Helicity and flow topology in three-dimensional anisotropic porous media

Flows showing complex topology are ubiquitous in natural systems. However, contrasting evidence exists on the helical nature of flow in porous media and on the occurrence of groundwater whirls. We analyze the topology of steady-state flow fields in porous media, highlighting the importance of considering the three-dimensionality of the flow field to properly capture the complexity of the system dynamics controlling the deformation of material surfaces, which is widely recognized as the main driver of mixing. We use the helicity density as topological measure and investigate the necessary and sufficient conditions to obtain non-zero helicity density for Darcy flow. We show that helical groundwater flow can develop in both homogeneous and heterogeneous porous media, provided that the hydraulic conductivity is anisotropic. In the homogeneous case, the additional condition of non-vanishing mixed second spatial derivatives of hydraulic head is required, while in heterogeneous media, helical flow may occur even when the hydraulic gradient is uniform. We present illustrative examples of complex flow topology in three-dimensional porous media and discuss the computed streamline patterns and their potential implications for mixing processes.