It is well known that transverse macroscopic dispersion cannot occur in two-dimensional (2D) steady heterogeneous Darcy flow as the fluid streamlines are confined to a plane, and hence cannot diverge or converge without bound. Conversely, it is widely believed that such constraints do not apply to steady three-dimensional (3D) flow as the fluid streamlines can now wander without bound throughout 3D space. However, the special form of the isotropic Darcy flow equation imposes specific constraints upon fluid motion which are not immediately obvious. The pioneering work of Sposito [1] shows that steady 3D isotropic heterogeneous Darcy flow has identically zero helicity, defined as the product of fluid velocity and vorticity, hence fluid elements cannot make helical motions as they travel along streamlines.
We show a further consequence of this constraint is that fluid streamlines in 3D space are confined to an orthogonal set of material surfaces, and as these surfaces are topologically simple and flat (in the Gaussian sense), then the same constraints as for 2D flow apply locally to these surfaces. As such, macroscopic transverse dispersion cannot arise in steady isotropic Darcy flow. We show that the majority of Darcy flow numerical codes do not explicitly enforce the helicity-free condition, and so macroscopic transverse dispersion calculated by these methods is purely a numerical artefact. We develop an explicitly helicity-free Darcy solver and show that indeed it does recover zero macroscopic dispersion.
These results have significant implications for the study of transport and dispersion in steady Darcy flow, and indicate that caution must be exercised when designing and interpreting numerical simulations of transport in Darcy flow. Moreover, as macroscopic dispersion is a well-observed phenomenon in field studies and laboratory experiments, these results show that the isotropic Darcy flow equation has limitations for capturing dispersion phenomena.