A generalization of the fractal/facies model

Abstract

In order to generalize the fractal/facies concept presented by Lu et al. (2002), a new stochastic fractal model for ln(K) (K = hydraulic conductivity) increment probability density functions (PDFs) is presented that produces non-Gaussian behavior at smaller measurement lags and converges to Gaussian behavior at larger lags, a property that is observed in data sets. The model is based on the classical Laplace PDF and its generalizations. In analogy with its Gaussian counterparts, the new stochastic fractal family is called fractional Laplace motion (fLam) having stationary increments called fractional Laplace noise (fLan). This fractal is different because the character of the underlying increment PDFs change dramatically with lag size, which leads to lack of self-similarity and self-affinity as they are traditionally defined. Data also appear to display this characteristic. In the larger lag size ranges, however, approximate self-affinity does hold. The basic field procedure for further testing of the fractional Laplace theory is to measure ln(K) increment distributions along transects, calculate frequency distributions from the data, and compare results to various members of the auto-correlated fLan family. The variances of the frequency distributions should also change with lag size (scale) in a prescribed manner. There are mathematical reasons, such as the geometric central limit theorem, for surmising that fLam/fLan may be more fundamental than other approaches that have been proposed for modeling ln(K) frequency distributions, such as the flexible scaling model of Painter (2001). If this turns out not to be the case, then other approaches may be comparable or preferable. (Less)