Frac Pa Q User Guide V2.2 RELEASE
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Fracture Pattern Quantification
User Guide
Version 2.2, February 2018
David Healy & Roberto Rizzo
d.healy@abdn.ac.uk
With contributions from Nick Timms, Matthew Belshaw & Daniel Knight
FracPaQ User Guide
Contents
1.
2.
3.
4.
5.
6.
Summary of changes in FracPaQ version 2.2
Introduction
System requirements
Obtaining & installing the code
Starting FracPaQ
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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Version 2.2, February 2018
FracPaQ User Guide
1.
Version 2.2, February 2018
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:
•
•
2.
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.
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.
2.
3.
4.
5.
Download the source code as a .zip or .tar.gz file, and extract all of the files.
Put all of these files into a single folder.
Start MATLAB™.
Set the current working folder in MATLAB™ to the same folder you installed the code in.
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.
Version 2.2, February 2018
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
Version 2.2, February 2018
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.
Version 2.2, February 2018
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 lognormal 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 midpoints (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
Version 2.2, February 2018
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 avalues 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 1015 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.
Version 2.2, February 2018
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!!!!"# 𝑆 𝑥 − 𝐹(𝑥)
(Eq. 1)
where 𝑆 𝑥 and 𝐹 𝑥 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 𝐹 𝑥 , and the theoretical CDF 𝑆 𝑥 . 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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a)
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.
b)
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
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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 nullhypothesis 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).
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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-segmentintersection
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-perceptuallyuniform-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.
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