Estimating Gas Relative Permeability of Shales from Pore Size Distribution

Abstract Modeling shale gas relative permeability, krg, has numerous practical applictaions, particularly in gas exploration and production in unconventional reservoirs. krg is a key petrophysical quantity for accurately determining recovery factor and production rate. In the literature, a few theoretical models developed to estimate krg are based upon either a "bundle of capillary tubes" conceptual approach or a combination of universal scaling laws e.g., from percolation theory. The former is a severely distorted idealization of porous rocks, while the latter is, generally speaking, valid near the percolation threshold and/or in rocks with narrow pore-throat size distribution. Although the effective medium approximation has been successfully applied to model wetting- and nonwetting-phase relative permeabilities in conventional rocks, to the best of the author's knowledge, it has never been used to estimate krg in unconventional reservoirs. Therefore, the main objective of this study is to develop a theoretical model based on the effective-medium approximation, an upscaling technique from statistical physics, to estimate shale gas relative permeability from pore-throat size distribution. In this study, we presumed that pore-throat sizes conform to a truncated log-normal probability density function. We further presumed that gas flow under variably-saturated conditions is mainly controlled by two mechanisms contributing in parallel: (1) molecular flow and (2) hydraulic flow. The total conductance of a single pore (gt), therefore, was equal to the summation of the molecular flow conductance, gm, and the hydraulic flow conductance, gh (i.e., gt = gm + gh). We then invoked the governing equation from the effective-medium approximation to determine effective conductances and, accordingly, gas relative permeabilities at various saturations. Results showed that krg varies as the log-normal distribution parameters standard deviation (σ) and geometric mean pore radius (rm) alter. By comparison with two- and three-dimensional pore-network model simulations, we found that the proposed model estimated gas relative permeability accurately, particularly in three dimensions. We also estimated krg from the pore-throat size distribution derived from measured mercury intrusion capillary pressure (MICP) curve for three experiments (i.e., Eagle Ford, Pierre, and Barnnet) and found that krg of Barnnet was remarkably less than that of Pierre and Eagle Ford.