Generating Heterogeneity Spectra from Spatially Resolved Measurements

Abstract

A new method is presented for describing spatial heterogeneity in materials that simultaneously present heterogeneity on a broad distribution of length scales. The method involves fitting the experimental variogram to a nested structure consisting of many model functions, each with a single range. Experimental regularization is rigorously considered, resulting in a description of the data on point-support. The output of this method is a plot called the heterogeneity spectrum. The heterogeneity spectrum shows the extent of spatial heterogeneity as a function of length scale, without bias from finite experimental resolution and sample size, covering all length scales to which the measurement is sensitive. As an example, the heterogeneity spectra of carbonate rocks measured by magnetic resonance imaging are shown. These experiments are sensitive to heterogeneity at length scales in the range 10−0.5–102 mm, and the heterogeneity spectra reveal relevant and otherwise unavailable information about the structure of these carbonate rocks. The geostatistical methodology behind constructing the heterogeneity spectra is described here, while full details on the magnetic resonance imaging aspects are provided in a companion paper (Pomerantz, Tilke and Song in J. Magn. Reson. 193:243–250, 2008).