4. Both isotherms can be modelled by the Langmuir model with reasonable
Fig. 4. Methane and CO2 adsorption results.
Table 2 Langmuir constants. Component
VL (std m3/tonne)
PL (MPa)
Methane CO2
27.0 53.8
2.96 5.20
accuracy. For this coal, the amount of CO2 adsorbed is about twice that of methane adsorbed over the measured pressure range. The Langmuir constants for the two gases are listed in Table 2. 3.2. Permeability results Permeability measured using helium with respect to pore pressure at a constant pressure difference between confining and pore pressures of 3.0 MPa is shown in Fig. 5. The pressure difference is defined in Eq. (24). These helium permeability measurements show a slight permeability decline with increasing pore pressure. Because helium has a very low adsorption capacity to coal, the matrix swelling and associated change of the cleat apertures are regarded as negligible. Thus the decline in permeability can be partly attributed to the Klinkenberg (1941) effect, especially at low pressure region and partly attribute to effective stress difference when m is less than unity. Effective stress is defined in Eq. (25). However, permeability decline with Helium is not as significant compared to the permeability results measured by methane and CO2, which will be shown below. ΔP = Pc −Pp
ð24Þ
σ = Pc −mPp
ð25Þ
Permeability measured using methane is shown in Fig. 6. It can be seen that the permeability measured by methane under a constant pressure difference decreases with increasing pore pressure. This permeability decrease attributes to three factors: (1) the Klinkenberg (1941) effect, especially low pressure region, (2) cleat volume decrease due to matrix swelling under triaxial stress conditions, and (3) effective stress increase under constant pressure difference condition. As expected the impact of the effective stress on permeability is also significant, which can be seen from the permeability results at constant pore pressure but with varying confining pressures. Since at constant pore pressure, the change in confining pressure is equal to the change in effective stress and the change in pressure difference. The permeability decreases almost 50% from pressure difference of 2.0 MPa to 6.0 MPa at pore pressure of 0.9 MPa. However, the magnitude of the decrease reduced to around 25% from pressure difference of 2.0 MPa to 6.0 MPa at pore pressure of 12.8 MPa. Permeability measured using methane is lower than that measured using helium, due to partly the Klinkenberg effect, which
Fig. 5. Permeability measured using helium.
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
Fig. 6. Permeability measured using methane.
257
Fig. 9. Relationship between permeability using methane and effective stress.
suggests that permeability measured by different gas species differs, and partly coal swelling by methane adsorption to reduce cleat opening under triaxial conditions. Permeability measured using CO2 is presented in Fig. 7. The permeability measured by CO2 is slightly lower than that measured by methane at the same pore pressure. This is also attributed to the different Klinkenberg effect and different swelling behaviour. Moreover, the permeability decreases more than 50% from pore pressure of 3.0 MPa to 13.0 MPa at constant pressure difference of 2 MPa and the permeability decreases more than 70% from pore pressure of 3.0 MPa to 13.0 MPa at pressure difference of 6 MPa. The declines with CO2 are larger than those with methane, possibly due to higher swelling effect with CO2 adsorption in coal.
Fig. 10. Relationship between permeability using CO2 and effective stress.
3.3. Cleat compressibility
Fig. 7. Permeability measured using CO2.
The relationship between pressure difference and permeability measured using helium, methane and CO2 are shown in Figs. 8–10, respectively. Cleat compressibility can be calculated using (16) and is listed in Table 3 and also plotted in Fig. 11. It can be seen from Fig. 11 that the cleat compressibility slightly decreases with increasing pore pressure for methane and increases with increasing pore pressure for CO2. The permeability measurement by helium with pore pressure of 2.1 MPa was the first series of permeability, thus the core may not be well consolidated. Hence, the compressibility determined using these permeability measurements may be over estimated. The cleat compressibility by helium with pore pressure of 10.1 MPa is about 0.05, which is comparable to that determined by methane. It should also be noted that cleat volume change due to changing effective stress may change the Klinkenberg effect on the gas permeability, although the pore pressure remains constant. However, the gas pressures in these measurements are relatively high. Thus the Klinkenberg effect should be relatively small. Hence, the dominant
Table 3 Cleat compressibility summary. Pore pressure (MPa)
Pore pressure Cf Pore pressure Cf Cf (MPa− 1) (MPa) (MPa− 1) (MPa) (MPa− 1)
Helium 2.1 10.1 Fig. 8. Relationship between permeability using helium and effective stress.
