Fractal Geometry of Faults and Fractures

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Mathematical Geology [mg] PP135-301052 April 23, 2001 20:1 Style file version June 30, 1999

Mathematical Geology, Vol. 33, No. 5, 2001

Scale Transitions in Fracture and ActiveFault Networks1

Thomas H. Wilson2

Detailed box counting analysis was conducted of (1) fractures observed in exposures of the DevonianShale in the central Appalachians Valley and Ridge Province of West Virginia, (2) several fracturepatterns presented in the literature, and (3) active faults mapped throughout the main island (Honshu)of Japan. Box curves reveal, with few exceptions, that most naturally occurring fracture patternsare characterized by nonfractal behavior. In many cases, two linear regions separated by an abrupttransition are observed in the logN/ logr box curves. The small-scale (largerr) features generally havehigher fractal dimension than do the larger scale features in the pattern. Transitions from one regionto another are usually abrupt. These transitions are not associated with sampling problems or otherdata limitations. In some cases three or more linear regions may appear. Box counting analysis ofmodel fracture patterns indicate that transitions are related to the dominant spacing of individualsets or to the dominant fragment size in the network. This study provides detailed documentation ofscale invariant features in natural fracture and active fault patterns. Although the relationship of thegeometrical properties of a pattern to the location of transitions is understood in terms of the models,to understand the physical mechanisms responsible for these transitions deserves further study.

KEY WORDS: fractals, fractures, active faults, box counting, scale transitions.

INTRODUCTION

This study presents observational and experimental evaluations of size-scalingrelationships in fracture and active fault networks that span areas extending from0.1 m2 to approximately 5000 km2. Several studies (e.g., Barton, 1995; Barton andHsieh, 1989; Turcotte, 1989; Hirata, 1989; Wilson, 1999, 2000) conclude that frac-ture networks can be characterized by a fractal scaling relationship. The conceptis intuitively appealing since the scale of rock outcrops portrayed in photographsoften cannot be accurately guessed unless a reference to actual scale is included.In the photograph of fractures shown in Figure 1A, for example, plants and grassprovide some reference to the actual scale of the exposure. In the close-up view

1Received 17 September 1999; accepted 21 June 2000.2Dept. Geology and Geography, West Virginia University, Morgantown, West Virginia 26506-6300.e-mail: [email protected]

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Figure 1. The above photographs represent A, distant and B, close views of the same rock exposure.Location of the large-scale view B is outlined in A. The size of the area photographed in B is one-tenth that photographed in A. Line drawing interpretations of the fracture traces interpreted in eachimage are presented in C and D.

(Fig. 1B) there is no reference to scale and less certainty regarding the absolutescale of features in the photograph. The lack of scale makes it difficult to appreciateactual differences in fracture spacing and length observed in the two photographs.Although it is not possible to identify the scale in either of the photographs, thereare obvious differences in the overall characteristics of the pattern of fracturesin these two photos. The fracture pattern observed in the smaller area (Fig. 1B)consists, in a relative sense, of more widely spaced and penetrative fractures thanthose characteristic of the larger area (Fig. 1A). The small area view (Fig. 1B)focuses on the white rectangular area shown in Figure 1A. The scale in Figure 1Bis 10 times that of Figure 1A. The fracture patterns interpreted at these two scalesare distinctly different.

Several studies find limitations in the fractal model. Chiles (1988), for ex-ample, notes the presence of slight nonlinearity in the box curve constructed fora 10× 10 m fracture pattern, implying that the fracture network is not a scaling



