Klinkenberg correction

Abstract: The permeability of Westerly granite was measured as a function of effective pressure to 4 kb. A transient method was used, in which the decay of a small incremental change of pressure was observed; decay characteristics, when combined with dimensions of the sample and compressibility and viscosity of the fluid (water or argon) yielded permeability, k. k of the granite ranged from 350 nd (nanodarcy = 10−17 cm2) at 100-bar pressure to 4 nd at 4000 bars. Based on linear decay characteristics, Darcy's law apparently held even at this lowest value. Both k and electrical resistivity, ρs, of Westerly granite vary markedly with pressure, and the two are closely related by k = Cρs−1.5±0.1, where C is a constant. With this relationship, an extrapolated value of k at 10-kb pressure would be about 0.5 nd. This value is roughly equivalent to flow rates involved in solute diffusion but is still a great deal more rapid than volume diffusion. Measured permeability and porosity enable hydraulic radius and, hence, the shape of pore spaces in the granite to be estimated. The shapes (flat slits at low pressure, equidimensional pores at high pressure) are consistent with those deduced from elastic characteristics of the rock. From the strong dependence of k on effective pressure, rocks subject to high pore pressure will probably be relatively permeable.

Abstract: Gas-producing mudrock systems are playing an important role in the volatile energy industry in North America and will soon play an equally important role in Europe. Mudrocks are composed of very fine grained particles, and their pores are very small, at the scale of nanometers. Gas production from these strata is much greater than what is anticipated given their very low Darcy permeability. In this paper, images of nanopores obtained by Atomic Force Microscopy (AFM) are presented for the first time. Gas flow in nanopores cannot be described simply by the Darcy equation. Processes such as Knudsen diffusion and slip flow at the solid matrix separate gas flow behaviour from Darcy-type flow. We present a formulation for gas flow in the nanopores of mudrocks based on Knudsen diffusion and slip flow. By comparing this new gas flow formulation and Darcy flow for compressible gas, we introduce an apparent permeability term that includes the complexity of flow in nanopores, and it takes the form of the Darcy equation so that it can easily be implemented in reservoir simulators. Results show that the ratio of apparent permeability to Darcy permeability increases sharply as pore sizes reduce to smaller than 100 nm. Also, Knudsen diffusion's contributions to flow increase as pores become smaller. Unlike Darcy permeability, which is a characteristic of the rock only, permeation of gas in nanopores of mudrocks depends on rock, gas type and operating conditions.

Abstract: Gaseous flow regimes through tight porous media are described by rigorous application of a unified Hagen–Poiseuille-type equation. Proper implementation is accomplished based on the realization of the preferential flow paths in porous media as a bundle of tortuous capillary tubes. Improved formulations and methodology presented here are shown to provide accurate and meaningful correlations of data considering the effect of the characteristic parameters of porous media including intrinsic permeability, porosity, and tortuosity on the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor.

Abstract: We study the gas flow processes in ultra-tight porous media in which the matrix pore network is composed of nanometre- to micrometre-size pores. We formulate a pressure-dependent permeability function, referred to as the apparent permeability function (APF), assuming that Knudsen diffusion and slip flow (the Klinkenberg effect) are the main contributors to the overall flow in porous media. The APF predicts that in nanometre-size pores, gas permeability values are as much as 10 times greater than results obtained by continuum hydrodynamics predictions, and with increasing pore size (i.e. of the order of the micrometre), gas permeability converges to continuum hydrodynamics values. In addition, the APF predicts that an increase in the fractal dimension of the pore surface leads to a decrease in Knudsen diffusion. Using the homogenization method, a rigorous analysis is performed to examine whether the APF is preserved throughout the process of upscaling from local scale to large scale. We use the well-known pulse-decay experiment to estimate the main parameter of the APF, which is Darcy permeability. Our newly derived late-transient analytical solution and the late-transient numerical solution consistently match the pressure decay data and yield approximately the same estimated value for Darcy permeability at the typical core-sample initial pressure range and pressure difference. Other parameters of the APF may be determined from independent laboratory experiments; however, a pulse-decay experiment can be used to estimate the unknown parameters of the APF if multiple tests are performed and/or the parameters are strictly constrained by upper and lower bounds.