The Power‐Law Relation Between Inclusion Aspect Ratio and Porosity: Implications for Electrical and Elastic Modeling

Geophysicists depend on rock physics relationships to interpret resistivity and seismic velocity in terms of rock porosity, but it has proven difficult to capture the effect of pore geometry on such relations through simple and easy to apply formulae. Inclusion modeling relates moduli to porosity through an equivalent grain or pore aspect ratio but often fails to account for observed trends, whereas empirical relations can be hard to extrapolate beyond their range of validity, often giving incorrect results in the low and high porosity limits. We show that introducing a power‐law relationship between porosity and equivalent grain or pore aspect ratio allows inclusion models to reproduce five published empirical resistivity‐porosity and velocity‐porosity relationships, providing a first principles basis for extrapolation to other cases of interest. We find that the deviation of resistivity from Archie's law in carbonates is related to a systematic change of grain shape with porosity, and we derive a new relation that fits carbonate resistivity data with similar accuracy to the Humble equation while being correct at high porosity. We then obtain an analog for the Castagna and Pickett relationships for wet, calcitic rocks, which is valid in the low and high porosity limits, giving rise to a new, physically derived V p / V s versus porosity model.