A self-similarity mathematical model of carbon isotopic flow fractionation during shale gas desorption

Abstract

The existence of nanosized pore systems differentiates isotopic gas transport inside a shale matrix from conventional continuum flow. In this study, a novel self-similarity mathematical model was developed to investigate the effects of gas flow transport (both slip flow and free molecular diffusion flow) on isotopic gas fractionation for four different shale samples (S1 and S2 from north Germany and S3 and S4 from Xiashiwan Field, Ordos Basin, China). In this model, the nonlinear permeability and diffusion coefficients were developed for the isotopologues (12CH4 and 13CH4), respectively. By selecting appropriate exponents of the pressure gradient for 12CH4 and 13CH4, respectively, the estimated isotopic methane concentration and production rate showed a good agreement with experimental data. The developed model shows that the gas concentration of the isotopologues in samples S1 and S2 increases with time following a power law. Similarly, the gas production rates of the isotopologues in samples S3 and S4 decay with time following a power law. Moreover, the exponents of the pressure gradient for the isotopologues are close to 4 for samples S1 and S2, indicating that the effect of slip flow on isotopic gas fractionation cannot be ignored. For samples S3 and S4, the exponents of the pressure gradient for the isotopologues increase with temperature rising, which shows the promotion of isotopic gas fractionation under higher heating temperatures. The slight difference between the exponents of the pressure gradient for the isotopologues for the same shale sample reveals that the isotopic gas fractionation of carbon is a slow process.