Comparison of the lognormal and beta distribution functions to describe the uncertainty in permeability

Abstract

The permeability of a single hydrostratigraphic unit is associated with considerable uncertainty due to measurement errors and significant spatial variability. Historically this uncertainty is characterized by a lognormal distribution. This distribution is generally heavy tailed, so using this distribution to describe the permeability has the limitation that all positive values of permeability for a given hydrostratigraphic unit have positive (not zero) probability of occurrence. The beta distribution, with its bounded domain, is explored as an alternative to the lognormal distribution in describing the uncertainty of permeability. The lognormal distribution and the beta distribution are both fit to the historic data provided by Jan Law in 1944 that was used to generalize the statement that the uncertainty in the permeability follows a lognormal distribution. The lognormal distribution and the beta distribution are also fit to an extensive permeability data set taken from regions within the Dakota Sandstone that have been shown to represent a single hydrostratigraphic unit. The results of this curve fitting exercise indicate that the beta distribution is a viable alternative to the lognormal distribution to characterize the uncertainty in permeability.