Application of the theory of micromixing to groundwater reactive transport models

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 11, 1313, doi: /2003wr002368, 2003 Application of the theory of micromixing to groundwater reactive transport models Bruce A. Robinson and Hari S. Viswanathan Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA Received 30 May 2003; accepted 30 July 2003; published 12 November [1] This study extends and applies the theory of micromixing, first introduced in the chemical reaction engineering literature, to the topic of reactive transport in porous media. For all but the simplest linear kinetic and sorption models the fate and transport of a reactive solute depends on the residence times and the details of small-scale mixing. The latter phenomenon, also called micromixing, is important because it brings into close proximity chemical species that react, and it controls the local concentrations in a flowing system. Solutes with reaction rates or sorption isotherms that depend on species concentration will therefore be affected by micromixing. Two models for micromixing are introduced, the minimum and maximum mixedness models, that provide bounds on the extent of reaction or retardation behavior within the constraints imposed by the residence time distribution (RTD) of a conservative solute in the same flow system. These mixing models prescribe the latest or earliest permissible mixing of parcels of fluid of different residence times, which in turn bounds the degree of reaction of a reactive solute for nonlinear rate laws or sorption isotherms. Simulation results using the bounding models show that micromixing effects are most important for nonlinear reaction curves, solute pulses of short duration, and systems with broad RTD curves. Use of these models is a straightforward and practical way to investigate the importance of a phenomenon for which data are seldom available and whose impact on groundwater reactive transport models has heretofore not been studied in a systematic, bounding manner. INDEX TERMS: 1803 Hydrology: Anthropogenic effects; 1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 1831 Hydrology: Groundwater quality; KEYWORDS: micromixing, reactive transport models, nonlinear reactions, mixedness models, groundwater, residence time distribution Citation: Robinson, B. A., and H. S. Viswanathan, Application of the theory of micromixing to groundwater reactive transport models, Water Resour. Res., 39(11), 1313, doi: /2003wr002368, Introduction [2] Reactive transport modeling codes have been developed by numerous researchers to simulate chemical transport processes such as aqueous reactions, biochemical reactions, and fluid-solid reactions [e.g., Yeh and Tripathi, 1989; Lichtner, 1996; Steefel and MacQuarrie, 1996; Yabusaki et al., 1998; Viswanathan et al., 1998; Chilakapati et al., 2000; Robinson et al., 2000; Xu et al., 2001; Mayer et al., 2001]. Different numerical formulations and approaches such as equilibrium reactions, kinetic formulations, and mixed formulations have been implemented, and the numerical methods have been tested and optimized to achieve rapid convergence. A fundamental assumption in the development of most reactive transport models is that an advective-dispersion equation (ADE) with chemical reaction sources and sinks can be written as a mass balance for each chemical component or species. The ADE for a given steady state or transient flow field and dispersion parameters is then solved to simulate the evolution of chemical species in the model. This paper is not subject to U.S. copyright. Published in 2003 by the American Geophysical Union. SBH 2-1 [3] These models can, in principle, be used in two- and three-dimensional flow and transport simulations in which the flow system is captured in detail. However, despite the advent of high performance computing hardware and software techniques, there are compelling reasons to expect that, for the foreseeable future, fundamental limitations will remain in applications such as the migration of contaminant plumes in the subsurface. First, prevailing theory and experimental observations point to the fact that transverse dispersion coefficients in groundwater transport are quite small [Gelhar, 1997]. Thus highly specialized methods are required to reduce numerical dispersion of plumes simulated using continuum-based solutions to the ADE [e.g., Cirpka et al., 1999]. Alternatively, particle tracking approaches can be employed, but the numerical techniques for combining particle tracking and reactive chemical transport such as that developed by Fabriol et al. [1993] and Sun [1999] are only now being developed. More fundamentally, reactive transport models will remain inherently nonunique due to the complex and uncertain processes of groundwater flow and transport, reaction mechanisms and effective parameters, and mixing effects. [4] In response to these challenges, stream tube and other similarly constructed models have been developed that enable the current suite of reactive transport codes to be

2 SBH 2-2 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING utilized to model contaminant migration in groundwater [Rainwater et al., 1987; Simmons et al., 1995; Yabusaki et al., 1998; Cirpka and Kitanidis, 2000a; Ginn, 2001]. This approach, outlined by Rainwater et al. [1987] for the problem of tracer test interpretation, assumes that a groundwater transport system can be represented by one or more independent one-dimensional curvilinear pathways of different travel time. An ensemble of such pathways is constructed to describe the basic nonreactive transport properties such as travel times and mixing processes. Then, reactive transport within each of these paths is computed in one dimension, and the overall behavior of the system is reconstructed by superimposing the results of the individual paths. Depending on the application, the distribution of travel times can be obtained directly from tracer experiments, by applying a theoretical technique for flow in heterogeneous porous