Mixing processes significantly affect reactive solute transport in fluids. For example, contaminant degradation in environmental aquatic systems can be limited either by the availability of one or more reactants, brought into contact by physical mixing, or by the kinetics of the (bio)chemical transformations. Appropriate metrics are needed to accurately quantify the interplay between mixing and reactive processes. The exponential of the Shannon entropy of the concentration probability distribution has been proposed and applied to quantify the dilution of conservative solutes either in a given volume (dilution index) or in a given water flux (flux-related dilution index). In this work we derive the transport equation for the entropy of a reactive solute. Adopting a flux-related framework, we show that the degree of uniformity of the solute mass flux distribution for a reactive species and its rate of change are informative measures of physical and (bio)chemical processes and their complex interaction.
 Quantifying the interplay between mixing and reactions is critical to deepening our understanding of reactive solute transport in geophysical flows [e.g., Weiss and Provenzale, 2008]. In the case of transport in porous media these processes are important in implementing effective engineered or natural remediation strategies for contaminated groundwater and in performing risk assessment analysis [Sanchez-Vila et al., 2007; Edery et al., 2009; Bellin et al., 2011; de Barros et al., 2012]. In subsurface environments mixing is very slow and, therefore, of key relevance since it often constitutes the main limiting mechanism for a reaction to occur.
 Appropriate measures are required to identify and quantitatively describe the interaction between transport mechanisms and reaction kinetics [e.g., Dentz et al., 2011]. Mixing of conservative solutes can be effectively quantified using metrics such as the scalar dissipation rate [e.g., Le Borgne et al., 2010; Bolster et al., 2011a] and the dilution index. In this work we focus on the dilution index, which, for a conservative species i with concentration Ci [ML−3], is defined as: [L3] [Kitanidis, 1994]. This metric has been successfully applied in many contexts to measure the degree of dilution and to effectively distinguish between spreading and dilution [e.g., Cao and Kitanidis, 1998; Beckie, 1998; Tartakovsky et al., 2009; Bolster et al., 2011b]. The dilution index represents the exponential of the Shannon entropy of a concentration probability distribution p [L−3] defined over a volume V [L3], where [L−3]. A formally similar measure of entropy was also applied by Rolle et al. [2009, 2012] and Chiogna et al. [2011a, 2011b] describing dilution as the act of distributing a given solute mass flux over a larger water flux: [L3T−1]; where [L−3 T] is the flux-related probability density function of the speciesi and qx [LT−1] is the component of the specific discharge in the principal flow direction, normal to the cross-sectional area Ω [L2]. EQhas been called the flux-related dilution index byRolle et al. and quantifies an effective volumetric flux transporting the solute mass flux at a given cross-section.
 We focus on reactive solute transport in both homogeneous and heterogeneous porous formations and we distinguish between dilution and reactive mixing. We work in a flux-related framework which has been shown to be effective in quantifying mixing in heterogeneous velocity fields [e.g.,Cirpka et al., 2011]. In such a framework, dilution refers to the distribution of the solute concentration over a larger water flux with increasing distance from the source; consequently, peak concentration is reduced. With the term reactive mixing we consider the condition where initially segregated reactants are distributed over the same water flux, thus allowing reactions to occur. For both conservative and reactive solutes, the Shannon entropy measures the distribution of a solute within a domain (i.e., the water flux in our setup). For conservative species, the solute concentration becomes distributed over a larger water discharge. For a reactive species, though, the reaction consumes part of the reactant mass and may tend to sustain non-uniformity in the reactant distribution. Consequently, due to the reaction, the reactant does not necessarily become distributed over a larger water flux as occurs for a conservative tracer. The approach we propose is based on the study of the entropy of a reactive species (e.g., a contaminant released in the system) and its rate of change. The investigation of these properties allows us to infer useful information on the kinetics of the transformation processes and to identify conditions for which reactive transport is dominated by dilution and conditions for which reactive mixing represents the dominant mechanism.
2. Transport Equations
 For the sake of simplicity we illustrate the approach for steady-state reactive transport in porous media. We focus on groundwater organic contaminant plumes, originating from continuous sources (e.g., NAPL spills), which typically reach a dynamic equilibrium between the contaminant mass released from a source and its destruction by (bio)degradation processes. However, the proposed methodology is general and can be extended to transient transport problems. Under the aforementioned assumptions and considering a divergence-free flow field, the transport equation of the speciesi involved in the reaction is:
where v [LT−1] is the flow velocity, D [L2T−1] is the dispersion tensor, and ri [ML−3 T−1] is the reaction term.
