Technical Notes - Calculations of Unsteady-State Gas Flow Through Porous Media, Corrected for Klinkenberg Effect

Mathematical equations have been derived to show the effect of the slippage phenomomenon (Klinkenberg effect) on unsteady-state gas flow through porous media. The assumption is made that the slippage may be represented by an equation of the type It is pointed out that most of the existing data on unsteady gas flow may be corrected for slippage by a simple change in definition of the dimen-sionless groups. INTRODUCTION Several solutions to the various problems associated with the unsteady-state flow of gas through porous media have been recently presented in the literature.','" The cited publications have resented solutions for flow under ideal conditjons. The principal assumptions involved include: (a) the density of the gas obeys the perfect gas law; (b) the viscosity of the gas is constant; and (c) the gas permeability is constant. Few, if any, actual gases under reservoir conditions are believed to conform with the above, but the ideal solutions do provide the basis for making improved estimates on the performance of gas reservoirs. The purpose of this paper is to show the performance characteristics when assumption (c) is corrected for the slippage phenomenon or the Klinkenberg effect. This is believed to be a step nearer to reality. MATHEMATICAL DEVELOPMENT Case 1. Gas Flow — Ideal Conditions The equations describing the flow of gas under ideal conditions have been clearly presented in the cited literature; however, for completeness of this work they are repeated here. The basic differential equation describing the flow of gas is Equation (6). It may be derived as follows: The continuity equation, the perfect gas law and Darcy's law are combined to give: Manuscript received in the Petroleum Branch office Aug. 3, 1963 or If — is constant, Equation (5) is rewritten to form the basic Equation (6). or for the linear case A dimensionless equation is obtained by dividing through by (pl)z and making the substitutions shown below: The resulting equation is A similar procedure for the radial case leads to the following equations: where and Equations (11), (12), and (13) are those used by Bruce, et al., and Jenkins and Aronofsky for their computational work.