Fractals in Rock Physics

Abstract

The concept of fractals, introduced by Mandelbrot ( 1 982), has proven to be a highly useful way to describe the statistics of naturally occurring geometries. Fluid flow, whether it be the rise and fall of rivers (Bridge & Jarvis 1 982), turbulence in air or water (Nelkin 1989), or precipitation (Mandelbrot & Wallis 1 968), is found to follow self-similar patterns in time and space. Natural objects, from mountains and coastlines (Mandel­ brot 1982) to the perimeters of forests (Loehle 1 983), are found to have boundaries best described as self-similar, appearing the same on all length scales and having a measured size that depends on the scale ofthe measure­ ment. Microscopic processes of diffusion (Orbach 1 989) and chemical reaction kinetics (Kopelman 1988) can lead to fractal structures, while the scale independence of natural processes can lead to the ubiquitous I/! noise and stretched exponential relaxation (West & Shlesinger 1 990). Many review articles and books have been written on fractals [see Hurd (1 988) for a bibliography], and reviews (Wong 1 988) and conference proceedings (Scholz & Mandelbrot 1 989, Aharony & Feder 1 989, Fleischmann et al 1 989, Avnir 1 989) have appeared on various aspects of fractal applications in geophysics. I do not attempt to review here basic fractal concepts as they are covered in the cited literature but rather concentrate on fractal structure at the microscopic scale in sedimentary porous rock and on the geometry of flowing fluids. One objective of this work is to develop models for rock pore geometry that will distinguish rocks from other porous materials. The length scales of interest include the submicron structure on pore walls,