Comment on “Plume spreading in groundwater by stretching and folding” by D. C. Mays and R. M. Neupauer


Corresponding author: D. R. Lester, CSIRO Mathematics, Informatics and Statistics, PO Box 56, Highett, Vic. 3190, Australia. (

[1] The recent interest in application of the principles of chaotic advection to groundwater hydrology has generated novel insights into the nature of fluid mixing and transport in the subsurface. These developments have led to significant advances regarding, for example, the nature of hydrodynamic dispersion and mixing [Sposito, 1994, 2006; Trefry et al., 2010] and have given rise to an array of novel groundwater engineering and intervention approaches [Bagtzoglou and Oates, 2007; Zhang et al., 2009; Trefry et al., 2012; Rolle et al., 2009; Lester et al., 2010; Metcalfe et al., 2010b] to control and enhance transport for improved soil remediation, geothermal energy recovery, in situ mining, carbon sequestration, and so on.

[2] The recent paper of Mays and Neupauer [2012] presents one such novel methodology for the acceleration of plume spreading with punctuated injection protocols that result in fluid stretching and folding which are the hallmarks of chaotic dynamics in continuous systems. This protocol shows excellent promise as a means to accelerate dispersion for groundwater remediation, recovery, and so on.

[3] Our comment regarding this paper does not directly pertain to this method but rather corrects an erroneous statement that has important implications for a wide class of intervention methods which involve “recycled” injection, whereby extracted fluid is reinjected elsewhere in the subsurface to impart continuous fluid flow and deformation. Mays and Neupauer state that the idealization of the timing and orientation of the reinjection of fluid particles render such studies [Jones and Aref, 1988; Metcalfe et al., 2010b; Sposito, 2006; Lester et al., 2009; Stremler et al., 2004] physically implausible due to neglect of both turbulent mixing and finite residence time in the reinjection plumbing and claim that these factors prohibit chaotic dynamics in subsurface flow.

[4] While the idealizations of instantaneous particle reinjection and orientation such that the stream function is preserved as used in various studies [Jones and Aref, 1988; Metcalfe et al., 2010b; Sposito, 2006; Lester et al., 2009; Stremler et al., 2004] are not physically relevant, these idealizations are utilized to aid visualization of the transport dynamics. In fact, the inclusion of random reinjection time and/or orientation preserves the chaotic dynamics and only alters the Lagrangian dynamics from deterministic chaos to stochastic chaos. We have explicitly considered the impact of finite reinjection time and random orientation for a periodically reoriented dipole flow in a previous study [Lester et al., 2009] (Figure 1) and demonstrate that while stochastic forcing does alter the chaotic dynamics in general, the relevant Lagrangian topology is preserved, and such forcing acts to accelerate dispersion by the addition of stochastic dynamics on the top of the underlying deterministic chaos.

Figure 1.

Poincaré section for the reoriented dipole flow with (a) deterministic reinjection and (b) stochastic reinjection. The thick white line in (b) bounds the coherent nonreinjected region Ω2, while the ergodic reinjected region Ω1 lies outside. Reproduced from Lester et al. [2009].

[5] For the reoriented dipole flow, the Lagrangian fluid domain Ω is divided into two topologically distinct regions Ω12 : Ω1 ∪ Ω2=Ω which, respectively, do or do not undergo reinjection, as reflected by the distinct regions in Figure 1b. These regions are invariant under changes in the particle reinjection time or the orientation and so persist for random reinjection protocols. This fact has been confirmed experimentally [Metcalfe et al., 2010a] for the periodically reoriented dipole flow as illustrated in Figure 2.

Figure 2.

Evolution of nonreinjected region Ω2 in the open periodically reoriented dipole flow from (a–d) experiments and (e–h) simulations. Reproduced from Metcalfe et al. [2010a].

[6] As such, the Lagrangian dynamics within the nonreinjected region Ω2 are not affected by variations in either the orientation or residence time of reinjected particles so long as the net reinjection flow is held constant. Conversely, for the reinjected region Ω1, random variations due to reorientation and/or residence time act to break down topological barriers within Ω1 between the chaotic and regular Kolmogorov-Arnold-Moser (KAM) regions such that the entire reinjected region is both chaotic and stochastic, as reflected by the breakdown of coherent structures in Ω1 in Figure 1b.

[7] Hence, the net effect of random reinjection is to render reinjection regions Ω1 globally chaotic and stochastic, while regions that do not undergo reinjection Ω2 are unchanged. These results mean that under physically plausible conditions of random reinjection confinement protocols [Trefry et al., 2012] are persistent, while mixing protocols [Jones and Aref, 1988; Metcalfe et al., 2010b; Sposito, 2006; Lester et al., 2009; Stremler et al., 2004] are accelerated. As such, this class of novel stirring protocols based upon reinjected flows that exhibit Lagrangian chaos does represent practical and feasible groundwater intervention and engineering tools.