The broad-band fractal nature of heterogeneity in the upper crust from petrophysical logs

Abstract

In situ measurements of petrophysical properties of the upper crust from wire-line logs provide a direct means of assessing fluctuations in these properties with depth, and thus allow for the statistical characterization of crustal heterogeneity. Wire-line logs from a cluster of nine boreholes (1000–1500 m deep) have been analysed using four different techniques: autocorrelation, semi-variogram, rescaled range and power spectra. Six of the boreholes are vertical, the remainder inclined. All penetrate clastic and pyroclastic rock. The analysis techniques have been tested on synthetic data, for which we have precise control on the scaling, in order to assess their robustness. The results, for the borehole data, show broad-band fractal scaling with a fractal dimension of 1.62 to 1.97 (autocorrelation), 1.59 to 1.79 (rescaled range) and 1.61 to 2.0 (power spectra). The use of different techniques to estimate the fractal dimension provides an excellent constraint on the results. Furthermore, it allows us to determine which model for crustal heterogeneity best describes the data: a 'band-limited' von Kármán function or 'unbounded' fractal scaling. In order to test the possible anisotropy in the scaling parameters, the horizontal scaling properties for this area have been calculated by correlation-spectra analysis of neighbouring wells. This suggests that upper-crustal correlation lengths are of the order of kilometres and anisotropic, being over four times greater in the horizontal direction. The origins of the observed power-law scaling are complex. Analysis of long-term correlations between logs and televiewer data points towards the influence of fracture porosity within the pyroclastics. But a fractal distribution of pore spaces within the clastics also controls the log fluctuations. Separating the logs into clastic and pyroclastic subsets reveals slight differences in fractal scaling, as determined by autocorrelation and semi-variogram. These differences are attributed to the dominance of either primary or fracture porosity within each geological unit.