  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 \emph{Lagrangian kinematics} of these flows: namely how fluid particle
trajectories (which form a dynamical system) are organized 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 organize solute transport and
mixing. Novel experimental methods are only recently permitting visualization
of LCSs are 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 ...
