A Mathematical Model of Coupled Gas Flow and Coal Deformation with Gas Diffusion and Klinkenberg Effects

Abstract

The influence of gas diffusion behavior on gas flow and permeability evolution in coal seams is evaluated in this paper. Coalbed methane (CBM) reservoirs differ from conventional porous media and fractured gas reservoirs due to certain unique features, which lead to two distinct gas pressures: one in fractures and the other in the coal matrix. The latter pressure, also known as the sorption pressure, will be used in calculating sorption-based volume changes. The effective stress laws for single-porosity media is not suitable for CBM reservoirs, and the effective stress laws for multi-porosity media need to be applied. The realization of the above two points is based on the study of the two-phase state of gas migration (involving Fickian diffusion and Darcy flow) in a coal seam. Then, a general porosity and permeability model based on the P-M model is proposed to fit this phenomenon. Moreover, the Klinkenberg effect has been taken into account and set as a reference object. Finally, a coupled gas flow and coal deformation model is proposed and solved by using a finite element method. The numerical results indicate that the effects of gas diffusion behavior and Klinkenberg behavior can have a critical influence on the gas pressure, residual gas content, and permeability evolution during the entire methane degasification period, and the impacts of the two effects are of the same order of magnitude. Without considering the gas diffusion effect, the gas pressure and residual gas content will be underestimated, and the permeability will be overestimated.

Appendices

Appendix A

The common conceptual model applied to coal is that it is a dual-porosity reservoir, where gas is mostly stored in the coal matrix and Darcy fluid flow occurs in the natural fracture system (the contribution of flow in the coal matrix to Darcy flow can be neglected) (Purl et al. 1991; Pan and Connell 2012). The permeability of a coalbed is a function of its fracture system. Thus, many typical CBM reservoir models treat the CBM system as a dual-porosity, single-permeability scheme, with the implication that the coal matrix serves as storage and provides a source for the fracture network, while the fluids can only flow through the fractures (Thararoop et al. 2012; Palmer 2009; Pan and Connell 2012; Liu et al. 2011). In this paper, the CBM reservoir has also been treated as a dual-porosity, single-permeability scheme.

The derivation of P-M model starts from the following equation of linear elasticity for strain changes in porous rock (Palmer and Mansoori 1998):

$$d\varepsilon_{\text{p}} = \frac{{d\varepsilon_{\text{r}} }}{\phi } - \left( {\frac{1 - \phi }{\phi }} \right)d\varepsilon_{\text{g}}$$

(A1)

where $d\varepsilon_{\text{g}}$ is the incremental grain volume strain, $d\varepsilon_{\text{r}}$ is the incremental rock volume strain, $d\varepsilon_{\text{p}}$ is the incremental pore volume strain, and $\phi$ is the natural fracture porosity [see the nomenclature of Palmer and Mansoori (1998)]. In this paper, $\phi_{\text{f}}$ is used to denote the fracture porosity in order to distinguish between it and the matrix porosity, so in the following, $\phi$ is replaced by $\phi_{\text{f}} .$

The theory of single-porosity poroelasticity has been used by Palmer and Mansoori (1998) to achieve the relation between porosity change and effective stress change:

$$d\phi_{\text{f}} = \left[ {\frac{1}{M} - (1 - \phi_{\text{f}} )f\gamma } \right](d\sigma - dp) + \left[ {\frac{K}{M} - (1 - \phi_{\text{f}} )} \right]\gamma dp - \left[ {\frac{K}{M} - (1 - \phi_{\text{f}} )} \right]\alpha dT$$

(A2)

where $f$ is a fraction ranging from 0 to1, $\gamma$ is the grain compressibility, Pa⁻¹, $\alpha$ is the grain thermal expansively,  °F⁻¹, and $(d\sigma - dp)$ denotes the change of the effective stress ($\sigma_{\text{e}} = \sigma - \beta_{\text{p}} p ,$ where $\beta_{\text{p}}$ ranges from 0 to 1, positive in compression), and it was obtained by using the effective stress law for a single-porosity system. As analyzed in the previous section, the effective stress law for dual-porosity media [see Eq. (16)] is more suitable for obtaining the effective stress changes; thus, the relation between porosity change and effective stress change can be obtained based on the dual-porosity poroelastic theory as:

$$- d\phi_{\text{f}} = \left[ {\frac{1}{M} - (1 - \phi_{\text{f}} )f\gamma } \right](d\sigma - \beta_{\text{f}} dp_{\text{f}} - \beta_{\text{m}} dp_{\text{m}} ) + \left[ {\frac{K}{M} - (1 - \phi_{\text{f}} )} \right]\gamma (\beta_{\text{f}} dp_{\text{f}} + \beta_{\text{m}} dp_{\text{m}} ) - \left[ {\frac{K}{M} - (1 - \phi_{\text{f}} )} \right]\alpha dT$$

(A3)

