Shale Gas Correction to Klinkenberg Slip Theory
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AbstractUsing lattice-Boltzmann simulation of steady-state gas flow we show that 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 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. It creates a molecular streaming effect that amplifies velocity profile developing across the diameter of capillary. In a sense, it is not only the molecules interacting with the wall that slip but also the bulk fluid, i.e., double slip. The double-slip effect is shown using measured permeability data of a crushed nanoporous material with a uniform pore size (10>h>40 nm) at varying pore pressures. Using the simulation results we propose a modification to the Klinkenberg equation. New double-slip Klinkenberg equation includes a characteristic length scale (LKE) that is proportional to the kinetic energy per capillary cross-sectional area of the bouncing-back molecules by the walls. The new length scale of the molecular kinetic effects of 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/LKE <<1, and converges to the absolute permeability value at high pressures. Due to nanoporous nature of coals and organic-rich shales, the double-slip effect is likely to be significant near hydraulic fractures and production wells in depleted reservoirs.The presented theoretical work brings new insights into the measurement of crushed particle permeability using gas uptake or release experiments. Although they are fast low-pressure laboratory measurements, there are widely-recognized uncertainties associated with them. In this paper we show that dramatic variation in the measured permeability is possible due to small changes in the measurement cell pressure. Hence, even when the crushed sample is considered to be representative of the reservoir, the measured permeability may not quantitatively be related to flow under the reservoir conditions. The double-slip permeability concept and the modified Klinkenberg theory eliminate some of the uncertainties associated with the laboratory measurement of permeability and lead to development of new measurement protocols for the unconventional resources industry. A double-slip Klinkenberg chart is developed including the effects of nano-capillaries on the permeability, and two procedures are presented with example calculations on how to use the chart to predict permeability of shale sample from its effective pore size and shape.
