zero-order monod model for christensen and larsen method

4
Zero-Order Kinetics Model for the Christensen-Larsen Method for Fugitive Fuel Age Estimates by Yakov Galperin and Isaac R. Kaplan* Abstract Estimating time of a middle distillate fuel release in soil can be performed under certain restricting environmental condi- tions using the Christensen-Larsen method (CLM). This method is based on the linear correlation between the time since a diesel fuel release and the corresponding value of n-heptadecane to pristane ratio (n-C 17 /Pr) but requires knowledge of the initial ratio value. The empirical nature of this method does not, however, allow accounting for variance in the initial fuel ðn-C 0 17 Þ=ðPr 0 Þ value used by CLM. Based on the zero-order approximation of the Monod model, we have deduced a general- ized equation that can be used for estimating release ages of middle distillate fuels with different initial ðn-C 0 17 Þ=ðPr 0 Þ val- ues. When combined with other site-specific factors, this equation provides a useful tool for the time of release estimates. Introduction Because legal and financial responsibility for hydrocar- bon contamination is often decided based on the date of a fuel release, the importance of reliable tools for deter- mining residence time of fuel in the subsurface environ- ment is undisputable. One of the age-dating approaches uses a linear correlation between the time since occurrence of a diesel fuel release (in years) and the corresponding values of n-heptadecane (n-C 17 ) to pristane (Pr) ratio (Christensen and Larsen 1993), which can be approximated (Hurst and Schmidt 2005) as follows: T = 9:47 ðn-C 17 =PrÞþ 20:5 (1) The observed decrease in the n-C 17 /Pr ratio over a time period of approximately 20 years follows a well-estab- lished path of the preferential biodegradation of n-alkanes relative to isoalkanes (Atlas 1981; Kennicutt 1988; Glazer 1991). Statistical analysis of 13 data points led Christensen and Larsen (1993) to conclude that for a specified set of environmental conditions (e.g., geographic location, hydro- carbon concentration, sampling depth, and paved soil), the age of a diesel fuel spill in soil up to 20 years in age can be estimated with an accuracy of 62 years at a 95% confi- dence level. The applicability of this correlation for estimating the age of petroleum product releases has been extensively discussed in the scientific literature (Kaplan et al. 1996; Wade 2002; Stout et al. 2002; Kaplan 2003; Oudijk et al. 2006). This age-dating approach can be a useful tool but should be restricted to sites where a sudden, single release of fuel has occurred. A slow continuous fuel release or con- secutive fuel releases at the same area will invalidate the method (Christensen and Larsen 1993; Stout et al. 2002). Among other parameters affecting applicability of the age-dating approach are selected soil characteristics, site hydrology, temperature, moisture content, and availability of oxygen and nutrients (Singer and Finnerty 1984; Foght and Westlake 1987; Atlas and Bartha 1992; Yang et al. 1995; Coates et al. 1997; Kaplan et al. 1997). In some envi- ronments, these parameters will favor fuel preservation; in others, local environmental conditions will strongly pro- mote weathering of the fuel and cause it to degrade at a higher rate. In this case, n-C 17 will be completely de- pleted faster or slower than the 20.5 years predicted by Christensen-Larsen method (CLM). In most environmental investigations, the uncertainties in these parameters typi- cally allow for only semiquantitative estimating age of a fuel release. In addition to the site-specific environmental condi- tions, application of the CLM is confounded by uncertain- ties introduced by variability of the initial ðn-C 0 17 Þ=ðPr 0 Þ ratio. As stated in Christensen and Larsen (1993), the origi- nal linear regression is based on the low variability of this ratio in dispensed diesel fuels sold in northern Europe dur- ing the early 1970s through the early 1990s. They have not, however, attempted to extend applicability of the method to fuels from different countries and continents that exhibit Copyright ª 2008 The Author(s) Journal compilation ª 2008 National Ground Water Association. 94 Ground Water Monitoring & Remediation 28, no. 2/ Spring 2008/pages 94–97

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Page 1: Zero-Order Monod Model for Christensen and Larsen Method

Zero-Order Kinetics Model for theChristensen-Larsen Method for Fugitive

Fuel Age Estimatesby Yakov Galperin and Isaac R. Kaplan*

AbstractEstimating time of a middle distillate fuel release in soil can be performed under certain restricting environmental condi-

tions using the Christensen-Larsen method (CLM). This method is based on the linear correlation between the time sincea diesel fuel release and the corresponding value of n-heptadecane to pristane ratio (n-C17/Pr) but requires knowledge of theinitial ratio value. The empirical nature of this method does not, however, allow accounting for variance in the initial fuelðn-C017Þ=ðPr0Þ value used by CLM. Based on the zero-order approximation of the Monod model, we have deduced a general-ized equation that can be used for estimating release ages of middle distillate fuels with different initial ðn-C017Þ=ðPr0Þ val-ues. When combined with other site-specific factors, this equation provides a useful tool for the time of release estimates.

