Skip to main content
SpringerLink
Log in

An Efficient Implementation of the Method of Lines for Multicomponent Reactive Transport Equations

  • Published:
Water, Air, & Soil Pollution Aims and scope Submit manuscript

Abstract

Modeling reactive transport with chemical equilibrium reactions requires solution of coupled partial differential and algebraic equations. In this work, two formulations are developed to combine the method of lines (MOL) with the global implicit approach. The first formulation has a non-conservative form and leads to a nonlinear system of ordinary differential equations with a reduced number of unknowns. The second formulation presents better conservation properties but leads to a nonlinear system of differential algebraic equations with a large number of unknowns. In both formulations, the resulting systems are integrated in time using the DLSODIS time solver which adapts both the order of the time integration and the time step size to provide the necessary accuracy. Numerical experiments show that higher-order time integration is effective for solving the non-conservative formulation and point out the high benefit of the MOL for solving reactive transport problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Amir, L., & Kern, M. (2009). A global method for coupling transport with chemistry in heterogeneous porous media. Computers and Geosciences. doi:10.1007/s10596-009-9162-x.

    Google Scholar 

  • Barry, D. A., Bajracharya, K., & Miller, C. T. (1996). Alternative split-operator approach for solving chemical reaction/groundwater transports models. Advances in Water Resources, 19(5), 261–275.

    Article  Google Scholar 

  • Barry, D. A., Miller, C. T., Culligan, P. J., & Bajracharya, K. (1997). Analysis of split operator methods for nonlinear and multispecies groundwater chemical transport models. Mathematics and Computers in Simulation, 43, 331–341.

    Article  Google Scholar 

  • Carrayrou, J., Mosé, R., & Behra, P. (2004). Operator-splitting procedures for reactive transport and comparison of mass balance errors. Journal of Contaminant Hydrology, 68, 239–268.

    Article  CAS  Google Scholar 

  • Curtis, A. R., Powell, M. J. D., & Reid, J. K. (1974). On the Estimation of Sparse Jacobian Matrices. Journal of the Institute of Mathematical Applications, 13, 117–119.

    Google Scholar 

  • De Dieuleveult, C., Erhel, J., & Kern, M. (2009). A global strategy for solving reactive transport equations. Jouranl of Computational Physics, 228(17), 6395–6410.

    Article  Google Scholar 

  • DeSimone, L. A., Howesh, B. L., & Barlowa, P. M. (1997). Mass-balance analysis of reactive transport and cation exchange in a plume of wastewater-contaminated groundwater. Journal of Hydrology, 203, 228–249.

    Article  Google Scholar 

  • Diersch, H. J. (1988). Finite element modelling of recirculating density driven saltwater intrusion processes in groundwater. Advances in Water Resources, 11(1), 25–43.

    Article  Google Scholar 

  • Diersch, H. J., & Kolditz, O. (1998). Coupled groundwater flow and transport: 2. Thermohaline and 3D convection systems. Advances in Water Resources, 21(5), 401–425.

    Article  Google Scholar 

  • Eisenstat, S. C., Gursky, M. C., Schultz, M. H., & Sherman, A. H. (1977). Yale sparse matrix package: II. The nonsymmetric codes. Research Report No. 114, Dept. of Computer Sciences, Yale University.

  • Eisenstat, S. C., Gursky, M. C., Schultz, M. H., & Sherman, A. H. (1982). Yale sparse matrix package: I. The symmetric codes. International Journal for Numerical Methods in Engineering, 18, 1145–1151.

    Article  Google Scholar 

  • Engesgaard, P., & Kipp, K. L. (1992). A geochemical transport model for redox-controlled movement of mineral front in groundwater flow systems: A case of nitrate removal by oxidation of pyrite. Water Resources Research, 28(10), 2829–2843.

    Article  CAS  Google Scholar 

  • Fahs, M., Carrayrou, J., Younes, A., & Ackerer, P. (2008). On the efficiency of the direct substitution approach for reactive transport problems in porous media. Water, Air, and Soil Pollution, 193, 1–4.

    Article  Google Scholar 

  • Fahs, M., Younes, A., & Lehmann, F. (2009). An easy and efficient combination of the mixed finite element method and the method of lines for the resolution of Richards’ Equation. Environmental Modelling & Software, 24, 1122–1126.

    Article  Google Scholar 

  • Farthing, M. W., Kees, C. E., & Miller, C. T. (2003a). Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Advances in Water Resources, 26, 373–94.

    Article  Google Scholar 

  • Farthing, M. W., Kees, C. E., Coffey, T. S., Kelley, C. T., & Miller, C. T. (2003b). Efficient steady-state solution techniques for variably saturated groundwater flow. Advances in water resources, 26, 833–49.

    Article  CAS  Google Scholar 

  • Grindrod, P., & Takase, H. (1996). Reactive chemical transport within engineered barriers. Journal of Contaminant Hydrology, 21(1–4), 283–296.

    Article  CAS  Google Scholar 

  • Hammond, G. E., Valocchi, A. J., & Lichnter, P. C. (2005). Application of Jacobian-free Newton–Krylov with physics based preconditioning to biogeochemical transport. Advances in Water Resources, 28(4), 359–376.

    Article  CAS  Google Scholar 

  • Herzer, J., & Kinzelbach, W. (1989). Coupling of transport and chemical processes in numerical transport models. Geoderma, 44, 115–127.

    Article  CAS  Google Scholar 

  • Hindmarsh, A. C. (1980). LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, 15(4), 10–11.

    Article  Google Scholar 

  • Hindmarsh, A. C. (1982). Large ordinary differential equation systems and software. IEEE Control System Magazine, 2, 24–30.

    Article  Google Scholar 

  • Jennings, A. A., Kirkner, D. J., & Theis, T. L. (1982). Multicomponent equilibrium chemistry in groundwater quality models. Water Resources Research, 18, 1089–1096.

    Article  CAS  Google Scholar 

  • Kanney, J. F., Miller, C. T., & Kelley, C. T. (2003a). Convergence of iterative split operator for approximating non linear reactive transport problem. Advances in Water Resources, 26, 247–261.

    Article  Google Scholar 

  • Kanney, J. F., Miller, C. T., & Barry, D. A. (2003b). Comparison of fully coupled approaches for approximating nonlinear transport and reaction problems. Advances in Water Resources, 26, 353–372.

    Article  CAS  Google Scholar 

  • Kees, C. E., & Miller, C. T. (2002). Higher order time integration methods for two-phase flow. Advances in Water Resources, 25, 159–77.

    Article  Google Scholar 

  • Kräutle, S., & Knabner, P. (2005). A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media. Water Resources Research, 41, W09414. doi:10.1029/2004WR003624.

    Article  Google Scholar 

  • Kräutle, S., & Knabner, P. (2007). A reduction scheme for coupled multicomponent transport-reaction problems in porous media: Generalization to problems with heterogeneous equilibrium reactions. Water Resources Research, 43, W03429.

    Article  Google Scholar 

  • Kräutle, S., Hoffmann, J., & Knabner, P. (2007). Reactions with minerals treated as complementarity problems and solved by the Semi smooth Newton Method. Santa Fe, USA: SIAM Conference on Mathematical and Computational Issues in the Geosciences.

    Google Scholar 

  • Li, H., Farthing, M. W., Dawson, C. N., & Miller, C. T. (2007). Local discontinuous Galerkin approximations to Richards’ equation. Advances in Water Resources, 30, 555–75.

    Article  Google Scholar 

  • Miller, C. T., & Rabideau, A. J. (1993). Development of split operator, Petrov–Galerkin methods to simulate transport and diffusion problems. Water Resources Research, 29(7), 2227–2240.

    Article  CAS  Google Scholar 

  • Miller, C. T., Williams, G. A., Kelly, C. T., & Tocci, M. D. (1998). Robust solution of Richards’ equation for nonuniform porous media. Water Resources Research, 34, 2599–610.

    Article  CAS  Google Scholar 

  • Miller, C. T., Abhishek, C., & Farthing, M. (2006). A spatially and temporally adaptive solution of Richards’ equation. Advances in Water Resources, 29, 525–45.

    Article  Google Scholar 

  • Radhakrishnan, K., & Hindmarsh, A. C. (1993). Description and use of LSODE, the Livermore solver for ordinary differential equations. LLNL report UCRL-ID-113855.

