An analytical coal permeability model for tri-axial strain and stress conditions
Research Highlights
►An improved theoretical interpretation of coal permeability behaviour during pressure and gas content changes that builds on the work of Shi and Durucan (2004) and Cui and Bustin (2005). ►A coal permeability model that is representative of hydrostatic tri-axial testing conditions. ►An improved representation of the role of coal matrix shrinkage/swelling as a result of changes in gas content by recognizing the distinct influence of pore volume sorption strain on coal permeability. ►Application of the developed hydrostatic permeability model to laboratory measurements of coal permeability with respect to pore and confining pressure and for a number of different gases.
Introduction
The permeability of coal is a key attribute in determining coal seam methane production and CO2 storage in coal seam reservoirs. In coal the permeability is often determined by regular sets of fractures called cleats, with the aperture of the cleats being a key property in the magnitude of the permeability. This cleat aperture is sensitive to the effective stress, with increased effective stress acting to decrease the cleat aperture and thus permeability. Gas in coal is largely stored by adsorption which introduces another complication in the understanding of coal permeability behaviour; as gas desorbs from coal the coal matrix shrinks, with gas adsorption the matrix swells. In this paper this shrinkage or swelling will be referred to as sorption strain. Thus there are two competing effects on coal permeability; lowering the pore pressure (such as during primary production) acts to increase the effective stress and thus reduces the permeability due to cleat compression. However the drawdown also results in desorption of methane leading to matrix shrinkage and increased coal cleat apertures and thus permeability. Conversely, raising the pore pressure and gas content (such as during CO2 storage to enhance coal bed methane recovery) will reverse the processes described in the preceding sentence. Thus, the permeability of coal is not a monotonically increasing or decreasing function of reservoir pressure. Instead, it may have a minimum, corresponding to a specific pressure, called the permeability rebound pressure.
Gray (1987) presented a coal permeability model which represents the effects of the matrix shrinkage and pore pressure changes on coal permeability. Various other models have been presented, including Harpalani and Zhao, 1989, Sawyer et al., 1990, Seidle et al., 1992, Seidle and Huitt, 1995, Palmer and Mansoori, 1998, Gilman and Beckie, 2000, Shi and Durucan, 2004, Shi and Durucan, 2005, Palmer (2009), etc., where both the shrinkage and pore pressure effects are included. Recently, Liu and Rutgvist (2009) have developed a new coal permeability model in the form of the combination of cubic and exponential representations. Liu et al. (2010) presented a coal permeability model based on a different interpretation of the coal structure to those dervied from the Seidle et al. (1992) bundled match-stick concept. Among these models, the Palmer–Mansoori (P–M) model (1998) and the Shi–Durucan (S–D) model (2004, 2005) are currently two popular choices used in reservoir simulation of gas migration.
Two assumptions are applied with the above-mentioned models in order to simplify their derivation and provide a concise equation convenient for representing the permeability behaviour; these are uni-axial strain and constant overburden or confining stress. However these conditions may not always be satisfied within the reservoir as discussed by Durucan and Edwards (1986) and more recently investigated using coupled modelling by Connell and Detournay, 2009, Connell, 2009. An important example regarding this issue is in relation to laboratory testing of core samples in tri-axial cells. This testing is used for the characterisation of permeability and in core flooding. For these tests (details of which will be further illustrated below) the coal sample is in a hydrostatic stress state and allowed to undergo tri-axial strain (for example, see Durucan and Edwards, 1986). However the existing coal reservoir permeability models are based on assuming uni-axial strain and constant vertical stress, conditions that are more difficult to replicate in the laboratory. Pan et al. (2010) presented a method for laboratory characterisation of coal permeability under tri-axial condition. Measurements of geomechanical properties, sorption strain and cleat compressibility with respect to confining pressure or pore pressure variation are also presented. Although these measurements can be applied to permeability models developed assuming uni-axial conditions, these models cannot represent the permeability behaviour with respect to confining and pore pressure changes under hydrostatic, tri-axial conditions. Thus, in order to more readily represent the routine conditions for laboratory testing, a new model is needed as these strain and stress assumptions can have a significant impact on the permeability.
This current paper presents two new analytical model representations; one is of an exponential form and the other a cubic form, in a manner consistent with Shi–Durucan and Palmer–Mansoori coal permeability models. Both models accommodate the two effects discussed above of sensitivity to effective stress and sorption strain. In the derivation of the models it is found that sorption strain needs to be partitioned into bulk, pore and matrix strains in contrast to existing approaches. Several different forms of the permeability models are derived for the distinct geometric and mechanical arrangements that can be encountered with laboratory testing. The approach employed could be extended to more general cases including possible field applications. A discussion of the two new models is presented, and they are then applied to a set of laboratory experimental data where the core permeability had been measured and the various geomechanical and permeability properties determined through a series of independent measurements.
Section snippets
Two general model representations
In this section the theoretical basis for the models developed in this paper is presented. In the next section the model derivations are presented for laboratory testing with tri-axial deformation and cylindrical geometry of core samples.
The volumetric balance between the volume of bulk rock (Vb), the grain or matrix volume (Vm), and the pore (or the cleat for coal) volume (Vp), is Vb = Vp + Vm. Since coal has a dual porosity structure there are two porosity systems involved. It is commonly assumed
Tri-axial laboratory experiments
The presented model representations contain the following material/model parameters Cpc(M), K (or E and ν), k0, ϕ0, ε1, pε, and α (when non-hydrostatic constraints are applied). In this paper these properties were determined from a series of independent laboratory tests and then, in the following section, used in the permeability models Eqs. (36), (19) and compared with measured permeability behaviour in order to test the accuracy of the presented approach.
The tests were performed on a 67 mm
Model validation
In this section three sets of laboratory permeability test data are used to validate the proposed exponential and cubic model representations incorporating the model parameters values obtained in the last section. These two are: 1) permeability with methane under variable pore pressures; 2) permeability with carbon dioxide under variable differential stresses; and under variable pore pressures.
- 1.
Prediction of methane permeability with respect to pore pressure
In Fig. 11 the permeability calculated
Conclusions
Coal permeability is complex in that it is sensitive to the effective stress and is affected by coal shrinkage with gas desorption and swelling with adsorption. It therefore varies as the pore pressure and gas content is decreased during gas production. Understanding coal permeability is important in order to reliably predict gas production or consider other reservoir gas migration issues. Various models have been proposed for this coal permeability behaviour however an important question in
Nomenclature
- Cpc(M)
compressibility of pore (cleat), Pa− 1
- E
Young's modulus, Pa
- G
shear modulus of rock, Pa
- k
permeability, Darcy
- k0
permeability at a reference state, Darcy
- K
bulk modulus of rock, Pa
- Km
bulk modulus of the matrix, Pa
- p
pore pressure, Pa
- pp0
pore pressure at a reference state, Pa
- pε
Langmuir-type matrix swell/shrinkage constant, Pa
- prb
permeability rebound pressure, Pa
- p⁎
confining stress, Pa
- pr*, pz*
confining stress in r- and z-direction, Pa
- Δpc*
differential pressure, Pa
- r, z, θ
radial, axial, and circular direction
Acknowledgements
The authors gratefully acknowledge the support of the CSIRO Coal Technology Portfolio.
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