SCHAEFFER, Malcolm F., Duke Engineering & Services, Inc., P.O. Box 1004, Charlotte, NC 28201-1004 (mfschaef@dukeengineering.com) and SCHAEFFER, Joseph M., California Institute of Technology, MSC778, Pasadena, CA 91126-0778

Four geometric properties of joints, 1) orientation, 2) intensity, 3) trace length, and 4) spacing, were determined at six locations in crystalline rock (biotite gneiss, granitic gneiss, quartz-feldspar gneiss, felsic metavolcanic rocks, and metagranite) within the Piedmont Province of the Carolinas (Figure 1). The data for this study were extracted from detailed joint trace mapping of bedrock exposures at scales of 1 cm = 0.6 m and 1 cm = 1.2 m. The bedrock exposures range in area from 545 m² to 5035 m².

The mapped joint traces were separated into sets for the analysis of intensity, trace length, and spacing. Joint sets at each site were determined by examination of the field exposures, the joint trace map, a rose diagram using all joint traces from the trace map, and the visual clustering of poles to joint planes (measured orientations) on a Schmidt net. The primary procedure for the determination of joint sets partitions the projective hemisphere into non-overlapping regions and treats the poles within each region as corresponding to an individual set of joints. Based on that procedure, the mapped joint traces are separated into their respective sets for the analysis of intensity, trace length, and spacing.

The joint systems at the sites have from two to five sets of joints (Table 1). The sets determined at the sites can be placed into six regional joint sets; Set A, N59E-N68E, Set B, N4W-N19W, Set C, N81W-N83E, Set D, N28E-N30E, Set E, N34W-N46W, and Set F, N68W. Joint intensity for each set of joints was calculated by summing the lengths of all joints in a set and dividing by the total area mapped. Total joint intensity is the sum of the intensities for each joint set at a site (Table 1). Appreciable differences in intensity exist between joint sets at and between sites and in total intensity between sites (Table 1).
 
 

SITE [Total Intensity]	Comparison of JOINT INTENSITY (m-1) and JOINT TRACE LENGTH (mean spacing, m) for JOINT SETS (Regional)
	Set A N59E-N68E	Set B N4W-N19W	Set C N81W-N83E	Set D N28E-N30E	Set E N34W-N46W	Set F N68W
Bridgewater [4.89 m-1]	N59E 1.50 m-1 2.47 m				N34W 3.39 m-1 1.91 m
Rhodhiss [0.42 m-1]		N19W 0.18 m-1 1.80 m	EW 0.24 m-1 2.47 m
Lookout Shoals [1.95 m-1]	N67E 0.88 m-1 3.30 m	N13W 0.57 m-1 1.71 m	N88E 0.50 m-1 2.66 m
Rocky Creek [2.96 m-1]	N68E 0.79 m-1 2.32 m	N4W 1.28 m-1 2.11 m	N86W 0.28 m-1 2.69 m	N30E 0.35 m-1 2.00 m		N68W 0.26 m-1 3.00 m
Gaston Shoals [1.24 m-1]	N63E 0.11 m-1 3.01 m		N83E 0.43 m-1 6.25 m	N30E 0.14 m-1 4.31 m	N46W 0.56 m-1 6.50 m
Ninety-Nine Islands [2.98 m-1]			N81W 0.45 m-1 1.72 m	N28E 2.53 m-1 1.94 m

The trace length data was truncated at 0.25 m (lower limits of geologic mapping) and corrected for censoring sample bias. Censoring refers to the intersection of some joint traces with the boundary of the mapped area so that their true length is not represented on the map. Histograms of the trace length data were constructed for each joint set at the sites and the statistics of the distributions were calculated. The trace length data generally fits a log normal distribution even when the truncation bias is taken into account. Goodness-of-fit tests have not been performed on the trace length distributions to date. Other workers have noted different distributions for trace length including exponential, power-law, and gamma.