media, or through direct numerical simulation of a flow field in a heterogeneous model. [5] Recently, it has become apparent that these and other reactive transport models have complexities and potential limitations that arise for nonlinear reaction systems. For example, Simmons et al. [1995] illustrated that for transport with a Michaelis-Menton nonlinear kinetics reaction, their stochastic-convective reactive transport method yielded a different extent of reaction in the breakthrough curve than is obtained using the standard ADE, despite the fact the distribution of residence times for the reactants was, by design, the same for the two cases. Kapoor et al. [1997], Miralles-Wilhelm et al. [1997], and Cirpka and Kitanidis [2000a] showed that for situations in which the rate of reaction depends on the concentrations of reactants, such as a bimolecular reaction mechanism, details of the mixing process down to the molecular level are important. These studies illustrate that for kinetically controlled, nonlinear reactions, the macroscale dispersive spreading of the reactants alone is insufficient to predict the extent of reaction; the details of the small-scale mixing process yield different transport behavior of the reactive system. Experimental studies in laboratory model systems have also been conducted to investigate this issue [Raje and Kapoor, 2000; Gramling et al., 2002]. [6] Extensions to the original stream tube models have been developed to capture the smaller-scale mixing process that is necessary to properly characterize the mixing of parcels of fluid containing reactants of differing concentration. Cirpka and Kitanidis [2000a] and Ginn [2001] are examples of this refined approach, which incorporates longitudinal dispersion within each stream tube, while ensuring that the desired distribution of travel times of solute mass is reproduced. Theoretical approaches have been proposed to describe the nature of the concentration fluctuations within the region defined by the macroscopic dimensions of a solute plume [Kitanidis, 1994; Kapoor and Gelhar, 1994; Cirpka and Kitanidis, 2000b]. However, Cirpka and Kitanidis [2000b] conclude that current methods and data limitations may yield rather uncertain model predictions of reactive transport behavior when the reactions are controlled by mixing. [7] In the present study we apply concepts from the field of chemical reaction engineering to the general topic of mixing and chemical reaction in groundwater reactive transport models in an attempt to provide a means for bounding the role of mixing on reactivity. Researchers studying chemical reactor design have investigated many of the same basic issues albeit in a somewhat different setting (reaction in flow-through chemical reactors) and for a different purpose (optimizing reactor performance [e.g., Nauman, 1981; Nauman and Buffham, 1983]). Here we employ and extend an approach originating in the chemical engineering literature for dealing with reaction in flowing systems to the study of reactive transport in groundwater. [8] The chemical reaction engineering analysis of continuous-flow systems such as flowing chemical reactors began with the pioneering work of Danckwerts [1953], who defined the residence time distribution (RTD) for fluid leaving a steady state flowing system. Danckwerts [1953] and subsequent studies by Danckwerts [1958] and Zwietering [1959] established that for all but the simplest reaction systems (first-order reactions) the RTD alone provides insufficient information about the flow system to predict the extent of reaction at the outlet. The fundamental idea is the concept of micromixing. The RTD is a measure of the distribution of residence times of molecules in the outlet stream, but does not fully characterize the mixing processes that govern the reaction rates, which occur at the molecular level. Therefore the problem is nonunique, in that different internal mixing models can provide equally good models for the RTD yet yield different reaction behavior. The term micromixing was coined to describe the molecular-scale mixing of molecules having different residence times, as these packets of fluid can have different reactant concentrations and hence different reaction rates. Micromixing theory in chemical reaction engineering was therefore cast in terms of the earliness or lateness of back mixing. With this formulation, we use the distribution of residence times, typically measured using a conservative tracer, to describe the reaction times resulting from the movement of mass through the system, and apply a model of micromixing to characterize the molecular-scale mixing processes that are critical to chemical reactions. [9] In groundwater reactive transport models, the micromixing construct is useful because fluid packets of different age are likely to have different concentrations, and therefore the extent of chemical reaction will depend on micromixing. Figure 1 illustrates this point. The inset on the left describes the case where the plume is thoroughly mixed and homogeneous, whereas the other inset details the case where complete mixing has not taken place, resulting in a heterogeneous concentration profile. Ginn [2001] cast his stream tube theory in the context of micromixing, and in so doing, demonstrated its importance. In the present study, we advance this concept by extending the bounding models of Zwietering [1959] to groundwater reactive transport problems. Using the residence time distribution, referred to herein as macromixing, as a constraint, we propose bounding models prescribing the details of mixing. We will show that the use of this theory allows us to establish robust theoretical bounds on the behavior of reacting systems with known RTD curves but unknown degrees of micromixing. In some cases, these bounds will be relatively narrow, whereas in other instances micromixing will be shown to be critical to the prediction of reaction extent. With this theoretical prediction, researchers can then assess whether additional effort is needed to reduce this uncertainty.

