A simplified numerical algorithm for oxygen- and nitrate-based biodegradation of hydrocarbons using...

Ž .Journal of Contaminant Hydrology 40 1999 53–77www.elsevier.comrlocaterjconhyd

A simplified numerical algorithm for oxygen- andnitrate-based biodegradation of hydrocarbons using

Monod expressions

Mamun Rashid, Jagath J. Kaluarachchi )

Department of CiÕil and EnÕironmental Engineering and Utah Water Research Laboratory, Utah StateUniÕersity, Logan, UT 84322-8200, USA

Received 18 August 1998; received in revised form 3 March 1999; accepted 17 March 1999

Abstract

Bioremediation has become an important remediation technology during the past few years.However, limited understanding of processes, data limitations and parameter uncertainty, andcomputationally intensive numerical solutions have reduced the interest to perform detailedfield-scale modeling, especially with non-linear Monod expressions. In this paper, the focus wascomputational efficiency where a simplified numerical algorithm was developed to solve thenon-linear reactions between the hydrocarbon, oxygen, nitrate, and a heterotrophic facultativemicrobial population. The numerical algorithm was presented using a commonly adapted concep-tual model of reaction kinetics between different species with multi-term Monod expressions. Theaccuracy and efficiency of the algorithm were tested with an iterative solution using theRunge–Kutta method. Simulation results obtained from reaction only and one-dimensionaladvection–dispersion-reaction transport examples showed good accuracy while maintaining nu-merical stability. In general, the accuracy increased with contaminant concentrations relativelyhigher than the corresponding half-saturation constant. Also, the simplified algorithm is found tobe much more computationally efficient than the iterative technique. The gain in computationalefficiency was at least threefold and this gain increased with the increase of time step. Future workwill explore the efficiency and accuracy of the proposed simplified algorithm with field-scalescenarios related to intrinsic and enhanced bioremediation. q 1999 Elsevier Science B.V. Allrights reserved.

Keywords: Groundwater; Biodegradation; Monod kinetics; Transport; Hydrocarbons

) Corresponding author. Tel.: q1-435-797-3918; Fax: q1-435-797-3663; E-mail: [email protected]

0169-7722r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-7722 99 00032-7

( )M. Rashid, J.J. KaluarachchirJournal of Contaminant Hydrology 40 1999 53–7754

1. Introduction

Modeling of biodegradation has gained popularity during the past decade for itsapplicability in bioremediation design. A wide variety of biodegradation models withfirst-order decay coefficients to complex Monod expressions has been developed inrecent years. However, the debate on how to best represent the biodegradation processes

Ž .in the subsurface is ongoing. According to Molz et al. 1986 , most biodegradationmodels fall into three distinct conceptual approaches. In the first approach, the works of

Ž . Ž . Ž .Rittmann et al. 1980 , Bouwer and McCarty 1984 , and Bouwer and Cobb 1987 arecharacterized by the biofilm theory where it is assumed that solid particles are uniformlycovered by a thin biofilm. This biofilm provides the necessary media for consumption of

Ž .the substrate in the presence of electron acceptor s and nutrients. In this approach, agood understanding of biofilm kinetics in groundwater is essential to understand the

Ž .different biodegradation processes McCarty et al., 1984 . The implementation ofbiofilm kinetics is more involved than Monod expression-based kinetics due to simulta-neous reaction and diffusion within the biofilm and external mass transport.

The second approach, as opposed to the biofilm theory, considers bacterial growth insmall discrete colonies or ‘microcolonies’ attached to the particle surface. Growthdynamics can be modeled as the result of substrate and electron acceptor utilizationŽ . Ž .Watson and Gardner, 1986 . Molz et al. 1986 proposed allowing the number of

Ž .microcolonies each having uniform and constant dimensions per unit volume ofaquifer to increase or decrease depending on the substrate and electron acceptorutilization rates. Biofilm and microcolony approaches have not been applied to a field

Ž .site Bedient et al., 1994 . Both biofilm and microcolony models utilize one of thefollowing three reaction kinetics: first-order decay, instantaneous reaction kinetics, andmulti-term Monod expressions.

The third approach deviates from ‘biofilm’ or ‘microcolony’ theories completely andmakes no assumption as to the microscopic configuration and distribution of the porespaces or the biological media. This macroscopic approach, as opposed to the micro-scopic approach employed in the biofilm and microcolony models, has been adopted by

Ž .soil microbiologists over the last few decades Bazin et al., 1976 , as well as byŽ . Ž .Corapcioglu and Haridas 1984; 1985 , Borden and Bedient 1986 , and Celia and

Ž .Kindred 1987 . In this approach, the assumption of partitioning between free flowingŽ .and adsorbed microorganisms is made Corapcioglu and Haridas, 1984, 1985 , but their

Ždistribution and interaction play no role in depicting the growth dynamics Baveye and.Valocchi, 1989 . In most cases of this approach, Monod expression is assumed, as

Monod function makes no assumption on the distribution of the microorganism popula-Ž .tion within the pore space Odencrantz et al., 1990 .

Most biodegradation models developed to date are based on numerical approaches.Analytical models describing biodegradation kinetics are scarce because of the complex-

Ž .ity of the resulting non-linear and coupled partial differential equations PDEs . Existinganalytical transport models treat biodegradation as first-order kinetics for simplicity. For

Ž .example, Domenico 1987 developed an analytical model for multi-dimensional trans-port of a decaying contaminant using a first-order decay coefficient. On the other hand,numerical models employ simple instantaneous reaction kinetics to complex multiple-

( )M. Rashid, J.J. KaluarachchirJournal of Contaminant Hydrology 40 1999 53–77 55

term Monod functions. In numerical models using Monod functions, the solutionschemes are complex and require enormous computational effort.

Ž .The kinetic models, commonly utilized in biological decay, include the following: aŽ . Ž .zero- and first-order reaction kinetics; b instantaneous reaction kinetics; and c Monod

or Michaelis–Menten kinetics. In zero-order reactions, the rate of transformation of anorganic constituent is unaffected by changes in the constituent concentration because the

Žreaction rate is determined by factors other than the constituent concentration Sims et.al., 1989 . Zero-order decay rate is simple and easy to use. However, as this reaction

does not consider the changes in the constituent concentration, the use of zero-orderreaction kinetics is limited.

First-order reaction kinetics represents an exponential decay and is the simplest formto use for representing biological decay. However, there are difficulties associated withthe use of first-order reaction kinetics. First-order degradation rates may be difficult tofind in the literature for specific field scenarios. Field conditions are site-specific andhighly variable, and first-order degradation rates measured in field conditions sometimesinclude the effects of dilution and sorption. Also, the use of a single degradation rate foran entire aquifer without considering the hydrogeological and biochemical conditionsmay not reflect the field conditions accurately. Sometimes, laboratory-measured first-order degradation rates are used in modeling, but extrapolation of laboratory data to fieldconditions is far from state of the art. Another major limitation is that first-order decaydoes not distinguish between aerobic and anaerobic conditions and, therefore, the resultsmay produce incorrect degradation values.

