Environmental Modeling & Assessment

, Volume 17, Issue 3, pp 275–288 | Cite as

Mathematical Modeling and Numerical Algorithms for Simulation of Oil Pollution

  • Quang A. Dang
  • Matthias Ehrhardt
  • Gia Lich Tran
  • Duc Le


This paper deals with the mathematical modeling and algorithms for the problem of oil pollution. For solving this task, we derive the adjoint problem for the advection–diffusion equation describing the propagation of oil slick after an accident, which we call the main problem. We prove a fundamental equality between the solutions of the main and the adjoint problems. Based on this equality, we propose a novel method for the identification of the pollution source location and the accident time of oil emission. This approach is illustrated on an example for an accident in the offshore of the central part of the Vietnamese coast. Numerical simulations demonstrate the effectiveness of the proposed method. Besides, the method is verified for 1D model of substance propagation.


Oil transport problems Oil spilling Weathering Adjoint equation approach  Pollution source identification 

Mathematics Subject Classifications (2010)

62P12 65M06 65M32 



This first two authors were supported partially by the bilateral German–Vietnamese project OILPOLL: Mathematical Modelling and Numerical Algorithms for Simulation of Oil Pollution, financed by the International Buro of the BMBF. The third author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).


  1. 1.
    Bagtzoglou, A. C., & Atmadja, J. (2005). Mathematical methods for hydrologic inversion: The case of pollution source identification. Handbook of Environmental Chemistry, 5, 65–96.CrossRefGoogle Scholar
  2. 2.
    Cacuci, D. G., & Schlesinger, M. E. (1994). On the application of the adjoint method of sensitivity analysis to problems in the atmospheric sciences. Atmósfera, 7, 47–59.Google Scholar
  3. 3.
    Cheng, W. P., & Jia, Y. (2010). Identification of contaminant point source in surface waters based on backward location probability density function method. Advanced Water Resources, 33, 397–410.CrossRefGoogle Scholar
  4. 4.
    Dang, Q. A. (2002). Monotone difference schemes for solving some problems of air pollution. Advances in Natural Sciences, 4, 297–307.Google Scholar
  5. 5.
    Dang, Q. A., & Ehrhardt, M. (2006). Adequate numerical solution of air pollution problems by positive difference schemes on unbounded domains. Mathematical and Computer Modelling, 44, 834–856.CrossRefGoogle Scholar
  6. 6.
    Dang, Q. A., Ehrhardt, M., Tran, G. J., & Le, D. (2007). On the numerical solution of some problems of environmental pollution. In C. B. Bodine (Ed.), Air pollution research advances (pp. 171–200). Hauppauge: Nova Science.Google Scholar
  7. 7.
    Dimov, I., & Zlatev, Z. (2002). Optimization problems in air-pollution modeling. In P. M. Pardalos, M. G. C. Resende (Eds.), Handbook on applied optimization. Oxford: Oxford University Press.Google Scholar
  8. 8.
    Doerffer, J. W. (1992). Oil spill response in the marine environment. Oxford: Pergamon.Google Scholar
  9. 9.
    Fay, J. A. (1971). Physical processes in the spread of oil on a water surface. In Proc. conf. prevention and control of oil spills (pp. 463–467). Washington, D.C.: American Petroleum Institute.Google Scholar
  10. 10.
    Kreiss, H.-O., & Lorenz, J. (1989). Initial-boundary value problems and the Navier–Stokes equations. New York: Academic.Google Scholar
  11. 11.
    Lehr, W. J., Fraga, R. J., Belen M. S., & Cekirge, H. M. (1984). A new technique to estimate initial spill size using a modified fay-type spreading formula. Marine Pollution Bulletin, 15, 326–329.CrossRefGoogle Scholar