3 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-3 key relationships and concepts that are used subsequently with respect to micromixing. Danckwerts [1953] introduced the concept of the residence time distribution (RTD), a distribution function to account for the fact that molecules entering a flow system do not all leave at the same time. Let t = the residence time, the total time a molecule spends in the system; a = the age, the time since a molecule entered the system; and l = the residual lifetime, the remaining time a molecule will stay in the system. It follows that t ¼ a þ l ð1þ Figure 1. Schematic of a contaminant plume which is within the capture zone of a pumping well. The pumping well measures a distribution of travel times that can be used to determine the experimental RTD just as the outlet ports of a chemical reactor can be used to determine the RTD. [10] The organization of this paper is as follows. First, we review the theory of residence time distributions in continuous flow systems to introduce the required concepts used in the subsequent theoretical development. Next, we introduce the concept of minimum and maximum mixedness, illustrating how relatively simple flow and transport models can be constructed to achieve the bounds on micromixing for a given distribution of residence times. Then, we present a series of model simulations showing how these bounding models can be used to establish the limits of behavior of reactive systems. Our examples include both homogeneous, aqueous-phase reactions and heterogeneous sorption reactions. We then examine the one-dimensional ADE in the context of micromixing, to provide further theoretical understanding of this commonly used model when applied to complex reaction systems. Finally, we discuss the implications, strengths, and limitations of the approaches developed in this study, suggesting potential areas for future research and applications of the theory of micromixing in groundwater reactive transport modeling. 2. Summary of Residence Time Distribution and Micromixing Theory 2.1. Macroscopic Treatment of Mixing [11] In this section we introduce theoretical results related to flowing systems and distributions of residence times in steady state flow systems, for the purpose of establishing It is well-established that the distributions of ages or residence times can be determined experimentally for steady state flow systems using inlet-outlet tracer techniques. The distributions of interest in the present study and in the field of groundwater transport in general are the following. (1) Residence time distribution f (t), f (t)dt is the fraction of molecules exiting the system with residence times between t and t + dt (also referred to as the exit age distribution to denote that the inventory is performed on molecules leaving the system). (2) Internal age distribution I(a), I(a)d a is the fraction of molecules inside the system with ages between a and a + da. (3) Internal residual lifetime distribution J(l), J(l)d l is the fraction of molecules inside the system with residual lifetimes between l and l + d l. (4) Internal residence time distribution c(t), c(t)dt = the fraction of molecules inside the system with eventual residence times between t and t + dt. These distributions refer to different temporal variables attributable to a molecule (residence time, age, residual lifetime) and apply to different inventories of molecules (the molecules in the exit stream versus molecules inside the system). The most tractable theory related to distributions of ages and residence times has been those developed for steady state, flow-through systems with a single inlet and outlet. Under those constraints, f(t) can be measured as the outlet response to a pulse of tracer injected at time 0: fðþ¼ t QCðÞ t where Q is the volumetric flow rate through the system, C(t) is the concentration measured in the outlet stream, and m p is the mass of tracer injected. Alternatively, fðþ¼ t R 1 0 m p Ct ðþ Ct ðþdt In practice, tracer response curves often have long tails, and experiments are terminated before tracer recovery is complete. Because the denominator in equation (3) can only be approximated, methods such as those presented by Robinson and Tester [1986] are required to fully capture the distribution of residence times in real systems. [12] Cumulative distributions are commonly defined for these probability density functions, such as the cumulative residence time distribution function f(t): Z t Ft ðþ¼ 0 fðþdt t ð2þ ð3þ ð4þ

4 SBH 2-4 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING Figure 2. time. The parallel plug flow (PPF) reactor model. Each path transmits fluid of a given residence By definition, F(t) is the fraction of fluid in the exit stream with residence times of t or less. The integral of equation (4) at infinity is 1, in keeping with the definition of f(t) asa probability density function. One other relation relevant to the subsequent presentation is the expression for the mean residence time. Danckwerts [1953] showed that in any steady state flow-through system, the following expression holds: t Z 1 0 tf ðþdt t ¼ V Q where V is the fluid volume of the system. Thus for steady state flow the first temporal moment is related to the volume and flow rate in a straightforward manner, regardless of the nature of the f(t) distribution. [13] Next, we review the relation between f(t) and the internal age, residual lifetime, and internal residence time distribution functions, I(a), J(l), and c(t), respectively. For I(a), Danckwerts [1953] showed that IðaÞ ¼ 1 FðaÞ t After this derivation was made, Zwietering [1959] proved that the distribution of residual lifetimes J(l) is given by the identical expression given in equation (6): Jl ð Þ ¼ 1 FðlÞ t In the groundwater literature, equation (6) for the age distribution provides the theoretical basis for studies in which the ages of groundwater in an aquifer can be interpreted [e.g., Eriksson, 1961; Bolin and Rodhe, 1973; Goode, 1996; Varni and Carrera, 1998; Ginn, 1999; Bethke et al., 2000; Etcheverry and Perrochet, 2000]. ð5þ ð6þ ð7þ [14] An expression for the internal residence time distribution was derived by Buffham [1983]: cðþ¼ t tf ðþ t t Robinson and Tester [1986] present an intuitive explanation of equation (8) based on the internal volumes of the collection of molecules of a given eventual residence time t. For this, they used an arrangement of the contents of the flowing