Numerical Solution of the Mathematical Model for Constant Pressure Gas Desorption in a Coal Matrix

Abstract

An in-depth understanding of gas diffusion behavior in a coal matrix is essential for the prevention of gas hazards and the prediction of coalbed methane (CBM) recovery. There is still no consensus on the mechanism of coal matrix gas diffusion and further research is needed. In this paper, a mathematical model of coal matrix gas desorption and diffusion driven by the free-gas density gradient (FGDG) was proposed and developed, solved by finite-difference numerical methods, and validated with experiment data regarding gas desorption under a constant pressure condition. The dependence of key coefficients (permeability coefficient and microchannel diffusion coefficient) on pressure was analyzed, and the accuracy and rationality of Fick's model, Darcy's model, and the FGDG model were analyzed. The accuracy and rationality of the Fick, Darcy, and FGDG models were discussed. The results show that the predicted results of FGDG and Darcy models are consistent with the experimental data, and the permeability and microchannel diffusion coefficients do not vary with time. However, when the diffusion coefficient in the Fick model is a constant value, the simulated results are not consistent with the experimental data in the whole desorption time. The permeability coefficient shows a negative exponential decrease with the initial gas pressure, while the microchannel diffusion coefficient is subject to very limited fluctuation with gas pressure. Therefore, the gas desorption diffusion flow process in the coal matrix is more consistent with the free-gas density gradient-driven mechanism, which is an upgraded version of the Fick and Darcy models. This paper enriches the research ideas and thoughts of a gas diffusion model in a coal matrix.

Discrimination Method of Gas Pressure Measurement in Coal Seams: Physical Experiment and Model Development

“…Gas generally exists in coal in the adsorption state and free state, and the gas in the two states is transformed into each other. The gas in the pore system of the coal matrix flows in the form of diffusion under the concentration gradient, 35 which conforms to Fick’s law where J m denotes the diffusion flux, m 3 /(m 2 s); D m is the diffusion coefficient, m 2 /s; and C m is the diffused gas content, m 3 /m 2 . Liu and Lin found that the diffusion coefficient D of gas in the matrix varies with the diffusion time, 36 and the functions of the dynamic diffusion coefficient and time are as follows …”

“…Gas generally exists in coal in the adsorption state and free state, and the gas in the two states is transformed into each other. The gas in the pore system of the coal matrix flows in the form of diffusion under the concentration gradient, 35 which conforms to Fick's law…”

Study on the Solution and Variation Law of Diffusion Coefficient Based on the Numerical Simulation Optimization Method

“…Yi et al held that the diffusion coefficient declines with the decrease of the equilibrium pressure . Wu et al consider that the diffusion coefficient is unrelated to the gas pressure . In Figure (a), the diffusion coefficient substantially lies near 1.062 × 10 –10 m 2 /s.…”

“…Diffusion of gas is essentially driven by the pressure gradient of free gas, and the flow velocity is proportional to the pressure gradient but not directly related to the gas content. Qin et al derived the flow equation of diffusion driven by the density gradient of free gas: , J = − D grad ⁡ ρ g where J is the gas mass passing through unit coal particle per unit time, g/(cm 2 ·s) and ρ g refers to free gas density in coal particles, g/cm 2 .…”

“…Encountered with complex theoretical models, such as constant volume gas desorption, gas desorption under the assumption of a cube coal matrix and diffusion coefficient is assumed to have time-related characteristics, etc., analytical solutions will be more difficult to solve. When the analytical solution cannot be obtained or is difficult to seek, it can be obtained by numerical analysis methods, including numerical approximation, the difference method, the interpolation method, etc. ,,, But the calculation process is also cumbersome. According to the characteristic and law of experimental data of gas desorption from coal particle, there appeared simplified analytical empirical formulas such as the t model, , Barrer’s power function formula, Sun Zhongxu’s formula, and Winter’s formula .…”