According to Palmer and Mansoori (1998), the fracture porosity, $\phi_{\text{f}} \ll 1 ,$ as is the case in coalbeds, and there is no change in overburden stress under uniaxial strain conditions, $d\sigma = 0 ,$ and grain compressibility is set to zero, $\gamma = 0$ [see Table 1 of Palmer and Mansoori (1998)]. The term $dT =$ is a temperature expansion/contraction term; this is directly analogous to matrix shrinkage. And the pressure in coal matrix, $p_{\text{m}} ,$ rather than the pressure $p$ (without considering the effects of diffusion behavior), is used to calculate the matrix shrinkage. By direct analogy, for incremental rock volume strain (i.e., increase in strain per unit temperature or pressure change), we have:

$$\alpha dT \equiv \frac{d}{{dp_{\text{m}} }}\left( {\frac{{\varepsilon_{\text{L} } p_{\text{m}} }}{{p_{\text{m}} + P_{\text{L} } }}} \right)dp_{\text{m}}$$

(A4)

and:

$$- d\phi_{\text{f}} = - \frac{1}{M}(\beta_{\text{f}} dp_{\text{f}} + \beta_{\text{m}} dp_{\text{m}} ) - \left( {\frac{K}{M} - 1} \right)\frac{d}{{dp_{\text{m}} }}\left( {\frac{{\varepsilon_{\text{L}} p_{\text{m}} }}{{p_{\text{m}} + P_{\text{L}} }}} \right)dp_{\text{m}}$$

(A5)

Equation (A5) can be easily converted to a total differential form:

$$d\phi_{\text{f}} (p_{\text{f}} ,p_{\text{m}} ) = \frac{{\beta_{\text{f}} }}{M}dp_{\text{f}} + \left[ {\frac{{\beta_{\text{m}} }}{M} + \left( {\frac{K}{M} - 1} \right)\frac{d}{{dp_{\text{m}} }}\left( {\frac{{\varepsilon_{\text{L}} p_{\text{m}} }}{{p_{\text{m}} + P_{\text{L}} }}} \right)} \right]dp_{\text{m}}$$

(A6)

Based on the theory of multivariable differential calculus (Kriz and Pultr 2013), we have:

$$\phi_{\text{f}} - \phi_{\text{f}0} = \frac{{\beta_{\text{f}} }}{M}(p_{\text{f}} - p_{\text{f}0} ) + \frac{{\beta_{\text{m}} }}{M}(p_{\text{m}} - p_{\text{m}0} ) + \left( {\frac{K}{M} - 1} \right)\left[ {\frac{{\varepsilon_{L} p_{\text{m}} }}{{p_{\text{m}} + P_{\text{L}} }} - \frac{{\varepsilon_{\text{L}} p_{\text{m}0} }}{{p_{\text{m}0} + P_{\text{L}} }}} \right]$$

(A7)

Dividing by $\phi_{\text{f}0}$ leads to:

$$\frac{{\phi_{\text{f}} }}{{\phi_{\text{f}0} }} = 1 + \frac{1}{{M\phi_{\text{f}0} }}[\beta_{\text{f}} (p_{\text{f}} - p_{\text{f}0} ) + \beta_{\text{m}} (p_{\text{m}} - p_{\text{m}0} )] + \frac{{\varepsilon_{\text{L}} }}{{\phi_{\text{f}0} }}\left( {\frac{K}{M} - 1} \right)\left( {\frac{{p_{\text{m}} }}{{P_{\text{L} } + p_{\text{m}} }} - \frac{{p_{\text{m}0} }}{{P_{\text{L} } + p_{\text{m}0} }}} \right)$$

(A8)

Equation (A8) is the same as Eq. (23). It is clear that there is a relationship between fracture porosity, permeability, and the grain size distribution in porous media. Chilingar (1964) defined this relationship as:

$$k_{\infty } = \frac{{d_{\text{e}}^{2} \phi_{\text{f}}^{3} }}{{72(1 - \phi_{\text{f}} )}}$$

(A9)

where $d_{\text{e}}$ is the effective diameter of grains. Based on this equation, we obtain the widely used cubic law as:

$$\frac{{k_{\infty } }}{{k_{\infty 0} }} = \left( {\frac{{\phi_{\text{f}} }}{{\phi_{\text{f}0} }}} \right)^{3} \left( {\frac{{1 - \phi_{\text{f}0} }}{{1 - \phi_{\text{f}} }}} \right)^{2}$$

(A10)

where $k_{\infty }$ and $k_{\infty 0}$ are the absolute permeability and initial absolute permeability, respectively. For the fracture porosity, $\phi_{\text{f}} \ll 1 ,$ as is the case in coalbeds, we obtain:

$$\frac{{k_{\infty } }}{{k_{\infty 0} }} = \left( {\frac{{\phi_{\text{f}} }}{{\phi_{\text{f}0} }}} \right)^{3} = \left\{ {1 + \frac{1}{{M\phi_{\text{f}0} }}[\beta_{\text{f}} (p_{\text{f}} - p_{\text{f}0} ) + \beta_{\text{m}} (p_{\text{m}} - p_{\text{m}0} )] + \frac{{\varepsilon_{\text{L}} }}{{\phi_{\text{f}0} }}\left( {\frac{K}{M} - 1} \right)\left( {\frac{{p_{\text{m}} }}{{P_{\text{L} } + p_{\text{m}} }} - \frac{{p_{\text{m}0} }}{{P_{\text{L} } + p_{\text{m}0} }}} \right)} \right\}^{3}$$

(A11)

Equation (A11) is the same as Eq. (24).

Appendix B

See Table 5.

Table 5 Input parameters required for the models and possible ranges of values