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Petroleum Geoscience

Quantified fracture (joint) clustering in Archean basement,

Wyoming: Application of Normalized Correlation Count

method

Qiqi Wang, S.E. Laubach, J.F.W. Gale & M.J. Ramos

DOI: https://doi.org/10.1144/petgeo2018-146

Received 12 October 2018

Revised 12 March 2019

Accepted 13 May 2019

© 2019 The Author(s). This is an Open Access article distributed under the terms of the Creative

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Quantified fracture (joint) clustering in Archean basement, Wyoming:

Application of Normalized Correlation Count method

Qiqi Wang1,2*, S.E. Laubach2, J.F.W. Gale2, M.J. Ramos1,2

1The University of Texas at Austin, Department of Geological Sciences, Austin, TX 78712

2The University of Texas at Austin, Bureau of Economic Geology, 10100 Burnet Road, Austin, TX 78713

*Corresponding author (e-mail: [email protected])

ABSTRACT: We demonstrate statistically significant self-organized clustering over a length scale

range from 10-2 to 101 m for north-striking opening-mode fractures (joints) in Late Archean Mount

Owen Quartz Monzonite. Spatial arrangement is a critical fracture network attribute that until recently

has only been assessed qualitatively. We use normalized fracture intensity plots and the Normalized

Correlation Count (NCC) method of Marrett et al. to discriminate clustered from randomly placed or

evenly spaced patterns quantitatively over a wide range of length scales and to test the statistical

significance of the resulting patterns. We propose a procedure for interpreting cluster patterns on NCC

diagrams generated by the freely available spatial analysis software CorrCount. Results illustrate the

efficacy of NCC to measure fracture clustering patterns in texturally homogeneous Archean granitic

rock in a setting distant (>2 km) from folds or faults. In their current geological setting, these regional

fractures are conduits for water flow and their patterns—and the NCC approach to defining clusters—

may be useful guides to spatial arrangement style and clustering magnitude of conductive fractures in

other, less accessible fractured basement rocks.

Fracture clustering in basement rocks

Fracture spatial arrangement is a key aspect of the structural heterogeneity of the brittle crust (Bear et

al. 2012; Laubach et al. 2018a). Understanding the spatial arrangement of opening-mode fractures

(joints and veins) has great practical importance. For example, fracture-enhanced permeability is

important in aquifers (De Dreuzy et al. 2001; Henriksen & Braathen 2006; Maillot et al. 2016), waste

repositories (Barton & Hsieh 1989; Carey et al. 2015), hydrothermal systems (Sanderson et al. 1994;

Cox et al. 2001; Hobbs & Ord 2017), and petroleum reservoirs (Laubach 2003; Gale et al. 2014)

including those in basement rocks (Cuong & Warren 2009; Belaidi et al. 2016; Bonter et al. 2018).

Typical of fracture-controlled permeability is tremendous heterogeneity and anisotropy in fluid flow

(e.g. Halihan et al. 2000; Solano et al. 2011). Although fracture network permeability is governed by

many factors including aperture and length distributions and connectivity, and by whether fractures

are open or sealed, a common cause of permeability variability is heterogeneous spatial arrangement.

Here we use a recently introduced method to describe fracture spatial arrangement in a large exposure

of texturally uniform Precambrian granitic rock in Wyoming.

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mailto:[email protected]


Spatial arrangement describes fracture positions, including whether fractures are clustered,

randomly placed or evenly spaced (Laubach et al. 2018). In conventional usage, spacing is the

perpendicular distance between two fractures in the same set (Priest & Hudson 1976; Ladeira & Price

1981; Narr & Suppe 1991), but the prevalence of highly clustered fractures is increasingly being

recognized, and what constitutes a useful quantitative measure of clustering is a matter of current

interest (Questiaux et al. 2010; Roy et al. 2014; Sanderson & Peacock 2019). For a set of co-planar

fractures, a common data collection method is to measure spacing along a straight line of observation

(1D scanline) orthogonal to fracture strike. The Normalized Correlation Count method (NCC, Marrett

et al. 2018) accounts for sequences of spacings, including non-neighbor spacings. The technique was

implemented with the freely available software CorrCount, accessible via the open-access Marrett et

al. paper. We compare results with conventional spacing analysis methods: descriptive statistics and

ratio of standard deviation to mean or coefficient of variation (Cv).

Here we demonstrate statistically significant self-organized clustering over a length scale range

from 10-2 to 101 m in joints in texturally homogeneous Archean granitic rock in a setting distant (>2

km) from folds or faults. These joints are currently conduits for water flow, and their patterns may be

useful guides to the spatial arrangement of fractures capable of transmitting fluids in other, less

accessible fractured basement rocks. Our example illustrates how NCC can quantify spatial

arrangement over a wide range of length scales, measure the pattern and degree of clustering, and test

the statistical significance of the results. We use the same method to assess spatial arrangement

patterns in six other previously published fractured granite data sets, finding a range of degrees of

clustering. Ours is the first application of the NCC technique to fractures in granitic rocks. Results

from our example and analysis of data sets from the literature show that opening-mode joints in

granites have patterns that range from power-law clustering to indistinguishable from random.

Geological setting

The Teton Range in northwestern Wyoming is a north-northeast-trending normal-fault-bounded

block of Precambrian crystalline rocks and dominantly west-dipping Cambrian through Mesozoic

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sedimentary rocks (Love et al. 1992; Roberts & Burbank 1993; Smith & Siegel 2000) (Fig. 1). To the

east the range is bounded by the east-dipping Teton normal fault (Love & Reed 1968; Love et al.

1992), a structure having Cenozoic to recent movement (Byrd et al. 1994; Smith et al. 1993; Leopold

et al. 2007). Within the range are north- to northwest-striking reverse faults of probable early Tertiary

(Laramide) age (Love et al. 1978; Lageson 1987).

Basement rocks in the Teton Range (Bradley 1956; Love et al. 1992; Reed & Zartman 1993;

Zartman & Reed 1998) include complexly deformed Archean gneiss, migmatite, and metasedimentary

rocks of the Wyoming Province (Chamberlain et al. 2003), Late Archean, mostly unfoliated granitic

intrusive rocks (Mount Owen Quartz Monzonite, Reed 1973), and east-west-striking diabase dikes.

Our data are from the upper part of Teton Canyon, in granitic rocks mapped as Mount Owen Quartz

Monzonite (Reed 1973; Love et al. 1992) (Fig. 1). The unit is a silica-rich peraluminous leucogranite

(Bagdonas et al. 2016; Frost et al. 2018) forming a discordant pluton (2.55 Ga Mount Owen batholith,

Frost et al. 2018), with margins marked by irregular bodies and dikes of pegmatite and aplite and

angular wall-rock inclusions (xenoliths). The exposures we studied are unfoliated, medium to fine,

texturally uniform light colored granite comprising quartz, microcline, and sodic plagioclase with

trace biotite and muscovite.