When microbial parameters of the subsurface have little or no effect on thecontaminant distribution due to high rates of microbial growth relative to the groundwa-

Ž .ter flow Borden et al., 1984 , contaminant degradation by microorganisms can beapproximated by an instantaneous reaction between the electron acceptor and thecontaminant. The instantaneous reaction model assumes that the reaction between thecontaminant and the electron acceptor mediated by the microorganisms is fast or almostinstantaneous. However, a number of difficulties exist with the use of instantaneousreaction kinetics as well. Instantaneous consumption of the contaminant may not besupported by existing hydrogeological conditions. Also, some contaminants can be highto moderately biodegradable while others may be completely non-biodegradable, butinstantaneous reaction kinetics does not have a mechanism to account for thesedifferences. Therefore, the use of instantaneous reaction kinetics is limited to highly

Ž .degradable contaminants with large Damkohler numbers Rifai and Bedient, 1990 . The¨Damkohler number is defined as m LrV, where m is the maximum contaminant¨ max max

utilization rate per unit microbial mass; L is the characteristic length of the domain; andV is the pore-water velocity. Otherwise, instantaneous reaction kinetics can underpredictthe concentration profile by producing high biodegradation rates. In addition, theinstantaneous reaction may not be applicable to all electron acceptors. However,oxygen-based biodegradation is relatively fast as indicated by Borden and BedientŽ .1986 and, therefore, the instantaneous reaction model may be practical. The instanta-neous model may also be applicable for nitrate for two reasons: first, nitrate has a redoxpotential of q740 mV at pHs7 and Ts258C which is much closer to oxygen atq820 mV than any other electron acceptor; also, oxygen and nitrate are thermodynami-


Ž .cally equally energetic Newell et al., 1995 . Second, the same microorganism popula-tion responsible for aerobic metabolism can switch to nitrate or another form of oxidized

Žnitrogen under anaerobasis or low to zero oxygen concentrations Widdowson et al.,.1988 .

One advantage of Monod function-based kinetics is minimizing the number of modelŽparameters needed to characterize the problem Borden et al., 1984; Borden and Bedient,

.1986 . The multiple-term Monod expression is commonly used when it is unknownŽwhich of the species i.e., substrate, electron acceptor, or nutrient such as assimilatory

.nitrogen, or all three simultaneously is rate-limiting to avoid unnecessary analysis in thenumerical solution to find the limiting species. The Monod function contains first-order,mixed-order, and zero-order regions. When the substrate concentration far exceeds thehalf-saturation constant of the substrate, the Monod function reduces to a zero-orderdecay. On the other hand, when the half-saturation constant of the substrate far exceedsthe substrate concentration, the Monod function simplifies to a first-order decay. If theMonod function cannot be simplified to either a first- or a zero-order reaction, then thecontaminant advection–dispersion mass transport equation is non-linear due to the

Ž .coupling with the growth dynamics of the microorganisms MacQuarrie et al., 1990 .Ž .A simple comparison scheme by Bedient et al. 1994 showed that Monod function is

the most conservative followed by the instantaneous reaction model and the first-orderdecay model. The first-order decay model can underpredict the concentration becausethe amount of electron acceptor available may not be stoichiometrically sufficient tosupport the predicted reduction. Therefore, it is important to recognize that the first-orderexpression does not incorporate the electron acceptor limitation, and thus care should be

Ž .taken when using this expression. Bekins et al. 1997 compared zero- and first-orderapproximations to the Monod function-based kinetic model. The results suggested thatfor substrate concentrations above the half-saturation constant, the first-order model maynot be valid and either the zero-order model or the Monod model should be used. It wasalso suggested that Monod function-based kinetics are capable of capturing both zero-and first-order regions of the biodegradation process.

As discussed earlier, Monod expressions have certain advantages over the other twomodels. First, Monod expressions take into account the field, contaminant, and biophaseconditions and, therefore, more closely reflect the biodegradation process. Second,Monod expressions are generic in nature because it can be used to model most organiccontaminants with any suitable electron acceptor by varying appropriate biochemicalparameters. Third, Monod function-based models are easily calibrated due to theflexibility introduced via the biokinetic parameters.

With its superiority over the instantaneous, first-, and zero-order reaction kinetics,Monod functions also have disadvantages. Use of Monod function originated from thewastewater industry within the context of a batch reactor model where complete mixingcan be assumed. On the other hand, the application of Monod function in groundwatercan be limited by diffusion and mixing of the species. Also, numerous forms of Monodexpressions are available in the literature and, therefore, care should be taken in

Ž .selecting the form that is suitable for a particular application Widdowson et al., 1988 .Monod expressions require a substantial amount of data which may not be available infield cases due to cost considerations. Since the data are scarce, the use of Monod


function-based kinetics in field cases, especially under heterogeneous conditions, maynot be feasible.

In addition to the data limitation, the use of Monod function is computationallyintensive. Utilization of one electron acceptor such as oxygen involves solving threecoupled, non-linear PDEs corresponding to the contaminant, electron acceptor, andmicroorganism. These equations require an iterative solution technique that usuallydemands high computational resources. The problem can be complex by orders ofmagnitude if additional electron acceptors are simulated. When this degree of complex-ity is used in solving large-scale field problems, the computational resources can be toodemanding and in most cases, the remediation designs are performed in the absence ofdetailed modeling. Therefore, both data limitation and computational resources makeMonod function-based kinetics in field-scale modeling unattractive to most designers.

In light of the above discussion, it is evident that multi-term Monod functions are animportant and valid representation of biological decay under a variety of geological andbiochemical conditions. Therefore, an efficient numerical algorithm that can solve thenon-linear, coupled Monod functions accurately can provide a contribution in field-scaleapplications of biodegradation models. The objective of this paper is to describe thedevelopment of a computationally efficient and accurate solution scheme for thesecomplex Monod expressions in biodegradation and to demonstrate the applicability. Thismanuscript will describe the development of the proposed simplified algorithm for thereaction model for contaminant, oxygen, nitrate, and microorganisms using a macro-scopic-scale conceptual model and applications to reaction only and one-dimensionaladvection–dispersion-reaction scenarios.

2. Theoretical derivations

As mentioned earlier, this manuscript will focus on the development of the numericalalgorithm, hereafter called the ‘simplified algorithm,’ for the reaction equations only.The approach taken in developing the simplified algorithm was first to study thequalitative behavior of Monod expression-based reaction kinetics and second, to imple-ment the behavior using empirically derived numerical links that eventually assemblethe complete algorithm.

In general, biodegradation of an organic contaminant in the subsurface consists of aseries of complex biochemical and physical processes that is not yet completelyunderstood. Common problems associated with the analysis of multiple-speciesbiodegradation include limited understanding of dominant processes, limited knowledgeon field-scale biokinetic parameters, uncertainty and spatial variability of parameters,and complex solution schemes requiring substantial computational efforts. In light ofthese difficulties, this manuscript will focus on one of those key issues, mainly thecomputational efficiency. Therefore, the proposed conceptual model will be based in

Žline with the work of previous researchers Corapcioglu and Haridas, 1984, 1985;.Borden and Bedient, 1986; Celia and Kindred, 1987 using a commonly adopted

macroscopic-scale conceptual model subject to a number of assumptions. In thisconceptual approach, the actual distribution of the organism population in the subsurfaceis not considered, and therefore, the mass transfer limitations including diffusion-limited


transfer to the biophase are ignored. As the typical solution approach taken in thisŽ .conceptual model is the iterative operator-splitting technique OST , the reaction equa-

Ž .tions are solved as a set of coupled, non-linear ordinary differential equations ODEs .Therefore, this manuscript will focus on the solution to the system of ODEs which, inprinciple, are equal to a batch reactor model in a physical system.

