Rock fractures organize as networks, exhibiting natural variation in their
spatial arrangements. Therefore, identifying, quantifying, and comparing
variations in spatial arrangements within network geometries are of interest
when explicit fracture representations or discrete fracture network models are
chosen to capture the influence of fractures on bulk rock behaviour. Treating
fracture networks as spatial graphs, we introduce a novel approach to quantify
spatial variation. The method combines graph similarity measures with
hierarchical clustering and is applied to investigate the spatial variation
within large-scale 2-D fracture networks digitized from the well-known Lilstock
limestone pavements, Bristol Channel, UK. We consider three large, fractured
regions, comprising nearly 300 000 fractures spread over
14 200

Fracture networks in rocks develop due to loading paths that vary over geological timescale

An important property of natural fracture networks is that of spatial organization, which means that the arrangements are not random but follow a statistically discernable pattern. One can view the spatial arrangement of fractures as a set of objects within a geographical reference system. Within such a framework, fracture objects are either regularly spaced, irregularly spaced with statistically significant regions of close spacing, and irregularly spaced with statistically insignificant regions of close spacing

Quantifying variations in spatial arrangements of fractures involves the sampling of fracture data. Such quantifications can be in the form of 1-D (using scanline methods, borehole sampling), in 2-D (fracture trace maps from outcrop imagery), or 3-D (ground-penetrating radar, microseismic). 1-D scanlines provide a method to quantify arrangements and variation, and several statistical measures have been proposed, such as fracture spacing

2-D fracture trace maps are especially useful, as this type of data combines both geometric and topological information in the form of a network. Recent advances in unmanned aerial vehicle (UAV)-photogrammetry

From a geostatistical perspective, the concept of spatial variability describes how a measurable attribute varies across a spatial domain

Many authors have suggested using graph theory for the characterization of fracture networks (such as

Comparing primal and dual forms of a fracture network from data published by

We depict an example of a fracture network in its primal form (see Fig.

In the case of the alternate representation, referred to as dual graphs by

In graph representations, weights can be assigned to edges that are proportional to the importance of that edge. In the case of fracture networks in the primal form, this can be the Euclidean distance between the nodes (or fracture edge intersections). The weight may also be the direction cosine of the particular edge that indicates orientation. In the dual graph representation, intersections between fractures represent the edges. Therefore, the edge weight may be specified in terms of intersection angle. Graphs may also be directed with a specific direction to edges. In the case of spatial graphs derived from fracture networks, an undirected but weighted representation is sufficient. Figure

Several graph similarity measures exist within the graph theory literature to compare graphs (see

Since we are interested in quantifying spatial variability, we may recast the problem as that of identifying clusters within the network. Clustering is also referred to as unsupervised classification and is a process of finding groups within a set of objects with an assigned measurement

A simple example of hierarchical clustering using Euclidean
distance:

In the existing literature on fracture networks, assigning labels to specific perceived archetypal networks (or end-members) is standard. These typologies include terms such as orthogonal, nested, ladder-like, conjugated, polygonal, corridors, etc. (

Hierarchical clustering (HC) is an unsupervised statistical clustering method

To validate the proposed approach based on graph distance metrics and hierarchical clustering, we utilize a 2-D joint fracture dataset from the Lilstock pavement in the Bristol Channel, UK

Overview of fracture networks corresponding to the three considered regions. This map is derived from an open image dataset published by

The proposed explanations include proximity and influence of faults explained by fluid-driven radial-jointing emanating from asperities within fault (e.g.

Comparison of the three regions in terms of networks, orientations, and length distributions. Map dimensions are in metres. This image has been modified from

Summary statistics for the three regions.

From this dataset, we utilize fracture networks corresponding to three contiguous regions. Figure

The detailed resolution, topological accuracy, and spatial extent of the
traced networks make the dataset appropriate for a detailed analysis of
spatial variation in fracturing. The networks have significant intra- and
inter-network variability in
fracturing. Figures

Correlation between sum of strike differences of fracture segments constituting tip-to-tip fractures versus total fracture length for the three regions.

Cutout from Region 2 depicting the detailed resolution of the fracture dataset.

We circularly sample the fracture networks on a cartesian grid with a subgraph extracted within a circular region centred at each grid point. The grid spacing to circle diameter is maintained such that neighbouring subgraphs share some portion of the area (see Fig.

