Nonempirical Apparent Permeability of Shale

Abstract

Summary Physics of fluid flow in shale reservoirs cannot be predicted from standard flow or mass-transfer models because of the presence of nanopores, ranging in size from one to hundreds of nanometers, in shales. Conventional continuum-flow equations, such as Darcy's law, greatly underestimate the fluid-flow rate when applied to nanopore-bearing shale reservoirs. As a result of the existence of nanopores in shales, the molecular mean free path becomes comparable with the characteristic geometric scale, and we hypothesize that under this condition, Knudsen diffusion, in addition to correction for the slip boundary condition, becomes the dominant mechanism. Recently, a few models have been developed that use various empirical parameters to account for these modifications (Javadpour 2009; Civan 2010; Darabi et al. 2012). This paper aims to provide a different approach to modeling apparent permeability in shale reservoirs. The proposed model is analytical, free of any empirical coefficients, and has been derived without invoking the assumption of slip flow at the pore wall. Our model of apparent permeability represented by a single analytical equation, depends only on pore size, pore geometry, temperature, gas properties, and average reservoir pressure. The proposed model is valid for Knudsen numbers less than unity and it stands up under the complete operating conditions of a shale reservoir. Our model reasonably predicts results as reported by other models. Finally, the model shows that pore-surface roughness and mineralogy have a negligible influence on gas-flow rate, whereas pore geometry and pore size play a significant role in the proportion of diffusion in total flow rate. Our study shows that a combination of Darcy flow and Knudsen flow—ignoring the Klinkenberg effect—can describe gas flow for a range of Knudsen flow applicable to a shale-gas system.

Correction of source-rock permeability measurements owing to slip flow and Knudsen diffusion: a method and its evaluation

“…Gas diffusion is really a result of thermal motion of gas molecules. Based on the classic statistical physics, the gas diffusion coefficient is given by (Singh et al 2014)…”

“…The latter two parameters are given by (Civan 2010;Ziarani and Aguilera 2012;Singh et al 2014;Liu and Cai 2014) k…”

“…(1) should be replaced by the latter. In this case, the gas diffusion coefficient becomes (Singh et al 2014)…”

“…One approach is based on a fundamental assumption that the total gas flow rate through a micro-tube is a summation of the following two components (Roy et al 2003;Singh et al 2014). One corresponds to viscous flow (determined with classic pipe flow formulations) and the other one to Knudsen diffusion (determined with diffusion coefficient given in Eq.…”

“…On the other hand, the relationships between measured permeability and intrinsic permeability have been well developed for micro/nano-tubes partially as a result of the simplicity of their flow-path geometries (Beskok and Karniadakis 1999). Then efforts have been made in using the relationships for micro/nano-tubes to approximate those for porous media (Civan 2010;Ziarani and Aguilera 2012;Singh et al 2014). However, the usefulness of these approximations is not totally clear because comparisons between results calculated with these approximations and laboratory measurements are very rare in the literature.…”