A mathematical diffusion model of carbon isotopic reversals inside ultra-tight Longmaxi shale matrixes

Abstract

Developing mathematical models for high Knudsen number (Kn) flow for isotopic gas fractionation in tight sedimentary rocks is still challenging. In this study, carbon isotopic reversals (δ13C1 > δ13C2) were found for four Longmaxi shale samples based on gas degassing experiments. Gas in shale with higher gas content exhibits larger reversal. Then, a mathematical model was developed to simulate the carbon isotopic reversals of methane and ethane. This model is based on these hypotheses: (i) diffusion flow is dominating during gas transport process; (ii) diffusion coefficients are nonlinear depending on concentration gradient. Our model not only shows a good agreement with isotopic reversals, but also well predicts gas production rates by selecting appropriate exponents m and m∗ of gas pressure gradient, where m is for 12C and m∗ is for 13C. Moreover, the (m−m∗) value has a positive correlation with fractionation level. (m1−m1∗) of methane are much higher than that of ethane. Finally, the predicted carbon isotopic reversal magnitude (δ13C1−δ13C2) exhibits a positive relationship with total gas content since gas in shale with higher gas content experiences a more extensive high Kn number diffusion flow. As a result, our model demonstrates an impressive agreement with the experimental carbon isotopic reversal data.