system in the configuration of the parallel plugflow (PPF) model depicted in Figure 2, which we note is identical to the stochastic-convective transport model presented by Simmons et al. [1995]. Consider the path that contains all of the fluid within the system that will have an eventual residence time t. According to the definition of f(t), the flow rate through this path is Qf(t)dt. To achieve the residence time of t, the volume of this path must be its flow rate times its residence time, or tqf(t)dt. The internal residence time distribution function is, by definition, a volume distribution function prescribing the relative amounts of the contents of a given eventual residence time, found by dividing the volume of a path by the total volume V: cðþdt t ¼ tqf ðþdt t V Making use of equation (5) relating the mean residence time t to eliminate Q and V, we obtain equation (8) directly. [15] The point of this exercise is to establish that once the residence time distribution function is specified, the structure of a model such as the stochastic-convective (or PPF) model is completely determined: that is, the flow rates and volumes of the paths must be set according to the distribution functions and interrelationships described above. Of course, in an actual model, the number of paths will not be infinite, so that finite time interval t is used instead of dt, ð8þ ð9þ

5 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-5 Figure 3. Reaction rate versus concentration for homogeneous kinetic rate laws of different reaction orders. but the same constraints still apply in order to reproduce a given residence time distribution. Later, we introduce an alternative implementation of the PPF model that yields transport results more efficiently in numerical simulations, but the same basic point holds Microscopic Treatment of Mixing [16] The key insight from the theory of micromixing is that the distribution of residence times (macromixing) constrains but is insufficient to fully characterize the extent of reaction except in certain special cases. The RTD is an inventory of residence times, but does not account for the details of mixing at a scale smaller than the scale at which the RTD is determined. The term micromixing refers to the detailed mixing process among molecules of different ages or concentrations. [17] One special case for which the RTD alone is sufficient to predict the steady state, outlet concentration is that of a homogeneous, irreversible, first-order reaction: C ¼ Z 1 0 e kt fðþdt t ð10þ where k is the first-order rate constant of the reaction. Equation (10) can be obtained using a stream tube model because the extent of reaction at the end of a uniform stream tube of residence time t is identical to the extent of reaction in a batch system at time t. However, for non-first-order reactions, Danckwerts [1953, 1958] proved that the earliness or lateness of the molecular scale mixing among fluid elements of different concentration has an impact on the extent of reaction. The concept of mixing among molecules of different age was termed micromixing. The reason that age was chosen as the parameter of interest is that fluid of different ages are most likely to possess different concentrations. Therefore, inasmuch as the extent of reaction within an element of fluid is a function of the age of the molecules, age and concentration play interchangeable roles. [18] Chauhan et al. [1972] provide a conceptually straightforward explanation of the impact of micromixing processes for kinetic reactions. Consider two elements of fluid, labeled 1 and 2, of different concentrations. Without loss of generality, assume these elements have the same volume. Figure 3 shows the reaction rate (R(C) = kc n ) versus concentration C for various reaction orders. Examining the second-order reaction (n = 2), if the two fluid elements are unmixed, rates symbolized by the filled circles are obtained, yielding an average reaction rate denoted by the open box symbol. Contrast this rate with the rate obtained by first mixing the contents of the two fluid elements (the filled circle of the n = 2 curve). Clearly, when fluid elements remain unmixed, faster reaction rates and greater overall reaction will result. By contrast, for the half-order reaction, mixing enhances the reaction rate, whereas for the first-order reaction, mixing does not influence the reaction rate. Nauman [1981] generalized this result for reaction systems of arbitrary complexity by considering the shape of the reaction rate versus concentration curve. Mixing enhances reaction rates for concave-down reaction curves (@ 2 R/@C 2 < 0), and inhibits reaction rates for concave-up reaction curves (@ 2 R/@C 2 >0) Minimum and Maximum Mixedness Models [19] This straightforward explanation of the role of micromixing begs the question: is it possible to bound the extent of reaction or other measures of reactive transport behavior in a flowing system of known RTD? The principle means for establishing such bounds was developed by Zwietering [1959], who provides a theoretical basis and a mathematical model for the micromixing extremes. The use of the PPF model of Figure 2 was clearly recognized from the outset as the minimum mixedness extreme, as molecules of different age mix with each other only at the outlet manifold, outside of the flow system in which the reaction takes place. Zwietering s contribution was to define the other extreme, called the maximum mixedness model, which prescribes the earliest permissible mixing of molecules of different age for a system within the constraints imposed by the RTD. The key requirement, that all molecules in the system with the same residual lifetime l mix with one another and travel together to the outlet, is achieved in the plug flow model with side entrances, shown in Figure 4a. A manifold of entrances of negligible volume is used to introduce fluid of given concentration at various locations along the flow path. After the inlet fluid immediately mixes with the internal contents at that location, the fluid exhibits pure plug flow with no longitudinal dispersion. Therefore the residual lifetime l of the molecules decreases to the right, and reaches 0 at the outlet. Entering molecules immediately mix with those with which they eventually leave. [20] Zwietering [1959] showed that the stipulation of the RTD for the maximum mixedness model completely fixes the construction of the flow rates and positions of the side entrances. Here we reconstruct Zwietering s derivation as the first step toward extending it for use in groundwater reactive transport problems. Consider a material balance for a differential portion of the reactor from l to l + d l

