Adding to chaos underground could help manage polluted water, according to CSIRO physicist Guy Metcalfe.

His team has been working on “chaotic advection”, which describes the motion of particles carried in a flow—from smoke drifting in the air, to the mixing of the milk into your morning coffee.

The same principle can be used to trap contaminated groundwater, and keep it continually rotating within a defined area, CSIRO researchers have calculated. Or it could be applied to encourage reactions between fluids and rocks that remove pollutants.

## Further information:

### GeoChaos: Towards Enhanced Subsurface Scalar Transport Using Chaotic Advection

Guy Metcalfe, 1 CSIRO Materials Science and Engineering – Box 56, Highett VIC 3194, Australia

Mike Trefry, CSIRO Land & Water Private Bag 5, Wembley, WA 6913, Australia

Daniel Lester, CSIRO Mathematics, Informatics and Statistics, Private Bag 33, South Clayton MDC, VIC 3169, Australia

Western Australian Geothermal Centre of Excellence, 26 Dick Perry Avenue, Kensington, WA 6151, Australia

### Abstract summary:

Many remediation activities in the terrestrial subsurface need to recover/emplace scalar quantities (dissolved phase concentration or heat) from/in volumes of saturated porous media. Scalars can be targeted by pump-and-treat technologies, where target fluids are abstracted from the porous medium, or by amendment technologies, were specific chemicals or substrates are injected into the porous medium. Application examples include solution mining, recovery of contaminant plumes, or harvesting geothermal heat energy. Transient switching of the pressure at different wells can intimately control subsurface flow, generating a range of “programmed’’ flows. While some programs produce chaotic mixing and rapid transport, others create encapsulating flows confining pollutants for in situ treatment. In a simplified model of an aquifer subject to balanced injection and extraction pumping, chaotic flow topologies have been predicted theoretically and verified experimentally. Mixing enhancement due to chaotic advection and kinematic confinement of aquifer volumes are key features of the chaotic dynamics. Understanding these phenomena may form the basis for new efficient technologies for groundwater remediation or amendment.

### Abstract:

I. INTRODUCTION

Groundwater remediation is a common component of the rehabilitation of contaminated sites, and a prime intervention for the protection of sensitive groundwater-dependent ecosystems. Costs of groundwater remediation can be significant, so improvements in the efficiency of subsurface remediation technologies are welcome. In this work we discuss chaotic advection, a new concept that holds potential for manipulating distributions of scalar quantities in flows within the subsurface. A growing body of work exists on the mathematical basis of chaotic advection and its application to non-turbulent flows in porous media [1—8]. In the present paper, we discuss chaotic advection in groundwater systems, highlighting two key processes that are of particular interest to remediation: enhanced mixing and confinement. Conceptual tools for understanding these advective regimes are introduced, and the potential impact of key physical processes, including dispersion and heterogeneity, are discussed.

II. CHAOTIC ADVECTION

The motion of passive tracers advecting in a fluid velocity field can potentially display chaotic dynamics, even in the limit of small Reynolds number, e.g. laminar flows. This phenomenon is termed chaotic advection. Understanding the basis of chaotic advection provides valuable insight into fluid mixing processes, including dispersion [9]. Mixing is promoted by chaotic dynamics, which can arise through repetitions of stretching and folding motions of the pathlines of transported fluid (Fig. 1), even when no such stretching and folding motion exists in the streamlines of the velocity field.

A. Time Dependence and Chaos in Porous Media

For groundwater contamination, the chaotic dynamics of Darcian flow regimes are of prime interest. Steady Darcy flows are irrotational, and hence cannot mix [1]. However, chaotic flows can be induced in transient Darcian systems through the crossing of streamlines in time [2—3]. This has led to the concept of the rotated potential mixing (RPM) flow generated by assemblies of dipole pumping pairs, where only two diametrically opposed wells are active at any time (one injecting and one extracting, forming a dipole), as discussed in [5—6] with a Hele-Shaw experimental study. Re-orientation

of the dipole flow is parameterized by the rotation angle Θ and the pumping time τ, normalized by the characteristic domain emptying time tb. As shown in Fig. 2 for an eighteen-well arrangement, key concepts of this work include the development of kinematic boundaries that separate chaotic and non-chaotic regions in the flow, and enhanced mixing performance [8].

III. POTENTIAL PRACTICAL IMPLEMENTATION

RPM dynamics can be used to generate two fundamental processes: confinement and enhanced mixing. We illustrate these processes by first presenting a simple schematic remediation problem, then discussing possible areas for further development.

