# Storage Coefficients and Permeability Functions for Coal-Bed Methane Production Under Uniaxial Strain Conditions

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## Abstract

The porosity and permeability of coal change with pore pressure, due to changes in effective stress and matrix swelling due to gas adsorption. Three analytical models to describe porosity and permeability change in this context have been presented in the literature, all of which are based on poroelastic theory and uniaxial strain conditions. However, each of the three models provides different results. Review articles have attributed these differences to the use of stress formulations or strain formulations. In this article, the three aforementioned porosity models are used to derive three associated expressions for the storage coefficient. A single mathematical equation for the storage coefficient in an aquifer under uniaxial strain conditions is well established. The storage coefficient represents the volume of fluid released per unit volume of a porous rock following a unit decline in pore pressure. It is shown that only one of the aforementioned three coal-bed methane porosity models leads to the correct equation for the uniaxial strain storage coefficient in the absence of gas sorption-induced strain.

## Keywords

Coal-bed methane Permeability Porosity Storage coefficient Matrix shrinkage Rock mechanics## List of Symbols

- \(C_\mathrm{g}\)
Compressibility of the gaseous methane (M\(^{-1}\)LT\(^{2}\))

- \(C_\mathrm{p}\)
Seidle’s compressibility parameter (M\(^{-1}\)LT\(^2\))

*E*Young’s modulus (ML\(^{-1}\)T\(^{-2}\))

*f*Palmer and Mansoori’s model parameter (−)

*g*Gravitational acceleration (LT\(^{-2}\))

*G*Shear modulus (ML\(^{-1}\)T\(^{-2}\))

- \(\mathbf k\)
Permeability tensor (L\(^2\))

*k*Isotropic permeability (L\(^2\))

*K*Bulk modulus (ML\(^{-1}\)T\(^{-2}\))

- \(k_0\)
Reference permeability (L\(^2\))

- \(M_\mathrm{g}\)
Mass of gaseous methane (M)

- \(m_\mathrm{g}\)
Mass of gaseous methane per unit bulk volume of coal (ML\(^{-3}\))

- \(M_s\)
Mass of adsorbed methane (M)

- \(P_\mathrm{p0}\)
Reference pore pressure (ML\(^{-1}\)T\(^{-2}\))

- \(P_\mathrm{L}\)
Langmuir isotherm pressure (ML\(^{-1}\)T\(^{-2}\))

- \(P_\mathrm{m}\)
Partial pressure of the adsorbed methane (ML\(^{-1}\)T\(^{-2}\))

- \(P_\mathrm{p}\)
Pore pressure (ML\(^{-1}\)T\(^{-2}\))

*S*Storage coefficient (M\(^{-1}\)LT\(^{2}\))

*T*Temperature (\(\Theta \))

*t*Time (T)

- \(V_\mathrm{b}\)
Bulk volume (L\(^3\))

- \(V_\mathrm{m}\)
Volume of coal mineral (L\(^3\))

- \(V_\mathrm{p}\)
Pore volume (L\(^3\))

*z*Elevation (L)

- \(\alpha \)
Biot coefficient (−)

- \({\varvec{\varepsilon }}\)
Strain tensor (−)

- \(\varepsilon _\mathrm{b}\)
Bulk volume strain (−)

- \(\varepsilon _\mathrm{L}\)
Langmuir isotherm strain (−)

- \(\varepsilon _\mathrm{m}\)
Mineral volume strain (−)

- \(\varepsilon _\mathrm{s}\)
Shrinkage strain associated with methane adsorption (−)

- \(\eta \)
Permeability exponent (−)

- \(\lambda \)
Lamé parameter (ML\(^{-1}\)T\(^{-2}\))

- \(\mu _\mathrm{g}\)
Dynamic viscosity of gaseous methane (ML\(^{-1}\)T\(^{-1}\))

- \(\nu \)
Poisson’s ratio (−)

- \(\rho _\mathrm{g}\)
Density of gaseous methane (ML\(^{-3}\))