The intersection of a joint set with a perpendicular scanline appears as a sequence of points that can be either regularly or irregularly located along the scanline. Fracture spacing is the distance between the points corrected for the dip of the joint set. Spacing data was collected along lines superimposed on the joint trace maps and the distances measured by the computer. A single distribution, such as exponential, does not fit all the data. Means of spacing data for joint sets at the six sites are given in Table 2. In this study, negative exponential, power-law, and log normal distributions have been fitted to data from various scanlines. All results are preliminary, as goodness-of-fit tests have not been completed.

The application of fractal geometry to joint set spacing starts with the realization that the intersections of the joint trace with the scanline appear in some manner to be similar when viewed at different scales. Spacing measurements that include spacing values with a high frequency of short spacing values occurring within clusters and a low frequency of long spacing values occurring between clusters suggest self-similar properties. Another example of self-similarity is spacing values that tend to show a decrease in spacing values adjacent to a nearby increase and show an increase adjacent to a nearby decrease. If a fractal relationship exists within a sequence of intersections (which form a fractal or Cantor dust), it will follow the relation N(r) = r ^-D, where N(r) is the number of events (the number of intervals of length r which contain at least one fracture intersection with the scanline), r is the dimension of the measurement (an interval r in this presentation), and D is the fractal dimension.

A modified box-counting technique is used to calculate the fractal dimension of the spacing intersections (Velde et al., 1990). The fractal dimension calculated is a characterization of the joint set in terms of the dimension of the set of points formed by the intersection of a scanline perpendicular to the trace of the joint set being measured. The scanline is divided into increasing smaller intervals, r and the number of intervals of length r containing at least one joint intersection is counted, N(r). N(r) is plotted versus r on a log-log plot and a straight line fitted. The slope of the line D is the fractal (box-counting) dimension and has a value between zero and one.

The fractal dimension estimated on a log-log plot using this approach may not always equal the Hausdorff-Besicovitch dimension due to size range limitations (Mandelbrot, 1983). For an ideal fractal set, a straight line on a log-log plot will yield the fractal dimension. In this study, the joint spacings are measured along scanlines of finite length and the spacing measurements fall within a range of r_min and r_max, the lower and upper limits, respectively. The use of measurement intervals r greater or less than these values should be avoided. When the interval r is greater than r_max, all intervals in the analysis are intersected by at least one fracture and when the interval is less than r_min only one intersection will occur in any interval. The slope determination is therefore limited for a range of r values and thus the fractal dimension is defined for a range. For reasonable values of D over a range of r_min < r < r_max, a suggested rule is that r_min and r_max should span one decade or more of r values (such as 0.1 m to 1.1 m; Boadu and Long, 1994).

Spacing data was collected along lines superimposed on the joint trace maps and the distances measured by the computer. The boxing and counting procedure is done by a computer program developed for this study. The fractal dimension was calculated for each scanline. The fractal dimensions are shown in Table 2 along with the mean (average) of spacing data for the various joint sets and scanlines.

The log N(r)/log r slope is obtained using a linear least-squaresregression. Values of R² greater than or equal to 0.95 suggest the dust formed by the intersections of the fractures with the scanline is fractal. However, a high correlation coefficient in itself does not indicate a fractal structure. Only 19 of the 44 scanline determinations yield R² values greater than of equal to 0.95. In the cases where lower regression coefficients were obtained, the points lying farthest from the regression line were at the ends, where the measuring interval r approaches r_max and r_min for that particular scanline. This may be an artifact of the finite length of the scanline due to the entire pattern not being sampled. An alternate interpretation is the dust formed by the intersection of fractures with the
 