 Considering the transport operator L = v · ∇ − ∇ · (D ∇) applied to a given function f(Ci) of the concentration, we obtain:
where the right hand side takes the form of sink/source terms. Equation (2) can be applied to obtain two interesting results that have been derived in the transport in porous media literature. The first result concerns the transport equation of f(Ci) = − pQ ln pQ, which is the entropy density of a conservative solute [Kitanidis, 1994]. In a flux-related framework, this can be written as [Chiogna et al., 2011a]:
The resulting rate of increase of the entropy in the mean flow direction x, which corresponds to the rate of increase of the natural logarithm of the flux-related dilution index, is obtained by integratingequation (3)over a cross-sectional area Ω perpendicular to the mean flow direction [Kitanidis, 1994; Chiogna et al., 2011a]:
The second relevant case is the linear combination of the equations of two reactants A and B (e.g., CU = CA − CB) undergoing a bimolecular reaction under certain constraints, such as the same local dispersion tensor for the different compounds [e.g., Chiogna et al., 2011b]. The quantity CU is hence described by a conservative transport equation (i.e., rU = rA − rB = 0). Considering chemical equilibrium we can then express CA as a function of CU (i.e., CA = CA(CU)) and therefore equation (2) can be written as:
 The transport equation for the entropy density of a reactive species undergoing advective and dispersive transport and subject to degradation expressed by a general reactive term ri (i.e., satisfying equation (1)) can be derived by combining equation (2) with equation (1):
where and the function g(Ci,ri) is defined as:
Note that the mass flux of the reactant is not conserved as in the nonreactive case since it diminishes according to the reaction. However, pQ still represents a probability density and . Inspection of equations (3) and (6) reveals that the entropy balance for a reactive solute involves two terms: a positive source term, defined as the dilution term, which is the only contribution in case of a nonreactive solute, and an additional term, which appears in equation (6) and describes the behavior of a reactive solute. This additional term represents the contribution of reactive mixing and can act as the only possible sink term for the entropy. The interplay between the reactive mixing term and the dilution term provides valuable insights on the mechanisms controlling transport of a reactive compound and will be illustrated in the following section through a few applications.
 We consider a two-dimensional planar velocity field. We assume a continuous injection from a source of finite width in the direction perpendicular to the main flow direction of a soluteA (i.e., the contaminant in a typical problem of groundwater pollution) with dimensionless inlet concentration Ain [−], which reacts with a solute B (i.e., the electron acceptor or another substrate for the contaminant degradation) with normalized ambient concentration Bamb [−]. We apply bimolecular reactions between the two species, which can be either instantaneous, CACB = 0, or described by double Monod reaction kinetics, , where fi (i = A,B) are the stoichiometric coefficients, kmax is the maximum degradation rate, Ki (i = A,B) are the half-saturation Monod coefficients, and the biomass (CBIO) is considered at steady state, resulting from the balance between a growth term and a linear decay (kdecCBIO) term. For simplicity, the stoichiometric coefficients are set to unity. The parameters for the two double Monod reaction kinetics considered are representative of a fast degradation reaction, as that for a pure aerobic strain reported by Rolle et al. , and of a slow degradation reaction, such as that observed for a petroleum hydrocarbons plume under sulfate-reducing conditions [Prommer et al., 2009], respectively.
4.1. Reactive Transport in Homogeneous Porous Media
 In this first illustrative example we consider reactive transport in a homogeneous laboratory-scale porous medium with different transverse dispersion coefficients to account for different degrees of mixing. We use the analytical solutions proposed byCirpka and Valocchi  to derive the concentrations of the reactants simply by applying algebraic equations to the concentration distribution of a conservative solute. Figure 1represents the flux-related dilution index of the reactantA as a function of the distance (Figure 1a) and as a function of the flux-related dilution index of a conservative tracer injected from the same source and with the same inlet mass flux (Figure 1b). Both the dimensionless inlet Ain and ambient Bamb concentrations are set to 1.