A. Schematic Problem: Remedication of a Dissolved Phase Plume

Consider a finite square plume advecting in regional groundwater flow, as shown in Fig. 3. Here we use a surrounding network of 16 wells, where the active dipole is effectively rotated in space through angle Θ by switching one dipole off, then switching a neighbouring dipole on, etc. The dipole pumping time τ and angle Θ are system parameters. Appropriate selection of τ and Θ leads to the formation of a kinematic boundary around the plume, providing indefinite confinement and preventing migration of the plume. Over many pumping cycles, hydrodynamic dispersion will eventually mix the confined volume. Also, as discussed in [8], RPM parameters can be chosen to induce chaotic flows within the target volume, which provides the potential for significant enhancement of mixing rate within the overall confinement of the plume.

Β. Future Developments: Regional Flows, Dispersion, Heterogeneity

Practical considerations are critical for implementation. In groundwater systems, regional flow gradients are usually present; these must be addressed for kinematic confinement to be sustainable, either by using large dipole pumping rates (to dominate regional fluxes) or by programming biases into the dipole pumping scheme in a time-averaged sense. Dispersion is unavoidable, and will ensure that confinement is not perfect through transverse dispersion across the kinematic boundary. The rate of dispersive release can be controlled by adjusting the radius of the kinematic boundary (for a given plume). Heterogeneity is ever-present in the subsurface and may provide perturbations to the dynamical flow structures. However, RPM flow topologies are stable to modest perturbations (Fig. 2, bottom row), so the domain of applicability is still likely to be significant for systems displaying modest heterogeneity. Research on chaotic dynamics in these areas is under way.

IV. CONCLUSIONS

Chaotic advection can be induced in groundwater flows by imposing time-dependent dynamics that lead to crossing of streamlines in the transient sense. Periodic re-orientations of dipole flows have been shown to exhibit phenomena of confinement and of enhanced mixing. These two phenomena are fundamental tools that are important to the design and operation of new subsurface remediation technologies.

Acknowledgements. This work was partly supported by the CSIRO Minerals Down Under National Research Flagship and by the Western Australian Geothermal Centre of Excellence.

REFERENCES

[1] Sposito, G. (1994) Steady groundwater flow as a dynamical system. Water Resour. Res. 30(8), 2935-2401.

[2] Metcalfe, G., Lester, D., Trefry, M. & Ord, A. (2007) Transport in a partially open porous media flow. Proc. SPIE 6802, Article 68020I. doi:10.1117/12.769319.

[3] Lester, D. R., Metcalfe, G., Trefry, M. G., Ord, A., Hobbs, B. & Rudman, M. (2009) Lagrangian topology of a periodically reoriented potential flow: Symmetry, optimization and mixing. Phyical Review E 80(3), Article 036208, doi:10.1103/PhysRevE.80.036208.

[4] Zhang, P., Devries, S. L., Dathe, A. D. & Bagtzoglou, A. C. (2009) Enhanced mixing and plume containment under time-dependent oscillatory flow. Environ. Sci. Tech. 43, 6283-6288.

[5] Metcalfe, G., Lester, D., Ord, A., Kulkarni, P., Trefry, M., Hobbs, B., Regenauer-Lieb, K. & Morris, J. (2010) A partially open porous media flow towards visualization of coupled velocity, concentration and deformation fields. Phil. Trans. R. Soc. A.368, 217-230, doi:10.1098/rsta.2009.0198.

[6] Metcalfe, G., Lester, D. R., Ord, A., Kulkarni, P., Rudman, M., Trefry, M., Hobbs, B., Regenauer-Lieb, K. & Morris, J., (2010) An experimental and theoretical study of the mixing characteristics of a periodically reoriented irrotational flow. Phil. Trans. R. Soc. A 368, 2147-2162, doi:10.1098/rsta.2010.0037.

[7] Trefry, M. G., McLaughlin, D., Metcalfe, G. Lester, D., Ord, A. Regenauer-Lieb, K. & Hobbs, B. (2010) On oscillating flows in randomly heterogeneous porous media. Phil. Trans. R. Soc. A. 368, 197-216, doi:10.1098/rsta.2009.0186.

[8] Lester, D. R., Rudman, M., Metcalfe, G., Trefry, M. G., Ord, A. & Hobbs, B. (2010) Scalar dispersion in a periodically reoriented potential flow: acceleration via Lagrangian chaos. Physical Review E 81(4), Article 046319, (doi:10.1103/PhysRevE.81.046319).

[9] Tang, X. Z. & Boozer, A. H. (1999) A Lagrangian analysis of advection-diffusion equation for a three dimensional chaotic flow. Phys. Fluids 11(6), 1418-1434.

[10] Metcalfe, G., Rudman, M., Brydon, A., Graham, L. J. W. & Hamilton, R. (2006) Composing chaos: an experimental and numerical study of an open duct mixing flow. AIChE J. 52(1), 9-28.

**Contact:**

Guy Metcalfe, guy.metcalfe@csiro.au