, Volume 24, Issue 2, pp 153–163 | Cite as

Mathematical modeling of the biodegradation of residual hydrocarbon in a variably-saturated sand column

Original Paper


The biodegradation of heptadecane in five sand columns was modeled using a multiplicative Monod approach. Each column contained 1.0 kg of sand and 2 g of heptadecane, and was supplied with an artificial seawater solution containing nutrients at a flow rate that resulted in unsaturated flow through the column. All nutrients were provided in excess with the exception of nitrate whose influent concentration was 0.1, 0.5, 1.0, 2.5, or 5.0 mg N/L. The experiment was run around 912 h until no measurable oxygen consumption or CO2 production was observed. The residual mass of heptadecane was measured at the end of the experiments and the biodegradation was monitored based on oxygen consumption and CO2 production. Biodegradation kinetic parameters were estimated by fitting the model to experimental data of oxygen, CO2, and residual mass of heptadecane obtained from the two columns having influent nitrate–N concentration of 0.5 and 2.5 mg/L. Noting that the oxygen and CO2 measurements leveled off at around 450 h, we fitted the model to these data for that range. The estimated parameters fell in within the range reported in the literature. In particular, the half-saturation constant for nitrate utilization, \( K_{\text{N}} \), was estimated to be 0.45 mg N/L, and the yield coefficient was found to be 0.15 mg biomass/mg heptadecane. Using these values, the rest of experimental data from the five columns was predicted, and the model agreed with the observations. There were some consistent discrepancies at large times between the model simulation and observed data in the cases with higher nitrate concentration. One plausible explanation for these differences could be limitation of biodegradation by reduction of the heptadecane–water interfacial area in these columns while the model uses a constant interfacial area.


Biodegradation Multiple-Monod kinetics Parameter estimation Mathematical model 


\( \mu \)

Gross growth of the active biomass (day−1)

\( \mu_{\max } \)

Maximum growth rate (day−1)

\( \rho_{\text{sand}} \)

True density of the sand (mg/cm3)

\( \sigma^{2} \)

Squared error of estimation

\( \varepsilon \)

Constant fraction of the decayed biomass

\( \omega_{ij} \)


\( \varphi \)

Shape factor

\( A_{s} \)

Specific surface area of sand (cm2/g)


CO2 production

\( d_{\text{dvg}} \)

Average particle size (mm)

\( F \)

Objective function

\( H \)

Hessian matrix

\( k_{d} \)

Endogenous biomass decay rate (day−1)

\( K_S \)

Half-saturation concentration (mg/cm2)

\( K_N \)

Half-saturation concentration for nitrogen consumption (mg N/L of pore water)


Number of observations

\( N \)

Nitrate concentration (mg N/L of pore water)

\( N_{K} \)

Number of measurements of dependent variable




Number of estimated parameters

\( S \)


\( S_{\text{area}} \)

Surface-area normalized concentration of heptadecane (mg/cm2)

\( S_{\text{mass}} \)

Concentration of heptadecane (mg/kg of dry sand)

\( X \)

Active biomass concentration (mg/cm2)

\( X_i \)

Inert biomass concentration (mg/cm2)

\( X_{0} \)

Initial biomass concentration (mg/cm2)

\( Y_{CS} \)

Stoichiometric coefficient for CO2 production from substrate (mg C/mg S)

\( Y_{CX} \)

Stoichiometric coefficient for CO2 produced during complete mineralization of biomass (mg C/mg X)

\( Y_{OS} \)

Stoichiometric coefficient for oxygen consumption based on complete mineralization of substrate (mg of O2/mg of S)

\( Y_{OX} \)

Stoichiometric coefficient for oxygen consumption during the complete mineralization of biomass (mg of O2/mg of X)

\( Y_X \)

Biomass yield coefficient for growth on substrate (mg X/mg S)

\( u_S \)

Simulated result for the dependent variable

\( u_O \)

Observed result for the dependent variable

\( V_x \)

Covariance matrix



This work was supported in part by funding from the Exxon Valdez Trustee Council under project no. 11100836. This article does not necessarily reflect the views of the funding agency and no official endorsement should be inferred.


