Lagrangian topology of a periodically reoriented potential flow: symmetry, optimization, and mixing

Controlling pore-scale processes to tame subsurface biomineralization

Microorganisms capable of biomineralization can catalyze mineral precipitation by modifying local physical and chemical conditions. In porous media, such as soil and rock, these microorganisms live and function in highly heterogeneous physical, chemical and ecological microenvironments, with strong local gradients created by both microbial activity and the pore-scale structure of the subsurface. Here, we focus on extracellular bacterial biomineralization, which is sensitive to external heterogeneity, and review the pore-scale processes controlling microbial biomineralization in natural and engineered porous media. We discuss how individual physical, chemical and ecological factors integrate to affect the spatial and temporal control of biomineralization, and how each of these factors contributes to a quantitative understanding of biomineralization in porous media. We find that an improved understanding of microbial behavior in heterogeneous microenvironments would promote understanding of natural systems and output in diverse technological applications, including improved representation and control of fluid mixing from pore to field scales. We suggest a range of directions by which future work can build from existing tools to advance each of these areas to improve understanding and predictability of biomineralization science and technology.

Use of steady-state hydraulic tomography to inform the selection of a chaotic advection system

The concept of chaotic advection is a novel approach that has the potential to overcome some of the challenges associated with mixing of reagents that commonly occur when injection based in situ treatment techniques are used. The rotated potential mixing (RPM) flow system is one configuration which has been theorized to achieve chaotic advection in porous media, and enhance reagent mixing by periodically re-oriented dipole pumping at a series of radial wells. Prior to field implementation of chaotic advection, the selection of an RPM flow protocol will likely require a numerical model that can adequately represent groundwater flow within the zone of interest. As expected, the hydraulic conductivity (K) field is the most critical input requirement for the selected groundwater flow model. Hydraulic tomography (HT) is an innovative characterization approach that has shown potential to provide information on a K field. In this investigation, we explored whether the same well system required to invoke chaotic advection can also be applied in a HT analysis, and evaluated the use of the generated K tomogram for the selection of RPM flow parameters that can enhance reagent mixing. A series of dipole pumping tests were conducted within an area of interest as defined by the limits of the circular network of eight injection/extraction wells used to invoke chaotic advection. Hydraulic head data collected from independent dipole pumping tests were used in an inverse model to perform steady-state hydraulic tomography (SSHT) analysis to generate a K tomogram. Both the K tomogram and an effective parameter approach (i.e., a single K value assigned across the entire spatial domain as determined by single well pumping and slug tests) produced estimates of hydraulic head that closely resembled those observed due to the relative homogeneous nature of the aquifer and the small spatial scale of the area of interest. In contrast, particle tracking results showed that incorporating a heterogeneous K field significantly enhanced the spatial distribution of particle trajectories indicative of reagent mixing. These findings support the hypothesis that the same well system used to invoke chaotic advection can be combined with SSHT analysis as a viable site characterization tool for delineating the spatial variability of K. Incorporating this K tomogram in a groundwater flow model with a particle tracking engine can be used as a design tool to aid in the selection of a site-specific RPM flow protocol to achieve enhanced reagent mixing.

Lagrangian approach to analysis and engineering of two generic transport problems in enhanced subsurface flows

Scope is scalar transport in enhanced subsurface flows driven by injection and extraction wells. Two transport problems of great practical relevance, viz. (i) rapid scalar extraction and (ii) contained in situ processing, are investigated in terms of the thermal transport in a two-dimensional circular subsurface reservoir with a well-driven Darcy-type flow. Lagrangian (fluid) transport is key to these problems and thus the notion of Lagrangian coherent structures (LCSs) is adopted for analysis and engineering purposes. Analysis by this approach of heat (scalar) extraction reveals that a basic pumping scheme involving a static injector-extractor pair invariably outperforms more elaborate schemes using time-periodic actuation of multiple well pairs due to the formation of "special" LCSs that retard heat release. Such LCSs are, on the other hand, well-suited for contained in situ processing. LCSs namely are fundamentally embedded in the thermal (scalar) transport and, given they admit rapid and accurate control by the pumping scheme, thus naturally emerge as "internal actuators" for the creation of (dynamic) processing zones and reaction fronts. This principle has been developed into an LCS-based in situ processing strategy. To this end the non-trivial link between thermal and Lagrangian transport is rigorously established via methods from dynamical-systems theory so as to enable systematic demarcation and characterisation of confinement zones and their interaction with the wells for the generic case of diffusion. The framework thus developed is demonstrated by way of examples.

Deceleration of one-dimensional mixing by discontinuous mappings

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equally sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes nonmonotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counterintuitively predicts a deceleration in the asymptotic mixing rate with an increasing diffusivity rate. The implications of the results on finite time mixing are discussed.

Predicting mixing via resonances: Application to spherical piecewise isometries

We present an analytic method to find the areas of nonmixing regions in orientation-preserving spherical piecewise isometries (PWIs), and apply it to determine the mixing efficacy of a class of spherical PWIs derived from granular flow in a biaxial tumbler. We show that mixing efficacy has a complex distribution across the protocol space, with local minima in mixing efficacy, termed resonances, that can be determined analytically. These resonances are caused by the interaction of two mode-locking-like phenomena.

Localized shear generates three-dimensional transport

Understanding the mechanisms that control three-dimensional (3D) fluid transport is central to many processes, including mixing, chemical reaction, and biological activity. Here a novel mechanism for 3D transport is uncovered where fluid particles are kicked between streamlines near a localized shear, which occurs in many flows and materials. This results in 3D transport similar to Resonance Induced Dispersion (RID); however, this new mechanism is more rapid and mutually incompatible with RID. We explore its governing impact with both an abstract 2-action flow and a model fluid flow. We show that transitions from one-dimensional (1D) to two-dimensional (2D) and 2D to 3D transport occur based on the relative magnitudes of streamline jumps in two transverse directions.

Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle-centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.

Mixing of discontinuously deforming media

Mixing of materials is fundamental to many natural phenomena and engineering applications. The presence of discontinuous deformations-such as shear banding or wall slip-creates new mechanisms for mixing and transport beyond those predicted by classical dynamical systems theory. Here, we show how a novel mixing mechanism combining stretching with cutting and shuffling yields exponential mixing rates, quantified by a positive Lyapunov exponent, an impossibility for systems with cutting and shuffling alone or bounded systems with stretching alone, and demonstrate it in a fluid flow. While dynamical systems theory provides a framework for understanding mixing in smoothly deforming media, a theory of discontinuous mixing is yet to be fully developed. New methods are needed to systematize, explain, and extrapolate measurements on systems with discontinuous deformations. Here, we investigate "webs" of Lagrangian discontinuities and show that they provide a template for the overall transport dynamics. Considering slip deformations as the asymptotic limit of increasingly localised smooth shear, we also demonstrate exactly how some of the new structures introduced by discontinuous deformations are analogous to structures in smoothly deforming systems.

Toward enhanced subsurface intervention methods using chaotic advection

Many intervention activities in the terrestrial subsurface involve the need to recover/emplace distributions of scalar quantities (e.g. dissolved phase concentrations or heat) from/in volumes of saturated porous media. These scalars can be targeted by pump-and-treat methods or by amendment technologies. Application examples include in-situ leaching for metals, recovery of dissolved contaminant plumes, or utilizing heat energy in geothermal reservoirs. While conventional pumping methods work reasonably well, costs associated with maintaining pumping schedules are high and improvements in efficiency would be welcome. In this paper we discuss how transient switching of the pressure at different wells can intimately control subsurface flow, generating a range of "programmed" flows with various beneficial characteristics. Some programs produce chaotic flows which accelerate mixing, while others create encapsulating flows which can isolate fluid zones for lengthy periods. In a simplified model of an aquifer subject to balanced pumping, chaotic flow topologies have been predicted theoretically and verified experimentally using Hele-Shaw cells. Here, a survey of the key characteristics of chaotic advection is presented. Mathematical methods are used to show how these characteristics may translate into practical situations involving regional flows and heterogeneity. The results are robust to perturbations, and withstand significant aquifer heterogeneity. It is proposed that chaotic advection may form the basis of new efficient technologies for groundwater interventions.

Logarithmic nonlinear Schrödinger equation and irrotational, compressible flows: an exact solution

A class of irrotational, isentropic, and compressible flows is studied theoretically by formulating the density and the velocity potential in a Madelung transformation. The resulting nonlinear Schrödinger equation is solved in terms of similarity variables. One particular family of exact solutions, valid for any ratio of the specific heat capacities of the gas, permits explicit expressions of the fluid properties and velocities in terms of time and spatial coordinates. Analytically, the density is a Gaussian function of the similarity variable, while the temperature is a function of time only. This method is applicable in one (1D), two, and three dimensional geometries. As a simple example, a 1D gas column, with mass injection on one side and a steadily translating wall on the other, can be formulated exactly. The connection with the evolution of an unsteady velocity potential will also be examined.

Chaotic mixing via streamline jumping in quasi-two-dimensional tumbled granular flows

We study, numerically and analytically, the singular limit of a vanishing flowing layer in tumbled granular flows in quasi-two-dimensional rotating containers. The limiting behavior is found to be identical under the two versions of the kinematic continuum model of such flows, and the transition to the limiting dynamics is analyzed in detail. In particular, we formulate the no-shear-layer dynamical system as a piecewise isometry. It is shown how such a discontinuous map, through the concordant mechanism of streamline jumping, leads to the physical mixing of granular matter. The dependence of the dynamics of Lagrangian particle trajectories on the tumbler fill fraction is also established through Poincaré sections, and, in the special case of a half-full tumbler, chaotic behavior is shown to disappear completely in the singular limit. At other fill levels, stretching in the sense of shear strain is replaced by spreading due to streamline jumping. Finally, we use finite-time Lyapunov exponents to establish the manifold structure and understand "how chaotic" the limiting piecewise isometry is.

Scalar dispersion in a periodically reoriented potential flow: acceleration via Lagrangian chaos

Although potential flows are irrotational, Lagrangian chaos can occur when these are unsteady, with rapid global mixing observed upon flow parameter optimization. What is unknown is whether Lagrangian chaos in potential flows results in accelerated scalar dispersion, to what magnitude, how robustly, and via what mechanisms. We consider scalar dispersion in a model unsteady potential flow, the Lagrangian topology of which is well understood. The asymptotic scalar dispersion rate q and corresponding scalar distribution (strange eigenmode) are calculated over the flow parameter space Q for Peclét numbers Pe=10{1}-10{4}. The richness of solutions over Q increases with Pe, with pattern mode locking, symmetry breaking transitions to chaos and fractally distributed maxima observed. Such behavior suggests detailed global resolution of Q is necessary for robust optimization, however localization of local optima to bifurcations between periodic and subharmonic eigenmodes suggests novel efficient means of optimization. Acceleration rates of 150 fold at Pe=10{4} are observed; significantly greater than corresponding values for chaotic Stokes flows, suggesting significant scope for dispersion acceleration in potential flows in general.