Coal cleat permeability for gas movement under triaxial, non-zero lateral strain condition: A theoretical and experimental study
Introduction
The process of carbon dioxide (CO2) sequestration in deep un-minable coal seams has been identified as an economical approach to the reduction of greenhouse gas emissions into the atmosphere, as it has a beneficial by-product, methane (CH4), obtained through enhanced coal bed methane recovery. The main storage mechanism of CO2 in the coal mass is adsorption [1]. However, CO2 adsorption into the coal matrix causes the coal to swell, leading potentially to permeability reduction. Permeability reduction and loss of injectivity are the main difficulties encountered in field applications to date and require further studies [2], [3]. According to Balan and Gumrah [4], cleat permeability is the most important parameter which determines the CO2 sequestration potential of coal seams. It represents the contribution of effective stress and matrix shrinkage and swelling. Coal mass has two types of cleats, face cleats and butt cleats, and they are normally orthogonal to each other. Of these two types, face cleats are the governing fractures for gas movement as they are more continuous and extensive, and normally make up the connected fracture network for fluid flow. Permeability along the face cleat direction is usually much higher than along the butt cleat direction. It is not uncommon that the permeability can be more than ten times higher [5].
Theoretical and empirical models play a very important role in the prediction of coal permeability as they obviate laboratory experimentation and speed the identification of coal mass properties in deep coal seams. Many research papers can be found related to the effect of stress on coal permeability. One of the earliest studies was by Somerton [6]. According to their study, stress history greatly affects permeability measurements and neither the loading sequence nor the maximum principal stress application directions have much effect on permeability. They presented the relationship among permeability and effective principal stress as follows:where k is the permeability under stress σm (md), ko is the permeability under zero stress (md) and σm is mean stress (psi). Durucan and Edwards [7] investigated the radial stress effect on coal permeability for fractured coal and found a common expression for permeability in any type of coal as follows:where σ3 is the radial stress (MPa), k is the permeability (md), C and ki are constants. Here, Cc is defined as the compressibility factor and depends on the volatile matter content of the coal. Durucan and Edwards [7] defined the ki as the relative incidence of existing fissures and fractures of coal. In this research they maintained the relationship of σ1 = 3σ3 and therefore, the determined permeability values have been also affected by the major principal stress (σ1). Gray [8] suggested the use of Darcy’s law for gas permeability calculation in coal:where k is the permeability (m2), q is the flow rate (m3/s), pout is the outlet pressure (Pa), pin is the inlet pressure (Pa), L is the length of core (m), μ is the viscosity of the fluid (Pa s), and A is the cross-sectional area of the core sample (m2). Seidle [9] proposed a new relationship among the permeability and the effective stress as follows:where kf is the cleat permeability (md), Cf is cleat volume compressibility (kPa−1), and σh is hydrostatic stress (kPa). Then, in 1995, Seidle and Huitt [10] investigated the gas desorption effect on coal matrix shrinkage and presented the following equation:where permeability is assumed to follow the cubic law for fracture flow (in the following equation):where ϕ is the coal bed porosity after sorption or desorption of gases, ϕo is the initial coal bed porosity, and ε1 and b are the Langmuir type matrix shrinkage constants. This model considers only the matrix shrinkage effect and not the effective stress effect. Sawyer et al. [11] proposed a new model for coal permeability as a function of both of these effects (ARI model).where Cp is the pore volume compressibility and Cm is the matrix shrinkage compressibility. According to McKee et al. [12], Cm = ϕ Cp. Δpi/Δci is the pressure change for Δci concentration variation and co is the initial gas concentration. In this equation, the first term describes the pressure effect and the second term describes the matrix shrinkage effect. The permeability can be found using Eq. (6). In 1998, Palmer and Mansoori [13] proposed a theoretical formula for coal permeability as a function of both matrix shrinkage and effective stress:wherewhere dϕ is change in porosity, dp is change in pore pressure (md), ε1 is the Langmuir volume, b is the Langmuir constant, f is a fraction (0–1), βg is the grain compressibility, and K and M are the bulk and the constrained axial modulus, respectively. They are given as follows:where ϕo, po and Cm are the initial porosity and pressure (MPa) and matrix shrinkage compressibility (MPa−1), respectively. If the pore volume compressibility factor Cm is constant,
However, Eq. (12) can be used only under conditions of constant applied pressure (only flow effect), and if there is a pressure gradient, it is necessary to consider the stress effect also. Therefore, the original equation should be used (Eq. (8)). In 2002, Pekot and Reeves [14] proposed Eq. (13) to calculate coal permeability. The equation contains a new term to account for the differential shrinkage effect of coal mass due to CO2 adsorption because, compared to some other gases such as CH4, CO2 causes a greater degree of swelling, resulting in greater reduction in associated permeability.where CK is the differential swelling coefficient. In 2000, Gilman and Beckie [5] developed theoretical models for coal matrix and cleat permeabilities for methane gas movement in a coal mass. Methane gas flow in fractures or cleats was modelled using Darcy’s law and in the matrix by assuming that Knudsen diffusion applies (coupled with the ideal gas behaviour). They proposed the following equation for cleat permeability of coal based on two basic assumptions: that coal mass behaves as an elastic medium and lateral strain is zero:where kf is the cleat permeability and kfo is the initial cleat permeability, ϑ is the matrix Poisson’s ratio, p is the pore pressure, E is the matrix Young’s modulus, Ef is the effective Young’s modulus of the sample with fractures, ΔS is the change of adsorbed gas mass and α is the volumetric swelling coefficient. Wang et al. [15] proposed a modified version of Eq. (14) to describe the variation of CO2 permeability in coal cleats due to adsorption:where τf is the tortuosity, So and S are the initial and final adsorbed mass, and k(so) and k(s) are the initial and final permeability values.
