## 1. Introduction

[2] Treatment solutions containing chemical amendments (oxidants, electron donors, or nutrients) are frequently injected into aquifers for in situ remediation of contaminated groundwater [*Domenico and Schwartz*, 1998], but this approach suffers from a fundamental technical problem: Contaminant degradation requires mixing of the treatment solution and the contaminated groundwater [*MacDonald and Kitanidis*, 1993], which is difficult because mixing in porous media is generally very poor [*National Research Council*, 2009]. Because flow in porous media is laminar, it lacks the turbulent eddies that generate most of the mixing in open channels and engineered reactors. As a result, in situ degradation reactions are confined to a narrow fluid interface between the injected plume of treatment solution and the contaminated groundwater [*National Research Council*, 2000], across which pore-scale dispersion is the rate-limiting step. This is a fundamental technical problem that has received considerable attention from the groundwater research community.

[3] The literature on mixing in aquifers can be grouped into two broad categories, (1) heterogeneity effects, and (2) chaotic advection. Both groups generally conceptualize mixing as a two-step process with a transport step and a dispersion step. Under this paradigm, the term*mixing*is used in a restrictive sense for the dispersion step resulting from pore-scale processes and causing dilution, while the terms stirring, stretching, or spreading refer to the transport step resulting from advection, including heterogeneity effects manifested as macrodispersion, causing plumes to become more contorted but without dilution [*Dentz et al.*, 2011; *Kapoor and Gelhar*, 1994a; *Kitanidis*, 1994; *Le Borgne et al.*, 2010]. A common metric to quantify mixing is the local mixing factor , where *C*(** x**,

*t*) is the scalar concentration of interest, and

**D**is the dispersion tensor. For example, a recent review shows that three other mixing metrics are proportional to this local mixing factor: the rate of decay of concentration variance, the rate of growth of concentration entropy, and the rate of chemical reactions at high Damköhler number [

*Dentz et al.*, 2011]. The volume integral of the local mixing factor is the scalar dissipation rate, which has been used to quantify mixing not only in porous media [e.g.,

*Le Borgne et al.*, 2010] but also in other branches of fluid mechanics [e.g.,

*Pope*, 2000]. In essentially all cases relevant to groundwater remediation, mixing is distinct from spreading, which can instead be quantified by the spatial moment of

*C*(

**,**

*x**t*) [

*Le Borgne et al.*, 2010] or by the plume interface length [

*Zhang et al.*, 2009], both of which increase with time as the plume becomes more contorted. The latter metric is particularly appropriate for the limiting case in which pore-scale dispersion is neglected [

*Dagan*, 1989, section 4.3.5;

*Kapoor and Gelhar*, 1994b]. In particular, plumes in incompressible flows without dispersion maintain their initial concentration and volume [

*Dagan*, 1989, section 4.3.5], so in this simplified case the plume interface length directly records the degree of spreading. The present study neglects dispersion in order to focus entirely on spreading, with the understanding that mixing (i.e., pore-scale dispersion) provides the final link required to increase chemical reaction rates and consequently the effectiveness of remediating groundwater by injecting treatment solutions.

[4] A review of the literature on mixing in aquifers highlights two key findings. In the first category of literature that discusses heterogeneity effects, the key finding is that spreading depends on the structure of the flow [*Dentz and Carrera*, 2005; *Finn et al.*, 2004], consistent with the literature on turbulence [e.g., *Crimaldi et al.*, 2008]. The structure of the flow in geologic porous media largely results from heterogeneous permeability [*Dagan*, 1989, section 4.3.5; *Kapoor and Gelhar*, 1994a, 1994b; *Kitanidis*, 1994; *Rolle et al.*, 2009]. In the usual conceptual model, where flow results from an imposed hydraulic head gradient in a certain primary flow direction, background flow and heterogeneous permeability are required to generate heterogeneous velocity in the porous media. Recently, *Le Borgne et al.* [2010] generalized the role of heterogeneity by emphasizing that spreading depends on heterogeneous velocity, which could result from heterogeneous permeability, or from other mechanisms such as temporal fluctuations in the fluid velocity [*Dentz and Carrera*, 2005]. Once achieved, spreading enhances mixing through increased transverse dispersion [*Cirpka*, 2005; *Cirpka et al.*, 2011]. In sum, the literature on heterogeneity effects articulates the need for spreading, and emphasizes that spreading depends on the structure of the flow, namely, heterogeneous velocity. It does not, however, provide guidance on what the structure of the flow can or should be to promote spreading. Accordingly, the present study proposes a new way to generate heterogeneous velocity, using extractions and injections at wells, that does not require background flow or heterogeneous permeability.