IntroductionBecause legal and financial responsibility for hydrocar-

bon contamination is often decided based on the date ofa fuel release, the importance of reliable tools for deter-mining residence time of fuel in the subsurface environ-ment is undisputable. One of the age-dating approachesuses a linear correlation between the time since occurrenceof a diesel fuel release (in years) and the correspondingvalues of n-heptadecane (n-C17) to pristane (Pr) ratio(Christensen and Larsen 1993), which can be approximated(Hurst and Schmidt 2005) as follows:

T = � 9:47 ðn-C17=PrÞ þ 20:5 (1)

The observed decrease in the n-C17/Pr ratio over a timeperiod of approximately 20 years follows a well-estab-lished path of the preferential biodegradation of n-alkanesrelative to isoalkanes (Atlas 1981; Kennicutt 1988; Glazer1991). Statistical analysis of 13 data points led Christensenand Larsen (1993) to conclude that for a specified set ofenvironmental conditions (e.g., geographic location, hydro-carbon concentration, sampling depth, and paved soil), theage of a diesel fuel spill in soil up to 20 years in age can beestimated with an accuracy of 62 years at a 95% confi-dence level.

The applicability of this correlation for estimating theage of petroleum product releases has been extensivelydiscussed in the scientific literature (Kaplan et al. 1996;

Wade 2002; Stout et al. 2002; Kaplan 2003; Oudijk et al.2006). This age-dating approach can be a useful tool butshould be restricted to sites where a sudden, single releaseof fuel has occurred. A slow continuous fuel release or con-secutive fuel releases at the same area will invalidate themethod (Christensen and Larsen 1993; Stout et al. 2002).Among other parameters affecting applicability of theage-dating approach are selected soil characteristics, sitehydrology, temperature, moisture content, and availabilityof oxygen and nutrients (Singer and Finnerty 1984; Foghtand Westlake 1987; Atlas and Bartha 1992; Yang et al.1995; Coates et al. 1997; Kaplan et al. 1997). In some envi-ronments, these parameters will favor fuel preservation; inothers, local environmental conditions will strongly pro-mote weathering of the fuel and cause it to degrade ata higher rate. In this case, n-C17 will be completely de-pleted faster or slower than the 20.5 years predicted byChristensen-Larsen method (CLM). In most environmentalinvestigations, the uncertainties in these parameters typi-cally allow for only semiquantitative estimating age ofa fuel release.

In addition to the site-specific environmental condi-tions, application of the CLM is confounded by uncertain-ties introduced by variability of the initial ðn-C017Þ=ðPr0Þratio. As stated in Christensen and Larsen (1993), the origi-nal linear regression is based on the low variability of thisratio in dispensed diesel fuels sold in northern Europe dur-ing the early 1970s through the early 1990s. They have not,however, attempted to extend applicability of the method tofuels from different countries and continents that exhibit

Copyright ª 2008 The Author(s)Journal compilationª 2008National GroundWater Association.

94 Ground Water Monitoring & Remediation 28, no. 2/ Spring 2008/pages 94–97

Page 2: Zero-Order Monod Model for Christensen and Larsen Method

a broad range of ðn-C017Þ=ðPr0Þ values. To evaluate theeffect of the ðn-C017Þ=ðPr0Þ variance on the calculation ofthe age of a fuel release, we developed a zero-order kinet-ics model and a generalized equation that can be appliedfor any value of the initial ðn-C017Þ=ðPr0Þ ratio within theenvironmental constrains set by the CLM.

