A Fast and Accurate Method for Parameter Estimation of Archie Saturation Equation
Authors: El-khatib Noaman 
Published: December 2007

Abstract
A modified regression analysis method is presented for the estimation of Archie equation parameters, a, m and n. The method is a weighted least squares with weights chosen as to approximate the actual residuals of the independent variable which are nonlinear by the residuals of the logarithms of the dependent variable which are linear. The method thus combines the simplicity of linear regression on the logarithms with the physical meaningfulness at the nonlinear regression on the actual variables. The method is found to give almost identical results with the nonlinear regression method. Better results in terms of data fitting is obtained if the Archie parameter a is allowed to be estimated from regression analysis rather than fixing its value at 1. Other forms of regression analysis including sequential parameter estimation and straight line regression analysis are discussed.

Introduction
Porosity and fluid saturations are among the most important reservoir properties used in reserve estimates of oil and gas reservoirs. Because of the heterogeneity of most of the reservoirs, a continuous recording of these properties vs. depth is essential for accurate estimations. Since a complete coring and core analysis of the entire pay zone is impractical, well logging appears to be the most plausible mean to obtain such information. To achieve this task, different logs including resistivity, acoustic and radioactive logs are recorded in the well and well log interpretation is used to obtain the required properties.
Porosity and lithology of the formation is usually obtained from a combination of logs called the porosity logs. These include the neutron, sonic and density logs. Fluid saturations are usually obtained from resistivity logs. Different resistivity logs with variable radii of investigation are recorded to obtains fluid saturations at different distances from the wellbore A micro resistivity log is used to obtain the resistivity in the flushed zone behind the wellbore while a deep resistivity device (induction or laterolog) is used to obtain the resistivity in the virgin (uninvaded) zone.
Fluid saturations are estimated from resistivity measurements by the use of Archie equation. This equation relates the resistivity of the formation to the porosity, water saturation and resistivity of the water saturating the formation
(1)
with
(2)
Eq. [1] is usually written in the form
(3)
Archie equation is used to determine the water saturation Sw provided the coefficients a, m and n are known and is obtained from independent porosity logs. Originally Archie used a value of a= 1 in his equation but it was found that values other than one fits better the data for some formation. So, Winsauer et al. introduced the coefficient a in Eq. (2) which may differ from unity. For the purpose of estimating S, Archie equation is written as
(4)
Equation (4) can be used to estimate the saturation in the original zone or in the flushed zone. When used with resistivity devices with deep radius of influence, it provides a value of the original (connate) water saturation S. P. 541^