Abstract - Characterisation of the spatial patterns of fault arrays poses both conceptual and practical problems. The former concern definition and characterisation of the independent attributes of individual faults and of fault arrays which contribute to and influence the 'pattern', e.g. size distribution, density, linkage, orientation. The practical problems concern mainly the availability of fault datasets which are valid in respect of quantified truncation criteria and which include all relevant attributes, e.g. displacement. A further problem concerns the possibility that spatial distribution systematics may vary with truncation. As fault arrays are 3-dimensional structures with no plane or line of symmetry a fault dataset should also be 3-dimensional. In practice, 2-dimensional datasets must suffice.
We have constructed a 2-D fault dataset with an unusually wide range of displacement values, 2.5 orders of magnitude, between upper and lower truncation values. The map area, 87km2, is large relative to the trace lengths of most of the included faults so censoring effects are relatively slight. The map has been constructed from 1:2500 seam plans of the South Yorkshire coalfield which record fault throws in the range 10cm to 180m, with an effective lower truncation value of 60cm. This database is used, together with conventional fault maps of hydrocarbon reservoirs, to examine the effects of truncation on spatial distributions and on the independent attributes which contribute to the distributions. Truncation values are changed by filtering out fault traces, or parts of fault traces, on which the throw is less than the specified value. The systematics of separate fault sets are individually characterised.
Fault size populations often provide power-law distributions with slopes systematically varying with sampling dimension (1-D or 2-D) and with the particular measure of fault size (displacement, maximum displacement or length). Fault displacement and maximum displacement populations are relatively robust and insensitive to truncation but trace length populations are severely degraded, always providing population slopes which underestimate the relative numbers of small to large faults. Spatial analyses of fault patterns using conventional box counting techniques indicate non-fractal geometries in which fractal dimension varies continuously from 2 to 1. Other box counting algorithms (e.g. correlation dimension and information dimension), which incorporate fault densities and displacements, provide a means of quantifying strain at different scales and offer better prospects for spatial characterisation of fault patterns. Although these methods are also sensitive to resolution effects, the results are meaningful when account is taken of truncation values.
Abstract of talk given to:
Quantification and modelling of spatial patterns in permeable rocks Institute of Mathematics and its Applications meeting, Scarborough, March 1995