The smoothly undulating, nearly complete exposures we studied retain a fine, striated glacial

polish except adjacent to some joints, where postglacial erosion has removed the polished surface

(Figs. 1-5). Locally visible are southeast- and northwest-facing crescent-shaped chatter mark

(fracture) arrays caused by subglacial deformation and linear features on glaciated surfaces caused by

postglacial to recent rock fall (Fig. 2a, b). The glacial and recent geomorphic history of the range is

summarized by Foster et al. (2010).

Methods

Our fracture orientation and relative timing measurements were made in an outcrop that is

approximately 0.1 km2 (Fig. 1). The scanline was measured with compass and 30-meter plastic tape

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(centimeter graduations). We oriented the scanline 259 degrees in a direction that maximized the

extent of vegetation- and debris-free outcrop where measurements could be made at approximately

uniform elevation (Fig. 3). The scanline is normal to the strike of a main, steeply dipping joint set, and

our analysis focuses on these fractures. Scanline location was based on the largest extent of well

exposed rock, not to maximize numbers of intersections. Because fractures we measured have long

trace lengths relative to outcrop size, moving the scanline around or adding scanlines within the

window of good exposure would yield no significant differences the resulting pattern. Although

across the outcrop, the strike of this set locally has as much as a 10-degree range, the configuration of

the scanline relative to fracture strike means that Terzaghi (1965) corrections of spacing are small.

Measurement imprecision of scanline length and fracture placement arises from compliance of the

tape, non-uniform tension under field conditions, and the non-planar, undulating character of the

outcrop (Figs. 3, 4). We judge these errors to be small compared to the scale of the pattern captured in

the scanline. We measured kinematic aperture (opening displacement) for representative fractures

using a comparator (Ortega et al. 2006) and measured or visually estimated trace lengths for

representative fractures.

The Normalized Correlation Count technique (NCC, Marrett et al. 2018) accounts for the sequence

of fracture spacings along a scanline. NCC uses distances between all pairs of fractures including non-

nearest neighbors. The technique provides a quantitative analysis of the degree to which fractures are

clustered, and can distinguish between even (or periodic) spacing, clusters arising due to random

arrangement, and clustering that is stronger than a random signal. NCC is based on correlation sum, or

the two-point correlation function (Bonnet et al. 2001), that calculates the proportion of fracture pairs

in a set, including pairs of non-neighboring fractures, separated by a distance less than each given

length scale λk in a logarithmically or linearly graduated series of length scales. A correlation count

assigned for a given λk is defined as the fraction of all fracture pairs for which the pair’s spacing falls

between λk+m and λk-m (Marrett et al. 2018), essentially the difference between the correlation sum

of λk+m and that of λk-m.

We used the NCC computer program, CorrCount, which provides analytical and Monte Carlo

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solutions for randomized input spacings and a 95% confidence interval constructed for the

randomized sequence (Marrett et al. 2018). The frequencies are normalized against the expected

frequency for a randomized sequence of the same fracture spacings at each length scale. Where a

length scale’s corresponding correlation count falls outside the upper or lower confidence limits, the

corresponding fracture spacing can be interpreted to be statistically significant.

We also report conventional fracture spacing statistics (Table 1) and a standard measure of spatial

heterogeneity: the coefficient of variation, Cv (e.g. Gillespie et al. 1999). Cv is σ/μ, where σ is the

standard deviation of spacings and μ is the mean.

Results

Fracture types and patterns

Our outcrop primarily contains opening-mode fractures, including quartz veins (Fig. 5b),

apparently non-mineral lined and locally open joints (Figs. 4-6), and fractures associated with

subglacial chatter marks (Fig. 2). One east-west striking right lateral fault with 4 cm displacement was

found. The quartz veins have opening displacements (widths) of from less than 0.1 mm to as much as

1 mm. They are fully sealed with quartz and lack porosity, and have no cement textures evident at

hand specimen scale. These veins have long traces relative to their widths. Aspect ratios locally

approach those of the joints, although some veins are markedly wider and shorter than joints.

Although locally abundant in the outcrop, veins are rare along the scanline trace. Veins dip steeply

and strike north-northwest, north, and east-northeast. Along the scanline, veins differ in strike from

the north-striking joint set we measured, but elsewhere in the southwest quadrant of the exposure,

north-striking joints are subparallel to veins. The quartz veins along the scanline are crosscut by joints

and in general in this outcrop, veins are the oldest fractures. Veins are not included in our spacing

analysis.

Joints are common, although there are large areas of the outcrop that lack fractures of any type

(Fig. 3). Joints crosscut and are readily distinguished from veins. Joints have narrow opening

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displacement, typically near or below values that reliably can be measured with the Ortega et al.

(2006) comparator and hand lens. They are mostly less than 0.1 mm wide (Fig. 4d). Although joint

apertures may have been increased by exhumation or exposure, our measurements put an upper limit

on cumulative aperture sizes along the scanline. Many joint traces are marked by dark seams that are

likely clay-mineral or iron-oxide fills. Some of these have a faintly foliated appearance at hand-lens

scale. These textures may mark minute shear offset. Joints otherwise have no visible mineral fill.

Some joints are surrounded by narrow (5-20 cm) irregular halos of red iron oxide stain, marking past

fluid flow. And locally along joints, direct evidence of modern fluid flow is apparent (Fig. 5d).

Such narrow fractures are visible mainly because of their great trace lengths. Although most joint

traces are long relative to outcrop size, traces visible at hand lens scale range from a few to many tens

of centimeters to fractures that cross the entire outcrop (>50 m). Short joint traces are undersampled in

our 1D analysis. Near our scanline, trace length distributions are censored by outcrop size. About 1

km southeast of our outcrop, one east-northeast-striking fracture zone in basement rocks visible on

Google Earth (centered at 43°41’38’’N, 110°50’40’’W) extends 2 km and can be traced up section

into overlying Paleozoic carbonate rocks. In our outcrop fracture heights cannot be systematically

measured, but fracture traces and outcrop topography imply heights of tens of meters or more for the

longest fractures and fracture zones.

All joint sets in our outcrop have steep dips. Although fracture strike dispersion across the outcrop

is considerable (Fig. 6), at least four preferred joint strikes are present: northwest, north-south, east-

northeast, and west-northwest to east-west (Fig. 6; sets 1-4). Additionally, in nearby exposures, Love

et al. (1992) mapped east-northeast-striking fracture zones (Fig. 6c). The prevalence of various joint

orientations varies across our outcrop. Focusing on fractures near our scanline, we designate the two

most prominent strike directions set 1 (northwest strikes; blue traces on Fig. 5) and set 2 (north

strikes; red traces on Fig. 5). For all sets, many fractures are discontinuous and do not intersect other

fractures. Where fractures intersect, abutting, stepping, and crossing relations (e.g. Hancock 1985)

suggest that east-west-striking joints are locally late. Some abutting relations, as we note below, imply

set 1 fractures predate set 2. Although sets 1 and 2 joints are close to vertical, they have a slight

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tendency to dip steeply east to east-northeast (Figs. 5a, 6a). Otherwise, relative timing between the

joint sets is mostly ambiguous.

