Mathematical Model Establishment for the Multi-Scale Permeability of Coal Reservoirs and Its Engineering Significance

Permeability is a critical parameter governing the gas flow behavior of the coalbed methane (CBM) reservoir during the exploration and exploitation of CBM, as well as the geological storage of CO2 in the coalbeds. It is strongly associated with the multi-scale fractures developed in coal. Based on the distribution characteristics of micro-fractures, a multi-scale permeability model for coal reservoirs was established by introducing the permeability tensor, which comprehensively considers adsorption-induced coal swelling, pore pressure, effective stress, and micro-fractures. Further, the dynamic evolution law and mechanism of multi-scale permeability of coal reservoirs under different adsorption pressures were discussed. The results indicate that the increase in effective stress on the coal caused by adsorption-induced swelling essentially leads to a decrease in the equivalent multi-scale permeability of coal. Two key indicators, namely equilibrium pressure and rebound pressure, were defined to quantitatively characterize the evolution law of the equivalent multi-scale permeability during gas adsorption or desorption processes. The effective stress generated by the CO2 adsorption-induced swelling effect in the low-rank coal is 1.47 times that in the middle-rank coal and 2.51 times that in the high-rank coal. Additionally, the effective stress generated by the CO2 adsorption-induced swelling effect in the low-rank coal is 5.15 times that generated by N2, while this level is 4.32 times higher than that in the middle-rank coal. Therefore, compared with the low- and middle-rank coal, the high-rank coal exhibits a smaller decrease in multi-scale permeability due to its weaker adsorption-induced swelling effect. During N2 adsorption, the pore pressure effect dominates over the adsorption-induced swelling effect, resulting in a decrease in the effective stress on the coal with increasing gas pressure. Consequently, the equivalent multi-scale permeability of coal will increase much more significantly with an increase in injected N2 pressure than with an increase in CO2 pressure. By accounting for the differences between the effects of adsorption-induced swelling and pore compression on the equivalent multi-scale permeability of coal reservoir, the injectivity of CO2 can be improved by mixing it with N2.