CO2
CH4 0.0848 0.0485
0.9 3.4 7.5 12.8
0.0507 0.0472 0.0468 0.0366
3.0 6.4 9.8 13.3
0.0606 0.0654 0.1046 0.1211
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Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261 Table 4 Model parameters for Pan and Connell swelling model. ρS (g/cc)
Es (GPa)
υs
x
1.6
1.08
0.4
0.5
Table 5 Model parameters for Langmuir-like swelling model.
Fig. 11. Cleat compressibility by helium, methane and CO2.
3.4. Swelling strain Methane and CO2 adsorption-induced coal swelling strain results are shown in Fig. 12. It can be seen from the figure that there are slight anisotropy in the measurements of swelling behaviour for both methane and CO2. The Pan and Connell (2007) swelling model was applied to the swelling measurements and the results are also shown in Fig. 12. The Pan and Connell swelling model applies an energy balance approach, which assumes that the surface energy change caused by adsorption is equal to the elastic energy change of the coal solid. Thus swelling strain can be described by adsorption isotherm and coal geomechanical properties. A simplified version of the Pan and Connell model incorporating a Langmuir adsorption isotherm can be written as: ρs P f ðx; υs Þ− ð1−2υs Þ Es Es
ð26Þ
where ε is linear strain, R is the universal gas constant, T is temperature, L and B are Langmuir adsorption constants, ρS is the coal solid density, ES and υS are the Young's modulus and Poisson's ratio for coal solid, x is the coal structure parameter, and f is a structure function. Details of this model can be found in the literature (Pan and Connell, 2007). The values of the model parameters are listed in Table 4. The model represents the swelling stain with reasonable accuracy. The swelling strain can also be fitted by Langmuir-like equation. The parameters are listed in Table 5.
Fig. 12. Methane and CO2 adsorption-induced swelling.
εV (−)
Pε (MPa)
0.030 0.052
12.0 16.0
3.5. Modulus and Poisson's ratio
effect on permeability in these measurements is the effective stress when the pore pressure is constant.
ε = RTL lnð1 + BPÞ
Component Methane CO2
The bulk modulus measurements are shown in Fig. 13 where the values under methane and CO2 adsorption are very similar. The only point which is an outlier is that measured at a CO2 pore pressure of 10 MPa which is about 10%–20% lower than the other values. As suggested by some researchers (e.g., White et al., 2005), coal softens with CO2 adsorption. However, there is no obvious coal softening for the coal sample studied. This suggests that each coal may have a unique behaviour with respect to softening with CO2 adsorption. On average, the bulk modulus for this coal is about 1620 MPa. Three load and strain tests were performed on the core and these results are presented in Table 6. Poisson's ratio was calculated in these tests from the bulk modulus and Young's modulus. In this case, the average bulk modulus of 1620 MPa and average Young's modulus of 791 MPa were used which led to a Poisson's ratio of 0.418. 4. Application of laboratory measurements to permeability models From the above measurement, significant cleat compressibility change with pressure and gas type was observed. To identify the impact of variable cleat compressibility on permeability, two simple assessments were performed using the S&D permeability model and the measured data by defining the cf as a function of the gas composition, as defined by the following relationship: cf = yCO2 cf;CO2 + yCH4 cf;CH4
ð27Þ
where yCO2 and yCH4 are the CO2 and methane molar fractions in the gas phase, cf,CO2 and cf,CH4 are the compressibilities using CO2 and
Fig. 13. Bulk modulus measured at each pore pressure.
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
259
Table 6 Load and strain test results. Stress (MPa)
Axial Strain
E (MPa)
1.825 1.864 1.787
0.00232 0.00231 0.00229 Average:
786.6 807.0 780.0 791.2
methane at the same pressure as the mixture. The mixture swelling was predicted by the Pan and Connell swelling model, which can predict mixed-gas adsorption-induced swelling strain from pure-gas adsorption-induced swelling (Pan and Connell, 2007):
ε = RT
N ρs P f ðx; υs Þ ∑ Li lnð1 + Bi yi PÞ− ð1−2υs Þ Es Es i=1
ð28Þ
where i is the ith gas component, yi is the molar fraction of the ith gas component in the gas phase. The linear swelling/shrinkage strain is approximately one third of volumetric swelling/shrinkage strain. Eq. (27) is then applied to (8) and Eq. (28) is applied to Eq. (7) assuming that linear swelling/shrinkage strain is approximately one third of volumetric swelling/shrinkage strain to investigate permeability change due to swelling and cleat compressibility change. Fig. 14 presents an example using the S&D permeability model for a constant pore pressure of 9 MPa, with the gas composition changing from pure methane to pure CO2. Fig. 14 compares the permeability calculated with a variable cf using (27) and a constant cf of 0.05. The results are shown in Fig. 14, in which it can be seen that permeability declines more significantly with variable cleat compressibility. The difference is larger at higher CO2 compositions due to the larger difference in cleat compressibility. In the second example the pore pressure varies from 9 MPa to 14 MPa and at the same time the methane composition in the gas phase decreases linearly from 1 to 0. The results are presented in Fig. 15. The permeability decline using variable cleat compressibility is much larger than that of using constant cleat compressibility. Similar analyses using the P&M permeability model can also be performed to investigate permeability change due to swelling and cleat compressibility change, by assuming an initial cleat porosity, which cannot be directly measured in the laboratory. However the calculated permeability behaviour using the P&M model is very sensitive to the initial cleat porosity. An arbitrary cleat porosity could be misleading to the interpretation of the permeability behaviour.
