Fractional Levy motion

Painter, S. 1996. Stochastic interpolation of aquifer properties using fractional Levy motion. Water Resour. Res. 32 1323-1332. 

UPSCALING OF NORMAL STRESS-PERMEABILITY RELATIONSHIPS FOR FRACTURE NETWORKS OBEYING FRACTIONAL LEVY MOTION... 

T-H-M-C processes are significantly affected by subsurface heterogeneity, which results in scale-dependence of the related parameters. To handle this scale-dependent behavior, we need to characterize this heterogeneity and consider its effects at different scales. In this study, we demonstrate that the measured permeability data from Sellafield site, UK, are very well described by fractional Levy motion (fLm), a stochastic fractal. This finding has important implications for modeling large-scale coupled processes in heterogeneous fractured rocks. 

The parameters characterizing fractional Levy motion include a, C, Co, and H. To determine these parameters from observed data, we used the quantile-based estimators for a and C for a given lag h (Fama and Roll, 1972). The scaling parameter H is calculated by fitting Equation 3 to the estimated C as a function of lag h. 

Liu, H.H., Bodvarsson, G. S., Molz F. J. Lu, S. 2003. A generalized successive random additions algorithm for simulating fractional Levy motion. Mathematical Geology (in review). 

A differential solute transport equation derived for Levy motions would facilitate solute transport studies in the same way that the ADE facilitated applications of the Brownian motion model. Recently, Zaslavsky (1994) suggested a procedure to derive such an equation using fractional derivatives that in effect account for the memory of solute particles. Zaichev and Zaslavsky (1997), Benson (1998), and Chaves (1998) modified Zaslavsky s procedure to account properly for mathematical properties of fractional derivatives in the one-dimensional case. The simplest form of the one-dimensional equation assumes symmetrical dispersion ... 

Benson, D.A., R. Schumer, M.M. Meerschaert, and S.W. Wheatcraft. 1999. Fractional dispersion, Levy motion, and the MADE tracer tests. Desert Res. Inst., Reno, NV. 

Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M. The fractional-order governing equation of Levy motion. Water Resour. Res. 36(6), 1413—1424 (2(XX)). http //www.agu.org/ pubs/crossref/2000/2000WR900032. shtml... 

R. Gorenflo and F. Mainardi, Feller fractional diffusion and Levy stable motion. Preprint 1999. 

The main objective of this chapter is to establish the relation between the macroscopic equations like (3.1) and (3.5), the mesoscopic equations (3.2) and (3.3), etc., and the underlying microscopic movement of particles. We will show how to derive mesoscopic reaction-transport equations like (3.2) and (3.3) from microscopic random walk models. In particular, we will discuss the scaling procedures that lead to macroscopic reaction-transport equations. As an example, let us mention that the macroscopic reaction-diffusion equation (3.1) occurs as a result of the convergence of the random microscopic movement of particles to Brownian motion, while the macroscopic fractional equation (3.5) is closely related to the convergence of random walks with heavy-tailed jump PDFs to a-stable random processes or Levy flights. 

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