  1. Aichinger G, Grady CPL, Tabak HH (1992) Application of respirometric biodegradability testing protocol to slightly soluble soluble organic compounds. Wat Env Res 64(7):890–900CrossRefGoogle Scholar
  2. Alvarez PJJ, Anid PJ, Vogel TM (1991) Kinetics of aerobic biodegradation of benzene and toluene in sandy aquifer material. Biodegradation 2:43–51PubMedCrossRefGoogle Scholar
  3. Atkinson KE (1978) An introduction to numerical analysis. Wiley, New York, p 587Google Scholar
  4. Atlas RM, Hazen TC (2011) Oil biodegradation and bioremediation: a tale of the two worst spills in US history. Environ Sci Technol 45(16):6709–6715PubMedCrossRefGoogle Scholar
  5. Bailey JE, Ollis DF (1986) Biochemical engineering fundamentals. McGraw-Hill, New YorkGoogle Scholar
  6. Bard Y (1974) Nonlinear parameter estimation. Academic Press, New YorkGoogle Scholar
  7. Borden RC, Bedient PB (1986) Transport of dissolved hydrocarbons influenced by oxygen-limited biodegradation. 1. Theoretical development. Water Resour Res 22(13):1973–1982CrossRefGoogle Scholar
  8. Boufadel MC (1998) Unit hydrographs derived from the Nash model. J Am Water Resour Assoc 34:167–177CrossRefGoogle Scholar
  9. Boufadel MC, Suidan MT, Venosa AD, Rauch CH, Biswas P (1998) 2D variably saturated flows: physical scaling and Bayesian estimation. J Hydrol Eng 3(4):223–231CrossRefGoogle Scholar
  10. Boufadel MC, Reeser P, Suidan MT, Wrenn BA, Cheng J, Du X, Venosa AD (1999) Optimal nitrate concentration for the biodegradation of n-heptadecane in a variably-saturated sand column. Environ Technol 20(2):191–199CrossRefGoogle Scholar
  11. Boufadel MC, Sharifi Y, Van Aken B, Wrenn BA, Lee K (2010) Nutrient and oxygen concentrations within the sediments of an Alaskan beach polluted with the Exxon Valdez oil spill. Environ Sci Technol 44(19):7418–7424PubMedCrossRefGoogle Scholar
  12. Boufadel MC, Wrenn BA, Moore BE, Boda KJ, Michel J (2011) A biodegradation assessment tool for decision on beach response. In: Proceeding of the 2011 International Oil Spill Conference. Portland, pp 2011–2348Google Scholar
  13. Bragg JR, Prince RC, Harner EJ, Atlas RM (1994) Effectiveness of bioremediation of the Exxon Valdez oil spill. Nature 368:413–418CrossRefGoogle Scholar
  14. Cao YS, Alaerts GJ (1996) A model for oxygen consumption in aerobic heterotrophic biodegradation in dual-phase drainage systems. Water Res 30(4):1010–1022CrossRefGoogle Scholar
  15. Chen YM, Abriola LM, Alvarez PJJ, Anid PJ, Vogel TM (1992) Modeling transport and biodegradation of benzene and toluene in sandy aquifer material-comparisons with experimental measurements. Water Resour Res 28(7):1833–1847CrossRefGoogle Scholar
  16. Clark A (1970) The theory of adsorption and catalysis. Academic Press, New YorkGoogle Scholar
  17. Du XM, Reeser P, Suidan MT, Huang TH, Moteleb M, Boufadel MC, Venosa AD (1999) Optimum nitrogen concentration supporting maximum crude oil biodegradation microcosms. In: Proceedings 1999 International oil spill conference, American Petroleum Institute, Washington, DC, pp 485–488Google Scholar
  18. Esler D (2010) Exxon Valdez effect goes. Trac-Trend Anal Chem 29(SI6):V–VIGoogle Scholar
  19. Essaid HI, Bekins BA (1997) BIOMOC, a multispecies solute-transport model with biodegradation. US Geological Survey, Water-Resources Investigations Report 97–4022Google Scholar
  20. Essaid HI, Bekins BA, Godsy EM, Warren E, Baedecker MJ, Cozzarelli IM (1995) Simulation of aerobic and anaerobic biodegradation processes at a crude oil spill site. Water Resour Res 31(12):3309–3327CrossRefGoogle Scholar