[5] The second category of literature, pioneered by *Aref* [1984] and *Ottino* [1989], falls under the heading of chaotic advection, which is known to optimize spreading in laminar flows [*Ottino et al.*, 1994]. Chaotic advection exists when fluid particles exhibit sensitive dependence on initial conditions, which requires at least two-dimensional (2-D) unsteady flow or three-dimensional (3-D) steady flow [*Ottino*, 1989, section 4.7]. In particular, certain velocity fields are manifestations of the horseshoe map, a mathematical expression of stretching and folding that implies the presence of chaotic advection [*Ottino et al.*, 1994]. Since chaotic advection optimizes spreading in laminar flows, and since stretching and folding can indicate chaotic advection, the key finding in this second category of literature is that stretching and folding can optimize spreading in laminar flows [*Chakravarthy and Ottino*, 1995; *Ottino et al.*, 1994]. For example, in a theoretical study and analysis of data from the Borden site, *Weeks and Sposito* [1998] found that spreading is more effective if chaotic flow leading to both stretching and folding can be induced. Indeed, the importance of stretching and folding for spreading in laminar flows was first recognized more than a century ago by *Reynolds* [1894]. One can also grasp the importance of folding from a practical perspective, since only a certain degree of stretching can be accomplished within a bounded domain before folding becomes necessary to achieve continued stretching [*Aref*, 2002]. In the present application, the bounded domain is a finite region of an aquifer, to which it is beneficial to constrain remediation activity for reasons of waste containment.

[6] There are several conceptual models for chaotic advection in laminar flows, e.g., (1) the blinking vortex [*Aref*, 1984], (2) eccentric cylinders [*Funakoshi*, 2008; *Muzzio et al.*, 1992; *Swanson and Ottino*, 1990], and (3) the pulsed dipole [*Jones and Aref*, 1988; *Sposito*, 2006; *Stremler et al.*, 2004; *Tel et al.*, 2000] and similar approaches [*Bagtzoglou and Oates*, 2007; *Trefry et al.*, 2012]. Of these, the pulsed dipole merits special attention because it has found several applications, including spreading in porous media, since its introduction by *Jones and Aref* [1988]. *Tel et al.* [2000]applied the pulsed dipole to 2-D open flows, and investigated the effects of chaotic advection on chemical and biological processes in such flows.*Stremler et al.* [2004] investigated chaotic advection with a single pulsed dipole and with two pulsed dipoles, with applications to porous media and to microfluidics. Extending the work of *Jones and Aref* [1988] and *Stremler et al.* [2004], *Sposito* [2006] applied the pulsed dipole in the specific context of in situ groundwater remediation.

[7] Several other authors have reported methods similar to the pulsed dipole. *Bagtzoglou and Oates* [2007]numerically simulated transport in groundwater due to random extraction and injection at three wells in a triangular pattern, with no net extraction of water from the aquifer. Their results showed that this oscillating well triplet led to enhanced plume spreading. In a follow-up experimental study,*Zhang et al.* [2009] found that the oscillating well triplet increased the plume interface length more than a nonoscillating well triplet. *Trefry et al.* [2012] presented the rotating dipole based on a circular array of wells around a contaminant plume, of which two wells on opposite sides of the circle form a dipole at any given time. When the wells are operated such that the dipole pair proceeds around the circle, *Trefry et al.* [2012] showed the system to exhibit chaotic advection, leading to enhanced plume spreading.

[8] Although it has been applied to chaotic advection in porous media, the pulsed dipole [*Jones and Aref*, 1988; *Sposito*, 2006; *Stremler et al.*, 2004; *Tel et al.*, 2000] includes assumptions about the timing and orientation of reinjection of fluid particles that are not physically realistic in groundwater applications, where the timing and orientation of reinjection is random because of turbulent mixing of fluid particles in the pumps and because of dispersion in the piping between wells. Chaotic advection cannot be achieved in the pulsed dipole unless certain reinjection rules are used; in particular, *Radabaugh* [2011] showed a lack of chaotic advection, and hence poor spreading, on a single pass from the injection to the extraction well. Accordingly the unrealistic reinjection rules in the pulsed dipole are particularly problematic in the context of groundwater remediation. Moreover, reinjection is a regulatory concern, because injecting contaminated water into an aquifer is generally forbidden, even in the context of groundwater remediation.

[9] The pulsed dipole also suffers from a practical limitation in groundwater applications. Because the pulsed dipole extracts both the treatment solution and the contaminated groundwater simultaneously, they will undergo turbulent mixing in the pumps, leading to reactions in the wells, rather than spatially extensive reactions throughout the aquifer itself. Such reactions near wells frequently generate clogging [*Bagtzoglou and Oates*, 2007; *Li et al.*, 2009, 2010; *MacDonald et al.*, 1999], leading to difficulty in well operations and uncertainty in the structure of the flow field. For this reason, a method that can produce spreading while minimizing plume extraction would be preferable.

[10] The goal of the present paper is to present a new approach to the hydraulics of plume spreading in the context of in situ groundwater remediation, using an engineered sequence of extractions and injections of clean water at an array of wells surrounding the injected plume of treatment solution. This engineered sequence of injections and extractions generates plume spreading by stretching and folding the fluid interface between an injected treatment solution and contaminated groundwater. The paper begins with an overview of the analytical techniques—Poincaré sections, periodic points, stable and unstable manifolds, heteroclinic points, and Lyapunov exponents—used to analyze chaotic advection. These analytical techniques are used to demonstrate chaotic advection in the engineered sequence of injections and extractions for the idealized case of homogeneous permeability. Numerical simulations are then presented to show the relative contribution of spreading resulting from stretching and folding compared to spreading resulting from aquifer heterogeneity.