Multifractal Sierpinski carpets: Theory and application to upscaling effective saturated hydraulic conductivity

Abstract

Recent analyses of field data suggest that saturated hydraulic conductivity, K, distributions of rocks and soils are multifractal in nature. Most previous attempts at generating multifractal K fields for flow and transport simulations have focused on stochastic approaches. Geometrical multifractals, in contrast, are grid-based and thus better able to simulate distinct facies or horizons. We present a theoretical framework for generating two-dimensional geometrical multifractal K fields. Construction of monofractal Sierpinski carpets using the homogenous and heterogeneous algorithms is recalled. Averaging multiple, non-spatially randomized, heterogeneous Sierpinski carpet generators yields a new generator with variable mass fractions determined by the truncated binomial probability distribution. Repeated application of this generator onto itself results in a multiplicative cascade of mass fractions or multifractal . The generalized moments, M i ( q), of these structures scale as M i ( q) = (1/ b i ) ( q−1) D q , where b is the scale factor, i is the iteration level and D q is the q−th order generalized dimension, with q being any integer between − ∞ and ∞. This theoretical approach is applied to the problem of aquifer heterogeneity by equating the mass fractions with K. An approximate analytical expression is derived for the effective hydraulic conductivity, K eff, of multifractal K fields, and K eff is shown to increase as a function of increasing length scale in power law fashion, with an exponent determined by D q→∞ . Numerical simulations of flow in b = 3, D q→∞ = 1.878 and i = 1 though 5 multifractal K fields produced similar increases in K eff with increasing length scale. Extension of this approach to three dimensions appears to be relatively straightforward.