The Fundamental Flaws of the Waxman-Smits and Dual-Water Formulations, Attempted Remedies, and New Revelations From Historical Laboratory Complex Conductivity Measurements
Authors:   Consultant, Rasmus John , Kennedy David ,   QED Petrophysics, Homan Dean ,   SLB
Published: November 2024

The Waxman-Smits formula1 was introduced in 1967 as a parallel conductance model to improve previous models. A careful inspection of Waxman’s and Smits’ model reveals that it is not a parallel conduction model by the conventional definition. First, Waxman-Smits assumed that “the electrical current transported by the counterions associated with the clay travels along the same tortuous path as the current attributed to the ions in the pore water” (Waxman-Smits, 1967), removing an essential feature of a parallel conduction model, that there be two separate conductors. Based upon this assumption, they assign the same geometrical factor to both current paths. The geometrical factor is defined as the reciprocal of the formation resistivity factor (1/F or jm). Waxman-Smits found experimentally that a shaly sand appeared to have an F that was larger than a clean sand and introduced F* to account for this. Therefore, the tortuosity of the current paths through the clay and the pore water was deemed to be equivalent, with both tortuosities increasing equally as the clay content increased. Second, a parallel model requires that the bulk conductivity of a volume be weighted by the fractional volumes of the separate clay and interstitial water current paths. Clavier, Coates, and Dumanoir2,3 discovered during the field testing of the new 1.1-GHZ electromagnetic propagation tool that there existed a volume of clay water of near-constant salinity in shales. These two concepts are not accounted for in the Waxman-Smits model. A re-evaluation of the Waxman-Smits database by Clavier et al. revealed that the F* increase was primarily due to the Waxman-Smits model not accounting for the physical presence of the volume of the clay-bound water. The inclusion of the clay-bound water volume in the dual-water model produces a true parallel conductivity model. However, like Waxman-Smits, it assigns the same tortuosity to both the clay and pore water current paths. Theoretically and experimentally, the value of the clay water conductivity (Ccw) at room temperature was found to be 6.8 S/m. Therefore, for a pore water conductivity (Cw) less than 6.8, the clay adds to the rock conductivity relative to an Archie rock, as written in the Waxman-Smits model. However, when Cw is greater than 6.8, the clay water subtracts from the rock conductivity relative to an Archie rock. This cannot be accommodated by the Waxman-Smits formulation. To correct for this model deficiency, B was made a function of salinity and temperature when, theoretically, it is a function of temperature only. Thirdly, neither model accurately predicts the rock conductivity at pore water salinities below approximately 0.5 S/m. Having a proper model at these lower salinities is important for geothermal evaluations, waterflooded reservoirs, and naturally occurring freshwater reservoirs. We have re-evaluated the Waxman-Smits database and the Vinegar-Waxman induced polarization database and supplemented these with broadband complex conductivity measurements of various clay minerals mixed with glass beads at various salinities. The assumption of equivalent tortuosities seems dubious based on observations of scanning electron microscope (SEM) photos showing actual clay morphologies. Our recent broadband complex conductivity laboratory experiments of pure clay and 250 µm diameter glass beads have allowed us to quantify tortuosity changes due to the introduction of clay into an otherwise pure glass bead environment. For the inaccuracies of both models at low salinities, we propose a correction method based on our knowledge gained from the study of the quadrature conductivity measurement from cores and recent laboratory measurements. We have fully described the parameters of the Waxman-Smits and dual-water formulations in terms of petrophysical and electrochemical concepts and proven that the Waxman-Smits B factor salinity corrections are an empirical correction that is unnecessary in the dual-water formulation. The dual-water formulation is proven to be a truly parallel conductance model, while the Waxman-Smits model is not. Finally, we provide an electrochemical-based correction to correct the dual-water model for the low-salinity conductivities, which is based on our recently gained knowledge of the quadrature conductivity behavior.