Abstract. 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 m2 from the Lilstock pavements.
Using a moving-window sampling approach, we first subsample the large networks
into subgraphs. Four graph similarity measures – fingerprint distance, D-measure,
Network Laplacian spectral descriptor (NetLSD), and portrait
divergence – that encapsulate topological relationships and geometry of fracture
networks are then used to compute pair-wise subgraph distances serving as input
for the statistical hierarchical clustering technique. In the form of hierarchical
dendrograms and derived spatial variation maps, the results indicate spatial
autocorrelation with localized spatial clusters that gradually vary over
distances of tens of metres with visually discernable and quantifiable boundaries.
Fractures within the identified clusters exhibit differences in fracture
orientations and topology. The comparison of graph similarity-derived
clusters with fracture persistence measures indicates an intra-network
spatial variation that is not immediately obvious from the ubiquitous fracture
intensity and density maps. The proposed method provides a quantitative way to
identify spatial variations in fracture networks, guiding stochastic and
geostatistical approaches to fracture network modelling.
Evaluation of the Gower coefficient modifications in hierarchical clustering