Dispersion and Mixing in Three‐Dimensional Discrete Fracture Networks: Nonlinear Interplay Between Structural and Hydraulic Heterogeneity

Abstract

We investigate the relative impact of topological, geometric, and hydraulic heterogeneity on transport processes in three-dimensional fracture networks. Focusing on the two largest scales of heterogeneity in these systems, individual fracture and network structure, we compare transport through analogous structured and disordered three-dimensional fracture networks with varying degrees of hydraulic heterogeneity. For the moderate levels of hydraulic heterogeneity we consider, network structure is the dominant control of transport through the networks. Less dispersion, both longitudinal and transverse, is observed in structured networks than in disordered networks, due in part to the higher connectivity in the former, independent of the level of hydraulic heterogeneity. However, increases in dispersion with higher hydraulic heterogeneity are larger in the disordered networks than in the structured networks, thereby indicating that the interplay between structural and hydraulic heterogeneity is nonlinear. We propose a measure of disorder in fracture networks by computing the Shannon entropy of the spectrum of the Laplacian of a weighted graph representation of the networks, where the weights are given by a combination of topological, geometric, and hydraulic properties. This metric, as a relative indicator by comparison between two networks, is a first approach to the dispersion potential and ''mixing capacity'' of a fracture network. Key Points:For moderate levels of hydraulic heterogeneity, fracture network structure is the principal control of transport times and dispersion The influence of the interplay between structural and hydraulic heterogeneity is nonlinear As hydraulic heterogeneity increased, both longitudinal and transverse dispersion also increased Supporting Information: Supporting Information S1Correspondence to:

Linking Structural and Transport Properties in Three‐Dimensional Fracture Networks

“…For every fracture that intersects the inflow boundary an edge is added between the vertex in the graph corresponding to that fracture and the vertex representing the inflow boundary; likewise for the outflow boundary. Similar graph theoretical approaches have been used for a variety of studies concerning DFN including topological characterization (Andresen et al, ; Huseby et al, ; Hope et al, ; Hyman & Jiménez‐Martínez, ) and backbone identification (Hyman et al, ; Valera et al, ). The utility of a graph theoretical approach is that topological properties of the networks can be queried and characterized in a formal mathematical framework.…”

“…In a fracture network, larger features can play a more dominant role than in-fracture aperture variability in determining the structure of the fluid velocity field (Bisdom et al, 2016;de Dreuzy et al, 2012;Karra et al, 2015;Makedonska et al, 2016). While it is understood that macroscale network traits influence the arrangement of the fluid flow field within a fracture network (Edery et al, 2016;Hyman & Jiménez-Martínez, 2018), a direct link between geometric and topological properties of the fracture network and upscaled transport observables is still lacking. With such a wide range of relevant length scales, several orders of magnitude (Bonnet et al, 2001;Davy et al, 2013;Hardebol et al, 2015), it is challenging to identify which features of a fracture network influence which flow and transport properties.…”

Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves

“…Each set contains 10 networks composed of uniformly sized square fractures with edge lengths of 1 m within cuboid domain with dimensions of 20 m × 10 m × 10 m. Note that power law tails in advective travel time distributions does not require a power law distribution of fracture lengths (Hyman and Jiménez-Martínez, 2018;Painter et al, 2002). Each set contains 10 networks composed of uniformly sized square fractures with edge lengths of 1 m within cuboid domain with dimensions of 20 m × 10 m × 10 m. Note that power law tails in advective travel time distributions does not require a power law distribution of fracture lengths (Hyman and Jiménez-Martínez, 2018;Painter et al, 2002).…”

Advancing Graph‐Based Algorithms for Predicting Flow and Transport in Fractured Rock

“…Sævik and Nixon () included topological properties of two‐dimensional fracture networks into analytic expressions for permeability to account for network structure. Hyman and Jiménez‐Marttínez () used this graph representation to aid in the investigation of the relative impact of connectivity, geometric, and hydraulic heterogeneity on transport processes. Santiago et al, (, , ) proposed a method of topological analysis that measured centrality properties of nodes in the graph, which describes characteristics such as the number of shortest paths through a given node.…”

“…In the second, each intersection between fractures is represented by a vertex and fractures are represented by a collection of edges. Hyman et al (2018) recently provided a rigorous mathematical exposition of these graph representations and showed that they are projections of a more general bipartite graph representation. Then we demonstrate the utility of the proposed method by using these graph representations to efficiently reproduce various features of the breakthrough curve, for example, first arrival, peak arrival, and scaling behavior in the tail.…”