Quantified fracture (joint) clustering in Archean basementWyoming application of the normalized correlation count method

Qiqi Wang12 S E Laubach2 J F W Gale2 amp M J Ramos121 Department of Geological Sciences University of Texas at Austin 2275 Speedway Stop C9000 Austin TX 78712 USA2 Bureau of Economic Geology University of Texas at Austin 10100 Burnet Road Austin TX 78713 USA

SEL 0000-0003-2511-9414 MJR 0000-0003-1457-2125Correspondence wangqiqiutexasedu

Abstract We demonstrate statistically significant self-organized clustering over a length scale range from 10minus2 to 101 m fornorth-striking opening-mode fractures ( joints) in Late Archean Mount Owen Quartz Monzonite Spatial arrangement is acritical fracture network attribute that until recently has only been assessed qualitatively We use normalized fracture intensityplots and the normalized correlation count (NCC) method of Marrett et al to discriminate clustered from randomly placed orevenly spaced patterns quantitatively over a wide range of length scales and to test the statistical significance of the resultingpatterns We propose a procedure for interpreting cluster patterns on NCC diagrams generated by the freely available spatialanalysis software CorrCount Results illustrate the efficacy of NCC to measure fracture clustering patterns in texturallyhomogeneous Archean granitic rock in a setting distant (gt2 km) from folds or faults In their current geological setting theseregional fractures are conduits for water flow and their patterns ndash and the NCC approach to defining clusters ndash may be usefulguides to the spatial arrangement style and clustering magnitude of conductive fractures in other less accessible fracturedbasement rocks

Received 12 October 2018 revised 12 March 2019 accepted 13 May 2019

Fracture spatial arrangement is a key aspect of the structuralheterogeneity of the brittle crust (Bear et al 2012 Laubach et al2018a) Understanding the spatial arrangement of opening-modefractures ( joints and veins) has great practical importance Forexample fracture-enhanced permeability is important in aquifers(De Dreuzy et al 2001 Henriksen amp Braathen 2006 Maillot et al2016) waste repositories (Barton amp Hsieh 1989 Carey et al 2015)hydrothermal systems (Sanderson et al 1994 Cox et al 2001Hobbs amp Ord 2018) and petroleum reservoirs (Laubach 2003 Galeet al 2014) including those in basement rocks (Cuong amp Warren2009 Belaidi et al 2018 Bonter et al 2018) Typical of fracture-controlled permeability is tremendous heterogeneity and anisotropyin fluid flow (eg Halihan et al 2000 Solano et al 2011) Althoughfracture network permeability is governed by many factorsincluding aperture and length distributions and connectivity andby whether fractures are open or sealed a common cause ofpermeability variability is heterogeneous spatial arrangement Herewe use a recently introduced method to describe fracture spatialarrangement in a large exposure of texturally uniform Precambriangranitic rock in Wyoming

lsquoSpatial arrangementrsquo describes fracture positions includingwhether fractures are clustered randomly placed or evenly spaced(Laubach et al 2018a) In conventional usage spacing is theperpendicular distance between two fractures in the same set (Priestamp Hudson 1976 Ladeira amp Price 1981 Narr amp Suppe 1991Gillespie et al 1993) but the prevalence of highly clusteredfractures is increasingly being recognized and what constitutes auseful quantitative measure of clustering is a matter of currentinterest (Questiaux et al 2010 Roy et al 2014 Sanderson ampPeacock 2019) For a set of co-planar fractures a common datacollection method is to measure spacing along a straight line ofobservation (1D scanline) orthogonal to fracture strike Thenormalized correlation count (NCC) method (Marrett et al 2018)accounts for sequences of spacings including non-neighbourspacings The technique was implemented with the freely available

software CorrCount accessible via the open-access Marrett et al(2018) paper We compare results with conventional spacinganalysis methods descriptive statistics and the ratio of standarddeviation to mean or coefficient of variation (Cv)

Here we demonstrate statistically significant self-organizedclustering over a length scale range from 10minus2 to 101 m in jointsin texturally homogeneous Archean granitic rock in a setting distant(gt2 km) from folds or faults These joints are currently conduits forwater flow and their patterns may be useful guides to the spatialarrangement of fractures capable of transmitting fluids in other lessaccessible fractured basement rocks Our example illustrates howNCC can quantify spatial arrangement over a wide range of lengthscales measure the pattern and degree of clustering and test thestatistical significance of the results We use the same method toassess spatial arrangement patterns in six other previously publishedfractured granite datasets finding a range of degrees of clusteringOurs is the first application of the NCC technique to fractures ingranitic rocks Results from our example and analysis of datasetsfrom the literature show that opening-mode joints in granites havepatterns that range from power-law clustering to indistinguishablefrom random

Geological setting

The Teton Range in NWWyoming is a NNE-trending normal-fault-bounded block of Precambrian crystalline rocks and dominantlywest-dipping CambrianndashMesozoic sedimentary rocks (Love et al1992RobertsampBurbank1993 SmithampSiegel 2000) (Fig 1) To theeast the range is bounded by the east-dipping Teton normal fault(Love et al 1992) a structure having Cenozoicndashrecent movement(Smith et al 1993 Byrd et al 1994 Leopold et al 2007) Within therange are north- to NW-striking reverse faults of probable earlyTertiary (Laramide) age (Love et al 1978 Lageson 1987)

Basement rocks in the Teton Range (Bradley 1956 Love et al1992 Reed amp Zartman 1973 Zartman amp Reed 1998) include

copy 2019 The Author(s) This is an Open Access article distributed under the terms of the Creative Commons Attribution 40 License (httpcreativecommonsorglicensesby40) Published by The Geological Society of London for GSL and EAGE Publishing disclaimer wwwgeolsocorgukpub_ethics

Thematic setNaturally fractured reservoirs Petroleum Geoscience

Published Online First httpsdoiorg101144petgeo2018-146

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complexly deformed Archean gneiss migmatite and metasedimen-tary rocks of theWyoming Province (Chamberlain et al 2003) LateArchean mostly unfoliated granitic intrusive rocks (Mount OwenQuartz Monzonite Reed amp Zartman 1973) and eastndashwest-strikingdiabase dykes Our data are from the upper part of Teton Canyon ingranitic rocks mapped as Mount Owen Quartz Monzonite (Reed ampZartman 1973 Love et al 1992) (Fig 1) The unit is a silica-richperaluminous leucogranite (Bagdonas et al 2016 Frost et al 2018)forming a discordant pluton (255 Ga Mount Owen batholith Frost

et al 2018) with margins marked by irregular bodies and dykes ofpegmatite and aplite and angular wall-rock inclusions (xenoliths)The exposures we studied are unfoliated medium to fine texturallyuniform light-coloured granite comprising quartz microcline andsodic plagioclase with trace biotite and muscovite

The smoothly undulating nearly complete exposures we studiedretain a fine striated glacial polish except when adjacent to somejoints where post-glacial erosion has removed the polished surface(Figs 1ndash5) Locally visible are SE- and NW-facing crescent-shaped

Fig 1 Location of the scanline within the Teton Range NW Wyoming (a) Geology highlighting the Late Archean Mount Owen Quartz Monzonite (Wg)and the location of the scanline (inset b) Modified after Love et al (1992) and Zartman amp Reed (1998) TF Teton Fault BMF Buck Mountain reversefault FPRF Forellen Peak reverse fault The elevation of the outcrop is 2589 m about 80 m vertically below the basal Cambrian unconformity (b) UpperTeton Canyon GoogleEarth image showing the large exposure (around the red diamond labelled lsquoScanlinersquo) north of the Teton Canyon trail (dash-dot line)(location a and inset b) Our example of clustering is in a readily accessible exposure (c) View of the outcrop looking SE the prominent peak in the centreof the image is Buck Mountain P polished surface of the granodiorite W outcrop with the polished surface weathered away The field of view of thepavement is c 40 m

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chatter mark (fracture) arrays caused by subglacial deformation andlinear features on glaciated surfaces caused by post-glacial to recentrock fall (Fig 2a b) The glacial and recent geomorphic history ofthe range is summarized by Foster et al (2010)

Methods

Our fracture orientation and relative timing measurements weremade in an outcrop that is c 01 km2 (Fig 1) The scanline wasmeasured with a compass and 30 m plastic tape (centimetregraduations) We orientated the scanline 259deg in a direction thatmaximized the extent of a vegetation- and debris-free outcrop wheremeasurements could be made at approximately uniform elevation(Fig 3) The scanline is normal to the strike of a main steeplydipping joint set and our analysis focuses on these fracturesScanline location was based on the largest extent of well-exposedrock not to maximize the number of intersections Becausefractures we measured have long trace lengths relative to outcropsize moving the scanline around or adding scanlines within thewindow of good exposure would yield no significant differences inthe resulting pattern Although across the outcrop the strike of thisset locally has as much as a 10deg range the configuration of thescanline relative to fracture strike means that Terzaghi (1965)corrections of spacing are small Measurement imprecision of

scanline length and fracture placement arises from compliance ofthe tape non-uniform tension under field conditions and the non-planar undulating character of the outcrop (Figs 3 and 4) We judgethese errors to be small compared to the scale of the pattern capturedin the scanline We measured kinematic aperture (openingdisplacement) for representative fractures using a comparator(Ortega et al 2006) and measured or visually estimated tracelengths for representative fractures

