Correction to Klinkenberg slip theory for gas flow in nano-capillaries
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Using a lattice Boltzmann (coarse grain) simulation of gas dynamics we show that the apparent permeability values of nano-scale capillaries could be significantly higher than those predicted by the Klinkenberg slip theory. The difference is due to kinetic effects of gas molecules that have gone through inelastic collisions with the capillary walls on those molecules that make up the bulk fluid in the capillary. The kinetic energy that the bouncing-back molecules have and the associated momentum carried to the bulk fluid is not a trivial matter in capillaries with diameter, h, less than 100nm. Momentum carried by bouncing-back molecules amplifies the velocity profile developing across the diameter of the capillary. In a sense, it is not only the molecules interacting with the capillary wall that slip but also those interacting with the bulk fluid, i.e., double-slip. The double-slip effect is shown using measured permeability data of two crushed nano-porous samples, Pyrophyllite, and three different shale samples at varying pore pressures. Using the simulation results, we propose a modification to the Klinkenberg equation. Our new double-slip Klinkenberg equation includes a characteristic length scale (L Ke) that is proportional to the kinetic energy per capillary cross-sectional area of the bouncing-back molecules by the capillary walls. The new length scale of the molecular kinetic effects in nano-capillaries is larger than the mean free path of the molecules. The double-slip Klinkenberg equation reduces to the classical equation for slip flow in large capillaries, i.e., h/L Ke>>1, and converges to the absolute permeability value at high pressure. 2012 Elsevier B.V.
