Frac Pa Q User Guide V2.2 RELEASE
User Manual: Pdf
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Fracture!Pattern!Quantification!!
!
User!Guide!!
Version!2.2,!February!2018!!
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!
David!Healy!&!Roberto!Rizzo!
d.healy@abdn.ac.uk!!
With!contributions!from!Nick!Timms,!Matthew!Belshaw!&!Daniel!Knight!!!
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FracPaQ!User!Guide!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Version!2.2,!February!2018!
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Contents!
1. Summary!of!changes!in!FracPaQ!version!2.2!!
2. Introduction!
3. System!requirements!!
4. Obtaining!&!installing!the!code!!!
5. Starting!FracPaQ!
6. Choosing!an!input!file!!
6.1. Image!file!input!!
6.2. Node!file!input!!
7. Building!an!input!file!in!.svg!format!
8. Scaling!!
9. North!correction!!
10. Outputs!
10.1. Maps!!
10.1.1. Traces,!segments!
10.1.2. Estimated!Intensity!!
10.1.3. Estimated!Density!!
10.2. Lengths!&!sizes!
10.2.1. Histogram!of!lengths!!
10.2.2. Log-log!lengths!!
10.2.3. MLE!analysis!
10.2.4. Variogram!of!segment!lengths!!
10.2.5. Cross!plot!lengths!v!angles!
10.2.6. Block!sizes!
10.3. Crack!tensor!!
10.4. Orientations!!
10.4.1. Histogram!of!angles!
10.4.2. Rose!diagram!
10.5. Fluid!flow!!
10.5.1. I-Y-X!connectivity!!
10.5.2. Permeability!ellipse!!
10.6. Wavelet!analysis!
11. Multi-coloured!fracture!maps!
12. Statistical!estimation!using!MLE!analysis!
13. Utilities!
14. Acknowledgements!!
15. Bibliography!
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1.!!Summary!of!changes!in!FracPaQ!version!2.2!!
In!this!new!release,!we!have!added:!
• 2D! wavelet! analysis! of! fracture! maps! to! determine! scale! transitions! in! fracture!
orientations;!!
• a! new! rose! diagram! with! segment! orientations! colour-coded! by! strike,! matching! the!
segment!strike!map;!!
• a!new!(semi-)variogram!plot!of!segment!lengths!using!their!calculated!centroids;!!
• 2!new!command!line!utilities!to!help!convert!image!files!into!8-bit!binary!format!and!to!
convert!trace!maps!from!Move™!into!FracPaQ!format.!!!
We!have!modified:!
• the!MLE!plots!of!fracture!lengths!to!fix!reported!bugs;!!!
• this!User!Guide!to!provide!more!detail!on!preparing!image!files!for!input!and!details!of!the!
new!features!listed!above.!!!
!
2.!Introduction!!
FracPaQ!is!an!open!source!toolbox!written!in!MATLAB™!and!publicly!available!on!GitHub!and!
the!Mathworks™!FileExchange.!FracPaQ!is!designed!to!quantify!fracture!patterns!in!rock!or!
other!materials!from!2D!images.!The!user!supplies!either!an!image!file!of!fractured!material,!
or!a!text!file!of!traced!fractures!and!their!(x,y)!coordinates.!From!either!kind!of!input,!the!code!
calculates!fracture!lengths,!angles!and!connectivity.!These!are!displayed!as!maps!and!graphs,!
and!saved!as!*.tif!files.!Estimates!are!made!of!fracture!intensity!(P21)!&!density!(P20),!and!
permeability!in!2D!using!a!simple!parallel-plate!model.!!!
FracPaQ! is! written! in! MATLAB™! and! totals! over! 4,000! lines! of! code.! As! in! any! software!
project!of!this!scale,!there!will!be!‘bugs’!–!i.e.!coding!errors.!If!you!encounter!a!bug,!please!let!
us! know,! through! GitHub,! Mathworks™! FileExchange! or! by! e-mail! (d.healy@abdn.ac.uk);!
please!provide!as!many!details!as!you!can,!including!(where!possible)!a!screen!shot!of!
the!error,!the!input!file!you!were!using!at!the!time,!and!the!MATLAB™!version.!!
FracPaQ!uses! code! written! by! others:! lineSegmentIntersect.m!by! U.! Murat! Erdem,!
readtext.m! by! Peder! Axensten,! cmocean.m!by! Kristen! Thyng,! variogram.m!by! Wolfgang!
Schwangart! (all!available! on! Mathworks! FileExchange),! and! various! wavelet! routines! by!
Roseanna!Neupauer!and!Kaye!Powell.!The!handling!of!image!file!input!in!FracPaQ!also!uses!
functions!from!the!Image!Processing!Toolbox!(version!9.4),!a!MATLAB™!add-on.!!!!
We! hope! that! the! code! will! enable! researchers! to! quantify! fracture! patterns! in! an! open,!
objective,!and!consistent!way.!We! also! hope! that! people! will! contribute! new! functions!
and!new!tools,!and!report!any!bugs!J!!
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Figure! 1.!Example!of!fracture!trace!maps!from!FracPaQ!version!2.2.!Left!side!shows!a!map!
with!segments!coloured!by!length!and!the!right!side!shows!a!map!with!segments!coloured!by!
strike!(assuming!North!is!parallel!to!the!y-axis).!!
!
3.!! System!requirements!!
FracPaQ!runs!on!any!computer!with!MATLAB™!installed!–!Microsoft!Windows™,!Mac!OS!X™!
or! various! flavours! of! Linux.! The! current! version! of! FracPaQ!is! 2.2! (February!2018).! We!
developed!FracPaQ!version!2.2!using!MATLAB™!version!9,!R2016a.!!
!
4.! Obtaining!&!installing!the!code!!
You!can!get!all!the!source!code!and!run!it!directly!from!the!MATLAB™!command!window.!!
The!source!code!and!this!User!Guide!are!available!on!GitHub:!
http://davehealy-aberdeen.github.io/FracPaQ/!
and!on!the!Mathworks™!FileExchange:!
https://uk.mathworks.com/matlabcentral/fileexchange/58860-davehealy-aberdeen-fracpaq!
Step-by-step!guide:!
1. Download!the!source!code!as!a!.zip!or!.tar.gz!file,!and!extract!all!of!the!files.!
2. Put!all!of!these!files!into!a!single!folder.!!
3. Start!MATLAB™.!!
4. Set!the!current!working!folder!in!MATLAB™!to!the!same!folder!you!installed!the!code!in.!
5. At!the!MATLAB™!command!prompt!type!‘guiFracPaQ2D’!and!hit!Enter.!!
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The!output!files!(i.e.!the!graphs,!maps!and!data)!will!appear!as!*.tif!and!*.txt!files!in!the!same!
folder!as!the!code.!!!!!
!
5.! Starting!FracPaQ!
There! is! one! main! window! in! the! FracPaQ!application! (see! below).! The! window! can! be!
minimized! or! closed! using! the! standard! GUI! controls! (top! left! in! the! Mac! OS! X™! version!
shown).!Input!parameters!are!on!the!left!side,!and!output!options!are!on!the!right.!!The!central!
area!is!for!previewing!the!input!data!as!a!fracture!trace!map.!!!