The NCC technique (Marrett et al 2018) accounts for thesequence of fracture spacings along a scanline NCC uses distancesbetween all pairs of fractures including non-nearest neighboursThe technique provides a quantitative analysis of the degree towhich fractures are clustered and can distinguish between even (orperiodic) spacing clusters arising due to random arrangement andclustering that is stronger than a random signal NCC is based on thecorrelation sum or the two-point correlation function (Bonnet et al2001) that calculates the proportion of fracture pairs in a setincluding pairs of non-neighbouring fractures separated by adistance less than each given length scale λk in a logarithmically orlinearly graduated series of length scales A correlation countassigned for a given λk is defined as the fraction of all fracture pairsfor which the pairrsquos spacing falls between λk +m and λkminusm(Marrett et al 2018) essentially the difference between thecorrelation sum of λk +m and that of λkminusm

We used the NCC computer program CorrCount which providesanalytical andMonte Carlo solutions for randomized input spacingsand a 95 confidence interval constructed for the randomizedsequence (Marrett et al 2018) The frequencies are normalizedagainst the expected frequency for a randomized sequence of thesame fracture spacings at each length scale Where a length scalersquoscorresponding correlation count falls outside the upper or lowerconfidence limits the corresponding fracture spacing can beinterpreted to be statistically significant

We also report conventional fracture spacing statistics (Table 1)and a standard measure of spatial heterogeneity the coefficient ofvariation Cv (eg Gillespie et al 1999) Cv is σμ where σ is thestandard deviation of spacings and μ is the mean

Results

Fracture types and patterns

Our outcrop primarily contains opening-mode fractures includingquartz veins (Fig 5b) apparently non-mineral lined and locallyopen joints (Figs 4ndash6) and fractures associated with subglacial

Fig 2 Linear and curvilinear surficial features (a) Linear scratches (Sc)caused by debris (Cl) falling on the outcrop (b) Subglacial featuresCurved chatter mark fractures (CM) and striations (ST) on glaciallypolished surfaces These features locally superficially resemble and mustbe distinguished from veins and joints

Fig 3 Scanline outcrop and fracture occurrence v distance (stick) plot The scanline is bearing 259deg and is measured from ENE to WSW Start location43deg 42prime 153Prime N 110deg 52prime 125Prime W north of the Teton Canyon trail (Fig 1) The scanline was offset 30 m NNW parallel to itself at 90 m to keep linewithin a continuous outcrop (a) Panoramic outcrop view looking north and east The measured scanline extends beyond the rocks visible (b) For the entirescanline fracture occurrence v distance Two clusters are apparent A and B Fracture indicator 1 means that a fracture is present For a few fractures thatsystematically are at about 60deg to the scanline from 51 to 53 m a lack of correction introduces slight inaccuracies in the spacing values The overallscanline uncertainty is low (Santos et al 2015) (c) Histogram of fracture spacings

Fracture clustering in basement rocks

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chatter marks (Fig 2) One eastndashwest-striking right-lateral fault with4 cm displacement was found The quartz veins have openingdisplacements (widths) of from lt01 mm to as much as 1 mmThey are fully sealed with quartz and lack porosity and have nocement textures evident at hand-specimen scale These veins havelong traces relative to their widths Aspect ratios locally approachthose of the joints although some veins are markedly wider andshorter than joints Although locally abundant in the outcrop veinsare rare along the scanline trace Veins dip steeply and strike NNWnorth and ENE Along the scanline veins differ in strike from thenorth-striking joint set we measured elsewhere in the SW quadrantof the exposure north-striking joints are subparallel to veins Thequartz veins along the scanline are cross-cut by joints and in generalin this outcrop veins are the oldest fractures Veins are not includedin our spacing analysis

Joints are common although there are large areas of the outcropthat lack fractures of any type (Fig 3) Joints cross-cut and arereadily distinguished from veins Joints have narrow openingdisplacement typically near or below values that reliably can bemeasured with the Ortega et al (2006) comparator and hand lensThey are mostly lt01 mm wide (Fig 4d) Although joint aperturesmay have been increased by exhumation or exposure our

measurements put an upper limit on cumulative aperture sizesalong the scanline Many joint traces are marked by dark seams thatare likely to be clay-mineral or iron-oxide fills Some of these have afaintly foliated appearance at the hand-lens scale These texturesmay mark minute shear offset Joints otherwise have no visiblemineral fill Some joints are surrounded by narrow (5ndash20 cm)irregular halos of red iron oxide stain marking past fluid flow Inaddition locally along joints direct evidence of modern fluid flowis apparent (Fig 5d)

Such narrow fractures are visible mainly because of their greattrace lengths Although most joint traces are long relative to theoutcrop size traces visible at hand-lens scale range from a few tomany tens of centimetres to fractures that cross the entire outcrop(gt50 m) Short joint traces are undersampled in our 1D analysisNear our scanline trace length distributions are censored by outcropsize About 1 km SE of our outcrop one ENE-striking fracture zonein basement rocks visible on Google Earth (centred at 43deg 41prime 38PrimeN110deg 50prime 40PrimeW) extends for 2 km and can be traced upsection intooverlying Paleozoic carbonate rocks In our outcrop fracture heightscannot be systematically measured but fracture traces and outcroptopography imply heights of tens of metres or more for the longestfractures and fracture zones

Fig 4 Joints showing clustering (Fz) at various scales (a) View SW near cluster B Fz marks the zone of closely spaced fractures (b) Narrow zone ofclosely spaced fractures (Fz) near the NW edge of cluster A View NE (c) Isolated fracture (F) and three closely spaced fractures (box Fz) between clustersA and B (d) Closely spaced fractures (Fz) The arrow marks the western edge of the zone Scale centimetres

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All joint sets in our outcrop have steep dips Although fracturestrike dispersion across the outcrop is considerable (Fig 6) at leastfour preferred joint strikes are present NW northndashsouth ENE andWNW to eastndashwest (Fig 6 sets 1ndash4) In addition in nearbyexposures Love et al (1992) mapped ENE-striking fracture zones(Fig 6c) The prevalence of various joint orientations varies acrossour outcrop Focusing on fractures near our scanline we designatethe two most prominent strike directions set 1 (NW strikes bluetraces in Fig 5) and set 2 (north strikes red traces in Fig 5) For allsets many fractures are discontinuous and do not intersect otherfractures Where fractures intersect abutting stepping and crossingrelationships (eg Hancock 1985) suggest that eastndashwest-strikingjoints are locally late Some abutting relationships as we notebelow imply set 1 fractures predate set 2 Although sets 1 and 2joints are close to vertical they have a slight tendency to dip steeplyeast to ENE (Figs 5a and 6a) Otherwise the relative timing betweenthe joint sets is mostly ambiguous

Many joints in both sets 1 and 2 have isolated traces or are inpatterns where two or more fractures have parallel traces over greatdistances (many metres) En echelon and left- and right-steppingpatterns are also apparent Overlapping tip traces for macroscopicfractures tend to be straight rather than sharply hooked but closeinspection of individual traces shows that they are composed in partof overlapping gently curved and locally hooked segmentsmarking small fractures that linked to form longer fractures

(eg Olson amp Pollard 1989 Lamarche et al 2018) Tracesconsequently are not everywhere fully interconnected along theirintersection with the outcrop surface making length definitionambiguous and in some instances the apparent continuity dependson the observation scale Some en echelon set 1 fractures areconnected at their tips by set 2 fractures (Fig 5e) resulting in a localzigzag pattern of fracture occurrence

Wing-crack arrays are common and important components ofbrittle deformation (Willemse amp Pollard 1998) Although no offsetmarkers are evident some set 1 joints are likely to have minute faultdisplacements based on set 1 in parent configurations relative to set2 wing cracks (Fig 5c d) Fractures in wing-crack configurations(set 2 relative to set 1 in Fig 5c d) abut against and extend at ashallow angle from the tips of other fractures (set 1 in Fig 5c d)Thus some set 1 fractures striking between 285deg and 310deg havenorth-striking set 2 splays that abruptly diverge from near their tipsat angles of 40degndash60deg This pattern can arise by right slip on weakplanes such as pre-existing joints Wing-crack arrays with patternscompatible with left slip are present on some fractures that strikeNNE (Fig 6) Such patterns are likely to mark slip along pre-existing fractures from which the splays diverge (eg Willemse ampPollard 1998) The wing cracks propagate in the extensionalquadrant of the fault tip In our pavement we found littlecorroborating evidence of slip such as widespread striations (oneexample was observed) and no macroscopic evidence of shortening

Fig 5 Fracture trace patterns(annotated photographs) (a) Set 2 incross-section view towards the NWNote the steep NE dip (nd) Fz cluster(b) Set 1 joints in parallel arrays Qvquartz veins cross-cut by joints slscanline (c) Wing-crack-like jointarrays (WC) of set 2 (strike c 0deg to10deg) splaying off set 1 (strike c 290deg)sl scanline View NNW (350deg)(d) Wing-crack-like joint arrays (WC)of set 2 extending from set 1 ViewNW Dense set 1 array is adjacent tocluster B but off the scanline (e) Jointwith copious water flow The exampleis from a joint in the gneiss atSnowdrift Lake (43deg 24prime 30Prime N110deg 49prime 16Prime W)