Many joints in both sets 1 and set 2 have isolated traces, or are in patterns where two or more

fractures have parallel traces over great distances (many meters). En echelon and left- and right-

stepping patterns are also apparent. Overlapping tip traces for macroscopic fractures tend to be

straight rather than sharply hooked, but close inspection of individual traces shows they are composed

in part of overlapping, gently curved and locally hooked segments, marking small fractures that linked

to form longer fractures (e.g. Olson & Pollard 1989; Lamarche et al. 2018). Traces consequently are

not everywhere fully interconnected along their intersection with the outcrop surface, making length

definition ambiguous and in some instances apparent continuity depends on observation scale. Some

en echelon set 1 fractures are connected at their tips by set 2 fractures (Fig. 5e), resulting in a local

zig-zag pattern of fracture occurrence.

Wing-crack arrays are common and important components of brittle deformation (Willemse &

Pollard 1998). Although no offset markers are evident, some set 1 joints likely have minute fault

displacements based on set 1 in parent configurations relative to set 2 wing cracks (Fig. 5c, d).

Fractures in wing-crack configuration (set 2 relative to set 1 in Fig. 5c, d) abut against and extend at a

shallow angle from the tips of other fractures (set 1 in Fig. 5c, d). Thus, some set 1 fractures striking

between 285 to 310 degrees have north-striking set 2 splays that abruptly diverge from near their tips

at angles of 40 to 60 degrees. This pattern can arise by right slip on weak planes, such as pre-existing

joints. Wing-crack arrays with patterns compatible with left slip are present on some fractures that

strike north-northeast (Fig. 6). Such patterns likely mark slip along pre-existing fractures from which

the splays diverge (e.g. Willemse & Pollard 1998). The wing cracks propagate in the extensional

quadrant of the fault tip. In our pavement, we found little corroborating evidence of slip such as

widespread striations (one example was observed) and no macroscopic evidence of shortening, such

as stylolites, in the contractional quadrants. Based on the configurations, we interpret set 1 to predate

set 2, with some set 2 joints formed as wing cracks, implying a component of right slip on set 1 joints.

The concentration of wing-cracks and other connecting fractures is apparent at a wide range of scales.

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Linking set 2 fractures account for locally dense fracture occurrences (Fig. 5e).

Overall, the prevalence of wing-crack arrays, fracture trace patterns that resemble Riedel shear

configurations (R1, R2, X) (e.g. Tchalenko 1970; Dresen 1991), smaller fractures near joint traces at

acute angles to the main trace, and faintly foliated, pinnate texture, and fine-scale undulations along

many joint traces, and, locally, striations, suggest that many joints in this outcrop have been sheared.

Possibly populations of pre-existing joints as well as some quartz veins were sheared in a consistent

pattern compatible with north-south shortening or east-west extension (Fig. 6). Unsheared north-

striking opening-mode set 2 joints may have formed as part of this deformation.

Our scanline was aligned to cross set 2 because this set is normal to the long dimension of the

outcrop, allowing the longest scanline. Our spatial analysis focused on this set. Set 2 fractures consist

of isolated, parallel fractures and fractures in wing-crack arrays. Fracture strain is low (Table 1). We

did not assess heterogeneity of strain (i.e. Putz-Perrier & Sanderson 2008) because for these

uniformly very narrow joints we do not have systematic aperture size population data.

Near joints the glacially polished surface is commonly eroded (Fig. 3), compatible with weathering

and fluid flow along open joints. However, joints are also present in glacially polished areas. Here,

joint traces appear to abut polished surfaces. We interpret joints to be truncated—or cut by—intact

polished surfaces (Fig. 5b-d). If this interpretation is correct, the joints predate glaciation. This

inference of relative timing is consistent with the lack of parallelism between joint orientations and

current topography and lack of evidence of joint concentrations or alignments relative to past ice flow

directions marked by glacial striations. The youngest fractures in the outcrop are chatter marks due to

glacial action, readily distinguished from joints and veins (Fig. 2). Also present are linear surface

marks (artifacts) caused by post-glacial weathering and erosion (notably, rock fall). These subglacial

and late features characteristically damage the glacial polish that marks much of the surface of the

outcrop.

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Fracture spatial arrangement

We measured 420 fractures over a scanline distance of 180.4 m. For set 2, spacings have a wide

range from less than 0.005 m to more than 34 m. Average set 2 spacing for the entire scanline is 0.43

m., but values are strongly skewed toward narrow spacings (Fig. 3c). A measure of spacing

heterogeneity is coefficient of variation (Cv) where Cv is σ/μ, σ is the standard deviation of spacings

and μ is the mean. For the overall scanline, Cv is high, 4.86. Occurrence versus distance (‘stick’) plots

show anomalously closely spaced fractures separated by large areas where fractures are sparse or

absent (Figs. 3, 7). Figure 7 shows clusters at expanded scale and Figures 8 and 9 show normalized

intensity and NCC plots for the scanlines as a whole and for clusters A and B. Descriptive statistics

and Cv values for the entire scanline and for subdivisions of the scanline are summarized in Table 1

and Figures 8 and 9.

CorrCount outputs include normalized fracture intensity (Fig. 8a) and NCC (Fig. 8b) plots. A

useful procedure for interpreting CorrCount outputs is to first inspect the normalized intensity plot

then analyze the NCC diagram. On one hand, for highly clustered fractures, the intensity plot reliably

but qualitatively marks regions having fracture concentrations. On the other hand, the value of the

NCC plot is such that the degree of clustering can be examined quantitatively across a range of length

scales. The scales that can be examined depend of course on the dataset, but for outcrop and

subsurface fracture and fault data ranges are typically from millimeters or less to several hundred

meters.

We first inspect the normalized intensity plot. For our scanline, two major clusters within the

scanline are apparent (Fig. 8a). Several statistically significant peaks above the upper 95% confidence

limit occur at approximately 30 m, 50 m, 145 m, and 175 m from the beginning of the scanline. The

intensity signals for the rest of the fractures along the scanline lie well within the 95% confidence

interval and their distribution is indistinguishable from random. The most statistically significant

trough (dips beneath the lower 95% confidence limit) is from 50 m to 140 m. Minor troughs occur at

the beginning of the scanline, from 0 m to 25 m, around 44 m, 150 m and 165 m.

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A step-wise procedure for interpreting NCC diagrams is as follows (Fig. 8b). By comparison with

Marrett et al. their fig 12, determine which category of NCC pattern is present. Recognizing that Fig.

