A Generalized Resistivity-Porosity Crossplot Concept
Authors: Sanyal Subir K. 
Published: December 2007

Abstract
This paper expands and generalizes the approach to crossplotting formation resistivity (Rt) and porosity () on log-log paper to derive more porosity () on log-log paper to derive more information than has so far been attempted from such a crossplot. Normally, an Rt - crossplot is used to estimate values of the cementation factor (m) and the product, where is the formation water resistivity and is the parameter in Archie's formula: F =. There is no existing technique to determine the value of n from resistivity or porosity logs. Improper assumption of the n value may cause considerable error in reserve estimates. Common assumptions of n=2 or n=m are often inaccurate; laboratory derived n values are rarely reported. The generalized Rt - crossplot concept presented in this paper may yield a value of n for a clean sand or carbonate in many cases. The approach presented here can be used to calculate, m, n and bulk volume water fraction for irreducible zones (C) from logs; it has several checks and balances to assess internal consistency of data. The use of this approach should improve calculation of hydrocarbon reserves. The estimated parameter C is a useful reservoir characteristic and can be used in exploration. Besides these parameters, the plot readily yields values of Sw and Swi for any zone whose porosity is known. Two field examples are provided. This technique will not work for shaly sands or formations where is not constant. Even in such cases, the crossplot may provide insight into some matrix and fluid characteristics. Possible modifications of this approach for shaly sands and other situations are indicated.

Introduction
In order to calculate the hydrocarbon reserves in a formation, one needs to know porosity and water saturation. These are usually calculated from well logs. Porosity () is calculated from the various porosity Porosity () is calculated from the various porosity logs with or without the use of core analysis information. Water saturation (Sw) is normally calculated from electrical resistivity of the formation (obtained from single or multiple resistivity logs), porosity, and the formation water resistivity (estimated from logs or measured from a formation water sample). In calculating SW, one also needs values of several rock-fluid parameters, which are usually estimated. These parameters are associated with the two well known parameters are associated with the two well known equations due to Archie:
(1)
and (2)
where F is the formation resistivity factor (defined as equal to Ro/Rw), Rt is the formation resistivity, Ro is the resistivity of the formation when saturated with the formation water of resistivity Rw, and a, m, and n are empirical parameters. The parameter 'm' is called the "cementation factor" and is usually assumed to have a value of 2. However, m is known to vary between 1.3 (for unconsolidated sands) to 2.2 (for highly cemented rocks). Values of m even lower than 1 have been reported in the literature. The parameter 'a' is usually assumed to have a value of unity; however, other values of 'a', such as 0.81, are sometimes used. The parameter 'n' is known as the "saturation exponent" and is usually assumed to be 2, although it is known to vary.
Sw is calculated from Archie's equation as follows:
From (2), (3)
because F = Ro/Rw by definition. Substituting equation (1) in equation (3),
(4)