Do Heterogeneous Sediment Properties and Turbulent Velocity Fluctuations Have Something in Common? Some History and a New Stochastic Process

Abstract

It is increasingly apparent that sediment property distributions on sufficiently small scales are probably irregular. This has led to the development of stochastic theory in subsurface hydrology, including statistically heterogeneous concepts based mainly on the Gaussian and Lévy-stable probability density functions (PDFs), the mathematical basis for stochastic fractals. Gaussian and Levy-stable stochastic fractals have been applied both in the field of turbulence and subsurface hydrology. However, measurements have shown that the increment frequency distributions do not always follow Gaussian or Lévy-stable PDFs. Provided herein is an overview of the origin and development of a new non-stationary stochastic process, called fractional Laplace motion (flam) with stationary, correlated, increments called fractional Laplace noise (fLan). It is based on the Laplace PDF and known generalizations, and does not display self-similarity. Uncorrelated versions are equivalent to a Brownian motion subordinated to the gamma process. In analogy to the development of fractional Brownian motion (fBm) from Brownian motion, fLam is equivalent to fBm subordinated to a gamma process. The new stochastic fractal has increment PDFs that compare better with measurements, the moments of the PDF family remain bounded, and decay of the increment distribution tails vary from being slower than exponential through exponential and on to a Gaussian decay as the lag size increases. This leads to increasingly more intermittent fluctuations as the lag size decreases. It may be that the geometric central limit theorem, and possible generalizations, will play an important role in connecting the abstract mathematics to the physics underlying applications. 1 1. HISTORICAL OVERVIEW Development of TheoryDuring the past 3 decades, it has become increasingly apparent that heterogeneity in natural sediments is pervasive, with the implication that property distributions on sufficiently small scales are probably irregular. (Within this chapter, the term "irregular" will mean continuous functions with discontinuous first derivatives or discontinuous functions.) As this concept developed, various individuals were motivated to characterize heterogeneous property distributions, such as ln(K); K = hydraulic conductivity, using stochastic concepts. In subsurface hydrology, a significant step forward was taken by Freeze [1975], who began to apply time-series

Fracture alteration by precipitation resulting from thermal gradients: Upscaled mean aperture‐effective transmissivity relationship

“…Although fractal/power law spectral density functions have been proposed for fracture surfaces, observations suggest a low‐wave number cutoff, below which power law behavior is not observed for aperture spectra [e.g., Brown , 1995]. A review paper by Molz et al [2005] also notes that power law behavior is seldom observed over a range greater than about two decades in wave number space. Power law spectra with low wave number cutoffs are essentially stationary random fields.…”