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such as stylolites in the contractional quadrants Based on theconfigurations we interpret set 1 to predate set 2 with some set 2joints formed as wing cracks implying a component of right slip onset 1 joints The concentration of wing cracks and other connectingfractures is apparent at a wide range of scales Linking set 2 fracturesaccount for locally dense fracture occurrences (Fig 5e)

Overall the prevalence of wing-crack arrays fracture tracepatterns that resemble Riedel shear configurations (R1 R2 X)(eg Tchalenko 1970 Dresen 1991) smaller fractures near jointtraces at acute angles to the main trace and faintly foliated pinnatetexture and fine-scale undulations along many joint traces andlocally striations suggest that many joints in this outcrop have beensheared Possibly populations of pre-existing joints as well as somequartz veins were sheared in a consistent pattern compatible withnorthndashsouth shortening or eastndashwest extension (Fig 6) Unshearednorth-striking opening-mode set 2 joints may have formed as part ofthis deformation

Our scanline was aligned to cross set 2 because this set is normalto the long dimension of the outcrop allowing the longest scanlineOur spatial analysis focused on this set Set 2 fractures consist ofisolated parallel fractures and fractures in wing-crack arraysFracture strain is low (Table 1) We did not assess the heterogeneityof strain (ie Putz-Perrier amp Sanderson 2008) because for theseuniformly very narrow joints we do not have systematic aperturesize population data

Near joints the glacially polished surface is commonly eroded(Fig 3) compatible with weathering and fluid flow along openjoints However joints are also present in glacially polished areasHere joint traces appear to abut polished surfaces We interpretjoints to be truncated ndash or cut by ndash intact polished surfaces (Fig 5bndashd) If this interpretation is correct the joints predate glaciation Thisinference of relative timing is consistent with the lack of parallelismbetween joint orientations and current topography and the lack ofevidence of joint concentrations or alignments relative to past ice-flow directions marked by glacial striations The youngest fracturesin the outcrop are chatter marks due to glacial action readilydistinguished from joints and veins (Fig 2) Also present are linearsurface marks (artefacts) caused by post-glacial weathering anderosion (notably rock fall) These subglacial and late features

characteristically damaged the glacial polish that marks much of thesurface of the outcrop

Fracture spatial arrangement

We measured 420 fractures over a scanline distance of 1804 m Forset 2 spacings have a wide range from lt0005 m to more than34 m Average set 2 spacing for the entire scanline is 043 m butvalues are strongly skewed towards narrow spacings (Fig 3c)A measure of spacing heterogeneity is the coefficient of variation(Cv) where Cv is σμ σ is the standard deviation of spacings andμ is the mean For the overall scanline Cv is high 486 Occurrencev distance (lsquostickrsquo) plots show anomalously closely spaced fracturesseparated by large areas where fractures are sparse or absent (Figs 3and 7) Figure 7 shows clusters at expanded scale and Figures 8 and9 show normalized intensity and NCC plots for the scanlines as awhole and for clusters A and B Descriptive statistics and Cv valuesfor the entire scanline and for subdivisions of the scanline aresummarized in Table 1 and in Figures 8 and 9

CorrCount outputs include normalized fracture intensity (Fig 8a)and NCC (Fig 8b) plots A useful procedure for interpretingCorrCount outputs is to first inspect the normalized intensity plotand then analyse the NCC diagram On the one hand for highlyclustered fractures the intensity plot reliably but qualitatively marksregions with fracture concentrations On the other hand the value ofthe NCC plot is such that the degree of clustering can be examinedquantitatively across a range of length scales The scales that can beexamined depend of course on the dataset however for outcropand subsurface fractures and faults data ranges are typically frommillimetres or less to several hundred metres

We first inspect the normalized intensity plot For ourscanline two major clusters within the scanline are apparent(Fig 8a) Several statistically significant peaks above the upper 95confidence limit occur at c 30 50 145 and 175 m from thebeginning of the scanline The intensity signals for the rest of thefractures along the scanline lie well within the 95 confidenceinterval and their distribution is indistinguishable from random Themost statistically significant trough (dips beneath the lower 95confidence limit) is from 50 to 140 m Minor troughs occur at the

Table 1 Fracture statistics and comparison with other fracture patterns

Scanlinelength (m) Number

Spacing (m)

Strain Cv NCC typea

Power law at smalllength scale

Clusterwidth (m) QualbMax Min Avg Exponent Coefficient

Teton scanline 1804 420 34 0005 043 2 times 10minus6c 486 12h minus027 328 24 1Cluster A 50 262 953 0005 019 52 times 10minus3c 323 12h minus024 196 c 5 2Cluster B 35 143 312 001 024 41 times 10minus3c 199 12h d d c 2 2Comparison datasetsPedernales1 59 916 226 000008 0064 53 times 10minus3 243 12h minus027 15 17 2Core2 18 40 gt1 c 01 042 22 times 10minus3 165 12e h nce nc nc 3Image log2 450 370 gt5 c 05 115 82 times 10minus4 42 12h minus17 066 35 3Faults3 1248 431 nc nc nc 35 times 10minus3 237 12e minus66 094 14 4Microfractures4 f 4ndash499 f f f 1 times 10minus4ndash2 times 10minus1 054ndash369 nc nc nc nc 4Sandstone5 3626 254 054 545 84 nc 180 nc nc nc nc 1Granite 16 32 353 084 001 011 nc 093g 12e h minus143 310 4h 3Granite 26 37 217 34 001 021 nc 189g 12h minus097 905 6 5Granite 36 27 191 096 001 014 nc 114g 12h minus114 47 3 5Granite 46 13 207 04 001 006 nc 116g 12h minus147 081 6h 5Granite 56 40 200 158 001 02 nc 124g 12h minus103 831 4h 3Granite 66 74 367 553 001 021 nc 210g 12h minus101 1403 5 1

aMarrett et al (2018 fig 12) bQualitative clustering degree 1 high 5 low cAssuming all apertures are 0001 mm calculated as (sum of apertures)(sum of spaces) dClose toindistinguishable from random enc not calculated f59 scanlines gCv calculated by CorrCount hWeak signal Comparison datasets 1Carbonate rock (Marrett et al 2018) 2Core andimage log data Cretaceous sandstone set 1 (Li et al 2018) 3Faults Miocene sandstone (Laubach et al 2018b) 4Microfractures various sandstones (Hooker et al 2018) average(avg) Cv 14 5Outcrop Cretaceous sandstone (Laubach 1991) 6Outcrop various granites (Ehlen 2000)

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

A stepwise procedure for interpreting NCC diagrams is as follows(Fig 8b) By comparison with Marrett et al (2018 fig 12)determine which category of NCC pattern is present Recognizingthat Figure 8b resembles the lsquofractal clusterrsquo pattern of Marrett et al(2018 fig 12h) we note the following features for our dataset(1ndash5 in Fig 8b) discussed in order of increasing length scaleFigure 8b 1 in the NCC plot a broad statistically significantelevated section extends from submetre length scales to c 21 m atwhich point it falls beneath the upper confidence limit and intersectsthe randomized data curve with a spatial correlation of 1 at near24 m The elevated section of the NCC curve indicates that spacingsat these length scales are more common than they would be forrandomized spacings Because these are small spacings clusteringis implied In this section the NCC values decrease as a power lawwith increasing length scale on the logarithmic NCC plotsuggesting a fractal pattern of spacings within the clusters Theslope and intercept of the best-fit line are characteristics of thepattern

In the NCC plot (Fig 8b) the far left of the diagram could be ahigh-level plateau at small length scales (c lt01 m of length scale)indicating that the spacing of the fractures inside the cluster is notorganized although our interpretation is that the pattern marks apower-law decay The elevated section exhibits multiple peaksaveraging around a slight and gradual negative slope before the roll-off point at around 15 m where the curve rapidly decreases to crossthe black line for randomized data The subsequent trough occurscentred on 54 m The curve remains under the randomized datacurve with a spatial correlation of 1 between 24 and 100 m the latterexceeds the half-length of the scanline The range of length scales issmall to make a reliable interpretation and close to the point whereNCC cannot be calculated because there are no spacings that fitwithin the window of length scales over which NCC is calculated

In Figure 8b 2 the elevated section of the NCC eventuallycrosses the NCC = 0 value at a length scale of c 24 m marking thepoint at which spacings are comparable with the randomized data(Marrett et al 2018) This length scale of 24 m is a measure ofcluster width In other words the NCC intercept near 24 mcorrelates with the width of the single clusters In Figure 8b 3 NCCvalues for spacings at length scales of gt24 m (up to 100 m) lie wellbelow the NCC = 0 line forming a trough centred on 54 m Thiszone marks the gap between clusters where spacings at those lengthscales are less common than for the randomized data