2.1. Assumptions

The following assumptions are made in the development of the coupled biodegrada-tion equations.

Ž .1 Sequential electron acceptor utilization during substrate consumption. After theŽ .oxygen concentration reaches a threshold concentration O , the electron acceptorT

switches from oxygen to nitrate. After the switch to nitrate, the use of oxygen ceases. Aslight modification of the simplified algorithm can accommodate a flexible switching

Ž .function such as the one proposed by Kinzelbach et al. 1991 where both oxygen andŽnitrate are used in the neighborhood of O with oxygen at a decreasing rate and nitrateT

.at an increasing rate during the transition period. Thereafter, the use of oxygen ceasescompletely and the use of nitrate takes full control.

Ž .2 There is no lag period between switching from oxygen to nitrate.Ž .3 Dead microbial population is not used, and there is no nutrient limitation.Ž .4 Microbial population responsible for aerobic- and nitrate-based respiration is

heterotrophic, facultative bacteria. Under this assumption, microorganisms responsiblefor aerobic respiration can switch to nitrate respiration under favorable conditions.

Ž . Ž .5 Metabolism is controlled by the lack of the substrate or the electron acceptor s .Ž .6 No limitations due to mixing or diffusion of species, or bioavailability.

2.2. GoÕerning equations of reaction using Monod expressions

Based on the earlier discussion on the conceptual model, the coupled, non-linearODEs describing the reactions between the contaminant, oxygen, nitrate, and microbialpopulation using multi-term Monod expressions can be written as follows:

dC m C O m C NO Nsy X 1yS O y XPS O ,Ž . Ž .Ž .

d t Y K qC K qO Y K qC K qNO CO O N CN N

1Ž .dO m C OO

sy X 1yS O F , 2Ž . Ž .Ž . Od t Y K qC K qOO CO O

dN m C NNsy XPS O PF , 3Ž . Ž .Nd t Y K qC K qNN CN N

d X C Osm X 1yS OŽ .Ž .Od t K qC K qOCO O

C Nqm XPS O yl X , 4Ž . Ž .N K qC K qNCN N


w x w y3 xwhere t is the time T ; C is the dissolved contaminant concentration ML ; O is thew y3 x w y3 xdissolved oxygen concentration ML ; N is the dissolved nitrate concentration ML ;

w y3 x Ž .X is the total microbial concentration ML ; S O is the oxygen-dependent switchingfunction; O is the threshold oxygen concentration for switching from oxygen to nitrateTw y3 x w y1 xML ; m is the maximum microbial growth rate for aerobic metabolism T ; m isO N

w y1 xthe maximum microbial growth rate for denitrification T ; l is the first-orderw y1 xmicrobial decay coefficient T ; Y is the yield coefficient for aerobic metabolismO

w y1 x w y1 xMM ; Y is the yield coefficient for denitrification MM ; K is the half-satura-N COw y3 xtion constant of the contaminant for aerobic metabolism ML ; K is the half-satura-CN

w y3 xtion constant of the contaminant for denitrification ML ; K is the half-saturationOw y3 xconstant of oxygen for aerobic metabolism ML ; K is the half-saturation constantN

w y3 xof nitrate for denitrification ML ; F is the mass ratio of oxygen consumed per unitOŽ .mass of contaminant consumed without allowance for cell synthesis ; and F is theN

Žmass ratio of nitrate consumed per unit mass of contaminant consumed without.allowance for cell synthesis .Ž .The switching function S O is defined as:

S O s0 for O)OŽ . T. 5Ž .s1 for OFOT

In the subsequent discussion related to the simplified algorithm, there is no restriction tospecify a zero for O , if needed, whereas common values are in the range of 0.2 mgrl.T

In the case of nitrate, there is no threshold concentration and consumption can occuruntil all nitrate are exhausted from the system.

Ž . Ž .Eqs. 1 – 5 describe the reaction model for contaminant, oxygen, nitrate, andmicrobes. These four non-linear, coupled ODEs can be solved simultaneously using theRunge–Kutta method where appropriate Fortran routines are available from the IMSLLibrary. This solution scheme can produce the correct solution to the multi-term Monodexpressions, hereafter referred to as the ‘exact solution.’ Due to the iterative nature ofthe solution scheme, the non-linear, coupled ODEs require substantial computationaleffort, especially when linked with multi-dimensional advection–dispersion models. Thesimplified algorithm described in Section 2.3 will solve the same system of ODEsefficiently and to a comparable accuracy to overcome the previously described computa-tional burden.

2.3. Simplified algorithm for solution of reaction terms

The proposed algorithm is applied at each time step of the numerical solution. Inorder to evaluate the consumption pattern over a given time step, several initialcalculations are performed. These computations are to determine suitable linear interpo-lations to non-linear Monod expressions. The preliminary tasks associated with thesimplified algorithm are described as follows.

Ž .1 Calculate the amount of contaminant that can be degraded by the first availableelectron acceptor. In the proposed model, oxygen is the first electron acceptor. In caseno oxygen is available at a given time, then nitrate will be the first available electronacceptor and all calculations will pertain to nitrate.


Ž .2 Predict the time required to consume the available electron acceptor by linearapproximation. Since the algorithm is developed empirically, this step requires severaladjustments and these adjustments will be discussed later.

Ž .Fig. 1. Schematic plots of concentration vs. time using Monod expressions: a Case A where O -6 or1Ž .C - K q1 and b Case B where O G6 and C G K q1.1 CO 1 1 CO


Ž .3 Calculate the new microbial concentration by linear approximation.Ž .4 Knowing the initial and final C, O, and X concentrations, and the time

Ž .approximated in Step 2, the rates of consumption or growth for X for each species canbe calculated. These slopes can be used to find the concentration of any given species atany time.

Ž .5 Next, the time step predicted in Step 2 will be compared with the stipulated timestep of the numerical solution. Depending on the difference between the two time steps,new concentrations of the species will be computed using the slopes computed in Step 4andror another cycle of computations from Steps 1 through 5 will be commenced.

In the next paragraphs, the simplified algorithm in relation to Steps 1 through 5 willbe discussed using Fig. 1, where it shows the decay of a contaminant during biodegrada-tion as a function of time. The algorithm is discussed using oxygen as the electronacceptor. As mentioned in the previous paragraph, once the use of oxygen is accom-plished in a given time step and, if time remains, then use of the next electron acceptor,in this case nitrate, will follow. The calculations with nitrate will be the same asdescribed below for oxygen except that oxygen has to be replaced by nitrate; i.e., all

Ž .biokinetic parameters i.e., K , m , Y , F , etc. pertaining to oxygen will be replacedCO O O O

by biokinetic parameters of nitrate.The proposed simplified algorithm branches into two major sections based on the

contaminant and electron acceptor concentrations, and the half-saturation constant of thecontaminant. The branching is due to the fact that biodegradation rates can vary fromslow to fast, depending on the existing conditions. The two conditions separating thedifferent biodegradation rates were obtained empirically using the observations derivedfrom a series of exact simulations under a variety of conditions.

