22. Lacunarity Analysis of Fracture Intensity Maps: Are they Multifractals?

Lacunarity (L) is a scale (r)-dependent parameter that was developed for quantifying clustering in fractals and has subsequently been employed to characterize various natural patterns. It has been further analytically proved that lacunarity analysis invoking the gliding-box algorithm can find the correlation dimension, D ₂ of multifractals. The present research empirically tests this on a set of multifractal models generated with known D ₂ values. The log-transformed lacunarityvalues of these models, log L were plotted as a function of the log-transformed box-size, log r. The slopes of these linear relations, estimated using regression analysis, were then used to calculate D ₂ values that gave an approximately 1:1 relationship with the known values. Multifractal behavior can therefore, be checked without having to compute the whole spectrum of non-integer dimensions, D _q (-∞ < q < +∞) that typically characterize a multifractal. The technique thus developed was applied to fracture intensity maps generated from a set of nested fracture networks from the Devonian Sandstone, Hornelen Basin, Norway in order to test if such intensity maps display multifractal behavior.