Heavy-tailed log hydraulic conductivity distributions imply heavy-tailed log velocity distributions

[ 1] Equations of contaminant transport describing non-Gaussian dispersion of solute in heterogeneous porous media have been developed by several authors ( e. g., Berkowitz and Scher ( 1995, 1998), Benson ( 1998, 2001), Berkowitz et al. ( 2000), and Baeumer et al. ( 2005)). The equations assume that solute particle movement in time and/or space can be described by heavy-tailed probability distributions ( e. g., Levy alpha stable distributions). This assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy-tailed probability distributions. This work investigates whether a heavy-tailed Levy alpha stable distribution of increments in log hydraulic conductivity results in a heavy-tailed distribution of increments in log velocity; a heavy-tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy-tailed probability distribution. A computer algorithm was written to generate realistic hydraulic conductivity fields. The algorithm modifies the spectral synthesis method for generating fractional Brownian motion ( i.e., Saupe ( 1988) and Brewer and Wheatcraft ( 1994)) and produces self-similar hydraulic conductivity fields with a heavy-tailed Levy alpha stable distribution of increments in log hydraulic conductivity. A Monte Carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of alpha: 0.8, 1.1, 1.4, and 1.7. The USGS finite difference groundwater code MODFLOW was used to calculate the velocity fields, and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy-tailed Levy alpha stable probability density function (PDF). After determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a Levy alpha stable PDF, the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the Nolan ( 1997) maximum likelihood estimator. The relationship between the two tail parameters was then determined. There are three conclusions of this research. First, Mandelbrot and PP plots confirm that Levy alpha stable distributions of increments in log hydraulic conductivity give rise to Levy alpha stable distributions of increments in log velocity. Second, the tail parameter alpha for the increments in log velocity is typically smaller, indicating a heavier probability tail, than the corresponding tail parameter for the increments in log hydraulic conductivity. Third, there is a positive statistical relation between the two tail parameters.