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fractal pattern. Walsh and Watterson (1993) note a gradual change of slope ap-pearing in box counting data for one of the patterns presented by Barton and Hsieh(1989). They suggest that nonlinearity in the log–log representation of the boxcounting data is the result of having large unmapped areas included within theinitial starting box. Walsh and Watterson (1993) reduce the nonlinearity by usingan irregular-size initial box to circumscribe the fracture map. Gillespie and others(1993) document significant nonlinearity in the box counting curves obtained forseveral joint and fault patterns and for numerous synthetic fracture maps. Walshand Watterson (1993) recommend that the analysis of box counting curves andestimation ofD be made only over a range of box sizes extending from the largestto smallest fragment size of fracture spacings within the pattern. It is in this rangeof box sizes that the rate of box filling provides information about the distributionof fragment size. Outside this range the slope of the curve tends to−2 for largerboxes and to 0 for the smaller boxes. Odling (1992) also notes the presence of slopebreaks in his analysis of a fracture pattern mapped in a Devonian age sandstoneexposure in Norway. Odling (1992) observed a break from a slope of−2 associatedwith the larger box coverings and noted that the break occurred approximately atthe average spacing of the fractures in the pattern. Wilson, Dominic, and Halverson(1997) note similar nonlinearity in the box curves for several patterns observed inDevonian Shale outcrops in the central Appalachians.

Berkowitz and Hadad (1997) emphasized the disagreement that is obtainedin parallel analyses of the same data set (e.g., Barton and Hsieh, 1989; Barton,1995; Walsh and Watterson, 1993). Berkowitz and Hadad noted the presence oftwo distinct fractal regimes in their evaluation of two of the 17 patterns presentedby Barton (1995) and suggested that this was from the presence of two distinctfamilies of fractures with possibly different mechanisms of origin and histories.Gonzato, Mulargia, and Marzocchi (1998) identify two linear regions in theiranalysis of fault patterns from the central Apennines, which they attribute largelyto a lack of sufficient data points. Wilson (1999, 2000) notes that these variationsof slope are a common feature of Barton’s (1995) and other fracture networks, andsuggests that it may be more appropriate to refer to the negative slopes of the boxcurve as scaling parameters rather than fractal dimensions.

Scholz (1995) addressed the issue of scale-limited self-similarity in termsof fractal transitions that are either gradual or abrupt. The appearance of slopebreaks (Odling, 1992) or fractal regimes (Berkowitz and Haddad, 1997) representabrupt transitions in the scaling properties of the fracture network. Scholz (1995)concentrated on discussions of abrupt transitions, noting that they correspondto characteristic lengths in the system that may provide clues to the underlyingphysics. Scholz (1995) also notes that abrupt transitions occur in the spectrum ofseafloor topography and the San Andreas fault trace.

The potential link between transitions (Scholz, 1995) or subranges (Berkowitzand Haddad, 1997) to basic physics or fracturing mechanism enriches the potential



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outgrowths of continued research into the identification and significance of thesetransitions. In the case of the San Andreas Fault, Scholz (1995) identified abrupttransitions at 0.5 mm and 10 km scales. He associates the 0.5 mm break withgrain size and the 10 km break with the seismogenic thickness of the crust. Scholz(1995) recommends that scaling response (e.g., power spectral or box counting)should be examined carefully for the presence of characteristic lengths. He notesthat the fractal nature of the object in question should be expected to be differentat scales above and below characteristic lengths.

In the following study we carefully examine several fracture patterns andactive fault trace patterns for the presence of fractal transitions, and present modelstudies to relate the occurrence of abrupt transitions and characteristic lengths tothe spatial properties of fracture networks.

A SIMPLE TEST OF SCALE INVARIANCE

The following analysis relies on box counting to evaluate the scaling attributesof fracture and active fault patterns. Box counting is a simple procedure thatinvolves covering the pattern in question with boxes of varying size (r ) and countingthe number of boxes (N) that are required to cover the pattern (e.g., Turcotte, 1989).If the number (N) of occupied (r -sized) boxes increases in such a manner thatN ∝ r−D then the pattern is considered to be fractal and to have fractal dimensionD. Log transformation of this simple power law yields a straight line with slope−D. D is constant over all scales (r ). Features of a fractal pattern at one scalecannot be distinguished from those observed at other scales.

