Advertisement

Computational Geosciences

, Volume 14, Issue 3, pp 483–502 | Cite as

Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems—the MoMaS benchmark case

  • Jérôme Carrayrou
  • Joachim Hoffmann
  • Peter Knabner
  • Serge Kräutle
  • Caroline de Dieuleveult
  • Jocelyne Erhel
  • Jan Van der Lee
  • V. Lagneau
  • K. Ulrich Mayer
  • Kerry T. B. MacQuarrie
Original Paper

Abstract

Although multicomponent reactive transport modeling is gaining wider application in various geoscience fields, it continues to present significant mathematical and computational challenges. There is a need to solve and compare the solutions to complex benchmark problems, using a variety of codes, because such intercomparisons can reveal promising numerical solution approaches and increase confidence in the application of reactive transport codes. In this contribution, the results and performance of five current reactive transport codes are compared for the 1D and 2D subproblems of the so-called easy test case of the MoMaS benchmark (Carrayrou et al., Comput Geosci, 2009, this issue). This benchmark presents a simple fictitious reactive transport problem that highlights the main numerical difficulties encountered in real reactive transport problems. As a group, the codes include iterative and noniterative operator splitting and global implicit solution approaches. The 1D easy advective and 1D easy diffusive scenarios were solved using all codes, and, in general, there was a good agreement, with solution discrepancies limited to regions with rapid concentration changes. Computational demands were typically consistent with what was expected for the various solution approaches. The differences between solutions given by the three codes solving the 2D problem are more important. The very high computing effort required by the 2D problem illustrates the importance of parallel computations. The most important outcome of the benchmark exercise is that all codes are able to generate comparable results for problems of significant complexity and computational difficulty.

