Modelling electrical conductivity for earth media with macroscopic fluid‐filled fractures

Abstract

ABSTRACTEffective‐medium theories for either highly conductive or more resistive electrical inclusions in a moderately conducting background medium are presented for modelling macroscopic (i.e., large‐scale) fluid‐filled fractures or cracks in a potential reservoir rock or granular medium. Conductive fluids are most often brine and the resistive fluids of interest are oil, gas, air and/or CO2. Novel features of the presentation for conductive fluids include results for both non‐interacting inclusions (using a Maxwell approximation) and for interacting inclusions (via a self‐consistent effective‐medium scheme). The anisotropic analysis is specifically designed to handle reservoirs with multiple orientations (usually three orthogonal sets) of oblate spheroidal cracks/fractures, while also having arbitrary aspect ratios. But these aspect ratios are strictly <1, thus excluding spherical pores and simple granular media – both already widely studied by others. Results show that the self‐consistent approximation depends on fracture aspect ratio α and that this approximation becomes important when fracture porosity is about φ= 1% for aspect ratio α≃ 0.05, or φ= 3% for aspect ratio α≃ 0.10. It is shown that the self‐consistent analysis is most important when the fractures have a very small aspect ratio — the inferred reason being that the fracture (or crack) number density (ρc≡φ/α) then becomes very high and the fracture relative spacing correspondingly very small for any fixed value of porosity (but with decreasing values of the aspect ratio). Hybrid methods (combining self‐consistent and non‐self consistent formulas) are also developed to deal with high volume fractions and multiple sets of fractures having different aspect ratios. Whenever possible and appropriate, the results are also compared to rigorous bounds, including the Wiener bounds and the Hashin‐Shtrikman bounds, in order to provide one type of partial validation of the methods being developed.