To!get!started,!click!Browse…!to!see!a!list!of!possible!input!files!in!the!current!working!folder.!!
Then! click! Preview! to! open! the! data! file! and! view! the! traces! in! the! main! window;! basic!
statistics!on!the!pattern!are!also!shown!(lower!left!text!box).!!
!
Figure!2.!Screenshot!of!the!graphical!user!interface!(GUI)!in!FracPaQ!version!2.2.!!This!is!the!
only!input!window.!!Each!output!(map,!graph!etc.)!is!directed!to!a!separate!MATLAB™!figure!
window!and!saved!to!a!separate!graphics!file!in!the!current!folder.!!!
Select! the! Outputs! you! need! (right! hand! side),! and! then! click! Run!to! produce! the! selected!
maps!and!graphs.!Each!output!map!or!plot!is!shown!in!a!separate!figure!window,!and!saved!as!
a!separate!.tif!image!file!in!the!current!folder.!!
Click!Exit!to!quit!FracPaQ.!!
!
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6.! Choosing!an!input!file!!
Click!Browse…!to!select!an!input!file.!!The!file!types!shown!are!filtered!to!file!extensions!*.txt,!
*.svg,!*.jpeg,!*.jpg,!*.tiff!and!*.tif.!!!
6.1$ Image$file$input$$
If! you! choose! a! graphic! file! format! (*.jpeg,! *.jpg,! *.tiff,! *.tif),! the! Input! file! type! changes! to!
‘Image!file’.!!This!file!will!then!be!processed!by!the!Hough!transform!method!to!build!fracture!
traces!(see!below).!NB!input!image!files!must!be!8-bit!and!binary!(i.e.!grayscale).!You!can!
check! the! format! of! your! any! potential! image! file! before! loading! into! FracPaQ! using! the!
MATLAB™! function! imfinfo().! At! the! MATLAB™! command! line,! type!
'imfinfo('this_is_my_file.tif')'! and! hit! Enter.! MATLAB™! will! display! a! range! of! file! format!
information.! Check! out! the! fields! for! BitDepth! and! ColorType.! BitDepth! should! be! 8! and!
ColorType!needs!to!be!'grayscale'.!You!can!modify!existing!image!files!-!e.g.!photographs!of!
outcrops!taken!on!digital!cameras!using!your!own!preferred!software!(e.g.!Adobe!Photoshop™!
or!ImageJ),!or!you!can!use!our!new!command!line!utility,!FracPaQ2D_imagePreprocess.m!
(see!section!13).!!
Image!files!are!processed!by!a!Hough!transform!method!which!looks!for!co-linear!pixels!and!
builds!straight!line!fracture!traces.!With!this!method!each!fracture!has!one!trace!and!only!one!
segment!per!trace.!The!user!can!choose!the!values!at!which!these!lines!are!merged!(if!they!are!
close! enough! together)! or! discarded! (if! they! are! too! short).! More! details! on! the! Hough!
transform!method! can!be!found!in! the!MATLAB™!Help! documentation.!The!key!parameters!
that! affect! the! numbers! and! lengths! of! fracture! traces! are! the! number! of! peaks!and! the!
threshold.!Using!the!Hough!transform!is!a!‘trial!and!error’!procedure:!load!an!image!file!and!
click!Preview.!Then!use!the!graphs!and!the!trace!map!displayed!in!the!main!window!to!tune!
the! Hough! transform! parameters.! Set! the! threshold!to! a! value! between! 0.0! and! 1.0! that!
captures!only!the!significant!peaks!(in!orange/yellow!on!the!graphs),!0.33!is!a!good!start.!!
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Figure! 3.! Output! from! Hough! transform! applied! to! the! sample! image! input! file!
MacduffBinary.tif.! Set! the! Threshold! parameter,! between! 0.0! and! 1.0,! to! a! value! that! just!
captures!the!maxima!-!i.e.!the!yellow/orange!peaks.!Values!of!0.3-0.4!often!work!well.!!!
6.2$ Node$file$input$$
If! you! select! a! *.txt! or! *.svg! file! of! (x,y)! fracture! trace! nodes,! the! Input! file! type! changes!
automatically!to!‘Node!file’.!!!
!
Figure!4.!Example!of!a!.txt!node!file.!Tab-delimited!(x,y)!pairs,!one!fracture!per!line.!!
A!node!file!contains!(x,y)!coordinates!of!nodes!along!each!fracture!trace.!!The!(x,y)!data!are!
tab-delimited,!and!each!fracture!trace!is!on!one!line.!There!is!a!minimum!of!4!columns!for!
each!line!(i.e.!each!fracture!trace):!(x1,!y1)!for!node!1,!and!(x2,!y2)!for!node!2.!A!fracture!trace!
can!be!made!of!many!segments.!In!the!example!shown!above!the!longest!trace!(line!7)!has!5!
segments,!bounded!by!6!nodes!–!i.e.!12!columns!=!6!x!2!(x,y)!pairs.!!
!
7.! Building!a!node!input!file!in!.svg!format!
As!noted!above,!one!option!for!the!Node!file!input!is!to!use!a!file!format!of!.svg!–!or!Scalable!
Vector!Graphics!(an!open!standard!based!on!XML).!!Graphics!packages!like!Adobe!Illustrator™!
and!Corel!Draw™!(and!many!others)!can!produce!.svg!files!from!images!of!fracture!traces.!!
Open! your! chosen! graphics! package! and! import! your! image! of! fractures! –! this! could! be! a!
photograph! from! an! outcrop! or! thin! section,! satellite! or! aerial! imagery! from! Google™! or!
Bing™,!or!a!scan!of!a!geological!map.!Before!spending!time!tracing!fractures!it!makes!sense!to!
assess!the!appropriate!level!of!magnification!to!use.!Choose!a!sub-area!of!your!fracture!image!
and!increase!the!magnification!(zoom!in)!until!the!fracture!traces!pixelate.!Typically,!we!have!
found! that! the! best! compromise! between! resolution! and! time! spent! tracing! is! around! 4-5!
levels!of!zoom!below!the!maximum!(i.e.!the!level!at!which!the!fractures!pixelate).!However,!
each!project!is!different!and!you!must!decide!on!your!own!preferences.!In!selecting!a!sub-area!
from!an!imported!image!of!fractured!rock!it!is!best!to!avoid!areas!with!significant!amounts!of!
vegetation!cover!or!scree!etc.!!!!!
1. Import!the!base!image!containing!the!‘raw’!fracture!traces.!!!
2. Select!an!area!of!the!image!and!draw!a!bounding!rectangle!(Rectangle!tool).!!
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3. Create!a!new!layer!(see!Figure!5!for!Adobe!Illustrator™).!!Lock!the!base!layer.!!
4. Use!the!Pen!tool!(Adobe!Illustrator™)!or!Polyline!tool!(Corel!Draw™)!to!trace!fractures!on!
this!new!layer.!Click!along!the!fracture!trace!at!intervals!where!the!orientation!changes.!!
