Abstract
Biofilm forming microbes have complex effects on the flow properties of natural porous media. Subsurface biofilms have the potential for the formation of biobarriers to inhibit contaminant migration in groundwater. Another example of beneficial microbial effects is the biotransformation of organic contaminants to less harmful forms, thereby providing an in situ method for treatment of contaminated groundwater supplies.
Mathematical models that describe contaminant transport with biodegradation involve a set of coupled convection-dispersion equations with non-linear reactions. The reactive solute transport equation is one for which numerical solution procedures continue to exhibit significant limitations for certain problems of groundwater hydrology interest. Accurate numerical simulations are crucial to the development of contaminant remediation strategies.
A new numerical method is developed for simulation of reactive bacterial transport in porous media. The non-standard numerical approach is based on the ideas of the ‘exact’ time-stepping scheme. It leads to solutions free from the numerical instabilities that arise from incorrect modeling of derivatives and reaction terms. Applications to different biofilm models are examined and numerical results are presented to demonstrate the performance of the proposed new method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allen, M. B., M. B. Allen III, G. A. Behie and J. A. Trangenstein (1988). Basic mechanics of oil reservoir flows, in Multiphase Flow in Porous Media, Lecture Notes in Engineering 34, C. A. Brebia and S. A. Orszag (Eds), New York: Springer-Verlag, pp. 1–81.
Allen, M. B. and M. C. Curran (1992). A multigrid-based solver for the mixed finite-element approximations to groundwater flow, in Computational Methods in Water Resources IX, Vol. 1: Numerical Methods in Water Resources, T. F. Russell et al. (Eds), London: Elsevier Applied Science Publishers, pp. 579–585.
Allen, M. B., R. E. Ewing and P. Lu (1992). Well conditioned iterative schemes for mixed finite-element models of porous-media flows. SIAM J. Sci. Stat. Comput. 13, 794–814.
Allen, M. B. and B. Liu (1995). A modified method of characteristics incorporating streamline diffusion. Numer. Methods Partial Diff. Eqns. 11, 155–174.
Allen, M. B. and Z. Wang (1994). A multigrid-based solver for transient groundwater flows using cell-centered differences, in Computational Methods in Water Resources X, A. Peters et al. (Eds), the Netherlands: Kluwer Academic Publishers, pp. 1375–1382.
Bailey, J. E. and D. F. Ollis (1986). Biochemical Engineering Fundamentals, New York: McGraw-Hill.
Celia, M. A., T. F. Russell, I. Herrera and R. E. Ewing (1990). An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation. Adv. Water. Resour. 13, 187–206.
Characklis, W. G. and K. C. Marshall (1990). Biofilms, New York: John Wiley and Sons, Inc.
Clement, T. P., B. S. Hooker and R. S. Skeen (1996). Microscopic models for predicting changes in the saturated porous media properties caused by microbial growth. Ground Water 34, 934–942.
Cunningham, A. B., W. G. Characklis, F. Abedeen and D. Crawford (1991). Influence of the biofilm accumulation on porous media hydrodynamics. Environ. Sci. Technol. 25, 1305–1311.
Currie, I. G. (1993). Fundamental Mechanics of Fluids, 2nd edition, New York: McGraw-Hill.
Douglas, J. Jr. and T. F. Russell (1982). Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885.
Ewing, R. E., R. D. Lazarov, P. Lu and P. S. Vassilevski (1990). Preconditioning indefinite systems arising from mixed finite element discretization of second-order elliptic systems, in Preconditioned Conjugate Gradient Methods, O. Axelsson and L. Kolotilina (Eds), Lecture Notes in Mathematics 1457, Berlin: Springer-Verlag, pp. 280–343.
Ewing, R. E. and T. F. Russell (1982). Efficient time-stepping methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 19, 1–66.
Gaudy, A. and E. Gaudy (1980). Microbiology for the Environmental Scientists and Engineers, New York: McGraw-Hill Series in Water Resources and Environmental Engineering.
Gray, W. G. and G. F. Pinder (1976). An analysis of the numerical solution of the transport equation. Water Resour. Res. 12, 547–555.
Healy, R. W. and T. F. Russell (1993). A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation. Water Resour. Res. 29, 2399–2413.
Huyakorn, P. S. and G. F. Pinder (1983). Computational Methods in Subsurface Flow, New York: Academic Press.
James, G. A., B. K. Warwood, A. B. Cunningham, P. J. Sturman, R. Hiebert and J. W. Costerton (1995). Evaluation of subsurface biobarrier formation and persistence, in Proceedings of the 10th Annual Conference on Hazardous Waste Research, Great Plains/Rocky Mountain Hazardous Substance Research Center, pp. 82–91.
John, F. (1991). Partial Differential Equations, New York: Springer-Verlag.
Johnson, C. and J. Saranen (1986). Streamline diffusion methods for the incompressible euler and Navier-Stokes equations. Math. Comp. 47, 1–18.
Kojouharov, H. V. and B. M. Chen (1998a). Non-standard methods for the convective transport equation with nonlinear reactions. Numer. Methods Partial Diff. Eqns 14, 467–485.
Kojouharov, H. V. and B. M. Chen (1998b). Non-standard methods for the convective-dispersive transport equation with nonlinear reactions, submitted to Numer. Methods Partial Diff. Eqns.
Liu, B., M. B. Allen, H. Kojouharov and B. Chen (1996). Finite-element solution of reaction-diffusion equations with advection, in Computational Methods in Water Resources XI, Vol. 1: Computational Methods in Subsurface Flow and Transport Problems, A. A. Aldama et al. (Eds), Southampton Boston: Computational Mechanics Publications, pp. 3–12.
Mickens, R. E. (1989). Exact solutions to a finite-difference model of nonlinear reaction-advection equation: implications for numerical analysis. Numer. Methods Partial Diff. Eqns 5, 313–325.
Mitchell, A. R. and D. F. Griffiths (1980). The Finite Difference Method in Partial Differential Equations, New York: John Wiley & Sons.
Murray, J. D. (1993). Mathematical Biology, Berlin: Springer.
Raviart, P. A. and J. M. Thomas (1977). A mixed finite-element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes (Eds), Lecture Notes in Mathematics 606, Berlin: Springer-Verlag, pp. 292–315.
Russell, T. F. and M. F. Wheeler (1983). Finite element and finite difference methods for continuous flows in porous media, in Frontiers in Applied Mathematics, Vol. 1: The Mathematics of Reservoir Simulation, R. E. Ewing (Ed.), Philadelphia, PA: SIAM, pp. 35–106.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, B.M., Kojouharov, H.V. Non-standard numerical methods applied to subsurface biobarrier formation models in porous media. Bull. Math. Biol. 61, 779–798 (1999). https://doi.org/10.1006/bulm.1999.0113
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1006/bulm.1999.0113