Science Inventory

HOW WELL ARE HYDRAULIC CONDUCTIVITY VARIATIONS APPROXIMATED BY ADDITIVE STABLE PROCESSES? (R826171)

Citation:

Lu, S. AND F. J. Molz. HOW WELL ARE HYDRAULIC CONDUCTIVITY VARIATIONS APPROXIMATED BY ADDITIVE STABLE PROCESSES? (R826171). ADVANCES IN ENVIRONMENTAL RESEARCH. American Geophysical Union, Washington, DC, 5(1):39-45, (2001).

Description:

Abstract

Analysis of the higher statistical moments of a hydraulic conductivity (K) and an intrinsic permeability (k) data set leads to the conclusion that the increments of the data and the logs of the data are not governed by Levy-stable or Gaussian distributions. The distribution tails appear to display a Pareto-like power-law decay (i.e. Prob(|x|>s|) proportional to s^- as s rightwards arrow infinity ), with small alpha, Greek values of approximate 2.5 and 3.6, respectively. Unlike the calculations of Liu and Molz (1997), values were largely independent of lag size. These results suggest that the Levy model does not fit the tail behavior of the data well, even prior to the need for truncation in order to keep the statistical moments of simulated K distributions from becoming unrealistically large. It is suggested also that the fractional diffusion equation, based on an underlying Levy motion rather than the usual Brownian motion, might be better justified if porosity variations, as well as K variations, were considered. For the past 10 years, hydrologists and petroleum scientists have explored the use of non-stationary stochastic processes with stationary increments as models for log hydraulic conductivity distributions _¯¯ the so-called scaling fractal models. Initially, Gaussian processes were used based on fractional Brownian motion. Later, a non-Gaussian model, fractional Levy motion, was suggested as a more realistic alternative. Even more recently, Levy multi-fractals have been proposed as direct models for K variations. In the on-going effort to arrive at the most practical and realistic model for K or log(K) increments, the present communication attempts to develop useful information by presenting a careful analysis of the tail behavior of K and log(K) data sets. It is concluded that although more realistic than the Gaussian model, the Levy model over-estimates the variability inherent in the two carefully measured data sets that were studied.