Zhao et al. [16] proposed new models for fracture (Eq. (16)) and matrix permeabilities (Eq. (17)) by coupling the solid deformation and gas seepage as follows:where kf is the fracture permeability, km is the matrix permeability, g is the gravitational acceleration, μL is the lame constant, which is relevant to the solid deformation of rock matrix, do is the initial opening of the fracture, p is the pore pressure, σn is the normal stress, αf is the connective coefficient of the fracture, αb is Biot’s coefficient of the fracture, Θ is the volumetric stress, bs is the gas adsorption coefficient, kn is the normal stiffness and ao and N are test coefficients.
According to Xu et al. [17], rock permeability changes significantly during loading due to the load-induced damage in the rock matrix, where rock permeability increases during the elastic deformation due to expansion and decreases due to compaction. In addition, closer to peak load, rock permeability may suddenly increase with the formation of large numbers of micro-cracks. If the load is compression and is further increased after the failure, permeability starts to reduce. On the other hand, if the load is tension and is further increased after the failure, permeability continuously increases. Therefore, the following two equations were proposed by Xu et al. [17] to calculate coal permeability under these conditions for uniaxial compression (Eq. (18)) and tension (Eq. (19)) tests:where k is the permeability, ko is the initial permeability, p is the pore pressure, αp is the coefficient of pore pressure, βp is the stress to pore pressure coupling factor, ξ is the coefficient for the sudden jump of the gas permeability for loaded elements in compression and, ξ′ is the coefficient for the sudden jump of the gas permeability for failed elements in tension, σ1 is the major principle stress and σ3 is the minor principle stress, and D is the damage variable, respectively.
All the models described above have been developed based on the assumption that coal mass has zero lateral strain condition in relation to the in situ stress condition in coal beds. However, this assumption does not apply to laboratory experiments such as triaxial tests, as they clearly produce considerable lateral strain ([15], [18]). Since triaxial testing has been identified as the most suitable laboratory experiment for permeability testing [19], it is important to develop an appropriate theoretical model to investigate permeability variation under triaxial test conditions. Therefore, the main objective of this study is to develop a new model for cleat permeability under non-zero lateral strain condition, which can be used for triaxial laboratory experiments.
Section snippets
Model development
In order to develop the model, a coal sample under triaxial test conditions was considered (Fig. 1) and the following basic assumptions were made:
- (1)
Flow is only in the z direction.
- (2)
Face cleats are the governing fractures for gas transport in z direction; hence the fractures are parallel to the flow direction.
- (3)
The system is elastic and isothermal.
In Fig. 1, “e” is the fracture aperture and “h” is the fracture spacing, “σC” is the confining stress, “σA” is the axial stress and “” is the CO2
Experimental methodology
In order to verify the model, triaxial experiments were conducted on two different naturally-fractured black coal samples. Both selected samples had almost parallel cleat systems along the flow direction and one sample had minor fractures and the other had major fractures. Coal samples were taken from the Appin mine of the Bulli coal seam, Southern Sydney basin and were cut into a standard size of 38 mm diameter by 76 mm height in the Monash Civil Engineering department. The natural cleat systems
Results and discussion
First, the potential of the developed model (Eq. (39)) to predict the effect of CO2 injection pressure on coal permeability was investigated using the experimentally-determined permeability data for sample 01. Fig. 3 shows the accuracy of the developed model in predicting cleat permeability for different injecting pressures under a particular confining pressure. According to the figure, the developed model can predict quite accurately the effect of gas injecting pressure on cleat permeability.
Conclusions
Coal cleat permeability under non-zero lateral strain triaxial test conditions can be successfully represented by a theoretical model containing four main parameters to describe: (1) the injecting pressure effect, (2) the confining pressure effect, (3) the axial load effect, and (4) the gas adsorption effect. Following the model’s verification with experimental data of two samples with different natural cleats, it has been shown that the new model can fairly accurately combine the effective
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