Comparison of Joint Set Spacing and Fractal Dimension
Joint Spacing [Line #, (# of measurements, Length of Line (m))] Mean Spacing D = Fractal Dimension (Cantor Dust Method)
Sites	Set A N59E-N68E	Set B N4W-N19W	Set C N81W-N83E	Set D N28E-N30E	Set E N34W-N46W	Set F N68W
Bridgewater	N59E Line 1 (36, 17.4) 0.48 m D=0.3957 Line 2 (37,19.5) 0.53 m D=0.3696 Line 3 (58, 21.9) 0.38 m D=0.2354 Line 4 (26, 20.3) 0.78 m D=0.3179				N34W Line 1 (75, 13.4) 0.18 m D=0.3330 Line 2 (132, 42.5) 0.32 m D=0.3364 Line 3 (120, 43.7)  0.36 m D=0.3256
Rhodhiss		N19W Line 1 (53, 77.9) 4.82 m D=0.2529	EW Line 1 (36, 172.5) 4.79 m D=0.2227
Lookout Shoals	N67E Line 1 (39, 48.8) 1.20 m D=0.2802 Line 2 (27, 39.6) 1.47 m D=0.1644 Line 3 (48, 47.3) 0.99 m D=0.1869	N13W Line 1 (27, 52.4) 1.94 m D=0.2886 Line 2 (35, 51.6) 1.47 m D=0.3288	N88E Line 1 (32, 47.4) 1.48 m D=0.3870 Line 2 (24, 50.1) 2.09 m D=0.2327 Line 3 (39, 44.2) 1.1.9 m D=0.3759
Rocky Creek	N68E Line 1 (46, 34.3) 0.75 m D=0.4029 Line 2 (65, 79.5) 1.22 m D=0.3983	N4W Line 1 (16, 19.2) 1.20 m D=0.2871 Line 2 (43, 27.2) 0.63 m D=0.3783 Line 3 (48, 37.1) 0.77 m D=0.3580	N86W Line 1 (11, 38.7) 3.51 m D=0.2652 Line 2 (17, 44.2) 2.60 m D=0.2620	N30E Line 1 (22, 38.9) 1.77 m D=0.3375 Line 2 (23, 41.0) 1.80 m D=0.1839		N68W Line 1 (13, 43.0) 3.30 m D=0.1720 Line 2 (13, 37.4) 2.88 m D=0.2137
Gaston Shoals	N63E Line 1 (10, 45.0) 4.50 m D=0.3178		N83E Line 1 (70, 32.0) 0.46 m D=0.3841 Line 2 (18, 35.8) 1.99 m D=0.3643 Line 3 (21, 37.0) 1.76 m D=0.3795	N30E Line 1 (11, 26.4) 2.40 m D=0.3178 Line 2 (8, 24.4) 3.04 m D=0.2736	N46W Line 1 (18, 33.5) 1.86 m D=0.3228 Line 2 (24, 23.7) 0.99 m D=0.2002 Line 3 (47, 35.9) 0.76 m D=0.3278 Line 4 (56, 50.8) 0.91 m D=0.2572
Ninety-Nine Islands			N81W Line 1 (8) 1.69 m Line 2 (13) 1.86 m Lines 1 & 2 (37.7) D=0.3627	N28E Line 1a (81, 30.2) 0.37 m D=0.4903 Line 1b (79, 29.6) 0.37 m D=0.5143 Line 1c (86, 30.4) 0.35 m D=0.6457 Line 2 (31, 13.4) 0.43 m D=0.4375 Line 3 (34, 14.0) 0.41 m D=0.5627

 

scanline is not fractal (power-law distribution). Other distributions are possible for spacing measurements including regular distributions with constant spacing, random point process distributions (negative exponential frequency distribution), Kolmogorov fragmentation process distributions (log normal frequency distributions), and others.

Other fractal techniques are applicable to spacing data and future work with these data sets will utilize and explore these techniques. The work on joint spacing in crystalline rocks is still underway and we can not make any conclusions at this time concerning the distribution of spacing measurements or of the potential fractal nature of the spacing data. The different joint sets, their orientation, trace lengths, and spacing in each set are characteristics or parameters which represents a portion of the entire joint/fracture network. We expect, in the end, some consistent relationship between all the various parameters.
 
 

References Cited:

Boadu, F. K. and Long, L. T., 1994, The fractal character of fracture spacing and RQD: International Journal of Rock Mechanics and Geomechanical Abstracts, v. 31, no. 2, p.127-134.

Mandelbrot, B. B., 1983, The fractal geometry of nature: Freeman, New York.Velde, B., Dubois, J., Touchard, G., and Badri, A., 1990, Fractal analysis of fractures in rocks: the Cantor’s Dust method: Tectonophysics, v. 179, p. 345-352.