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

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Historically, the dominant conceptual paradigm of porous media flow, solute mixing and transport was based on steady two-dimensional flows in heterogeneous porous media. Although it is now well recognised that novel transport phenomena can arise in unsteady and/or three-dimensional flows at both the pore or Darcy scales, appropriate methods for analysis and understanding of these more complex flows have not been widely employed. In this primer, we advocate for methods borrowed from dynamical systems (chaos) theory, which aim to uncover the Lagrangian kinematics of these flows: namely how fluid particle trajectories (which form a dynamical system) are organised and interact and the associated impacts on solute transport and mixing. This dynamical systems approach to transport is inherently Lagrangian, and the Lagrangian kinematics form Lagrangian coherent structures (LCSs), special sets of trajectories that divide the Lagrangian frame into chaotic mixing regions, poorly mixing hold-up regions (and in some cases non-mixing “islands”) and the transport barriers that organise these regions. Hence, the dynamical systems approach provides insights into flows that may exhibit chaotic, regular (non-chaotic) or mixed Lagrangian kinematics, and also into how LCSs organise solute transport and mixing. Novel experimental methods are only recently permitting visualisation of LCSs in porous media flows. In this primer, we review the dynamical systems approach to porous media flow and transport and connect the associated tools and techniques with the latest research findings from pore to Darcy scales. This primer provides an introduction to the methods and tools of dynamical systems theory. Once familiar with these approaches, porous media researchers will be better positioned to know when to expect complex Lagrangian kinematics, how to uncover and understand LCSs and their impacts on solute transport, and how to exploit these dynamics to control solute transport in porous media flows.

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GM thanks the Lorenz Center for travel support and hospitality. Thoughtful and constructive comments of two reviewers helped improve this manuscript.


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Metcalfe, G., Lester, D. & Trefry, M. A Primer on the Dynamical Systems Approach to Transport in Porous Media. Transp Porous Med 146, 55–84 (2023).

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