Discrete Fracture Network (DFN) User Guide

This guide covers the Discrete Fracture Network (DFN) modeling capabilities in VRGS, including generation, visualization, and analysis features.

Building a DFN from interpreted data?

If you have interpreted trace polylines or structural measurements and want the DFN to match them — derived Fisher statistics, observation-driven P32, placement biased toward your outcrop, validation against observed traces — see the companion DFN Interpretation Matching guide.

Overview

A Discrete Fracture Network (DFN) is a stochastic representation of fracture systems in rock masses. VRGS allows you to:

• Generate statistically-controlled fracture networks from orientation data
• Compute fracture-fracture intersections and fracture traces on outcrop meshes
• Analyze network connectivity and topology using graph theory
• Model stress-controlled fracture propagation
• Create cross-sections through the network
• Filter fractures by proximity to outcrops
• Colour fractures by computed geometric, connectivity, topology, and stress attributes

Creating a DFN

A DFN is created from one or more orientation (plane attitude) groups:

1. In the Interpretation tree, select the orientation group(s) you want to base the network on (each group becomes one fracture set)
2. Right-click and choose Create DFN from Selected Groups

For each group, VRGS derives the set's mean orientation and Fisher concentration (K) from the measured poles. If the group's planes carry trace geometry, a size distribution is also fitted to the trace lengths (see Distribution Fitting); otherwise sensible defaults are used. The source of each derived value is shown in the properties panel (P32 Source, Size Distribution Source).

The names of the source orientation groups are stored with the DFN, so you can later right-click the DFN and choose Re-derive from Source Orientation Groups to re-run the derivation against updated data.

Setting the Domain

The DFN domain defines the 3D bounding box where fractures are generated. It is set from the data extents at creation and can be edited in the properties panel under Domain (Min/Max X, Y, Z).

Fracture Sets

A DFN consists of one or more fracture sets, each with distinct statistical properties. Sets correspond to the orientation groups the DFN was created from; each can be enabled/disabled individually (Enable Set).

Fracture Set Parameters

All parameters are edited in the properties panel under Fracture Sets.

Orientation (Fisher Distribution)

Parameter	Description
Mean Dip	Average dip angle (0-90 degrees from horizontal)
Mean Azimuth	Average dip direction (0-360 degrees, 0=North)
Fisher K	Fisher concentration parameter. Higher values = tighter clustering around mean

• Fisher K ≈ 1-5: Strongly dispersed orientations
• Fisher K = 10-20: Moderate clustering
• Fisher K = 50+: Tight clustering around mean

When derived from measurements, K is estimated from the resultant length of the pole population using the spherical (3D) Fisher maximum-likelihood estimator and clamped to the range [0.1, 1000].

Size Distribution

Parameter	Description
Size Distribution	Power Law, Lognormal, Exponential, or Uniform
Min Radius	Minimum fracture radius
Max Radius	Maximum fracture radius
Exponent	For power law distribution (typically 2.0-3.5)
Lognormal Mean / Stddev	Parameters of ln(r) for the lognormal distribution. Sampled by exact inverse-CDF truncation to [Min Radius, Max Radius]
Exponential Rate	Rate λ of P(r) ~ exp(-λ·r) for the exponential distribution. Larger λ means more small fractures. Leave at 0 to derive it from the size bounds; fitting a size distribution from measurements sets it automatically
Aspect Ratio	Major/minor axis ratio for elliptical fractures (1.0 = circular)

Deriving these from measurements

Rather than typing sizes and intensities, Derive DFN Statistics… (right-click the DFN) lets each set take each statistic from a named source — an orientation group, a trace group, a scanline, or a P21/P32 attribute layer — and cross-checks the intensity routes against each other.

It also handles the fact that a mapped trace is a chord, not a fracture diameter, so sizes fitted from traces are not systematically too small. See DFN — Matching Interpretations §4.

Intensity (P32)

Parameter	Description
P32 Intensity	Fracture area per unit volume (m²/m³)

P32 is the most robust intensity measure as it is scale-independent. The generator calibrates the fracture count from the expected fracture area of the size distribution (not the mean radius), computed in closed form, so the realized P32 matches the target even for broad power-law size distributions.

