Logging and Log Interpretation - Further Discussion of the Significance of Particle Shape in Formation Resistivity Factor-Porosity Relationships

The boundary conditions F = 1.0 and 4 = 1.0 (100 per cent) are dictated by the definition of F in the equation 1/F = aFp. These conditions have physical significance as the "limiting conditions" for the observed behavior in systems of dispersed particles. The other boundary conditions, cited by Gardner in his Eq. 3 as applying to this system, have no significance in the systems we studied. Maxwell's equation (Eq. 2) cited in Gardner's comments was developed for and limited to a regular, rather than random, dispersion of spheres which are sufficiently removed from each other so that they do not interfere with current flow about their neighbors. These restrictions also apply to Rayleigh's analysis. The title of Rayleigh's paper cited by Gardner is "On the Influence of Obstacles Arrayed in Rectangular Order upon the Properties of a Medium". He is not discussing randomly dispersed spheres. It might be inferred from Gardner's comments that Rayleigh specifically considered the differential equation (Gardner's Eq. 3). This is not the case. Mathematically, Gardner's Eq. 3 is exactly true for the system to which it applies, however, as both Rayleigh and Maxwell note that their equation for F in terms of F is not for randomly dispersed spheres. At the time we wrote our paper, we were aware of the work of Wyllie and Gregory as cited by Gardner. We cited their work in Ref. 5 of our paper. Our comment on page 287 of the paper about the validity of an m value of 1.3 for spheres with a porosity of 38 per cent remains as our answer to Gardner's question on this point. Our data, at 38 per cent porosity, show that the condition a = 1 and m = 1.3 holds for spheres near 40 per cent porosity.