On the role of microgeometry in conductivity substitution

Conductivity substitution is the process of predicting the change in the effective electrical conductivity of a rock upon a change in conductivity of the mineral or fluid phase. Conductivity substitution is nonunique — only a range of conductivities can be predicted from knowledge of the initial effective conductivity, the porosity, and the initial and final compositions. The precise change depends strongly on the rock microstructure, which is seldom adequately known. Rigorous bounds on the change in effective conductivity upon changes in the phase conductivities for two-phase isotropic composites are used to gain insights into the roles of microgeometry and phase conductivity contrast. When the conductivity contrast between phases is high, the conductivity substitution predicted by Archie’s law corresponds approximately to the upper bound on the change of conductivity upon substitution. Inclusion modeling suggests that vuggy, highly tortuous, or partially disconnected pore space could account for conductivity changes smaller than those predicted by Archie’s law. Substitution behavior computed analytically for known microgeometries correlates with measures of microgeometry, including the fraction of connected fluid phase and variance of electric field strength in each phase. Comparison of the conductivity substitution bounds with brine-saturated sandstone data reveals that the position of measured data with respect to the conductivity substitution bounds can be indicative of the effective clay content. The bounds provide a template for better prediction of effective conductivity if we have at least some knowledge of the pore microstructure. Similarly, multiple conductivity measurements on the same rock might be used to extract more information about the rock and pore space properties than is possible with only a single measurement.