Development of the Zero-Order Kinetics ModelTo examine biochemical processes that could justify

the linear correlation represented by Equation 1, we con-sider the following basic model. Biodegradation of hydro-carbon fuel in a soil system is an enzyme-catalyzedtransformation of the fuel components. The rate of this cat-alytic reaction can be described by the Monod equation asfollows (Alexander 1994):

�dS

dt=lmaxSMt

Ks þ S(2)

where S is contaminant concentration, Mt is total activemicrobial cells concentration, lmax is maximum specificcontaminant utilization rate, and Ks is Monod constant(contaminant concentration when l = 1

2lmax).Experimental data in Christensen and Larsen (1993)

show that biodegradation of n-C17 relative to Pr can beapproximated by a linear equation. This is typical of micro-bial degradation of long-chain hydrocarbons (>C12) in dis-solved phase. Unlike the more soluble organic substrates,biodegradation rates of these low water soluble (<0.01 mg/L) hydrocarbons do not display the dependence on concen-tration (Leahy and Colwell 1990; Atlas and Bartha 1992;Scow 1993), which is indicative of zero-order kinetics.Zero-order kinetics imply excess of substrate, or in termsof the Monod equation, S >> Ks. The Christensen andLarsen (1993) data also show that the time required forcomplete depletion of n-C17 in a soil system is independentof its initial concentration. This means that shortly after thehydrocarbon release into the soil system, the cell density ofthe n-alkane degraders reaches its specific level pro-portional to the initial concentration of hydrocarbon, Mt ¼aS0 (a is the proportionality constant), and remains con-stant afterward. This interpretation is consistent with theMonod model assumption that in certain conditions, theyield of active microorganisms per unit amount of substratetransformed does not vary with time and substrate concen-tration (Simkins and Alexander 1984, 1985). Under theseconditions, Equation 2 can be reduced to:

�dS

dt= lmaxaS0 (3)

or in integrated form,

S = S0 � lmaxaS0t (4)

Now, for the biodegradation of n-C17 and Pr, we canwrite:

ðn-C17Þ = ðn-C017Þ � lmaxa17ðn-C017Þt�Pr

�=�Pr0

�� lmaxPraPr

�Pr0

�t

8<: (5)

where (n-C17) and (Pr) are concentrations of n-C17 and Prat time t, respectively; (n-C017 ) and (Pr0) are initial concen-trations of n-C17 and Pr at time t ¼ 0, respectively.

Because lmax17 >> lmaxPr , we assume that lmaxPr = 0, sothat

ðn-C17ÞðPrÞ =

ðn-C017ÞðPr0Þ

� lmax17a17ðn-C017ÞðPr0Þ

t: (6)

Solving Equation 6 for t, we obtain:

t = � ðPr0Þlmax17a17

�n-C017

� ðn-C17ÞðPrÞ þ 1

lmax17a17(7)

The parameters of Equation 7 can be estimated usingthe following coefficients of the Christensen and Larsen(1993) linear regression (Table 1 in Hurst and Schmidt2005):

ðn-C017Þ=ðPr0Þ = 2:16 (8)

t0 =1

lmax17a17= 20:5

so that Equation 7 is reduced to:

t = � 9:49ðn-C17ÞðPrÞ þ 20:5 (9)

which is equivalent to the Christensen and Larsen methodin Equation 1 shown previously.

Further, to account for deviations from the initialðn-C017Þ=ðPr0Þ = 2:16, new initial values ðn-C#017Þ=ðPr#0Þcan be expressed through the coefficient q as follows:

q =

�n-C#017

ðPr#0Þ=ðn-C017ÞðPr0Þ

=

�n-C#017

ðPr#0Þ=2:16 (10)

so that Equation 7 can now be presented as:

t = � 9:49

q

ðn-C17ÞðPrÞ þ 20:5 (11)

This is a generalized equation that can be applied forany value of the initial ðn-C017Þ=ðPr0Þ ratio, provided thatthe CLM stated environmental conditions pertain.

Uncertainties Related to Variation of the Initialn-C17/Pr Ratio

Using a broader data base (35 data points), Hurstand Schmidt (2005) proposed the middle distillate degrada-tion (MDD) model and obtained a slightly different linearcorrelation:

T = � 9:76 ðn-C17=PrÞ þ 20:7; (12)

corresponding to a new value for the initial ratioðn-C017Þ=ðPr0Þ = 2:12. In addition, Hurst and Schmidt(2005) stated that a deviation of the ðn-C017Þ=ðPr0Þ ratiovalue from 2.12 will cause a proportional change in theintercept of the Equation 12 and, therefore, in the time re-quired for the complete removal of n-C17. This interpretation

Y. Galperin and I.R. Kaplan/ Ground Water Monitoring & Remediation 28, no. 2: 94–97 95

Page 3: Zero-Order Monod Model for Christensen and Larsen Method

is incorrect as shown by applying the generalized Equation11 to evaluate two examples discussed in Hurst and Schmidt(2005), when ðn-C017Þ=ðPr0Þ equals 4.0 and 1.5, respectively.