Each!click!defines!a!new!node,!and!the!nodes!define!the!single!straight-line!segments!that!
make!up!a!fracture!trace.!Take! care! not! to! drag!the!cursor!when!clicking!along!a! trace.!!
When!you!reach!the!end!of!a!trace!click!Enter.!Ensure!‘Snap!to!Grid’!(in!the!View!menu)!is!
turned!off,!to!ensure!optimum!accuracy!of!fracture!node!positions.!!!
!
!
Figure! 5.!Screenshot!of!Adobe!Illustrator™!showing!how!to!create!a!new!layer!on!top!of!an!
imported!base!image!of!fractured!rock.!!!
5. Trace!all!visible!fractures!at!the!selected!magnification;!do!not!zoom!in!and!out!as!this!can!
create!a!sampling!bias.!Take!care!not!to!accidentally!join!(or!merge)!two!distinct!fracture!
traces,!as!FracPaQ!will!read!this!as!a!single!fracture!trace.!!
6. If!you!want!to! separate! distinct! sets!of!fracture!traces!(e.g.!by!mineral! fill,! inferred! age,!
orientation,!or!some!other!criterion)!then!you!can!add!a!new!layer!and!choose!a!new!Line!
and!Fill!colour.!Then!use!the!Pen!or!Polyline!tool!to!trace!this!fracture!set!on!the!new!layer,!
with!as!many!layers!as!there!are!fracture!sets.!!
7. Save!the!file!in!.ai!format!(Adobe!Illustrator™)!or!.cdr!format!(Corel!Draw™).!!
8. Delete!the!layer!containing!the!base!level!image.!!
9. [If! the! area! traced! is! in! the! top! left! corner! of! the! document! page! then! skip! this! step.]!!
Unlock!all!layers;!select!all!fractures!(Ctrl+A!then!Ctrl+C)!using!the!Selection!Tool.!Drag!all!
the!fractures!to!the!top!left!corner!of!the!document.!
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10. Using!‘Save!As…’!save!the!file!in!.svg!format,!with!the!following!options:!SVG!version!1.1!
and!CSS! Properties!=!‘Presentation!Attributes’!(Adobe!Illustrator™)!or! Styling!Options!=!
‘Presentation!Attributes’!(Corel!Draw™).!!
8.!! Scaling!!
The!default!unit!of!length!in!FracPaQ!is!the!pixel.!For!any!input!file,!the!x-!and!y-coordinates!
on!the!maps!and!all!length!data!are!reported!in!pixels.!However,!if!you!wish!to!change!this!to!
SI!units!of!length!(i.e.!metres)!enter!a!scaling!factor!in!the!‘Scaling’!text!box.!For!example,!if!
you!used!a!binary!image!photograph!as!input!with!a!width!of!100!pixels,!and!you!know!the!
true!field!of!view!is!1.0!metre,!then!enter!100.0!in!the!‘Scaling’!text!box!before!you!click!Run!to!
generate!output!maps!or!plots.!!!
!
9.!North!correction!
By! default,! FracPaQ! assumes! that! fracture! orientations! (i.e.! their! strikes)! are! measured!
clockwise!from!the!positive!Y-axis,!assumed!to!be!pointing!North.!If!this!is!not!the!case,!you!
can!enter!an!angle!(0-180°)!between!the!Y-axis!and!true!North!for!the!mapped!data!in!the!text!
box! marked! ‘North! correction! for! Y-axis,! °’.! This! angle! is! then! added!to!all! orientations! as!
plotted! in! the! selected! outputs,! including! the! rose! plot,! the! segment! strike! map,! the! crack!
tensor!plot!and!the!permeability!ellipses.!If!you!want!to!remove!an!angle!then!enter!a!negative!
value!(<!−180°)!in!the!text!box.!!!!!
!!
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10.!Outputs!
10.1.$ Maps$–$Traces,$segments$
Tick!the!check!box!‘Traces,!segments’!in!the!Maps!section!of!Outputs!to!generate!four!maps!of!
fracture! traces! and! segments.! These! are:! the! basic! single-colour! map! of! all! traces! and!
segments;!a!map!with!traces!colour-coded!by!trace!length;!a!map!with!segments!colour-coded!
by!segment!length;!and!a!map!with!segments!colour-coded!by!segment!strike!(NB!if!you!have!
entered! a! non-zero! value! in! the! text! box! ‘North! correction! for! Y-axis,! °’,! then! this! angle! is!
added!to!these!colour!coded!strike!values,!see!below).!
!
!
Figure!6.!Left!side!shows!the!basic!trace!map,!and!right!side!shows!the!trace!map!with!‘Show!
nodes’! option! checked.! Nodes! are! shown! with! black! circles,! segment! mid-points! with! red!
squares!and!trace!mid-points!with!red!circles.!!!
Maps$–$Traces,$segments;$Show$nodes$
FracPaQ!is!based!on!coordinate!geometry!in!2D.!Therefore,!when!the!traces!and!segments!are!
loaded!into!the!application,!we!also!calculate!the!mid-points!(or!centroids)!of!each!trace!and!
each!segment.!To!display!these!data!on!a!trace!map,!simply!check!the!‘Show!nodes’!check!box!
and!click!Run.!The!mid-points,!together!with!the!nodes!located!at!the!ends!of!all!traces!and!
segments,!are!displayed!on!the!map.!The!node!and!centroid!data!are!stored!in!the!underlying!
traces!data!structure!(see!guiFracPaQ2D_list.m),!and!may!be!useful!for!new!maps!and!plots!
exploring!the!spatial!statistics!of!fracture!patterns.!!!
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Figure!7.!Left!side!shows!the!trace!map,!with!traces!coloured!by!length.!Right!side!shows!the!
trace!map!with!segments!coloured!by!length.!!
!
Figure!8.!Left!side!shows!a!trace!map,!with!segments!coloured!by!strike.!Right!side!shows!the!
same!map!data,!but!now!with!a!North!correction!of!−30°!applied.!Insets!show!the!equal!area!
rose!plots!superimposed.!!
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Maps$–$Estimated$Intensity,$P21$
FracPaQ!uses! the! circular! scan! window! method! of! Mauldon! et! al.! (2001)! to! calculate! an!
estimate!of!fracture!intensity!(units!of!L-1).!Click!on!the!check!box!‘Estimated!Intensity,!P21’!to!
generate!this!map.!Estimates!of!intensity!are!made!on!a!regular!grid!pattern!using!the!number!
of!scan!circles!specified!in!the!text!box!‘Number!of!scan!circles’.!If!the!map!area!is!not!square,!
then!this!value!is!the!minimum!number!of!scan!circles!used!along!the!shorter!side!of!the!area.!!
Maps$–$Estimated$Density,$P20$$
FracPaQ! uses! the! circular! scan! window! method! of! Mauldon! et! al.! (2001)! to! calculate! an!
estimate!of!fracture!intensity!(units!of!L-2).!Click!on!the!check!box!‘Estimated!Density,!P20’!to!
generate!this!map.!Estimates!of!density!are!made!on!a!regular!grid!pattern!using!the!number!
of!scan!circles!specified!in!the!text!box!‘Number!of!scan!circles’.!If!the!map!area!is!not!square,!
then!this!value!is!the!minimum!number!of!scan!circles!used!along!the!shorter!side!of!the!area.!!