 The flux-related dilution index of a conservative quantity is a monotonically increasing function of the distancex (equation (4)) and it can be applied to quantify the distance from the source in terms of the dilution occurred to a conservative tracer. In particular, notice that while the representation of the flux-related dilution index ofA in spatial coordinates leads to different curves depending on the value of the transverse dispersion coefficient, when EQ(A) is plotted as a function of the flux-related dilution index of a conservative solute, the curves collapse on the same line, depending on the reaction kinetics but independent of the transverse dispersion coefficient. Hence, the representation inFigure 1b can be useful since the behavior of the entropy of reactant A yields a characteristic pattern dependent on the reaction kinetics. In the instantaneous case the entropy is monotonically decreasing. A different behavior can be observed for the double Monod slow case, where the degradation of A cannot counteract the transverse dispersive fluxes which lead to an increase in dilution and in the entropy of the reactive plume. The double Monod case with fast kinetics shows an initially decreasing trend of EQ(A) as in the instantaneous case; however, the small concentration values cannot be degraded since the transformation rate becomes negligible when the reactant concentrations are significantly lower than the half-saturation constants. The turning point of the curves, for this specific setup, was found to be at a value of dilutionEQcons = 5.7 × 10−7 m3s−1, and is dependent on the kinetics and stoichiometry of the reaction. Therefore, first the entropy of the reactant A decreases, similarly to the case of instantaneous reaction, and, successively, the entropy increases since the compound eventually tends to behave as a conservative species. It is worth noting that the same value of entropy does not represent the same value of the concentration, but just the same relative distribution with respect to the total water flux through a cross section.
 The behavior observed in Figure 1b, where the scenarios characterized by different transverse dispersion collapse on the same line, directly stems from the mapping between the longitudinal spatial coordinate (Figure 1a) and the “dilution coordinate” represented by the flux-related dilution index of a conservative compound,EQcons. In the latter case, the distribution of the mass flux of a reactive species over the water flux (EQ(A)) is represented as a function of the equivalent amount of dilution of a conservative tracer. This amount of dilution is, of course, reached at different distances in the domain for the considered scenarios with different intensities of transverse mixing (i.e., different Dt). However, a given value of dilution (EQcons) corresponds to the same degree of uniformity of the reactive solute mass flux distribution (i.e. EQ(A)), hence the merging of the results using different Dt onto the same line. In the case of instantaneous reaction, this issue has been discussed by Chiogna et al. [2011a], who formally identified a value of dilution (the “Critical Dilution Index”) at which a reactive plume ends, which is independent of the transverse dispersion coefficient.
 Another advantage of mapping the flux-related dilution index of the reactantA is the identification of zones of the domain where dilution is dominant and zones where reactive mixing prevails. This property is further illustrated in the example shown in Figure 2. In this case we consider instantaneous reactions between compounds A and B, and we vary the inlet concentration value Ain. An important quantity is the mixing ratio X, which represents the volumetric fraction of the source water (i.e., the water introduced during injection of A) in the mixture with the ambient solution. A change of Ain results in a variation of the critical mixing ratio, Xcrit = fACBamb/(fBCAin + fACBamb), which defines the particular value of X at which the concentrations of A and B are in the stoichiometric ratio of the reaction [Cirpka and Valocchi, 2007]. Therefore, the profile at X = Xcrit identifies the contour line at the fringe of the plume, where the concentration of A and B are both equal to 0. At low values of Ain, EQ(A) monotonically decreases (i.e., the compound A is readily consumed by B present at much higher concentration in the ambient groundwater). At high values of Ain, the flux-related dilution index increases monotonically, thus showing a behavior similar to the conservative case, since the high concentration and transverse mass flux ofA overwhelms that of B. Between these two extreme cases there are intermediate situations where EQ(A) monotonically decreases but at a slower rate or it initially increases and successively decreases in different zones of the domain. The inspection of the spatial derivative of the natural logarithm of EQ(A) (i.e., the rate of change of the Shannon entropy) provides interesting insights on whether transport is dominated by dilution or by reactive mixing. Where the derivative of the natural logarithm of EQ(A) is negative, the entropy of the plume is decreasing and reactive mixing is dominating. On the contrary, where this derivative is positive, the dilution process represents the dominant mechanism. The interplay between these two processes and the spatial variability of the derivative of the natural logarithm of the flux-related dilution index show that even for instantaneous reaction kinetics in a homogeneous porous medium the relation between dilution and reaction is not trivial. It depends on parameters such as the stoichiometry of the reaction and the inlet and ambient concentration of the reactants.