To highlight the possible exceptions of the fractal model, a simple experimentis considered. This experiment consists of photographing an outcrop at two scalesthat differ by an order of magnitude, such as those shown in Figures 1A and 1B.Fracture patterns observed in both views (Figs. 1C and 1D) were digitized from thephotographs. Box counting analysis was then performed on each pattern (Fig. 2).The logN vs logr box curve (Fig. 2A) of the small-scale pattern (Fig. 1C) variesalmost linearly over the 0.6–0.06 m range. However, the box curve has slightlysteeper slope for largerr and shallower slope for smallr causing the curve tocross slightly above the fitted line and then back beneath it. The fractal dimensionof 1.12± 0.024 (Fig. 2A) derived for the pattern of fractures observed at smallscale (Fig. 1C) should equalD computed for the large scale view of this pattern(Fig. 1B). However, the value ofD derived for the large-scale analysis (Fig. 2B)is only 1.02± 0.01. The estimates ofD are statistically different. Clearly, thelarge-scale view of the fracture pattern (Fig. 1B) is less complex than the small-scale pattern (Fig. 1A). Predictions based on the small-scale analysis incorrectlyrepresent the complexity of the pattern observed at larger scale.

The analysis in this simple experiment was conducted carefully to avoiderrors associated with sampling and box saturation (e.g., Walsh and Watterson,




Figure 2. LogN/logr plots illustrate the results of box-counting fracture patterns observed in A,small-scale and B, large scale views shown in Figure 1.

1993; Wilson, Dominic, and Halverson, 1997). The smallest box sizes used in theanalysis are much larger than the sample interval along individual traces. Large,completely filled boxes have also been excluded from the analysis. The first boxcovering consists of 16 square boxes with length equal to one-quarter the lengthof the analysis area. Larger boxes are often occupied even for simple patterns andthis leads to anomalously high slope on the large-r end of the box counting plot.

Analysis of the fracture patterns in Figure 1 suggests that the fracture networkis scale variant. Features observed in the box curve (Fig. 2A) for the small-scalepattern reveal a transitional response. An abrupt transition is interpreted arounda scale ofr = 0.25 m (Fig. 3A). Separate computation of the logN/logr slopeson the large and smallr ends of the curve yields statistically different slopes of−1.28± 0.08 and−0.93± 0.02, respectively (Fig. 3A). The value ofD (0.93)derived from the small-r region of the curve (Fig. 3A) might be a better predictorof the larger scale characteristics of the pattern. ComputingD from the large-scale (smallr ) pattern observed in the close-up view (Fig. 1B and 1D) checks theaccuracy of the prediction. Visual inspection of the pattern of fractures at small-scale (Fig. 1C) suggests that the complexity of the pattern decreases. However,the D (0.93) derived in the smallr region of the box curve (Fig. 3A) does notaccurately predictD derived from the large-scale analysis (1.02 in Figure 2B).

It might be argued that there is also a transition in the large-scale box curve(Fig. 2B). A slight break in slope appears aroundr = 0.03 m. Separate computa-tion of slope yieldsD of 1.15± 0.09 and 0.99± 0.01 on the high and low sides,respectively, of the interpreted transition at 0.03 m (Fig. 3B). These slopes are notstatistically different.D of 0.93 obtained from the small-scale analysis (Fig. 3A)disagrees even more with theD of 1.15 obtained over the adjacent range ofr(Fig. 3B).



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Figure 3. Slope variations are illustrated in the logN/logr plots of the A, small scale and B, large scalefeatures observed in distant and close-up views of the fracture patterns shown in Figure 1.

Analysis of the large-scale pattern (Figs. 2B and 3B) indicates that this patternhas a fractal dimension almost equal to 1. Analysis of the small-scale pattern(Fig. 3A) suggests that the larger scale features in the pattern has fractal dimensionsignificantly less than 1 (0.93± 0.02). This is what we would expect since theoverall features of the small-scale fracture pattern (Fig. 1A and 1C) are fragmented.The fractal dimension of such patterns is analogous to that obtained for the Cantordust (see Turcotte, 1989). The fractal dimension in this case is related to thedistribution of fractures rather than to their geometry (see Hirata, 1989). The small-scale prediction is that the pattern will continue to be fragmented at larger scale,while the large-scale observation reveals that the fracture patterns are, instead,largely continuous.

