Multifractal Sierpinski carpets: Theory and application to upscaling effective saturated hydraulic conductivity E. Perfect a, ⁎ , R.W. Gentry b , M.C. Sukop c , J.E. Lawson a a Department of Earth and Planetary Sciences, University of Tennessee, Knoxville, TN 37996-1410, USA b Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996-2010, USA c Department of Earth Sciences, Florida International University, Miami, Fl 33199, USA Available online 19 April 2006 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)Dq , 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. © 2006 Elsevier B.V. All rights reserved. 1. Introduction How to describe, predict and simulate heterogeneity are pervasive issues in the fields of hydrogeology, petroleum engineering, and soil physics. Heterogene- ities can occur in chemical and physical properties, both spatially and temporally. We are concerned with the spatial variation in physical properties, specifically the saturated hydraulic conductivity, K, of different geolog- ical facies or soil horizons. Such variations impact flow and transport in the subsurface, and thus have practical significance for the design and operation of pumping wells for human water use, oil production, and the spreading of contaminants in polluted soils and aquifers. Increasingly, fractal-based models are being used to describe, predict and simulate aquifer heterogeneity (see for example the recent reviews by Neuman and Di Federico, 2003; Molz et al., 2004). Fractals are spatial or temporal patterns that repeat themselves at increasingly finer (or coarser) scales of resolution (Mandlebrot, Geoderma 134 (2006) 240 – 252 www.elsevier.com/locate/geoderma ⁎ Corresponding author. Tel.: +1 865 974 6017; fax: +1 865 974 2368. E-mail address: eperfect@utk.edu (E. Perfect). 0016-7061/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.geoderma.2006.03.001