Relations between porosity, saturation, and resistivity are essential when estimating in-place hydrocarbon volumes. The most common relation is the Archie formula. During the last decade, several new coarse-scale electromagnetic techniques for resistivity measurements have been developed. Applications of coarse-scale resistivity must consider the underlying geologic variability of porosity and saturation to be used in such relations. I have investigated the effect of small-scale (logging-scale) variability in porosity and water saturation on coarse-scale resistivity for Archie relations. An analytical expression for coarse-scale resistivity is derived based on the assumption of lognormal distributed porosity and saturation. Similarly, coarse-scale water saturation is derived assuming lognormal distributed porosity and resistivity. The results suggest that the coarse-scale equivalent Archie resistivity is normally less than the geometric average and larger than the harmonic average of the heterogeneous resistivity distribution. An important aspect of the analytical derived relations is the quantification of the effect of the variability (variance and covariance) on the coarse-scale results. Another coarse-scale average resistivity based on macroscopic anisotropy could be a practical approximation of the coarse-scale equivalent Archie resistivity in many cases.

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