Under boundary truncation, fractures centred outside the domain that cross into it are generated too — for every spatial model, not just Poisson. Without that buffer, clipping removes fracture area near the faces with nothing flowing in to replace it, and the realized intensity lands below target (measured at about 11% low for the clustered, stratified and region-mask models before this was added).

The Set Statistics to CSV export reports target against realized P32 per set, and the validation report shows the same comparison graphically.

Spatial Model

Model	Description
Poisson	Completely random (uniform) fracture centres
Clustered	Neyman-Scott clustering: a reduced set of parent centres each spawns several fractures within Cluster Radius; Fractures per Cluster sets the mean number of fractures per parent
Stratified	Grid-stratified sampling — more even spatial coverage than Poisson
Region Mask	Placement biased toward a mask mesh with a Gaussian falloff (Region Mask Falloff); see the Interpretation Matching guide

Termination Rule

Rule	Description
None	Fractures extend beyond the domain unmodified
Truncate at Boundary	Fracture polygons are clipped to the domain box
Abutting	Younger fractures terminate against older ones with probability Abutting Probability

Modelled surfaces are handled separately from this rule — see Model Surfaces below. A set normally wants boundary truncation and bed confinement at the same time, so they are independent settings rather than alternatives.

Model Surfaces

Surfaces produced by the surface gridding can be attached to a DFN to give it a shape other than a box. Right-click the DFN and choose Model Surfaces….

Each attached surface has a role:

Role	Effect
Upper limit	The model exists below the surface. Fractures are clipped at it.
Lower limit	The model exists above the surface.
Internal barrier	A fracture reaching the surface either crosses it or stops, decided by the crossing probability.

An upper and a lower limit together define a bed, and a set generated inside it comes out with fractures as high as the bed rather than as high as its fitted size distribution. That is the point: bed-confined (stratabound) fracturing is one of the commonest fabrics in outcrop, and a box cannot express it.

Crossing probability

An internal barrier's crossing probability is the chance that a fracture reaching it carries on through:

• 1.0 — transparent. The surface changes nothing. This is the default, so attaching a surface never alters a model until you tell it to.
• 0.0 — impermeable. No fracture crosses; every set is confined between barriers.
• Anything between — that fraction of fractures crosses.

Each fracture set also has a Surface Crossing Probability in the property bar, which overrides the surface's own value:

Set value	Meaning
-1 (default)	Use whatever each surface says
0	This set is bed-confined, whatever the surfaces say
1	This set cuts straight through every surface

The set wins rather than the two being multiplied together, because a product could never express a through-going master joint set cutting a strong bedding plane — and that is a real and common fabric. With an override, a weak bedding plane can stop the bed-confined sets while the master joints cut it.

Coverage, and where a surface is not mapped

A modelled surface rarely spans the whole domain footprint. Where it does not, it is simply not there: fractures pass through freely and bounding surfaces do not clip. VRGS never extrapolates a surface beyond the data it was gridded from.

That is the honest behaviour, but it is invisible in the result — a bed boundary covering a third of the model produces a network that looks perfectly reasonable and is only bed-confined in a third of its volume. The dialog therefore shows a Coverage column for every surface. If it reads 35%, only 35% of your model is constrained by it.

The volume, and what P32 now means

caution

P32 is fracture area per unit volume, and bounding surfaces change the volume.

Once upper/lower limits are attached, intensity is measured against the bounded volume — the rock between the surfaces — not against the domain box. The dialog states the bounded volume and its share of the box for exactly this reason.

This is the right behaviour: a target P32 of 2 m²/m³ in a 10 m bed means 2 m² of fracture per cubic metre of that bed, which is what a geologist means by it. But it does mean that attaching a bounding surface to an existing model changes what its intensity numbers refer to, so re-check any P32 you derived before.

Target and realised intensity both use the bounded volume, so the two stay comparable and the validation report's intensity panel stays meaningful.