6 SBH 2-6 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING Figure 4. model. Bounding models for micromixing. (a) Maximum mixedness model. (b) Minimum mixedness (Figure 4a). The solute mass flux into and out of the element are: Source fluxðfrom side entranceþ ðfluid fluxþðinlet conc: Þ ¼ Qf ½ ðlþdlšc in ð11þ Flux in ðupstream fluid fluxþðupstream conc: Þ ¼ Q½1 Fðl þ dlþšcðl þ dlþ ð12þ Flux out ðdownstream fluid fluxþðdownstream conc: Þ ¼ Q½1 FðlÞŠCðlÞ ð13þ Reaction rate ðrate=volumeþðvolume of elementþ ¼ RC ð ÞV½1 FðlÞŠdl=t ð14þ One can demonstrate that these expressions follow from the definitions and relations introduced in section 2.1 for macromixing. Combining these terms, Zwietering [1959] obtained the following result for steady state outlet concentration for a constant injection concentration C in : 0 ¼ RC þ f ðlþ ð 1 FðlÞ C C inþ ð15þ To summarize this derivation, the incremental path volumes, inlet flow locations, and inlet flow rates of the side entrances can be uniquely determined to reproduce an arbitrary distribution of residence times for a system, in a manner that prescribes the condition of maximum mixedness for the model. [21] A final result attributable to Zwietering [1959] is the assertion that an alternate form of the minimum mixedness model that maintains the same RTD can be formulated simply by reversing the flow rates in the maximum mixedness model. As shown in Figure 4b, the right end of the model becomes a single inlet, and a distributed set of outlets issuing from the one-dimensional path are mixed to obtain the mixed outlet concentration. This result follows from the correspondence between the distributions I(a) and J(l) (see equations (6) and (7)), recognizing that l of the maximum mixedness model becomes a for the minimum mixedness model. Note that the mathematical representation of the reactive transport model of Rainwater et al. [1987] or Dagan and Cvetkovic [1996] in essence takes the same approach as the model of Figure 4b by applying a single characteristic solution to the transport equation and convolving the solution over the distribution of travel times. Figure 4b does this using a physically based model with one entrance and many side exits. [22] Although equation (15) is a steady state transport expression, it is based on a physical model that can be used to simulate the transient transport behavior (that is, timevarying solute mass flux within the steady state flow system and at the outlet) that is more relevant to transport in groundwater. To do this, we translate the residual lifetime l to a path length variable x along the one-dimensional path model, and solve the transient advection-reaction equation without dispersion within the model and at ¼ RC ð Þ ux þ Sx ðþ ð16þ where u(x) is the linear velocity, S(x) represents the solute source term associated with the time-varying injection

7 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-7 concentration C in (t) and the spatially varying injection flow rate. Within the flow path, a reactive transport model can be included to study the role of micromixing. [23] From the derivation of the maximum mixedness model geometry above, we now outline a simple method for building a one-dimensional model with side entrances that can be used with a reactive transport code to study micromixing. Assume that the flow path has a constant volumetric water content f and cross sectional area A x.to transform the residual lifetime along the path to an axial distance x, we use the fluid flow rate from equation (12), and sum the contributions from residual lifetime of 0 to l: xðlþ ¼ Q fa x Z l 0 ½1 FðlÞŠdl ð17þ This expression poses no difficulty as l!1, as can be demonstrated by integrating both sides of equation (6) from 0to1 to obtain the total length L: xðl!1þ ¼ L ¼ Q fa x Z 1 0 ½1 FðlÞŠdl ¼ Qt fa x Z 1 0 IðlÞdl ¼ Qt fa x ð18þ The inlet fluid flux q(l) at any location x(l) is obtained from the RTD function f (l): qðlþ ¼ Qf ðlþdl ð19þ and this inlet flow placed at a location given by equation (17). The time-varying concentration is input at each side entrance, and the outlet concentration versus time is that simulated at the exit of the path. [24] In practice, a one-dimensional path constructed for use with an existing reactive transport code requires a grid with discretization sufficient to minimize numerical dispersion. Assuming that the discretization of the RTD curve is not sufficient to obtain accurate solutions using a one-to-one matching of grid points and the RTD curve points, the RTD curve can be subdivided into more points by interpolation. Recall also that the minimum mixedness model is obtained by performing the transport simulation in the same domain with the flow rates reversed. This is considerably simpler and more efficient than constructing a series of one-dimensional pathways as in Figure 2. In essence, the plug flow system with side exits (Figure 4b) takes advantage of the identical chemical profiles of concentrations within each plug flow path from the entrance to any age, and only performs those calculations once, rather than within each path. In this aspect, it is effectively the same as the Lagrangian reactive transport model of Dagan and Cvetkovic [1996]. [25] Finally, we note that these models should be understood as abstract representations of mixing processes, rather than as real plumbing systems that reproduce a subsurface flow configuration. This is, of course, true of most groundwater models, since details of subsurface flow paths are almost always lacking. Regarding the maximum mixedness model, side entrances can be viewed a device that enforces mixing between fluid that has been in the system for a long time with fluid that has just entered. A subsurface condition that this system would represent well is one in which a reactive species is introduced into both a preferential flow path and adjacent low-permeability material. Mass entering the low-permeability material slowly migrates to the preferential path; this mass, with considerable age, mixes with the rapidly moving fluid of young age. This mixing occurs near the entry location of the solute, thereby resulting in early mixing of fluids of vastly different age. The appeal of the approach