Calculation of Klinkenberg permeability, slip factor and turbulence factor of core plugs via nonlinear regression ☆ Fernando A. Pazos a , Amit Bhaya a, ⁎, André Luiz Martins Compan b a Department of Electrical Engineering, Federal Univ of Rio de Janeiro, PEE/COPPE/UFRJ, PO Box 68504, Rio de Janeiro 21945970, Brazil b Petrobras Research Center, CENPES/PDP/TRA, Room 2097, Rio de Janeiro, Brazil abstract article info Article history: Received 19 June 2008 Accepted 26 May 2009 Keywords: Klinkenberg permeability Klinkenberg slip factor Forchheimer turbulence factor core plugs nonlinear regression transient state permeameter iterative algorithm convergence gas pressure decay measurements Published methods to determine the Klinkenberg permeability, Klinkenberg slip factor and Forchheimer turbulence factor of core plugs can exhibit considerable error. Jones presented a technique based on gas pressure decay measurements during the transient state, where, with a single run, an algorithm can calculate the parameters with precision. However, this paper shows that Jones' method is based on a linear regression to find a fixed point of a nonlinear error function, and is presented without theoretical justification or convergence conditions. This paper proposes a simple algorithm, based on nonlinear regression, to calculate the unknown parameters, and has the advantage of theoretical justification as well as weaker requirements for convergence. In addition, a strategy to calculate the unknown physical parameters when the measurements are noisy is presented. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Permeability of core plugs is usually measured in steady state with air at mean pressure just above 1 atm. This determination is rapid, but it can lead to serious errors (Jones, 1972). Correction factors are available, but the corrected, low pressure measurements can still exhibit considerable error. Jones (1972) developed a transient permeability technique which allows the determination of the liquid permeability, the Forchheimer turbulence factor and the Klinkenberg slip factor of a core plug from a single run, where measurements of gas pressure flowing through the core plug are made along time. This method does not require an empirical correlation using cores of known permeability to construct calibration curves. Jones' method is based on making a change of variables that allows the rewriting of the slip-corrected algebraic Forchheimer equation in a “linear” form in which the unknown physical parameters (perme- ability, turbulence and slip factors) appear nonlinearly in the coefficients of the equation. Linear regression is then used to find the coefficients, based on an initial guess, which is updated by recalculating the parameters iteratively, until convergence occurs. The potential drawbacks of Jones' proposal are: 1) The parameters of the linear Forchheimer equation are not independent, and the unknown variables to be calculated depend nonlinearly on these parameters. The change of variables proposed by (Jones, 1972) could increase the uncertainty of the unknown parameters. 2) The iterative algorithm proposed by Jones can be viewed as an algorithm to find a fixed point of a so called linear equation. This algorithm has restrictive convergence conditions, which are not analysed by (Jones, 1972). The change of variables to rewrite the algebraic slip-corrected nonlinear Forchheimer equation into a linear form is unnecessary, because methods using nonlinear regression to calculate parameters that minimize errors in a nonlinear equation are well known. For example, Finsterle and Persoff (1997) propose the use of inverse modeling to determine the Klinkenberg slip factor and the permeability, from an experiment where pressure decay is measured. The errors are minimized using the Levenberg–Marquardt modification of the Gauss–Newton algorithm (see the description of this and other algorithms to solve nonlinear least squares problems in Madsen et al. (2004)). However, Finsterle and Persoff (1997) do not consider the Forchheimer effect, since they use Darcy's law, and also do not present the equations necessary to implement this method, just discussing the theoretical procedure. This paper shows that Jones' method is a fixed point iteration using linear regression for a nonlinear problem and therefore proposes the Journal of Petroleum Science and Engineering 67 (2009) 159–167 ☆ Submitted to the Journal of Petroleum Science and Engineering. ⁎ Corresponding author. Tel.: +55 21 2562 8078; fax: +55 21 2562 8080. E-mail addresses: quini@ort.org.br (F.A. Pazos), amit@nacad.ufrj.br (A. Bhaya), andrecompan@petrobras.com.br (A.L.M. Compan). 0920-4105/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2009.05.012 Contents lists available at ScienceDirect Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol