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Mixing, entropy and reactive solute transport Gabriele Chiogna, 1 David L. Hochstetler, 2 Alberto Bellin, 1 Peter K. Kitanidis, 2 and Massimo Rolle 2,3 Received 26 July 2012; revised 18 September 2012; accepted 19 September 2012; published 23 October 2012. [1] 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 accu- rately 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 reac- tive species and its rate of change are informative measures of physical and (bio)chemical processes and their complex interaction. Citation: Chiogna, G., D. L. Hochstetler, A. Bellin, P. K. Kitanidis, and M. Rolle (2012), Mixing, entropy and reactive solute transport, Geophys. Res. Lett., 39, L20405, doi:10.1029/ 2012GL053295. 1. Introduction [2] Quantifying the interplay between mixing and reac- tions 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 contami- nated groundwater and in performing risk assessment anal- ysis [Sanchez-Vila et al., 2007; Edery et al., 2009; Bellin et al., 2011; de Barros et al., 2012]. In subsurface environ- ments mixing is very slow and, therefore, of key relevance since it often constitutes the main limiting mechanism for a reaction to occur. [3] 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 C i [ML 3 ], is defined as: E ¼ exp Z V p ln pdV [L 3 ][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 [L 3 ], where p ¼ C i Z V C i dV [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: E Q ¼ exp Z W p Q ln p Q q x dW [L 3 T 1 ]; where p Q ¼ C i Z W C i q x dW [L 3 T] is the flux-related probability den- sity function of the species i and q x [LT 1 ] is the component of the specific discharge in the principal flow direction, normal to the cross-sectional area W [L 2 ]. E Q has been called the flux-related dilution index by Rolle et al. [2009] and quantifies an effective volumetric flux transporting the sol- ute mass flux at a given cross-section. [4] We focus on reactive solute transport in both homo- geneous and heterogeneous porous formations and we dis- tinguish between dilution and reactive mixing. We work in a flux-related framework which has been shown to be effec- tive 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 distri- bution 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 1 Dipartimento di Ingegneria Civile ed Ambientale, Università di Trento, Trento, Italy. 2 Department of Civil and Environmental Engineering, Stanford University, Stanford, California, USA. 3 Center for Applied Geoscience, University of Tübingen, Tübingen, Germany. Corresponding author: M. Rolle, Department of Civil and Environmental Engineering, Stanford University, 473 Via Ortega, Stanford, CA 94305, USA. ([email protected]) ©2012. American Geophysical Union. All Rights Reserved. 0094-8276/12/2012GL053295 GEOPHYSICAL RESEARCH LETTERS, VOL. 39, L20405, doi:10.1029/2012GL053295, 2012 L20405 1 of 6

Mixing, entropy and reactive solute transport

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Mixing, entropy and reactive solute transport

Gabriele Chiogna,1 David L. Hochstetler,2 Alberto Bellin,1 Peter K. Kitanidis,2

and Massimo Rolle2,3

Received 26 July 2012; revised 18 September 2012; accepted 19 September 2012; published 23 October 2012.

[1] Mixing processes significantly affect reactive solutetransport in fluids. For example, contaminant degradationin environmental aquatic systems can be limited either bythe availability of one or more reactants, brought into contactby physical mixing, or by the kinetics of the (bio)chemicaltransformations. Appropriate metrics are needed to accu-rately quantify the interplay between mixing and reactiveprocesses. The exponential of the Shannon entropy of theconcentration probability distribution has been proposedand applied to quantify the dilution of conservative soluteseither in a given volume (dilution index) or in a given waterflux (flux-related dilution index). In this work we derive thetransport equation for the entropy of a reactive solute.Adopting a flux-related framework, we show that the degreeof uniformity of the solute mass flux distribution for a reac-tive species and its rate of change are informative measuresof physical and (bio)chemical processes and their complexinteraction. Citation: Chiogna, G., D. L. Hochstetler, A. Bellin,P. K. Kitanidis, and M. Rolle (2012), Mixing, entropy and reactivesolute transport, Geophys. Res. Lett., 39, L20405, doi:10.1029/2012GL053295.

1. Introduction

[2] Quantifying the interplay between mixing and reac-tions is critical to deepening our understanding of reactivesolute transport in geophysical flows [e.g., Weiss andProvenzale, 2008]. In the case of transport in porous mediathese processes are important in implementing effectiveengineered or natural remediation strategies for contami-nated groundwater and in performing risk assessment anal-ysis [Sanchez-Vila et al., 2007; Edery et al., 2009; Bellinet al., 2011; de Barros et al., 2012]. In subsurface environ-ments mixing is very slow and, therefore, of key relevancesince it often constitutes the main limiting mechanism for areaction to occur.[3] Appropriate measures are required to identify and

quantitatively describe the interaction between transportmechanisms and reaction kinetics [e.g., Dentz et al., 2011].Mixing of conservative solutes can be effectively quantifiedusing metrics such as the scalar dissipation rate [e.g.,

Le Borgne et al., 2010; Bolster et al., 2011a] and the dilutionindex. In this work we focus on the dilution index, which,for a conservative species i with concentration Ci [ML�3],

is defined as: E ¼ exp

��

ZV

p ln pdV

�[L3] [Kitanidis,

1994]. This metric has been successfully applied in manycontexts to measure the degree of dilution and to effectivelydistinguish between spreading and dilution [e.g., Cao andKitanidis, 1998; Beckie, 1998; Tartakovsky et al., 2009;Bolster et al., 2011b]. The dilution index represents theexponential of the Shannon entropy of a concentrationprobability distribution p [L�3] defined over a volume V

[L3], where p ¼ CiZV

Ci dV[L�3]. A formally similar measure

of entropy was also applied by Rolle et al. [2009, 2012] andChiogna et al. [2011a, 2011b] describing dilution as the actof distributing a given solute mass flux over a larger water

flux: EQ ¼ exp

��

ZW

pQ ln pQ� �

qxdW�

[L3T�1]; where

pQ ¼ CiZW

CiqxdW[L�3 T] is the flux-related probability den-

sity function of the species i and qx [LT�1] is the component

of the specific discharge in the principal flow direction,normal to the cross-sectional area W [L2]. EQ has been calledthe flux-related dilution index by Rolle et al. [2009] andquantifies an effective volumetric flux transporting the sol-ute mass flux at a given cross-section.[4] We focus on reactive solute transport in both homo-

geneous and heterogeneous porous formations and we dis-tinguish between dilution and reactive mixing. We work in aflux-related framework which has been shown to be effec-tive in quantifying mixing in heterogeneous velocity fields[e.g., Cirpka et al., 2011]. In such a framework, dilutionrefers to the distribution of the solute concentration over alarger water flux with increasing distance from the source;consequently, peak concentration is reduced. With the termreactive mixing we consider the condition where initiallysegregated reactants are distributed over the same water flux,thus allowing reactions to occur. For both conservative andreactive solutes, the Shannon entropy measures the distri-bution of a solute within a domain (i.e., the water flux in oursetup). For conservative species, the solute concentrationbecomes distributed over a larger water discharge. For areactive species, though, the reaction consumes part of thereactant mass and may tend to sustain non-uniformity inthe reactant distribution. Consequently, due to the reaction,the reactant does not necessarily become distributed over alarger water flux as occurs for a conservative tracer. The

1Dipartimento di Ingegneria Civile ed Ambientale, Università di Trento,Trento, Italy.

2Department of Civil and Environmental Engineering, StanfordUniversity, Stanford, California, USA.

3Center for Applied Geoscience, University of Tübingen, Tübingen,Germany.

Corresponding author: M. Rolle, Department of Civil andEnvironmental Engineering, Stanford University, 473 Via Ortega, Stanford,CA 94305, USA. ([email protected])

©2012. American Geophysical Union. All Rights Reserved.0094-8276/12/2012GL053295

GEOPHYSICAL RESEARCH LETTERS, VOL. 39, L20405, doi:10.1029/2012GL053295, 2012

L20405 1 of 6

approach we propose is based on the study of the entropy ofa reactive species (e.g., a contaminant released in thesystem) and its rate of change. The investigation of theseproperties allows us to infer useful information onthe kinetics of the transformation processes and to identifyconditions for which reactive transport is dominated bydilution and conditions for which reactive mixing representsthe dominant mechanism.

