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.Plain Language Summary The spreading of solutes, colloids and pollutants in groundwater flow is key to both risk assessment and a fundamental understanding of chemical, biological and geophysical transport in the subsurface. In many scenarios the spreading of solutes transverse to the mean flow direction is dominated by hydrodynamic transport from diverging streamlines and the contribution from local dispersion is relatively small. Although it is well understood that such hydrodynamic transport (termed transverse macrodispersion) is zero in steady two-dimensional groundwater flow for purely advective transport, there has been much debate in recent years as to whether transverse macrodispersion can occur in steady three-dimensional groundwater flows. In this study we address this question by considering the kinematics associated with Darcy flow of heterogeneous porous with different conductivity structures and identify the structures that generate non-zero transverse macrodispersion in the absence of local dispersion. We show that numerical predictions of transverse macrodispersion can be highly sensitive to the details of the numerical solvers used and propose a novel numerical method to mitigate against spurious predictions of transverse macrodispersion. These findings are used to reconcile seemingly contradictory results in the literature and provide insights into the suitability of commonly used models of groundwater flow and transport.