A Mathematical Representation Of The Carr, Kobayashi And Burrows Natural Gas Viscosity Correlations
Authors: Dranchuk P.M. , Islam M.R. , Bentsen R.G. 
Published: March 2010

Abstract
Viscosities of sour natural gases at reservoir conditions are usually estimated by means of the Carr, Kobayashi and Burrows correlations. There are currently two methods which convert these correlations to forms suitable to machine calculation. The first known as the Dempsey-Standing method, uses polynomial expressions for calculating µ, the viscosity at reservoir temperature and atmospheric pressure, and the sour gas corrections, as well as a mixed power polynomial to calculate the viscosity ratio. The second method, developed by Energy Resources Conservation Board of Alberta staff members and herein referred to as the ERCB method, also uses polynomials to generate µ, and corrections, but uses a table look-up, Lagrangian interpolation scheme to generate viscosity ratios. The first method is applicable in a restricted region whereas the second requires data storage and tedious interpolation.
This paper presents a method wherein µ and the corrections are generated as in the first method but the viscosity ratio is generated from an orthogonal polynomial fitting of the original surface. This method is valid over the same range as the ERCB method, does not require data storage or interpolation and is more accurate than the second method. Comparison at some 2000 different reduced pressures and temperatures indicates a mean absolute error of less than 0.95%.

Introduction
In the petroleum industry, natural gas viscosities are most commonly estimated by applying the Carr, Kobayashi and Burrows correlations(l). The method consists of a simple two-step procedure. In the first step, µ, the viscosity at atmospheric pressure and temperature TR, is estimated using the apparent molecular weight or gravity of the gas and the temperature at which the viscosity is desired. If the gas contains N2, H2S and CO2 the µ, value is corrected for each, depending upon its concentration and the gas gravity. In the second step the ratio of the viscosity of the gas at the desired temperature and pressure to µ, is obtained using the reduced pressure and temperature of the gas at the conditions required.
Since this method employs graphical representations, they could not be used in machine calculations in their original form. To overcome this difficulty Dempsey(2) expressed the graphical representations of µ, and µ/ µ by means of polynomial expressions using multiple regression to generate the necessary coefficients. However, the mathematical expressions proposed make no provision for correcting µ for the presence of non-hydrocarbon components. Furthermore, the range of validity of this, µ/ µ, representation is reduced from
Equation (available in full paper)
in the original to
Equation (available in full paper)
A computer subroutine for determining natural gas viscosities incorporating corrections for H2S, CO2 and N2 was developed in 1968 by A.B. Dick of the Energy Resources Conservation Board of Alberta and was later modified by D.H. Burger (1972) and J.M. Ethier (1975)(3). The method consists of polynomial expressions for µ and each of the H2S, CO2 and N2 corrections, plus a table look-up and Lagrangian interpolation scheme of a digitized version of the original graphical representation of the µ/ µ, ratio(4).