8b resembles the ‘fractal cluster’ pattern of Marrett et al., their fig. 12h, we note the following

features for our dataset—1 through 5—on Fig. 8b, discussed in order of increasing length scale. 1: In

the NCC plot (Fig. 8b), a broad, statistically significant elevated section extends from submeter length

scales to approximately 21 m, at which point it falls beneath the upper confidence limit and intersects

the randomized data curve with a spatial correlation of 1 at near 24 m. The elevated section of the

NCC curve indicates spacings at these length scales are more common than they would be for

randomized spacings. Because these are small spacings then clustering is implied. In this section the

NCC values decrease as a power law with increasing length scale on the logarithmic NCC plot,

suggesting a fractal pattern of spacings within the clusters. The slope and intercept of the best-fit line

are characteristics of the pattern.

In the NCC plot (Fig. 8b), the far left of the diagram could be a high-level plateau at small length

scales (~<0.1 m of length scale) indicating that the spacing of fractures inside the cluster is not

organized, although our interpretation is that the pattern marks a power-law decay. The elevated

section exhibits multiple peaks averaging around a slight and gradual negative slope before the roll-

off point at around 15 m, where the curve rapidly decreases to cross the black line for randomized

data. The subsequent trough occurs centered on 54 m. The curve remains under the randomized data

curve with a spatial correlation of 1 between 24 m and 100 m, the latter exceeds the half-length of the

scanline. The range of length scales is small to make a reliable interpretation and close to the point

where NCC cannot be calculated because there are no spacings that fit within the window of length

scales over which NCC is calculated.

In Fig. 8b, 2: the elevated section of the NCC eventually crosses the NCC = 0 value at a length

scale of ~24 m marking the point at which spacings are comparable with the randomized data (Marrett

et al. 2018). This length scale of 24 m is a measure of cluster width. In other words, the NCC

intercept at near 24 m correlates with the width of the single clusters. 3: NCC values for spacings at

length scales greater than 24 m (up to 100 m) lie well below the NCC = 0 line, forming a trough

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centered on 54 m. This zone marks the gap between clusters where spacings at those length scales are

less common than for the randomized data.

The distance between cluster A and cluster B is marked by the right edge of the major trough

where the curve intersects the randomized data curve at around 100 m (cf. L in Fig. 8a). Fig. 8b, 4:

this peak is a measure of cluster spacing and corresponds to the distance between cluster A and B

(Fig. 8a). Because the clusters themselves have a width (rather than a peak at a single length scale) the

peak at point 4 will also tend to be quite broad (from 100 to ~150 m), and in some datasets there may

be several harmonics if multiple clusters are observed (cf. Marrett et al. their fig 12d, f, h).

Although cluster spacing can be detected using logarithmic graduations of length scales, plots of

NCC with linear graduations for the entire scanline more readily reveal if the pattern contains

evidence of ‘regularly-spaced fractal clusters’. NCC peaks marked below for logarithmically

graduated Figures 9b and 9d were obtained using linear graduations of length scales because of the

scarcity of length scales at high values when using logarithmic graduations. Linear plots of length

scale also more clearly show spacing patterns within the clusters. Fig. 8b, 5: nested peaks repeating at

increasing length scale are a mark of clusters-within-clusters. In this example, this signal is noisy, but

the fact that there are apparently nested peaks over ~3 orders of magnitude, overall showing a power-

law decay in NCC, suggests a fractal pattern. Smaller variations like those represented by 5 in Figure

8b can be an artifact of the number of length scales chosen for plotting. This is apparent if one

increases the number of length-scales used to calculate NCC; small variations tend to disappear but

the trend above the 95% confidence interval does not.

In order to assess the robustness of the technique for a given dataset it is necessary to inspect the

randomness curves; the upper and lower confidence limits. At some length scale they ‘pinch’ in

toward the 0 spatial concentration line. This marks the half-length of the scanline (Figs. 2, 8b). NCC

values beyond this point are compromised because less than half of the scanline is available for

counting the number of fracture spacings at those length scales. The limit is the length of the scanline

itself. The overall length of the scanline governs what part of the spacing signal is interpretable. In our

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dataset the cluster spacing is wider than the half-length of the scanline because the clusters, A and B,

are encountered near the beginning and end of the scanline. The effect of this configuration is to

accentuate the amplitude of the trough and peak at 3 and 4 respectively.

Normalized correlation count analysis can be applied to parts, or subdivisions, of the data set (Fig.

9). In the normalized intensity plot for cluster A (Fig. 9a), four neighboring yet distinct, statistically

significant peaks occur near 26 m, 30 m, 32.5 m, and 37.5 m, which form a sub-cluster within cluster

A. Two neighboring peaks occur at near 46 m and 47.5 m, with the former being the largest of all

peaks in the plot, form the other sub-cluster within cluster A. Major troughs showing intervals where

very few fractures were identified occur along the scanline near the beginning of the scanline, 43 m,

and 50-70 m.

The NNC plot for cluster A has a broadly elevated section from submeter length scales to ~3.5 m

indicating fractal spatial organization within clusters, as for the whole scanline but with a weaker

signal, and at a smaller length scale representing clusters within clusters (nested clusters) discussed

for Fig. 8b, point 5. The series of peaks and troughs that pass above and below the randomized data

confidence limits, from 3.5 to ~21 m represent distances between the peaks and sub-clusters in Fig 9a.

The curve dips below NCC = 0 at length scales beyond 21 m because this is longer that the distance

between the first and last sub-cluster (26 m to 47.5 m), that is, the width of cluster A.

In the normalized intensity plot for cluster B (Fig. 9c), peaks are not as distinct and statistically

significant as that in cluster A. The highest peak occur nears 174 m, close to the end of the scanline. A

neighboring peak at around 168 m is also statistically significant. At 143 m to 147 m and 158 m to

163 m intervals, there are two clusters of peaks that are very narrow, barely above or below the 95%

confidence limit, which indicates weak clustering at those intervals. Four troughs occur at near 142 m,

150 m, 159 m, and 171m. Figure 9d may represent fractal clustering close to indistinguishable from

random, although the plot lacks a well-defined low NCC trough (point 4 in Fig. 8b) between cluster

width (point 2 in Figure 8b) and cluster spacing (obtained using linear graduations of length scale).

The area above the 95% confidence interval is not continuous and where it is above the 95%

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confidence interval it is only for a small range of length scales. The pattern overall is suggestive of

fractal clusters but we rate it as nearly indistinguishable from random (Table 1).

Ehlen (2000) described 1D fracture spacing from a large number of different granites. We used

data reported by Ehlen for CorrCount analysis. Results of our analysis for a subset of Ehlen’s

datasets are listed in Table 1 and examples are illustrated in Figure 10.