This simple experiment reveals (1) the presence of a transition that separatesstatistically different slopes in the box count data, and (2) that even when transitionsare taken into account, predictions made over one range of scales do not matchobservations at the predicted scale.

ADDITIONAL EXAMPLES OF SCALE VARIANCEAND ABRUPT TRANSITIONS

To emphasize that the preceding analysis does not represent an isolated in-stance of disagreement in the characterization of network complexity using fractalanalysis, box counting analysis is repeated in this section, using patterns fromthe literature (e.g., Barton, 1995). In addition, analysis is undertaken of numerouspatterns observed in exposures of the Devonian Shale in the central Appalachiansand also of the active fault networks in Japan.




Fracture patterns shown in Figure 4 were taken from Barton (1995). Boxcounting was confined to the square areas highlighted in each pattern to avoid theinfluence of map edges. The logN/logr plots for all three of these patterns reveal adecrease in slope (decrease ofD) with decrease inr . In one example (Fig. 4B) thelogN/logr plot contains three linear segments and two transitions. Box countingof fracture patterns observed in Devonian Shale exposures in the Valley and RidgeProvince (Figs. 5 and 6) reveal similar complexity in the box counting curves.The patterns displayed in Figure 5 show undifferentiated (Fig. 5A) and system-atic (reinterpreted in Fig. 5B) fracture sets observed at one of these Valley andRidge exposures. The logN/logr plots for both the undifferentiated and system-atic fracture patterns (Figs. 5C and 5D) decrease in slope across abrupt transitionsasr decreases. The combined pattern of systematic and nonsystematic fractures(Fig. 5A) is much more complex than the pattern of systematic fractures takenalone (Fig. 5B) and this complexity is reflected in the higher fractal dimensionsobserved at each scale in Figure 5C. Although the patterns are quite different,abrupt transitions occur at similar scale, which suggests some underlying simi-larity between the two patterns. The example shown in Figure 6 presents anothertest of the predictability from one scale of observation to another. Both distant andnear photographs were taken of another Devonian Shale outcrop. The location ofthe close-up area (Fig. 6B) is outlined in Figure 6A.

Box curves at these two scales (Figs. 6C and 6D) reveal a decrease in patterncomplexity across abrupt transition points. The small-scale analysis (Fig. 6C)extends fromr of 1.3–0.13 m. The response reveals slope transitions (and patterncomplexity) atr of 0.7 and 0.26 meters. Asr drops below 0.26 meters,D falls to1.47. In the large-scale analysis (Fig. 6D) slope reductions continue with abrupttransitions occurring at values ofr equal to 0.09 and 0.035 meters.

Again, prediction of large-scale behavior based on the small-scale analysis(Fig. 6C) does not match results actually observed at larger scale (Fig. 6D). How-ever, the general prediction of a reduction in the complexity of the pattern at largerscale is confirmed in the large-scale analysis. The analysis is limited in that thelarge-scale evaluation is conducted on only one portion of the smaller scale image.

Examination of the active fault networks in Japan provides an illustrationof size-scaling properties on a regional tectonic scale. Previous studies of theactive faults in Japan (Fig. 7) suggest that active fault networks like outcrop-scalefracture networks are fractal. Hirata (1989) used the box counting technique toevaluate the fractal properties of Japan’s active fault pattern. Hirata conductedhis analysis on 1◦ longitude by 40′ latitude (roughly 90 by 74 km) active-faultsheet maps of Japan published by the Research Group for Active Faults of Japan(1980). Matsumoto, Yomogoda, and Honda (1992) analyzed relatively narrowstrips along the Median Tectonic Line and the Izu Peninsula using data from theResearch Group for Active Faults of Japan (1980) and also the Research Groupfor Active Tectonic Structures in Kyushu (1989). Hirata (1989) and Matsumoto



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Figure 4. LogN/ logr plots computed for patterns a (A), c (B), and g (C) (from Barton, 1995)are presented. Box counting was carried out over the square-shaped regions highlighted ineach pattern.




