Graph theory and the analysis of fracture networks

摘要

Two-dimensional exposures of fracture networks can be represented as large planar graphs that comprise a series of branches (B) representing the fracture traces and nodes (N) representing their terminations and linkages. The nodes and branches may link to form connected components (K), which may contain fracture-bounded regions (R) or blocks. The proportions of node types provide a basis for characterizing the topology of the network. The average degree d relates the number of branches (vertical bar B vertical bar) and nodes (vertical bar N vertical bar) and Euler's formula establishes a link between all four elements of the graph with vertical bar N vertical bar - vertical bar B vertical bar + vertical bar R vertical bar - vertical bar K vertical bar = 0.Treating a set of fractures as a graph returns the focus of description to the underlying relationships between the fractures and, hence, to the network rather that its constitutive elements. Graph theory provides a wide range of applicable theorems and well-tested algorithms that can be used in the analysis of fault and fracture systems. We discuss a range of applications to two-dimensional fracture and fault networks, and briefly discuss application to three-dimensions.