Surfaces that cannot be used

Source	How its side is decided
Gridded surface (Grid, from surface gridding)	The mesh is rasterised into a height field in the surface's own frame — its normal, not vertical, so a steep surface works
Implicit surface (from the 3D potential field)	The potential field itself, carried straight through. A level set has no preferred axis, so an overturned or recumbent fold works

Overturned folds therefore bound a DFN correctly. That comes from using the potential rather than the triangles: the sign is stratigraphic, not geometric, so "below the upper limit" means below it in the sequence. On an overturned limb that is the only reading that makes sense — a vertical line crosses the same horizon three times, and a geometric "below" would get the middle crossing wrong.

Two cases are still refused, both with a message rather than a silent approximation:

• A gridded surface that folds back on itself. The gridding cannot produce one, but an imported or hand-edited mesh can. Use an implicit surface instead.
• An implicit surface that has not been generated yet. After reopening a project the generator is re-attached but not re-solved. Regenerate the surface and the DFN will pick it up.

Beyond the region the implicit surface was solved over, the potential is extrapolation rather than data, so the constraint lapses there exactly as a gridded surface lapses outside its footprint — visible in the Coverage column.

Generating the Network

1. Right-click on the DFN in the Interpretation tree
2. Select Regenerate

Each regeneration uses a fresh random seed, producing a new statistical realization with the same parameters. The Grow menu item regenerates and then runs fracture propagation (see below).

Visualization

Visibility Options

• Visible: Toggle the fracture network on/off
• Show Vertices: Toggle rendering of fracture intersection points

Render Mode

The Render Mode selector in the properties panel controls how fractures are drawn:

Mode	Description
Disks	Filled fracture polygons, coloured by fracture set
Outlines	Polygon edges only
Colour by Property	Filled polygons coloured by the active attribute layer through the colour map

Generating an attribute automatically switches to Colour by Property so the result is visible immediately; switch back to Disks at any time without losing the attribute layer.

Stereonet and Rose Diagram

A visible DFN plots on any stereonet window (Window → New Stereonet View) just like measured orientation data:

• Poles: each fracture's pole is plotted, coloured by fracture set (a fixed per-set palette, distinct from the bedding/fracture/cleavage colours used for measured data). Very large networks are uniformly decimated to keep the display responsive while remaining statistically representative.
• Density contours: the Kamb/Schmidt contouring includes DFN poles.
• Rose diagram: fracture azimuths (or strikes, with Rose as Strike) contribute to the rose petals, density heat ring, and azimuth dots.
• SVG export includes the DFN poles, mirroring the on-screen display.

To validate a generated network against its source data, check both the DFN and its source orientation groups in the stereonet window — generated and observed poles plot together with distinguishable colours.

Slicers and proximity filtering

See Slicers and Proximity Filtering below.

Intersection Analysis

Fracture-Fracture Intersections

Compute where fractures intersect each other:

1. Right-click on the DFN
2. Select Fracture Intersections

This creates line segments at all fracture-fracture intersections and automatically builds the connectivity graph from them.

Mesh Intersections

Compute where fractures intersect mesh surfaces (e.g., outcrop models):

1. Ensure the target mesh(es) are visible
2. Right-click on the DFN
3. Select Mesh Intersections

This creates polylines representing the fracture traces on the mesh surface. Each fracture's intersection segments are automatically merged into continuous polylines.

It also records which fractures reached the mesh, so they can be displayed on their own — see Outcrop-Constrained Display below.

Outcrop-Constrained Display

Show Outcrop-Constrained Fractures Only (right-click menu) hides every fracture that does not cut the outcrop, leaving only the ones that should correspond to mapped traces. Most of a network never reaches the exposure, so this is the difference between eyeballing a comparable subset and eyeballing a cloud.

The subset comes from the last cut against a mesh (Mesh Intersections, or Score DFN against Selected Traces…), so run one of those first. Regenerating the DFN clears it — fracture identities are reused on rebuild, so a stale subset would name the wrong fractures.

Full guide: DFN — Validating Against Outcrop.

