Generalization of the Maxwell Equation for Formation Resistivity Factors(includes associated papers 6556 and 6557 )
- C. Perez-Rosales (Instituto Mexicano del Petreleo)
- Document ID
- Society of Petroleum Engineers
- Journal of Petroleum Technology
- Publication Date
- July 1976
- Document Type
- Journal Paper
- 819 - 824
- 1976. Society of Petroleum Engineers
- 2.4.3 Sand/Solids Control, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 4.2 Pipelines, Flowlines and Risers, 5.3.1 Flow in Porous Media
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A formula relating porosity and formation resistivity factor is presented. This equation is applicable not only to consolidated and unconsolidated materials, but also to dispersive systems. A comparison of calculated values with experimental data shows the equation yields satisfactory results.
The formation resistivity factor of a porous sample has been defined as the ratio of the resistivity of the sample when completely saturated with an electrolyte to the resistivity of the saturating electrolyte. One of the theoretical expressions relating the formation resistivity factor, F R, to porosity, phi, is known as the Maxwell equation. Its form is
3 - F = ---------- ...........................(1) R 2
This equation can be applied to dispersive systems of spheres, where electrical interference among the elements is negligible.
In practice, little attention has been paid to Eq. 1, mainly because its application is limited to idealized systems. However, contrary to other empirical expressions currently used, it has the virtue of having a rigorous theoretical deduction.
In view of the potential importance of Eq. 1, and because the idealizations made in its derivation are not well known, a theoretical development is presented in the Appendix, according to the ideas suggested by Maxwell.
Outstanding among the attempts to generalize the Maxwell equation is the work of Fricke, who theoretically 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 of the axial ratio of the spheroids, and whose value is less than 2. When x = 2, Eq. 2 reduces to Eq. 1.
Fricke confirmed Eq. 2 by using experimental data on the conductivity of blood. For this purpose, he treated the red corpuscles as oblate spheroids. Furthermore, he made use of the fact that the red corpuscles behave as perfect insulators for direct or low-frequency current; this behavior results from polarization effects.
It is interesting to note that Maxwell's formula for spheres and Fricke's equation for spheroids are equilateral 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 parameter whose value becomes larger as the sphericity becomes smaller. Similarly, it has been found that, in the case of two-dimensional dispersive systems, a relationship of the form of Eq. 3 is also satisfied. However, it should be noted that idealizations have been made in its derivation that, in principle, do not allow its application to those cases where the elements are in contact or near each 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 of the flowlines in the neighborhood of two spheres in contact.
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