  12. 12.
    Marchuk, G. I. (1986). Mathematical models in environmental problems. New York: Elsevier.Google Scholar
  13. 13.
    Marchuk, G. I. (1995). Adjoint equations and analysis of complex systems. Dordrecht: Kluwer.Google Scholar
  14. 14.
    Milnes, E., & Perrochet, P. (2007). Simultaneous identification of a single pollution point-source location and contamination time under known flow field conditions. Advances in Water Resources, 30, 2439–2446.CrossRefGoogle Scholar
  15. 15.
    Neupauer, R. M., & Wilson, J. L. (1999). Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant. Water Resources Research, 35, 3389–3398.CrossRefGoogle Scholar
  16. 16.
    Pudykiewicz, J. A. (1998). Application of adjoint tracer transport equations for evaluating source parameters. Atmospheric Environment, 32, 3039–3050.CrossRefGoogle Scholar
  17. 17.
    Reed, M., Johansen, Ø., Brandvik, P. J., Daling, P., Lewis, A., Fiocco, R., et al. (1999). Oil spill modeling towards the close of the 20th century: Overview of the state of the art. Spill Science & Technology Bulletin, 5, 3–16.Google Scholar
  18. 18.
    Reed, M. (2001). Technical description and verification tests of OSCAR2000, a multi-component 3-dimensional oil spill contingency and response model. SINTEF Applied Chemistry Report.Google Scholar
  19. 19.
    Samarskii, A. A. (2001). The theory of difference schemes. New York: Dekker.CrossRefGoogle Scholar
  20. 20.
    Skiba, Y. N. (1995). Direct and adjoint estimates in the oil spill problem. Revista Internacional de Contaminación Ambiental, 11, 69–75.Google Scholar
  21. 21.
    Skiba, Y. N. (1996). Dual oil concentration estimates in ecologically sensitive zones. Environmental Monitoring and Assessment, 43, 139–151.CrossRefGoogle Scholar
  22. 22.
    Skiba, Y. N. (1996). The derivation and applications of the adjoint solutions of a simple thermodynamic limited area model of the atmosphere–ocean–soil system. World Resource Review, 8, 98–113.Google Scholar
  23. 23.
    Skiba, Y. N. (1999). Direct and adjoint oil spill estimates. Environmental Monitoring and Assessment, 59, 95–109.CrossRefGoogle Scholar
  24. 24.
    Skiba, Y. N., & Parra-Guevara, D. (1999). Mathematics of oil spills: Existence, uniqueness, and stability of solutions. Geofísica Internacional, 38, 117–124.Google Scholar
  25. 25.
    Skiba, Y. N. (2003). On a method of detecting the industrial plants which violate prescribed emission rates. Ecological Modelling, 159, 125–132.CrossRefGoogle Scholar
  26. 26.
    Skiba, Y. N., Parra-Guevara, D., & Belitskaya, D. V. (2005). Air quality assessment and control of emission rates. Environmental Monitoring and Assessment, 111, 89–112.CrossRefGoogle Scholar
  27. 27.
    Wang, H. Q. & Lacroix, M. (1997). Optimal weighting in the finite difference solution of the convection–dispersion equation. Journal of Hydrology, 200, 228–242.CrossRefGoogle Scholar
  28. 28.
    Wang, S.-D., Shen, Y.-M., & Zheng, Y.H. (2005). Two-dimensional numerical simulation for transport and fate of oil spills in seas. Ocean Engineering, 32, 1556–1571.CrossRefGoogle Scholar
  29. 29.
    Yanenko, N. N. (1971). The method of fractional steps. New York: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Quang A. Dang
    • 1
  • Matthias Ehrhardt
    • 2
  • Gia Lich Tran
    • 3
  • Duc Le
    • 4
  1. 1.Institute of Information TechnologyHa noiVietnam
  2. 2.Lehrstuhl für Angewandte Mathematik und Numerische Analysis, Fachbereich C Mathematik und NaturwissenschaftenBergische Universität WuppertalWuppertalGermany
  3. 3.Institute of MathematicsHa noiVietnam
  4. 4.Buro of Hydrology and MeteorologyDang Thai ThanHa noiVietnam

Personalised recommendations