Cutting and shuffling control mixing in three-dimensional discrete fracture networks

Mixing in fracture networks plays a central role in many environmental and geological processes, influencing contaminant dispersion, dilution and biogeochemical reactions. Despite this, fundamental aspects are not well understood, including how the network topology and large fracture aspect ratio governs the stretching, folding, cutting and rearrangement of fluid elements, which represent the advective components of mixing. In this study, we focus on three-dimensional (3-D) discrete fracture networks (DFNs) governed by steady isotropic Darcy flow. We develop a theoretical framework for advective mixing and uncover two distinct mixing mechanisms at different length scales – fracture mixing and intersection mixing – that arise due to streamline routing in fractures and intersections. We show that the large fracture aspect ratio enforces discontinuous mixing, involving cutting and shuffling (CS) of fluid elements due to streamline routing. Mixing is controlled by combined CS and fluctuating fluid deformation, forming a piecewise smooth (PWS) transform and weak ergodic mixing. An efficient graph-based representation of this process is developed via a mixing graph GM that efficiently encodes the fracture network topology while capturing mixing dynamics. Numerical predictions of mixing from GM agree to high precision with those from direct numerical simulations. This framework provides new insights into the processes which govern mixing in DFNs, and forms a rigorous basis for further application to fluid borne processes such as solute mixing and dispersion, colloidal transport and deposition, chemical reactions, and biological activity.