The Fundamental Flaws of the Waxman-Smits and Dual Water Formulations, Attempted Remedies, and New Revelations from Historical Laboratory Complex Conductivity Measurements
Authors:   Interfacial Polarization Dielectric Petrophysics (IDPD), Rasmus John , Kennedy David ,   Retired, Homan Dean ,   SLB
Published: February 2024

The Waxman-Smits formula was introduced in 1968 as a parallel conductance model to improve previous models. A careful inspection of the Waxman and Smits model reveals 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 and Smits, 1968), removing an essential feature of a parallel conduction model, that there be two separate conductors. Based on 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 σm). 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 were deemed to be equivalent, with both tortuosities increasing equally as the clay content increased. Second, a parallel model requires the bulk conductivity of a volume to be weighted by the fractional volumes of the separate clay and interstitial water current paths. Clavier et al. (1977) 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. (1977, 1984) revealed the F* increase was primarily due to the Waxman-Smits model not accounting for the physical presence of the volume of the clay water. The inclusion of the clay 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. This assumption seems dubious based on observations of scanning electron microscope (SEM) photos showing actual clay morphologies. Laboratory measurements of pure clay and glass beads would allow one to quantify tortuosity changes due to the introduction of clay into an otherwise pure glass bead environment. 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 to 1 S/m. Having a proper model at these lower salinities is important for geothermal evaluations, waterflooded reservoirs, and naturally occurring freshwater reservoirs. We propose a correction method based on our knowledge gained from the study of the quadrature conductivity measurement from cores and recent laboratory measurements.