A practical gas permeability equation for tight and ultra-tight rocks

Gas rarefaction phenomenon during gas flows in tight pore spaces makes the gas permeability of tight and ultra-tight rocks higher than the intrinsic permeability. Gas permeability equations in the literature usually assume a pre-defined pore morphology (or geometry) which may not reflect the real pore structure of rocks. This study aims to develop a practical gas permeability equation that considers the pore morphology variance in different tight/ultra-tight rocks. We build an analytical equation for describing the full-Knudsen-number-ranged rarefied gas flows through single channels. By approximating this equation within a Knudsen number range of (0, 1), we develop a second-order equation of channel permeability vs. Knudsen number. In the literature, most of the second-order equations of channel permeability vs. Knudsen number were built for a 2D channel. Herein, we establish such second-order relation with different sets of parameter values valid for the rarefied gas flows in a cylinder, in a triangular prism, and in a duct, respectively. After checking the relation between a Knudsen number and real gas properties, we further transform the second-order equation in terms of Knudsen number into a second-order equation in terms of the reciprocal of gas pressure. We then quantify how a channel morphology influences the parameter values in the permeability equation. Accordingly, a constraint on the parameters of the permeability equation is proposed by considering the most observed pore morphologies in tight/ultra-tight rocks. The second-order permeability equation with the constraint is utilized to well fit Lattice Boltzmann simulations on reconstructed rock samples and multiple sets of permeability data that are measured in the literature. When being used to fit the measured permeability data of gas flows at pressures of above 0.5 MPa in the rocks with an intrinsic permeability of below 0.1 μD, the proposed second-order permeability equation demonstrates superiority to the permeability equations selected from the literature. The regressed values of the parameters in the proposed permeability equation suggest that the dominant pores in tight and ultra-tight rocks would normally have a rectangular cross-section, a circular cross-section, or a triangular cross-section. The proposed equation with the constraint offers a simpler quantitative description of the interplay among pore morphology, gas properties, and tight/ultra-tight rock permeability than the permeability equations in the literature.