Figure 5. Tracings of A, undifferentiated and B, systematic fractures were interpreted independently;C, LogN/ logr plot for the fracture traces shown in A; and D, logN/ logr plot for the fracture tracesshown in B.

and others (1992) employed different methods to estimate the fractal dimensionsof active fault distributions. Hirata used box counting (e.g., Turcotte, 1989, 1992),while Matsumoto, Yomogoda, and Honda (1992) used the method of Okubo andAki (1987), which is based on counts of the number of circles of varying radiusneeded to cover the fault traces.

Hirata (1989) was the first to conduct a detailed analysis of spatial variationsin the fractal dimension of the active fault network throughout Japan. Hirata’scomputations were carried out using base 2 decreases in box size, which didnot provide enough detail in the box curve to recognize abrupt scale transitions.Hirata’s (1989) analysis of box-count data was restricted approximately to the2.34–18.7 km range of scales, while the results of Matsumoto, Yomogoda, andHonda (1992) were restricted to the 1–10 km range of scales. Based on outcrop



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Figure 6. A, Small scale and B, large scale views of a Devonian Shale fracture patterns; C and D,LogN/ logr plots for the small A and large B scale views.

scale studies of fracture patterns in the North Izu Peninsula area, Hirata (1989)suggests that the fractal characteristics of fractures on the 10−1 to 10−2 m scaleare similar to the smaller scale (103 to 104) m active fault patterns.

Active faults analyzed in this study (Fig. 7) were digitized from revised edi-tions of active fault maps analyzed by Hirata (1989). The revisions include thoseof the Research Group for Active Faults of Japan (1991) and also updates by theGeological Survey of Japan. Initially, four 70× 70 km test areas (see highlightedareas in Fig. 7) were selected for detailed analysis. The active fault patterns in thetest areas are representative of the variety of patterns present in the active faultnetwork. In the present study, estimates ofD are made over the 17.5–2 km range.The largest box size (17.5 km) corresponds to one-fourth the 70 km length ofthe analysis region. The 17.5–2 km range of box sizes was spanned in 25 equallogarithmic-size steps.

Fault patterns in the 70× 70 km test areas and their box curves are shown inFigures 8 and 9. Abrupt scale transitions are observed in each of the box curves.On average, scale transitions occur at 7.8 km. In the Iida area (Fig. 8A) a second




Figure 7. Active fault map of Japan. Detailed box counting tests were conducted of the patterns in theHiroshima, Iida, Nikko, and NEJ areas highlighted above. Locations of analysis lines 1–3 are plotted.

transition is observed at 3.5 km. Based on the average (7.8 km) transition, theslopes of the box curves were computed throughout the database along three linesover the 2–7.75 and 8.5–17.5 km. The gap from 7.75 to 8.5 km is associatedwith one logarithmic step. Analysis was carried out for a total of 115 overlapping70× 70 km boxes, which were spaced at 20 km intervals along each line. Thedifference inD derived from the 2–7.75 and 8.5–17.5 km range is on averageabout 0.32. Standard deviations on the estimates ofD are on average 0.09 for thelarger boxes and 0.03 for the smaller boxes. The differences between these twoestimates are generally significant at the 95% confidence level.

The presence of an abrupt transition in the distribution of faults about a char-acteristic length of 8 km may be related to differences in the tectonic origins of



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Figure 8. Fault patterns and box counting plots for active faults in the A and B, Hiroshima and Cand D, Iida test areas.

faults above and below the 8 km scale. The relationship of fault dimension to theseismic moment of earthquakes occurring on them, for example, is a fundamentalrelationship that varies with scale (e.g., Shimazaki, 1986; Scholz, 1990). Kanamoriand Anderson (1975) recognized that except for very large earthquakes, the seismicmoment of an earthquake is proportional to the cube of fault dimension. Scholz(1982) noted that seismic moment scales differently for earthquakes with rup-ture dimensions exceeding the seismogenic thickness of the crust. Scholz (1997,1998) showed that the size distribution of earthquakes bears a simple relation-ship to the size distribution of faults and subfaults. Scale changes associated withthe seismogenic thickness of the earth’s crust are also inferred from a transition