The distance between cluster A and cluster B is marked by theright edge of the major trough where the curve intersects therandomized data curve at around 100 m (cf L in Fig 8a) Figure 8b4 this peak is a measure of cluster spacing and corresponds to thedistance between cluster A and B (Fig 8a) Because the clustersthemselves 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 c150 m) and in some datasets there may be several harmonics ifmultiple clusters are observed (cf Marrett et al 2018 fig 12d f h)

Although cluster spacing can be detected using logarithmicgraduations of length scales plots of NCC with linear graduationsfor the entire scanline more readily reveal whether the patterncontains evidence of lsquoregularly-spaced fractal clustersrsquo NCC peaksmarked below for logarithmically graduated Figure 9b and d wereobtained using linear graduations of length scales because of thescarcity of length scales at high values when using logarithmicgraduations Linear plots of length scale also show more clearly thespacing patterns within the clusters Figure 8b 5 nested peaksrepeating at increasing length scale are a mark of clusters withinclusters In this example this signal is noisy but the fact that thereare apparently nested peaks over approximately three orders ofmagnitude overall showing a power-law decay in NCC suggests afractal pattern Smaller variations like those represented by 5 inFigure 8b can be an artefact of the number of length scales chosenfor plotting This is apparent if one increases the number of lengthscales used to calculate NCC small variations tend to disappear butthe trend above the 95 confidence interval does not

In order to assess the robustness of the technique for a given datasetit is necessary to inspect the randomness curves the upper and lowerconfidence limits At some length scales they lsquopinchrsquo in towards the 0spatial concentration line This marks the half-length of the scanline(Figs 2 and 8b) NCC values beyond this point are compromisedbecause less than half of the scanline is available for counting thenumber of fracture spacings at those length scales The limit is thelength of the scanline itself The overall length of the scanline governswhich part of the spacing signal is interpretable In our dataset thecluster spacing is wider than the half-length of the scanline becausethe clusters A and B are encountered near the beginning and end ofthe scanline The effect of this configuration is to accentuate theamplitude of the trough and peak at 3 and 4 respectively

Normalized correlation count analysis can be applied to parts orsubdivisions of the dataset (Fig 9) In the normalized intensity plot

Fig 6 Fracture orientations lower-hemisphere equal-angle plots(a) Quartz veins n = 6 (b) Vein (v) cut by joints s1 and s2 (c) Prominentjoints in the outcrop surrounding the scanline Set 1 WNW (285degndash320deg)and set 2 striking broadly northndashsouth (350degndash015deg) are common eastndashwest- and NE-striking joints are also present n = 94 (d) Selected joints inwing-crack arrays n = 7 Map patterns of subsidiary WNW- and ENE-striking joint arrays are compatible with left and right slip but measurablestriations were not found (e) Rose diagram showing ENE strikes of linearfeatures ndash probable fractures or fracture zones ndash extracted from the Loveet al (1992) map of Mount Owen granite outcrops in the upper TetonCanyon

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for cluster A (Fig 9a) four neighbouring yet distinct statisticallysignificant peaks occur near 26 30 325 and 375 m which form asubcluster within cluster A Two neighbouring peaks occur near46 and 475 m with the former being the largest of all peaks in theplot and form the other subcluster within cluster A Major troughsshowing intervals where very few fractures were identified occuralong the scanline near the beginning of the scanline and at 43 and50ndash70 m

The NNC plot for cluster A has a broadly elevated section fromsubmetre length scales to c 35 m indicating a fractal spatialorganization within clusters as for the whole scanline but with aweaker signal and at a smaller length scale representing clusterswithin clusters (nested clusters) discussed for Figure 8b point5 The series of peaks and troughs that pass above and below therandomized data confidence limits from 35 to c 21 m representdistances between the peaks and subclusters in Figure 9a The curvedips below NCC = 0 at length scales beyond 21 m because this islonger than the distance between the first and last subcluster(26ndash475 m) that is the width of cluster A

In the normalized intensity plot for cluster B (Fig 9c) peaks arenot as distinct and statistically significant as that in cluster A Thehighest peak occurs nears 174 m close to the end of the scanline Aneighbouring peak at around 168 m is also statistically significantAt the 143ndash147 m and 158ndash163 m intervals there are two clustersof peaks that are very narrow barely above or below the 95confidence limit which indicates weak clustering at those intervalsFour troughs occur at near 142 150 159 and 171 m Figure 9d mayrepresent fractal clustering close to indistinguishable from randomalthough the plot lacks a well-defined low NCC trough (point 4 inFig 8b) between cluster width (point 2 in Fig 8b) and clusterspacing (obtained using linear graduations of length scale) The area

above the 95 confidence interval is not continuous and where it isabove the 95 confidence interval it is only for a small range oflength scales The pattern overall is suggestive of fractal clusters butwe rate it as nearly indistinguishable from random (Table 1)

Ehlen (2000) described 1D fracture spacing from a large numberof different granites We used data reported by Ehlen for CorrCountanalysis Results of our analysis for a subset of Ehlenrsquos datasets arelisted 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 occurin clusters is evident from visual inspection (Figs 3 and 4)Anomalously closely spaced fractures in narrow arrays are calledfracture swarms or corridors (Laubach et al 1995 Questiaux et al2010 Gabrielsen amp Braathen 2014 Miranda et al 2018Sanderson amp Peacock 2019) but how lsquoanomalouslyrsquo closelyspaced or how narrow or sharply defined an array needs to be toconstitute a corridor and what constitutes a swarm or corridorboundary have been qualitative assessments Fracture patterns thatare indistinguishable from random will show some degree ofclustering (eg Laubach et al 2018a fig 1a) The spacing histogramfor our dataset (Fig 3c) merely shows that very closely spacedfractures are common (mean spacing of 043 m) an uninformativeresult that is one of the drawbacks of using frequency distributions tointerpret spacing patterns (eg McGinnis et al 2015) NCC providesa way to rigorously identify and quantify corridors

The ratio of standard deviation of spacings to the mean thecoefficient of variation (Cv) increases with increasing irregularity

Fig 7 Fracture occurrence v distance for subdivisions of the scanline at expanded scale set 2 (a) Cluster A from 20 to 70 m n = 264 The location ofone subsidiary cluster within this interval is marked (cluster A-c) (b) Cluster B from 140 to 175 m n = 152 The location of one subsidiary cluster withinthis 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 ofcluster B-d at expanded scale 158ndash162 m n = 17 Fine-scale clustering is evident

Q Wang et al

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of spacing since periodic spacing produces Cv values of zero(Gillespie et al 1999 2001) and random locations produce a Cvnear 1 (Hooker et al 2018) For our scanline the Cv is 486compatible with markedly uneven spacing (Table 1) The markedcontrast in spacing within and outside clusters increases the standarddeviation Our Cv value is higher than those reported for fractures insome other granites (Table 1) Ehlen (2000) reported mean spacingsand standard deviations for six scanlines in several granites with amean Cv of 141 The Cv for her longest scanline however is closeto the value we found for the cluster A segment of our scanlineInspection of the spacing patterns for the Ehlen (2000) scanlinessuggests however that the Cv as expected is not fullydocumenting the variations in clustering within these examplesThe averaged values do not capture mixtures of highly clustered andmore regularly space patterns along the scanline

Our results are the first to use NCC to describe fracture spatialarrangement in granitic rocks Direct NCC comparison withpublished NCC results for fractured granites is impossible but inTable 1 we compare the Wyoming example with NCC values wecalculated for spacing datasets in granite reported by Ehlen (2000)Table 1 also reports the Cv a measure of spacing heterogeneityAlthough only recently introduced the NCC method has been usedon fractured carbonate rock outcrops (Marrett et al 2018) tight gassandstone horizontal core and image-log data and outcrops (Li et al2018) and faults in sedimentary rocks near detachment faults(Laubach et al 2018b) Compared with other examples our datasetis more clustered than those in most sedimentary and crystallinerock examples described so far (Table 1)

Interpreting our NCC curve using the Marrett et al (2018)terminology we find that of the eight characteristic patterns (theirfig 12) ours resembles that of fractal clusters where locally closespacings contain smaller self-similar spacing patterns NCC patternsfor the entire scanline (Fig 8a b) show a systematic power-lawvariation of spatial correlation with length scale which suggests thatself-organized clustering arose across a wide range of scales Withinthe elevated section repeated patterns of peaks troughs and

plateaus of similar width on the log scale (Fig 8b point 5) markfractal clustering

Above the 95 confidence line the slope of the power-lawpattern (ie exponent of power law) provides a scale-independentmeasure of the degree of clustering (Marrett et al 2018 fig 12)with steeper slopes indicating more intense clustering (ie morefractures in clusters at the expense of intercluster regions) Here theslope is relatively high compared to other datasets (Table 1) Thismeans more clustering at small length scales suggesting there areclusters within clusters a result supported by the analysis ofindividual clusters A and B (Fig 9)

Similar but weaker patterns are evident for subdivisions of thescanline within the two main clusters (Fig 9) Cluster spacingwithin cluster A is marked by the peaks at about 3 m length scaleCluster spacing within cluster B is marked by the peaks at about 5 mlength scale Multiples of the dominant spacings are marked byadditional peaks separated by intervening troughs (6 m for clusterA 10 and 15 m for cluster B) Qualitatively the flatter moreplateau-like within-cluster patterns in cluster B suggest that theorganizational style differs Overall this subpattern is nearlyindistinguishable from random which may mean that the entiredataset has fractal clustering that is less intense than for example inthe Pedernales dataset (Marrett et al 2018)