Subsampling of a fracture graph corresponding to full region into subgraphs of 7.5

Treating isolated nodes and dangling edges that arise due to circular sampling:

Number of subgraphs obtained per region.

For

We use the following four graph similarity measures to compare the subgraphs:

fingerprint distance

D-measure

Network Laplacian spectral descriptor (NetLSD)

portrait divergence

The performance of these similarity measures have been validated previously by

The fingerprint distance introduced by

The value of

An example of a “fingerprint”, so named by

Denoting

As per

We have attached our MATLAB implementation of the fingerprint distance in the code Supplement. We computed the distance matrix for all subgraphs corresponding to the three regions using this implementation.

The D-measure introduced by

D-measure components for the two example fracture graphs comparing

As per

The third term in Eq. (

The portrait divergence similarity score derives from network portraits introduced by

The portrait divergence measure provides a single value

Heatmap representations of network portrait sparse matrices (

The NetLSD distance was introduced by

Figure

Comparing heat trace signature vectors for the two example fracture graphs computed using NetLSD.

The values of graph similarity computed using the four metrics described by Eqs. (

Summary of graph similarities computed for example fracture networks.

After subsampling the fracture networks (see
Sect.

We first show region-wise results of graph property computations. Intra-region spatial clustering resulting from the combined application of graph similarity measures with HC is then discussed. We use the following abbreviations for brevity throughout the section: FP – fingerprint distance, DM – D-measure, LSD – NetLSD, PD – portrait divergence.

Fingerprints pertaining to the regions are depicted in Fig.

Region-wise graph properties:

Region-wise properties used to compute the D-measure represented as ensemble plots of

Intra-region spatial variation results can be presented as distance matrix heatmaps corresponding to each graph similarity metric. Dendrograms depict the hierarchical organization of the subgraphs corresponding to similarity entries within the distance matrix entries. The intra-regional variation is more intuitively illustrated spatially by showing subgraphs using an appropriate colour scheme that groups similar clusters under colours picked within a linear spectrum. This section presents the clustering results for all three regions using a combination of dendrograms, spatial cluster maps, and heatmaps.

The spatial distribution of clusters pertaining to the four distance metrics overlain over the network is shown in Fig.

Summary of subgraphs within each cluster of Region 1 for

We can observe that spatial autocorrelation exists for the FP (Fig.

Figure

We briefly describe the characteristics of the clustering results prefixing “

A similar variation is observable from the result of DM (see Fig.

The results of PD also depict N–S variation (see Fig.

Hierarchical clustering results for Region 1 depicting the top 10 clusters using

Variation in fracture orientations and topological summary for Region 1 corresponding to

Spatial distribution along with dendrograms of top 10 clusters pertaining to the four graph similarity measures for Region 2 is depicted in Fig.

Summary of subgraphs within each cluster of Region 2 for

Node degree histograms and rose plots depict the differences in network topology and fracture orientations between the identified clusters pertaining to FP (Fig.

From FP clustering results (see Fig.

Hierarchical clustering results for Region 2 depicting the top 10 clusters using

Variation in fracture orientations and topological summary for Region 2 corresponding to

The spatial distribution along with dendrograms of the top 10 clusters pertaining to the four graph similarity measures for Region 3 is depicted in Fig.

Summary of subgraphs within each cluster of Region 3 for

Node degree histograms and rose plots depict the differences in network topology and fracture orientations between the identified clusters relating to FP (Fig.

From the FP clustering results (Fig.

Hierarchical clustering results for Region 3 depicting the top 10 clusters using

Variation in fracture orientations and topological summary for Region 3 corresponding to

Within the structural geology literature, the quantitative fracture persistence measures of

In this contribution, we treat 2-D fracture networks as planar graph structures and apply graph similarity measures to quantitatively compare subsampling within large fracture networks and discover clusters of similarity. The statistical technique of HC was used along with graph distance metrics to extract spatial clusters. Subgraphs within a spatial cluster are more similar to each other than other clusters. A hierarchy of patterns is derived based on similarity scores, which can be examined at deeper levels.

One can argue that variation exists at multiple length scales, and more granular inquiry would lead to different clusters. While our choices of grid spacing and subsampling of graphs were to keep computational requirements in mind, it is possible to do more dense subsampling than what we have already achieved to further highlight spatial variations within a given network. The clusters that we have depicted are particular to the spacing and sampling diameters that we have chosen. In this section, we discuss some additional perspectives and issues related to our methodology and results.