Truncating Against the Boundary

Truncate Against Boundary (right-click menu) clips all existing fracture polygons to the current domain box without regenerating the network.

Connectivity Graph

The fracture graph represents network topology using nodes (intersection points) and edges (intersection-line segments between nodes). It is built automatically when you run Fracture Intersections.

Topology classification

Nodes are classified following Sanderson & Nixon (2015):

• I-nodes — free ends of intersection lines (degree 1)
• Y-nodes — abutments, where the end of one intersection line meets the interior of another (degree 3)
• X-nodes — crossings of two intersection lines (degree 4)

The reported proportions P_I, P_Y, P_X are computed over the classified nodes and sum to 1. Connections per line (C_L = 2(N_Y + N_X)/N_lines) and connections per branch (C_B) are also computed.

Graph Analysis

The graph provides:

• Connected component identification
• Node degree analysis (I, Y, X nodes)
• Per-fracture topology and connectivity attributes (see Attributes)

Fracture Propagation

VRGS includes a stress-controlled fracture propagation model that extends fracture tips incrementally.

Propagation Parameters

Parameter	Description
Step Length	Distance each tip advances per step
Max Total Length	Maximum length a fracture can grow
Max Steps	Maximum propagation iterations
Abutting Distance	Distance at which tips terminate against other fractures
Abutting Probability	Probability of termination when within abutting distance
Stress Shadow Distance	Zone of influence around existing fractures
Stress Shadow Angle	Parallelism threshold: a tip growing within this angular tolerance of a nearby fracture's orientation (and within the shadow distance) terminates
Direction Noise	Random variation in growth direction (degrees)

Stress Tensor

The propagation direction is influenced by the regional stress field:

• Sigma1: Maximum principal stress direction — in-plane growth is deflected toward it, weighted by the stress anisotropy (σ1 − σ3)/σ1
• Sigma2: Intermediate principal stress
• Sigma3: Minimum principal stress (fractures propagate perpendicular to this)

Running Propagation

From the DFN's Attribute menu, under Propagation:

1. Initialize from Tips — creates two growth tips per fracture at the ends of its strike axis
2. Either:
   • Step: Single growth iteration
   • Run to Completion: Run until all tips terminate or Max Steps is reached

The spatial index is refreshed as fractures grow, so abutting and stress-shadow checks always see the current geometry.

Slicers (Cross-Sections)

Create 2D cross-sections through the DFN to visualize fracture traces on orthogonal planes.

Enabling Slicers

In the DFN properties panel:

1. Enable X Axis Slicer, Y Axis Slicer, or Z Axis Slicer
2. Adjust the slider position (0-100% of domain extent)

Slicer Display

When a slicer is active:

• Fracture disks are hidden
• Intersection lines with the slicer plane are displayed

Multiple slicers can be active simultaneously.

Proximity Filtering

Filter fractures based on their distance to outcrop data (meshes or point clouds).

Enabling Proximity Filter

1. In DFN properties, enable Enable Filter under Proximity Filter
2. Set the Distance Threshold (meters)
3. Optionally enable Invert Filter to show only distant fractures

How It Works

• Distances are computed from fracture centroids to the nearest visible mesh/point cloud
• A distance grid accelerates lookups for large datasets
• Filtering is performed in the GPU shader for smooth interaction

Attributes

DFN fractures support computed attributes for analysis and colour mapping. Generate them from the DFN's Attribute right-click menu; the fracture display is coloured by the most recently generated attribute.