2. Transport Equations

[5] For the sake of simplicity we illustrate the approachfor steady-state reactive transport in porous media. We focuson groundwater organic contaminant plumes, originatingfrom continuous sources (e.g., NAPL spills), which typicallyreach a dynamic equilibrium between the contaminant massreleased from a source and its destruction by (bio)degrada-tion processes. However, the proposed methodology isgeneral and can be extended to transient transport problems.Under the aforementioned assumptions and considering adivergence-free flow field, the transport equation of thespecies i involved in the reaction is:

v � rCi �r � DrCið Þ ¼ ri ð1Þ

where v [LT�1] is the flow velocity, D [L2T�1] is the dis-persion tensor, and ri [ML�3 T�1] is the reaction term.[6] Considering the transport operator L = v � r�r � (Dr)

applied to a given function f (Ci) of the concentration, weobtain:

v � rf Cið Þ � r � Drf Cið Þð Þ ¼ ∂f Cið Þ∂Ci

v � rCi �r � DrCið Þð Þ

� ∂2f Cið Þ∂Ci

2 rCiTDrCi ð2Þ

where the right hand side takes the form of sink/sourceterms. Equation (2) can be applied to obtain two interestingresults that have been derived in the transport in porousmedia literature. The first result concerns the transportequation of f (Ci) = � pQ ln pQ, which is the entropy densityof a conservative solute [Kitanidis, 1994]. In a flux-relatedframework, this can be written as [Chiogna et al., 2011a]:

v � r �pQ ln pQ� ��r � Dr �pQ ln pQ

� �� � ¼ 1

pQrpQ

TDrpQ:

ð3Þ

The resulting rate of increase of the entropy in the mean flowdirection x, which corresponds to the rate of increase of thenatural logarithm of the flux-related dilution index, isobtained by integrating equation (3) over a cross-sectionalarea W perpendicular to the mean flow direction [Kitanidis,1994; Chiogna et al., 2011a]:

d ln EQ

� �dx

¼ZW

1

pQrpQ

TDrpQdW ð4Þ

The second relevant case is the linear combination of theequations 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 differentcompounds [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 canthen express CA as a function of CU (i.e., CA = CA(CU)) andtherefore equation (2) can be written as:

v � rCA �r � DrCAð Þ ¼ � ∂2CA

∂CU2 rCU

TDrCU ð5Þ

which corresponds to the result obtained by De Simoni et al.[2005] and further generalized by Sanchez-Vila et al. [2007].

3. Transport of the Entropy of a Reactive Solute

[7] The transport equation for the entropy density of areactive species undergoing advective and dispersive trans-port and subject to degradation expressed by a generalreactive term ri (i.e., satisfying equation (1)) can be derivedby combining equation (2) with equation (1):

v � r �pQ ln pQ� ��r � Dr �pQ ln pQ

� �� � ¼ � 1þ ln pQ� �

r∗i|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}reactive mixing term

þ 1

pQrpQ

TDrpQ|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}dilution term

ð6Þ

where r∗i ¼ g Ci; rið Þri and the function g(Ci,ri) is defined as:

g Ci; rið Þ ¼ ∂pQ∂Ci

� 1

ri

∂2pQ∂C2

i

rCiTDrCi ð7Þ

Note that the mass flux of the reactantZW

CiqxdW is not

conserved as in the nonreactive case since it diminishesaccording to the reaction. However, pQ still represents a

probability density andZW

pQqxdW ¼ 1. Inspection of

equations (3) and (6) reveals that the entropy balance for areactive solute involves two terms: a positive source term,defined as the dilution term, which is the only contributionin case of a nonreactive solute, and an additional term, whichappears in equation (6) and describes the behavior of areactive solute. This additional term represents the contri-bution of reactive mixing and can act as the only possiblesink term for the entropy. The interplay between the reactivemixing term and the dilution term provides valuable insightson the mechanisms controlling transport of a reactive com-pound and will be illustrated in the following sectionthrough a few applications.

