REPRESENTATION AND INDUCTIVE RESPONSE OF FRACTAL RESISTIVITY DISTRIBUTIONS

Abstract

A long standing approximation in EM logging for fractures has been to assume that a fracture is an extended thin sheet, perhaps a half-plane. This approximation can be useful and is theoretically and numerically convenient. However, examination of core, FMS logs, and outcrop reveals that fracture zones can be geometrically self-similar over dimensional scales of less than millimeters to kilometers. In such a case, a strict geophysical model would have to account for such a fractal structure to truly represent the resistivity structure of the earth. The purpose of this work is to suggest some methods for forward and inverse EM modeling of geophysical fractal distributions.

We model fractal distributions of conductivity with band-limited Weierstrass functions. The conductivity is then discretized over some averaging window to give thin isotropic or thicker anisotropic layers, whose response can be calculated by traditional means. This representation gives a facile means of scaling small alternating sequences of open fracture and matrix rock, as observed in a borehole, to fracture packets. Parallel and transverse resistivities as functions of averaging window are easily calculated from such a distribution assuming that dip of the bedding with respect to the borehole is known - as would be the case with the auxiliary use of a FMS log, for example.