Variations in the pattern from one cluster to the next might becompatible with self-organized clustering modified locally by someexternal forcing mechanism such as the localization of wing-crackarrays by pre-existing set 1 fractures Clusters also have a range ofshapes from uniform lsquocorridorrsquo-like widths to those that varymarkedly in width along strike CorrCount can take fracture size(and other attributes) into account but we lack the length or aperturemeasurements that would allow this analysis Joints here have anarrow range of aperture sizes and small differences in lengthcompared to the scale of the exposure the spatial organization islikely to reflect the sequence of fracture spacings The scale of ouranalysis has limits imposed by finite scanline length and possiblylimited resolution for small fractures

Fig 8 CorrCount results spatial arrangement analysis for opening-mode fractures (set 2 joints) entire scanline (a) Normalized intensity Cluster A and Bindicated L area with values less common than random (see b and text) (b) Normalized correlation count (NCC) Highlighted areas mark parts of thecurve exceeding 95 confidence interval Numbers 1ndash5 refer to comments in the text For 1 the slope is indicated by the black line offset above the curvefor clarity and by the dotted red line on the curve For 5 grey circles mark three of the elevated sections of the curve The scanline ends at 1804 m lengthscales beyond half scanline length or 90 m are expected to have statistically lower correlation counts An estimated cluster spacing implies a typical valueof cluster spacing and the peak labelled 4 in (b) could be suggestive of that Cluster spacing detected at length scales larger than half of the scanline mightbe the result of two clusters but could nevertheless be statistically significant clustering has to be more clear (less random) to be detected A reason forclassifying the NCC pattern in (b) as a lsquofractal clusterrsquo is not just that the values are above the 95 confidence interval but that the pattern is systematicand follows a well-defined power law for c 3 orders of magnitude with a well-defined intercept of the NCC curve with NCC = 1 A cluster near the end of ascanline does not capture the space around the cluster leading to an edge artefact Cluster width wider than half length of the scanline is likely not to becaptured leading to an effect similar to censoring on cluster width Cluster width on the NCC plot is likely to be shifted left due to this artefact marked bypeak at the right-hand end of the NCC plot

Fracture clustering in basement rocks

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The spacing data reported by Ehlen (2000) allow us to calculateNCC values for her crystalline rock examples (Table 1 Fig 10)Results show a range of patterns Many of the arrangements areindistinguishable from random but some have evidence ofhierarchical clustering (Fig 10a b) Our example and those ofEhlen (2000) show opening-mode joints in granites having patternsthat range from indistinguishable from random to power-lawclustering This comparison shows that our Wyoming examplefits within the spectrum of clustering found in these other graniticrock scanlines

Our example and the observations in Ehlen (2000) show whylarge exposures and methods sensitive to spacing hierarchy aresometimes needed to adequately document the spatial arrangementThe longest scanline in Ehlen (2000) was 74 m and the rest are 40 mlong or less Yet Ehlen (2000) noted that along her scanlines ndashqualitatively ndash patterns varied from regularly spaced to clusteredBecause our outcrop is exceptionally long (gt180 m) and free ofobscuring cover marked differences in fracture intensity andspacing patterns are apparent If only part of the scanline wereexposed even in fairly large outcrops this heterogeneous patternwould be obscured For example although from 0 to 50 m fracturesare highly clustered and have an NCC pattern that resembles that ofthe scanline as a whole from 160 to 175 m the pattern appears moreregularly spaced From 60 to 140 m the rock is largely unfracturedthis interval gives a misleading view of the overall intensityTogether with Ehlenrsquos (2000) scanline data and spacing patternsand evidence in other granite exposures (eg Escuder Viruete et al2001) both long scanlines and sensitive spatial analysis methodsare needed to unravel heterogeneous hierarchical patterns The Cvdoes not account for sequences of spacings and so is an inadequate

measure of spacing irregularity or pattern and cannot capture thishierarchy of patterns

Correlation count has advantages over other methods that use acumulative departure from uniform For example the correlationsum and Kuiperrsquos method using maximum departures from auniform distribution (eg Sanderson amp Peacock 2019) do not reflectthe extent to which the pattern is non-random at that locationAlthough these values can be computed with CorrCount Marrettet al (2018) recommended the parameter correlation count becausethe count allows a signal to be captured at each length scale chosenrather than up to a given length scale Correlation count providesmore information because it tells the analyst how much morelesscommon spacings are than random at a given length scale Thuscluster spacings and widths and different patterns at different scalesfor the same dataset can be identified Another advantage of theMarrett et al (2018) method implemented in CorrCount is theability to normalize the correlation integral or correlation count to arandom rather than a uniform distribution with the randomdistribution calculated analytically or modelled with a MonteCarlo method

Parallel anomalously closely spaced fractures can arise fromseveral processes that depend on mechanical properties and loadinghistories (Olson 2004) Without clear evidence of fracture relativetiming or spacing variation with strain (ie Putz-Perrier ampSanderson 2008 Hooker et al 2018) the origin of clustering inour outcrop cannot be pinned down In unstratified rock stressconcentrations near fracture tips can stimulate the development ofnearby fractures (Segall amp Pollard 1983 Olson 2004) Numericalmodels show that propagating fracture tips cause nearby offsetmicrofractures to grow producing tabular clustered patterns (Olson

Fig 9 CorrCount results spatial arrangement analysis for opening-mode fractures (set 2 joints) subsets of scanline (a) Normalized intensity clusterA (b) Normalized correlation count (NCC) cluster A (c) Normalized intensity cluster B (d) Normalized correlation count cluster B Highlighted areasmark parts of the curve exceeding the 95 confidence interval The peak at around 30 m could also be interpreted as a measure of cluster spacing

Q Wang et al

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1993 2004) Such a mechanism is compatible with the segmentedpattern of fracture traces in our outcrop

This process of growth and interaction could lead to self-organization For granitic rock probably possessing high subcriticalcrack index values (eg Nara et al 2018) and comprising a thickmechanical unit densely fractured widely spaced clusters could beexpected Another process that could produce clusters are simplefault zones formed as oblique dilatant fractures (splay fractures orwing-crack arrays) that link non-coplanar faults side to side and endto end Closely spaced set 2 fractures connecting tips of set 1fractures could have formed in this way (Fig 5d) A similar patternof progressive joint and joint-fault development has been describedfrom other granitic rocks in the western USA (Martel 1990Mazurek et al 2003)

The origins of joints in the Wyoming outcrop are ambiguousFractures primarily accommodating opening displacement propa-gate along a plane of zero shear stress in isotropic rock specificallythe plane perpendicular to the least compressive principal stress(Pollard ampAydin 1988) Unfortunately these outcrops provide littlebasis for determining the timing of joint formation or for inferringtheir causative stress fields Cross-cutting relationships with quartzveins show that joints formed well after consolidation of the graniteand overlap by glacial polish is compatible with joints formed priorto glaciation Steep east to NE set 2 dips could result from jointformation in rocks prior to post-Paleozoic gentle (c 10degndash15deg)westwards tilting of the range (Roberts amp Burbank 1993 Leopoldet al 2007) although the dip pattern is not well defined and could inany case result from other factors in an igneous mass of unknowndimensions

These joint sets can be classified as lsquoregionalrsquo (Hancock 1985)since they are in otherwise undeformed rocks more than 2 km fromthe nearest map-scale folds and faults (Fig 1) Contrasting setorientations and relative timing information imply progressive orepisodic joint development with shifting stress orientations Wing-crack arrays are consistent with some early formed set 1 jointsreactivating in right slip The lack of slip direction indicators anduncertain relative ages of fractures having right and left slipmeans thatthe kinematic compatibility of the ensemble of fractures is uncertain

Without evidence of fracture age this information still leavesmany possibilities for the joint origins A wide range of loadingpaths might contribute (eg Engelder 1985) including thermo-elastic contraction implicated in the formation of some joints in otherwestern USA uplifts (English amp Laubach 2017) The shearing of thejoint pattern is compatible with northndashsouth shortening or eastndashwestextension Deformation in these brittle rocks might result from mildbending The range experienced tilting and subtle open folds(flexures) are evident elsewhere in the western Tetons Regionalcorrelates for such deformation are uncertain Attested regional LateCretaceousndashEocene shortening directions are eastndashwest NE (065deg)and later broadly eastndashwest extension (Powers 1982)

Joints formed in the currently active Cordilleran extensionalstress province (Heidbach et al 2018) and kinematically compatiblewith the active generally north- and NNE-striking Teton normalfault would be expected to strike northndashsouth to north to NNERecent deformation might also be responsible for slip on older NW-and NE-striking joints On the other hand joint sets in the granitebroadly overlap with the strikes of undated joints and veins inoverlying Paleozoic rocks Temperature patterns of fluid-inclusion

