Macroscopic transport models calibrated by flux-averaged breakthrough curves of conservative compounds do not necessarily characterize mixing well because such breakthrough curves do not provide information on fluctuations of concentration within the solute flux, which may influence mean reaction rates. We numerically examine the validity of macroscopic transport models, which are capable of describing all details of flux-averaged breakthrough curves, for predicting a mixing-controlled bimolecular precipitation reaction in heterogeneous media. We consider a homogeneous, isotropic medium with an elliptical, low-permeability inclusion and random heterogeneous fields. For the single-inclusion case, slow advection through the inclusion results in a multimodal breakthrough curve with enhanced tailing. We vary the hydraulic conductivity contrast and Peclet number to investigate the performance of a “perfect” macroscopic transport model for predicting the total precipitated mass within the domain and the peak concentration difference between the conservative and reactive cases at the outflow boundary. The results indicate that such a model may perform well in media with either very small or very high permeability contrast or at low Peclet number. In the high-contrast case, most flow takes place in preferential flow paths, resulting in a small variance of the flux-weighted concentration, even though the offset in the breakthrough between the slow and fast travel paths is substantial. Maximum relative errors in terms of total precipitated mass and the peak concentration difference between the conservative and reactive cases occur at intermediate permeability contrasts and large Peclet numbers. Numerical simulations on random heterogeneous fields confirm the finding of the single-inclusion case. Thus, in cases with intermediate hydraulic conductivity contrast, making macroscopic models fit flux-averaged concentration breakthrough curves better may not improve the prediction of mixing-controlled reactive transport, and it becomes necessary to quantify and account for the variability of conservative concentrations in the flux in order to formulate an appropriate macroscopic transport model that predicts mixing-controlled reactive transport.
 Reactive transport models are essential tools for predicting contaminant fate and transport in the subsurface and designing effective remediation strategies. Spatial variability and uncertainty of hydraulic parameters have been recognized as a major challenge in predicting the fate and transport of contaminants in heterogeneous aquifers. Macroscopic models rely on upscaling the transport equation and deriving effective parameters. A particular challenge lies in predicting mixing-controlled reactive transport in heterogeneous domains. The common procedure is to determine physical transport parameters by fitting model results to flux-averaged breakthrough curves of a conservative tracer and to determine the reactive parameters in perfectly mixed laboratory experiments. However, such macroscopic models may inaccurately predict breakthrough curves of reactive species in which the reaction is controlled by solute mixing [Raje and Kapoor, 2000; Gramling et al., 2002]. In particular cases of mixing-controlled reactive transport, the reactive species concentrations can directly be inferred from concentrations of conservative, linearly mixing components [e.g., Ham et al., 2004; De Simoni et al., 2005, 2007; Cirpka and Valocchi, 2007, 2009]. The relationships are inherently nonlinear and are, in principle, valid only on the scale at which reactions occur, which may be addressed as the point scale for our purposes. If the concentrations of the conservative components vary within the water flux, the flux-averaged concentrations are representative only of the conservative tracer but cannot be used to predict reactive species. For the latter purpose, information about the concentration fluctuations caused by incomplete mixing and causing segregation of reactive species becomes necessary [Kapoor et al., 1997; Miralles-Wilhem et al., 1997; Cirpka and Kitanidis, 2000b; Raje and Kapoor, 2000; Cirpka, 2002; Gramling et al., 2002]. For example, the widely applied macroscopic advection-dispersion equation (ADE) may adequately describe the flux-averaged concentration of a conservative tracer crossing a control plane in a mildly heterogeneous domain located at a sufficient distance from the injection. However, such models coupled with nonlinear chemical reactions have proven unsuccessful in forecasting mixing-controlled reactive transport because macrodispersion implicitly assumes that plume spreading controlled by spatial variations in hydraulic conductivity is equivalent to mixing [Molz and Widdowson, 1988; MacQuarrie and Sudicky, 1990; Kitanidis, 1994; Kapoor et al., 1997; Kapoor and Kitanidis, 1998; Cirpka and Kitanidis, 2000a; Raje and Kapoor, 2000; Cirpka, 2002; Gramling et al., 2002; Dentz and Carrera, 2007; Luo et al., 2008].