- \(\rho _\mathrm{s}\)
Density of the adsorbed methane (ML\(^{-3}\))

- \(\varvec{\tau }\)
Stress tensor (ML\(^{-1}\)T\(^{-2}\))

- \(\tau _\mathrm{h}\)
Hydrostatic stress (ML\(^{-1}\)T\(^{-2}\))

- \(\tau _\mathrm{m}\)
Mean stress (ML\(^{-1}\)T\(^{-2}\))

- \(\phi \)
Porosity (−)

- \(\phi _0\)
Reference porosity (−)

## 1 Introduction

There are several previously published derivations of analytical models to describe how the porosity and permeability of coal change due to changes in pore pressure. Knowledge about how porosity and permeability evolve is important to help estimate the productivity of coal-bed methane production wells (Liu and Harpalani 2013) in addition to forecasting potential hydromechanical impacts on surrounding geological formations during coal-bed methane production (Wu et al. 2018). Permeability of coal is mostly attributed to the cleats within a given coal formation. As pore pressure is reduced, the effective stress is increased, which leads to a reduction in coal cleat apertures and hence a reduction in coal porosity and permeability. However, reductions in pore pressure also lead to desorption of gas from the coal matrix, which in turn leads to shrinkage of the coal matrix, an increase in coal cleat apertures, and an increase in coal cleat permeability. Of interest in the present work are the differences between the associated analytical models of Palmer and Mansoori (1998), Shi and Durucan (2004), and Cui and Bustin (2005). These three models all claim to satisfy uniaxial strain conditions, with gas sorption-induced strain (GSIS) treated as analogous to thermal expansion. The fact that the three models provide different results has been attributed to Palmer and Mansoori (1998) adopting a strain formulation, whereas Shi and Durucan (2004) and Cui and Bustin (2005) adopted a stress formulation (Gu and Chalaturnyk 2006; Palmer 2009; Liu and Harpalani 2013; Li et al. 2017). This explanation is unsatisfactory, firstly because it does not explicitly explain the difference between the models of Shi and Durucan (2004) and Cui and Bustin (2005), and secondly, because, if the same theoretical assumptions have been made, the final results should be the same regardless of whether a stress-based or strain-based formulation is used.

Analytical solutions for porosity change due to pore pressure under uniaxial strain conditions have also been derived in the context of fluid production/injection in aquifers (e.g., Gambolati et al. 2000; Jaeger et al. 2007; Zimmerman 2017; Andersen et al. 2017). Associated authors presented a single common equation for the storage coefficient, which they rigorously derived from poroelastic theory. The storage coefficient represents the volume of fluid released per unit volume of a porous rock following a unit decline in pore pressure. In this article, we derive equations for this aforementioned storage coefficient using the three different porosity models of Cui and Bustin (2005), Palmer and Mansoori (1998), and Shi and Durucan (2004). We look at the limits of these equations for when there is no GSIS and compare these with the widely accepted uniaxial strain storage coefficient associated with fluid movement in aquifers (e.g., Gambolati et al. 2000; Jaeger et al. 2007; Zimmerman 2017; Andersen et al. 2017).

The outline of the article is as follows. A modified form of Hooke’s law is presented, which incorporates pore pressure and GSIS. It is then shown how to relate GSIS to the mass of adsorbed gas within the coal. A mass conservation equation is derived for gas migration in coal, which leads to a general expression for the aforementioned storage coefficient in terms of fluid compressibility, porosity, bulk strain, and GSIS. A general equation is developed to describe the associated change in porosity. Expressions for stress and bulk strain are derived assuming uniaxial strain conditions. Finally, three storage coefficients are derived using the porosity models of Cui and Bustin (2005), Palmer and Mansoori (1998), and Shi and Durucan (2004). Implications of the results are then discussed in the context of permeability modeling.