Fig 10 CorrCount results for scanline spacing data in crystalline rock reported in Ehlen (2000) (a) Normalized intensity Ehlen joint set 1 (b) Normalizedcorrelation count Ehlen joint set 1 The pattern resembles clustered fractures (Fig 8) (c) Normalized intensity Ehlen joint set 2 (d) Normalized correlationcount Ehlen joint set 2 The pattern shows arrangements indistinguishable from random Scanline datasets labelled as in Ehlen (2000) Results for all NCCanalysis of datasets reported by Ehlen (2000) are in Table 1 In our scanline and in Ehlenrsquos (2000) data over different parts of the scanlines patterns rangefrom fractal clustering to indistinguishable from random These are examples of potentially meaningful signals that can be found from interrogating patternsby position analysis that can be extended to include feature size age or other attributes if observations are available

Fracture clustering in basement rocks

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assemblages in quartz deposits in joint-like fractures in nearbyCambrian Flathead Sandstone compared with temperatures inferredfrom burial history are compatible with north- and NW-strikingfractures in those rocks forming prior to and during emplacement ofthe Laramide Buck Mountain reverse fault (Forstner et al 2018) sosimilarly orientated fractures in the underlying granite may haveformed contemporaneously

Ramifications for fractured basement fluid flow

Regular spacing typifies many fracture arrays (eg Narr amp Suppe1991 Bai et al 2000) and regular spacing is sometimes a usefulassumption for practical applications (eg Wehunt et al 2017) Butirregular to highly clustered fracture patterns are increasinglyrecognized as important for subsurface fractures (Questiaux et al2010 Li et al 2018) Clustered fractures are viewed as a key to thehydromechanical properties of crystalline rocks (Gabrielsen ampBraathen 2014 Torabi et al 2018) Clustered patterns typify jointpatterns in many granites (Genter amp Castaing 1997 Ehlen 2000) Ingranite both closely spaced joint arrays and patterns arising fromthe development of wing-cracks are known (Martel 1990 Mazureket al 2003) Documenting and quantifying such clustered patterns isa useful objective

Our exposures may be an analogue for fracture zones insubsurface granites where the effects of fractures on fluid flowand rock strength are issues For example the Lancaster reservoir offShetland produces oil from fractured granite gneiss and other high-grade metamorphic rocks (Slightam 2014 Belaidi et al 2018)Basement rock types here closely resemble Wyoming Provincecrystalline rocks Moreover the size and shape of the off-Shetlandfractured mass is comparable to that of the Teton Range (cfSlightam 2014 and Fig 1) For the off-Shetland example seismicdata supplemented with outcrop observations reveal fracture zonesin a range of orientations Some are 15ndash75 m wide comparable tothe cluster widths we found Our example or the NCC methodapplied to NW Scotland analogues could provide insight into thehierarchical arrangement of fracture clusters that may exist at andbelow the resolution limits of seismic methods

Microfracture populations are present in many granites (Anderset al 2014) In some granites a wide range of lengths can bedescribed with power laws (Bertrand et al 2015) We cannot ruleout that microfractures are present in our outcrop although thefracture set we measured appears to have a narrow aperture sizerange Indeed the narrow aperture size of visible fractures in this setfalls within some definitions of microfractures although thegenerally long trace lengths do not In our outcrop the lack ofstaining away from large joints suggests that little fluid flow hasoccurred through microfractures (those small enough to requiremicroscopy to detect Anders et al 2014) a pattern that contrastswith the Lancaster Field where microfractures are conduits for flowand storage (Belaidi et al 2018)

For assessing fracture networks 1D observations are frequentlyall that can be obtained from the subsurface and 1D data are usefulelements in pattern assessment and as a link between subsurfaceobservations and the 2D and 3D descriptions and topologicalanalysis of geometrical as well as spatial characteristics ofnetworks including connectivity (Sanderson amp Nixon 2015) or toseismic observations (Hu et al 2018) Our scanline site is notablefor providing a clear and readily accessible view of joint clusteringin granitic rocks and as a demonstration of the NCC methodFracture patterns in this exposure and in other granites have aconsiderable 2D and 3D complexity over a range of scales(Le Garzic et al 2011 Bertrand et al 2015) and so our 1Dscanline results are only a first step towards 3D descriptions wherecluster dimensions and shapes can be rigorously defined (egdiscussion in Marrett et al 2018) Statistically defined cluster

boundaries such as those noted in Figure 8 provide non-arbitrarycriteria for delineating dimensions of clusters (swarms corridors)

Conclusions

Self-organized clustering over a length scale range from 10minus2 to101 m is indicated for north-striking opening-mode fractures( joints) in Late Archean Mount Owen Quartz Monzonite byanalysis with the recently developed CorrCount software and theNCCmethod These are regional joint patterns in a structural settinggt2 km from folds and faults Using a protocol for interpreting NCCand intensity diagrams we find that joint clusters are a few to tens ofmetres wide and are separated by areas as much as 70 m wide thatare largely devoid of fractures Within large clusters are smaller self-similar spacing patterns

By normalizing frequencies against the expected frequency for arandomized sequence of the same spacings at each length scale thedegree of clustering and the margins of clusters (or corridors) ofanomalously closely spaced narrow arrays are rigorously definedCompared with a few clustered patterns in sedimentary rocks thathave been described with the NCC method out example ismarkedly more clustered as shown by high exponents and steeperslopes above the 95 confidence limits Our pattern may reflect thepropagation of corridor-like process-zone joint clusters by themechanism described by Olson (2004) and local wing-crackpatterns forced near the tips of pre-existing fractures a processcommon in other joint arrays in granite (eg Martel 1990)

Our NCC analysis of spacing datasets from other fracturedgranites based on measurements previously published by Ehlen(2000) finds patterns that range from indistinguishable fromrandom to marked hierarchical clustering similar to those in theWyoming examplewe report These examples illustrate the utility ofthe NCC method over other approaches for describing the spatialarrangement quantifying clustering and identifying patterns that arestatistically more clustered than random

Acknowledgements We are grateful for research access to the JedidiahSmith Wilderness to Diane Wheeler and Jay Pence the United States ForestService and to Grand Teton National Park for permission under National ParkService permits GRTE-2014-SCI-0021 and GRTE-2017-SCI-0072 StephanieR Forstner and Finn Tierney assisted with fieldwork IAM Laubach ECLaubach and AM Laubach provided essential field support We appreciateanonymous reviewer comments on the manuscript comments on the researchfrom L Gomez and JN Hooker and insights from members of the 2018 trans-Tetons field trip including Sergio Sarmiento Tiago Miranda Thiago FalcatildeoMarcus Ebner Rodrigo Correia and Lane Boyer

Funding Our research on fracture pattern development is funded by grantDE-FG02-03ER15430 from Chemical Sciences Geosciences and BiosciencesDivision Office of Basic Energy Sciences Office of Science United StatesDepartment of Energy and by the Fracture Research and ApplicationConsortium

ReferencesAnders MH Laubach SE amp Scholz CH 2014 Microfractures a review

Journal of Structural Geology 69B 377ndash394 httpsdoiorg101016jjsg201405011

Bagdonas DA Frost CD amp Fanning CM 2016 The origin of extensiveNeoarchean high-silica batholiths and the nature of intrusive complements tosilicic ignimbrites Insights from theWyoming batholith USA The AmericanMineralogist 101 1332ndash1347 httpsdoiorg102138am-2016-5512

Bai T Pollard DD amp Gao H 2000 Explanation for fracture spacing in layeredmaterials Nature 403 753ndash756 httpsdoiorg10103835001550

Barton CC amp Hsieh PA 1989 Physical and Hydrologic Flow Properties ofFractures American Geophysical Union Field Trip Guidebook T385

Bear J Tsang CF amp DeMarsily G 2012 Flow and Contaminant Transport inFractured Rock Academic Press San Diego CA

Belaidi A Bonter DA Slightam C amp Trice RC 2018 The Lancaster Fieldprogress in opening the UKrsquos fractured basement play In Bowman M ampLevell B (eds) Petroleum Geology of NW Europe 50 Years of Learning ndashProceedings of the 8th Petroleum Geology Conference Geological SocietyLondon 385ndash398 httpsdoiorg101144PGC820

Q Wang et al

by guest on December 10 2020httppglyellcollectionorgDownloaded from

Bertrand L Geacuteraud Y Le Garzic E Place J Diraison M Walter B ampHaffen S 2015 A multiscale analysis of a fracture pattern in granite A casestudy of the Tamariu granite Catalunya Spain Journal of Structural Geology78 52ndash66 httpsdoiorg101016jjsg201505013

Bonnet E Bour O Odling NE Davy P Main I Cowie P amp Berkowitz B2001 Scaling of fracture systems in geological media Reviews of Geophysics29 347ndash383 httpsdoiorg1010291999RG000074

Bonter D Trice R Cavalleri C Delius H amp Singh K 2018 Giant oildiscovery west of Shetland ndash challenges for fractured basement formationevaluation Paper presented at the SPWLA 59th Annual Logging Symposium2ndash6 June 2018 London UK

Bradley CC 1956 The Precambrian complex of Grand Teton National ParkWyoming In Berg RR (ed) Jackson Hole 11th Annual Field ConferenceGuidebook Wyoming Geological Association Casper WY 34ndash42