Geometry Attributes

Attribute	Description
Dip	Dip angle (0-90°)
Azimuth	Dip direction (0-360°)
Width	Fracture extent along strike
Height	Fracture extent down dip
Area	Fracture polygon area
Distance	Distance from the fracture to the nearest visible outcrop

Connectivity Attributes

Attribute	Description
Fracture Intersections	Number of fractures intersecting this one
Connected Groups	Connected-component ID of the fracture
Component Size	Number of fractures in this fracture's connected component
Total Intersection Length	Summed length of this fracture's intersection lines
Intersection Density	Intersection length per unit fracture area
Mean Intersection Angle	Mean dihedral angle with intersecting fractures

Topology Attributes

Attribute	Description
Node Count	Graph nodes on this fracture
I-Node Count (Tips)	Free ends (degree 1)
Y-Node Count (Abutments)	Abutting junctions (degree 3)
X-Node Count (Crossings)	Crossing junctions (degree 4)
Branch Count	Graph edges on this fracture
Connections Per Line	Y+X junction count for this fracture

Stress Attributes

Stress attributes are resolved on each fracture plane from the full 3D stress tensor configured in the project's stress settings (principal magnitudes and plunge/azimuth directions). Three failure parameters are configurable per DFN in the properties panel under Stress Parameters:

Parameter	Default	Description
Pore Pressure (Pp)	0 MPa	Pore fluid pressure for effective-stress calculations
Friction Coefficient (μ)	0.6	Coulomb friction coefficient
Cohesion (C0)	0 MPa	Cohesive strength in the Coulomb criterion

Attribute	Formula	Description
Normal Stress (σn)	t·n with t = S·n	Stress normal to the fracture plane (MPa)
Shear Stress (τ)	√(|t|² − σn²)	Shear stress on the fracture plane (MPa)
Slip Tendency (Ts)	τ / σn	Shear reactivation potential (Morris et al. 1996)
Dilation Tendency (Td)	(σ1 − σn) / (σ1 − σ3)	Tendency to dilate and conduct fluid
Fracture Susceptibility	(σn − Pp) − (τ − C0)/μ	Pore-pressure increase required to reach Coulomb failure (MPa). Negative values indicate critically stressed fractures
Critically Stressed	τ ≥ C0 + μ·(σn − Pp)	Boolean Mohr-Coulomb failure flag

Interpretation

• High Slip Tendency (Ts > 0.6): Fracture likely to slip under the current stress
• High Dilation Tendency (Td > 0.6): Fracture likely open/conductive
• Low or negative Fracture Susceptibility: Fracture at or near the failure envelope — candidate for reactivation and induced seismicity risk

Export

All exports live under the DFN's right-click Export submenu.

Fractures to FAB...

Writes the fracture geometry in FracMan ASCII (.fab) format: one record per active fracture (polygon vertices plus the plane normal), with fracture-set names in the SETS section and each fracture tagged with its set number. The file imports directly into FracMan and other DFN tools that read the format (e.g. Itasca codes, porepy).

Set Statistics to CSV...

One row per fracture set: orientation statistics (mean dip/azimuth, Fisher K), size-distribution parameters, spatial model and termination settings, target vs realized P32, total fracture area, fracture count, and the provenance tags shown in the properties panel.

Fractures to CSV...

One row per fracture: id, set, centroid coordinates, dip, azimuth, radius, polygon area, vertex count, and active flag — ready for external statistical analysis.

Intersection Lines as Polylines

Creates a polyline group in the project tree (DFN-Intersection Lines) from the fracture-fracture intersection segments, with each fracture pair's segments merged into continuous chains. Run Fracture Intersections first. The polylines are ordinary project objects, so they can be exported onward via the standard polyline export.

Analysis

The DFN's right-click Analysis submenu provides quantitative network analyses.

Virtual Scanlines to CSV...

Samples the network along three virtual scanlines through the domain centre (one per axis) — the numerical equivalent of a scanline survey or vertical borehole. The CSV report contains:

• Per-set summary: intersection count, raw P10 (fractures/m), and Terzaghi-corrected P10 for each scanline. The Terzaghi correction weights each intersection by 1/|cos θ| (θ = angle between the scanline and the fracture pole) to compensate for the orientation sampling bias; weights are capped at 7 (≈8° blind zone, Priest 1993).
• Per-intersection detail: fracture id, set, position, distance along the line, cos θ, and weight — ready for spacing analysis.

Compare the Terzaghi-corrected P10 of a vertical scanline against real borehole fracture frequencies to validate intensity.

