THE FRACTAL/FACIES HYPOTHESIS: A POTENTIAL FRAMEWORK FOR UNDERSTANDING AQUIFER HETEROGENEITY

Where is the order in the hydraulic properties of natural, heterogeneous sedimentary materials, upon which an understanding can be built? Early efforts at understanding used deterministic concepts built around assumed homogeneity within units, or else smooth, gradual variation. This was followed by the rise of stochastic theory in hydrogeology, based initially on treating heterogeneous property distributions as stationary, correlated, random processes. Here the order was assumed to be in the statistical characterization (probability density functions (PDFs), means, variances, etc.) of the properties themselves or the logarithms of the properties. This approach has had limited success because property distributions based on these statistically homogeneous concepts were too regular. The next step in the search for order was to consider statistically heterogeneous concepts that conceptualize heterogeneity in terms of non-stationary stochastic processes with stationary increments, the mathematical basis for stochastic fractals. The hoped for order then lies in the property or log(property) increment PDFs that usually display characteristics of the Levy distribution. However, there are problems with the Levy approach, such as the extreme variability displayed by these PDFs. A possible explanation for the appearance of Levy PDFs is that they actually represent a superposition of several Gaussian distributions of different variance. This leads to a working hypothesis, called the fractal/facies hypothesis, that increment distributions of heterogeneous properties, such as log(permeability) increments, are Gaussian within a single facies. This hypothesis is supported by previous applications to an alluvial fan environment and by permeability measurements from an outcrop of the Upper Cretaceous Straight Cliffs Formation near Escalante, Utah. Analysis of individual facies data suggests that Gaussian stochastic fractal structure is present. The fractal/facies hypothesis is a melding of facies sedimentology with fractal theory for generating distributions of sediment properties. It can be thought of as bringing deterministic concepts (facies structure) back into stochastic hydrogeology, and may serve as a more realistic basis for simulating flow and transport in heterogeneous aquifers.