developed in the present study, depicted in Figure 4, is that complex, multidimensional mixing situations whose details are unknown can be bracketed by simple, reduced one-dimensional models that capture the range of mixing behaviors possible in a system of known RTD. 3. Mixing Model Results [26] In this section we apply the minimum and maximum mixedness models to a series of simulations examining the influence of various reaction situations, RTD curves, and solute input functions on reactive transport. The results presented in this section are designed to provide insight into the role of micromixing in groundwater reactive transport systems and to illustrate the potential usefulness of this approach for more complex reaction schemes Model Setup [27] In the present study, theoretical RTD curves are generated using the one-dimensional ADE. The use of this model to prescribe a certain level of macromixing is in no way a requirement of the micromixing modeling approach, nor is it a statement of the validity of the advectivedispersion equation with constant coefficients for simulating field-scale transport processes. We chose this macromixing model mainly due to the ease of use, its familiarity to most researchers, and because of its use as a building block within the advective-dispersive stream tube approach in recent studies [Cirpka and Kitanidis, 2000a; Ginn, 2001]. Alternatively, other types of residence time distribution models, such as those developed from theoretical considerations, could have been used [e.g., Dagan et al., 1992; Bellin et al., 1994; Harvey and Gorelick, 1995], or explicit heterogeneous numerical models could be used to generate the RTD curves, as in the work by Cirpka and Kitanidis [2000a] and Ginn [2001]. Alternatively, field conservative tracer data can also be used to establish the RTD [e.g., Rainwater et al., 1987]. In the present study, to illustrate the concepts, RTD curves were generated from the one-dimensional ADE for two values of the Peclet number Pe = L/a L, 10 and 100, where L is the flow path length and a L is the longitudinal dispersivity. [28] From the breakthrough curve predicted by the onedimensional model, a one-dimensional model with side entrances is constructed to simulate the maximum mixedness model as outlined above. The computer code FEHM [Zyvoloski et al., 1997; Robinson et al., 2000] was used to perform the flow and reactive transport calculations. A separate model grid was generated for each Pe with a prescribed variable grid spacing and inlet flow rates along the flow path, as shown in Figure 5. Proceeding from left to right (higher to lower residual lifetimes), velocities increase. The details of this flow pattern are not important, except in

8 SBH 2-8 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING Figure 5. Schematic diagrams for the maximum mixedness model, showing inlet flow rates and velocities within the flow path. (left) Pe = 100. (right) Pe = 10. the sense that this model setup reproduces the RTD curve in a manner that prescribes the maximum (or minimum) mixedness model. For systems with greater macroscopic dispersion (the Pe = 10 case), inlet flow is distributed along a longer stretch of the flow path, including locations closer to the exit. It can also be shown that systems with significant tailing yield a larger region of the path with long residual lifetimes (locations to the left of the path). The flow simulation consists of a preliminary steady state flow model run using the specified source fluid flux distribution along the path, and a specified pressure boundary condition at the outlet end. [29] To demonstrate that the model yields the correct distribution of residence times, Figure 6 compares the original RTD curves with conservative tracer model runs using the maximum mixedness model, for a pulse injection of solute. The close comparison illustrates that for the Pe = 10 and 100 cases, the maximum mixedness model constructed in this way can very accurately match the original RTD curve specified. More complex RTD curves, such as those with multiple peaks or long tails, can be reproduced with equal accuracy This ability to reproduce an arbitrary RTD ensures that the subsequent comparisons of the minimum and maximum mixedness models for reactive transport reflect only micromixing effects. [30] For the minimum mixedness case, the same grid is used, the specified inlet flow rates are switched to outlet flow rates of the same magnitude, and the specified-pressure boundary condition at the end of the model now reverses to become an inlet rather than an outlet. The outlet concentration at any time is then the flux-weighted average concentration, computed by summing all of the individual contributions. Comparisons (not shown) of the original RTD curves with the minimum mixedness model also yielded excellent agreement Homogeneous Reaction Simulations: Kinetically Controlled Transformation [31] The purpose of the simulations in this section is to examine behavior of the mixedness models for homogeneous, aqueous-phase reactions for different reaction types, RTDs, and solute injection scenarios. The reactions consist of a series of simple, kinetic reactions of individual species, and a bimolecular reaction system. First, we present results for a simple homogeneous reaction, with an irreversible, kinetically controlled reaction (Figure 3). This reaction is a surrogate for more complex reaction schemes, such as a reaction between two species, in which one of the reactants is present in great excess, or its concentration is buffered through other equilibrium reactions. For example, a reaction such as 2A + B! products, with the concentration of B held constant, has an effective reaction rate of k eff [A] 2, where k eff = k[b]. Recalling the qualitative explanation of Chauhan et al. [1972] described earlier, we expect that for concave-up rate laws (that is, n > 1), the minimum mixedness model should result in greater extent of reaction and lower outlet concentrations. The converse is true for n < 1. [32] Figures 7a and 7b show the results of a series of runs for the Pe = 100 macromixing case for n = 2. Each set of calculations examines a shorter duration of injection of the reactant, while the total mass of reactant is held constant from run to run. Therefore, for pulse duration t p = t, C in =1, Figure 6. Comparisons of breakthrough curves for the ADE model and the maximum mixedness model for a conservative solute.