Percolation Analysis

Builds connected clusters from the fracture-fracture intersections (run Fracture Intersections first) and reports:

• Cluster count and the number of isolated (non-intersecting) fractures
• The largest-cluster fraction of fractures and of area — the percolation order parameter
• Whether any cluster spans the domain along X, Y, and Z (its geometry reaches both opposite faces)

A network that spans the domain is above the percolation threshold and can support large-scale flow; details for the top clusters are written to the log.

Oda Permeability Tensor...

Computes the upscaled equivalent permeability tensor (Oda 1985) for a user-specified hydraulic aperture, treating each fracture as a parallel-plate conduit (cubic law):

k_ij = (a³ / 12V) · Σ A_k (δ_ij − n_i n_j)

The report gives the principal permeabilities k1 ≥ k2 ≥ k3 (m²) with their plunge/trend directions, the fracture porosity (a·P32), and the full tensor in the log. Note this assumes every fracture is hydraulically connected — an upper bound; combine with the Percolation Analysis for context.

Validating against mapped traces

Where the DFN is meant to represent a real outcrop, Score DFN against Selected Traces… cuts the network with the outcrop mesh, compares the resulting synthetic traces against your interpretation, and writes a visual report covering length distribution, orientation fabric and intensity — per fracture set as well as pooled.

See DFN — Validating Against Outcrop.

Best Practices

For Realistic Networks

1. Base parameters on field measurements when available — use Create DFN from Selected Groups so orientation statistics come from your data
2. Use multiple fracture sets to represent different deformation events
3. Validate P32 intensity against borehole or scanline data
4. Compare generated orientation distributions with stereonet data
5. Use the Match Quality panel (see the Interpretation Matching guide) to score the DFN against observed traces

For Performance

1. Start with low P32 during parameter tuning
2. Use proximity filtering to focus on relevant fractures
3. Run Fracture Intersections only when needed — intersection and graph computation scale with fracture count

For Analysis

1. Run Fracture Intersections before generating connectivity or topology attributes
2. Use attributes for quantitative comparisons between realizations

Troubleshooting

No fractures generated

• Check that domain bounds are set correctly
• Verify P32 intensity is greater than 0
• Ensure at least one fracture set is enabled

Slow intersection computation

• Reduce fracture count (lower P32 or smaller domain)
• Use proximity filtering to limit active fractures

Mesh intersections disconnected

• Segments from the same fracture are automatically merged
• If gaps exist, the mesh may have holes or the fracture doesn't fully intersect

Slicers not showing

• Ensure slicer is enabled AND position slider is moved
• Check that fractures exist within the slice plane

Advanced Statistical Analysis

VRGS includes statistical tools for analyzing fracture data, based on techniques from the literature and FracPaQ. Distribution fitting runs automatically when a DFN is created from groups that carry trace geometry; the Fisher statistics below underpin set derivation and the match-quality scoring.

Distribution Fitting

Fit statistical distributions to fracture size data using Maximum Likelihood Estimation (MLE) with Kolmogorov-Smirnov goodness-of-fit testing.

Supported Distributions

Distribution	Parameters	Use Case
Power Law	α (exponent), x_min	Scale-invariant fracture lengths (most common)
Lognormal	μ, σ, x_min	Natural log-normal processes
Exponential	λ (rate), x_min	Constant spacing/fragmentation

How It Works

1. MLE Fitting: Parameters estimated by maximizing likelihood
2. x_min Optimization: Lower bound automatically selected to minimize the K-S statistic; all theoretical CDFs are conditioned on x ≥ x_min so the three families compete fairly
3. Bootstrap K-S Testing: The selected fit's p-value comes from a semi-parametric bootstrap (Clauset et al. 2009) — synthetic datasets are drawn from the fitted model and refitted, so the p-value correctly accounts for the parameters having been estimated from the same data. p > 0.05 indicates an acceptable fit

Power Law MLE

For power law P(x) ~ x^(-α):

α = 1 + n / Σ[ln(x_i / x_min)]

The finite-size correction is applied for small samples.