Fig. 15. Permeability prediction with changing pressure using S&D model.
Thus analyses using the P&M permeability model were not included in this work. 5. Conclusions This work has presented a methodology of laboratory characterisation of coal core for CBM, ECBM and CO2 sequestration in coal. The parameters used in the popular P&M and S&D permeability models can be obtained through a series of permeability and strain measurements after gas adsorption equilibration at various pore and confining pressures. An example of this procedure is presented for an Australian coal sample from the southern Sydney Basin. The behaviour of permeability with respect to effective stress is well described using the accepted exponential relationship. As expected the measurements show permeability declines with increasing pore pressure at constant effective stress in response to coal swelling with gas adsorption with the magnitude of the decline depending on the gas type. Simultaneous measurements of coal strain found that the coal sample swells about 2.4% in CO2 at 13.5 MPa and about 1.6% in methane at 12.8 MPa. The swelling strains were modelled by Pan and Connell swelling model with reasonable accuracy and can be explained with empirical Langmuir-like equations too. The bulk modulus was measured with gas adsorption and no evidence was found of coal softening with CO2 adsorption for this coal. The measurements also show that the cleat compressibility changes with pore pressure and gas type, which suggests the cleat compressibility should not be regarded as constant for some coals. In addition, variable cleat compressibility may have significant impact on permeability predictions suggested by the simple analysis conducted in this work. Acknowledgement Financial support for this work was provided by CSIRO Energy Transformed Flagship. Appendix A. Permeability–stress relation in triaxial condition
Fig. 14. Permeability prediction at constant pressure using S&D model.
Compared with the experimental work by Seidle et al. (1992), gases instead of water were used to measure the coal permeability. To avoid possible permeability change due to gas adsorption-induced coal swelling, permeability was measured at constant pore pressure with changing confining pressure to determine the permeability– effective stress relationship. In Seidle et al.'s work (1992), the relationship between permeability and stress was determined by permeability measurements under various pore pressures with a
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Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
constant confining pressure. In this appendix, the permeability–stress relation was derived for the experimental conditions applied in this work. Meanwhile, the permeability–stress relation derived by Seidle et al. (1992) was revisited and compared. Following Seidle et al.'s work (1992) assuming the matchstick geometrical model of coal porosity, permeability can be related to the cleat porosity by: 1 2 3 a ϕf 48
k=
ðA 1Þ
Simplifying (A-10) and using (A-1), (A-5) and (A-9): ∂k 2ð1−2υÞ 3 ∂ϕf =k − E mϕf ∂Pp ∂σ
ðA 3Þ
where ε is the strain. Because the sample is in hydrostatic stress: ε=
1−2υ σ E
∂ϕf ∂Pp
1 cf = ϕf
! ðA 12Þ Pc
3 ≪m cf For coal, 2ð12υÞ E
Thus, (A-11) can be integrated to: ðA 2Þ
where σ is the hydrostatic stress. Again following Seidle et al.'s work (1992), ∂a ∂ε =a ∂σ ∂σ
ðA 11Þ
Cleat volume compressibility, an important parameter in coal reservoir permeability, is defined as:
where k is the permeability, a is the cleat spacing, ϕf is the cleat porosity. Differentiating (A-1) with respect to hydrostatic stress: ∂k 2aϕ3f ∂a 3a2 ϕ2f ∂ϕf = + 48 ∂σ 48 ∂σ ∂σ
!