In the first example, ðn-C017Þ=ðPr0Þ = 4:0, q ¼ 1.85, and

t = � 5:13ðn-C17ÞðPrÞ þ 20:5 (13)

which differs by both the slope and the intercept fromEquation 14 suggested by the MDD model (Hurst andSchmidt 2005) as follows:

t = � 9:76ðn-C17ÞðPrÞ þ 39 (14)

In the second example, ðn-C017Þ=ðPr0Þ ¼ 1.5, q ¼ 0.69,and

t = � 13:75ðn-C17ÞðPrÞ þ 20:5 (15)

which again differs by both the slope and the interceptfrom the equation suggested by Hurst and Schmidt (2005):

t = � 9:76ðn-C17ÞðPrÞ þ 14:6 (16)

For illustration, Equation 9 and Equations 13–16 areplotted in Figure 1. The Hurst and Schmidt (2005) in-terpretation suggests dependence between the initial con-centrations of alkanes and the time required for theirbiodegradation. However, the linearity of the CLM plotdemonstrates the independence of these parameters, whichis characteristic of zero-order kinetics.

ConclusionsThe empirical linear relationship between the time

since the diesel fuel release in the subsurface and the

corresponding values of n-C17/Pr can be explained in termsof the Monod model. Using the zero-order kinetics approx-imation and the slope and intercept coefficients derivedfrom the Christensen and Larsen (1993) data, we estab-lished a generalized equation that can be used for any valueof the initial ðn-C017Þ=ðPr0Þ ratio. The fact that depletion ofn-C17 in a soil system follows the Monod model with zero-order kinetics is a result of microbial degradation of a low-solubility substrate in dissolved phase.

The application of this equation is limited to certainenvironmental conditions of soil contamination by a sud-den, single release of a middle distillate fuel and can beused for approximation of age of a fuel release with thecaveat that the n-C17/Pr ratio is only one of many otherfactors that typically need to be reconciled for a reliableestimate of a fuel release age at a given site.

ReferencesAlexander, M. 1994. Biodegradation and Bioremediation. New

York: Academic Press.Atlas, R.M. 1981. Microbial degradation of petroleum hydro-

carbons: An environmental perspective. MicrobiologicalReview 45, no. 1: 180–209.

Atlas, R.M., and R. Bartha. 1992. Hydrocarbon biodegradationand oil spill bioremediation. In Advances in Microbial Ecol-ogy, vol. 12, ed. K.C. Marshall, 287–338. New York: PlenumPress.

Coates, J.D., J. Woodward, J. Allen, P. Philip, and D.R. Lovley.1997. Anaerobic degradation of polycyclic hydrocarbons andalkanes in petroleum-contaminated marine harbor sediments.Applied Environmental Microbiology 63, no. 9: 3589–3593.

Christensen, L.B., and T.H. Larsen. 1993. Method for determiningthe age of diesel oil spills in the soil. Ground Water Monitor-ing and Remediation 23, no. 4: 142–149.

Foght, J.M., and D.W.S. Westlake. 1987. Biodegradation ofhydrocarbons in freshwater. In Oil in Freshwater: Chemistry,

t=-9.49(n-C17 )/(Pr)+20.5, eq.9

t=-9.76(n-C17 )/(Pr)+39, eq.14

t=-5.13(n-C17)/(Pr)+20.5, eq.13

t=-13.75(n-C17 )/(Pr)+20.5, eq.15

t=-9.76(n-C17 )/(Pr)+14.6, eq.16

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

(n-C17)/(Pr)

Ag

e o

f re

leas

e (y

ears

)

Figure 1. Plots demonstrating effect of variation of the initial ðn-C017Þ=ðPr0Þ on the age of release equation. Solid lines (Equations9, 13, and 15) correspond to generalized Equation 11 with initial ratio ðn-C017Þ=ðPr0Þ values of 2.16, 4.0, and 1.5, respectively. Dash-ed lines (Equations 14 and 16) correspond to Hurst and Schmidt (2005) interpretation for initial ratio values of 4.0 and 1.5, respec-tively.

Y. Galperin and I.R. Kaplan/ Ground Water Monitoring & Remediation 28, no. 2: 94–9796

Page 4: Zero-Order Monod Model for Christensen and Larsen Method

Biology, Countermeasure Technology, ed. J.H. Vandermeulenand S.E. Hrudey, 217–230. New York: Pergamon Press.