Discussion

Interpretation of spatial arrangement patterns

That the fractures along our scanline are unevenly spaced and occur in clusters is evident from

visual inspection (Figs. 3, 4). Anomalously closely spaced fractures in narrow arrays are called

fracture swarms or corridors (Laubach et al. 1995; Questiaux et al. 2010; Gabrielsen & Braathan

2014; Miranda et al. 2018; Sanderson & Peacock 2019) but how ‘anomalously’ closely spaced, or

how narrow or sharply defined an array needs to be to constitute a corridor and what constitutes a

swarm or corridor boundary have been qualitative assessments. Fracture patterns that are

indistinguishable from random will show some degree of clustering (e.g. Laubach et al. 2018a, their

fig. 1a). The spacing histogram for our data set (Fig. 3c) merely shows that very closely spaced

fractures are common (mean spacing, 0.43 m), an uninformative result that is one of the drawbacks of

using frequency distributions to interpret spacing patterns (e.g., McGinnis et al. 2015). NCC provides

a way to rigorously identify and quantify corridors.

The ratio of standard deviation of spacings to the mean, the coefficient of variation (Cv) increases

with increasing irregularity of spacing, since periodic spacing produces Cv of zero (Gillespie et al.

1999; 2001) and random locations produce a Cv near 1 (Hooker et al. 2018). For our scanline, Cv is

4.86, compatible with markedly uneven spacing (Table 1). The marked contrast in spacing within and

outside clusters increases the standard deviation. Our Cv value is higher than those of reported for

fractures in some other granites (Table 1). Ehlen (2000) reported mean spacings and standard

deviations for six scanlines in several granites having mean Cv of 1.41. The Cv for her longest

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scanline, however, is close to the value we found for the Cluster A segment of our scanline.

Inspection of the spacing patterns for Ehlen’s scanlines suggests, however, that Cv as expected is not

fully documenting variations in clustering within these examples. The averaged values do not capture

mixtures of highly clustered and more regularly space patterns along the scanline.

Our results are the first to use NCC to describe fracture spatial arrangement in granitic rocks.

Direct NCC comparison with published NCC results for fractured granites is impossible, but in Table

1 we compare the Wyoming example with NCC values we calculated for spacing data sets in granite

reported by Ehlen (2000). Table 1 also reports coefficient of variation (Cv), a measure of spacing

heterogeneity. Although only recently introduced, the NCC method has been used on fractured

carbonate rock outcrops (Marrett et al. 2018), tight gas sandstone horizontal core and image-log data

and outcrops (Li et al. 2018), and faults in sedimentary rocks near detachment faults (Laubach et al.

2018b). Compared with other examples, our dataset is more clustered than those in most sedimentary

and crystalline rock examples described so far (Table 1).

Interpreting our NCC curve using the Marrett et al. terminology, we find that of the eight

characteristic patterns (their fig. 12), ours resembles that of fractal clusters, where locally close

spacings contain smaller self-similar spacing patterns. NCC patterns for the entire scanline (Fig. 8a, b)

show a systematic power-law variation of spatial correlation with length scale that suggest self-

organized clustering arose across a wide range of scales. Within the elevated section, repeated

patterns of peaks, troughs and plateaus of similar width on the log scale (Fig. 8b, point 5) mark fractal

clustering.

Above the 95% confidence line, the slope of the power-law pattern (i.e. exponent of power law)

provides a scale-independent measure of the degree of clustering (Marrett et al. 2018, their fig. 12),

with steeper slopes indicating more intense clustering (i.e. more fractures in clusters at the expense of

inter-cluster regions). Here the slope is relatively high compared to other data sets (Table 1). This

means more clustering at small length scales, suggesting there are clusters within clusters, a result

supported by analysis of individual clusters A and B (Fig. 9).

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Similar, but weaker patterns are evident for subdivisions of the scanline within the two main

clusters (Fig. 9). Cluster spacing within the cluster A is marked by the peaks at around 3 m length

scale. Cluster spacing within the cluster B is marked by the peaks at around 5 m length scale.

Multiples of the dominant spacing are marked by additional peaks separated by intervening troughs (6

m for cluster A; 10 m and 15 m for cluster B). Qualitatively, the flatter, more plateau-like within-

cluster patterns in cluster B suggests that organizational style differs. Overall this sub pattern is nearly

indistinguishable from random, which may mean that the entire dataset has fractal clustering less

intense than for example in the Pedernales data set (Marrett et al. 2018).

Variations in the pattern from one cluster to the next might be compatible with self-organized

clustering modified locally by some external forcing mechanism, such as localization of wing-crack

arrays by preexisting set 1 fractures. Clusters also have a range of shapes, from uniform, ‘corridor’ –

like widths, to those that vary markedly in width along strike. CorrCount can take fracture size (and

other attributes) into account, but we lack the length or aperture measurements that would allow this

analysis. Joints here have a narrow range of aperture sizes, and small differences in length compared

to the scale of the exposure; the spatial organization likely reflects the sequence of fracture spacings.

The scale of our analysis has limits imposed by finite scanline length and, possibly, limited resolution

for small fractures.

The spacing data reported by Ehlen allow us to calculate NCC values for her crystalline rock

examples (Table 1; Fig. 10). Results show a range of patterns. Many of the arrangements are

indistinguishable from random, but some have evidence of hierarchical clustering (Fig. 10a,b). Our

example and Ehlen’s show opening-mode joints in granites having patterns that range from

indistinguishable from random to power-law clustering. This comparison shows that our Wyoming

example fits within the spectrum of clustering found in these other granitic rock scanlines.

Our example and Ehlen’s (2000) observations show why large exposures and methods sensitive to

spacing hierarchy are sometimes needed to adequately document spatial arrangement. Ehlen’s longest

scanline was 74 m and the rest are 40 m long or less. Yet Ehlen noted that along her scanlines—

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qualitatively—patterns varied from regularly spaced to clustered. Because our outcrop is

exceptionally long (>180 m) and free of obscuring cover, marked differences in fracture intensity and

spacing patterns are apparent. If only part of the scanline were exposed, even in fairly large outcrops,

this heterogeneous pattern would be obscured. For example, although from 0 to 50 m fractures are

highly clustered and have an NCC pattern that resembles that of the scanline as a whole, from 160-

175 m the pattern appears more regularly spaced. From 60-140 m the rock is largely unfractured; this

interval gives a misleading view of overall intensity. Together with Ehlen’s scanline data and spacing

patterns and evidence in other granite exposures (e.g. Escuder Viruete et al. 2001), both long

scanlines and sensitive spatial analysis methods are needed to unravel heterogeneous, hierarchical

patterns. Cv does not account for sequences of spacings, and so is an inadequate measure of spacing

irregularity or pattern and cannot capture this hierarchy of patterns.

Correlation count has advantages over other methods that use a cumulative departure from

uniform. For example, the correlation sum and Kuiper’s method using maximum departures from a

uniform distribution (e.g., Sanderson & Peacock 2019) do not reflect the extent to which the pattern is

non-random at that location. Although these values can be computed with CorrCount, Marrett et al.