Maps$–$Estimated$Intensity$or$Density;$Show$scan$circles$
Click!on!the!check!box!‘Show!scan!circles’!to!generate!a!trace!map!showing!the!location!and!
size!of!the!scan!circles!used!to!estimate!Intensity!or!Density.!
!
!
!
Figure! 9.!Left! side! shows! a! map! of! Estimated! Intensity! (P21,! m-1).! Centre!shows!a!map! of!
Estimated!Density!(P20,!m-2).!!Right!side!shows!the!positions!and!sizes!of!the!scan!circles!used!
in!the!estimations.!
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10.2.$ Lengths$&$sizes$–$Histogram$of$lengths$$
Tick! the! check! box! ‘Histogram! of! lengths’! to! generate! two! frequency-size! plots! for! trace!
lengths!and!segments!lengths.!You!can!change!the!number!of!bins!used!in!these!histograms!by!
picking!a!different!value!in!the!dropdown!list!box!‘Number!of!bins’!(default!is!20).!Lengths!are!
reported!in!pixels,!unless!a!non-zero!value!is!entered!in!the!text!box!‘Scaling!(pixels/metre)’.!
!
!
Figure! 10.!Left! side! shows! frequency-size! histogram! for! trace! lengths.! Right! side! shows!
frequency-size!histogram!for!segment! lengths.! Red! lines! show! the! minimum!and!maximum!
lengths.!!
Lengths$&$sizes$–$Log-log$lengths$
Tick!the!check!box!‘Log-log!lengths’!to!generate!two!cumulative!frequency!versus!length!plots!
on!log-log!scales!for!trace!lengths!and!segment!lengths.!These!can!be!useful!for!assessing!the!
scaling!of!fracture!lengths.!Tick!the!check!box!‘Censor!lengths!at!edges’!to!remove!any!traces!
or! segments! from! the! plots! that! intersect! the! boundaries! of! the! mapped! area! (i.e.! those!
fractures!for!which!the!true!length!is!unknown).!!
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!
!
Figure!11.!Left!side!shows!a!log-log!plot!of!cumulative!frequency!versus!trace!lengths.!Right!
side!shows!log-log!plot!of!cumulative!frequency!versus!segment!lengths.!The!blue!solid!lines!
show!the!minimum!and!maximum!lengths,!and!the!dashed!line!shows!the!maximum!possible!
length!for!this!area!(the!longest!diagonal!of!the!rectangular!bounding!box).!
Lengths$&$sizes$–$Maximum$Likelihood$Estimators$(MLE$analysis)$$
Tick!the!check!box!‘MLE!analysis’!to!calculate!the!best-fitting!power!law,!exponential!and!log-
normal!scaling!distributions!for!the!trace!and!segment!lengths.!Six!plots!are!produced,!three!
for!trace!lengths!(power!law,!exponential,!log-normal)!and!three!for!segment!lengths!(power!
law,! exponential,! log-normal).! The! key! parameters! for! each! of! the! best-fit! distributions! are!
displayed!in!a!text!boxes!within!the!plots.!The!%!degree!of!fit!for!each!distribution!for!each!of!
the!trace!and!segment!length!distributions!is!printed!in!the!MATLAB™!Command!window.!!
!
!
!
Figure!12.!MLE!analysis!for!segment!lengths!(trace!length!plots!not!shown).!Left!side!shows!
the!power-law!scaling!estimate,!centre!shows!the!exponential!scaling!estimate,!right!shows!
the!log-normal!scaling!estimate.!In!each!case,!the!key!parameters!of!the!best-fit!distribution!
are!provided!in!the!inset!text!box.!Note:!different!X-!and!Y-axis!scales!as!appropriate!for!these!
plots!(see!Rizzo!et!al.,!2017a!for!more!details).!!!
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$
Figure! 13.! MLE! analysis! for! segment! lengths! with! applied! upper! and! lower! cut-offs,!
respectively!of!10%!and!5%,!of!the!initial!dataset!(marked!with!red!dashed!lines).!Compared!
with!the!previous!figure!(Figure!10),!note!the!changes!in!the!estimated!statistical!parameters!
(see!inset!text!boxes).!
Lengths$&$sizes$–$Variogram$of$segment$lengths$
From!version!2.2!we!include!a!plot!showing!the!variogram!(or!semi-variogram)!of!segment!
lengths!(see! Bohling! (2005)! for! an! introduction).! We! calculate! gamma! as! a! function! of! the!
separation! of! two! segments! from! the! stored! coordinate! positions! of! all! the! segment! mid-
points!(centroids)!for!every!pair!of!segments.!In!future!versions!we!plan!to!add!curve!fitting!to!
this!initial!raw!data!plot,!and!allow!for!anisotropy!(directional!dependence)!in!the!data.!!
!
!
Figure!14.!Variogram!of!segment!lengths!(left)!for!the!fracture!map!shown!with!nodes!(right).!!!
$
$
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Lengths$&$sizes$–$Cross$plot$lengths$v$angles$
Tick!the!‘Cross-plot!lengths!v!angles’!check!box!to!generate!a!scatter!plot!of!segment!lengths!
versus!their!angles.!This!may!be!useful!to!assess!any!correlation!between!fracture!size!and!
orientation.!!!!
!
!
Figure! 15.!Cross!plot!of!segment!lengths!and!angles.!The!red!lines!mark!the!minimum!and!
maximum!lengths.!!
Lengths$&$sizes$–$Block$sizes$
Click!on!the!check!box!‘Block!size!graphs’!to!generate!data!on!block!sizes!bound!by!fractures!
in!the! loaded! pattern.!!This!produces! three! output! figures:! a!map!showing!the! locations! of!
each! block! centroid,! a! log-log! plot! of! cumulative! frequency! versus! block! area! to! assess! the!
scaling!relationship,!and!a!simple!histogram!of!block!sizes!(areas).!!!
!
!
!
Figure!16.!Output!for!block!sizes.!Left!shows!the!centroid!map,!with!block!centroids!shown!as!
red!dots!overlain!on!the!fracture!trace!map.!Centre!shows!the!log-log!frequency-size!(area)!
graph.!Right!shows!a!simple!histogram!of!block!sizes!(areas).!
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10.3.$ Crack$tensor$plot$$
Click! the! check! box! ‘Crack! tensor! plot’! to! generate! 2-dimensional! crack! tensors! from! the!
fracture!traces!(see!Oda!et!al.,!1987!and!Suzuki!et!al.,!1997!for!more!details).!The!crack!tensor!
combines! data! from! the! orientation! distribution! (angles)! with! the! sizes! (lengths)! of! the!
fractures! and! their! spatial! density! to! provide! a! single! dimensionless! measure! of! a! crack!
pattern.!The! crack! tensors! of! 0th,! 2nd,! 4th!and! 8th!rank! can! be! related! to! other! physical!
properties,!such!as!bulk!permeability!(see!below),!bulk!elasticity!and!the!acoustic!velocities!