ðA 4Þ
3c − f ðσ−σ0 Þ k = k0 e m
ðA 13Þ
Using Eq. (A-6) and since Pc is constant, (A-13) can be rewritten as: ð3cf ðPp −Pp;0 ÞÞ
k = k0 e
ðA 14Þ
In this work, Pp was kept constant. Differentiating (A-6) with respect to σ, it yields: ∂Pc =1 ∂σ PP
ðA 15Þ
where E is the Young's modulus and υ is the Poisson's ratio. Differentiating (A-4) with respective to hydrostatic stress and then (A-3) can be rewritten as:
Following Seidle et al.'s work, using the chain rule, the derivative of cleat porosity with respect to hydrostatic stress can be rewritten as:
∂a 1−2υ =a E ∂σ
∂ϕf ∂ϕf ∂Pc = ∂σ ∂Pc ∂σ
ðA 5Þ
Hydrostatic stress or effective stress can be defined by (see e.g. Zimmerman et al., 1986): σ = Pc −mPp
ðA 6Þ
where Pc is the confining pressure, Pp is the pore pressure and m is the effective stress coefficient. In Seidle et al. (1992)'s work, Pc was kept constant and m was assumed to be 1. Here we revisited Seidle et al's work with a more general scenario where m is less than 1. Differentiating (A-6) with respect to σ, it yields: ∂Pp ∂σ
!
Simplifying (A-10) and using (A-1), (A-5) and (A-16), it yields: ∂k 2ð1−2υÞ 3 ∂ϕf =k + E ϕf ∂Pc ∂σ
ðA 17Þ
Define cfc as: cfc = −
1 ϕf
∂ϕf ∂Pc
ðA 18Þ Pp
Thus, (A-17) can be rewritten as: =−
Pc
ðA 16Þ
1 m
ðA 7Þ
Following Seidle et al.'s work, using the chain rule, the derivative of cleat porosity with respect to hydrostatic stress can be rewritten as: ∂ϕf ∂ϕf ∂Pp = ∂σ ∂Pp ∂σ
ðA 8Þ
ðA 9Þ
For coal, 2ð1−2υÞ ≪3cf E Thus, (A-19) can be integrated to: ð−3cf ðσ−σ0 ÞÞ
k = k0 e
ðA 20Þ
ð−3cfc ðPc −Pc;0 ÞÞ
k = k0 e
ðA 21Þ
The following is to demonstrate that cf and cfc are equal. Porosity of the cleat is defined as:
Thus (A-2) can be rewritten as: ∂k 2aϕ3f að1−2υÞ 3a2 ϕ2f ∂ϕf = + 48 48 ∂σ E ∂σ
ðA 19Þ
Since in this work, Pp is kept constant, Using Eq. (A-6), (A-20) can be rewritten as:
Using (A-7), (A-8) can be rewritten as: ∂ϕf 1 ∂ϕf =− m ∂Pp ∂σ
∂k 2ð1−2υÞ =k −3cfc E ∂σ
ðA 10Þ
ϕf =
Vp Vb
ðA 22Þ
Z. Pan et al. / International Journal of Coal Geology 82 (2010) 252–261
where Vp is the pore volume of cleat, Vb is the bulk volume. Then (A-12) can be rewritten as: 0 cf =
1 Vp Vb
1
V ∂ p @ Vb A
∂Pp
ðA 23Þ
Pc
Simplifying (A-23) yields: ∂Vp
1 cf = Vp
∂Pp
! Pc
1 − Vb
∂Vb ∂Pp
! ðA 24Þ Pc
Define compressibilities cpp = V1p Thus (A-24) can be rewritten as:
∂Vp ∂Pp P c
and cbp =
1 Vb
cf = cpp −cbp
∂Vb ∂Pp P c
ðA 25Þ
Meantime, using (A-22), (A-18) can be rewritten as: 0 V1 p 1 @∂ Vb A cfc = − V p ∂Pc Vb
ðA 26Þ Pp
Simplifying (A-26) and yields: 1 cfc = − Vp
∂Vp ∂Pc
! + Pp
1 ∂Vb Vb ∂Pc Pp
Defining compressibilities cpc = − V1p Then (A-27) can be rewritten as:
ðA 27Þ
∂Vp ∂Pc P p
and cbc = − V1b
cfc = cpc −cbc
∂Vb ∂Pc P p
ðA 28Þ
From Zimmerman et al.'s (1986) work, cpp = cpc −cr and cbp = cbc −cr
ðA 29Þ
where cr is the solid or grain compressibility. Applying (A-29), (A-28) can be rewritten as: cf = cpc −cr −ðcbc −cr Þ = cpc −cbc
ðA 30Þ
Comparing (A-28) and (A-30), the relationship below can be obtained: cfc = cf
ðA 31Þ
Thus (A-14) and (A-21) are equivalent to obtain cleat compressibility from the two different experimental procedures. However, using gases to measure permeability enables us to investigate impact of gas species on cleat compressibility.
261
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