Fig. 1a and b show the concentration distribution of the contaminant with time for theproposed branches. Here at time T , the known concentrations of contaminant, oxygen,1

and microbes are C , O , and X , respectively. The goal will be to solve the correct1 1 1

concentrations at time T sT qD t where D t is the time step of the numericalnum 1 num numŽ .solution. The time T qD t may fall either before or after the time T sT qD t1 num 2 1 st

given in Fig. 1 and the simplified algorithm needs to find the correct solution subject tothis condition. Here, D t is the time required to consume all of the first availablest

electron acceptor. In order to achieve this goal, the simplified algorithm seeks a linearapproximation to the non-linear Monod expressions.

Now the details related to these two limiting conditions are described in Sections2.3.1 and 2.3.2 and the units of concentration are in milligrams per liter.

2.3.1. Case A: O -6 or C -K q11 1 CO

Through our detailed simulations for a variety of conditions, it was observed that forrelatively low oxygen andror contaminant concentrations satisfying O -6 or C -1 1

Ž .K q1, the time required to consume O –O can be approximated reasonablyCO 1 T

accurate. First, the amount of contaminant concentration, DC , that can be stoichio-st

metrically degraded by oxygen, is calculated by the following expression:

O yO1 TDC s . 6Ž .st FO


Second, the time required to consume DC , defined as D t , can be approximatedst stŽ .through the rearrangement of Eq. 1 as follows:

DCstD t s 7Ž .st m C O yO XŽ .O 1 1 T 1

Y K qC K q O yOŽ . Ž .Ž .O CO 1 O 1 T

In Fig. 1a, these values correspond to point 2 where C sC yDC and T sT qD t .2 1 st 2 1 stŽ .It was found from detailed numerical simulations that D t in Eq. 7 may overestimatest

Ž .the actual time required to decay DC in Eq. 6 and, therefore, D t needs to best st

adjusted. The detailed numerical simulations indicated that this overestimation can becorrected by using a correction factor corresponding to 67% of the computed D t .st

Although this correction factor was obtained through careful analysis of results obtainedfrom a series of simulations, there may be some dependence of this factor on theparameters. However, we will use this value of 67% irrespective of this possibility as itproduced acceptable results under a variety of reasonable scenarios and parametervalues.

If C -K q1 and DC )C , then D t needs to be calculated alternatively as:1 CO st 1 st

C1D t s . 8Ž .st m C O yO XŽ .O 1 1 T 1


Ž .The new microbial concentration corresponding to T , which is denoted as X qD X ,2 1 st

can be linearly approximated by using the following equation for D X :st

D X sY PDC yl X PD t , 9Ž .st O st 1 st

Ž . Ž .which was obtained by manipulating Eqs. 1 and 4 and substituting dCsyDC ,st

d tsD t , and d XsD X .st st

Now that the initial concentrations of species, C , O , X , and the concentrations1 1 1

after time T are known, the consumption or production rates can be calculated.2

Referring to Fig. 1a, the slope of the line through points 1 and 2 for contaminant,oxygen, and microbes, b , b , and b , respectively, are:C O X

DCstb sy , 10aŽ .C

D tst

b sb F , 10bŽ .O C O

D Xstb s . 10cŽ .X

D tst

Ž . Ž . Ž .Now, knowing the slopes given by Eqs. 10a , 10b and 10c , it is possible to computethe concentrations of the three species at any other given time between T and T .1 2

At this stage, it is important to compare the new time corresponding to the numericalŽ . Ž .solution, which is T qD t with T qD t . In the perfect scenario, D t sD t1 num 1 st num st

and the solution can proceed to the next time step. Otherwise, there are two possibilitiescorresponding to D t )D t or D t -D t , and each case will be discussed innum st num st

Sections 2.3.1.1 and 2.3.1.2.


2.3.1.1. Dt )Dt . In this case, the numerical time step is greater than the timenum st

required to consume the available electron acceptor, and therefore, there will beadditional time left from the numerical time step. Also, the first electron acceptor will becompletely consumed; up to O in the case of oxygen or zero for nitrate. The additionalT

Ž .time of D t yD t will be used by the next available electron acceptor, and the samenum st

procedure is repeated again with the next electron acceptor for a numerical time step ofŽ .D t yD t ; e.g., if the first electron acceptor was oxygen, then nitrate consumptionnum st

Ž .will commence for a period of D t yD t or if the first electron acceptor was nitratenum st

due to previous depletion of oxygen, then no more degradation will occur.

2.3.1.2. Dt -Dt . In this case, the specified numerical time step is inadequate fornum st

complete consumption of the first available electron acceptor and therefore, there will beŽ .partial decay of the contaminant. In this case, the slopes computed through Eqs. 10a ,

Ž . Ž .10b and 10c will be used to compute the concentrations of the three speciescorresponding to the time T qD t .1 num

2.3.2. Case B: O G6 and C GK q11 1 CO

In this case, oxygen and contaminant concentrations will be high. However, it shouldbe noted that oxygen or contaminant concentrations cannot be too high to causebreakdown of biodegradation; e.g., high concentrations of contaminant can be toxic oroxygen concentrations cannot exceed commonly supplied values during enhancedremediation. Although there is no restriction in the proposed algorithm, such high valueswill not be sustained in actual field conditions.

Fig. 1b shows the qualitative behavior of Case B which usually falls below the curveof Case A due to rapid depletion of the contaminant. Also, the curvature of thecontaminant decay curve given in Fig. 1b is opposite to that of Fig. 1a. When O G61

and C GK q1, the contaminant can decay fast and the linear approximation of time1 COŽ . Ž .D t by Eq. 7 or Eq. 8 is typically greater than the actual time required. Therefore, anst

empirical oxygen concentration, O , is defined as the concentration corresponding to theC

time, T , to calculate the actual time accurately. The motivation for this approach is that2

if this empirical oxygen concentration is used in calculating the time D t , a closest

approximation to the actual time can be obtained.A series of simulations was performed with different oxygen concentrations, and the

times to completely degrade oxygen were observed. As mentioned earlier, the timerequired for degradation was found to be smaller than time D t . Knowing this smallerst

time, an empirical oxygen concentration, g , where later g will be modified to obtainO , is calculated. The relationship between the actual oxygen concentration and theC

empirical oxygen concentration, thus calculated, is observed for a wide range ofcontaminant and oxygen concentrations and half-saturation constants. A third-degreepolynomial is then used to describe the relationship as follows:

gsy5.0=10y6 O3 q0.0027 O2 q0.4336 O y2.4704. 11Ž .1 1 1

Ž .Eq. 11 is valid for oxygen concentrations much greater than typical oxygen concentra-tions found in groundwater, thereby making it useful even for enhanced bioremediation


Table 1Ž . Ž .Adjustments of g to obtain O in Eqs. 12 – 14C

Condition OC

Ž .1 C G K q61 CO

O G8.0 g1Ž .7.5FO -8.0 5r7 g1Ž .7.0FO -7.5 4r7 g1Ž .6.5FO -7.0 3r7 g1Ž .6.0FO -6.5 2r7 g1

