Is the Applicability of Fractal Statistics to Sedimentary Structures the Result of Scale-Invariant Stochastic Processes or Deterministic Chaos?

Abstract

Fractal statistics are the only statistics that are scale invariant. Examples in tectonics include distributions of faults, displacements on faults, distributions and permeabilities of fractures, and distributions of folds. Many aspects of sedimentology are also fractal including distributions of sedimentary sequences, variations in permeability, and shapes of boundaries. Since the underlying processes are likely to be scale invariant, it is reasonable to conclude that the number-size statistics of oil fields will be fractal. Log-normal statistics are often applied; they are not scale invariant. Two explanations for fractal statistics can be given. They may be the result of scale-invariant stochastic processes. Random walk (Brownian noise) is one example. Topography generally resembles Brownian noise, a power-law spectrum with fractal dimension D = 1.5. Alternatively fractal statistics can be the result of deterministic chaos. Turbulent flows are examples of deterministic chaos, the governing equations are deterministic but the resulting flows are statistical. Tectonic displacements can be shown to be the result of deterministic chaos; it is likely that erosion is another example.