Under What Conditions Does Transverse Macrodispersion Exist in Groundwater Flow?

In recent years there has been vigorous debate whether asymptotic transverse macrodispersion exists in steady three-dimensional (3D) groundwater flows in the purely advective limit. This question is tied to the topology of 3D flow paths (termed the Lagrangian kinematics), specifically whether streamlines can undergo braiding motions or can wander freely in the transverse direction. In this study we determine which Darcy flows do admit asymptotic transverse macrodispersion for purely advective transport on the basis of the conductivity structure. We prove that porous media with smooth, locally isotropic hydraulic conductivity exhibit zero transverse macrodispersion under pure advection due to constraints on the Lagrangian kinematics of these flows, whereas either non-smooth or locally anisotropic conductivity fields can generate transverse macrodispersion. This has implications for upscaling locally isotropic porous media to the block scale as this can result in a locally anisotropic conductivity, leading to non-zero macrodispersion at the block scale that is spurious in that it does not arise for the fully resolved Darcy scale flow. We also show that conventional numerical methods for computation of particle trajectories do not explicitly preserve the kinematic constraints associated with locally isotropic Darcy flow, and propose a novel psuedo-symplectic method that preserves these constraints. These results provide insights into the mechanisms that govern transverse macrodispersion in groundwater flow, and unify seemingly contradictory results in the literature in a consistent framework. These insights call into question the ability of smooth, locally isotropic conductivity fields to represent flow and transport in real heterogeneous porous media.
The authors thank the reviewers for their constructive feedback which has improved the manuscript. M.D. acknowledges the support of the Spanish Research Agency (https://doi.org/10.13039/501100011033), Spanish Ministry of Science and Innovation and European Regional Development Fund “A way of making Europe” through Grants CEX2018-000794-S and HydroPore PID2019-106887GB-C31. Open access publishing facilitated by RMIT University, as part of the Wiley - RMIT University agreement via the Council of Australian University Librarians.