Ž . Ž .2 K q5FC - K q6 5r7 gCO 1 COŽ . Ž .3 K q4FC - K q5 4r7 gCO 1 COŽ . Ž .4 K q3FC - K q4 3r7 gCO 1 COŽ . Ž .5 K q2FC - K q3 2r7 gCO 1 COŽ . Ž .6 K q1FC - K q2 1r7 gCO 1 CO

simulations. The value of g , thus obtained, depends only on the initial value of oxygen.A slight modification to the value of g is needed to obtain the value of O whichC

depends both on C and O . Again, this modification was obtained through a systematic1 1

observation of a series of simulations. First, if C GK q6, then g is adjusted by a1 CO

factor based on the O value to obtain O . Second, for K q1FC -K q6, g is1 C CO 1 CO

adjusted based on comparison of C with the K . Table 1 shows the adjustments of g1 CO

required to obtain O .C

Once O is obtained, the remaining steps are similar to that of Case A where O isC C

replaced by O if O -O . Based on the newly found O , D t can be calculated asT C T C st

follows:

O yO1 CDC s , 12Ž .C FO

DCCD t s . 13Ž .st m C O yO XŽ .O 1 1 C 1

Y K qC K q O yOŽ . Ž .Ž .O CO 1 O 1 C

If C )K q6 and DC )C , then D t needs to be calculated alternatively as:1 CO C 1 st

C1D t s . 14Ž .st m C O yO XŽ .O 1 1 C 1

Y K qC K q O yOŽ . Ž .Ž .O CO 1 O 1 C

Now, the actual consumption of contaminant, DC , and the time TU can be given as:st 2

O yO1 TDC s , 15aŽ .st FO

DCstUT yT s . 15bŽ .2 1 m C O yO XŽ .O 1 1 T 1



The new microbial concentration, X qD X , corresponding to T and DC can be1 st 2 st

linearly approximated using the following equation for D X :st

D X sY PDC yl X PD t . 16Ž .st O st 1 st

Ž . Ž . Ž .As in Case A, Eq. 16 was obtained using Eqs. 1 and 4 . The rates of consumption orproduction can be computed as in Case A using the following equations:

DCstb sy , 17aŽ .C

D tst

b sb F , 17bŽ .O C O

D Xstb s . 17cŽ .X

D tst

Now the remaining computations such as comparison of the numerical time step withD t will be the same as in Case A, and the same approach will be utilized to update thest

concentrations at the new time of T qD t .1 num

It should be noted that the simplified algorithm has two main branches based on thecontaminant and electron acceptor concentrations and on the half-saturation constant ofthe contaminant, and is not influenced by the half-saturation constant of the electronacceptor. Finally, a Fortran 77 program written to implement this numerical algorithm isavailable from the authors and can be linked with any advection–dispersion transportmodel, as discussed in the second set of simulations.

3. Application of the simplified algorithm

The first set of simulations will evaluate the applicability of the simplified algorithmon an organic contaminant undergoing decay due to reaction only. The second set ofsimulations will include the reaction model described earlier with one-dimensionaladvection–dispersion mass transport.

3.1. Examples with reaction only

Three hypothetical examples were selected to demonstrate the applicability of thesimplified algorithm. The proposed simplified algorithm is independent of the organiccontaminant. However, since the model is proposed for both oxygen- and nitrate-basedbiodegradation, toluene was selected as the representative contaminant. Previous experi-mental and field studies have shown that benzene is not biodegradable by nitrate,

Žwhereas toluene is biodegradable by both oxygen and nitrate Arcangeli and Arvin,.1994; Hutchins and Wilson, 1994; Jensen and Arvin, 1994 . Stoichiometric ratios, FO

and F , are calculated by simple stoichiometry or balanced equations, with no account-N

ing for cell synthesis, as follows.Toluene oxidationraerobic respiration:

9O qC H ´7CO q4H O. 18aŽ .2 7 8 2,g 2


Toluene oxidationrdenitrification:

7NOy q6HqqC H ´7CO q7H Oq3.5N . 18bŽ .3 7 8 2,g 2 2,g

For a detailed discussion on balanced equations with cell synthesis, the reader can referŽ .to McCarty 1975 .

As mentioned earlier, the main focus of this work is to develop a simplifiednumerical algorithm for Monod expression-based biodegradation and to demonstrate theapplicability of the algorithm. The conceptual model used here was the same as themodel used by other researchers for the past decade or more. Similarly, the biokineticparameters used here will be almost the same as those used by others in previousstudies. It is well-understood that these biokinetic parameters can be difficult to obtainand are typically subjected to uncertainty and spatial variability. However, in this work,

Žwe will use a set of parameters adopted from previous studies Kinzelbach et al., 1991;.Bedient et al., 1994 in order to maintain consistency for future comparisons. The values

of K , K , K , and K are 8, 8, 0.2, and 0.2 mgrl, respectively. The yieldCO CN O N

coefficients, Y and Y , are 0.09 and 0.1, respectively. The growth and decay rates, m ,O N O

m , and l are 2.3148=10y5, 2.3148=10y5, and 2.3148=10y6 sy1, respectively.N

The threshold oxygen concentration, O , is fixed at 0.2 mgrl, and the initial uniformT

microbial concentration of 0.5 mgrl is assumed.

3.1.1. Example simulationsThree reaction only simulations are performed to evaluate the applicability of the

simplified algorithm. As discussed previously, the simplified algorithm branches intotwo sections based on the contaminant and electron acceptor concentrations, and on thehalf-saturation constant of the contaminant. Therefore, the range of concentrations of thecontaminant, electron acceptor, and microbes for the three simulations is chosen to coverthe different branches of the simplified algorithm and to show the applicability of thealgorithm. In addition, these examples are based on reaction kinetics only and, therefore,may not be suitable to show the efficiency of the simplified algorithm. However, thesecond set of simulations will demonstrate the attractive computational features and theapplicability. Therefore, these examples should be viewed as a demonstration of theaccuracy of the proposed algorithm in modeling complex Monod functions in oxygen-and nitrate-based biodegradation.

3.1.1.1. Case A when O -6 or C -K q1. This example was selected to demon-1 1 CO

strate the applicability of the proposed simplified algorithm to biokinetic parameterssatisfying Case A. As discussed earlier, this example refers to slower biodegradationwithin the context of the simplified algorithm. The initial concentrations of toluene,oxygen, nitrate, and microbes used were 20, 2, 20, and 0.5 mgrl, respectively. Thesimulation time was 2 days with a time step of 0.02 days.

The computational time for the simplified algorithm was approximately 1 s, whereasthe corresponding value for the exact solution was 12 s. Fig. 2 shows the distribution oftoluene, oxygen, nitrate, and microbial population with time. The results show goodagreement for oxygen and an acceptable agreement for nitrate, contaminant, andmicrobes. In all four species, the simplified algorithm showed an earlier arrival or


Ž .Fig. 2. Toluene, oxygen, nitrate, and microbial concentrations for the exact solution solid line and simplifiedŽ . Ž .algorithm dashed line for Case A C s20; O s2; N s20; X s0.5; K s8 .1 1 1 1 CO

depletion of a given species compared to the concentrations produced by the exactsolution. The solution using the simplified algorithm managed to capture the keyfeatures of the biodegradation process. For example, as the oxygen depleted, the nitrateconsumption commenced and during these reactions, the growth of the microorganismsoccurred. After both electron acceptors were exhausted, the microorganisms underwentnatural decay and the contaminant concentration remained constant.