9 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-9 Figure 7. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Homogeneous, kinetic reaction, n =2,Pe = 100. (a) t p = t and 0.1t; (b)t p = 0.01t and ADE solution. whereas for t p =0.1t, C in = 10, and for t p = 0.01t, C in = 100. In this and subsequent figures, each pair of curves with the same symbols have the same injection pulse duration, and the mixedness models are distinguished by the line type (solid for maximum mixedness and dotted for minimum mixedness). [33] For Pe = 100 and the longest injection duration (Figure 7a), the differences between the minimum and maximum mixedness models are relatively minor. However, shorter injection duration at higher injection concentrations accentuates the differences in the models (the t p =0.1t simulations in Figure 7a and the t p = 0.01t simulations in Figure 7b). Short injection of high-concentration reactant is immediately diluted in the maximum mixedness model, and local reaction rates are much lower than the minimum mixedness case, where high concentrations remain high, and reaction rates are elevated. Earliness or lateness of micromixing therefore is most important for nonlinear aqueous reactions when reactant injection durations are short. [34] Another characteristic of these results is that for shorter pulse duration (and higher injection concentrations), the maximum mixedness model achieves a limiting condition in which further sharpening of the pulse yields no further change in the breakthrough curve of the reactive solute. However, for the minimum mixedness case, sharpening the pulse and increasing the concentrations results in ever increasing reaction rates. This result shows that the minimum mixedness case, as represented by the stochasticconvective reactive transport model of Simmons et al. [1995] or the model results of Cvetkovic and Dagan [1996], can yield bounding behavior that is potentially far different than the maximum mixedness case. In relying on one or the other of these mixedness models, one could be neglecting an important uncertainty due to lack of knowledge of micromixing in a real system. We believe that the appropriate approach in the face of uncertainty is to bound the impact of micromixing by performing model calculations with both the minimum and maximum mixedness models. [35] A series of identical simulations for the Pe = 10 macromixing case are shown in Figures 8a and 8b. Although the results are similar, the differences in the two mixedness extremes are more pronounced, such that even the longest pulse injection shows a significant deviation between the minimum and maximum mixedness cases. Therefore broader distributions of residence times (the Pe = 10 case in this example) offer a greater possibility for mixing of fluids of vastly different residence times, and these fluids will have significantly different concentrations. [36] Turning to a half-order reaction (n = 1/2), Figures 9a and 9b show results for a single injection duration, namely, t p = 0.01t and C in = 100. The results show that when the reaction rate is concave-down, the situation with respect to micromixing reverses. That is, the minimum mixedness model now provides the upper bound on the outlet concentration (the lowest overall reaction rates), and the maximum mixedness model results in the lower bound on concentrations. As with the second-order reaction, the differences attributable to micromixing effects are larger for the more dispersed Pe = 10 case. [37] The results on the role of earliness or lateness of mixing for the simple reaction rate R = kc n correspond to the qualitative explanation of Figure 3, and the transport model results show that the differences can be quite large. Finally, the first-order reaction results in Figures 9a and 9b demonstrate that micromixing does not influence the extent of reaction for first-order systems, and that the RTD alone completely determines the extent of reaction for a given kinetics constant. [38] To illustrate the use of the mixing models for a bimolecular reaction, we choose a simple reaction scheme A + B! products, with R = k[a][b]. Reactant B is assumed

10 SBH 2-10 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING reaction (lower outlet concentrations). These two reactors represent theoretical bounds on mixing for the purpose of predicting the overall extent of reaction Heterogeneous Reaction Simulations: Sorption [39] There is an extensive literature on both kinetic and equilibrium sorption formulations used in groundwater Figure 8. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Homogeneous, kinetic reaction, n = 2, Pe = 10. (a) t p = t and 0.1t; (b)t p = 0.01t and ADE solution. to be present in the system and in the injection fluid at a concentration of one. Reactant A is initially not present, and is introduced as a pulse of duration t p = 0.01t and concentration of 100. Figure 10 shows the results for reactant A for the two micromixing models. With mixing occurring only at the outlet, reaction can only take place within the fluid within the pulse of solute injected, which contains both A and B. By contrast, the maximum mixedness model allows reactant A to mix with B within the system, yielding greater Figure 9. Breakthrough curves for the minimum and maximum mixedness models for the short solute injection pulse duration (t p = 0.01t). Homogeneous, kinetic reaction, n = 1/2 and n = 1. Minimum and maximum mixedness curves fall on top of each other for n =1.(a)Pe = 100; (b) Pe = 10.

