Helical flow in three-dimensional nonstationary anisotropic heterogeneous porous media

Characterizing the topology of three-dimensional steady-state flow fields is useful to describe the physical processes controlling the deformation of solute plumes and, consequently, obtain helpful information on mixing processes without solving the transport equation. In this work, we study the topology of flow in three-dimensional nonstationary anisotropic heterogeneous porous media. In particular, we apply a topological metric, i.e., the helicity density, and two complementary kinematic descriptors of mixing, i.e., stretching and folding, to investigate: (i) the flow field resulting from applying a uniform-in-the-average hydraulic gradient within a fully resolved heterogeneous three-dimensional porous medium with a nonstationary anisotropic covariance function of the locally isotropic hydraulic log conductivity; (ii) the flow field obtained by averaging a set of Monte Carlo realizations of the former field; (iii) the flow field obtained considering the blockwise uniform anisotropic effective conductivity tensor computed for the fully resolved case. While in the fully resolved case, the local helicity density is zero as a consequence of the local isotropy of hydraulic conductivity, it differs from zero in the other two cases. We show, therefore, that this topological metric is scale dependent and should be computed at the appropriate scale to be informative about the leading patterns of plume deformation. Indeed, streamlines are helical in all three cases at scales larger than the characteristic scale of spatial variability. We apply stretching and folding metrics to investigate the scales at which plume deformation is more influenced by helical motion than by the effect of small-scale spatial heterogeneity in the hydraulic-conductivity field. Under steady-state flow conditions, stretching, which quantifies the increasing length of an interface, dominates at short distances from a given starting plane, while folding, which describes how this interface is bent to fill a finite volume of space, dominates further downstream and can be correlated with the appearance of large-scale secondary motion. We conclude that three-dimensional flows in porous media may show a complex topology whose analysis is relevant for the description of plume deformation. These results have important implications for the understanding of mixing processes, as shown in detail in the companion paper focusing on solute transport.