The second example in Case A used initial concentrations of toluene, oxygen, nitrate,and microbes to be 2, 8, 20, and 0.5 mgrl, respectively. This example was selected as aspecial problem of Case A where the contaminant concentration is much less than in theprevious example. It is important to consider problems with low contaminant concentra-tions since in most organic-contaminated sites, the groundwater concentration of organ-ics may be small in some locations. In such cases, it is of interest to evaluate theapplicability of this algorithm in the presence of low oxygen concentrations. Thesimulation time was 3 days with a time step of 0.03 days.

In this example too, the simulation with the simplified algorithm used approximately1 s while the exact solution used 17 s. The additional time taken by the exact solutioncompared to previous examples is due to the longer simulation time. Since the simplifiedalgorithm uses linear approximation between a few selected points of interest, the lengthof the simulation is somewhat irrelevant and, therefore, performs the simulations moreefficiently. In fact, this is the major contributing factor for the efficiency of thesimplified algorithm.

Fig. 3 shows the concentration distribution of all four species with time. In compari-son to the previous example, the match between the simplified and the exact solution is


Ž .Fig. 3. Toluene, oxygen, nitrate, and microbial concentrations for the exact solution solid line and simplifiedŽ . Ž .algorithm dashed line for Case A C s2; O s8; N s20; X s0.5; K s8 .1 1 1 1 CO

poor for toluene. However, the accuracy of the other three species is comparable to theprevious examples. The toluene and oxygen concentrations used in this exampleproduced a limited and a slow growth of microorganisms. At low substrate and electronacceptor concentrations, the growth of biophase is slow and the non-linearity of Monodfunctions increases at these concentration levels. However, the simplified algorithm usesa linear approximation which results in a deviation from the true concentration profile.Another important point to notice in Fig. 3 is that the degradation is so slow that theswitch to nitrate did not occur within the simulation time of 3 days. As the concentrationlevel decreases, the rate of electron acceptor consumption decreases and it takes a longertime before the electron acceptor concentration reaches the threshold concentration forthe switch to occur.

3.1.1.2. Case B when O G6 and C GK q1. In this example, the initial concentra-1 1 CO

tions of toluene, oxygen, nitrate, and microbes were 20, 8, 20, and 0.5 mgrl,respectively, and corresponded to Case B discussed in the development of the simplifiedalgorithm. The simulation time was 2 days with a time step of 0.02 days. The simulationwith the simplified algorithm used approximately 1 s while the exact solution simulationused 12 s.

Fig. 4 shows the concentration distribution of all four species with time. Thecomparisons between the simplified algorithm and the exact solution for oxygen andtoluene are relatively good. There are some differences in the profiles for nitrate andmicrobial concentrations between the two simulations, especially towards the latter


Ž .Fig. 4. Toluene, oxygen, nitrate, and microbial concentrations for the exact solution solid line and simplifiedŽ . Ž .algorithm dashed line for Case B C s20; O s8; N s20; X s0.5; K s8 .1 1 1 1 CO

stages. However, compared to the overall computational efficiency, it can be consideredthat the simplified algorithm managed to capture the key features of the processaccurately. For example, the switching from oxygen to nitrate, when the oxygenconcentration reached the threshold concentration, was evident, although a little earlierthan the exact solution. Also, the growth of microorganisms continued when the electronacceptors were present, but when the electron acceptors were depleted, the microorgan-ism concentration declined.

The differences between the two methods may have significant consequences. If theuser is only interested in the final concentrations of the species at the end of thesimulation, the results are perfect, provided that simulation time is longer than the timeduring which the discrepancies persist. This time is approximately equal to the timerequired to consume the electron acceptor to the threshold concentration. This time is

Ž .approximate as it is obtained through a linear approximation of Eq. 1 . Due to thenon-linearity of Monod expressions, the actual time can be slightly different than TU for2

a given problem with associated half-saturation constants. For example, TU is small for2

a low half-saturation constant due to faster degradation and vice versa. On the otherhand, if the interest is in the concentration profile for the entire length of the simulation,discrepancies between the two methods may exist.

In the second set of simulations where advection–dispersion mass transport is linkedwith reaction, the discrepancies between the two methods are found to be small,provided the transport time step is larger than DT . A detailed discussion related to thesest

discrepancies will be presented with the second set of simulations. Finally, Fig. 4 shows


that the discrepancy of the microbes may not disappear. The reason is due to the factthat once the terminal electron acceptor is consumed, the growth of microbes is haltedand the decay continues with the existing discrepancy.

From the discussion of these three examples, it is evident that the simplifiedalgorithm works slightly better for high contaminant and electron acceptor concentra-tions relative to the half-saturation constants. From the comparison of computationaltimes for the three examples, it is evident that the simplified algorithm is much fasterthan the iterative exact solution. The gain in simulation times will be even greater withincreases in the number of equations due to the increased number of species, number ofnodes, andror number of time steps.

3.2. One-dimensional adÕection–dispersion-reactiÕe transport of toluene

In order to perform this simulation, the proposed simplified model solving thenon-linear reaction module was incorporated into the flow and transport model, SUTRAŽ . ŽVoss, 1984 using the alternate OST Morshed, 1994; Morshed and Kaluarachchi,

.1995 . Although the SUTRA model is capable of handling two-dimensional field-scalescenarios, the simulations in this work will be focused on a one-dimensional exampleonly. The reason for this approach is to show the applicability of the simplified model ina consistent manner from a simple reaction-only scenario to a detailed one-dimensionalcase. If the results of these simulations showed both good accuracy and computationalefficiency, then future work will further demonstrate the applicability to field-scalescenarios using the SUTRA-based simulations.

The problem domain consisted of a 20-ft long one-dimensional column discretized to20 layers of 1-ft each. The discretization was such that both the Peclet and Courantnumbers were -1. The initial toluene concentration was 20 mgrl in the first 5 ft of thecolumn and zero elsewhere. The initial oxygen and nitrate concentrations were 8 and 20mgrl, respectively, everywhere. The flow and transport parameters used in the simula-tion were the same as in the previous simulations. The steady pore water velocity,dispersivity, and porosity were 0.72 ftrday, 1 ft, and 0.3, respectively. The inlettransport boundary condition corresponds to incoming uncontaminated water, and theoutlet boundary condition corresponds to zero dispersive mass flux for all species. Thesimulations were run for 5 days with a base case time step of 0.01 days. The initialuniform microbial concentration was 0.5 mgrl.

Table 2Different combinations of half-saturation constants used in the one-dimensional example

Ž .Parameter mgrl Case

1 2 3 4 5

K 20 10 4 2 0.2O

K 20 10 4 2 0.2N

K 200 100 50 30 8CO

K 200 100 50 30 8CN


3.2.1. AccuracyThe first goal of this simulation was to assess the accuracy of the simplified

algorithm in solving the reaction terms when coupled with the advection–dispersionequations. In order to assess the accuracy, a series of simulations using differentcombinations of half-saturation constants was performed. A detailed description of thedifferent cases used in the simulations is given in Table 2. The predicted tolueneconcentration at 5 days is shown in Fig. 5 for both the exact and simplified solutionswhen Monod expressions are used, and the instantaneous reaction solution is also shownin the figure. The results clearly show a good match between the exact and thesimplified solution over a wide range of half-saturation values. In addition, when thehalf-saturation values are small such as in Case 5, the degradation is fast and behavesclose to the instantaneous reaction kinetics. On the contrary, the decay is almostnegligible and acts close to pure advection and dispersion only when the half-saturationvalues are large such as in Case 1.