(e.g.!P-!and!S-wave!anisotropy).!Note!that!the!orientations!used!in!the!crack!tensor!calculation!
are! the! poles! to! the! fracture! trace! segments;! and! therefore! the! magnitude! of! the! tensor! is!
largest! in! directions! perpendicular! to! those! shown! in! the! rose! diagram! for! the! same! data,!
which!uses!the!angles!of!the!trace!segments,!and!not!the!poles.!!
!
!
Figure! 17.!Crack!tensors!(left)!for!the!fracture!pattern!shown!in!the!map!(right).!0th,!2nd,!4th!
and!8th!rank!crack!tensors!are!shown!in!different!colours.!!!!
10.4.$ Orientations$–$Histogram$of$angles$
Click!the!check!box!‘Histogram!of!angles’!to!generate!a!simple!graph!of!segment!angles!and!
their!frequencies,!expressed!as!a!percentage!of!the!total.!The!user!can!adjust!the!number!of!
bins!used!in!the!histogram!using!the!dropdown!list!box!‘Number!of!bins’!(choices!of!10,!20,!30!
or!50).!
Orientations$–$Rose$diagram$
Click!the!check!box!‘Rose!diagram!(equal!area)’!to!generate!a!rose!plot!of!segment!angles.!An!
equal! area! format! is! preferred! over! linear! scaling! (Nemec,! 1988)! as! it! provides! a! more!
consistent!orientation!distribution.!Tick!the!box!labelled!‘Length!weighted’!to!weight!each!bin!
by!the!total!segment!length!that!lies!within!that!angular!arc.!Tick!the!check!box!‘Show!mean!
vector’!to!display!a!red!line!across!the!rose!plot!marking!circular!mean.!Note!that!this!measure!
is!really!only!valid!for!unimodal!distributions.!!The!user!can!adjust!the!angular!bin!size!using!
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the!dropdown!list!box!‘Bin!size’!(choices!of!5,!10,!20!and!30),!and!rotate!the!data!by!a!specified!
angle! in! cases! where! the! Y-axis! of! the! mapped! fracture! pattern! is! not! parallel! to! North! by!
supplying! a! value! in! the! text! box! labelled! ‘Rotate! Y-axis! from! N’! (see! the! section! ‘North!
Correction’,!above).!!
!
!
Figure!18.!A!histogram!of!segment!angles!(left)!and!an!equal!area!rose!diagram!(right)!of!the!
same!data.!!
!
!
!
Figure! 19.! An! equal! area! rose! diagram! colour-coded! by! segment! strike! (left)! and! the!
corresponding!segment!strike!map!(right).!!
!! !
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10.5$$ Fluid$flow$–$I-Y-X$connectivity$!
Click!on!the!check!box!‘I-Y-X!connectivity’!to!generate!a!ternary!plot!of!segment!connectivity.!
The!ternary!plot!is!from!Manzocchi!(2002)!and!shows!the!relative!proportions!of!isolated!(I),!
splay!or!abutment!(Y),!and!intersection!(X)!nodes!in!the!fracture!network.!Better!connected!
fracture!patterns!plot!towards!the!base!of!the!triangle!(i.e.!a!higher!proportion!of!X+Y!nodes).!
If!the!hydraulic!conductivity!of!the!fractures!differs!significantly!from!that!of!the!rock!matrix!
(either!higher!or!lower)!then!the!connectivity!of!the!network!has!implications!for!fluid!flow!
through!the!rock!mass.!
!
!
Figure! 20.! A! ternary! plot! of! fracture! segment! connectivity! (left)! for! the! fracture! pattern!
shown!in!the!map!(right).!Two!contour! lines!for!Connections!per! Line!(CL)!are!also! shown,!
using!indicative!values!described!by!Sanderson!&!Nixon!(2015).!!
Fluid$flow$–$Permeability$ellipses$$
Click!on!the!check!box!‘Permeability!ellipse’!to!generate!two!plots!of!permeability!anisotropy,!
one!in!the!direction!of!flow!and!one!in!the!direction!of!fluid!pressure!gradient!(Long!et!al.,!
1982).!Permeabilities!are!calculated!using!the!2nd!rank!crack!tensor!following!the!method!of!
Oda!et!al.!(1987)!and!Suzuki!et!al.!(1997).!Plotting!of!the!ellipses!follows!the!method!of!Long!
et!al.!(1982),!with!the!semi-axes!scaled!as!√k1!and!√k2!for!the!direction!of!flow,!and!scaled!as!
1/√k1!and!1/√k2!for!the!direction!of!gradient,!where!k1!is!the!maximum!permeability!and!k2!is!
the!minimum.!The!permeability!calculation!is!based!on!a!parallel!plate!model!of!fractures,!and!
a!constant!default!aperture!of!1!x!10-3!units!is!assumed.!For!the!example!shown,!this!equates!
to! 1!x! 10-3! metres! =! 1! mm,! a! reasonable! default! value.! Scaled! apertures! can! also! be! used,!
where!aperture!A!is!a!function!of!segment!length!according!to!the!general!equation!A!=!aLb,!
where!a!is!a!constant!factor!and!b!is!a!power!law!exponent!(>=!1).!!!
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!
Figure! 21.! Permeability! of! a! fracture! network,! plotted! as! 2-D! ellipses!to! visualise! the!
anisotropy!in!the!direction!of!flow!(left)!and!the!direction!of!pressure!gradient!(right).!!
10.6$$ Wavelet$analysis$
FracPaQ!version!2.2!implements!2D!wavelet!analysis!of!fracture!maps!(Rizzo!et!al.,!2017b),!
using!either!Morlet!or!Mexican!hat!wavelet!filters.!Wavelet!analysis!(WA)!is!based!on!scaling!
and!moving!a!filter,!the!selected!wavelet,!onto!a!signal!which!in!our!case!is!a!binary!fracture!
trace! map.! Compared! to! the! classical! Fourier! Transform,! WA! results! are! better! for! two!
reasons:!1)!the!size!of!the!wavelet!can!be!adjusted!according!to!the!scale!of!the!entity!under!
consideration;! 2)! critically,! the! wavelet! can! be! selected! to! best! match! the! geometry! of! the!
entity,! i.e.! cracks.! Building! on! the!work! of! Ouillon! et! al.! (1995),! we! have! recently!
demonstrated! the! superiority! of! the! Morlet! wavelet! over! the! Mexican! hat! wavelet! in!
quantifying!scale!transitions!in!2D!fracture!maps!(Rizzo!et!al.,!2017b).!!
!
!
Figure! 22.! The! new! Wavelets...! button! on! the! main! GUI! window! (left)! and! the! Wavelet!
analysis!parameter!dialog!(right).!!!