  21. Findlay RH, King GM, Watling L (1989) Efficacy of phospholipid analysis in determining microbial biomass in sediments. Appl Environ Microbiol 55:2888–2893PubMedGoogle Scholar
  22. Floodgate GD (1979) Nutrient limitation. In: Bourquin AW, Pritchard PH (eds) Microbial degradation of pollutants in marine environments. EPA-66019-79-012, Environmental Research Laboratory, Gulf Breeze, pp 107–119Google Scholar
  23. Frostegard A, Tunlid A, Baath E (1991) Microbial biomass measured as total lipid phosphate in soils of different organic content. J Microbiol Meth 14(3):151–163CrossRefGoogle Scholar
  24. Gibbs CF (1975) Quantitative studies on marine biodegradation of oil. 1. Nutrient limitation at 14 °C. Proc Royal Soc Lond B 188:61–82CrossRefGoogle Scholar
  25. Godeke S, Voqt C, Schirmer M (2008) Estimation of kinetic Monod parameters for anaerobic degradation of benzene in groundwater. Environ Geol 55(2):423–431CrossRefGoogle Scholar
  26. Grady CPL Jr., Lim HC (1980) Biological wastewater treatment: theory and applications. Mardet Dekker, New YorkGoogle Scholar
  27. Guo QN, Li HL, Boufadel MC, Sharifi Y (2010) Hydrodynamics in a gravel beach and its impact on the Exxon Valdez oil. J Geophys Res-oceans. doi: 10.1029/2010JC006169 Google Scholar
  28. Kindred JS, Celia MA (1989) Contaminant transport and biodegradation. 2. Conceptual model and test simulations. Water Resour Res 25(6):1149–1160Google Scholar
  29. Kopke B, Wilms R, Engelen B, Cypionka H, Sass H (2005) Microbial diversity in coastal subsurface sediments: a cultivation approach using various electron acceptors and substrate gradients. Appl Environ Microbiol 71(12):7819–7830PubMedCrossRefGoogle Scholar
  30. Köster M, Meyer-Reil L (2001) Characterization of carbon and microbial biomass pools in shallow water coastal sediments of the southern Baltic Sea (Nordrugensche Bodden). Mar Ecol-Prog Ser 214:25–41Google Scholar
  31. Lasdon LS, Warren AD, Jain A, Ratner M (1979) Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Trans Math Software 4:34–50CrossRefGoogle Scholar
  32. Lasdon LS, Warren AD, Jain A, Ratner M (1980) GRG2’s user’s guide report. Dept of General Business, University of Texas, AustinGoogle Scholar
  33. Laspidou CS, Rittmann BE (2002) A unified theory for extracellular polymeric substances, soluble microbial products, and active and inert biomass. Water Res 36(11):2711–2720PubMedCrossRefGoogle Scholar
  34. Li HL, Boufadel MC (2010) Long-term persistence of oil from the Exxon Valdez spill in two-layer beaches. Nat Geosci 3(2):96–99CrossRefGoogle Scholar
  35. MacQuarrie KTB, Sudicky EA, Frind EO (1990) Simulation of biodegradable organic contaminants in groundwater. 1. Numerical formulation in principal directions. Water Resour Res 26(2):207–222Google Scholar
  36. Malone DR, Kao CM, Borden RC (1993) Dissolution and biorestoration of nonaqueous phase hydrocarbons: model development and laboratory evaluation. Water Resour Res 29(7):2203–2213CrossRefGoogle Scholar
  37. Metcalf Eddy (1991) Wastewater engineering: treatment, disposal, and reuse, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  38. Mishra S, Parker JC (1989) Parameter estimation for coupled unsaturated flow and transport. Water Resour Res 25:385–396CrossRefGoogle Scholar
  39. Mohamed M, Hatfield K (2011) Dimensionless parameters to summarize the influence of microbial growth and inhibition on the bioremediation of groundwater contaminants. Biodegradation 22(5):877–896PubMedCrossRefGoogle Scholar