Figure 9. Fault patterns and box counting plots for active faults in the A and B, Nikko and C andD, NEJ test areas.

observed in the Gutenberg–Richter frequency magnitude relationship (Pacheco,Scholz, and Sykes, 1992). Shimazaki (1986) found that the seismic moment/ faultlength relationship for intraplate seismicity in Japan is best fit by two lines; onefor small earthquakes and the other for large earthquakes. The transition betweenthese two regions occurs at rupture lengths of approximately 15 km or, roughly,that equal to the thickness of the seismogenic zone in Japan (see also Scholz, 1990,Fig. 4.12).

The tectonic significance of the 8 km transition observed in box curves com-puted for the active fault patterns in Japan is not clear. It is little more than half thethickness of the seismogenic layer. Scholz (1998) notes that at scales of 1–10 km,



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and perhaps smaller, faults are not continuous surfaces but consist instead of amesh of strands or subfaults. Scholz (1998) also notes that the cumulative sizedistribution of subfaults for the San Andreas fault system in central California(Fig. 10) compiled by Wallace (1973) follow a power law with a transition near16 km. Reevaluation of Wallace’s (1973) data (Fig. 10) indicates that the lengthdistribution also has a transition at lengths of approximately 8 km. Exponentsof −1.85± 0.02 and−2.4± 0.18 were derived for segments of the curve over

Figure 10. Cumulative number (N) of subfaults on the San Andreas fault system vs.length (L). Data are from Wallace (1973).




2.6< L > 7.8 and 9< L > 17.3 km, respectively. Standard deviations and 95%confidence limits on the exponents indicate they are statistically different. Thetectonic origins of faults in Japan are generally dissimilar to the transform faultinteractions occurring along the San Andreas Fault. However, the presence of char-acteristic lengths less than the thickness of the seismogenic zone in both tectonicenvironments suggests that there may be some fundamental relationship of thistransition to the mechanics of faulting in crustal scale fault systems. In the follow-ing section, we explore the general relationship of transitions to the geometricalfeatures of the fracture trace network.

MODEL STUDIES

As shown above, the logN/ logr plots of fracture trace and active fault patternsoften contain abrupt slope transitions. The result indicates that the number ofoccupied boxes (N) does not follow a simple power law relationship tor . Whilethe results do not support a fractal model, the slope transitions provide someinformation about the pattern and about changes in its characteristics with scale.The significance of these transitions is examined here in a series of model studies.The model studies start with analysis of very simple fracture patterns consistingof a single penetrative fracture set with constant spacing and are expanded toincorporate additional pattern complexity. While the models are generally simplerby comparison to naturally occurring fracture trace patterns, they provide insightsinto the significance of transitions observed in the logN/ logr box curves.

Single Fracture Set

Single set fracture models (Figs. 11A and 11B) consist of a single systematicset of parallel fracture traces spaced at 1 cm intervals. In the single set modelof Figure 11A, the sides of the covering boxes are parallel or orthogonal to thefracture trace orientation. The logN/ logr plot for this model (Fig. 11C) revealslinear behavior over two scale ranges separated by an abrupt transition. The largerboxes tend to remain occupied as box size is reduced yielding a larger fractaldimension (power law slope) than is observed for smaller box sizes (smaller r). Inthis area of the plot a reduction ofr by a factor of 2 yields a factor of 4 increase inthe number of occupied boxes and power-law slope of 2 (Wilson, 1997). When thesize of the box becomes less than the spacing between traces, the fractal dimensiondrops to 1. Once the size of the box is less than the spacing between traces, decreasesof r by a factor of 2 only double the number of occupied boxes so that the power-law slope (or fractal dimension) is 1. In the second model (Fig. 11B), the fracturetraces have been rotated 45◦. The slope transition in this model occurs around0.74 cm. It is not untilr decreases to∼0.74 cm that the covering boxes begin to fitbetween individual fractures and become unoccupied. In this example (Fig. 11D)



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Figure 11. A, fracture model consisting of a single set of fractures with constant spacing of 1 cm; B,model in which fractures have been rotated 45◦ with respect to the covering box; C, LogN/ logr plotfor the model shown in A; D, LogN/ logr plot for model shown in B.

the transition atr ∼ 0.74, corresponds roughly to the cosine of 45◦ (i.e., 0.707).This relationship has been verified in general using test patterns with fracture traceorientations at intermediate angles.

