Loop-Component Index (LCI): A bounded topological metric for quantifying transition from component to loop-dominated fracture networks

Abstract

Topological characterization of fracture networks is essential for understanding their structural organization and large-scale hydraulic behavior. In this study, I introduce the Loop–Component Index (LCI), a bounded and scale-independent metric derived from the Euler number (E = β₀ − β₁), where β₀ represents the number of disconnected components and, β₁ denotes the number of enclosed loops. Disconnected components are isolated fracture clusters/fracture-traces with no topological connection to other clusters within the mapped domain, whereas loops are closed polygonal regions formed by intersecting fracture segments. The raw Euler number, however, strongly depends on map size—particularly given the limited availability of large, well-exposed outcrops across different locations—making direct comparison between datasets unreliable. Moreover, ‘E’ can theoretically range from −∞ to +∞ and therefore, unbounded in characteristics. This absence of fixed limits complicates its use in comparative or statistical analyses, especially when compared with other fracture-network descriptors that are typically normalized per unit area or length.

To overcome these limitations, ‘E’ is first normalized by map area and then transformed using a hyperbolic tangent function (tanh f(x)), yielding LCI values constrained between 0 and 1. In this framework, LCI → 0 indicates component-dominated networks with limited interconnection, whereas LCI → 1 reflects loop-dominated systems characterized by abundant interconnections. An intermediate value of LCI = 0.5 marks the critical topological transition (E = 0), where components and loops are balanced.

Analyses of synthetic fracture networks show that LCI effectively captures topology transitions driven by fracture density, trace length, and intersection angle. Percolation consistently occurs at LCI >0.5, and the positive correlation between LCI and equivalent permeability (k_eq) underscores its effectiveness as a bounded, scale-independent descriptor of fracture-network structural evolution.