 An interesting problem remaining unresolved concerns the capability of sophisticated macroscopic models that are capable of reproducing anomalous flux-averaged breakthrough to predict mixing-controlled reactive transport. Recently, several studies have reported successful applications of macroscopic nonlocal models with multirate mass transfer [Willmann et al., 2010] and the method of continuous time random walk [Edery et al., 2009] to the simulation of mixing-controlled reactive transport. However, characterizing anomalous behavior of a conservative tracer well does not necessarily mean that such models also characterize mixing well. To the best of our knowledge, the existing nonlocal transport equations are meant to reproduce mean concentrations; if necessary, they are calibrated by flux-averaged breakthrough curves, and they do not provide information about the variability of the flux concentration. As discussed, mean conservative concentrations without information about the variability should be insufficient to evaluate mean reaction rates. Conversely, successful applications exist. Thus, it remains unknown under which condition the models capable of reproducing anomalous transport of a conservative tracer are appropriate for simulating mixing-controlled reactive transport.
 In the present research, we examine the validity of macroscopic transport models that reproduce flux-averaged breakthrough curves of a conservative tracer for simulating mixing-controlled reactive transport. It is not the objective of our study to test the performance of various parameterizations for anomalous transport with respect to reproducing the flux-averaged conservative compound breakthrough curves. Instead, we directly work with the true flux-averaged conservative compound breakthrough curves, transfer those to breakthrough curves of reactive species according to the rules valid on the point scale, and compare the results to simulations in which the transfer from conservative to reactive concentrations is performed on the local scale before taking the flux average. We will present two numerical studies of mixing-controlled reactive transport in heterogeneous domains: one in a medium with a single, elliptical, low-permeability inclusion within a homogeneous, isotropic medium and the other in random heterogeneous media. We vary hydrogeological parameters to investigate how the performance of the macroscopic model changes and identify the conditions under which the simple macroscopic model yields satisfactory predictions of mixing-controlled reactive transport.
2. Numerical Models
2.1. Hydrogeological Settings
Figure 1 shows the first setup used in our simulations: An elliptical low-permeability inclusion is embedded in a rectangular two-dimensional homogeneous, isotropic domain. The hydraulic head is fixed at the left and right boundaries, whereas no flow crosses the top and bottom boundaries. The major and minor axes of the ellipse are half of the domain length and width, respectively. Table 1 summarizes the hydrogeological parameters used in the numerical case. Solute transport in this domain is essentially controlled by two dimensionless parameters:
where K1 and K2 are the hydraulic conductivity in the inclusion and matrix, respectively, Krel represents the hydraulic conductivity contrast, is the effective mean velocity within the entire domain, b is the half width of the elliptical inclusion, L is the domain length, Dt is the transverse dispersion coefficient, and Pe is the transverse Peclet number. In our studies on the validity of macroscopic models, we vary K2 and Dt so that Krel ranges across three orders of magnitude (100 ∼ 102) and Pe ranges across four orders of magnitude (100 ∼ 103). For each combination of Krel and Pe, we solve the steady state flow field and the reactive transport case described in section 2.2.
Table 1. Hydrogeologic Parameters for the Case With an Elliptical, Low-Permeability Inclusion
Dimension of domain
L × W
5 m × 2.5 m
Dimension of elliptical inclusion
2a × 2b
2.5 m × 1.25 m
0.005 m × 0.005 m
10−5 ∼ 10−3 m/s
Mean hydraulic gradient
10−9 ∼ 10−6 m2/s
 The second set of simulations is performed in random heterogeneous fields, which may be considered as a composition of many low- or high-permeability inclusions [Suribhatla et al., 2004]. We consider sets of two-dimensional heterogeneous fields in which the mean flow is in direction x. The length and width of the domain are 20 and 10 m, respectively. The variance of log hydraulic conductivity is varied to represent mildly to strongly heterogeneous fields. For each variance, we consider 100 realizations of log hydraulic conductivity following an isotropic, Gaussian covariance model. All hydrogeological parameters are listed in Table 2. The log conductivity fields are generated by the spectral method of Dykaar and Kitanidis  on a rectangular 1000 × 500 cell grid. The steady state flow field is solved for a mean hydraulic gradient of 0.01 in direction x. A streamline-oriented grid for transport with grid resolution identical to that of the rectangular grid is generated using the streamline method of Cirpka et al. [1999a, 1999b]. The flow rate in each stream tube is identical. The numerical schemes for solving the transport problem have been presented elsewhere [Cirpka et al., 1999a].