The analysis of spatial variation can assist in deciphering fracture timing. Given the temporal nature of network formation, it is desirable to delineate network evolution into relative episodes of fracturing. In previous analyses specific to the Lilstock dataset used in this contribution,

As may be observed from our results, the metrics highlight certain aspects of the fracture network while not considering others. For instance, the fingerprint distance only considers block area and shape factor distributions of the blocks and neglects orientations. The other three distances use graph properties directly, and hence orientation information (or the lack of it) is a consequence of how the spatial graph is defined. We used weighted graphs that incorporate Euclidean distance between nodes as edge weights for the similarity computations. However, each edge also has a striking attribute to completely describe its position in 2-D space (in the case of 3-D, it needs a dip). Ideally, the edge weight should then be a vector,

Regardless of the extrapolation method used, stationarity decisions have to be made based on hard data, and this is where our approach is helpful. We can use outcrop-derived networks to define and delineate stationarity's spatial boundaries and assign a particular type of network with due cognition of the inherent graph structure. Much literature exists on linking fracture patterns to high-deformation drivers such as folding, faulting, and diapirism, with the goal being to identify and correlate appropriate outcrop analogues to particular subsurface conditions. As our clustering results indicate, at the dimensional scales of sampling we have used, Tobler's first law of geography applies to fracture networks. Therefore, a representative network based on network similarity can be derived. The method can be applied to analogues for which data already exist. Further work is required to differentiate fluid-flow and transport responses of the identified cluster type.

This contribution presents a method to automatically identify spatial clusters and quantify intra-network spatial variation within 2-D fracture networks. We test the technique on 2-D trace data from a prominent limestone outcrop within the Lilstock pavements, located off the southern coast of the Bristol Channel, UK. The fracture network data that span three separate regions and cover over 14 200

Representing fracture networks as graphs enables combining hierarchical clustering and graph-distance metrics to reveal interesting intra-network spatial similarity patterns not otherwise discernable from existing global or local fracture network descriptors.

Organization of fracture network subgraphs based on pair-wise similarities into a hierarchical tree enables identification of spatial clustering at different dendrogram heights with newer and more granular cluster boundaries emerging at successively deeper levels of enquiry.

Spatial autocorrelation is more apparent with the fingerprint, D-measure, and the portrait divergence distances than the NetLSD, which yields speckled patterns with little or no spatial autocorrelation.

Spatial variation maps deriving from hierarchical clustering using the D-measure and portrait divergence identify similar spatial clusters and cluster boundaries. However, with the fingerprint distance, the cluster boundaries are different.

Fracture segment orientations show gradual variation in segment strikes across the identified clusters despite orientation not being explicitly considered and only Euclidean distance being used to weight spatial graph edges.

Combined symmetric heatmap of distance matrix and dendrograms, dendrograms, and sum-of-squares elbow plots for Region 1

Combined symmetric heatmap of distance matrix and dendrograms, dendrograms, and sum-of-squares elbow plots for Region 2

Combined symmetric heatmap of distance matrix and dendrograms, dendrograms, and sum-of-squares elbow plots for Region 3

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Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by fingerprint distance in Region 1. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by D-measure in Region 1. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by portrait divergence in Region 1. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by fingerprint distance in Region 2. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by D-measure in Region 2. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by portrait divergence in Region 2. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by fingerprint distance in Region 3. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by D-measure in Region 3. Coordinates of circular sample centres are below each subgraph example.

Circular subgraph samples depicting variation in fracturing style as identified in the 10 largest clusters by portrait divergence in Region 3. Coordinates of circular sample centres are below each subgraph example.

A MATLAB implementation to compute graph fingerprints and fingerprint distance is available on the GitHub repository

The circularly sampled fracture subgraphs are derived from the open fracture network dataset published by

RP wrote the code to convert shapefiles to graphs, sample subgraphs, and compute fingerprints and fingerprint distances; did the HC analysis; and wrote the manuscript with inputs from all authors. GB and JU contributed to development of methodology, structure of the manuscript, and discussion of results. DS provided funding and contributed to discussions on the results and methods that are utilized but are not limited to this paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Pierre-Olivier Bruna at TU Delft for useful discussions on spatial variation in fracturing. We would also like to thank David Sanderson and one other anonymous reviewer for comments that improved the quality of this contribution.

This paper was edited by David Healy and reviewed by David Sanderson and one anonymous referee.