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To! run! a! Wavelet! analysis,! click! on! the! Wavelets! button! on! the! main! GUI! window.! A! new!
smaller!dialog!window!appears!to!collect!the!parameters!needed!for!the!analysis.!The!list!of!a-
values!controls!the!sizes!of!the!wavelets!used.!Note!that!1a!=!2.2!pixels.!The!list!of!L-values!
controls!the!shapes!(anisotropies)!of!the!wavelets!used.!In!general,!it!is!best!to!use!the!same!
values!for!a!and!L,!and!a!common!pattern!is!to!use!a!list!such!as!2n,!n=1,2,3...!etc.!The!angular!
increment! controls! the! subdivision! of! orientations! used! in! the! analysis:! setting! a! smaller!
number!here!can!significantly!increase!run-times.!Wavelet!analysis!can!be!slow:!processing!of!
a!typical!fracture!map!for!3-4!values!of!a!and!L,!and!with!a!20°!angular!increment!can!take!10-
15!minutes!on!a!recent!(February!2018)!computer.!After!choosing!the!parameter!values,!click!
Run,!or!cancel!to!return!back!to!the!main!GUI!window.!
FracPaQ! outputs! from! the! Wavelet! analysis! are! shown! in! Figure! 23.! For! each! a-value!
(selected! length! scale),! 3! plots! are! produced:! the! Wavelet! Coefficient! (WC)! map,! the! rose!
diagram!of!optimal!orientations!and!the!histogram!of!WC!values.!!
!
!
Figure! 23.! Examples! of! output! from! Wavelet! analysis! in! FracPaQ.! Left! side! shows! results!
from!Morlet!wavelet!analysis!at!a!length!scale!of!a=4,!and!right!side!shows!same!analysis!for!a!
length! scale! a=8.! Note! how! the! orientation! distribution! (rose! diagrams)! change! at! bigger!
length!scales.!See!Rizzo!et!al.,!2017b!for!more!details.!!
!
!
!
!
!
! !
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11.!Multi-coloured!fracture!maps!
If! you! supply! a! multi-layer! .svg! node! file! as! input,! where!each! layer! has! traces! of! different!
colours,!then!FracPaQ!version!2.2!can!produce!a!separate!set!of!output!plots!for!each!fracture!
set!in!the!corresponding!colours.!!
!
!
!
!
!
!
!
!
!
Figure! 24.!Examples!of!multicolour!fracture!trace!support!in!FracPaQ!version!2.2.!Top!row!
shows! the! outputs! for! the! original! input! *.svg! file! containing! two! coloured! layers! for! 2!
different!fracture!sets!(in!this!cases!separated!on!the!basis!of!vein!fill).!The!middle!row!shows!
the!trace!map,!segment!lengths!and!rose!plot!for!the!green! fractures! only;! the! bottom! row!
shows!the!same!outputs!for!the!pink!fractures.!These!last!two!sets!of!output!were!generated!
by!inputting!the!‘*_colour*_converted.txt’!files!produced!from!the!first!step.!!
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On!reading!the!.svg!file,!FracPaQ!reads!in!the!colour!code!(in!RGB!format)!from!each!layer!
and! uses! this! colour! to! write! out! a! separate! .txt! node! file! for! each! fracture! set.! These! are!
named! in! the! pattern! ’name_colourHHHHHH_converted.txt’,! where! HHHHHH! is! the!
hexadecimal!colour!code!for!the!RGB!colour!used!in!the!original!.svg!file!for!that!set!of!fracture!
traces.! See! here! for! examples.! There! will! be! as! many! ‘*_converted.txt’! files! as! there! are!
different!colours!in!the!original!.svg!file.!If!you!now!input!each!of!these!files!into!FracPaQ,!all!
the!output!plots!will!be!coloured!according!to!the!fracture!set!colour.!!!
!
12.! Statistical!estimations!using!MLE!analysis!
FracPaQ! uses! Maximum! Likelihood! Estimators! (MLEs)! to! fit! a! given! dataset! of! fracture!
lengths!and!then!to!extract!the!corresponding!best-fit!distribution!parameters.!To!illustrate!
the!methodology!used!in!FracPaQ!for!fitting!a!given!dataset!and!extracting!the!corresponding!
distribution! parameters,! we! employ! a! synthetic! power-law! dataset.! Before! attempting! any!
fitting!or!parameter!estimation,!data!are!arranged!following!a!cumulative$distribution$function!
(CDF), 𝐹(𝑥),! where! the! fracture! trace! length! data! (𝑋=𝑥!,𝑥!,…,𝑥!)! are! plotted! against! the!
probability!that!𝑥!!(a!random!variable)!has!a!value!greater!than!or!equal!to!𝑥,!i.e.!𝑃𝑟(𝑥!≥𝑥)!
(see!Figure!19).!
After!this!stage,!FracPaQ!can!estimate!the!parameters!governing!the!distribution,!which,!for!a!
power-law! distribution,! are! the! 𝛼-value! (i.e.! the! scaling! parameter)! and! 𝑥!"# !(i.e.! the!
minimum!value!until!which!the!power-law!function!holds).!In!this!case,!for!estimating!𝛼!we!
need! to! have! information! about!𝑥!"#,! which! constitutes! the! most! challenging! part! of! the!
estimation!method.!This!involves!the!application!of!the!Kolmogorov-Smirnoff!(K-S)!statistic,!
through! which!𝑥!"#!is! found! minimising! the! value! of! the! K-S! statistic.! The! latter! simply!
requires!FracPaQ!to!calculate! the!maximum!distance!between! the!CDF!of!the!data!and!the!
fitted!model!(inset!in!Figure!19):!
𝐷= max!!!!"# 𝑆𝑥−𝐹(𝑥)!(Eq.!1)!
where! 𝑆𝑥!and! 𝐹𝑥!are,! respectively,! the! theoretical! probability! distribution! and! the!
empirical!distribution!(e.g.!measured!values!for!fracture!trace!lengths).!The!estimate!of!the!
lower!bound!of!the!observed!data,!𝑥!"#,!is!the!value!of!𝑥!that!minimises!𝐷.!This!approach!is!
completely! general! and! is! also! used! for! the! other! statistical! distribution! considered!in!
FracPaQ.!Practically,!to!find!the!best!possible!𝑥!"#!value,!FracPaQ!first!runs!through!all!the!
observed!data!and!uses!each!datum!as!the!‘true’!𝑥!"#;!then!it!truncates!the!dataset!to!include!
every!data!above!the!temporary!𝑥!"#,!and!finally!it!uses!these!data!to!compute!the!empirical!
cumulative! density! function!𝐹𝑥,! and! the! theoretical! CDF!𝑆𝑥.! Next,! FracPaQ!takes! the!
maximum! of! the! absolute! value! of! the! difference! between! each! pair! of! empirical! and!
theoretical!CDF!value!(Eq.!1):!this!computation!is!the!actual!K-S!statistic.!
!
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!
Figure!25.!Example!of!cumulative!density!function!probability!𝑃𝑟 (𝑥!≥𝑥)!for!MLE!plot.!Inset!
plot!is!a!closer!caption!of!the!respective!graph.!!Red!bar!is!simply!a!graphical!illustration!of!the!
calculation!of!the!K-S!statistic.!
FracPaQ!then!calculates!the!minimum!of!the!K-S!statistic,!finds!the!corresponding!𝑥!"#!value,!
and!assigns!it!to!be!the!most!likely!value!of!the!theoretical!CDF;!in!other!words,!the!new!‘true’!
