Generalizing the Geometrical Factor Theory

Archie developed his model based upon trends observed in formation resistivity factor - porosity and resistivity index-water saturation plots. The trends appear when the data is plotted on log-log graph pa-per. The trends are apparently straight lines on log-log paper, which means that they are “power laws” of the form F = 1/φm and I = 1/Swn where m and n are observed to distribute themselves around m and n = 2. Archie did not attempt to connect his empirical discovery to any physical first principles, nor did he discuss alternative trends that might have fit the data. However, Archie’s model filled a void in formation evaluation, and works well. Four decades would pass before researchers began to attempt to connect conductivity in rocks as described by the Archie model to first principles. Since 1980 several attempts to connect the Archie model to first principles were, and continue to be, made. There has been no generally accepted success in the attempts. Waxman and Smits (following M. R. J. Wyllie, 1952) changed the focus of attention from resistivity to conductivity in 1968. This was a step in the right direction, but they retained Archie’s model reformulated in conductivity terms; in that sense there was no change in thinking. The first rethinking of the problem appeared in 1993 by positing three first principles: bulk rock conductivity is proportional to (1) brine conductivity; (2) fractional volume of brine in the rock; (3) a geometrical factor. Brine conductivity and fractional volume of brine (i.e., porosity) are directly measurable, and the geometrical factor can be estimated from measurable quantities. From these “first principles” a model relating bulk rock conductivity to water saturation can be derived. The model, called the geometrical factor theory (GFT), is different from the Archie model, but the Archie model is contained within it as a limiting case. In 2007 a different model based upon a different formulation of the first principles, known as a pseudo-percolation threshold theory (PPTT), was developed. Although different from the GFT, PPTT also contains the Archie model as a limiting case. Since GFT and PPTT are different, it seems as if one of them must be wrong. However, I show that neither is wrong, but both are special cases of a more general model. Whereas Archie analyzed his data in terms of formation resistivity factor versus porosity and deduced a power law, an alternative analysis in terms of formation conductivity factor would have revealed that Archie’s data for both formation factor vs. porosity and resistivity index vs. water saturation is fit, arguably better, by a quadratic function. The quadrat-ic function follows directly from first principles, whereas Archie’s power law does not. I call the new model the Generalized GFT (GGFT), and show that it subsumes the GFT, PPTT, and Archie’s model as special cases. Interestingly, the shaly sand models can also be accommodated in GGFT. As an explanation of the Archie model from first principles has long been a Holy Grail of petrophysics, this is of significant interest.