4.2. Reactive Transport in Heterogeneous Porous Media
 As a further illustrative example we performed a 2-D simulation of reactive solute transport in a heterogeneous flow field with statistical properties consistent with the Columbus field site [Rehfeldt et al., 1992]. We assume that the natural logarithm of the hydraulic conductivity distribution follows a Gaussian distribution described by mean, variance, longitudinal and lateral integral scales of μlnK = −5.2 (i.e., K = 5.4 × 10−3 ms−1), = 2.7, lx = 4.8 m and ly = 0.8 m, respectively. The flow and transport problems were solved using the streamline approach of Cirpka et al. , considering as input values Ain = 5, Bamb = 1, and as reaction kinetics instantaneous and double Monod formulations, the latter using the parameters of the slow, sulfate-reducing case. The behavior of the computed flux-related dilution index for the three different scenarios is shown inFigure 3. The dilution of a conservative tracer is monotonically increasing as predicted by equation (4). Sudden jumps in the value of the flux-related dilution index for the conservative case indicate regions where the plume is focused in high permeability inclusions. This process enhances the dilution of the solute concentration over a larger water flux [e.g.,Willingham et al., 2008; Rolle et al., 2009]. Considering the behavior of the entropy of the reactive plumes notice that, although the value of EQ(A) is lower than that of the conservative case, the derivatives behave similarly to the conservative case up to the distance of 55 m from the source, where for the first time the derivative of the instantaneous reaction kinetics case is negative. In the gray shaded region, the plumes of the reactive species A are mainly diluted, and this process is dominant compared to reactive mixing mechanisms that decrease entropy. In this region, the interplay between local concentration gradients and flow focusing enhances the dilution term, which dominates the entropy balance of equation (6). Therefore, the flux-related dilution index of the reactant is not significantly affected by the reaction kinetics implying that both the double Monod and the instantaneous cases have similar values of entropy and the derivatives also follow a similar pattern. Outside the gray shaded region reactive mixing processes become increasingly dominant for the two reactive cases. In the instantaneous case, the entropy of the plume decreases, thus showing that reactive mixing is dominant. A direct comparison of the instantaneous and the conservative curves shows that where flow focusing occurs, the reactive plume is not diluted over a larger water flux but is mainly consumed by the reaction with compoundB, since the local enhancement of transverse mixing directly implies a reaction enhancement. In the double Monod case dilution remains the dominant process along the entire length of the simulation domain since the enhancement of the dispersive fluxes overwhelms the degradation potential of the reaction. The comparison with the conservative case shows that localized flow-focusing events are even more effective in diluting the double Monod reactive plume. In fact, at any given distance, the reactive plume is less diluted than the conservative one because of the degradation processes occurring at its fringe; therefore, flow focusing and the consequent enhanced dilution are more effective in distributing the reactant concentration over a larger water flux.
5. Summary and Conclusions
 In this work we presented a novel approach to quantify mixing in geophysical flows based on the study of the entropy of a reactive plume, for which we derived a transport equation. For the purpose of illustration we considered instantaneous and double Monod reaction kinetics in porous media, but the proposed approach is general and not limited to these specific cases. We show that the flux-related dilution index curve of a reactive compoundAin a homogeneous domain, expressed as a function of the dilution of a conservative solute, can be used as a useful indicator of the reaction process occurring in the domain. Furthermore, we show that if the derivative of the natural logarithm of the dilution index is negative, reactive mixing processes are dominant over dilution processes, while dilution is the dominant mechanism when the derivative is positive. In particular, the flux-related dilution index of a reactive solute is an increasing function of the distance from the source as long as the dispersive fluxes distribute the solute within the water flux more intensively than the mass-removal effect of the reaction term. On the contrary, when reactive mixing is the prevailing process, the flux-related dilution index decreases indicating that the reactive mixing term is dominant in the entropy balance ofequation (6). The field-scale heterogeneous application highlights interesting effects of flow focusing depending on the interplay between dilution and reactive mixing terms and on the reaction kinetics. It has been shown that flow focusing in heterogeneous porous formations can enhance both the dilution of a reacting plume by distributing the solute flux over a larger water flux, as well as reactive mixing between the reactants, thus leading to a faster degradation. The proposed approach is attractive since it provides important information on the interaction between mixing and reactive processes based on the quantification of the entropy of a single reactive species (e.g., a groundwater contaminant), without the need to simultaneously map the concentration of different reactants and/or the evolution of reaction rates. We think that this methodology has the potential to be extended to a wide variety of reactive solute transport problems.
 G.C. and A.B. acknowledge the support of the Collaborative Research Project CLIMB (Climate Induced Changes on the Hydrology of Mediterranean Basins, grant 244151) within the 7th European Community Framework Programme. D.L.H. and P.K.K. acknowledge the support of the NSF grant EAR-0738772 “Nonequilibrium Transport and Transport-Controlled Reactions” and the student funding from Government awarded by DOD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) fellowship, 32 CFR 168a. M.R. acknowledges the support of the Marie Curie International Outgoing Fellowship (DILREACT project) within the 7th European Community Framework Programme. The authors thank two anonymous reviewers for their constructive comments.
 The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.