11 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-11 Figure 10. Breakthrough curves for the minimum and maximum mixedness models and the ADE model for the short solute injection pulse duration (t p = 0.01t). Homogeneous, bimolecular kinetic reaction, Pe = 10. transport models, motivated primarily by the need to predict the migration of contaminants in aquifers. For solutes that are retarded with respect to a conservative tracer due to sorption, a range of models are used, from simple, equilibrium sorption to more mechanistic reactive transport models with kinetically controlled reactions. Although the computational tools for simulating these rock-solute interactions have been developed and used, the role of micromixing on the transport of sorbing solutes has not been studied previously. The simulations presented here are an initial attempt to investigate micromixing processes in detail for time-varying inputs of a sorbing solute. [40] In this study we investigate micromixing effects on the behavior of a solute undergoing equilibrium, reversible sorption with either a Langmuir or Freundlich isotherm: solute-rock interactions in a real geochemical system, and that heterogeneities can play an important role in the interpretation of the isotherm parameters [e.g., Wise, 1993]. [41] We begin with a simplified explanation of the role of micromixing analogous to that provided for kinetic homogeneous reactions. In Figure 11, because both the Langmuir and Freundlich isotherms (S versus C) are concave down, mixing of the packets of fluid results in an intermediate concentration yielding greater effective sorption (the closed circles) than when the two packets are unmixed (the open boxes). We therefore expect greater overall delay of solute mass for the maximum mixedness case for these isotherms. The linear model should show no influence of micromixing, and the breakthrough curve should be controlled completely by the macromixing RTD curve and the sorption coefficient. [42] To use the micromixing models to investigate breakthrough curves of sorbing solutes, we assume that the fundamental dispersion parameters in the groundwater system (dispersivity, diffusion coefficient, porosity) derived from a conservative solute RTD are assumed to be applicable for the sorbing solute. This implies that the RTD curve as measured by a nonsorbing tracer represents the distribution of travel times of the solute had it not undergone sorption. Under this restriction, equation (15) can be revised to include retardation: R þ Sx ðþ ð22þ where R f (C) is the retardation factor. Using the wellestablished derivation outlined by Fetter [1999] for incorporating equilibrium sorption into the transport equation, the following relationships are obtained: Langmuir R f " # ðcþ ¼ 1 þ r b K 1 f ð1 þ K 2 CÞ 2 ð23þ Langmuir Freundlich S ¼ K 1C 1 þ K 2 C S ¼ K f C n ð20þ ð21þ where S is the sorbed concentration, and K 1, K 2, K f, and n are constants determined by performing batch sorption tests at a series of aqueous concentrations. At low concentrations, the Langmuir model reverts to a linear sorption model, whereas at high concentrations, sorption sites become full and S reaches an asymptotic value. The Freundlich isotherm has a somewhat different functionality, with no linear isotherm range and no asymptote. In the present study, we treat these isotherms as experimentally determined relationships, recognizing that they may be proxies for more complex Figure 11. Sorbed concentration versus aqueous concentration for different linear and nonlinear sorption isotherms.

12 SBH 2-12 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING durations under the assumption that identical mass is injected in each case. For the shortest pulse injection, the minimum mixedness model yields earlier arrival at the outlet than the maximum mixedness case, in keeping with the qualitative explanation of Figure 11. Another feature is the presence of a sharp arrival front for the maximum mixedness case, compared to the minimum mixedness model. Self-sharpening fronts and long tails are a wellknown feature of transport of solutes with nonlinear iso- Figure 12. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Langmuir sorption isotherm, Pe = 100. (a) t p = t and 0.1t; (b)t p = 0.01t and ADE solution. Freundlich R f ðcþ ¼ 1 þ r bk f nc n 1 f ð24þ where r b is the bulk rock density. [43] Figures 12 and 13 show the predicted breakthrough curves for the Langmuir isotherm for the Pe = 100 and Pe = 10 cases, respectively. As in previous simulations, the impact of micromixing is displayed for a variety of pulse Figure 13. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Langmuir sorption isotherm, Pe = 10. (a) t p = t and 0.1t; (b)t p = 0.01t and ADE solution.

13 ROBINSON AND VISWANATHAN: REACTIVE TRANSPORT AND MICROMIXING SBH 2-13 [44] Figure 13, the high-dispersion Pe = 10 case, exhibits behavior similar to that in Figure 12, although the details (first arrival times, peak concentration, degree of front sharpening) are different due to the broader distribution of residence times. Like the Pe = 100 case, the most important feature of these results is the influence of micromixing on the arrival time distributions of a sorbing solute. [45] For the Freundlich isotherm, Figures 14 and 15 show a series of simulations analogous to those just presented for the Langmuir sorption case. Because the Freundlich iso- Figure 14. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Freundlich sorption isotherm, Pe = 100. (a) t p = t and 0.1t; (b)t p = 0.01t. therms [e.g., van der Zee, 1990]. Although these phenomena occur in both models, the minimum mixedness case retains a broader distribution of arrival times due to the configuration of the model as a series of independent flow segments with the original distribution of travel times. Therefore the degree to which front sharpening occurs for a nonlinear sorption isotherm such as the Langmuir isotherm depends on the mixedness model as well as the concentrations and shape of the sorption isotherm. Figure 15. Breakthrough curves for the minimum and maximum mixedness models for different solute injection pulse durations. Freundlich sorption isotherm, Pe = 10. (a) t p = t and 0.1t; (b)t p = 0.01t.