𝑥!"#.!Once!the!lower!bound!has!been!found,!FracPaQ!can!then!find!the!other!parameter!for!
the!distribution!of!interest.!
Despite! its! unequivocal! robustness! for! parameter! estimation! (e.g.! Rizzo! et! al.,! 2017a),! the!
maximum! likelihood! method! always! yields! a! value! of! the! MLE! relative! to! the! used! model!
function,!no!matter!which!data!one!is!using,!even!in!the!case!where!the!analysed!sample!does!
not!derive!from!that!theoretical!distribution.!Therefore,!no!quality!of!fit!is!guaranteed.!For!this!
reason,!a!goodness-of-fit!is!needed!and,!following!Clauset!et!al.!(2009),!FracPaQ!uses!the!K-S!
test.!
Computationally,!in!FracPaQ!this!K-S!test!is!performed!generating!a!large!number!of!random!
datasets,! all! following! the! theoretical! distribution! –! 𝑆(𝑥)!–! of! interest,! based! on! the!
corresponding! estimated! parameters.! Hence,! for! the! example! case! (i.e.! a! power-law!
distribution),! these! random! ‘new’!distributions! would! be! power-law! functions,! having! as!
parameter!those!estimated.!Each! of!these!random!‘new’!distributions!are!then! compared!to!
the!empirical!dataset!(𝐹(𝑥))!to!determine!if!the!two!populations!have!the!same!underlying!
function.!Intuitively,!if!𝐷!is!large!the!fit!is!poor,!whereas!if!𝐷!is!small!the!fit!can!be!considered!
good.!At!the!same!time,!the!relative!scale!of!𝐷!is!provided!by!its!own!probability!distribution,!
through! the! calculation! of! the!𝑝-value.! Under! the! hypothesis! that! the! data! follow! the!
theoretical!distribution,!with!the!parameter!obtained!from!the!previous!estimation!(this!is!the!
null-hypothesis),!the!𝑝-value!provides!the!probability!that!the!K-S!statistic!takes!a!value!larger!
a)
b)
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than! the! one! obtained! empirically.! This! mean! that,! if! the!𝑝-value! is! greater! than! a! set!
significance!level,!the!test!fails!to!reject!the!null-hypothesis!and!it!be!concluded!that!both!the!
datasets!follow!the!same!statistical!function;!i.e.!both!𝑆(𝑥)!and!𝐹(𝑥)!have!the!same!underlying!
distribution.! In! contrast,! if! the!𝑝-value! is! smaller! than! the! set! significance! level,! the! null-
hypothesis!is!rejected.!More!formally,!in!FracPaQ!if!
𝑝−value ≥0.05 ⇒accept the null −hypothesis!
otherwise,!
𝑝−value ≤0.05 ⇒reject the null −hypothesis!
Of! course,! the! significance! level! of!0.05!is! arbitrary! and! can! be! changed! to! another! value.!
Because! the!𝑝−value !derived! from! the! test! needs! to! be! comparable! with! the! chosen!
significance!level!(i.e.!0.05),!to!have!a!𝑝−value!correct!within!two!decimal!points!(Rizzo!et!al.,!
2017a;!Clauset!et!al.,!2009),!2500!K-S!tests!are!by!default!performed!when!running!the!MLE!
analysis!in!FracPaQ.!
!
13.!Utilities!
In!FracPaQ!v2.2,!we!provide!two!utility!MATLAB!scripts!that!may!help!users!with!common!
tasks!in!preparing!input!for!FracPaQ.!We!will!look!to!incorporate!these!within!the!main!body!
of!code!in!future!versions.!Run!these!scripts!from!the!MATLAB™!command!line.!!
13.1$ Move2FracPaQ.m$(contributed$by$X.$Ogaya$&$J.$Alcalde)$
This!script!converts!fracture!traces!from!Midland!Valley's!Move™!software!into!the!Node!file!
format! of! FracPaQ.! ! Full! instructions! for! use! are! included! in! the! header! comments.! of! this!
script.!!
13.2$ FracPaQ2D_imagePreprocess.m$
This!script!illustrates!some!of!the!functions!in!the!MATLAB™!Image!Processing!toolbox!that!
can! help! to! convert! a! 'raw'! digital! image,! e.g.! an! outcrop! photograph! taken! with! a! digital!
camera,!into!the! 8-bit!binary!format!needed!for!the!FracPaQ!Hough! transform!method.!An!
excellent!example!of!the!methods!that!can!be!employed!to!improve!'signal-to-noise'!ratios!in!
these!kinds!of!images!can!be!found!in!Griffiths!et!al.!(2017).!!!
NB!we!strongly!advise!that,!for!the!best!results!with!FracPaQ,!the!user!should!digitise!
their! images! (or! maps)! using! a! graphics! package! and! use! the! Node! file! input! format!
(see!section!7!above).!!!
!
!
! !
FracPaQ!User!Guide!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Version!2.2,!February!2018!
Page!26!of!28!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
14.! Acknowledgements!
We!have!used!public!domain!MATLAB™!code!to!enhance!FracPaQ:!
• lineSegmentIntersect.m! written! by! U.! Murat! Erdem;! available! on! Mathworks™!
FileExchange:!
http://uk.mathworks.com/matlabcentral/fileexchange/27205-fast-line-segment-
intersection!
• readtext.m!written!by!Peder!Axensten;!available!on!Mathworks™!FileExchange:!
http://uk.mathworks.com/matlabcentral/fileexchange/10946-readtext!
• cmocean.m! written! by! Kristen! Thyng!and! Chad! Greene;! available! on! Mathworks™!!
FileExchange:!
http://uk.mathworks.com/matlabcentral/fileexchange/57773-cmocean-perceptually-
uniform-colormaps!
• variogram.m!written!by!Wolfgang! Schwangart;!available!on!Mathworks™!FileExchange:!
https://uk.mathworks.com/matlabcentral/fileexchange/20355-experimental--semi---
variogram!
We! thank! Paul! Gillespie! (Statoil),! Mark! Fischer! (Northern! Illinois! University)! and! Bill!
Dunne! (University! of! Tennessee,! and! Editor! of! the! Journal! of! Structural! Geology)! for!
comments! on! our! original! manuscript,! which! led! to! major! improvements! in! the! code,! the!
published!paper!in!the!Journal!of!Structural!Geology!and!this!User!Guide.!!!
Matthew! Belshaw!and! Daniel! Knight! (both! MSc! students! at! the! University! of! Aberdeen,!
2016-2017)!contributed!to!the!testing!of!FracPaQ,!and!specific!details!for!producing!.svg!files!
in!Adobe!Illustrator™!and!CorelDraw™,!respectively.!!
If!you!use!FracPaQ!in!your!scientific!research,!please!include!a!citation!to!one!or!more!of!the!
following!papers,!as!appropriate:!
Healy,!D.,!Rizzo,!R.E.,!Cornwell,!D.C.,!Farrell,!N.J.C.,!Watkins,!H.,!Timms,!N.E.,!Gomez-Rivas,!E.!&!
Smith,!M.!(2017).!FracPaQ:!a!MATLAB™!toolbox!for!the!quantification!of!fracture!patterns.!
