Transport in Porous Media

, 89:487 | Cite as

Modeling Chemotactic Waves in Saturated Porous Media using Adaptive Mesh Refinement

  • Nicholas Dudley Ward
  • Samuel Falle
  • Mira Stone Olson
Article

Abstract

Bacterial transport is heavily influenced by chemical gradients and interfaces that exist in the subsurface. The main aim of this article is to describe a method of simulating the propagation of a traveling bacterial wave in a contaminated region and the resulting degradation of the contaminant. The presence of the chemotactic term and the relatively small bacterial diffusion means that the wave contains a very sharp wavefront. We, therefore, use an upwind conservative numerical scheme to obtain accurate and numerically stable solutions. The accuracy of the method is verified by comparisons with an exact one-dimensional solution of a simplified problem to give the same wavespeed. The method is then used to simulate the propagation of a realistic chemotactic wave in one dimension. We then use adaptive mesh refinement (AMR) to compute the propagation of chemotactic waves in two dimensions using the simplified model calibrated to give the same wavespeed as the full model.

Keywords

Chemotaxis Adaptive mesh refinement 

References

  1. Ahn I.S., Ghiorse W.C., Lion L.W., Shuler M.L.: Growth kinetics of Pseudomonas putida G7 on naphthalene and occurrence of naphthalene toxicity during nutrient deprivation. Biotechnol. Bioeng. 59(5), 587–594 (1998)CrossRefGoogle Scholar
  2. Armitage J.P.: Bacterial tactic responses. Adv. Microb. Physiol. 41, 229–289 (1999)CrossRefGoogle Scholar
  3. Bailey J.E., Ollis D.F.: Biochemical Engineering Fundamentals. 2nd edn. McGraw-Hill, New York (1986)Google Scholar
  4. Berger M.J., Collela P.L.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)CrossRefGoogle Scholar
  5. Falle S.A.E.G.: Self-similar jets. Mon. Not. R. Astron. Soc. 250, 581–596 (1991)Google Scholar
  6. Falle S.A.E.G.: AMR applied to non-linear elastodynamics. In: Plewa, T., Linde, T., Weirs, V.G.J (eds) Adaptive Mesh Refinement—Theory and Applications, pp. 235–253. Lect. Notes Comput. Sci. Eng., Springer (2005)CrossRefGoogle Scholar
  7. Ford R.M., Cummings P.T.: On the relationship between cell balance-equations for chemotactic cell-populations. SIAM J. Appl. Math. 52(5), 1426–1441 (1992)CrossRefGoogle Scholar
  8. Ford R.M., Harvey R.W.: Role of chemotaxis in the transport of bacteria through saturated porous media. Adv. Water Resour. 30(6-7), 1608–1617 (2007)CrossRefGoogle Scholar
  9. Keller E.F., Segel L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)CrossRefGoogle Scholar
  10. Landman K.A., Simpson M.J., Pettet G.J.: Tactically-driven nonmonotone travelling waves. Physica D 237, 678–691 (2008)CrossRefGoogle Scholar
  11. Law A.M., Aitken M.D.: Continuous-flow capillary assay for measuring bacterial chemotaxis. Appl. Environ. Microbiol. 71(6), 3137–3143 (2005)CrossRefGoogle Scholar
  12. Leveque R.: Finite volume methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  13. Long W., Hilpert M.: Analytical solutions for bacterial energy taxis (chemotaxis): traveling bacterial bands. Adv. Water Resour. 30, 2262–2270 (2007)CrossRefGoogle Scholar
  14. Long W., Hilpert M.: Lattice-Boltzmann modeling of contaminant degradation by chemotactic bacteria: exploring the formation and movement of bacterial bands. Water Resour. Res. 44, W09415 (2008). doi: 10.1029/2007/WR006129 CrossRefGoogle Scholar
  15. Marx R.B., Aitken M.D.: Quantification of chemotaxis to naphthalene by Pseudomonas putida G7. Appl. Environ. Microbiol. 65(7), 2847–2852 (1999)Google Scholar
  16. Olson M.S., Ford R.M., Smith J.A., Fernandez E.J.: Analysis of column tortuosity for MnCl 2 and bacterial diffusion using magnetic resonance imaging. Environ. Sci. Technol. 39, 149–154 (2005)CrossRefGoogle Scholar
  17. Olson M.S., Ford R.M., Smith J.A., Fernandez E.J.: Mathematical modeling of chemotactic bacterial transport through a two-dimensional heterogeneous porous medium. Bioremediat. J. 10, 13–23 (2006)CrossRefGoogle Scholar
  18. Pandy G., Jain R.K.: Bacterial chemotaxis toward environmental pollutants: role in Bioremediation. Appl. Environ. Microbiol. 68(12), 5789–5795 (2002)CrossRefGoogle Scholar
  19. Parales R.E., Ditty J.L., Harwood C.S.: Toluene-degrading bacteria are chemotactic towards the environmental pollutants benzene, toluene, and trichloroethylene. Appl. Environ. Microbiol. 66(9), 4098–4104 (2000)CrossRefGoogle Scholar
  20. Rivero M.A., Tranquillo R.T., Buettner H.M., Lauffenburger D.: Transport models for chemotactic cell populations based on individual cell behaviour. Chem. Eng. Sci. 44(12), 2881–2897 (1989)CrossRefGoogle Scholar
  21. Sethian J.A.: Level Set Methods. Cambridge University Press, Cambridge (1996)Google Scholar
  22. Sherwood J.L., Sung J.C., Ford R.M., Fernandez E.J., Maneval J.E., Smith J.A.: Analysis of bacterial random motility in a porous medium using magnetic resonance imaging and immunomagnetic labeling. Environ. Sci. Technol. 37, 781–785 (2003)CrossRefGoogle Scholar
  23. Singh, R., Olson, M.S.: Application of bacterial swimming and chemotaxis for enhanced bioremediation. In: Shah, V. ed. Emerging Environmental Technologies, pp. 149–172. Springer, the Netherlands (2008)Google Scholar
  24. Tros M.E., Schraa G., Zhender A.J.B.: Transformation of low concentrations of 3-chlorobenzoate by Pseudomonas sp. strain B13: kinetics and residual concentrations. Appl. Environ. Microbiol. 62(2), 437–442 (1996)Google Scholar
  25. Witt M.E., Dybas M.J., Worden R.M., Criddle C.S.: Motility-enhanced bioremediation of carbon tetrachloride-contaminated aquifer sediments. Environ. Sci. Technol. 33(17), 2958–2964 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Nicholas Dudley Ward
    • 1
  • Samuel Falle
    • 2
  • Mira Stone Olson
    • 3
  1. 1.Mantis NumericsKurowNew Zealand
  2. 2.Department of Applied MathematicsUniversity of LeedsLeedsUnited Kingdom
  3. 3.Department of Civil, Architectural and Environmental EngineeringDrexel UniversityPhiladelphiaUSA

Personalised recommendations