Table 2. Hydrogeologic Parameters for the Two-Dimensional Random Heterogeneous Case
Dimension of domain
L × W
20 m × 10 m
0.02 m × 0.02 m
Mean hydraulic conductivity
1.16 × 10−5 m/s
Variance of hydraulic conductivity
0.2, 0.5, 1, 2, …, 6
Ix × Iy
0.4 m × 0.4 m
Mean hydraulic gradient
2.2. Mixing-Controlled Reactive Transport
 We consider reactive transport of compounds undergoing an instantaneous bimolecular precipitation reaction with 1:1:1 stoichiometry:
where A and B are aqueous species (solutes) and C is a mineral present throughout the domain. This reaction is assumed to be fast compared to typical transport times, so that it can be treated as in local equilibrium. The concentrations of the aqueous species A and B satisfy
where cA and cB are the molar concentrations of the reactive species A and B, respectively, and Keq is the chemical equilibrium constant. The transport equations for the reactive species A and B are given by
where is porosity, t is time, q is the specific discharge or Darcy velocity, D is the local dispersion tensor, and r is the reaction rate (precipitation of C, if r is positive). The concentration cC of the precipitate is given as moles of compound C per volume of the solid phase; it is not explicitly computed in the following. A and B have the same reaction rate because of the stoichiometry balance of the bimolecular reaction. For simplicity, we assume local chemical equilibrium; that is, equation (4) is satisfied at all locations and times. This reactive transport case can be solved completely relying upon the mixing ratio of conservative transport [De Simoni et al., 2005, 2007].
 For the sake of completeness, we briefly summarize the methodology developed by De Simoni et al. . We define the conservative component
 Thus, cA and cB can be solely evaluated on the basis of the conservative component, i.e., cA,B(x, t) = cA,B[u(x, t)], for given chemical equilibrium constants. The specific algorithm is described in section 2.3.
2.3. Numerical Experiments
 As described, we conduct numerous simulations in which we apply different hydrogeological parameters in both general setups. Replacement simulations are considered for the bimolecular precipitation reaction; that is, we assume the domain is initially uniformly filled with a solution containing species A, and a solution of species A and B with a constant concentration is continuously injected into the domain at the inflow boundary. The transport parameters of A and B are assumed to be identical. For our numerical simulations, we define the initial and boundary conditions in terms of the conservative component u:
 Thus, the corresponding initial and boundary conditions for species A and B can be calculated by equations (10) and (11):
 We assume the chemical equilibrium constant to be Keq = 0.01 with a unit of squared concentration.
 Utilizing the concept of the mixing ratio, the numerical algorithm to evaluate the amount of reactant A missing due to the precipitation reaction is summarized as follows.
 1. Evaluate the mixing ratio X by numerically solving the conservative transport equation
 3. Evaluate the species concentration cA(u) by equation (10).
 4. Evaluate the missing concentration of A due to the precipitation:
where XcAin + (1 − X)cA0 is the concentration of compound A if there was no precipitation reaction and cA is the actual concentration. Integrated over the entire domain, quantifies the total amount of the precipitate C formed. Locally, does not quantify the mass of the precipitate at a given location and time but how much precipitate has cumulatively been formed somewhere on the way from the inflow to the outflow boundary, detected by the missing concentration of A.