## 2 Mathematical Model

### 2.1 Incorporation of Adsorption into Hooke’s law

*G*(ML\(^{-1}\)T\(^{-2}\)) is the shear modulus, \(\nu \) (−) is Poisson’s ratio, \(\lambda \) (ML\(^{-1}\)T\(^{-2}\)) is the Lamé parameter,

*K*(ML\(^{-1}\)T\(^{-2}\)) is the bulk modulus, \(\alpha \) (−) is the Biot coefficient, \(P_\mathrm{p}\) (ML\(^{-1}\)T\(^{-2}\)) is the pore pressure, and \(\varepsilon _\mathrm{s}\) (−) is a shrinkage strain associated with methane adsorption, typically described using a Langmuir isotherm of the form (e.g., Ye et al. 2014)

The value of \(\varepsilon _\mathrm{L}\) is negative in this context because \(\varepsilon \) is a positive compression strain. The absolute value of \(\varepsilon _\mathrm{L}\) represents the maximum possible volumetric expansion strain that can be incurred due to gas adsorption. The \(P_\mathrm{L}\) parameter represents the value of \(P_\mathrm{m}\) at which \(\varepsilon _\mathrm{s}=\varepsilon _\mathrm{L}/2\).

### 2.2 Linking Adsorption Mass to Adsorption Strain

*T*(\(\Theta \)) is temperature and \(\varepsilon _\mathrm{m}\) (−) and \(\varepsilon _\mathrm{b}\) (−) are mineral volume and bulk volume strains, respectively, found from

### 2.3 Mass Conservation Statement

*t*(T) is time and (consider Jaeger et al. 2007, p. 180)

*g*(LT\(^{-2}\)) is gravitational acceleration, and

*z*(L) is elevation.

*k*when \(\phi =\phi _0\), \(\phi _0\) is a reference value of the porosity and \(\eta \) (−) is an empirical exponent (Pyrak-Nolte et al. 1998), generally assumed to be three (due to an association with the so-called match stick model, Seidle et al. 1992).

*S*(M\(^{-1}\)LT\(^{2}\)), is found from (Green and Wang 1990)

### 2.4 Determining the Change in Porosity

### 2.5 Imposing Uniaxial Strain Conditions

### 2.6 Application of the Cui and Bustin Model

### 2.7 Application of the Palmer and Mansoori Model

*f*(−) is described as a “fraction” that ranges from zero to one. Based on a discussion by Moor et al. (2015), Zimmerman (2017) suggested that

*f*is thought to relate to the ratio of isotropic strain to deviatoric strain within the mineral grains of the rock of concern. Note that Palmer and Mansoori (1998) state that Eq. (31) can be derived by assuming uniaxial strain conditions, as described above.

*f*, Eqs. (33) and (34) only reduce to Eqs. (29) and (30) when \(\phi =0\) and \(\alpha =1\). It follows that Eq. (31) is invalid when \(\phi >0\) or \(\alpha <1\).

### 2.8 Application of the Shi and Durucan Model

## 3 Implications for Permeability Modeling

*E*(ML\(^{-1}\)T\(^{-2}\)) and \(\nu \) (−) are the Young’s modulus and Poisson’s ratio, respectively.

## 4 Conclusions

Theoretical equations have been derived to describe porosity change and storage coefficient in coal-bed methane systems under both general and uniaxial strain conditions. Three equations for the storage coefficient were then derived using the porosity models of Cui and Bustin (2005), Palmer and Mansoori (1998), and Shi and Durucan (2004). For the limiting case when there is no gas adsorption, only the storage coefficient derived using the Cui and Bustin (2005) porosity model correctly reduced to an established expression for the storage coefficient in an aquifer under uniaxial strain conditions. The storage coefficient derived using the Palmer and Mansoori (1998) porosity model was found to provide the correct limiting result only when the Biot coefficient is assumed to be one and the porosity is assumed to be zero. The storage coefficient derived using the Shi and Durucan (2004) porosity model is found not to have the correct limit because its derivation involves a mixture of hydrostatic and uniaxial stress assumptions. Out of the three models, only the Cui and Bustin (2005) porosity model is shown to rigourously satisfy poroelastic theory under uniaxial strain conditions.

## Notes

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