Hydraulic fracturing in shales is challenging because of the complicated stress status. The confining pressure imposed on a shale formation has a tremendous impact on the permeability of the rock. The correlation between confining pressure and rock permeability is complicated and might be nonlinear. Gas flow in low-permeability shales differs significantly from liquid flow because of the Klinkenberg effect, especially when the pore pressure is relatively low. The Klinkenberg effect results from gas molecule slip at the solid walls inside the nanopores, where the collision between gas molecules and solid surfaces is more frequent than the collision between gas molecules. This effect causes the increase of apparent permeability (i.e., the measured permeability). In this study, the simple effective stress law and the effective stress coefficient law were used to study the relationship between permeability and effective stress. In the simple effective stress law, the effective stress is calculated as the difference between confining pressure and pore pressure. The Klinkenberg coefficient and the effective mean pore radius can then be calculated. In the effective stress coefficient law, there is an effective stress coefficient (i.e., the Biot coefficient) which controls the influence of pore pressure on the effective stress. In this study, the effective stress coefficient was obtained by analyzing a large number of laboratory data measured under varying pore pressures and confining pressures. Specifically, the permeabilities of core samples extracted from four U.S. shale formations were measured using a pulse decay permeameter under varying combinations of confining and pore pressures. The samples were cored in the directions parallel to and perpendicular to the shale bedding planes, in order to test the role of bedding plane direction on the measured permeability. Laboratory results demonstrate that the permeabilities of all core samples fell in the range between 10-2 millidarcy (mD) and 10-4 mD. In the same formation, the permeabilities of the core samples in which the bedding planes were in the longitudinal direction were about one order of magnitude higher than the permeabilities of the core samples in which the bedding planes were in the transverse direction. Using the simple effective stress law, the Klinkenberg effect was observed, because the measured apparent permeability decreased with increasing pore pressure. Using the effective stress coefficient law, the effective stress coefficient was found around 0.5, which suggests that the pore pressure had a less influence on the effective stress compared to the confining pressure. Moreover, a multiphysical shale transport (MPST) model is built that accounts for fluid dynamics, geomechanics, and the Klinkenberg effect. The model fitting result is quite matched with PDP experimental results. These comprehensive laboratory experiments and model fitting demonstrate the role of confining pressure, Klinkenberg effect, and bedding plane direction on the gas flow in the nanoscale pore space in shales. These experimental data will be valuable in validating and calibrating pore- to core-scale numerical models of the flow and transport properties in shale formations.

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