 The mixing ratio X describes the volumetric fraction of the injected tracer solution in the mixture with the background water. Using this algorithm, we can solve the spatial concentration distributions for all species. Thus, the flux-averaged breakthrough curve for the missing part of the reactant A is given by
where the angle brackets represent flux-weighted averaging over the cross section S of the outflow boundary,
which is a function of time. The expressions for flux-weighted averages of other concentrations are analogous. The flux-weighted standard deviation of the mixing ratio reads
 In our virtual truth, we have access to the full spatiotemporal distribution of the mixing ratio within the outflow. This facilitates computing the same distribution of the dissolved reactive species A and B and the amount of the formed precipitate using the summarized algorithm [Cirpka et al., 2008]. In practice, such detailed information would not be available. Instead, we assume that only the flux-averaged mixing ratio in the outflow is known, which can be conveniently evaluated by conducting a conservative tracer test with a continuous step injection or a pulse injection with subsequent integration of the signal. We will examine how well we can predict the flux-averaged breakthrough curves of reactive species if we only know the mean breakthrough curve of the mixing ratio. We do this by applying the transfer rules from conservative to reactive species concentrations to the flux-averaged mixing ratio , despite the nonlinearity of the transfer rules. Such an approach is equivalent to calibrating a one-dimensional macroscopic transport model to the mean conservative tracer breakthrough curve and subsequently applying the model to the reactive case. We do not define a specific macroscopic model but actually assume a “perfect” macroscopic model that is capable of exactly reproducing all details of the flux-averaged breakthrough curve. Any calibrated one-dimensional model can only be worse. In this context, the difference between the reactive and conservative breakthrough curves predicted by the “perfect” macroscopic model is given by
is directly evaluated from the conservative breakthrough curve.
2.4. Measure of Goodness
 The validity of macroscopic mean models are examined by comparing with . The measures of goodness of the “perfect” macroscopic model include the relative errors in the total and peak differences in the mass of A between the conservative and reactive cases at the outflow boundary. The relative error between the true and predicted breakthrough curves can be evaluated by equations (24) and (27),
which is a function of time.
 The factors that may influence the relative error ɛ include the reaction kinetics, chemical constants, boundary and initial conditions, and the variability of the mixing ratio within the solute flux. Technically, ɛ becomes zero when , which implies that cA is a linear function of X (excluding the presence of mixing-controlled reactions) or the variation of X is negligible. The relative errors in total and peak mass difference between the conservative and reactive cases are defined as
3. Results and Discussion
3.1. Elliptical, Low-Permeability Inclusion
Figure 2 shows snapshots of concentration distributions for Krel = 10 and Krel = 100 within the domain at different times. It is noticeable that the concentration breakthrough in the low-permeability inclusion significantly lags behind, particularly in the case with high conductivity contrast. The stream tube passing through the inclusion is squeezed downstream from the inclusion. The larger the conductivity contrast, the thinner this downstream stream tube is exiting the inclusion. This has two consequences: First, only a very small fraction of the water flux samples the low-conductivity inclusions. Thus, even large differences in concentration do not contribute that much to the flux-weighted standard deviation of concentration. Second, downstream from the inclusion it is comparably simple to mix water, which has bypassed the inclusion, into the stream tube that went through the inclusion. This is because the decisive transverse diffusion length is comparably small. This implies that shortly after the inclusion the strong concentration difference between streams that went through and around the inclusion is significantly reduced, resulting in smaller variations.
Figure 3 shows the travel time distributions and associated variances at the entire outflow boundary for two specific cases with hydraulic conductivity contrasts of 10 and 100 at similar large Peclet numbers. Both breakthrough curves are bimodal, resulting from a fraction of the total flux passing through the low-conductivity inclusion while the remaining flux surpasses the inclusion. The case with larger conductivity contrast Krel shows a significantly enhanced tail after the first peak. Obviously, both cases could not be reproduced by a 1-D ADE model, and more sophisticated, most likely nonlocal, macroscopic models may be needed to capture the observed anomalous transport behavior in the conservative breakthrough curves. Variances show similar bimodal and enhanced tailing behavior as the travel time distributions.
Figure 4a shows the normalized flux-averaged concentration breakthrough curves of a conservative tracer at the outflow boundary, which are the time integrals of the travel time distributions shown in Figure 3a. Particularly, for the case with Krel = 100, it requires long-time monitoring with high detection accuracy to identify the enhanced tailing behavior. In practical applications, it would be tempting to erroneously set the full breakthrough concentration at the level reached between one and two pore volumes and fit the ADE to this modified breakthrough curve, thus ignoring the enhanced tail. Figure 4b shows the coefficient of variation (CV) of the breakthrough curves. The variability in concentration results mainly from different times (and shapes) of breakthrough between stream tubes passing through the low-permeability inclusion and those bypassing it. An early peak of CV occurs before major breakthrough at the outflow boundary. Because the mean concentration is nearly zero at these times, the early peak in CV does not necessarily indicate a large concentration variance. In addition, CV also shows long tailing. Specifically, the less heterogeneous case with Krel = 10 has much higher CV at late times, whereas the case with Krel = 100 has a longer tail of CV at smaller values. Thus, highly heterogeneous hydraulic conductivity fields (high contrast in this case) do not necessarily lead to high concentration variations because the majority of flow flux travels through the preferential zone, which dominates the evaluation of the flux-averaged concentration.
