evaluation of monod kinetic parameters in the subsurface using moment analysis: theory and numerical...

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Advances in Water Resources 30 (2007) 2034–2050

Evaluation of Monod kinetic parameters in the subsurfaceusing moment analysis: Theory and numerical testing

M. Mohamed a,*, K. Hatfield b, I.V. Perminova c

a Civil and Environmental Engineering Department, United Arab Emirates University, P.O. Box 17555, Alain, United Arab Emiratesb Civil and Coastal Engineering Department, University of Florida, P.O. Box 116580, Gainesville, FL 32611, USA

c Department of Chemistry, Lomonosov Moscow State University, Moscow 119992, Russia

Received 30 April 2006; received in revised form 16 April 2007; accepted 19 April 2007Available online 6 May 2007

Abstract

The spatial moments of a contaminant plume undergoing bio-attenuation are coupled to the moments of microbial populations effect-ing that attenuation. In this paper, a scalable inverse method is developed for estimating field-scale Monod parameters such as the max-imum microbial growth rate (lmax), the contaminant half saturation coefficient (Ks), and the contaminant yield coefficient (Ys). Themethod uses spatial moments that characterize the distribution of dissolved contaminant and active microbial biomass in the aquifer.A finite element model is used to generate hypothetical field-scale data to test the method under both homogeneous and heterogeneousaquifer conditions. Two general cases are examined. In the first, Monod parameters are estimated where it is assumed a microbial pop-ulation comprised of a single bacterial species is attenuating one contaminant (e.g., an electron donor and an electron acceptor). In asecond case, contaminant attenuation is attributed to a microbial consortium comprised of two microbial species, and Monod param-eters for both species are estimated. Results indicate the inverse method is only slightly sensitive to aquifer heterogeneity and that esti-mation errors decrease as the sampling time interval decreases with respect to the groundwater travel time between sample locations.Optimum conditions for applying the scalable inverse method in both space and time are investigated under both homogeneous and het-erogeneous aquifer conditions.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Groundwater; Scalable inverse method; Moment analysis; Biodegradation; Biological parameters

1. Introduction

Natural and enhanced bio-attenuation are frequentlyconsidered cost-effective remediation solutions to subsur-face contamination. It is well known contaminant attenua-tion varies with physical and chemical heterogeneities inporous media [1–13], chemical heterogeneities in the con-taminant plume [14–16], and biological heterogeneities rep-resented as spatial variations in microbial species, activebiomass, and Monod kinetic parameters [17–21]. Evalua-tion of natural and enhanced bio-attenuation as viableremediation strategies necessitates modeling [22], which in

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* Corresponding author. Tel.: +971 3 7133509.E-mail address: [email protected] (M. Mohamed).

turn requires representative or effective field-scale Monodkinetic parameters for field-scale simulations [23–26].

Initial attempts at estimating field-scale Monod param-eters involved upscaling results from batch or flow-throughlaboratory experiments [27,28]. This approach was latershown to overestimate field-scale attenuation [25,29]. Diffi-culties associated with generating unique estimates forMonod parameters at the bench-scale [22,30–33] com-pounded with uncertainties introduced by upscaling werefactors contributing to these early failures. In subsequentyears it was determined for bench-scale systems, that theuniqueness problem could be resolved. Ellis et al. [32] dem-onstrated that it was feasible to identify unique estimatesfor the maximum microbial growth rate (lmax) and the con-taminant half saturation coefficient (Ks), when the ratio ofinitial substrate concentration to Ks was greater than 1.0

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M. Mohamed et al. / Advances in Water Resources 30 (2007) 2034–2050 2035

and less feasible as this ratio decreased below 0.1. Whereas,Knightes and Peters [33] and Schirmer et al. [22] resolvedunique batch-scale Monod parameters using concentrationmeasurements of both active microbial biomass and sub-strate from multiple experiments conducted with differentinitial substrate concentrations.

A second approach to estimating field-scale Monodparameters required plume-scale modeling and availablefield concentrations for contaminants and associated elec-tron donor/acceptors [25,34–42]. Valid field measures ofactive biomass were generally not available; consequently,most studies assumed spatially uniform biomass concentra-tions [25,34–41]. As a result, biotic reactions were often notexpressed in terms of classic Monod parameters, but interms of quasi-Monod or lumped parameters [25] and zero-or first-order rate parameters [34–41]. For example, Chap-elle et al. [37] assumed first-order reactions and spatiallyuniform kinetics and biomass concentrations. With theseassumptions, they were able to combine laboratory andfield results and overcome the problem of estimatingunique kinetic parameters. Their approach was consideredreasonable; as it was well known Monod kinetics approachfirst-order rates for quasi-steady microbial populations insubstrate limited groundwater systems [35]. At about thesame time, Borden et al. [38] assumed first-order kineticsand presented a novel approach of using indirect measuresof MTBE and BTEX fluxes in the field to quantify intrinsicrates of contaminant biodegradation. And more recently,Basu et al. [39] estimated plume-scale first-order naturalattenuation rates for trichloroethylene (TCE), but usedpassive flux meters [43,44] to obtain direct measurementsof TCE fluxes. In other studies, zero-order rates wereassumed wherever contaminant concentrations were highwith respect to Ks such that microbes were attenuating con-taminant at the maximum potential rate (i.e, near the con-taminant source) [33].

Schirmer et al. [42] pursued a third inverse strategy andused extensive data from both laboratory and field studiesto develop a multi-component coupled numerical model ofa dissolved gasoline plume. Like previous inverse efforts,their method was computationally intensive; although, itshowed laboratory-derived Monod kinetic parameterscould predict field-scale attenuation provided the modelconsidered field-scale advection/dispersive transport ofmultiple electron donors and acceptors.

Absent in all previous inverse efforts was the use ofactive biomass concentrations or surrogate biotic mea-sures in analyses to characterize field-scale Monod param-eters. From Knightes and Peters [33] and Schirmer et al.[22,42] one would surmise concentrations of active micro-bial biomass (or a surrogate), contaminant, and electrondonor/acceptors would be needed to obtain uniqueparameters from inverse modeling. In the past it was dif-ficult to quantify active biomass from groundwater sam-ples [20]. Today, phospholipid fatty acid analyses arewidely conducted to quantify active microbial biomassand to characterize temporal and spatial variations in

subsurface microbial communities [19,21,45–49]. Still, thistype data has not been used to characterize Monod reac-tions in the subsurface.

Any transport model promoted to describe coupledmicrobial growth and contaminant degradation throughMonod reactions would be complex, nonlinear, and thenonly an approximation of the true system. During the1980s and 1990s analytical moment techniques were suc-cessfully used to simplify governing equations of complextransport problems and expedite analyses of multiple reac-tive processes affecting contaminant fate and transportacross multiple scales [7,50–60]. To the best of our knowl-edge, inverse models predicated on the spatial concentra-tion moments for contaminants, electron donor/acceptors,and active biomass (or surrogate biomarker) were notderived for estimating field-scale Monod parameters. Thenagain, calculating spatial concentration moments requiredconsiderable data [42,61], and until recently there werefew options but gather these data from spatially fixed sam-pling networks which was difficult and expensive.

Of late there has been an advent of various direct-pushtechnologies (e.g., cone penetrometer and the Geoprobe)that provide real-time, cost-effective, high-resolution depthdiscrete, physical and chemical information from the sub-surface [62–64]. This type of rapid sampling can be con-ducted easily at 20–30 horizontal locations over a shallowaquifer in a single day. The resultant sampling network ismuch more dynamic, such that installed wells are easilyremoved or used again for subsequent sampling.

In this paper, spatial moment analysis is used to developa scalable inverse method for estimating field-scale Monodkinetic parameters. The method uses measured concentra-tions of both contaminant and active microbial biomass(or surrogate biomarker) and measured contaminantfluxes; that is, data obtainable through direct-push technol-ogies. Two test cases are used to evaluate the parameterestimation method. In the first case, one microbial speciesis assumed to consume a single contaminant; while in thesecond a microbial consortium comprised of two bacterialspecies compete to degrade a single contaminant. The finiteelement transport model METABIOTRANS [65] is used tosimulate all test conditions that generate contaminant andmicrobial biomass concentration distributions in a hypo-thetical aquifer in both space and time. For each test prob-lem, the spatial moments are calculated at specific timeintervals for contaminant and active microbial biomassconcentrations and contaminant flux. Next, the derivedscalable inverse functions are employed to obtain estimatesof the Monod kinetic parameters such as the maximummicrobial specific growth rate (lmax), the contaminant halfsaturation coefficient (Ks), and the contaminant yield coef-ficient (Ys). Parameter estimation errors are calculatedfrom comparisons to original parameter values used in gen-erating the METABIOTRANS simulations. The behaviorof these errors in the context of other kinetic parametersand with respect the temporal/spatial sampling interval isstudied.

2036 M. Mohamed et al. / Advances in Water Resources 30 (2007) 2034–2050

The remainder of this paper is organized as follows. InSection 2, governing equations that describe the transportand biodegradation of contaminants in groundwater arepresented. Derived in Section 3 are several scalable inversefunctions that relate the spatial moments of contaminantflux and contaminant and active biomass concentrationsto various Monod kinetic parameters. In Section 4,research results are discussed. Finally, a summary ofresearch findings and conclusions is presented in Section 5.

2. Governing equations

2.1. Transport equations

Microbial activities associated with groundwater con-taminant attenuation have been conceptually modeledfrom at least three distinct perspectives [66]. The first,assumes solid particles are uniformly covered by a thin bio-film of microbes or layer-like aggregation of cells [67]. Thebiofilm model originates from the field of environmentalengineering where it is currently used to describe reactionkinetics in wastewater-treatment reactors. The second per-spective assumes bacteria grow in small colonies or micro-colonies attached to solid surfaces [68,69]. A microcolonyhas the form of cylindrical plate and is 10–100 cells of uni-form dimension [69]. The number of colonies per unit vol-ume of aquifer changes with time and the target substrate(the contaminant) passes through an imaginary diffusionlayer to account for external mass transport. Key assump-tions in the model are that active biomass is attached (notmoving with groundwater) and that internal diffusion isnegligible, because the volume of each microcolony isalways small. The third conceptual model is macroscopicfor which no assumptions are made regarding the micro-scopic configuration and distribution of biomass in thepore space [66]. The macroscopic conceptual model is byfar the most common one applied to subsurface environ-ments [70]. Here, Monod reaction expressions [71] are typ-ically applied because assumptions concerning thedistribution of the microbial population are not required.Instead, reaction kinetics are driven by the bulk concentra-tions of coupled electron donors and acceptors.

Baveye and Valocchi [66] discussed differences betweenthe three conceptual models and concluded little practicaldifference exists between biofilm and microcolony perspec-tives if the goal is to perform solute transport modeling.However, the macroscopic approach differs significantly,because mass transport to the surface is ignored. In the sec-tions that follow we assume the macroscopic conceptualmodel and present those elements of the model critical tothis analysis. Our intent here is to be concise and not min-imize the true complexity of microbial-mediated contami-nant degradation and microbial ecology in the subsurfaceenvironment. However, further details on microbial-medi-ated contaminant transformations and microbial modelingcan be found in Baveye and Valocchi [66], Molz et al. [69],Bedient et al. [70] and references therein.

Here we assume some contaminants are electron accep-tors whereby microbial-mediated transformations [e.g.,with the reduction of Cr(VI)] or degradation [e.g., with chlo-rinated solvents like TCE, PCE, etc.] are coupled to the oxi-dation of an electron donor. In the case of Cr(VI) and otherheavy metals auxiliary constituents such as Fe3+/Fe2+ andhumics are needed to facilitate the transfer of electrons[72–75]. Other contaminants [e.g., various petroleum hydro-carbons, carbohydrates, etc.] are electron donors and assuch require available electron acceptors [e.g., O2, NO�1

3 ,Fe+3, etc.] to undergo biodegradation. The following analy-sis is general, such that the substrate or contaminant ofinterest can be an electron donor or an acceptor.

Aqueous phase transport is described by the advection–dispersion equation for each substrate (electron donor and/or electron acceptor). These equations are coupled throughsource/sink terms for biotransformation. For an electrondonor s, the transport equation may be written as

oSs

ot¼ � o

oxiðJ sÞ þ Qbio

s and J s ¼ �DijoSs

oxjþ V iSs ð1Þ

where Ss is the aqueous phase concentration [MsL�3] for

electron donor s; Js is the dissolved phase mass flux[MsL

�2T�1] of electron donor s; Vi is the average porewater velocity in the ith direction [LT�1]; xi is the distancein the ith direction (i = 1, 2 or xi = x, y) [L]; Dij is the sec-ond-order tensor for hydrodynamic dispersion [L2T�1];Qbio

s is a biotransformation sink term [MsL�3T�1]; and t

is time [T].For an electron acceptor a, the transport equation can

be written as

oAa

ot¼ � o

oxiðJ aÞ þ Qbio

a and J a ¼ �DijoAa

oxjþ V iAa ð2Þ

where Aa is the aqueous phase concentration [MaL�3] ofelectron acceptor a; Ja is the dissolved phase mass flux[MaL�2T�1] of electron acceptor a; and Qbio

a is the biotrans-formation sink term for the electron acceptora[MaL�3T�1].

2.2. Source/sink equations

The biotransformation source/sink terms are evaluatedby summing the effects of all bacterial species on each elec-tron donor and/or acceptor:

Qbios ðSs;Aa;MbÞ ¼

�1

h

XNB

b¼1

Mbvb;s ð3Þ

Qbioa ðSs;Aa;MbÞ ¼

�1

h

XNB

b¼1

Mbvb;a ð4Þ

where Mb is the microbial biomass concentration [MbL�3]of bacterial species b = 1,2, . . .. NB (number of microbialspecies); h is the aquifer porosity [L0]; vb,s is the utilizationrate of electron donor s by the bacterial species b [T�1]; andvb,a is the utilization rate of electron acceptor a by the bac-terial species b [T�1].


2.3. Utilization equations

Utilization rates may be related to the specific growthrates (lb,s) for bacterial species b utilizing electron donors [T�1]. Utilization of electron donor s by bacterial speciesb can be written as

vb;s ¼lb;s

Y b;sð5Þ

and the utilization of electron acceptor a by bacterial spe-cies b can be written as

vb;a ¼ cb;alb;s ¼lb;s

Y b;að6Þ

where Yb,s, and Yb,a are the yield coefficient [dimensionless]representing the mass of bacterial species b produced perunit mass of electron donor s and electron acceptor a,respectively; and cb,a is the use coefficient [dimensionless]representing the mass of electron acceptor a reduced toproduce a unit mass of bacterial species b.

2.4. Microbial growth equations

Modified Monod kinetics [76] are used to describe spe-cific growth rate of the microbial species b utilizing electrondonor s and electron acceptor a and can be written as

lb;s ¼ lmaxb;s

Aa

ðKb;a þ AaÞ

� �Ss

Kb;s þ Ss

� �ð7Þ

where lmaxb;s is the maximum specific growth rate for microbial

species b[T�1]; Kb,a is the half saturation coefficient for elec-tron acceptor a for microbial species b[MaL�3]; and Kb,s isthe half saturation coefficient for electron donor s for micro-bial species b[MkL�3]. The mass balance equation for growthand death of microbial species b can be written as

dMb

dt¼ Mblb;s �MbBb ð8Þ

where Bb is the first-order decay rate of microbial species bwhich accounts for cell death.

3. Moment equations and scaling relations

3.1. Case 1: One microbial species and two solutes

3.1.1. Moment equations

For the simple case of one microbial species (b = 1), asingle electron donor ‘s’, and a single electron acceptor‘a’, Eqs. (1), (2), and (8) reduce to the following threeequations:

oSot¼ � o

oxiðJ sÞ �

M1

Y 1;sl1;s ð9Þ

oAot¼ � o

oxiðJ aÞ �

M1

Y 1;al1;s ð10Þ

dM1

dt¼ l1;sM1 � B1M1 ð11Þ

where M1 is the active biomass concentrations of microbialspecies 1, [ML�3]; S and A are aqueous phase concentra-tions of the electron donor s and electron acceptor a,respectively, [MsL

�3 and MaL�3] (note that subscripts havebeen dropped from S and A because one electron donorand one electron acceptor are considered); Js and Ja are dis-solved mass fluxes of the electron donor and electronacceptor, respectively, [MsL

�2T�1 or MaL�2T�1]; andY1,s and Y1,a (= 1/c1,a) are yield coefficients. To expressthe electron donor transport equation in terms of the zer-oth absolute spatial moment, direct integration of Eq. (9)is performed as follows:I

oSot

dxi ¼ �I

o

oxiðJ sÞdxi �

IM1

Y 1;sl1;sdxi ð12Þ

whereH

denotes integration over a control volume (CV) ofthe plume [e.g. xi = {(�1,1), (�1,1)} i.e. {�1 < x < 1,�1 < y < 1}]. For spatially uniform maximum specificgrowth rate lmax

1;s and yield coefficient Y1,s, Eq. (12) simpli-fies to:

o

ot

IS dxi ¼ �

IdJ s �

1

Y 1;s

IM1l1;s dxi ð13Þ

and then:

oðmo�SjCV Þot

¼ �ðJ sjxmaxi � J sjx

mini Þ � 1

Y 1;s

IM1l1;sdxi ð14Þ

where mo�S is the zeroth absolute spatial moment for theelectron donor S and is defined as mo�S ¼

HS dxi. Terms

J sjxmaxi and J sjx

mini represent electron donor fluxes at the

down-gradient and the up-gradient boundaries of themonitored aquifer located respectively at X max

i and X mini

along direction i. Multiple methods now exist for measur-ing/calculating dissolved constituent fluxes in the subsur-face, some of which are more effective than others inheterogeneous aquifers [43,44,77]. Assuming the fluxesfor the electron donor are measurable; Eq. (14) is approx-imated as

mo�S2�mo�S1

t2� t1

ffi�1

2ððJ sjx

maxi � J sjx

mini Þjt2 þðJ sjx

maxi � J sjx

mini Þjt1

Þ

� 1

2Y 1;s

XCV

M1l1;sjt2 þXCV

M1l1;sjt1

!DxDy

ð15Þ

where mo�S2and mo�S1

are zeroth absolute spatialmoments for electron donor S at times t2 and t1,respectively.

To solve Eq. (15) requires information about electrondonor fluxes at both the up- and down-gradient boundariesof the sampling network. For the special case of an aquiferof infinite volume [xi = {(�1,1), (�1,1), (�1,1)} i.e.{�1 < x <1, �1 < y <1, �1 < z <1}], the followingboundary conditions can be assumed

J sðx! �1; tÞ ¼ J aðx! �1; tÞ ¼ 0 ð16Þ


Thus, Eq. (15) simplifies as follows:

mo�S2�mo�S1

t2� t1

ffi� 1

2Y 1;s

XR

M1l1;sjt2þX

R

M1l1;sjt1

!DxDy

ð17Þwhere the right hand side (RHS) of Eq. (17) constitutes a dis-crete volume integration of the second term on the RHS ofEq. (12). For the electron acceptor, an equation analogousto (17) can be written, but is not shown. For this equationthe pertinent zeroth spatial moment is mo � A ¼

HAdxi.

To express the microbial growth in terms of the zerothabsolute spatial moment for biomass, Eq. (11) is integratedas follows:I

oM1

otdxi ¼

IðM1l1;s � B1M1Þdxi ð18Þ

The bacterial decay rate B1 is often neglected as a low orderprocess by many scientists (see Table 2). Making the sameassumption here, the following approximation is obtainedfor Eq. (18) by defining the absolute zeroth spatial momentfor biomass as mo�M1 ¼

HM1 dxi:

mo�M12� mo�M11

t2 � t1

ffi 1

2

XCV

M1l1;sjt2 þXCV

M1l1;sjt1

!DxDy

ð19Þwhere mo�M12

and mo�M11are zeroth absolute moments for

biomass distributions at times t2 and t1, respectively; andthe RHS of Eq. (19) represents the discrete volume integra-tion of the first term on the RHS of Eq. (11).

To express the microbial growth in terms of the firstabsolute spatial moment for biomass, Eq. (11) is again inte-grated as follows:I

oM1

otxdxi ¼

IðM1l1:s � B1M1Þxdxi ð20Þ

By defining the absolute first moment for biomass asm1�M1 ¼

HM1xi dxi, Eq. (20) reduces to:

m1�M12� m1�M11

t2 � t1

ffi 1

2

XCV

M1l1;sxjt2 þXCV

M1l1;sxjt1

!DxDy

ð21Þwhere mo�M12

and mo�M11are first absolute spatial mo-

ments for biomass distributions at times t2 and t1, respec-tively. It may be noted that the RHS of Eq. (21) containsthe longitudinal coordinate variable x in the discrete vol-ume integration.

3.1.2. Scalable inverse functions for case 1The system of moment equations obtained thus far can

be solved simultaneously to generate values of importantMonod parameters. Dividing Eq. (19) by Eq. (15) yields:

Y 1;s ffi �mo�M12

� mo�M11

mo�S2� mo�S1

þ DtDJ sð22Þ

where Dt = (t2 � t1) and DJ s ¼ 12ððJ sjx

maxi � J sjx

mini Þjt2

þðJ sjxmaxi � J sjx

mini Þjt1Þ. A similar equation to (22) can be

written for the electron acceptor. For an infinite domain,DJs = DJa = 0, Eq. (22) takes the form of an approximationused by Schirmer et al. [22] to evaluate Y1,s from the degrada-tion m-xylene (an electron donor) in batch reactors. On theother hand, dividing Eq. (21) by Eq. (15) yields:

m1�M12�m1�M11

mo�S2�mo�S1

þDtDJs

ffi�Y 1;s

PCV

M1S

K1;sþS

� �A

K1;aþA

� �X jt2 þ

PCV

M1S

K1;sþS

� �A

K1;aþA

� �X jt1

� �PCV

M1S

K1;sþS

� �A

K1;aþA

� �jt2 þ

PCV

M1S

K1;sþS

� �A

K1;aþA

� �jt1

� �ð23Þ

Again for an infinite domain, one can assume DJs = DJa = 0 in Eq. (23). For the electron acceptor, equationsanalogous to (22) and (23) can be written but are not shown.

To use the above inverse functions (Eqs. (17), (19), (21),(22), (23)) in a field application, spatial concentration dataare required for the electron donor, the electron acceptor,and microbial biomass. In addition, at the up and down-gradient boundaries of a finite sampling network, mass fluxmeasurements are needed for the electron donor and accep-tor. Data can be gathered as two or more snap-shots intime using direct-push technologies in a rapid and cost-effective manner [44,62–64]. Once collected, numericalspatial integration of these data are performed to calculateseveral pertinent absolute moments including: mo�A, mo�S,mo�M1, and m1�M1. Next, values of Y1,s and Y1,a are esti-mated from Eq. (22) (and the analogous equation forY1,a), after which half saturation coefficients K1,s and K1,a

are calculated from Eq. (23) (and the analogous equationfor the electron acceptor). Finally, the maximum specificbacterial growth rate is determined by substituting knownvalues of Y1,s, Y1,a, K1,s, and K1,a in either Eqs. (17), (19),or (21). Because the above analysis is predicated on modelswhich are simple approximations of a complex environ-mental system, care must be taken when interpretingresults. For example, a microbial consortium is oftenresponsible for contaminant degradation; consequently,inverse modeling would generate estimates of effectiveparameters for the consortium and not a single bacterialspecies.

3.2. Case 2: Two microbial species and one solute

3.2.1. Moment equations

For the simple case of two microbial species (b = 1,2)and a single contaminant ‘s’ (an electron donor in thiscase), Eqs. (1), (2), and (8) reduce to the following threeequations:

oSot¼ � o

oxiðJ sÞ �

l1;sM1

Y 1;s�

l2;sM2

Y 2;sð24Þ

dM1


dM2



where Y1,s and Y2,s are yield coefficients, representing themass of bacterial species 1 and 2 produced per unit massof substrate s, respectively, [dimensionless]; l1;sð¼ lmax

1;s

ð SK1;sþSÞÞ and l2;sð¼ lmax

2;s ð SK2;sþSÞÞ are specific growth rates

for microbial species 1 and 2, respectively, [T�1]; K1,s

and K2,s are half saturation coefficients for substrate swhen it is utilized by each microbial species (b = 1,2),[MsL

�3]; and B1 and B2 are first-order decay rates whichaccounts for cell death of each species. Because Eqs.(24)–(26) do not include an electron acceptor, they arestrictly valid as long as available electron acceptors arenot limiting. However, if our contaminant of concern isin fact an electron acceptor and the available supply ofelectron donors is not limiting, then Eqs. (24)–(26)(and the analysis that follows) are valid to model the fateand transport of the electron acceptor: simply substituteconcentrations A for S in the equations and this becomesevident.

Assuming spatially uniform maximum growth rateslmax

1;s , lmax2;s , and yield coefficients Y1,s,Y2,s, Eq. (24) is inte-

grated to produce:

mo�S2�mo�S1

t2� t1

ffi�DJ s�1

2Y 1;s

XCV

M1l1;sjt2 þXCV

M1l1;sjt1

!DxDy

� 1

2Y 2;s

XCV

M2l2;sjt2 þXCV

M2l2;sjt1

!DxDy

ð27Þ

where mo�S2and mo�S1

are the zeroth absolute moment ofsubstrate S at times t2 and t1, respectively; and DJ s ¼ 1

2

ððJ sjxmaxi � J sjx

mini Þjt2

þ ðJ sjxmaxi � J sjx

mini Þjt1Þ. Again, it is as-

sumed electron donor fluxes are measurable.To express the electron donor transport Eq. (24) in

terms of the first absolute moment, the following integra-tion is evaluated:I

oSot

xi dxi ¼ �I

o

oxiðJ sÞxi dxi �

Il1;sM1

Y 1;sxi dxi

�I

l2;sM2

Y 2;sxi dxi ð28Þ

Applying the same assumptions used to derive Eq. (27),one obtains:

o

ot

ISxi dxi ¼ �

Ixi dJ S �

1

Y 1;s

IM1l1;sxi dxi

� 1

Y 2;s

IM2l2;sxi dxi ð29Þ

which is rewritten as follows:

om1�S

ot¼ �ðxiJ sÞj

xmaxi

xminiþI

J s dxi �1

Y 1;s

IM1l1;sxi dxi

� 1

Y 2;s


or

om1�S

ot¼ �ðxmax

i J sjxmaxi � xmin

i J sjxmini Þ þ m0�JS

� 1

Y 1;s

IM1l1;sxi dxi �

1

Y 2;s


Here, m1�S ¼H

Sxi dxi is the first absolute moment of elec-tron donor (i.e., contaminant) s; and mo�JS ¼

HJ S dxi is

the absolute zeroth moment of the mass flux distributionfor electron donor s. Under convective dominated flowconditions in a relatively homogeneous aquifermo�JS � Vmo�S. The following is a numerical approxima-tion of Eq. (31):

m1�S2�m1�S1

t2� t1

ffi�DJ �s þmo�JS2

þmo�JS1

2

� 1

2Y 1;s

XCV

M1l1;sX jt2 þXCV

M1l1;sX jt1

!DxDy

� 1

2Y 2;s

XCV

M2l2;sX jt2 þXCV

M2l2;sX jt1

!DxDy

ð32Þ

where m1�S2and m1�S1

are first absolute spatial momentsfor substrate s at times t2 and t1; and DJ �s ¼ 1

2ððxmax

i

J sjxmaxi � xmin

i J sjxmini Þjt2

þ ðxmaxi J sjx

maxi � xmin

i J sjxmini Þjt1Þ. For the

case of infinite boundary conditions (Eq. (16)), DJ �s ¼ 0.Assuming again a zero rate for microbial death, and the

following definition for the zeroth absolute spatial momentfor the biomass distributions of microbial species 1,mo�M1 ¼

HM1 dxi, direct integration of Eq. (25) yields


t2 � t1

ffi 1

2

XCV

M1l1;sjt2 þXCV

M1l1;sjt1

!DxDy

ð33Þ

where mo�M12and mo�M11

are zeroth absolute moments ofthe bacterial species 1 biomass distribution at times t2 andt1, respectively.

The first absolute moment for the biomass distributionof microbial species 1 is defined as m1�M1 ¼

HM1xi dxi.

Direct integration of Eq. (25) yields the followingequation:

m1�M12�m1�M11

t2� t1

ffi 1

2

XCV

M1l1;sX jt2 þXCV

M1l1;sX jt1

!DxDy

ð34Þ

where m1�M12and m1�M11

are the first absolute spatial mo-ments for the biomass distribution of microbial species 1 attimes t2 and t1, respectively. Dividing Eq. (34) by Eq. (33),yields:

m1�M12� m1�M11


ffi

PCV

M1S

K1;sþS

� �X jt2þPCV

M1S

K1;sþS

� �X jt1

� �PCV

M1S

K1;sþS

� �jt2þPCV

M1S

K1;sþS

� �jt1

� �ð35Þ


For the second microbial species, zeroth and first absolutespatial biomass moments are respectively defined asmo�M2 ¼

HM2 dxi and m1�M2 ¼

HM2xi dxi. Eq. (26) is then

integrated as described above to generate analogous equa-tions to (33)–(35).

3.2.2. Scalable inverse functions for case 2

To use the above analysis in a field application, neededare concentration data characterizing spatial distributionsof electron donor and biomass of both microbial species.In addition, measured electron donor fluxes are requiredto calculate the zeroth moment of the flux distribution,which appears in Eq. (30). Again, these data can be gath-ered as two or more snap-shots in time using direct-pushtechnologies [44,62–64]. Spatial integration of this data isperformed numerically such that the following absolutemoments are calculated: mo�S, m1�S, mo�JS, mo�M1,mo�M2, m1�M1 and m1�M2. Next, the half saturation coef-ficient K1 is estimated using Eq. (35). It is then feasible tocalculate the maximum bacterial growth rate lmax

1;s for bac-terial species 1 using Eq. (33) or (34). Values of K2,s andlmax

2;s are obtained from analogous equations (not shown).Substituting Eq. (33) into Eq. (27) and multiplying by

the time interval Dt = (t2 � t1), yields:

mo�S2� mo�S1

þ DJ sDt ffi � 1

Y 1;sðmo�M12

� mo�M11Þ

� 1

Y 2;sðmo�M22

� mo�M21Þ ð36Þ

Similarly, by combining Eqs. (34) and (32) and then multi-plying by the time interval Dt, the following is obtained:

m1�S2� m1�S1

þ DJ �s Dt � mo�JS2þ mo�JS1

2Dt

ffi � 1

Y 1;sðm1�M12

� m1�M11Þ � 1

Y 2;sðm1�M22

� m1�M21Þ

ð37Þ

Values for yield coefficients Y1,s and Y2,s are then deter-mined from solving Eqs. (36) and (37) simultaneously. Gi-ven the form of (36) and (37), it is easy to see how a systemof equations can be written to include any number ofmicrobial species. Furthermore, it is evident Eq. (37) re-

0.00 10.00 20.000.00

5.00

10.00

15.00

20.00

25.00

Y(m

)

5.0 m

3.0

m

So

Xmin Xma

X

Fig. 1. Proble

duces to the same form as Eq. (22) when either microbeis absent.

4. Testing examples and sensitivity analysis

4.1. Example 1 (Case 1 – homogeneous aquifer – infinitevolume)

To test the scalable inverse functions, transient concen-trations of electron donor, acceptor, and microbial biomasswere simulated using METABIOTRANS [65] over a two-dimensional x–y hypothetical aquifer of homogeneous per-meability with dimensions 50 m · 25 m (Fig. 1). A constantvelocity of 0.5 m/day was assumed in the x-direction (Vx).Longitudinal and transverse dispersivities were chosen tobe 0.1 and 0.05 m, respectively, while porosity was assumedto be 0.3. An instantaneous discharge of 1 mg/l of contam-inant (assumed to be an electron donor) at time t = 0 wasreleased over an area with dimensions 5 m · 3 m (Fig. 1).The centroid of this initial contaminant release was locatedat (4, 12.5). The supply of electron acceptor was assumedto be not limiting, and therefore not considered further inthis analysis. The up- and down-gradient boundaries wereassigned constant head values to produce a constant veloc-ity of 0.5 m/day, whereas the top and bottom boundarieswere assumed impervious. The aquifer domain was repre-sented using a mesh of 31,250 rectangular elements(250 · 125) and 31,626 nodes. A spatially uniform concen-tration of 0.001 mg/l was used to define the initial micro-bial biomass distribution; however, upon contaminantrelease microbial growth produced a heterogeneous bio-mass distribution within the contaminant plume trajectory.

In this first example, the plume exists within an infinitevolume or well within specified coordinates of the modeldomain (i.e. Xmin = 0.0 m and Xmax = 50.0 m, Fig. 1).Seven test problems are employed to evaluate the scalableinverse functions for case 1. Values of Monod kineticparameters used in METABIOTRANS [65] to generatesimulated field data for all seven problems are given inTable 1. Literature values of these parameters vary overa wide range depending on both the contaminant (an elec-tron donor) and the microbial species (Table 2). Valuesfrom this analysis are close to those used by Schirmer

30.00 40.00 50.00

X (m)

No flow boundary

No flow boundary

x

Location of plume center at t1 and t2

m layout.

Table 1Biological input parameters used in Example 1

Y1,s K1,s (mg/l) lmax1;s (day�1)

Problem 1 0.01 5 0.5Problem 2 0.05 5 0.5Problem 3 0.1 5 0.5Problem 4 0.01 5 1.0

Problem 5 0.01 5 2.0

Problem 6 0.01 3 0.5Problem 7 0.01 7 0.5

Table 2Literature values for various Monod parameters

Reference Parameter Value/range

Schafer [78] lmax 0.1 day�1

Ks 0.2–70 mg/lYs 0.02–0.1

Brusseau et al. [79] lmax 0.1 day�1

Ks 1–3 mg/lYs 0.8B Neglected in most simulation, they

concluded that effect of B becomesmeasurable when B > 0.1lmax

Schafer et al. [80] lmax 0.96–4.8 day�1

Ks 0.003–1.25 mmol/lYs 0.02–0.1B 0.1lmax

Kindred and Celia [81] lmax 0.096–0.96 day�1

Ks 0.01–0.1 mg/l

Parlange et al. [82] andEssaid and Bekins[83]

lmax 4.8E�4 day�1

Ks 0.5 mmol/lB Neglected

Salvage and Yeh [84] lmax 0..9288–32.4 day�1

Ks 1E�4–2.2E�3 mmol/lKs 7.5 mg/lYs 1.3E�4–2.0B Neglected

Schirmer et al. [42] lmax 0.1992–4.128 day�1

Ks 0.1–2.0 mg/lB Neglected

Wang and Shen [85] lmax 0.6588–1.5443 day�1

Ks 5.43– 8.64 mg/lYs 1.39E11–4.80E11 (cell/mg)

Schirmer et al. [22] lmax 4.128–4.5 day�1

Ks 0.79–4.7 mg/lB Neglected


et al. [22,42] who studied the attenuation of various elec-tron donors, namely BTEX compounds.

The transport model [65] was used to simulate contam-inant and biomass concentrations at each node every 5days (t = 5,10, . . . , 70 days). Using these concentrations,zeroth moments for both the contaminant and the bio-mass distributions and the first moment for the biomassdistribution were calculated numerically at times(t = 5,10, . . . , 70 days) using simple two-dimensional dis-crete integral approximations based on a uniform sam-

pling network [86]. Scalable inverse functions (Eqs. (19),(22), and (23)) were then used to estimate the Monodparameters as described above using all possible combina-tions of t1 and t2. Parameter estimates for all combina-tions of t1 and t2 were then separated into groupsaccording to the magnitude of the sampling time interval(Dt = t2 � t1). The minimum time interval considered inthis analysis was 5 days, whereas, the maximum intervalwas limited to 30 days in order to generate a sufficientnumber of estimates to permit an analysis of parameterestimation errors.

Parameter estimation errors could evolve from numeri-cal approximations of differentials and integrals used toobtain Eqs. (19), (22), and (23); but they would also dependon the spatial density and variability of measured electrondonor/acceptor and biomass concentrations. The percenterror (eP) in estimating a given kinetic parameter was cal-culated according to Eq. (38); and, for each value of Dt, theexpected error (EP) for a given parameter was calculatedfrom Eq. (39). Thus, for each value of Dt

(5,10,15, . . . , 30) and for any biological parameter P

eP ¼ jP ðt1; t1 þ DtÞ �OPjOP

� 100; t1 ¼ 5; 10; 15; . . . days

ð38Þ

EP ¼PN

n¼1ePn

Nð39Þ

where P(t1, t1 + Dt) was the estimated value of the Monodparameter P(Y1,s, K1,s, or lmax

1;s ) from inverse functions eval-uated at t1 and t2 = t1 + Dt; OP was the original value of P

(input data, Table 1); eP was the error in estimating theparameter P; EP was the average percent error in the esti-mate of parameter P(EY1,s, EK1,s, or Elmax

1;s ) obtained forall evaluations at a given time interval Dt; and N was thetotal number of parameter evaluations for that time inter-val Dt(N = 13,12, . . . , 8 for Dt = 5,10, . . . , 30 days,respectively).

Fig. 2 illustrates for problem 1, simulated contaminant(electron donor) and microbial biomass plumes after 10,40, and 70 days. Contaminant and biomass concentrationdistributions are symmetric in the transverse directionbecause aquifer conductivity is uniform. However, in longi-tudinal direction both plumes are asymmetric due to micro-bial growth and contaminant attenuation: this isparticularly evident with microbial biomass. Fig. 3a revealsthe change in the averaged estimation error for the yieldcoefficient (EY1,s) with respect to Dt. From this figure, itcan be seen that the error is stable (around 3%) for almostall seven test problems, especially when the time interval isten days or less. Fig. 3a also shows that by increasing Y1,s

(problems 2 and 3) or increasing K1,s (problem 7) we areperturbing Monod parameters in a manner that reducesthe rate of contaminant attenuation per unit of active bio-mass such that the inverse function for specific yield exhib-its greater sensitivity to error. Increasing lmax

1;s (problem 5),on the other hand, appears to decrease this sensitivity and

0 5 10 150

5

10

15

20

25

Y (

m)

25 30 35 40 45 5015 20 25 30

X (m)

t = 10 days t = 40 days t = 70 days

0 5 100

5

10

15

20

25

Y (m

)

0.001

0.00102

0.00104

0.00106

0.00108

0 5 10 15 20 25 0 5 10 15 20 25 30 35 40 45 50

X (m)


Fig. 2. Example 1 results at days 10, 40, and 70: (a) contaminant concentrations and (b) microbial biomass concentrations.

2.5

3.5

4.5

10 15 20 25 30 35time interval (days)

EY

s

Test problem 1Test problem 2Test problem 3Test problem 4Test problem 5Test problem 6Test problem 7

0

4

8

12

10 15 20 25 30 35

time interval (days)

EK

s

0

20

40

0 5

0 5

0 5

10 15 20 25 30 35


Eµ

max

Fig. 3. Example 1, EPs versus sampling time interval; (a) EY1,s, (b) EK1,s,and (c) Elmax

1;s .


in turn produces values of EY1,s that are somewhat insen-sitive to the sampling interval Dt.

Increasing the sampling interval Dt appears to induce amild parabolic increase in estimation errors associated withthe half saturation coefficients K1,s (Fig. 3b). Errors rangefrom 1% to 5% for an interval of 5 days and increase to2–11% for 30 days. In addition, errors vary inversely tothe magnitude of K1,s. That is, EK1,s increases at all sam-pling intervals when K1,s decreases from 5 mg/l (problem1) to 3 mg/l (problem 6). Thus, it seems to be more difficultto estimate K1,s for microbes capable of thriving under lowsubstrate concentrations. This finding appears to contra-dict the claim of Ellis et al. [32] who found it difficult toobtain unique parameter in batch systems for lmax

1;s andK1,s for ratios of S/K1,s < 1.

In Fig. 3c, it is shown that Elmax1;s increases with respect

to Dt; however, the sensitivity to the time incrementincreases as lmax

1;s increases (problems 4 and 5) or as K1,s

is reduced (problem 6). Elmax1;s does decrease for larger val-

ues of Y1,s (problems 2 and 3). The observed behavior ofElmax

1;s is in part explained by examining Eqs. (17), (19),or (21) and noting calculations of lmax

1;s depend on actualdifferences in mass (contaminant or microbial) and notnormalized differences (as with K1,s and Y1,s ). In general,greater errors are observed as more of contaminant isdegraded. For example, in problem 5 the highest estima-tion error for the maximum microbial growth rate isobtained when biotic reactions have achieved 94% contam-inant attenuation as opposed to problem 3 where it is lim-ited to 9% (Fig. 3c).

Analyses were conducted to investigate the sensitivity ofinverse modeling results to perturbations in Monod param-eters used in forward model simulations. Kabala [87]reviewed basic concepts of sensitivity analysis pertinent togroundwater. He recognized several types of sensitivities


namely: traditional sensitivity, normalized sensitivity, andlogarithmic sensitivity. His discussion was based on typicalmodels that simulate the evolution of a system whose out-put depends on a number of input parameters. In thispaper, outputs of interest were the errors in estimatedMonod parameters EP (Eq. (39)) and inputs were the ori-ginal values of the biological parameters used in the for-ward simulation model (OP in Eq. (38) and Table 1). Themodel in this case included the software METABIO-TRANS, various scalable inverse functions, and the erroranalysis (Eqs. (38) and (39)). Traditional sensitivities ofoutputs (EPs) to inputs (OPs) were shown in Fig. 4.Elmax

1;s was the most sensitive output. It was particularlysensitive to the magnitude of lmax

1;s , such that increasingmaximums specific growth rate by a factor of two, alsodoubled Elmax

1;s for most Dt time increments. However, thesame percent increase in either Y1,s or K1,s reduced Elmax

1;s

substantially. Fig. 4 also showed EY1,s was not particularlysensitive to any one biological parameters. EK1,s was sensi-tive to variations in K1,s and perhaps most sensitive to Dt.Increasing K1,s from 3 to 7 mg/l reduced EK1,s by 20–30%.

4.2. Example 2 (Case 1 – heterogeneous aquifer – infinite

volume)

The purpose of this example is to test the scalableinverse method on a heterogeneous aquifer. For this pur-

2.6

3.6

4.6

0 0.05 0.1

Ys0 0

Y

EY

s

EY

s

2.6

3.1

3.6

3 5 7Ks

3

EY

s

EY

s

2.6

2.9

3.2

0 1 2µmax

0

EY

s

2

4

6

0

6

12

5 da15 d25 d

0

2.5

5

EK

s

Fig. 4. Example 1, sensitivity of syste

pose, a random field generator (RFG) is used to create spa-tially varying hydraulic conductivity (K) field. The RFG isbased on the Fast Fourier Degrad approach described andused by Hassan et al. [88,89]. To generate the heteroge-neous K distribution, a statistically homogeneous, isotro-pic, and second-order stationary random field withspatial correlation structure is assumed. The conductivityis assumed to have a lognormal distribution with an expo-nential covariance structure (Cov(r) = r2e�r/k), in which r2

is the process variance, r is the spatial lag, and k is the cor-relation length that describes the distance over which con-ductivity values are spatially correlated. To use the RFG inthis paper, a log K variance of 1.0 and a unit correlationlength are used.

Using the generated heterogeneous hydraulic conductiv-ity field, test problems 1 and 4 were re-simulated (andreferred to as problems 1H and 4H, respectively). Fig. 5illustrates for problem 1H, simulated contaminant (elec-tron donor) and microbial biomass plumes after 10, 40,and 70 days. Because the aquifer is assumed heterogeneous,both plumes depict asymmetric concentration distributionsin both longitudinal and transverse directions. Fig. 6a–cillustrate parameter estimation errors from problems 1Hand 4H together with results from problems 1 and 4. Theseresults support previous findings of others [13] who foundphysical heterogeneities had almost no influence on theefficacy of contaminant attenuation. Here it is seen that

.05 0.1

s0 0.05 0.1

Ys

5 7Ks

3 5 7Ks

1 2µmax

0 1 2µmax

ys 10 daysays 20 daysays 30 days

0

7

14

0

11

22

Eµ

max

Eµ

max

Eµ

max

0

22

44

m outputs (EPs) to inputs (OPs).

0 5 10 150

5

10

15

20

25

Y (

m)

10 15 20 25 30

X (m)

20 25 30 35 40 45 50


0 5 100

5

10

15

20

25

Y (

m)

0 5 10 15 20 250.001

0.00102

0.00104

0.00106

0.00108

0.0011

0.00112

0.00114

0 5 10 15 20 25 30 35 40 45 50

X (m)


Fig. 5. Example 2 results at days 10, 40, and 70: (a) contaminant concentrations and (b) microbial biomass concentrations.

2.00

4.00

6.00

1050

50

50

15 20 25 30 35time interval (days)

EY

s

Test problem 1 Test problem 4Test problem 1H Test problem 4HTest problem 1F Test problem 4F

1.00

3.00

5.00

7.00

10 15 20 25 30 35


EK

s

0

20

40

10 15 20 25 30 35


Eµ

max

a

b

c

Fig. 6. Examples 1, 2, and 3, EPs versus sampling time interval; (a) EY1,s,(b) EK1,s, and (c) Elmax

1;s .


physical heterogeneity produces minor increases on theaverage error of estimates for Y1,s and K1,s and even smal-ler increases for lmax

1;s . Fig. 6a shows aquifer heterogeneitymagnifies the effect of increasing lmax

1;s on EY1,s (comparefor example the relative closeness of EY1,s for problems 1and 4 to that of problems 1H and 4H); whereas, fromFig. 6b heterogeneity did not alter the influence of varyinglmax

1;s on EK1,s. Fig. 6c shows the linear trends observedbetween Elmax

1;s and Dt under homogeneous conditions arepreserved under conditions of spatially varying hydraulicconductivity.

4.3. Example 3 (Case 1 – homogenous aquifer – infinite

volume with coarse sampling network)

In all previous test problems the sampling network isconfined within the coordinates �5 6 x 6 50 and7.5 6 y 6 17.5. This physical range encompasses the spatialextent of all plumes simulated (concentration 0.02 isassumed to represent the edge of the plume). A uniformisotropic horizontal sampling interval DL characterizesmonitoring network. This sampling interval is incorporatedin the dimensionless parameter kð¼ V

lmax1;s

DLÞ representing the

ratio of time required for microbial growth over thehydraulic residence time. For problems 1 and 4, k equals5 and 2.5 respectively. This corresponds to a separation dis-tance of DL = 0.2 m between monitoring locations.

In this example, test problems 1 and 4 were re-simulatedusing coarse uniform sampling network (problems 1F and4F, respectively). k values for problems 1F and 4F were 0.5and 0.25 respectively, giving a consistent DL = 2.0 m, or anorder of magnitude greater than applied in problems 1 and

Table 3values of sampling area length and the corresponding values of Xmin andXmax

Dx 2 4 6 8 10 12 14 16 18 20Xmin 18 17 16 15 14 13 12 11 10 9Xmax 20 21 22 23 24 25 26 27 28 29

Table 4values of time interval and the corresponding values of X1 and X2

(V = 0.5 m/s)

Dt 2 4 6 8 10 12 14 16t1 29 28 27 26 25 24 23 22t2 31 32 33 34 35 36 37 38X1 18.5 18 17.5 17 16.5 16 15.5 15X2 19.5 20 20.5 21 21.5 22 22.5 23


4. This produced a sampling network comprised of 144sampling locations; although, under the assumption thata concentration 0.02 defines the extent of the plume, only32 were used with smaller plumes (t = 10 days) and 60 toevaluate moments of larger plumes (t = 70 days).

Fig. 6a–c illustrate parameter estimation errors fromproblems 1F and 4F compared to results from problems1 and 4. Using a sampling network that is an order of mag-nitude less dense produces minor increases in average esti-mation errors for Y1,s and K1,s and lmax

1;s . Maximum valuesof EY1,s or EK1,s are still less than 7% and linear trendsbetween Elmax

1;s and Dt are preserved between the differentnetworks. The inverse method appears quite robust evenwith fewer samples. Then again, the scalable inverse func-tions are based on lower-order moments (less than secondorder) which tend to be more accurate and stable acrosschanges in sampling network density.

4.4. Example 4 (Case 1 – homogenous aquifer – finite

volume)

Estimation of Monod parameters using data obtainedfrom a finite volume (sampling area) of the aquifer is pos-sible according to derived equations (Section 3). However,special care must be taken when selecting the position anddimensions of that finite volume to ensure a portion of theplume is interrogated within the finite monitoring networkfor two sampling campaigns separated by time interval(Dt).

In this example, scalable inverse functions use dataobtained from a finite sampling area. The transverse hori-zontal dimension of this finite sampling area is treated asa fixed parameter that is sufficiently large to encompassthe entire width of the plume (�15 m, Fig. 1) at all times.The magnitude of longitudinal horizontal dimension(Dx), is treated as a variable and is examined here for itseffect on the accuracy of the inverse functions. To ensurea sufficient portion of plume is encompassed within thefinite sampling area, Dx and Dt are selected to satisfy twoconditions. The first condition guarantees that the leadinghalf of the plume enters the finite sampling area at orbefore time t1 and that the trailing half does not leave thesampling transect before time t2. Assuming X1 = Vt1 andX2 = Vt2, this first condition can be written as

X 1 P X min and X 2 6 X max ð40Þwhere Xmin and Xmax represent the beginning and end loca-tions of the sampling area along the flow axis(Dx = Xmax � Xmin). The second condition is useful whendesigning or deploying small monitoring networks or whenusing large time intervals between sampling campaigns, itensures a large portion of the plume is monitored duringsampling events. This condition is not critical; however, itstates that the time-average location of the sampled portionof the plume coincide with the centroid of the finite sam-pling area (see Fig. 1). The second condition can be de-scribed mathematically as following:

X ave ¼X max þ X min

2¼ V ðt1 þ t2Þ

2or

2X min þ Dx ¼ V ð2t1 þ DtÞ ð41Þ

where Xave represents the center location of the samplingarea. Clearly, prior estimates of contaminant plume extentand groundwater velocity are needed to develop and locatea finite sampling network that satisfies conditions ex-pressed in Eqs. (40) and (41). For these purposes, historicalrecords and prior site characterization data are often suffi-cient to generate an initial network design. Then, as direct-push methods are used to gather current concentrationdata for inverse modeling, the network can be updateddynamically.

Studied here are the effects of both the time interval Dt

between sampling campaigns and the longitudinal extentDx of the monitoring network on the accuracy of the scal-able inverse method. Data from Example 1 are used againwith different network extents Dx and different samplingtime intervals as shown in Tables 3 and 4. Xave is main-tained at 19 m (15 m from the center of the source). Errorsin estimating each Monod parameter (EP) were obtainedfor all possible pairs of Dt and Dx shown in Tables 3 and4 and, all pairs satisfy condition 2.

Fig. 7a–c shows the effects of ranging both Dx and Dt onEY1,s, EK1,s and Elmax

1;s . In general, increasing Dx anddecreasing Dt improves the estimation of all three parame-ters. This is because all scalable inverse functions are pred-icated on numerical approximates of Eqs. (9) and (10),which for advection dominated flow systems tend to bemost valid when the dimensionless parameter w = Dx/VDt is greater than 1.0. This dimensionless parameter rep-resents the ratio of groundwater travel time between sam-ple locations to the sampling time interval Dt, and it isessentially the reciprocal of the Courant number [90]. Con-tours of w are presented in Fig. 7a–c showing (Dx, Dt) pairswhich satisfy different values of w and in turn different lev-els of accuracy. As the value of w increases, parameter esti-mation errors decrease. This is illustrated more clearly in

0

20

40

60

0 2 4 6 8 10 12 14 16 18 20 22dx

EY

s

dt=2 daysdt=4 daysdt=6 daysdt=8 daysdt=10 daysdt=12 daysdt=14 daysdt=16 dayscontour of w=1contour of w=2.5contour of w=5

Xmin=18 17 16 15 14 13 12 11 10 9

0

40

80

120

160

0 2 4 6 8 1 0 12 1 4 1 6 1 8 20 2 2dx

Eµ

max

Xmin=18 17 16 15 14 13 12 11 10 9

a

c

0

20

40

60

0 2 4 6 8 10 12 14 16 18 20 22dx

EK

s

Xmin=18 17 16 15 14 13 12 11 10 9b

Fig. 7. Example 4, EPs versus sampling time interval and transect lengthwith corresponding contours of w; (a) EY1,s, (b) EK1,s, and (c) Elmax

1;s .


Fig. 8 where EY1,s is shown to decreases exponentially withw.