4. Applications

[8] We consider a two-dimensional planar velocity field.We assume a continuous injection from a source of finitewidth in the direction perpendicular to the main flow direc-tion of a solute A (i.e., the contaminant in a typical problemof groundwater pollution) with dimensionless inlet concen-tration Ain [�], which reacts with a solute B (i.e., the electronacceptor or another substrate for the contaminant degrada-tion) with normalized ambient concentration Bamb [�]. Weapply bimolecular reactions between the two species, whichcan be either instantaneous, CACB = 0, or described by

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double Monod reaction kinetics, ri ¼ fi kmaxCA

CA þ KA�

CB

CB þ KBCBIO , 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 thebiomass (CBIO) is considered at steady state, resulting fromthe balance between a growth term and a linear decay(kdecCBIO) term. For simplicity, the stoichiometric coeffi-cients are set to unity. The parameters for the two doubleMonod reaction kinetics considered are representative of afast degradation reaction, as that for a pure aerobic strainreported by Rolle et al. [2010], and of a slow degradationreaction, such as that observed for a petroleum hydro-carbons plume under sulfate-reducing conditions [Prommeret al., 2009], respectively.

4.1. Reactive Transport in Homogeneous Porous Media

[9] In this first illustrative example we consider reactivetransport in a homogeneous laboratory-scale porous mediumwith different transverse dispersion coefficients to accountfor different degrees of mixing. We use the analytical solu-tions proposed by Cirpka and Valocchi [2007] to derive theconcentrations of the reactants simply by applying algebraicequations to the concentration distribution of a conservativesolute. Figure 1 represents the flux-related dilution index ofthe reactant A as a function of the distance (Figure 1a) and asa function of the flux-related dilution index of a conservativetracer injected from the same source and with the same inletmass flux (Figure 1b). Both the dimensionless inlet Ain andambient Bamb concentrations are set to 1.[10] The flux-related dilution index of a conservative

quantity is a monotonically increasing function of the dis-tance x (equation (4)) and it can be applied to quantify thedistance from the source in terms of the dilution occurred toa conservative tracer. In particular, notice that while therepresentation of the flux-related dilution index of A inspatial coordinates leads to different curves depending onthe value of the transverse dispersion coefficient, whenEQ(A) is plotted as a function of the flux-related dilutionindex of a conservative solute, the curves collapse on thesame line, depending on the reaction kinetics but

independent of the transverse dispersion coefficient. Hence,the representation in Figure 1b can be useful since thebehavior of the entropy of reactant A yields a characteristicpattern dependent on the reaction kinetics. In the instanta-neous case the entropy is monotonically decreasing. A dif-ferent behavior can be observed for the double Monod slowcase, where the degradation of A cannot counteract thetransverse dispersive fluxes which lead to an increase indilution and in the entropy of the reactive plume. The doubleMonod case with fast kinetics shows an initially decreasingtrend of EQ(A) as in the instantaneous case; however,the small concentration values cannot be degraded sincethe transformation rate becomes negligible when the reac-tant concentrations are significantly lower than the half-saturation constants. The turning point of the curves, for thisspecific setup, was found to be at a value of dilutionEQcons = 5.7 � 10�7 m3s�1, and is dependent on the kineticsand stoichiometry of the reaction. Therefore, first theentropy of the reactant A decreases, similarly to the case ofinstantaneous reaction, and, successively, the entropyincreases since the compound eventually tends to behave asa conservative species. It is worth noting that the samevalue of entropy does not represent the same value of theconcentration, but just the same relative distribution withrespect to the total water flux through a cross section.[11] The behavior observed in Figure 1b, where the sce-

narios characterized by different transverse dispersion col-lapse on the same line, directly stems from the mappingbetween the longitudinal spatial coordinate (Figure 1a) andthe “dilution coordinate” represented by the flux-relateddilution index of a conservative compound, EQcons. In thelatter case, the distribution of the mass flux of a reactivespecies over the water flux (EQ(A)) is represented as afunction of the equivalent amount of dilution of a conser-vative tracer. This amount of dilution is, of course, reachedat different distances in the domain for the considered sce-narios 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 reactivesolute mass flux distribution (i.e. EQ(A)), hence the mergingof the results using different Dt onto the same line. In thecase of instantaneous reaction, this issue has been discussed

Figure 1. Flux-related dilution index of reactant A as a function of the distance from the source (a) and as a function of theflux-related dilution index of a conservative species (b). The different lines indicate different values of transverse dispersioncoefficient, Dt.