Double Set

The model shown in Figure 12A consists of two sets of fractures, each or-thogonal to the other. The more extensive set of horizontal fractures have a 2.5 cmspacing while the vertical set of fractures are spaced at 1 cm intervals. Two tran-sitions appear in the box curve (Fig. 12B): one atr ∼ 2.4 cm and the other at




Figure 12. Model fracture network consisting of twosets of uniformly spaced fractures that are orthogonalto each other. The vertical fractures have 1 cm spacingwhile the horizontal fractures are more extensive andhave 2.5 cm spacing. LogN/ logr plot of the model.

r ∼ 0.92 cm. Their locations correspond approximately to the spacings in the twofracture sets. These two transitions divide the line into three linear regions eachwith different slope or fractal dimension. There is a consistent decrease in slopefrom large to smallr .

Discontinuous Fracture Set

Discontinuous fracture models (Figs. 13A and 13B) introduce gaps of con-stant size into a single set of vertically oriented fractures. In both models, thefracture traces are spaced at 1 cm intervals. In Figure 13A the fracture gaps are1 cm in length. Because the gap and spacing are the same, the logN/logr plothas a single slope transition at approximately 1 cm (Fig. 13C). In Figure 13B,the fracture gap is increased to approximately 2 cm. The logN/logr plot for thisfracture pattern (Fig. 13D) has three roughly linear regions separated by abrupt



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Figure 13. A, Model fracture network consisting of a single uniformly spaced (1 cm) set of fractures.Each fracture trace is interrupted by 1 cm gaps every 2 cm along the trend of the fracture; B, in thismodel the size of the gap has been increased to 2 cm; C, LogN/ logr plot of the model shown in A,D, LogN/ logr plot of the model shown in B.

transitions at 2 and 1 cm. The transition at 2 cm is associated with the gap and thetransition around 1 cm is associated with the spacing between fractures.

Variable Spacing Single Set and Clusters

Naturally occurring fracture networks generally have considerable variabil-ity in orientation, spacing, and length. The effect of such variability is assessedusing the models shown in Figure 14A and B. The set of fracture traces shown inFigure 14A incorporates variable spacing and orientation into the pattern, whiletheir length is held constant. The logN/ logr plot (Fig. 14C) is characterized by




Figure 14. A, Model fracture network consisting of a single set of fractures with variable spacing andorientation; B, model fracture network consisting of multiple clusters of a single set of fractures withsome variation in spacing and orientation; C, LogN/ logr plot of model shown in A; LogN/ logr plot ofmodel shown in B.

two transitions separating three linear segments. Although the individual frac-ture traces have variable orientation and thus variable spacing, the logN/ logr plot(Fig. 14C) suggests that, overall, there are two “dominant” spacings within the setof approximately 1.1 and 3.3 cm. Closer inspection of the fracture pattern revealsthat several pairs of lines have nearly 1 cm spacing while others have variable butlarger spacing that on average account for the larger 3.3 cm break (Fig. 14C).

The pattern illustrated in Figure 14B consists of a series of relatively isolatedzones or clusters of intense fracturing that is widely scattered across the area.The distribution leads to an unusual response in the logN/ logr plot (Fig. 14D) inwhich a linear region with smaller slope develops for larger . This segment does not



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Figure 15. A, Model fracture network obtained froma combination of the models presented in Figure 14.B, LogN/ logr plot of the composite fracture model.

extend over a large range ofr and is believed to result when boxes of size 3–5 cmbecome unoccupied within the spaces between clusters.D increases from about 1to 1.4 asr falls below 3 cm (Fig. 14D). In the regionr less than 3 cm the numberof occupied boxes (N) follows a power law relation withr . The spacing betweentraces in individual clusters is generally less than 0.5 cm so that the slope breakassociated with the dominant spacing of fractures in the clusters is never observed.