Journal$of$Structural$Geology,$95,!pp1-16.!!!
Rizzo,!R.E.,!Healy,!D.!&!De!Siena,!L.!(2017a).!Benefits!of!a!Maximum!Likelihood!Estimator!for!
fracture!attribute!analysis.!Journal$of$Structural$Geology,$95,!pp.17-31.!!
Rizzo,!R.!E.,!Healy,!D.,!Farrell,!N.!J.,!&!Heap,!M.!J.!(2017b).!Riding!the!right!wavelet:!Quantifying!
scale!transitions!in!fractured!rocks.!Geophysical$Research$Letters.!
!
NB!these!are!all!Gold!open!access.!!
!
! !
FracPaQ!User!Guide!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Version!2.2,!February!2018!
Page!27!of!28!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
15.!! Bibliography!
Belshaw,!M.!(2017).!Unpublished!MSc!thesis.!University!of!Aberdeen.!!
Bohling,! G.! (2005).! Introduction! to! geostatistics! and! variogram! analysis.! Kansas$ Geological$
Survey,!1,!1-20.!
Bonnet,!E.,!Bour,!O.,!Odling,!N.E.,!Davy,!P.,!Main,!I.,!Cowie,!P.!&!Berkowitz,!B.!(2001).!Scaling!of!
fracture!systems!in!geological!media.!Reviews$of$Geophysics,!39(3),!pp.347-383.!
Clauset,!A.,!Shalizi,!C.!R.!&!Newman,!M.!E.!J.!(2009).!Power-law!distributions!in!empirical!data.!
SIAM$Review,!51(4),!661.!!
Dershowitz,! W.S.! &! Herda,! H.H.! (1992).! Interpretation!of!fracture! spacing! and! intensity.! In:!
The!33th!US!Symposium!on!Rock!Mechanics!(USRMS).!American!Rock!Mechanics!Association.!
Farrell,!N.J.C.!(2015).!Unpublished!PhD!thesis.!!University!of!Aberdeen.!!
Gillespie,! P.A.,! Howard,! C.B.,! Walsh,! J.J.! &! Watterson,! J.! (1993).! Measurement! and!
characterisation!of!spatial!distributions!of!fractures.!Tectonophysics,!226(1-4),!pp.113-141.!
Griffiths,!L.,!Heap,!M.!J.,!Baud,!P.!&!Schmittbuhl,!J.!(2017).!Quantification!of!microcrack!
characteristics!and!implications!for!stiffness!and!strength!of!granite.!International$Journal$of$
Rock$Mechanics$and$Mining$Sciences,!100,!138-150.!
Healy,!D.,!Rizzo,!R.E.,!Cornwell,!D.C.,!Farrell,!N.J.C.,!Watkins,!H.,!Timms,!N.E.,!Gomez-Rivas,!E.!&!
Smith,!M.!(2017).!FracPaQ:!a!MATLAB™!toolbox!for!the!quantification!of!fracture!patterns.!
Journal$of$Structural$Geology,$95,!pp1-16.!!!
Knight,!D.!(2017).!Unpublished!MSc!thesis.!University!of!Aberdeen.!!
Long,!J.C.S.,!Remer,!J.S.,!Wilson,!C.R.!&!Witherspoon,!P.A.!(1982).!Porous!media!equivalents!for!
networks!of!discontinuous!fractures.!Water$Resources$Research,!18(3).!!
Manzocchi,!T.!(2002).!The!connectivity!of! two-dimensional!networks!of!spatially!correlated!
fractures.!Water$Resources$Research,!38(9).!
Mauldon,!M.,!Dunne,!W.M.!&!Rohrbaugh,!M.B.!(2001).!Circular!scanlines!and!circular!windows:!
new! tools! for! characterizing! the! geometry! of! fracture! traces.! Journal$ of$ Structural$ Geology,!
23(2),!pp.247-258.!
Middleton,!G.!V.!(2000).!Data$analysis$in$the$Earth$Sciences$using$MATLAB.!Prentice!Hall.!
Nemec,!W.!(1988).!The!shape!of!the!rose.!Sedimentary$Geology,!59(1-2),!pp.149-152.!
Neupauer,!R.!M.!&!Powell,!K.!L.!(2005).!A!fully-anisotropic!Morlet!wavelet!to!identify!dominant!
orientations!in!a!porous!medium.!Computers$&$geosciences,!31(4),!465-471.!
Newman,! M.! E.! J.! (2005).! Power! laws,! Pareto! distributions! and! Zipf’s! law.! Contemporary$
Physics,!46(5),!323–351.!
Oda,!M.,!Hatsuyama,!Y.!&!Ohnishi,!Y.!(1987).!Numerical!experiments!on!permeability!tensor!
and!its!application!to!jointed!granite!at!Stripa!mine,!Sweden.!Journal$of$Geophysical$Research:$
Solid$Earth,!92(B8),!pp.8037-8048.!
Ouillon,!G.,!Sornette,!D.,!&!Castaing,!C.!(1995).!Organisation!of!joints!and!faults!from!1-cm!to!
100-km!scales!revealed!by!optimized!anisotropic!wavelet!coefficient!method!and!multifractal!
analysis.!Nonlinear$Processes$in$Geophysics,!2(3/4),!158-177.!
FracPaQ!User!Guide!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Version!2.2,!February!2018!
Page!28!of!28!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
Rizzo,!R.E.,!Healy,!D.!&!De!Siena,!L.!(2017a).!Benefits!of!a!Maximum!Likelihood!Estimator!for!
Fracture!Attribute!Analysis.!Journal$of$Structural$Geology,$95,!pp.17-31.!!
Rizzo,!R.!E.,!Healy,!D.,!Farrell,!N.!J.,!&!Heap,!M.!J.!(2017b).!Riding!the!right!wavelet:!Quantifying!
scale!transitions!in!fractured!rocks.!Geophysical$Research$Letters.!
Sanderson,! D.J.! &! Nixon,! C.W.! (2015).! The! use! of! topology! in! fracture! network!
characterization.!Journal$of$Structural$Geology,!72,!pp.55-66.!
Smith,!M.!(2016).!Unpublished!MSc!thesis.!University!of!Aberdeen.!!
Suzuki,!K.,!Oda,!M.,!Yamazaki,!M.!&!Kuwahara,!T.!(1998).!Permeability!changes!in!granite!with!
crack! growth! during! immersion! in! hot! water.! International$ Journal$ of$ Rock$ Mechanics$ and$
Mining$Sciences,!35(7),!pp.907-921.!
Thyng,!K.!M.,!Greene,!C.!A.,!Hetland,!R.!D.,!Zimmerle,!H.!M.,!&!DiMarco,!S.!F.!(2016).!True!Colors!
of!Oceanography.!Oceanography,!29(3),!10.!
Vermilye,!J.!M.!&!Scholz,!C.!H.!(1995).!Relation!between!vein!length!and!aperture.!Journal$of$
structural$geology,!17(3),!423-434.!
Watkins,!H.!(2015).!!Unpublished!PhD!thesis.!!University!of!Aberdeen.!
!