  • Saaltink, M. W., Ayora, C., & Carrera, J. (1998). A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resources Research, 34(7), 1649–1656.

    Article  CAS  Google Scholar 

  • Saaltink, M. W., Carrera, J., & Ayora, C. (2001). On the behavior of approaches to simulate reactive transport. Journal of Contaminant Hydrology, 48, 213–235.

    Article  CAS  Google Scholar 

  • Saaltink, M. W., Carrera, J., & Olivella, S. (2004). Mass balance errors when solving the convective form of the transport equation in transient flow problems. Water Resources Research, 40, W05107. doi:10.1029/2003WR002866.

    Article  Google Scholar 

  • Seager, M. K., & Balsdon, S. (1982). LSODIS, a sparse implicit ODE solver. In Proceedings of the IMACS 10th World Congress, Montreal, August, pp 8–13.

  • Shen, H., & Nikolaidis, N. P. (1997). A direct substitution method for multicomponent solute transport in ground water. Ground Water, 35, 67–78.

    Article  CAS  Google Scholar 

  • Steefel, C. I., & MacQuarrie, K. T. B. (1996). Approaches to modeling reactive transport. In P. C. Lichtner, C. I. Steefel, & E. H. Oelkers (Eds.), Reactive transport in porous media. Reviews in mineralogy 34 (pp. 83–129). Washington, DC: Mineralogical Society of America.

    Google Scholar 

  • Steefel, C. I., & Yabusaki, S. B. (1995). OS3D/GIMRT, software for modeling multicomponent–multidimensional reactive transport. User manual and programmer’s guide. Richland, WA: Pacific Northwest Laboratory.

    Google Scholar 

  • Tocci, M. D., Kelly, C. T., & Miller, C. T. (1997). Accurate and economical solution of the pressure-head form of Richards’ equation by the method of lines. Advances in Water Resources, 20, 1–14.

    Article  Google Scholar 

  • Tocci, M. D., Kelly, C. T., Miller, C. T., & Kees, C. E. (1998). Inexact Newton methods and the method of lines for solving Richards’ equation in two space dimensions. Computers and Geosciences, 2, 291–309.

    Article  Google Scholar 

  • Valocchi, A. J., Street, R. L., & Roberts, P. V. (1981). Transport of ion-exchanging solutes in groundwater: Chromatographic theory and field simulation. Water Resources Research, 17, 1517–1527.

    Article  CAS  Google Scholar 

  • White, S. P. (1995). Multiphase nonisothermal transport of systems of reacting chemicals. Water Resources Research, 31(7), 1761–1772.

    Article  CAS  Google Scholar 

  • Xu, T., Samper, J., Ayor, C., Manzano, M., & Custodio, E. (1999). Modelling of non-isothermal multi-component reactive transport in field scale porous media flow systems. Journal of Hydrology, 214(1–4), 144–164.

    Article  CAS  Google Scholar 

  • Yeh, G. T., & Tripathi, V. S. (1989). A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resources Research, 25(1), 93–108.

    Article  CAS  Google Scholar 

  • Younes, A., Fahs, M., & Ahmed, S. (2009). Solving density driven flow problems with efficient spatial discretizations and higher-order time integration methods. Advances in Water Resources, 32, 340–352.

    Article  Google Scholar 

Download references

Acknowledgments

This work was partly supported by the GdR MoMas CNRS-2439 sponsored by ANDRA, BRGM, CEA and EDF whose support is gratefully acknowledged.

Author information

Authors and Affiliations

  1. School of Engineering, Lebanese International University, Beirut, Lebanon

    Marwan Fahs

  2. Laboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST, CNRS, 1 rue de Blessig, 67000, Strasbourg, France

    Anis Younes & Philippe Ackerer

Authors
  1. Marwan Fahs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anis Younes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Philippe Ackerer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anis Younes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fahs, M., Younes, A. & Ackerer, P. An Efficient Implementation of the Method of Lines for Multicomponent Reactive Transport Equations. Water Air Soil Pollut 215, 273–283 (2011). https://doi.org/10.1007/s11270-010-0477-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11270-010-0477-y

Keywords

Search

Navigation