Variable Spacing Combined Sets

The model shown in Figure 15A combines the fracture sets presented inFigures 14A and 14B. The logN/ logr plot (Fig. 15B) is interpreted to containtwo linear regions separated by a transition around 1.9 cm. Values ofD in thesetwo regions are 1.87± 0.03 and 1.5± 0.02, and are significantly different fromeach other. The break observed atr ∼ 2 cm suggests that it is associated with a“dominant” or “representative” spacing of the entire fracture network. The 2 cmtransition, observed in this box curve, is located approximately at the average ofthe transitions observed in the two models of Figure 14. This example servedto illustrate that the transitions in the box curve may obscure or average out thegeometrical attributes of individual fracture sets. However, this 2 cm break couldalso be interpreted as the approximate size of fragments bounded by fractures andclusters in this composite pattern.

CONCLUSIONS

Box curves of fracture networks from a variety of different tectonic settings arefrequently characterized by abrupt transitions between regions of roughly constantfractal dimension. In general, the complexity of a fracture network tends to decreasewith increased scale. The opposite may also occur as noted by Scholz (1995) forthe surface rupture pattern produced by the 1968 Dashte-e-Bayaz earthquake inIran. When viewed from a distance, an isolated fault may appear as a more-or-lesscontinuous break marked by gentle curves. However as one approaches the fault,a more complex pattern of strands and subfaults emerges.

The main conclusion of this study is that scale invariance in fracture networksappears to be the exception rather than the rule. Predictions of pattern complexityat scales other than the scale of investigation are generally incorrect. The complexbehavior observed in the box curve does, however, provide useful information.Model studies indicate that even the simplest of fracture patterns contain abrupttransitions between roughly linear regions in the box curve. These transitions areoften related to average spacings or fragment size at a given scale. Spacing dis-tributions presented by Wilson (2000) have modes, which often occur at spacingssimilar to the characteristic lengths of these scale transitions. In some instances,box curves may be characterized by constant slope if the range of scales is notlarge enough to include the dominant spacing between fracture traces or the averagelength of gaps between fracture segments. If linearity is observed in the box curveplot this possibility should be examined before concluding that the set has fractaldistribution.

The presence of an abrupt transition around 8 km was found in the box curvescomputed for the active fault network of Japan. A change in the distribution ofearthquake moment above and below fault dimensions of approximately 15 km was



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noted by Shimazaki (1986) for earthquakes in Japan. The 15 km scale correspondsapproximately to the thickness of the seismogenic zone in Japan. The physicalsignificance of the 8 km transition that is observed in the patterns of active faultingis unknown. However, a similar break at 8 km was found in the reevaluation(Fig. 10) of the length distribution of subfaults along the San Andreas fault systemreported by Wallace (1973).

The foregoing model studies indicate that scale transitions often have a simplerelationship to the geometrical characteristics of the pattern at a certain scale.However, questions regarding the physical significance of these transitions in termsof plate tectonic interactions or fracture mechanism are questions that requireadditional study.

ACKNOWLEDGMENTS

Special thanks to Osamu Nishizawa for hosting my 1998 visit to theGeological Survey of Japan, and to Osamu Nishizawa, Hirokazu Kato, and KojiWakita who, at various times, provided active fault maps of Japan. Reviews ofthe manuscript by Christopher Scholz and an anonymous reviewer were greatlyappreciated. Thanks are also extended to Eberhard Werner, who provided digitalversions of some of the fracture images. Discussions with Tom Mroz, Royal Watts,and Bill Gwilliams of the Morgantown National Energy Technology Lab were alsomuch appreciated. This research was supported through U.S. Department of En-ergy Grants DE-FG21-95MC32158 and DE-FG26-98FT40385. The analysis ofactive faults in Japan was supported through a Japan Science and TechnologyAgency Fellowship with the Geological Survey of Japan.

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Fractal Geometry of Faults and Fractures

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