Generalization of the Maxwell Equation for Formation Resistivity Factors(includes associated papers 6556 and 6557 )

Abstract

A formula relating porosity and formation resistivity factor is presented.This equation is applicable not only to consolidated and unconsolidatedmaterials, but also to dispersive systems. A comparison of calculatedvalues with experimental data shows the equation yields satisfactory results. Introduction The formation resistivity factor of a porous sample hasbeen defined as the ratio of the resistivity of the samplewhen completely saturated with an electrolyte to theresistivity of the saturating electrolyte. One of thetheoretical expressions relating the formation resistivityfactor, F R, to porosity, phi, is known as the Maxwellequation. Its form is 3 −F = ---------- ...........................(1)R 2 This equation can be applied to dispersive systems ofspheres, where electrical interference among theelements is negligible. In practice, little attention has been paid to Eq. 1, mainly because its application is limited to idealizedsystems. However, contrary to other empiricalexpressions currently used, it has the virtue of having arigorous theoretical deduction. In view of the potential importance of Eq. 1, andbecause the idealizations made in its derivation are notwell known, a theoretical development is presented inthe Appendix, according to the ideas suggested byMaxwell. Theory Outstanding among the attempts to generalize theMaxwell equation is the work of Fricke, whotheoretically demonstrated that for dispersive systems of oblate and prolate spheroids (x + 1) −F = -------------,.........................(2)R x where x is a geometric parameter that is a function ofthe axial ratio of the spheroids, and whose value is lessthan 2. When x = 2, Eq. 2 reduces to Eq. 1. Fricke confirmed Eq. 2 by using experimental dataon the conductivity of blood. For this purpose, hetreated the red corpuscles as oblate spheroids. Furthermore, he made use of the fact that the redcorpuscles behave as perfect insulators for director low-frequency current; this behavior results frompolarization effects. It is interesting to note that Maxwell's formula forspheres and Fricke's equation for spheroids areequilateral hyperbolas that can be written as P(1−) F = 1 + -----------,.......................(3)R where P takes on the values of 1.5 and (1 + x)/x, respectively. Furthermore, P is a geometric parameterwhose value becomes larger as the sphericity becomessmaller. Similarly, it has been found that, in the caseof two-dimensional dispersive systems, a relationship ofthe form of Eq. 3 is also satisfied. However, it shouldbe noted that idealizations have been made in itsderivation that, in principle, do not allow its applicationto those cases where the elements are in contact or neareach other, or when they have irregular shapes. Therefore, Eq. 3 should be modified if it is to be applied to real cases. For this purpose, consider a system of spheres, whether disperse or in contact. Fig. 1 shows some ofthe flowlines in the neighborhood of two spheres incontact. JPT P. 819^