Byrd JO Smith RB amp Geissman JW 1994 The Teton fault Wyomingtopographic signature neotectonics and mechanisms of deformation Journalof Geophysical Research Solid Earth 99 20 095ndash20 122 httpsdoiorg10102994JB00281

Carey JW Lei Z Rougier E Mori H amp Viswanathan H 2015 Fracturendashpermeability behavior of shale Journal of Unconventional Oil and GasResources 11 27ndash43 httpsdoiorg101016jjuogr201504003

Chamberlain KR Frost CD amp Frost BR 2003 Early Archean toMesoproterozoic evolution of the Wyoming Province Archean origins tomodern lithospheric architecture Canadian Journal of Earth Sciences 401357ndash1374 httpsdoiorg101139e03-054

Cox SF Knackstedt MA amp Braun J 2001 Principles of structural control onpermeability and fluid flow in hydrothermal systems Reviews in EconomicGeology 14 1ndash24 httpsdoiorg101130REG14-p1

Cuong TX amp Warren JK 2009 Back Ho Field a fractured granitic basementreservoir Cuu Long basin offshore SE Vietnam a lsquoburied-hillrsquo play Journalof Petroleum Geology 32 129ndash156 httpsdoiorg101111j1747-5457200900440x

De Dreuzy JR Davy P amp Bour O 2001 Hydraulic properties of two-dimensional random fracture networks following a power law lengthdistribution 1 Effective connectivity Water Resources Research 372065ndash2078 httpsdoiorg1010292001WR900011

Dresen G 1991 Stress distribution and the orientation of Riedel shearsTectonophysics 188 239ndash247 httpsdoiorg1010160040-1951(91)90458-5

Ehlen J 2000 Fractal analysis of joint patterns in granite International Journalof Rock Mechanics and Mining Sciences 37 909ndash922 httpsdoiorg101016S1365-1609(00)00027-7

Engelder T 1985 Loading paths to joint propagation during a tectonic cycle anexample from the Appalachian Plateau USA Journal of Structural Geology7 459ndash476 httpsdoiorg1010160191-8141(85)90049-5

English JM amp Laubach SE 2017 Opening-mode fracture systems ndash insightsfrom recent fluid inclusion microthermometry studies of crack-seal fracturecements In Turner JP Healy D Hillis RR amp Welch M (eds)Geomechanics and Geology Geological Society London SpecialPublications 458 257ndash272 httpsdoiorg101144SP4581

Escuder Viruete J Carbonell R Jurado MJ Martiacute D amp amp Peacuterez-Estauacuten A2001 Two dimensional geostatistical modeling and prediction of fracturesystem in the Albala granitic pluton SW Iberian massif Spain Journal ofStructural Geology 23 2011ndash2023 httpsdoiorg101016S0191-8141(01)00026-8

Forstner SR Laubach SE amp Fall A 2018 Evolution of deformation in theBuck Mountain fault damage zone Cambrian Flathead sandstone TetonRange Wyoming Pan-American Current Research on Fluid Inclusions 1447ndash48

Foster D Brocklehurst SH amp Gawthorpe RL 2010 Glacialndashtopographicinteractions in the Teton Range Wyoming Journal of Geophysical ResearchEarth Surface 115 F01007 httpsdoiorg1010292008JF001135

Frost BR Swapp SM Frost CD Bagdonas DA amp Chamberlain KR2018 Neoarchean tectonic history of the Teton Range Record of accretionagainst the present-day western margin of the Wyoming ProvinceGeosphere14 1008ndash1030 httpsdoiorg101130GES015591

Gabrielsen RH amp Braathen A 2014 Models of fracture lineaments ndash Jointswarms fracture corridors and faults in crystalline rocks and their geneticrelations Tectonophysics 628 26ndash44 httpsdoiorg101016jtecto201404022

Gale JFW Laubach SE Olson JE Eichhubl P amp Fall A 2014 Naturalfractures in shale a review and new observations AAPG Bulletin 982165ndash2216 httpsdoiorg10130608121413151

Genter A amp Castaing C 1997 Effets drsquoeacutechelle dans la fracturation des granites[Scale effects in the fracturing of granites] Tectonics 325 439ndash445 httpsdoiorg101016S1251-8050(97)81162-7

Gillespie PA Howard CB Walsh JJ amp Watterson J 1993 Measurementand characterization of spatial distributions of fractures Tectonophysics 226114ndash141 httpsdoiorg1010160040-1951(93)90114-Y

Gillespie PA Johnston JD Loriga MA McCaffrey KJW Walsh JJ ampWatterson J 1999 Influence of layering on vein systematics in line samplesIn McCaffrey KJW Walsh JJ amp Watterson J (eds) Fractures FluidFlow and Mineralization Geological Society London Special Publications155 35ndash56 httpsdoiorg101144GSLSP19991550105

Gillespie PA Walsh JJ Watterson J Bonson CG amp Manzocchi T 2001Scaling relationships of joint and vein arrays from The Burren Co Clare

Ireland Journal of Structural Geology 23 183ndash201 httpsdoiorg101016S0191-8141(00)00090-0

Halihan T Mace RE amp Sharp JMJ 2000 Flow in the San Antonio segmentof the Edwards aquifer matrix fractures or conduits In Sasowsky ID ampWicks CM (eds) Groundwater Flow and Contaminant Transport inCarbonate Aquifers Balkema Rotterdam The Netherlands 129ndash146

Hancock PL 1985 Brittle microtectonics principles and practice Journal ofStructural Geology 7 437ndash457 httpsdoiorg1010160191-8141(85)90048-3

Heidbach O Rajabi M et al 2018 The World Stress Map database release2016 crustal stress pattern across scales Tectonophysics 744 484ndash498httpsdoiorg101016jtecto201807007

Henriksen H amp Braathen A 2006 Effects of fracture lineaments and in-siturock stresses on groundwater flow in hard rocks a case study from Sunnfjordwestern Norway Hydrogeology Journal 14 444ndash461 httpsdoiorg101007s10040-005-0444-7

Hobbs BE ampOrd A 2018 Coupling of fluid flow to permeability developmentin mid- to upper crustal environments a tale of three pressures In Gessner KBlenkinsop TG amp Sorjonen-Ward P (eds) Characterization of Ore-Forming Systems from Geological Geochemical and Geophysical StudiesGeological Society London Special Publications 453 81ndash120 httpsdoiorg101144SP4539

Hooker JN Laubach SE amp Marrett R 2018 Microfracture spacingdistributions and the evolution of fracture patterns in sandstones Journal ofStructural Geology 108 66ndash79 httpsdoiorg101016jjsg201704001

Hu H Zheng Y Fang X amp Fehler MC 2018 3D seismic characterization offractures with random spacing using double-beam method Geophysics 83M63ndashM74 httpsdoiorg101190geo2017-07391

Ladeira FL amp Price NJ 1981 Relationship between fracture spacing and bedthickness Journal of Structural Geology 3 179ndash183 httpsdoiorg1010160191-8141(81)90013-4

Lageson DR 1987 Laramide uplift of the Gros Ventre Range and implicationsfor the origin of the Teton Fault Wyoming In Miller WR (ed) The ThrustBelt Revisited 38th Annual Field Conference Guidebook WyomingGeological Association Casper WY 79ndash89

Lamarche J Chabani A amp Gauthier BD 2018 Dimensional threshold forfracture linkage and hooking Journal of Structural Geology 108 171ndash179httpsdoiorg101016jjsg201711016

Laubach SE 1991 Fracture patterns in low-permeability-sandstone gasreservoir rocks in the Rocky Mountain region Paper presented at the LowPermeability Reservoirs Symposium April 15ndash17 1991 Denver CO USAhttpsdoiorg10211821853-MS

Laubach SE 2003 Practical approaches to identifying sealed and openfractures AAPG Bulletin 87 561ndash579 httpsdoiorg10130611060201106

Laubach SE Mace RE amp Nance HS 1995 Fault and joint swarms in anormal fault zone In Rossmanith H-P (ed) Mechanics of Jointed andFaulted Rock Balkema Rotterdam The Netherlands 305ndash309

Laubach SE Lamarche J Gauthier BDM Dunne WM amp Sanderson DJ2018a Spatial arrangement of faults and opening-mode fractures Journal ofStructural Geology 108 2ndash15 httpsdoiorg101016jjsg201708008

Laubach SE Hundley TH Hooker JN amp Marrett R 2018b Spatialarrangement and size distribution of normal faults Buckskin Detachmentupper plate Western Arizona Journal of Structural Geology 108 230ndash242httpsdoiorg101016jjsg201710001

Le Garzic E De LrsquoHamaide T et al 2011 Scaling and geometric properties ofextensional fracture systems in the proterozoic basement of Yemen Tectonicinterpretation and fluid flow implications Journal of Structural Geology 33519ndash536 httpsdoiorg101016jjsg201101012

Leopold EB Lu G Love JD amp Love DW 2007 Plio-Pleistocene climatictransition and the lifting of the Teton Range WyomingQuaternary Research67 1ndash11 httpsdoiorg101016jyqres200610006