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by Chiogna et al. [2011a], who formally identified a valueof dilution (the “Critical Dilution Index”) at which a reactiveplume ends, which is independent of the transverse disper-sion coefficient.[12] Another advantage of mapping the flux-related dilu-

tion index of the reactant A is the identification of zones ofthe domain where dilution is dominant and zones wherereactive mixing prevails. This property is further illustratedin the example shown in Figure 2. In this case we considerinstantaneous reactions between compounds A and B, andwe vary the inlet concentration value Ain. An importantquantity is the mixing ratio X, which represents the volu-metric fraction of the source water (i.e., the water introducedduring injection of A) in the mixture with the ambient solu-tion. A change of Ain results in a variation of the criticalmixing ratio, Xcrit = fACBamb/( fBCAin + fACBamb), whichdefines the particular value of X at which the concentrationsof A and B are in the stoichiometric ratio of the reaction[Cirpka and Valocchi, 2007]. Therefore, the profile atX = Xcrit identifies the contour line at the fringe of the plume,where the concentration of A and B are both equal to 0. Atlow values of Ain, EQ(A) monotonically decreases (i.e., thecompound A is readily consumed by B present at muchhigher concentration in the ambient groundwater). At highvalues of Ain, the flux-related dilution index increasesmonotonically, thus showing a behavior similar to the con-servative case, since the high concentration and transversemass flux of A overwhelms that of B. Between these twoextreme cases there are intermediate situations where EQ(A)monotonically decreases but at a slower rate or it initiallyincreases and successively decreases in different zones of thedomain. The inspection of the spatial derivative of the nat-ural logarithm of EQ(A) (i.e., the rate of change of theShannon entropy) provides interesting insights on whethertransport is dominated by dilution or by reactive mixing.Where the derivative of the natural logarithm of EQ(A) isnegative, the entropy of the plume is decreasing and reactivemixing is dominating. On the contrary, where this derivativeis positive, the dilution process represents the dominantmechanism. The interplay between these two processes and

the spatial variability of the derivative of the natural loga-rithm of the flux-related dilution index show that even forinstantaneous reaction kinetics in a homogeneous porousmedium the relation between dilution and reaction is nottrivial. It depends on parameters such as the stoichiometry ofthe reaction and the inlet and ambient concentration ofthe reactants.

4.2. Reactive Transport in HeterogeneousPorous Media

[13] As a further illustrative example we performed a 2-Dsimulation of reactive solute transport in a heterogeneousflow field with statistical properties consistent with theColumbus field site [Rehfeldt et al., 1992]. We assume thatthe natural logarithm of the hydraulic conductivity distribu-tion follows a Gaussian distribution described by mean,variance, longitudinal and lateral integral scales of mlnK =�5.2 (i.e., K = 5.4 � 10�3 ms�1), s2

lnK = 2.7, lx = 4.8 m andly = 0.8 m, respectively. The flow and transport problemswere solved using the streamline approach of Cirpka et al.[1999], considering as input values Ain = 5, Bamb = 1,and as reaction kinetics instantaneous and double Monodformulations, 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 isshown in Figure 3. The dilution of a conservative tracer ismonotonically increasing as predicted by equation (4).Sudden jumps in the value of the flux-related dilution indexfor the conservative case indicate regions where the plumeis focused in high permeability inclusions. This processenhances the dilution of the solute concentration over alarger water flux [e.g., Willingham et al., 2008; Rolle et al.,2009]. Considering the behavior of the entropy of the reac-tive plumes notice that, although the value of EQ(A) is lowerthan that of the conservative case, the derivatives behavesimilarly to the conservative case up to the distance of 55 mfrom the source, where for the first time the derivative of theinstantaneous reaction kinetics case is negative. In the grayshaded region, the plumes of the reactive species A aremainly diluted, and this process is dominant compared toreactive mixing mechanisms that decrease entropy. In thisregion, the interplay between local concentration gradientsand flow focusing enhances the dilution term, which dom-inates the entropy balance of equation (6). Therefore, theflux-related dilution index of the reactant is not significantlyaffected by the reaction kinetics implying that both thedouble Monod and the instantaneous cases have similarvalues of entropy and the derivatives also follow a similarpattern. Outside the gray shaded region reactive mixingprocesses become increasingly dominant for the two reactivecases. In the instantaneous case, the entropy of the plumedecreases, thus showing that reactive mixing is dominant. Adirect comparison of the instantaneous and the conservativecurves shows that where flow focusing occurs, the reactiveplume is not diluted over a larger water flux but is mainlyconsumed by the reaction with compound B, since the localenhancement of transverse mixing directly implies a reactionenhancement. In the double Monod case dilution remains thedominant process along the entire length of the simulationdomain since the enhancement of the dispersive fluxesoverwhelms the degradation potential of the reaction. Thecomparison with the conservative case shows that localizedflow-focusing events are even more effective in diluting the

