Effect Of Filtering On Autocorrelation, Flow, And Transport In Random Fractal Fields

Abstract

" Fractal" concepts have become the focus of much interest in the earth sciences during the last fifteen years. The term "fractal" is especially appealing from a semantic point of view in that Mandelbrot [ 1983] derived it from the Latin "fractus ", describing the appearance of a broken stone. In this report, we focus on issues of flow and contaminant transport in porous media. Here, fractal concepts have been widely associated with attempts to explain scale- effects such as the apparent growth of effective longitudinal dispersion with the scale of observation. However, a much broader range of topics has been explored in the literature on fractals, which can be roughly divided into two broad categories. The first category concerns a fractal description of medium geometry, over a given range of scales [Adler, 1992]. Within this category, the fractal geometry is considered to be either deterministic (self -similar) or random (statistically self -similar, or self -affine) [Voss 1985]. The second category views medium physical properties (porosity, log- conductivity) as random fields, most commonly with statistical self -similarity of second -order moments such as structure function ( variogram) or autocovariance. In this report, we focus on random fractal fields. We start with an introduction in Chapter 1 of isotropic random fractal fields and the scaling properties of corresponding power -law variogram and spectral densities in one, two, and three dimensions. We then derive new expressions for autocovariance functions corresponding to truncated power -law spectral densities; demonstrate that the power -law variogram and associated power spectra can be constructed as weighted integrals of exponential autocovariance functions and their spectra, representing an infinite hierarchy of unconelated homogeneous isotropic fields (modes); and analyze the effect of filtering out (truncating) high and low frequency modes from this hierarchy in the realand spectral domains. In Chapter 2, we derive first -order results relative to early preasymptotic, and late time asymptotic, transport in media characterized by a truncated log -conductivity power -law spectral density. In Chapter 3, we return to the multiscale log- conductivity fields constructed in Chapter 1; present some general results for early preasymptotic and late time asymptotic transport; and obtain complete first -order results for flow and transport, at preasymptotic and asymptotic stages, in two dimensions. In Chapter 4, we explore the multiscale behavior of conductivity from an aquifer in Mobile, Alabama, using different methods of data reduction. In Chapter 5, we summarize our main conclusions.