Figure 5 compares the true concentration difference of A between the conservative and reactive cases at the outflow boundary and the results predicted by the “perfect” macroscopic model for the two cases. The macroscopic model accurately predicts precipitation at early times and very late times, which corresponds to the cases of nearly zero mean concentration and nearly zero concentration variation, respectively. At intermediate times, the reaction rate is overestimated in both cases, yielding erroneously high precipitation, confirming that concentration variations cannot be neglected in the simulation of mixing-controlled reactive transport. It is also obvious that the macroscopic model performs better for the case with Krel = 100 than for Krel = 10 because of smaller flux-averaged concentration variations associated with the enhanced long tail of the mean concentration. These specific cases indicate that (1) in some cases, no matter how good the macroscopic model is in capturing details (peaks, tailing behavior, and others) of the flux-averaged conservative breakthrough curve, it may still fail in predicting mixing-controlled reactive transport and (2) macroscopic models may yield better predictions in media with higher hydraulic conductivity contrast than in those with intermediate contrasts.
Figure 6 shows the relative error of the macroscopic mean model in predicting the total precipitated mass as a function of the hydraulic conductivity contrast and the Peclet number. Accurate predictions only occur when Krel is close to unity or the Peclet number is small, which corresponds to a nearly homogeneous domain or the case of large transverse dispersion, respectively. At a given Krel, the relative error increases with the Peclet number. At a given Peclet number, the maximum error occurs at an intermediate value of Krel, which is about 10 for the given geometry, i.e., 1 order of magnitude difference between K1 and K2. The maximum error within the tested ranges of Krel and the Peclet number can be as much as 400%. For large Krel, the relative error decreases with Krel until the limiting case where the elliptical zone becomes completely impermeable. Thus, for heterogeneous media with an intermediate hydraulic conductivity contrast and a large Peclet number, macroscopic models may result in significant overestimation of the total precipitated mass.
Figure 7 shows the relative error of the peak concentration difference of compound A between the conservative and reactive cases. The peak concentration may play a significant role in risk assessment when peak concentration is the criterion. Similar to the total precipitated mass, accurate predictions are achieved only when the domain is homogeneous and the Peclet number is very small. At a given Krel, the error quickly increases with Pe until the Peclet number becomes greater than 100, at which point a steady value is reached. At a given Peclet number, the maximum error is found at an intermediate Krel, about 2 for the given geometry, and the maximum relative error of the peak concentration can be as much as 50% within the tested range of the Peclet number. In addition, unlike the total precipitated mass, the error of the peak concentration does not decrease at high conductivity contrasts. At the upper limit of tested Krel, the relative error is still about 10% at small Peclet numbers and about 20% at large Peclet numbers.
Figure 8 shows the normalized mean square error (NMSE) of the mixing ratio X within the solute flux, which clearly shows that the maximum concentration variation occurs at large Peclet numbers and at an intermediate hydraulic conductivity contrasts,
3.2. Random Case
Figure 9 shows the streamlines in four specific random fields with different variance. The streamlines are rather uniformly distributed in the case of a small log conductivity variance. With increasing variance, streamlines are more focused in the zones with large hydraulic conductivities and form narrow preferential paths. The majority of flow moves along these preferential paths, which accounts for the large proportion in evaluating flux-average breakthrough curves. In addition, it requires shorter transverse distance for the preferential flow to mix. Thus, concentration variations in highly heterogeneous fields may become smaller than those in intermediately heterogeneous fields.
Figure 10a quantifies the concentration variations by the NMSE of the mixing ratio X within the solute flux. Figures 10b and 10c show the mean relative error in the total precipitated mass and peak concentration difference of compound A between the conservative and reactive cases based on the Monte Carlo simulations for random heterogeneous fields with different values of the log conductivity variance. The four lines represent different transverse Peclet numbers, defined here as the ratio of characteristic time scales for transverse dispersion across the transverse integral scale versus advection across distance x. All plots show the following: (1) The maximum error occurs at an intermediate level of heterogeneity, i.e., macroscopic models perform better in highly than in mildly heterogeneous cases; on the basis of the simulations of 100 realizations for each tested variance, we have that the maximum mean error occurs around . (2) The prediction error decreases with increasing travel distance, which is consistent with the common understanding that mixing can catch up with spreading in the large-distance limit.