(2018) recommend the parameter correlation count because the count allows a signal to be captured at

each length scale chosen, rather than up to a given length scale. Correlation count provides more

information because it tells the analyst how much more/less common spacings are than random at a

given length scale. Thus, cluster spacings and widths and different patterns at different scales for the

same data set can be identified. Another advantage of the Marrett et al. (2018) method implemented

in CorrCount is the ability to normalize correlation integral or correlation count to a random rather

than a uniform distribution, with the random distribution calculated analytically or modeled with a

Monte Carlo method.

Parallel, anomalously closely spaced fractures can arise from several processes that depend on

mechanical properties and loading histories (Olson 2004). Without clear evidence of fracture relative

timing or spacing variation with strain (i.e. Putz-Perrier & Sanderson 2008; Hooker et al. 2018), the

origin of clustering in our outcrop cannot be pinned down. In unstratified rock, stress concentrations

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near fracture tips can stimulate development of nearby fractures (Segall & Pollard 1983; Olson 2004).

Numerical models show that propagating fracture tips cause nearby, offset microfractures to grow,

producing tabular, clustered patterns (Olson 1993; 2004). Such a mechanism is compatible with the

segmented pattern of fracture traces in our outcrop.

This process of growth and interaction could lead to self-organization. For granitic rock, likely

possessing high subcritical crack index values (e.g. Nara et al. 2018) and comprising a thick

mechanical unit, densely fractured, widely spaced clusters could be expected. Another process that

could produce clusters are simple fault zones formed as oblique dilatant fractures (splay fractures or

wing-crack arrays) that link non-coplanar faults side-to-side and end-to-end. Closely spaced set 2

fractures connecting tips of set 1 fractures could have formed in this way (Fig. 5d). A similar pattern

of progressive joint and joint-fault development has been described from other granitic rocks in the

western US (Martel 1990; Mazurek et al. 2003).

The origins of joints in the Wyoming outcrop are ambiguous. Fractures primarily accommodating

opening displacement propagate along a plane of zero shear stress in isotropic rock, specifically the

plane perpendicular to the least compressive principal stress (Pollard & Aydin 1988). Unfortunately,

these outcrops provide little basis for determining timing of joint formation or for inferring their

causative stress fields. Crosscutting relations with quartz veins show that joints formed well after

consolidation of the granite, and overlap by glacial polish is compatible with joints formed prior to

glaciation. Steep east to north-east set 2 dips could result from joint formation in rocks prior to post-

Paleozoic gentle (ca. 10-15 degrees) westward tilting of the range (Roberts & Burbank 1996; Leopold

et al. 2007), although the dip pattern is not well defined and could in any case result from other

factors in an igneous mass of unknown dimensions.

These joint sets can be classified as ‘regional’ (Hancock 1985) since they are in otherwise

undeformed rocks more than 2 km from the nearest map-scale folds and faults (Fig. 1). Contrasting

set orientations and relative timing information imply progressive or episodic joint development with

shifting stress orientations. Wing-crack arrays are consistent with some early formed set 1 joints

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reactivating in right slip. Lack of slip direction indicators and uncertain relative ages of fractures

having right- and left slip means the kinematic compatibility of the ensemble of fractures is uncertain.

Without evidence of fracture age, this information still leaves many possibilities for joint origins.

A wide range of loading paths might contribute (e.g. Engelder 1985), including thermoelastic

contraction implicated in formation of some joints in other western US uplifts (English & Laubach

2017). The shearing of the joint pattern is compatible with north-south shortening or east-west

extension. Deformation in these brittle rocks might result from mild bending. The range experienced

tilting, and subtle open folds (flexures) are evident elsewhere in the western Tetons. Regional

correlates for such deformation are uncertain. Attested regional Late Cretaceous to Eocene shortening

directions are east-west, northeast (065 degrees), and, later, broadly east-west extension (Powers

1982).

Joints formed in the currently active Cordilleran extensional stress province (Heidbach et al. 2018)

and kinematically compatible with the active generally north- and north-northeast striking Teton

normal fault would be expected to strike north-south to north-northeast. Recent deformation might

also be responsible for slip on older northwest- and northeast-striking joints. On the other hand, joint

sets in the granite broadly overlap with the strikes of undated joints and veins in overlying Paleozoic

rocks. Temperature patterns of fluid-inclusion assemblages in quartz deposits in joint-like fractures in

nearby Cambrian Flathead sandstone compared with temperatures inferred from burial history are

compatible with north- and northwest-striking fractures in those rocks forming prior to and during

emplacement of the Laramide Buck Mountain reverse fault (Forstner et al. 2018), so similarly

oriented fractures in the underlying granite may have formed contemporaneously.

Ramifications for fractured basement fluid flow

Regular spacing typifies many fracture arrays (e.g. Narr & Suppe 1991; Bai & Pollard 2000) and

regular spacing is sometimes a useful assumption for practical applications (e.g. Wehunt et al. 2017).

But irregular to highly clustered fracture patterns are increasingly recognized as important for

subsurface fractures (Questiaux et al. 2010; Li et al. 2018). Clustered fractures are viewed as a key to

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the hydromechanical properties of crystalline rocks (Gabrielsen & Braathan 2014; Torabi et al. 2018).

Clustered patterns typify joint patterns in many granites (Genter & Castaing 1997; Ehlen 2000). In

granite, both closely spaced joint arrays and patterns arising by development of wing-cracks are

known (Martel 1990; Mazurek et al. 2003). Documenting and quantifying such clustered patterns is a

useful objective.

Our exposures may be an analog for fracture zones in subsurface granites where the effects of

fractures on fluid flow and rock strength are issues. For example, the Lancaster reservoir off Shetland

produces oil from fractured granite, gneiss, and other high-grade metamorphic rocks (Slightam 2014;

Belaidi et al. 2016). Basement rock types here closely resemble Wyoming Province crystalline rocks.

Moreover, the size and shape of the off-Shetland fractured mass is comparable to that of the Teton

Range (cf. Sligtham 2014, and Fig. 1). For the off-Shetland example, seismic data supplemented with

outcrop observations reveals fracture zones in a range of orientations. Some are 15 to 75 m wide,

comparable to the cluster widths we found. Our example, or the NCC method applied to NW Scotland

analogs, could provide insight into the hierarchical arrangement of fracture clusters that may exist at

and below the resolution limits of seismic methods.

Microfracture populations are present in many granites (Anders et al. 2014). In some granites, a

wide range of lengths can be described with power laws (Bertrand et al. 2015). We cannot rule out

that microfractures are present in our outcrop, although the fracture set we measured appears to have a

narrow aperture size range. Indeed, the narrow aperture size of visible fractures in this set falls within

some definitions of microfracture although the generally long trace lengths do not. In our outcrop the

lack of staining away from large joints suggest little fluid flow through microfractures (those small

enough to require microscopy to detect; Anders et al. 2014), a pattern that contrasts with Lancaster

field, where microfractures are conduits for flow and storage (Belaidi et al. 2016).

