Analysis of Errors in Historical Use of Archie's Parameters
Authors: Fasesan O. , Awolusi F. , Heinze L.R. 
Published: March 2010

Abstract
While it is common knowledge that the values of Archie's parameters ‘a’, ‘m’ and ‘n’ may vary for sandstone reservoirs at different conditions ? consolidated/unconsolidated, water-wet/oil-wet, pore geometry, degree of sphericity, clay content and in-situ depositional environment ? the values of a = 1, m = 2 and n = 2 have been linked historically with Archie's equations. Statistically, these values were assumed as the population mean when used in calculations, but the magnitude of errors associated with their use is often neglected. This study shows how a sample of data drawn from experimental and analytical methods determine Archie's parameters. Well logs from sandstone reservoirs are used to draw statistical inferences about the population characteristics of Archie's parameters in sandstone reservoirs. This also shows the magnitude of relative error possible when m = 2 and n = 2 are used to compute water saturation and formation resistivity factors.

Introduction
The 1942 landmark publication by Gus Archie titled, "The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics "(1) introduced new parameters relevant to describing reservoirs using well logs and set standard parameters for the identification of permeable zones within a reservoir.
Basically, Archie's experiments involved measuring the porosity and electrical resistivity of numerous shale-free sandstone cores from the Gulf Coast by saturating them with brine of salinities ranging from 20 to 100,000 ppm of NaCl. Archie's work established the following relationships:
Equation (1) (Available In Full Paper)
where Ro is the resistivity of the rock fully saturated with brine and Rw is the formation water resistivity. F is termed the formation resistivity factor and is a measure of the effect of formation on the path of electrical current traveling through the electrolytic brine in the rock pore system. The plot of F against porosity (Φ) on log-log scales revealed a linear trend equivalent to:
Equation (2) (Available In Full Paper)
The ‘m’ parameter represents the trend's negative slope.
In deriving the above expression, Archie force-fitted a line to his formation resistivity factor against porosity data such that F = 1.0 at 100% porosity. A replica of Archie's original plot using clean sandstone cores is shown in Figure 1. This was however deemed unnecessary as other research works revealed that when a line is fitted to formation resistivity factor against porosity, the intercept at 100% porosity would not always derive one, but can be greater or less than one. Winsauer et al.(2), for example, duplicated Archie's experiments with sandstone cores from a wide range of reservoirs and arrived at:
Equation (3) (Available In Full Paper)
Thus, the general form of Archie's formation resistivity factor is expressed as:
Equation (4) (Available In Full Paper)
Winsauer first referred to Archie's ‘m’ parameter as the cementation factor, while ‘a’ would later be referred to as the tortuosity factor.
Archie, again, considered partially saturated hydrocarbon bearing shale-free sandstones and proposed a second factor called the resistivity index (I) that would further increase the rock's resistivity. He expressed this as:
Equation (5) (Available In Full Paper)
FIGURE 1: Plot of formation resistivity factor against porosity (Available In Full Paper)