Topological analysis of 3D fracture networks: Graph representation and percolation threshold

Abstract

This work presents a framework to analyze geometrical, topological and hydraulic properties of 3D Discrete Fracture Networks (DFN) and to estimate its properties near percolation threshold. A set of efficient algorithms has been developed to perform geometrical and topological analyses upon 3D networks of planar fractures with various shapes (mainly circular and elliptical fractures). This toolbox is capable of (i) calculating all possible intersections in the 3D network, and calculating also the resulting trace lengths, and other geometrical attributes of the DFN; (ii) extracting the percolating clusters and eliminating dead end clusters; (iii) constructing the corresponding graph of the 3D network of planar fractures. All the calculations implemented with these algorithms have been strictly validated by direct numerical simulations. The graph representation of the DFN enables the application of broad-purpose algorithms inspired by graph theory and percolation theory (among other applications). It is demonstrated that the use of our toolbox as a pre-treatment (extracting the percolation network, eliminating all dead-end fractures and clusters, and searching for shortest paths), considerably reduces the CPU time of flow/transport simulations. The gain on CPU time was of several orders of magnitude for networks containing thousands of fractures. The computational efficiency of the tools permits a broad study on the percolation in 3D fracture networks. The two main results presented in this paper concern two important issues: (i)a new procedure is developed for the numerical determination of the critical percolation density based on the marginal percolation concept; and (ii)an enhanced formula with low sensitivity to fracture shape, orientation, and size distribution, is proposed for estimating the critical percolation density.