Glazer, J.A. 1991. Nutrient-enhanced bioremediation of oil-contaminated shoreline: The Valdez experience. In In-Situ Bio-reclamation, ed. R.E. Hinchee and R.F. Olfenbuttel, 125–142.Oxford, UK: Butterworth-Heinemann.

Hurst, R.W., and G.W. Schmidt. 2005. Age significance of nC17/Pr ratios in forensic investigations of refined product and crudeoil releases. Environmental Geosciences 12, no. 3: 177–192.

Kaplan, I.R. 2003. Age dating of environmental organic residues.Environmental Forensics 4, no. 4: 95–141.

Kaplan, I.R., Y. Galperin, S.-T. Lu, and R.-P. Lee. 1997. Foren-sic environmental geochemistry: Differentiation of fuel-types, their sources and release time. Organic Geochemistry27, no. 5–6: 289–317.

Kaplan, I.R., Y. Galperin, H. Alimi, R.P. Lee, and S.T. Lu. 1996.Patterns of chemical changes during environmental alterationof hydrocarbon fuels. Groundwater Monitoring and Remedia-tion 16, no. 4: 113–124.

Kennicutt, M.C. 1988. The effect of biodegradation on crude oilbulk and molecular composition. Oil and Chemical Pollution4, 89–112.

Leahy, J.G., and R.R. Colwell. 1990. Microbial degradation ofhydrocarbons in the environment. Microbiology Reviews 53,no. 3: 305–315.

Oudijk, G., M. Obolensky, and K. Polidoro. 2006. The use of theChristensen-Larsen model to age date residential heating oilreleases: Conditions, limitations, and recommended practices.Environmental Claims 18, no. 3: 257–273.

Scow, K.M. 1993. Sorption and Degradation of Pesticides andOrganic Chemicals in Soil, ed. D.M. Linn, T.H. Carski, M.L.Brusseau, and F.H. Chang, 73–114. Madison, Wisconsin:American Society of Agronomy.

Simkins, S., and M. Alexander. 1985. Nonlinear estimation of theparameters of Monod kinetics that best describe mineraliza-tion of several substrate concentrations by dissimilar bacterialdensities. Applied Environmental Microbiology 50, no. 4:816–824.

Simkins, S., and M. Alexander. 1984. Models for mineralizationkinetics with the variables of substrate concentration and

population density. Applied Environmental Microbiology 47,no. 6: 1299–1306.

Singer, M.E., and W.R. Finnerty. 1984. Microbial metabolism ofstraight-chain and branched alkanes. In Petroleum Microbiol-ogy, ed. R. Atlas, 1–60. New York: MacMillan Publishing.

Stout, S.A., A.D. Uhler, K.J. McCarthy, and S.D. Emsbo-Mattingly. 2002. Invited commentary on the Christensen andLarsen technique. Environmental Forensics 3, no. 1: 9–11.

Wade, M.J. 2002. Invited commentary on the Christensen andLarsen technique. Environmental Forensics 3, no. 1: 13.

Yang, X.Y., L.E. Ericson, and L.T. Fan. 1995. A study of thedissolution rate-limited bioremediation of soil contaminatedby residual hydrocarbons. Journal of Hazardous Materials 41,no. 2–3: 299–313.

Biographical SketchesYakov Galperin, corresponding author, received an

engineering degree in chemistry (1972) and Ph.D. in physi-cal organic chemistry (1975) from Leningrad Institute ofTechnology, Russia. His current efforts are directed todevelopment and application of forensic geochemicalmethods. He can be reached at Environmental Geochem-istry Consulting, 13543 Bear Valley Road, Moorpark, CA93021; (805) 529-4423; fax (805) 523-2074; [email protected].

Isaac (Ian) R. Kaplan, received his B.S. and M.S.degrees in chemistry from Canterbury University, NewZealand, and a Ph.D. in biogeochemistry from the Uni-versity of Southern California in 1961. Dr. Kaplan receiveda faculty appointment at University of California, LosAngeles, in 1965 and is currently professor emeritus. Hecan be reached at the Department of Earth & Space Scien-ces and Institute of Geophysics & Planetary Physics,University of California, 595 Charles Young Drive, LosAngeles, CA 90094; [email protected].

*IGPP contribution 6301.

Y. Galperin and I.R. Kaplan/ Ground Water Monitoring & Remediation 28, no. 2: 94–97 97