  40. Molz FJ, Widdowson MA, Benefield LD (1986) Simulation of microbial growth dynamics coupled to nutrient and oxygen transport in porous media. Water Resour Res 22(8):1207–1216CrossRefGoogle Scholar
  41. Moussa MS, Hooijmans CM, Lubberding HJ, Gijzen HJ, Van Loosdrecht MCM (2005) Modeling nitrification, heterotrophic growth and predation in activated sludge. Water Res 39(20):5080–5098PubMedCrossRefGoogle Scholar
  42. Nicol JP, Wise WR, Molz FJ, Benefield LD (1994) Modeling biodegradation of residual petroleum in a saturated porous column. Water Resour Res 30:3313–3325CrossRefGoogle Scholar
  43. Paniconi C, Putti M (1994) A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems. Water Resour Res 30:3357–3374CrossRefGoogle Scholar
  44. Reynolds Tom D, Paul Ricards (1996) Unit operations and processes in environmental engineering. PWS, BostonGoogle Scholar
  45. Rittmann BE, McCarty PL (2001) Environmental biotechnology: principles and applications. McGraw-Hill, BostonGoogle Scholar
  46. Sarioglu MS, Copty NK (2008) Modeling the enhanced bioremediation of organic contaminants in pyrite-containing aquifers. Transport Porous Media 72(2):203–221Google Scholar
  47. Schirmer M, Butler BJ, Roy JW, Frind EO, Barker JF (1999) A relative-least-squares technique to determine unique Monod kinetic parameters of BTEX compounds using batch experiments. J Contam Hydrol 37:69–86CrossRefGoogle Scholar
  48. Schirmer M, Molson JW, Frind EO, Barker JF (2000) Biodegradation modeling of a dissolved gasoline plume applying independent laboratory and field parameters. J Contam Hydrol 46:339–374CrossRefGoogle Scholar
  49. Sperandio M, Paul E (1997) Determination of carbon dioxide evolution rate using on-line gas analysis during dynamic biodegradation experiments. Biotechnol Bioeng 53(3):243–252PubMedCrossRefGoogle Scholar
  50. Taylor E, Reimer D (2008) Oil persistence on beaches in Prince William Sound-A review of SCAT surveys conducted from 1989 to 2002. Mar Pollut Bull 56(3):458–474PubMedCrossRefGoogle Scholar
  51. Unver O, Mays LW (1984) Optimal determination of loss rate functions and unit hydrograph. Water Resour Res 20:203–214CrossRefGoogle Scholar
  52. Venosa AD, Suidan MT, Wrenn BA, Strohmeier KL, Haines JR, Eberhart BL, King D, Holder E (1996) Bioremediation of an experimental oil spill on the shoreline of Delaware bay. Environ Sci Technol 30(5):1764–1775CrossRefGoogle Scholar
  53. Venosa AD, Campo P, Suidan MT (2010) Biodegradability of lingering crude oil 19 years after the Exxon Valdez oil spill. Environ Sci Technol 44:7613–7621PubMedCrossRefGoogle Scholar
  54. Wrenn BA, Suidan MT, Strohmeier KL, Eberhart BL, Wilson GJ, Venosa AD (1997) Nutrient transport during bioremediation of contaminated beaches: evaluation with lithium as a conservative tracer. Water Res 31:515–524CrossRefGoogle Scholar
  55. Xia YQ, Li HL, Boufadel MC, Sharifi Y (2010) Hydrodynamic factors affecting the persistence of the Exxon Valdez oil in a shallow bedrock beach. Water Resour Res. doi: 10.1029/2010WR009179 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Xiaolong Geng
    • 1
    • 2
  • Michel C. Boufadel
    • 1
  • Brian Wrenn
    • 1
  1. 1.Department of Civil and Environmental Engineering, Center for Natural Resources Development and ProtectionNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Department of MathematicsAnshan Normal UniversityAnshanChina

Personalised recommendations