Multifractals and Local Singularity Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 429

space. In the first example, these objects are fractures in granite measured at the
surface. The second example deals with the geographical distribution of orebodies
in a study area. Two different kinds of edge effects will be considered in these
examples. Boundaries of study area have to be considered in both examples. In the
first case, the fractures could only be observed in areas where the granite is exposed
at the surface. The irregularities in pattern of rock exposures will be considered
to reduce bias in the mass-partition function to be estimated for the multifractal
modeling.

11.4.1 Lac du Bonnet Batholith Fractures Example

The Lac du Bonnet Batholith in the Winnipeg River Subprovince of the Archean
Superior Province, Canadian Shield, mainly consists of pink porphyritic and gray,
more equigranular granite; both contain layers of schlieric and xenolithic granite
(Brown et al. 1989 ). Total area of granite exposed at the surface measures about
1,500 km 2 . In 1980, Atomic Energy of Canada Limited (AECL) acquired a 21-year
lease on a 3.8 km 2 area of the batholith for construction of the Underground
Research Laboratory (URL) as part of geoscience research into the disposal of
nuclear fuel-waste in crystalline rocks. Detailed mapping of both lithology and
fracturing has been performed at surface, and extensive subsurface information was
made available for the URL excavations. About 130 boreholes were drilled, to
depths up to 1 l00 m; these are mainly cored boreholes, logged in detail for
lithological and fracture information. In 1992 a project was commenced to analyze
these surface and borehole data from a fractal/multifractal point of view, as it was of
interest to AECL to estimate three-dimensionally the relative frequencies of large
blocks of sparsely fractured granite. Most faults and mesoscopic fractures are either
subvertical or dip 10 -0 . Many subvertical joints die out about 100 m below the
surface. Low-intermediate-dipping (10 -30 ) fractures are associated with rela-
tively few well-defined fault zones in the subsurface extending to at least 800 m
depth (Agterberg et al. 1996a ).
Fractal modeling of fractures had been the subject of a number of studies
(Korvin 1992 ; Turcotte 1997 ; Ghosh and Daemen 1993 ). The primary purpose of
the 1992 study was to show that surface fractures can be modeled as multifractals.
Natural fault populations had been shown to possess multifractal scaling properties
by Cowie et al. ( 1995 ). Multifractal modeling provides a link between different
types of fractal measurements and the geostatistical approach using spatial covari-
ance functions or semivariograms. A geographic information system (SPANS,
cf . Chap. 5 ) was used to perform the measurements required for multifractal
modeling (Agterberg et al. 1996a ). Because most surface fractures die out with
depth, it is not possible to use the results of this study for downward extrapolation.
Figure 11.15 shows a relatively well-exposed, triangular test area ( 0.11 km 2 )at

the URL site where the surface fractures have been mapped in detail. The effect of
limited exposure at the surface on the statistical measures required special