A Fractal Interpretation of Archie-like Equation for Partially Saturated Porous Media

The complex mechanism by which the conducting phase in pore space influences the electrical conductivity of rocks has been a critical focus in geophysical exploration. In this study, the Pore-Solid Fractal model is used to accurately describe the fluid distribution within pore space, and the resistivity index follows a form analogous to Archie equation, being expressed in terms of the tortuosity fractal dimension, the pore fractal dimension, and saturation. The physically based fractal parameters then determine the saturation exponent. We find a reasonable prediction for experimental data with a feasible parameter linking the cementation exponent and the saturation exponent. Furthermore, the interrelationship between cementation exponent and saturation exponent is also analyzed under various conditions. An alternative expression for the exponents is introduced, exhibiting good agreement with experimental measurements. Additionally, a bounds model is presented in order to address the uncertainty of tortuosity due to uncertainties of the tortuosity fractal dimension during the saturating process. The boundary is a useful constraint on the range of the resistivity index, particularly in cases where structural parameters are insufficiently constrained.