The sensitivity of the solution to m , m , Y , Y , and l was not explored due to theO N O N

linear dependence of these parameters to the final solution. However, an important pointto notice about l is that if l is excessively high, then microbial concentration mayreach zero in some parts of the domain and no more biodegradation can occur in futuretime steps. Since transport of microbes is not simulated here except reaction, there is nopossibility for the microbes to grow at such locations. In addition to high l values,biophase concentration at a point can also reach zero if D t is too large. In most cases,D t is restricted by the Courant number and also by the OST. However, if the microbialconcentration still reaches zero, biophase can be conserved by assigning a minimum

Ž .Fig. 5. Comparison of toluene concentration at 5 days between the exact solution solid line and the simplifiedŽ .dashed line algorithm solution for different cases given in Table 2 for one-dimensional example.


Ž . Ž . Ž . Ž .Fig. 6. Toluene C , oxygen O , nitrate N , and microbial X distributions at 5 days for one-dimensionalŽ . Ž .example: exact solution solid line and simplified algorithm dashed line .

microbial concentration for future growth as substrate and electron acceptor becomeavailable.

Fig. 6 shows the concentration of the remaining species, oxygen, nitrate, andmicrobes at 5 days for the base case simulation. Once again, the agreement between theexact solution and the simplified algorithm-based solution is excellent.

Ž . Ž . Ž . Ž .Fig. 7. Toluene C , oxygen O , nitrate N , and microbial X concentrations with time at a location 5 ftŽ . Ž .from the inlet of one-dimensional example: exact solution solid line and simplified algorithm dashed line .


Previous results demonstrated the accuracy of the proposed simplified algorithm. Inorder to further verify those results, a close-up analysis of the solution was conducted toevaluate whether qualitative behavior of biodegradation was captured by the algorithm.The results in Fig. 7 show the concentration breakthrough curve at a location 5 ft fromthe inlet boundary. The results clearly show good accuracy and as the oxygen concentra-tion reached the threshold value, nitrate consumption commenced rapidly. Up to thetime of complete consumption of nitrate, the microbial population showed rapid growthand the toluene concentration decreased. After the complete consumption of all electronacceptors, the microbial concentration decreased due to natural decay. A similarbehavior was seen along the column length as shown in Fig. 6. The results of Figs. 6 and7 clearly show that the simplified algorithm managed to capture the important featuresof the degradation process, including switching of electron acceptors from oxygen tonitrate. It should be noted that an oxygen-based step function was used as the switchingfunction here and, therefore, sharp changes in oxygen and nitrate concentrations areshown in Fig. 7. However, if the switching function was based on both oxygen andnitrate, then there would be a region where both electron acceptors would be usedsimultaneously.

3.2.2. Numerical stabilityIn order to evaluate the stability of the solution obtained from the simplified

algorithm, the base case simulation was repeated with different time steps. The tolueneconcentration at 5 days, obtained from the simplified algorithm, is shown in Fig. 8where it is seen that the overall solution is stable even with large time steps. Forexample, when the time step was increased by one order of magnitude to 0.1 days, the

Fig. 8. Effect of different time step sizes on predicted toluene concentration at 5 days for one-dimensionalexample using the simplified algorithm.


solution remained consistent and stable. However, if the time step was increased beyond0.1 days, then the accuracy of the solution decreased. The time step size of up to 0.1days produced reasonably small Peclet and Courant numbers, indicating that any errorintroduced in the solution is probably due to the time-lag error caused by OST. Theexact solution too, although not shown here, obtained from the Runge–Kutta methodremained stable and consistent for the same time step sizes.

3.2.3. Computational efficiencyThe goal of this work was to develop a computationally efficient numerical algorithm

capable of representing Monod function-based kinetics while maintaining good accu-racy. In order to assess the computational efficiency under a wide selection of inputparameters, a series of simulations was conducted using different combinations of K ,O

K , K , and K . The computational efforts associated with these simulations areN CO CN

summarized in Table 3. The results clearly show that the simplified algorithm used lessthan one-third the CPU time required by the exact solution while maintaining goodaccuracy as indicated in previous plots. Since these results included the simulation timeof the advection–dispersion part, the base case was run without the reaction to determinethe net time taken by the advection–dispersion solution. The results showed that thissimulation time was 0.25 min and, therefore, the burden on the overall solution isminimal. In order to evaluate the competitiveness of the instantaneous solution, the base

Ž .case was run using the instantaneous reaction kinetics as discussed by Rifai et al. 1988 .As expected, this solution provided the smallest computational effort as the solutiondoes not explicitly consider the reaction terms in the governing equations. Also, this

Table 3Comparison of simulation times between the different solution schemes and input parameters for theone-dimensional example using a Pentium PC

Ž . Ž . Ž .K and K mgrl K and K mgrl CPU time minCO CN O N

Exact Simplified

0.5 0.01 11 4.52 0.05 12.5 4.58 0.2 14.5 4.530 2 15.5 4.5100 10 16.5 4.5200 20 17.5 4.5Advection–dispersion only 0.25 0.25Instantaneous reaction 2.5 2.5

Effect of time step for base case

Ž . Ž .Time step days CPU time min

Exact Simplified

0.01 14.5 4.50.05 10.5 10.1 10 0.50.5 10 0.25


solution cannot be effectively compared with the proposed work as the presentation ofthe reaction terms is totally different between the instantaneous kinetics and Monodexpression-based kinetics.

In order to evaluate the impact of different time step sizes, the computational effortsused in base case simulations, shown in Fig. 8, were analyzed. The results are shown inTable 3 also. Here, it is noted that as the time step was increased by one order ofmagnitude, the simplified algorithm showed nearly a one order of magnitude reductionin computational time. However, the exact solution showed a small reduction ofapproximately one-third.

4. Summary and conclusions

The objective of the work was to develop a simplified numerical algorithm to solvethe non-linear Monod expressions resulting from oxygen- and nitrate-based biodegrada-tion of hydrocarbons. The reaction equations are non-linear and coupled ODEs. Theapproach used in the numerical algorithm was to develop equivalent linear approxima-tions to the potential decay of the contaminant based on the availability of the electron

Ž .acceptor s and the relevant stoichiometric ratio.Three example simulations from reaction only and one detailed example using

one-dimensional advection–dispersion-reaction mass transport were performed todemonstrate the efficiency and accuracy of the simplified algorithm. The algorithm wasfound to be computationally efficient compared to the Runge–Kutta method-basediterative solution technique. The accuracy was found to be acceptable to good over awide range of concentrations of contaminant and electron acceptors. It was also foundthat the simplified algorithm provides a better accuracy with high contaminant concen-tration relative to the half-saturation constant. The gain in computational efficiencywhile maintaining acceptable accuracy with the simplified algorithm demonstrates thatMonod expression can be utilized for a variety of real-world scenarios. For example,more than one electron acceptor can be simulated by successive use of the algorithm.Also, the simplified algorithm can be incorporated into the advection–dispersion masstransport equations to simulate mass transport with biological decay. Future work willexplore the efficiency and accuracy issue of the proposed simplified algorithm in greaterdetail in the context of field-scale scenarios.

References

Arcangeli, J.P., Arvin, E., 1994. Biodegradation of BTEX compounds in a biofilm system under nitrate-reduc-ing conditions. Hydrocarbon Bioremediation. Lewis Publishers, Boca Raton, FL.

