Chaotic advection at the pore scale: Mechanisms, upscaling and implications for macroscopic transport

Abstract

The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the porescale generate chaotic advection-involving exponential stretching and folding of fluid elements-the mechanisms by which it arises and implications of microscopic chaos for macroscopic dispersion and mixing. Prohibited in steady 2D flow due to topological constraints, these phenomena are ubiquitous due to the topological complexity inherent to all 3D porous media. Consequently 3D porous media flows generate profoundly different fluid deformation and mixing processes to those of 2D flow. The interplay of chaotic advection and broad transit time distributions can be incorporated into a continuous-time random walk (CTRW) framework to predict macroscopic solute mixing and spreading. We show how these results may be generalised to real porous architectures via a CTRW model of fluid deformation, leading to stochastic models of macroscopic dispersion and mixing which both honour the pore-scale kinematics and are directly conditioned on the pore-scale architecture.

Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media

“…Recent studies [24][25][26] have also established that chaotic mixing is inherent to three-dimensional (3D) porous media for which the solid phase is continuous (i.e., not granular), such as open porous networks. While such chaotic mixing is driven by the topological complexity inherent to all porous media, it is unknown whether these concepts apply to discrete porous media as their topology is now contingent upon pointlike contacts between grains.…”

The Lagrangian kinematics of three-dimensional Darcy flow

“…Thus this explicit representation in terms of the velocity magnitude is not possible for non-integrable steady 3-D flows such as pore-scale 3-D Stokes flow or 3-D tensorial Darcy flow as the helicity density h is non-zero for these flows, and so there do not exist a pair of linearly independent invariants (ψ 1 , ψ 2 ). Indeed, this property is a necessary condition for pore-scale 3-D flow to exhibit chaotic advection (Lester, Trefry & Metcalfe 2016b).…”

“…Recent studies (Lester et al 2016b;Turuban et al 2018Turuban et al , 2019Souzy et al 2020) have established that chaotic advection is inherent to steady 3-D Stokes flow at the pore-scale, even in a medium that is homogeneous at the pore scale, providing a decisive link between pore-scale structural properties and the Lagrangian kinematics of porous media (Heyman et al 2020;Heyman, Lester & Le Borgne 2021). This result is not surprising as the Poincaré-Bendixson theorem states that only continuous systems with three or more degrees-of-freedom (DOFs) can admit a chaotic dynamics, hence, chaos is possible in three dimensions but not 2-D steady pore-scale flow.…”

A Primer on the Dynamical Systems Approach to Transport in Porous Media

“…compressible Darcian systems where engineered pumping activity is absent. Dynamical systems concepts have also been used to link the topology of pore-scale architectures with pore-scale solute mixing and macroscopic transport phenomena (Lester et al, 2013b(Lester et al, , 2014b(Lester et al, , 2016b, resulting in fundamental upscaling behaviours conditioned on the details of pore-scale Lagrangian structures. These mechanisms have recently been confirmed via direct experimental observations of pore-scale mixing (Souzy et al, 2020;Heyman et al, 2021Heyman et al, , 2020, leading to the highly striated solute distributions shown in Figure 1.…”