Deep borehole log evidence for fractal distribution of fractures in crystalline rock

Abstract

Sonic velocity and electrical resistivity logs run to a depth of 3.5 km in crystalline rock near the San Andreas fault at Cajon Pass in southern California correlate over scale-lengths both small (sub-metre) and large (tens to hundreds of metres). No such correlations are seen with the more lithologically sensitive natural gamma intensity log. The correlation between the sonic velocity and electrical resistivity logs suggests that a non-lithologic property of the crystalline rock controls fluctuations. In situ fracture intensity is a logical candidate for the controlling rock property. The fluctuations of the individual sonic velocity and electrical resistivity logs are examined with the Hurst rescaled range parameter over borehole log intervals 1.5 m < L < 1500 m. For log fluctuations arising from a scale-invariant physical process the Hurst rescaled range scales with data interval as LH, 0 < H < 1. A purely random sequence of in situ fractures produces a scaling exponent H= 0.50. Fluctuations in the Cajon Pass sonic velocity and electrical resistivity logs yield H~ 0.70 evidence that in situ fracture sets tend to occur in clusters rather than at purely random intervals. The tendency for fracture clustering over log intervals 1.5 m < L < 1500 m suggests that fracture formation is a fractal process independent of length-scale in which larger fracture intervals form from clustering of numerous smaller fracture intervals. Seismic reflectivity derived from the borehole sonic velocity log is also scale independent over the range of data intervals 1.5m < L < 1500 m with a Hurst exponent H= 0.21. If we associate fracture clustering with crustal fault formation, the Cajon Pass borehole sonic velocity and electrical resistivity logs predict that crustal faults scale fractally with fractal dimension D̃ 2.30. The equivalent b-value for earthquake size distribution is b~D/2~ 1.15. On this hypothesis observed b-values < 1.15 indicate a tendency for earthquakes to cluster on existing (weak?) faults. A scale-independent mechanics for crystalline rock fracture formation in which larger scale fractures form as clusters of smaller scale fractures provides a mechanical link between the small pervasive stress aligned flaws, cracks and microfractures which can impart anisotropy to crystalline rock and the larger scale fractures associated with finite strain and faulting. Thus in crustal regions of active but low strain, fracture anisotropy is observed to be aligned with the inferred maximum principal stress, while in actively faulted crust, fracture anisotropy is observed to be more nearly fault parallel as if the fractures are aligned by the finite strain faulting process.