For assessing fracture networks, 1D observations are frequently all that can be obtained from the

subsurface, and 1D data are a useful element in pattern assessment and as a link between subsurface

observations and the 2- and 3D descriptions and topological analysis of geometric as well as spatial

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characteristics of networks, including connectivity (Sanderson & Nixon 2015) or to seismic

observations (Hu et al. 2018). Our scanline site is notable for providing a clear and readily accessible

view of joint clustering in granitic rocks and as a demonstration of the NCC method. Fracture patterns

in this exposure and in other granites have considerable 2- and 3D complexity over a range of scales

(Le Garzic et al. 2011; Bertrand et al. 2015), so our 1D scanline results are only a first step toward 3D

descriptions where cluster dimensions and shapes can be rigorously defined (e.g., discussion in

Marrett et al. 2018). Statistically defined cluster boundaries such as those noted in Figure 8 provide

non-arbitrary criteria for delineating dimensions of clusters (swarms, corridors).

Conclusions

Self-organized clustering over a length scale range from 10-2 to 101 m is indicated for north-

striking opening-mode fractures (joints) in Late Archean Mount Owen Quartz Monzonite by analysis

with the recently developed CorrCount software and the NCC method. These are regional joint

patterns in a structural setting >2 km from folds and faults. Using a protocol for interpreting NCC and

intensity diagrams, we find joint clusters are a few to tens of meters wide, and are separated by areas

as much as 70 m wide that are largely devoid of fractures. Within large clusters are smaller self-

similar spacing patterns.

By normalizing frequencies against the expected frequency for a randomized sequence of the same

spacings at each length scale, the degree of clustering and margins of clusters (or corridors) of

anomalously’ closely spaced, narrow arrays are rigorously defined. Compared with a few clustered

patterns in sedimentary rocks that have been described with the NCC method, out example is

markedly more clustered, as shown by high exponents and steeper slopes above 95 percent confidence

limits. Our pattern may reflect propagation of corridor-like process-zone joint clusters by the

mechanism described by Olson (2004) and local wing-crack patterns forced near the tips of pre-

existing fractures, a process common in other joint arrays in granite (e.g. Martel 1990).

Our NCC analysis of spacing datasets from other fractured granites, based on measurements

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previously published by Ehlen (2000), finds patterns that range from indistinguishable from random

to marked hierarchical clustering, similar to those in the Wyoming example we report. These

examples illustrate the utility of the NCC method over other approaches for describing spatial

arrangement, quantifying clustering, and identifying patterns that are statistically more clustered than

random.

Acknowledgements Our research on fracture pattern development is funded by grant DE-FG02-

03ER15430 from Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy

Sciences, Office of Science, U.S. Department of Energy, and by the Fracture Research and

Application Consortium. We are grateful for research access to the Jedidiah Smith Wilderness to

Diane Wheeler and Jay Pence, U.S. Forest Service and to Grand Teton National Park for permission

under National Park Service permits GRTE-2014-SCI-0021 and GRTE-2017-SCI-0072. Stephanie R.

Forstner and Finn Tierney assisted with field work. I.A.M., E.C., and A.M. Laubach provided

essential field support. We appreciate anonymous reviewer comments on the manuscript, comments

on the research from L. Gomez and J.N. Hooker and insights from members of the 2018 trans-Tetons

field trip, including Sergio Sarmiento, Tiago Miranda, Thiago Falcão, Marcus Ebner, Rodrigo

Correia, and Lane Boyer.

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Figure captions

Fig. 1. Location of scanline within the Teton Range, northwestern Wyoming. (a) Geology,

highlighting Late Archean Mount Owen Quartz Monzonite (Wg) and location of scanline (inset, b).

Modified after Zartman & Reed 1998 and Love et al. 1992. TF, Teton fault; BMF, Buck Mountain

reverse fault; FPRF, Forellen Peak reverse fault. Elevation of outcrop is 2589 m, about 80 m vertically

below basal Cambrian unconformity. (b) Upper Teton Canyon, GoogleEarth image, showing large

exposure (around red diamond labeled scanline) north of Teton Canyon trail (dash-dot line) (location,

inset, Fig. 1a). Our example of clustering is in a readily accessible exposure. (c) View of outcrop

looking southeast; prominent peak in center of image is Buck Mountain. P, polished surface of

granodiorite. W, outcrop with polished surface weathered away. Field of view of pavement ~40 m.

Fig. 2. Linear and curvilinear surficial features. (a) Linear scratches (Sc) caused by debris (Cl) falling

on outcrop. (b) Subglacial features. Curved chatter mark fractures (CM) and striations (ST) on

glacially polished surfaces. These features locally superficially resemble and must be distinguished

from veins and joints.

Fig. 3. Scanline outcrop and fracture occurrence versus distance (stick) plot. Scanline bearing 259

degrees and measured from east-northeast to west-southwest. Start location 43°42‘15.3"N;

110°52‘12.5"W, north of Teton Canyon trail (Fig. 1). Scanline was offset 30 m north-northwest

parallel to itself at 90 m to keep line within continuous outcrop. (a) Panoramic outcrop view looking

north and east. Measured scanline extends beyond the rocks visible. (b) For entire scanline, fracture

occurrence versus distance. Two clusters are apparent: A, B. Fracture indicator, 1 means fracture is

present. For a few fractures that systematically are at about 60 degrees to the scanline, from 51 to 53

m, lack of correction introduces slight inaccuracies in spacing values. Overall scanline uncertainty is

low (Santos et al. 2015). (c) Histogram of fracture spacings.

Fig. 4. Joints showing clustering (Fz) at various scales. (a) View southwest near cluster B. Fz marks

zone of closely spaced fractures. (b) Narrow zone of closely spaced fractures (Fz) near northwest edge

of cluster A. View northeast. (c) Isolated fracture, F, and three closely spaced fractures (box, Fz)

between clusters A and B. (d) Closely spaced fractures, Fz. Arrow marks west edge of zone. Scale,

cm.

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Fig. 5. Fracture trace patterns (annotated photographs). (a) Set 2 in cross section, view northwestward.

Note steep northeast dip, nd. Fz, cluster. (b) Set 1 joints in parallel arrays. Qv, quartz veins cross cut

by joints. Scanline, s.l. (c) Wing-crack-like joint arrays, wc, of set 2 (strike ~0 to 10°) splaying off set

1 (~strike 290°). Scanline, s.l. View north-northwest (350°) (d) Wing-crack-like joint arrays, wc, of

set 2 extending from set 1. View northwest. Dense set 1 array is adjacent to Cluster B, but off

scanline. (e) Joint with copious water flow. Example is from joint in gneiss at Snowdrift Lake

(43°24‘30"N; 110°49‘16"W).