Baveye, P., Valocchi, A., 1989. An evaluation of mathematical models of the transport of biologically reactingŽ .solutes in saturated soils and aquifers. Water Resources Research 25 6 , 1413–1421.

Bazin, M.J., Saunders, P.T., Prosser, J.I., 1976. Models of microbial interactions in the soil. CRC CriticalReview of Microbiology 4, 463–498.

Bedient, P.B., Rifai, H.S., Newell, C.J., 1994. Groundwater Contamination: Transport and Remediation.Prentice-Hall, Englewood Cliffs, NJ, pp. 207–249.

Bekins, B.A., Warren, E., Godsy, M.E., 1997. Comparing zero- and first-order approximations to the Monod


model. Proceedings of the Fourth International In Situ and On-Site Bioremediation Symposium, NewOrleans, LA, Battelle Press, Vol. 5, pp. 547–552.

Borden, R.C., Bedient, P.B., 1986. Transport of dissolved hydrocarbons influenced by oxygen-limitedŽ .biodegradation: 1. Theoretical Development. Water Resources Research 22 13 , 1973–1982.

Borden, R.C., Lee, M.D., Wilson, J.T., Ward, C.H., Bedient, P.B., 1984. Modeling the migration andbiodegradation of hydrocarbons derived from a wood-creosoting process waste. Proceedings of theNational Water Well AssociationrAmerican Petroleum Institute Conference on Petroleum Hydrocarbonsand Organic Chemicals in Groundwater: Prevention, Detection and Restoration, Houston, TX, pp.130–143.

Bouwer, E.J., Cobb, G.D., 1987. Modeling of biological processes in the subsurface. Water ScienceTechnology 19, 769–779.

Bouwer, E.J., McCarty, P.L., 1984. Modeling of trace organics biotransformation in the subsurface. GroundWater 22, 433–440.

Celia, M.A., Kindred, J.S., 1987. Numerical simulation of subsurface contaminant transport with multiplenutrient biodegradation. Proceedings of the International Conference on the Impact of Physiochemistry onthe Study, Design, and Optimization of Processes in Natural Porous Media, Presses University de Nancy,Nancy, France.

Corapcioglu, M.Y., Haridas, A., 1984. Transport and fate of microorganisms in porous media: a theoreticalinvestigation. Journal of Hydrology 72, 149–169.

Corapcioglu, M.Y., Haridas, A., 1985. Microbial transport in soils and groundwater: a numerical model.Advances in Water Resources 8, 188–200.

Domenico, P.A., 1987. An analytical model for multidimensional transport of a decaying contaminant species.Journal of Hydrology 91, 49–58.

Hutchins, S.R., Wilson, J.T., 1994. Nitrate-based bioremediation of petroleum-contaminated aquifer at ParkCity, Kansas: site characterization and treatability study. Hydrocarbon Bioremediation. Lewis Publishers,Boca Raton, FL.

Jensen, B.K., Arvin, E., 1994. Aromatic hydrocarbon degradation specifically of an enriched denitrifyingmixed culture. Hydrocarbon Bioremediation. Lewis Publishers, Boca Raton, FL.

Kinzelbach, W., Schafer, W., Herzer, J., 1991. Numerical modeling of natural and enhanced denitrificationŽ .processes in aquifers. Water Resources Research 27 6 , 1123–1135.

MacQuarrie, K.T.B., Sudicky, E.A., Frind, E.O., 1990. Simulation of biodegradable organic contaminants inŽ .groundwater: 1. Numerical formulation in principal directions. Water Resources Research 26 2 , 207–222.

McCarty, P.L., 1975. Stoichiometry of Biological Reactions. Progress in Water Technology, Vol. 7. Perga-mon, London, UK, pp. 157-172.

McCarty, P.L., Rittmann, B.E., Bouwer, E.J., 1984. Microbiological processes affecting chemical transforma-Ž .tions in groundwater. In: Bitton, G., Gerba, C.P. Eds. , Groundwater Pollution Microbiology. Wiley, New

York, pp. 89–115.Molz, F.J., Widdowson, M.A., Benefield, L.D., 1986. Simulation of microbial growth dynamics coupled to

Ž .nutrient and oxygen transport in porous media. Water Resources Research 22 8 , 1207–1216.Morshed, J., 1994. Critical assessment of the operator-splitting technique in solving the advection–dispersion-

reaction problems. MS Thesis, Utah State University, Logan, UT, 162 pp.Morshed, J., Kaluarachchi, J.J., 1995. Critical assessment of the operator-splitting technique in solving the

advection–dispersion-reaction equation: 2. Monod kinetics and coupled transport. Advances in WaterŽ .Resources 18 2 , 101–110.

Newell, C.J., Winters, J.A., Rifai, H.S., Miller, R.N., Gonzales, J., Wiedemeier, T.H., 1995. Modeling intrinsicremediation with multiple electron acceptors: results from seven sites. Proceedings of the PetroleumHydrocarbons and Organic Chemicals in Groundwater: Prevention, Detection and Restoration, NationalWater Well Association, Houston, TX, pp. 33–47.

Odencrantz, J.E., Valocchi, A.J., Rittmann, B.E., 1990. Modeling two-dimensional solute transport withdifferent biodegradation kinetics. Proceedings of the Petroleum Hydrocarbons and Organic Chemicals inGroundwater: Prevention, Detection and Restoration, National Water Well Association, Houston, TX, pp.355–368.

Rifai, H.S., Bedient, P.B., 1990. Comparison of biodegradation kinetics with an instantaneous reaction modelfor groundwater. Water Resources Research 26, 637–645.


Rifai, H.S., Bedient, P.B., Wilson, J.T., Miller, K.M., Armstrong, J.M., 1988. Biodegradation modeling atŽ .aviation fuel spill site. Journal of Environmental Engineering 114 5 , 1007–1029.

Rittmann, B.E., McCarty, P.L., Roberts, P.V., 1980. Trace-organics biodegradation in aquifer recharge.Ground Water 18, 236–243.

Sims, J.L., Sims, R.C., Matthews, J.E., 1989. Bioremediation of contaminated surface soils. R.S. KerrEnvironmental Research Laboratory, U.S. Environmental Protection Agency, Ada, OK.

Voss, C., 1984. Saturated–unsaturated transport: a finite element simulation model for saturated–unsaturated,fluid-density dependent groundwater flow with energy transport or chemically reactive single-speciessolute transport. Water Resources Investigation Report 84-4369, U.S. Geological Survey.

Watson, J.E., Gardner, W.R., 1986. A mechanistic model of bacterial colony growth response to substratesupply. A paper presented at the Chapman Conference on Microbial Processes in the Transport, Fate, andIn Situ Treatment of Subsurface Contaminants, Snowbird, UT.

Widdowson, M.A., Molz, F.J., Benefield, L.D., 1988. A numerical transport model for oxygen- andnitrate-based respiration linked to substrate and nutrient availability in porous media. Water Resources

Ž .Research 24 9 , 1553–1565.

A simplified numerical algorithm for oxygen- and nitrate-based biodegradation of hydrocarbons using...

Documents

Transcript of A simplified numerical algorithm for oxygen- and nitrate-based biodegradation of hydrocarbons using...