Figure 2. (a) Flux-related dilution index of compound Aand (b) the spatial derivative of its natural logarithm, consid-ering different inlet concentration values (Ain) and, hence,different corresponding values for the critical mixing ratioXcrit.

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double Monod reactive plume. In fact, at any given distance,the reactive plume is less diluted than the conservative onebecause of the degradation processes occurring at its fringe;therefore, flow focusing and the consequent enhanced dilu-tion are more effective in distributing the reactant concen-tration over a larger water flux.

5. Summary and Conclusions

[14] In this work we presented a novel approach to quan-tify mixing in geophysical flows based on the study of theentropy of a reactive plume, for which we derived a transportequation. For the purpose of illustration we consideredinstantaneous and double Monod reaction kinetics in porousmedia, but the proposed approach is general and not limitedto these specific cases. We show that the flux-related dilu-tion index curve of a reactive compound A in a homoge-neous domain, expressed as a function of the dilution of aconservative solute, can be used as a useful indicator of thereaction process occurring in the domain. Furthermore, weshow that if the derivative of the natural logarithm of thedilution index is negative, reactive mixing processes aredominant over dilution processes, while dilution is thedominant mechanism when the derivative is positive. Inparticular, the flux-related dilution index of a reactive soluteis an increasing function of the distance from the source aslong as the dispersive fluxes distribute the solute within thewater flux more intensively than the mass-removal effect ofthe reaction term. On the contrary, when reactive mixing isthe prevailing process, the flux-related dilution indexdecreases indicating that the reactive mixing term is domi-nant in the entropy balance of equation (6). The field-scaleheterogeneous application highlights interesting effects offlow focusing depending on the interplay between dilutionand reactive mixing terms and on the reaction kinetics. It hasbeen shown that flow focusing in heterogeneous porousformations can enhance both the dilution of a reacting plumeby distributing the solute flux over a larger water flux, aswell as reactive mixing between the reactants, thus leadingto a faster degradation. The proposed approach is attractivesince it provides important information on the interaction

between mixing and reactive processes based on the quan-tification of the entropy of a single reactive species (e.g.,a groundwater contaminant), without the need to simulta-neously map the concentration of different reactants and/orthe evolution of reaction rates. We think that this method-ology has the potential to be extended to a wide variety ofreactive solute transport problems.

[15] Acknowledgments. G.C. and A.B. acknowledge the support ofthe Collaborative Research Project CLIMB (Climate Induced Changeson the Hydrology of Mediterranean Basins, grant 244151) within the7th European Community Framework Programme. D.L.H. and P.K.K.acknowledge the support of the NSF grant EAR-0738772 “NonequilibriumTransport and Transport-Controlled Reactions” and the student fundingfrom Government awarded by DOD, Air Force Office of ScientificResearch, National Defense Science and Engineering Graduate (NDSEG)fellowship, 32 CFR 168a. M.R. acknowledges the support of the MarieCurie International Outgoing Fellowship (DILREACT project) within the7th European Community Framework Programme. The authors thank twoanonymous reviewers for their constructive comments.[16] The Editor thanks the two anonymous reviewers for their assis-

tance in evaluating this paper.

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Figure 3. (a) Steady-state conservative plume. Flux-related dilution index of compound A for (b) the conservative and thetwo reactive cases and (c) the spatial derivative of its natural logarithm in a heterogeneous domain considering different reac-tion kinetics: red instantaneous, black double Monod and blue conservative.

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