Representation and Inductive Response of Fractal Resistivity Distributions

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. INTRODUCTION An extensive literature exists demonstrating a correspondence between electrical and fluid flow (see, e.g., the 1996 National Research Council Report). For this reason, inversion of EM data to conductivity or current flow patterns has been touted as a means of imaging fluid flow paths. This electric current flow fluid flow equivalence is a powerful construct, which is critically dependent on the notion that both flows have the same geometry. Hence, it is critical in modeling EM data that the scattering geometries not be simplified to excess. Fractal electrical conductivity distribution packets seem to be ubiquitous in geothermal reservoirs. Fracture spatial occurances, fracture surfaces, and fracture aperature widths all can have fractal geometries. (Baran et al., 1992; Barton, 1989; Russ, 1994; Thompson, 1991; National Research Council, 1996; Bruhn et al., 1997). The electromagnetic expression of fractally distributed fluid filled voids is fractal. Unfortunately, there do not seem to be any publications in the geophysical literature concerning the EM response of fractal materials. The purpose of this work is to suggest some methods for forward and inverse EM modeling of geophysical fractal distributions. REPRESENTATION OF FRACTAL CONDUCTIVITY Forward and inverse modeling of the EM response of a fractal conductivity distribution requires a set of basis functions which can represent the petrophysical conductivity distribution and whose EM response is readily calculated. Suppose that we have a borehole which penetrates a fractured formation. Then the resistivity log can have the character of a fractal function. There are several representations of such functions which facilitate EM modeling. Weierstrass function representation Resistivity logs in fractal material would be "spiky", and any interpolating function should be capable of reproducting the spiky behavior. One classical function which is "spiky" is the Weierstrass function F(x) = Σ b-ρn exp (ibn x) For 0<ρ<1 and b>1, this function is not differentiable, as discussed by Hardy (1916). Mandebrot (1982) extended the Weierstrass function to define the WeierstrassMandebrot function as an infinite summation F(x) = Σ [{1 exp (ibn x)}exp (iφn) /b(2-D)n ], where 1<D<2, and φn is arbitrary. The infinite number of terms limit the use of this function for computations. For these Jaggard and Kim (1987) and Jaggard (1990) define the bandlimited cosine Weierstrass function W(z) as: W(z) = η S Σ Tn (z) , where η2 is the variance of the function; S = sqrt [ 2 2b(2D-4) ] / sqrt [ b(2D-4)N1 b(2D-4)(N2 +1) ],