Fig. 6. Fracture orientations, lower hemisphere equal angle plots. (a) Quartz veins; n = 6. (b) Vein, v,

cut by joints s1, s2. (c) Prominent joints in outcrop surrounding scanline. Set 1 west-northwest (285-

320 degrees) and set 2 striking broadly north-south (350-015 degrees) are common; east-west and

northeast-striking joints are also present. n = 94. (d) Selected joints in wing-crack arrays, n = 7. Map

patterns of subsidiary west-northwest- and east-northeast-striking joint arrays are compatible with left

and right slip but measurable striations were not found. (e) Rose diagram, showing east-northeast

strikes of linear features—probable fractures or fracture zones—extracted from Love et al. (1992)

map in Mount Owen granite outcrops in upper Teton Canyon.

Fig. 7. Fracture occurrence versus distance for subdivisions of scanline, at expanded scale, set 2. (a)

Cluster A, from 20 to 70 m, n = 264. Location of one subsidiary cluster within this interval is marked

(Cluster A-c). (b) Cluster B, from 140 to 175 m, n = 152. Location of one subsidiary cluster within

this interval is marked (Cluster B-d). (c) Vicinity of cluster A-c, at expanded scale, 28 to 32 m; n =

50. Fine-scale clustering is evident. (d) Vicinity of cluster B-d, at expanded scale, 158-162 m, n = 17.

Fine-scale clustering is evident.

Fig. 8. CorrCount results, spatial arrangement analysis for opening-mode fractures (set 2 joints),

entire scanline. (a) Normalized intensity. Cluster A and B indicated. L, area with values less common

than random (see Fig. 8b and text). (b) Normalized correlation count (NCC). Highlighted areas mark

parts of curve exceeding 95 percent confidence interval. Numbers 1 – 5 refer to comments in text. For

1, slope is indicated by black line, offset above the curve for clarity, and by the dotted red line on the

curve. For 5, grey circles mark three of the elevated sections of the curve. Scanline ends at 180.4 m;

length scales beyond 0.5 scanline length, or 90 m, are expected to have statistically lower correlation

counts. An estimated cluster spacing implies a typical value of cluster spacing, and the peak labeled #

4 in 8b could be suggestive of that. Cluster spacing detected at length scales larger than half of the

scanline might be the result of two clusters but could nevertheless be statistically significant;

clustering has to be more clear (less random) to be detected. A reason for classifying the 8b NCC

pattern as a ‗fractal cluster‘ is not just that the values are above the 95% confidence interval, but that

the pattern is systematic, and follows a well-defined power law for ~3 orders of magnitude with a

well-defined intercept of the NCC curve with NCC=1. A cluster near the end of a scanline does not

capture the space around the cluster, leading to an edge artifact. Cluster width wider than half length

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of the scanline is likely not captured, leading to an effect similar to censoring on cluster width. Cluster

width on NCC plot is likely shifted left due to this artifact, marked by peak at right end of NCC plot.

Fig. 9. CorrCount results, spatial arrangement analysis for opening-mode fractures (set 2 joints),

subsets of scanline. (a) Normalized intensity, Cluster A. (b) Normalized correlation count (NCC),

Cluster A. (c) Normalized intensity, Cluster B. (d) Normalized correlation count, Cluster B.

Highlighted areas mark parts of curve exceeding 95 percent confidence interval. Peak at around 30 m

could also be interpreted as a measure of cluster spacing.

Fig. 10. CorrCount results for scanline spacing data in crystalline rock reported in Ehlen (2000). (a)

Normalized intensity, Ehlen joint set 1. (b) Normalized correlation count, Ehlen joint set 1. Pattern

resembles clustered fractures Fig. 8. (c) Normalized intensity, Ehlen joint set 2. (d) Normalized

correlation count, Ehlen joint set 2. Pattern shows arrangements indistinguishable from random.

Scanline data sets labeled as in Ehlen (2000). Results for all NCC analysis of data sets reported by

Ehlen are in Table 1. In our scanline and in Ehlen‘s (2000) data, over different parts of the scanlines,

patterns range from fractal clustering to indistinguishable from random. These are examples of

potentially meaningful signals that can be found from interrogating patterns by position, analysis that

can be extended to include feature size, age, or other attributes if observations are available.

Table 1. Fracture statistics and comparison with other fracture patterns

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Table 1. Fracture statistics and comparison with other fracture patterns

Scanline

length (m)

Number Spacing (m) Strain Cv NCC

type*

Power-law at small length scale

Cluster

width (m)

Qu

alµ Max. Min. Avg. Exponent Coefficient

Teton scanline 180.4 420 34 0.005 0.43 2×10-6† 4.86 12h -0.27 3.28 24 1

Cluster A 50 262 9.53 0.005 0.19 5.2×10-3† 3.23 12h -0.24 1.96 ca. 5 2

Cluster B 35 143 3.12 0.01 0.24 4.1×10-3† 1.99 12h *** *** ca. 2 2

Comparison datasets

Pedernales1 59 916 2.26 0.00008 0.064 5.3×10-3 2.43 12h -0.27 1.5 17 2

Core2 18 40 >1 ~0.1 0.42 2.2×10-3 1.65 12e, h n.c. n.c. n.c. 3

Image log2 450 370 >5 ~0.5 1.15 8.2×10-4 4.2 12h -1.7 0.66 35 3

Faults3 124.8 431 n.c.†† n.c. n.c. 3.5×10-3 2.37 12e -6.6 0.94 14 4

Microfractures4 * 4 to 499 * * * 1×10−4 to

2×10−1

0.54 -

3.69

n.c.†† n.c. n.c. n.c. 4

Sandstone 5 362.6 254 0.54 54.5 8.4 n.c. 1.80 n.c. n.c. n.c. n.c. 1

Granite 16 32 353 0.84 0.01 0.11 n.c. 0.93# 12e, h -1.43 3.10 4^ 3

Granite 26 37 217 3.4 0.01 0.21 n.c. 1.89# 12h -0.97 9.05 6 5

Granite 36 27 191 0.96 0.01 0.14 n.c. 1.14# 12h -1.14 4.7 3 5

Granite 46 13 207 0.4 0.01 0.06 n.c. 1.16# 12h -1.47 0.81 6^ 5

Granite 56 40 200 1.58 0.01 0.2 n.c. 1.24# 12h -1.03 8.31 4^ 3

Granite 66 74 367 5.53 0.01 0.21 n.c. 2.10# 12h -1.01 14.03 5 1

*Marrett et al., their fig. 12; †Assuming all apertures: 0.001 mm, calculated as (sum of apertures)/(sum of spaces); ††n.c., Not calculated. 1Carbonate rock, Marrett et al.

2018; 2Core & image log data, Cretaceous sandstone, set 1, Li et al., 2018; 3Faults, Miocene sandstone; Laubach et al. 2018b; 4Microfractures, various sandstones,

Hooker et al., 2018; avg. Cv 1.4; *59 scanlines; ***Close to indistinguishable from random; 5Outcrop, Cretaceous sandstone, Laubach, 1991; 6Outcrop, various granites,

Ehlen 2000; µQualitative clustering degree, 1, high; 5, low; #Cv calculated by CorrCount; ^weak signal

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