Hydrodynamic Dispersion and Lamb Surfaces in Darcy Flow

Transport processes such as the dispersion and mixing of solutes are governed by the interplay of advection and diffusion, where advection acts to organise fluid streamlines and diffusion acts to randomise solute molecules. Thus, the structure and organisation of streamlines, termed the Lagrangian kinematics of the flow, is central to the understanding and modelling of these transport processes. A key question is whether the streamlines in three-dimensional (3D) Darcy flows can wander freely through the fluid domain, or whether all streamlines of the flow are organised into a series of smooth, non-intersecting two-dimensional (2D) surfaces. The existence of such a foliation of surfaces constrains the Lagrangian kinematics in a manner similar to that of 2D flows, which in turn constrains the allowable transport processes. In a series of pioneering studies, Sposito (Water Resour. Res., 30(8):2395-2401, 1994; Adv. Water Resour., 24(7):793-801, 2001) argues that steady Darcy flow in locally isotropic media gives rise to Lamb surfaces, 2D material surfaces which are spanned by both the streamlines and vortex lines (field lines of the vorticity vector) of the flow. Hence, the existence of these surfaces renders the kinematics of such 3D steady Darcy flow as two dimensions. This topological constraint strongly affects transverse mixing and dispersion because 2D steady flow fields limit the rate of deformation of fluid elements and can only admit zero hydrodynamic transverse dispersion. In this study, however, we show that Lamb surfaces are not ubiquitous to all steady Darcy flows in locally isotropic media. We derive the conditions for when Lamb surfaces exist in such Darcy flows, and discuss the implications of these findings for the transport, mixing, and dispersion of solutes.