Second-order correction of Klinkenberg equation and its experimental verification on gas shale with respect to anisotropic stress

Shale is an extremely tight and fine-grained sedimentary rock with nanometer-scale pore sizes. The internal nanopore structure gives rise to not merely the ultra-low permeability of shale, but also the significant slip flow (non-Darcy) phenomenon. This study involves a second-order correction of the traditional Klinkenberg equation and its experimental verification, in consideration of the effect of anisotropic stress, using cube-sized shale samples sourced from a typical Chinese sedimentary basin. The effects of two gas slippage factors, Klinkenberg-corrected permeability, and anisotropic stress were also discussed. The results showed an increasing trend of apparent permeability with decreasing pore pressure when the pressures are relatively low, due to the gas slippage effect. The second-order model results in a lower Klinkenberg-corrected permeability compared with that from the traditional Klinkenberg equation and yields a better match with the experimental data. Analysis of the experimental data showed that both the first-order slippage factor A and the second-order slippage factor B rose as stress anisotropy increased, and that A was more sensitive to stress anisotropy compared with B. Interestingly, both A and B first increased slightly and then dramatically as the permeability declined. It is recommended that when the shale permeability is below 10 −3 mD, the second-order approach should be taken into consideration. Darcy's law starts to deviate when K n  > 0.01 and is invalid at high Knudsen numbers. The second-order approach alleviates the problem of permeability overestimation compared with the traditional Klinkenberg equation. When subjected to effective stress, shale pores become constricted and the degree of this constriction decreases gradually as the effective stress continues to increase.