Abstract. Existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability K as the variable of concern. We treat K as a multifractal and investigate its scaling and fractality using measured horizontal K data from two locations in the United States. The first data set was from a shoreline sandstone near Coalinga, California, and the second was from an eolian sandstone [Goggin, 1988]. By applying spectral analyses and computing the scaling of moments of various orders (using the double trace moment method [Lavallee, 1991; Lavallee et al., 1992]), we found that K is multiscaling (i.e., scaling and multifractal). We also found that the so-called universal multifractal (UM) [Schertzer and Lovejoy, 1987] model (essentially a log-Levy multifractal), was able to reproduce the multiscaling behavior reasonably well. The UM model has three parameters: a, rr, and H, representing the multifractality index, the codimension of the mean field, and the "distance" to stationary multifractal, respectively. We found (a = 1.7, rr = 0.23, H = 0.22) and (a = 1.6, rr = 0.11, H = 0.075) for the shoreline and eolian data sets, respectively. The fact that a values were less than 2 indicates that the underlying statistics are non-Gaussian. We generated stationary and nonstationary multifractals and illustrated the role of the UM parameters on simulated fields. Studies that treated Log K as the variable of concern have pointed out the necessity for large data records, especially when the underlying distribution is Levy-stable. Our investigation revealed that even larger data records are required when treating K as a multifractal, because Log K is less intermittent (or irregular) than K.•
[1] Based on an examination of K data from four different sites, a new stochastic fractal model, fractional Laplace motion, is proposed. This model is based on the assumption of spatially stationary ln(K) increments governed by the Laplace PDF, with the increments named fractional Laplace noise. Similar behavior has been reported for other increment processes (often called fluctuations) in the fields of finance and turbulence. The Laplace PDF serves as the basis for a stochastic fractal as a result of the geometric central limit theorem. All Laplace processes reduce to their Gaussian analogs for sufficiently large lags, which may explain the apparent contradiction between large-scale models based on fractional Brownian motion and nonGaussian behavior on smaller scales.
Modern measurement techniques have shown that property distributions in natural porous and fractured media appear highly irregular and nonstationary in a spatial statistical sense. This implies that direct statistical analyses of the property distributions are not appropriate, because the statistical measures developed will be dependent on position and therefore will be nonunique. An alternative, which has been explored to an increasing degree during the past 20 years, is to consider the class of functions known as nonstationary stochastic processes with spatially stationary increments. When such increment distributions are described by probability density functions (PDFs) of the Gaussian, Levy, or gamma class or PDFs that converge to one of these classes under additions, then one is also dealing with a so‐called stochastic fractal, the mathematical theory of which was developed during the first half of the last century. The scaling property associated with such fractals is called self‐affinity, which is more general that geometric self‐similarity. Herein we review the application of Gaussian and Levy stochastic fractals and multifractals in subsurface hydrology, mainly to porosity, hydraulic conductivity, and fracture roughness, along with the characteristics of flow and transport in such fields. Included are the development and application of fractal and multifractal concepts; a review of the measurement techniques, such as the borehole flowmeter and gas minipermeameter, that are motivating the use of fractal‐based theories; the idea of a spatial weighting function associated with a measuring instrument; how fractal fields are generated; and descriptions of the topography and aperture distributions of self‐affine fractures. In a somewhat different vein the last part of the review deals with fractal‐ and fragmentation‐based descriptions of fracture networks and the implications for transport in such networks. Broad conclusions include the implication that models based on increment distributions, while more realistic, are inherently less predictive than models based directly on stationary stochastic processes; that there is presently an unresolved ambiguity when a measurement is attempted in a medium that exhibits property variations on all scales; the strong possibility that log(property) increment distributions that appear to be described by the Levy PDF are actually superpositions of several PDFs of finite variance, one for each facies; that there are apparent similarities in the transport behavior of heterogeneous porous media and fractured rock at the field scale that appear to be related to the existence of a few preferential flow paths in both types of media; and finally, that additional carefully collected data sets are needed to clarify and advance the fractal‐based theories, particularly in the case of three‐dimensional fracture networks where few data are available. Further refinement is needed also in the understanding of instrument spatial weighting functions in heterogeneous media and how measuremen...
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