Conductivity models for Archie rocks

Abstract

The petroleum industry’s standard porosity-resistivity model (i.e., Archie’s law), although it is fit for its purpose, remains poorly understood after seven decades of use. This results from the choice of the graphical display and trend formula used to analyze Archie’s seminal porosity-resistivity data, taken in the Nacatoch sandstone, a petroliferous clastic formation in the Gulf of Mexico coastal area. Archie’s model accurately predicts the conductivity-brine volume trend for this sandstone. Not all rocks follow the same porosity-resistivity trends observed in the Nacatoch sandstone, but those that do are defined as Archie rocks. Archie’s Nacatoch sandstone data set has significant irreducible scatter, or noise. Data with significant scatter cannot be used to uniquely define a trend. Alternative graphical analyses of Archie’s Nacatoch sandstone data indicates that Archie could have analyzed these data differently had it occurred to him to do so. A physics-based porosity-conductivity model, a “geometrical factor theory” (GFT), is preferred as an alternative to the Archie model because it has a physical interpretation. In this model, the bulk conductivity of an Archie rock is the product of three factors: brine conductivity, fractional brine volume, and an explicit geometrical factor. The model is offered in the form of a theorem, proved in three steps, to make our arguments as explicit and transparent as possible. The model is developed through its culmination as a saturation equation to illustrate that it is a complete theory for Archie rocks. The predictive power of the Archie model and GFT are similar, but unlike the adjustable parameters of the Archie model ([Formula: see text], [Formula: see text], and [Formula: see text]), all of the parameters of GFT have a priori physical interpretations. Through a connection to site percolation theory, GFT has promise to connect porosity-conductivity interpretation to circuit theory first principles.