Interpretation

p-value	Interpretation
> 0.10	Good fit
0.05 - 0.10	Marginal fit
< 0.05	Poor fit - distribution may not apply

Fitting a tail vs fitting a generator

The x_min search answers "is some tail of this data a power law?" — the right question for testing a hypothesis, and the reason each family is free to choose its own cutoff.

It is the wrong question when the fit becomes a generator's size distribution, because the model is then used across the whole range rather than above a cutoff. Using the searched x_min as the generator's minimum size discards the body of the population, and shape parameters estimated on a tail extrapolate badly downwards.

So when a DFN is created from measurements, the families are re-fitted and re-selected over the full observed range, with the lower bound pinned to the smallest measurement. The generated size population then spans, and matches, what was measured.

References

• Clauset et al. (2009) "Power-Law Distributions in Empirical Data"
• Newman (2006) "Power Laws, Pareto Distributions and Zipf's Law"

Fisher Concentration from Pole Data

When orientation statistics are derived from measurements (set creation and match-quality scoring), the Fisher concentration is estimated from the mean resultant length R of the 3D pole population:

K ≈ R(3 − R²) / (1 − R²)

This is the spherical (3D) maximum-likelihood estimator — appropriate for poles on the sphere, unlike the 2D von Mises estimator used for circular (strike/azimuth-only) data, which would underestimate K by roughly half at high concentration. Estimates are clamped to [0.1, 1000].

R is computed axially, via the orientation tensor T = (1/n) Σ p pᵀ, whose principal eigenvector gives the mean axis; each pole is folded into that hemisphere before the resultant is summed. Averaging poles as plain vectors lets two descriptions of the same plane cancel — and because dip is reported in [0, 90], a subvertical set has exactly that problem: its two valid descriptions sit at opposite ends of one near-horizontal axis. Steep sets, which are the common case in outcrop, are the ones a vector average gets most wrong.

Circular Statistics

For circular (2D angular) data such as strikes and paleocurrent directions:

Axial vs Vectorial Data

Type	Range	Examples
Axial	0-180° (headless)	Fracture strikes, lineation trends
Vectorial	0-360° (directed)	Dip directions, current azimuths

For axial data, angles are doubled before analysis and results halved.

Computed Statistics

Statistic	Description
Mean Direction	Circular mean of angles
Resultant Length (R)	0-1, measures concentration (0 = uniform, 1 = identical)
Circular Variance	1 - R
Circular Std Dev	√(-2·ln(R))
Concentration (κ)	von Mises dispersion parameter (2D circular estimator)

Uniformity Tests

Test	Output
Rayleigh Test	Tests for unimodal concentration (Z = nR² against a critical value)
Watson U² Test	p-value for non-uniformity on the circle
Kuiper Test	Test statistic V_n (compare against published critical values)

References

• Fisher (1993) "Statistical Analysis of Circular Data"
• Mardia & Jupp (2000) "Directional Statistics"

Fabric Tensor Analysis

Quantify fracture network anisotropy using Oda's 2nd-rank fabric tensor (crack tensor).

Fabric Tensor (F2)

The tensor is computed as:

F2_ij = (1/V) * Σ[A_p * n_i * n_j]

Where:

• V = domain volume
• A_p = fracture area
• n = unit pole vector (normal to fracture)

Output Properties

Property	Description
F2 Matrix	3x3 symmetric tensor
Eigenvalues	Principal anisotropy magnitudes (λ1 ≥ λ2 ≥ λ3)
Eigenvectors	Principal anisotropy directions
Trace	λ1 + λ2 + λ3 (proxy for P32)
Anisotropy Ratio	λ1 / λ3 (1 = isotropic, higher = anisotropic)
Shape Factor	(λ1 - λ2) / (λ1 - λ3) (0 = prolate, 1 = oblate)

Interpretation

Anisotropy Ratio	Interpretation
1.0 - 1.5	Nearly isotropic
1.5 - 3.0	Moderately anisotropic
3.0 - 10.0	Strongly anisotropic
> 10.0	Highly anisotropic (single dominant set)

Reference

• Oda (1982) "Fabric tensor for discontinuous geological materials"