It is feasible that an entire plume could enter and exit afinite sampling area within period Dt and satisfy the secondcondition but fail the first. Sampling and subsequent anal-ysis would in turn yield poor parameter estimates. Allpoints appearing in Fig. 7 satisfy condition 2 (Eq. (41));but, only those points enclosed within the contour of

0

20

40

60

80

0 5 10 15 20

w = delta x / v delta t

EY

s

Fig. 8. Example 4, EY1,s versus w = Dx/(VDt).

w = 1.0 satisfy condition 1. It appears from Fig. 7a–c thatsatisfying condition 1 (Eq. (40)) is critical for minimizingparameter estimation errors. Thus, hydrologists tasked todevelop sampling schedules and networks should constrainselection of sampling variables Dx and Dt to satisfy Condi-tion 1.

Closer inspection of Fig. 7a–c shows Elmax1;s > EK1;s >

EY1;s for a given transect length Dx and for the same rea-sons discussed earlier. Fig. 7a–c can be used to design sam-pling networks given prior information about thegroundwater velocity and plume extent. For example, toobtain estimates of both Y1,s and K1,s with errors less than10%, a transect length is selected that is at least 2.5 timeslonger than the plume travel distance for a given time inter-val Dt between sampling events (Fig. 7a and b). In this case,however, Elmax

1;s exceeds 20% for Dt = 12 days and Dx =16 m; but is less than 20% elsewhere (Fig. 7c). To ensurethat Elmax

1;s is less than 10%, the length of the sampling areamust be five times longer than distance traveled by theplume over period Dt.

4.5. Example 5 (Case 2 – homogeneous aquifer – infinitevolume)

In this example two microbial species are assumed tocompete for the same contaminant. Initially, both speciesare uniformly distributed throughout the aquifer at a bio-mass concentration of 0.001 mg/l; and again, upon con-taminant release growth of both species occurs along thecontaminant flow path which in turn produces a heteroge-neous biomass distribution. The purpose of this example isto examine the scalable inverse functions derived for case 2(Section 3.2). Hence using Eqs. (24)–(26), predictions areexamined for the six Monod parameters between twomicrobial species (Y 1;s; Y 2;s;K1;s;K2;s;lmax

1;s , and lmax2;s ).

Four more test problems are simulated in this example(problems 8–11). Values for the six model kinetic parame-ters are given in Table 5. Half saturation coefficient K1,s isobtained by solving Eq. (35) (and K2,s using an analogousequation not shown). The maximum bacterial growth ratelmax

1;s (and lmax2;s ) is calculated using Eq. (33) (or (34)).

Finally, yield coefficients Y1,s and Y2,s are obtained usingEqs. (36) and (37).

Fig. 9a clearly shows Y1,s (and similarly Y2,s) were easilyestimated with less than 5% error which generally indicatesestimating the yield coefficient for both microbial species is

Table 5Biological input parameters used in Example 2

Microbial species 1 Microbial species 2

Y1,s K1,s

(mg/l)lmax

1;s(day�1)

Y2,s K2,s

(mg/l)lmax

2;s(day�1)

Problem 8 0.01 5 0.25 0.01 5 0.25Problem 9 0.01 5 0.15 0.01 5 0.25Problem 10 0.05 5 0.25 0.01 5 0.25Problem 11 0.01 7 0.25 0.01 5 0.25

0

1

2

3

0 5

0 5

0 5


EY

1

Test problem 8Test problem 9Test problem 10Test problem 11

1

3

5


EK

1

0

5

10


Eµ

max

1

a

b

c

Fig. 9. Example 5, EPs versus sampling time interval; (a) EY1,s, (b) EK1,s,and (c) Elmax

1;s .


stable and only slightly dependent on perturbations in anyof six estimated kinetic parameters. With more time per-mitted between sampling events there is in turn greater dif-ferences in measured biomass and contaminant mass whichappears to improve specific yield estimates. Consistent withprevious observations (recall Fig. 3a), the error in Y1 esti-mates increases if kinetic parameters are perturbed in adirection that reduces the rate of contaminant attenuationper unit of active biomass; this is evident when lmax

1;s (prob-lem 9) is decreased, or Y1,s (problem 10) or K1,s (problem11) are increased.

Fig. 9b shows EK1,s oscillates and does not show a dis-tinct relationship with the magnitude of the sampling inter-val Dt. Maximum magnitudes of both EK1,s and EK2,s (notshown) are small (less than 5% in all cases). Furthermore,similar results were observed with EK1,s when a singlemicrobial species was present (Fig. 3b), Fig. 9b shows thatEK1,s increases with Y1,s (problem 10) and inversely withlmax

1;s (problem 9) and K1,s (problem 11). Any perturbationof the half saturation coefficients (K1,s or K2,s) has recipro-cal effect on the average error calculated for that parameterand a proportional effect on the error of the half saturationcoefficient of the competing microbe.

Finally, Fig. 9c displays the effect of changing the valueof the time interval on the average error of lmax

1;s . This figureshows a linear relationship between this error and samplinginterval for time increments Dt > 10 days. Consistent withwhat is observed from Fig. 3c, it is evident from Fig. 9cthat Elmax

1;s is proportional to the magnitude of lmax1;s (prob-

lem 9) and inversely proportional to the magnitude of Y1,s

(problem 10) and K1,s (problem 11). For problems 9, 10and 11, similar proportional relationships (almost of thesame magnitude) were noted for Elmax

2;s . The combinedeffect of varying all three parameters was greater thanthe effect of increasing lmax

1;s and decreasing K1,s alone inthe same problem. Both Elmax

1;s and Elmax2;s are confined to

a small range from less than 4% for intervals up to 15 daysand less than 10% overall.

5. Summary and conclusions

Using spatial moment analysis, scalable inverse func-tions were derived to estimate Monod kinetic parametersat the field scale. These inverse functions used spatialmoments that characterize the distribution of dissolvedcontaminant and active microbial biomass in the aquifer.Two general cases were examined. In the first Monodparameters were estimated where it was assumed a micro-bial population comprised of a single bacterial specieswas attenuating a single contaminant (e.g., an electrondonor or an electron acceptor). In a second case, contam-inant attenuation was attributed to a microbial consor-tium comprised of two bacterial species, and Monodparameters for both species were estimated. Eleven two-dimensional horizontal test problems were simulated withthe purpose of examining the performance of the inversefunctions; seven concerned the first case (this includestests involving heterogeneous aquifers), while the remain-ing four focused on the second case. Parameter estima-tion errors were generally small for K1,s, K2,s, Y1,s, andY2,s and greater for lmax

1;s and lmax2;s . Results indicate the

inverse method was only slightly sensitive to aquifer het-erogeneity and that estimation errors decreased as thesampling time interval decreased with respect to thegroundwater travel time between sample locations. Opti-mum conditions for applying the scalable inverse methodin both space and time were investigated assuming a finitesampling area.

Future research could include testing scalable inversefunctions in the field. However, care should be taken wheninterpreting results, because the analysis was predicated onmodels representing simple approximations of a complexenvironmental system. Scalable inverse equations couldbe extended to three dimensions given available contami-nant and biomass concentrations in the vertical, and thiswould not alter the number of terms or the general formof each equation. Inverse functions predicated on temporalmoments could be developed to estimate Monod parame-ters using transient data from a limited number of temporalobservation points. This type of transient inverse analysis


could become advantageous as continuous monitoringbecomes both feasible and cost-effective.

Acknowledgements

This research was funded by the Natural and Acceler-ated Bioremediation Research (NABIR) program, Biolog-ical and Environmental Research (BER), US Departmentof Energy: (Grant Number DE-FG02-97ER62471) andthe Florida Water Resources Research Center under agrant from the US Department of Interior (06HQGR0079).

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