 In the simulations with the random fields, the prediction of reactive transport by the one-dimensional models based on the mean breakthrough curve of a conservative compound performed, in general, better than in the single-inclusion simulations. This can best be understood if the random case is conceptualized as a combination of many single inclusions, with both higher and lower permeability than the average. Each inclusion leads to a variation of the mixing ratio within the flux. However, the likelihood that an individual stream tube samples exclusively either low or high inclusions is fairly low. Most stream tubes sample a combination of low and high inclusions, limiting the range of variability in the mixing ratio.
4. Discussion and Conclusions
 We have presented numerical test cases of mixing-controlled reactive transport with a bimolecular precipitation reaction at local equilibrium in heterogeneous domains. The case of a single inclusion could be characterized by two dimensionless variables: the hydraulic conductivity contrast and the transverse Peclet number. In the simulations using random fields, the hydraulic conductivity contrast was replaced by the variance of log conductivity. The key objective was to analyze to what extent concentration fluctuations within the solute flux could be neglected in the transfer from breakthrough curves of conservative to reactive compounds. From a strictly theoretical standpoint, it is clear that neglecting such variations must lead to a mass balance error because the transfer from conservative to reactive compound concentrations is nonlinear. However, the studies of Edery et al.  and Willmann et al.  indicated good performance, despite the fact that their models could not account for concentration variations in the solute flux.
 Our results indicate the largest errors in macroscopic one-dimensional models for intermediate conductivity contrasts and high Peclet numbers. With respect to total mass balance errors, increasing the degree of heterogeneity beyond a critical value led to an improvement of the performance, whereas the prediction of the peak concentration difference between conservative and reactive simulations does not benefit that much from increasing the degree of heterogeneity. The comparably good performance in highly heterogeneous cases can be attributed to (1) small fractions of the water flux passing through low-conductivity inclusions so that, while the fronts lag extremely behind in such inclusions, their contribution to the overall breakthrough curve is not that big and (2) efficient mixing between water fluxes that have experienced low-conductivity zones and those that have not, caused by transverse dispersion over short diffusion lengths within preferential flow zones downstream from inclusions.
 These findings visualize that the conceptual model of mobile and immobile flow zones with diffusive mass transfer in between is not that bad, even in cases where the transfer through the low-conductivity zones is by slow advection rather than diffusion, provided that the degree of heterogeneity is high enough. In such cases, the diffusive mass transfer occurs downstream from the inclusions by transverse dispersion. For sufficient travel distance, the conceptual difference becomes irrelevant and does not result in large overall errors. The multirate mass transfer model, as a typical representative of nonlocal-in-time one-dimensional transport models, does not account for concentration variation within the solute flux. However, in the cases where such models are typically applied, namely, in highly heterogeneous domains exhibiting anomalous transport, the concentration variation within the flux is much smaller than the variation in the resident concentration, which is represented by different concentrations in different domains. That is, the error introduced by applying a model that ignores concentration variation within the flux remains limited. The comparably good performance of these models in mixing-controlled reactive transport in highly heterogeneous domains is not because they can separate between spreading and mixing but because they are applied to cases where this difference is not so pronounced.
 Our results also indicate that intermediately heterogeneous fields lead to transport behavior that should not be represented by upscaled models that ignore concentration variations in the solute flux when applied to mixing-controlled reactive transport. Without accounting for the concentration variations, it is inappropriate to directly apply the transport model calibrated by the flux-averaged conservative breakthrough curve to mixing-controlled reactive transport. We have suggested pseudo-one-dimensional models accounting for concentration variations in the past [Cirpka and Kitanidis, 2000b; Luo and Cirpka, 2008] but admit a certain lack of elegance in their derivation and application. We see a clear research need in deriving upscaled nonlocal transport formalisms that go beyond ensemble mean concentrations so that long tails or other anomalous features of the flux-averaged breakthrough curves can be captured and concentration variations can be quantified.
 We would like to thank T. Ginn, D. Bolster, and an anonymous reviewer for their constructive comments.