Li JZ Laubach SE Gale JFW amp Marrett R 2018 Quantifying opening-mode fracture spatial organization in horizontal wellbore image logs core andoutcrop application to Upper Cretaceous Frontier Formation tight gassandstones USA Journal of Structural Geology 108 137ndash156 httpsdoiorg101016jjsg201707005

Love JD Leopold EB amp Love DW 1978 Eocene Rocks Fossils andGeologic History Teton Range Northwestern Wyoming United StatesGeological Survey Professional Paper 932-B

Love JD Reed JC amp Christiansen AC 1992 Geologic Map of Grand TetonNational Park Teton County Wyoming (Scale 162 500) United StatesGeological Survey Miscellaneous Investigations Series Map 1-2031 (IMAP2031)

Maillot J Davy P Le Goc R Darcel C amp De Dreuzy JR 2016Connectivity permeability and channeling in randomly distributed andkinematically defined discrete fracture network models Water ResourcesResearch 52 8526ndash8545 httpsdoiorg1010022016WR018973

Marrett R Gale JFW Gomez L amp Laubach SE 2018 Correlation analysisof fracture arrangement in space Journal of Structural Geology 108 16ndash33httpsdoiorg101016jjsg201706012

Martel SJ 1990 Formation of compound strike-slip fault zones Mount Abbotquadrangle California Journal of Structural Geology 12 869ndash882 httpsdoiorg1010160191-8141(90)90060-C

Mazurek M Jakob A amp Bossart P 2003 Solute transport in crystalline rocks atAumlspouml ndash I Geological basis and model calibration Journal of ContaminantHydrology 61 157ndash174 httpsdoiorg101016S0169-7722(02)00137-7

Fracture clustering in basement rocks

by guest on December 10 2020httppglyellcollectionorgDownloaded from

McGinnis RN Ferrill DA Smart KJ Morris AP Higuera-Diaz C ampPrawika D 2015 Pitfalls of using entrenched fracture relationships Fracturesin bedded carbonates of the Hidden Valley Fault Zone Canyon Lake GorgeComal County Texas AAPG Bulletin 99 2221ndash2245 httpsdoiorg10130607061513012

Miranda TS Santos RF et al 2018 Quantifying aperture spacing and fractureintensity in a carbonate reservoir analogue Crato Formation NE BrazilMarine amp Petroleum Geology 97 556ndash567 httpsdoiorg101016jmarpetgeo201807019

Nara Y Harui T amp Kashiwaya K 2018 Influence of calcium ions onsubcritical crack growth in granite International Journal of Rock Mechanicsand Mining Sciences 102 71ndash77 httpsdoiorg101016jijrmms201801001

Narr W amp Suppe J 1991 Joint spacing in sedimentary rocks Journal ofStructural Geology 13 1037ndash1048 httpsdoiorg1010160191-8141(91)90055-N

Olson JE 1993 Joint pattern development effects of subcritical crack growthand mechanical interaction Journal of Geophysical Research 98 12 251ndash12265 httpsdoiorg10102993JB00779

Olson JE 2004 Predicting fracture swarms ndash the influence of subcritical crackgrowth and the crack-tip process zone on joint spacing in rock In CosgroveJWamp Engelder T (eds) The Initiation Propagation and Arrest of Joints andOther Fractures Geological Society London Special Publications 23173ndash88 httpsdoiorg101144GSLSP20042310105

Olson JE amp Pollard DD 1989 Inferring paleostresses from natural fracturepatterns a new method Geology 17 345ndash348 httpsdoiorg1011300091-7613(1989)017lt0345IPFNFPgt23CO2

Ortega O Marrett R amp Laubach SE 2006 A scale-independent approach tofracture intensity and average spacing measurement AAPG Bulletin 90193ndash208 httpsdoiorg10130608250505059

Pollard DD amp Aydin A 1988 Progress in understanding jointing over the pastcentury Geological Society of America Bulletin 100 1181ndash1204 httpsdoiorg1011300016-7606(1988)100lt1181PIUJOTgt23CO2

Powers RB (ed) 1982Geologic Studies of the Cordilleran Thrust Belt RockyMountain Association of Geologists Denver CO

Priest SD amp Hudson JA 1976 Discontinuity spacings in rock InternationalJournal of Rock Mechanics 13 135ndash148 httpsdoiorg1010160148-9062(76)90818-4

Putz-Perrier MW amp Sanderson DJ 2008 The distribution of faults andfractures and their importance in accommodating extensional strain atKimmeridge Bay Dorset UK In Wibberley CAJ Kurz WHoldsworth RE amp Collettini C (eds) The Internal Structure of FaultZones Implications for Mechanical and Fluid-Flow Properties GeologicalSociety London Special Publications 299 97ndash111 httpsdoiorg101144SP2996

Questiaux JM Couples GD amp Ruby N 2010 Fractured reservoirs withfracture corridors Geophysical Prospecting 58 279ndash295 httpsdoiorg101111j1365-2478200900810x

Reed JC Jr amp Zartman RE 1973 Geochronology of Precambrian rocks ofthe Teton Range Wyoming Geological Society of America Bulletin 84561ndash582 httpsdoiorg1011300016-7606(1973)84lt561GOPROTgt20CO2

Roberts SV amp Burbank DW 1993 Uplift and thermal history of the TetonRange (northwestern Wyoming) defined by apatite fission-track dating Earth

and Planetary Science Letters 118 295ndash309 httpsdoiorg1010160012-821X(93)90174-8

Roy A Perfect E Dunne WM ampMcKay LD 2014 A technique for revealingscale-dependent patterns in fracture spacing data Journal of GeophysicalResearch Solid Earth 119 5979ndash5986 httpsdoiorg1010022013JB010647

Sanderson DJ amp Nixon CW 2015 The use of topology in fracture networkcharacterization Journal of Structural Geology 72 55ndash66 httpsdoiorg101016jjsg201501005

Sanderson DJ amp Peacock DCP 2019 Line sampling of fracture swarms andcorridors Journal of Structural Geology 122 27ndash37 httpsdoiorg101016jjsg201902006

Sanderson DJ Roberts S amp Gumiel P 1994 A fractal relationship betweenvein thickness and gold grade in drill core from La Codosera Spain EconomicGeology 89 168ndash173 httpsdoiorg102113gsecongeo891168

Santos RF Miranda TS et al 2015 Characterization of natural fracturesystems analysis of uncertainty effects in linear scanline results AAPGBulletin 99 2203ndash2219 httpsdoiorg10130605211514104

Segall P amp Pollard DD 1983 Joint formation in granitic rock of the SierraNevada Geological Society of America Bulletin 94 563ndash575 httpsdoiorg1011300016-7606(1983)94lt563JFIGROgt20CO2

Slightam C 2014 Characterizing seismic-scale faults pre- and post-drillingLewisian Basement West of Shetlands UK In Spence GH Redfern JAguilera R Bevan TG Cosgrove JW Couples GD amp Daniel J-M(eds) Advances in the Study of Fractured Reservoirs Geological SocietyLondon Special Publications 374 311ndash331 httpsdoiorg101144SP3746

Smith RB amp Siegel L 2000 Windows into Earth The Geologic Story ofYellowstone and Grand Teton National Parks Oxford University PressOxford

Smith RB Byrd JOD amp Susong DD 1993 The Teton fault seismotec-tonics Quaternary history and earthquake hazards In Snoke AWSteidtmann J amp Roberts SM (eds) Geology of Wyoming GeologicalSurvey of Wyoming Memoirs 5 628ndash667

Solano N Zambrano L amp Aguilera R 2011 Cumulative-gas-productiondistribution on the Nikanassin Formation Alberta and British ColumbiaCanada SPE Reservoir Evaluation and Engineering 14 357ndash376 httpsdoiorg102118132923-PA

Tchalenko JS 1970 Similarities between shear zones of different magnitudesGeological Society of America Bulletin 81 1625ndash1640 httpsdoiorg1011300016-7606(1970)81[1625SBSZOD]20CO2

Terzaghi R 1965 Sources of error in joint surveys Geotechnique 15 287ndash297httpsdoiorg101680geot1965153287

Torabi A Alaei B amp Ellingsen TSS 2018 Faults and fractures in basementrocks their architecture petrophysical and mechanical properties Journal ofStructural Geology 117 256ndash263 httpsdoiorg101016jjsg201807001

Wehunt D Borovykh M amp Narr W 2017 Stochastic 2D well-pathassessments for naturally fractured carbonate reservoirs SPE ReservoirEvaluation amp Engineering 20 853ndash875 httpsdoiorg102118180468-PA

Willemse EJ amp Pollard DD 1998 On the orientation and patterns ofwing cracksand solution surfaces at the tips of a sliding flaw or fault Journal of GeophysicalResearch Solid Earth 103 2427ndash2438 httpsdoiorg10102997JB01587

Zartman RE amp Reed JC Jr 1998 Zircon geochronology of theWebb CanyonGneiss and the Mount Owen Quartz Monzonite Teton Range Wyomingsignificance to dating Late Archean metamorphism in the Wyoming cratonThe Mountain Geologist 35 71ndash77

Q Wang et al

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