Modified SLD model for coalbed methane adsorption under reservoir conditions

Abstract

In the process of coalbed methane (CBM) mining, with the decreasing of reservoir pressure, stress sensitivity and matrix desorption shrinkage, which could be called self-regulating effect comprehensively, possess great effect on the gas adsorption due to the changing of pore volume. Therefore, it is necessary to modify the simplified local density (SLD) model to investigate the gas adsorption under reservoir conditions for an accurate prediction of CBM production. For the utilization of SLD adsorption model, assume the pores as slit pore. When pressure is lower than critical desorption pressure and keeps falling, the deformations of specific surface area (SSA) and pore width could be studied by the chosen Shi and Durucan (S&D) model. Furthermore, based on the variation relationship and the modification of SLD model, a new adsorption predicting model could be derived with the consideration of stress sensitivity, and matrix desorption shrinkage. In the absence of stress sensitivity and matrix desorption shrinkage, the calculated consequence is relatively smaller than the actual field adsorption data. What’s more, the sensitivity analyses of Poisson’s ratio, pore volume compressibility and critical desorption pressure are conducted with the application of this new model. At the same reservoir pressure, Poisson’s ratio possesses negative relationship with modified adsorption, while pore volume compressibility and critical desorption pressure both possess positive influence on modified adsorption. The main reason is that Poisson’s ratio affects matrix desorption shrinkage negatively, while pore volume compressibility and critical desorption pressure affect matrix desorption shrinkage positively. In spite of the opposite effect of stress sensitivity and matrix desorption shrinkage on pore deformation, matrix desorption shrinkage exhibits a dominant role in self-regulating.

Appendix

Due to the positive relationship of relative matrix deformation and the surface energy variation, there is a relational equation

$$ \varepsilon =\frac{\rho_{\mathrm{c}} A\varDelta \gamma}{E} $$

(12)

where ε is the relative strain deformation and Δγ is the surface energy variation, J/m².

Based on Gibbs equation (Bangham 1937), the surface energy variation is given by

$$ \varDelta \gamma ={\int}_0^p\Gamma RTd\ln p $$

(13)

where Γ is the surface concentration of adsorbed gas, mol/m², which could be expressed as

$$ \Gamma =\frac{V}{V_0S} $$

(14)

where V is the gas adsorption at pressure p, m³/t, which could be expressed by Langmuir adsorption equation.

$$ V=\frac{V_{\mathrm{L}} bp}{1+ bp} $$

(15)

Combine Eq. (12) to (16); then the strain deformation caused by desorption

$$ \varDelta \varepsilon =\varepsilon (p)-\varepsilon \left({p}_{\mathrm{c}}\right)=\frac{V_{\mathrm{L}}{\rho}_c RT}{EV_0}\left[\ln \left(1+ bp\right)-\ln \left(1+{bp}_{\mathrm{c}}\right)\right] $$

(16)