Keywords

MoMaS Benchmark Code intercomparison Numerical methods for reactive transport Direct substitution approach (DSA) Differential and algebraic equations (DAE) Sequential iterative approach (SIA) Sequential noniterative approach (SNIA) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bain, J.G., Mayer, K.U., Molson, J.W.H., Blowes, D.W., Frind, E.O., Kahnt, R., Jenk, U.: Assessment of the suitability of reactive transport modelling for the evaluation of mine closure options. J. Contam. Hydrol. 52, 109–135 (2001)CrossRefGoogle Scholar
  2. 2.
    Barry, D.A., Miller, C.T., Culligan-Hensley, P.J.: Temporal discretisation errors in non-iterative split-operator approaches to solving chemical reaction/groundwater transport models. J. Contam. Hydrol. 22, 1–17 (1996)CrossRefGoogle Scholar
  3. 3.
    Barry, D.A., Miller, C.T., Culligan, P.J., Bajracharya, K.: Analysis of split operator methods for nonlinear and multispecies groundwater chemical transport models. Math. Comput. Simul. 43, 331–341 (1997)CrossRefGoogle Scholar
  4. 4.
    Bauer, R.D., Rolle, M., Bauer, S., Eberhardt, C., Grathwohl, P., Kolditz, O., Meckenstock, R.U., Griebler, C.: Enhanced biodegradation by hydraulic heterogeneities in petroleum hydrocarbon plumes. J. Contam. Hydrol. 105, 56–68 (2009)CrossRefGoogle Scholar
  5. 5.
    Carnahan, C.L., Remer, J.S.: Nonequilibrium and equilibrium sorption with a linear sorption isotherm during mass transport through an infinite porous medium: some analytical solutions. J. Hydrol. 73, 227–258 (1984)Google Scholar
  6. 6.
    Carrayrou, J.: Looking for some reference solutions for the reactive transport benchmark of MoMaS with SPECY. Comput. Geosci. (2010, this issue). doi: 10.1007/s10596-009-9161-y
  7. 7.
    Carrayrou, J., Mosé, R., Behra, Ph.: A new efficient algorithm for solving thermodynamic chemistry. AIChE. J. 48, 894–904 (2002)CrossRefGoogle Scholar
  8. 8.
    Carrayrou, J., Mosé, R., Behra, Ph.: Modélisation du transport réactif en milieu poreux : schéma itératif associé à une combinaison d’éléments finis discontinus et mixtes-hybrides. Comptes Rendus Ac. Sci Mécanique 331, 211–216 (2003)CrossRefGoogle Scholar
  9. 9.
    Carrayrou, J., Mosé, R., Behra, Ph.: Efficiency of operator splitting procedures for solving reactive transport equation. J. Contam. Hydrol. 68, 239–268 (2004)CrossRefGoogle Scholar
  10. 10.
    Carrayrou, J., Kern, M., Knabner, P.: Reactive transport benchmark of MoMaS. Comput. Geosci. (2010, this issue). doi: 10.1007/s10596-009-9157-7
  11. 11.
    de Dieuleveult, C.: Un modèle numérique global et performant pour le couplage géochimie-transport. Ph.D. thesis, University of Rennes 1 (2008)Google Scholar
  12. 12.
    de Dieuleveult, C., Erhel, J.: A global approach to reactive transport: application to the MoMaS benchmark. Comput. Geosci. (2010, this issue) doi: 10.1007/s10596-009-9163-9
  13. 13.
    de Dieuleveult, C., Erhel, J., Kern, M.: A global strategy for solving reactive transport equations. J. Comput. Phys. 228, 6395–6410 (2009)MATHCrossRefGoogle Scholar
  14. 14.
    De Windt, L., Burnol, A., Montarnal, P., van der Lee, J.: Intercomparison of reactive transport models applied to UO2 oxidative dissolution and uranium migration. J. Contam. Hydrol. 61, 303–312 (2003)CrossRefGoogle Scholar
  15. 15.
    De Windt, L., Schneider, H., Ferry, C., Catalette, H., Lagneau, V., Poinssot, C., Poulesquen, A., Jegou, C.: Modeling spent nuclear fuel alteration and radionuclide migration in disposal conditions. Radiochim. Acta 94, 787–794 (2006)CrossRefGoogle Scholar
  16. 16.
    Fahs, M., Carrayrou, J., Younes, A., Ackerer, P.: On the efficiency of the direct substitution approach for reactive transport problems in porous media. Water Air Soil Pollut. 193, 299–308 (2008)CrossRefGoogle Scholar
  17. 17.
    Freedman, V.L., Ibaraki, M.: Coupled reactive mass transport and fluid flow: issues in model verification. Adv. Water Resour. 26, 117–127 (2003)CrossRefGoogle Scholar
  18. 18.
    Henderson, T.H., Mayer, K.U., Parker, B.L., Al, T.A.: Three-dimensional density-dependent flow and multicomponent reactive transport modeling of chlorinated solvent oxidation by potassium permanganate. J. Contam. Hydrol. 106, 183–199 (2009)CrossRefGoogle Scholar
  19. 19.
    Hoffmann, J., Kräutle, S., Knabner, P.: A parallel global-implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMAS benchmark problem. Comput. Geosci. (2010, this issue). doi: 10.1007/s10596-009-9173-7
  20. 20.
    Kaluarachchi, J.J., Morshed, J.: Critical assessment of the operator-splitting technique in solving the advection–dispersion–reaction equation: 1. First-order reaction. Adv. Water Resour. 18, 89–100 (1995)CrossRefGoogle Scholar
  21. 21.
    Kanney, J.F., Miller, C.T., Kelley, C.T.: Convergence of iterative split-operator for approximating non-linear reactive transport problem. Adv. Water Resour. 26, 247–261 (2003)CrossRefGoogle Scholar
  22. 22.
    Kräutle, S., Knabner, P.: A new numerical reduction scheme for coupled multicomponent transport–reaction problems in porous media: generalization to problems with heterogeneous equilibrium reactions. Water Resour. Res. 43, W03429.1–W03429.15 (2007). doi: 10.1029/2005WR004465 CrossRefGoogle Scholar
  23. 23.
    Lagneau, V., van der Lee, J.: HYTEC results of the MoMas reactive transport benchmark. Comput. Geosci. (2010, this issue). doi: 10.1007/s10596-009-9159-5
  24. 24.
    Leeming, G.J.S., Mayer, K.U., Simpson, R.B.: Effects of chemical reactions on iterative methods for implicit time stepping. Adv. Water Resour. 22, 333–347 (1998)CrossRefGoogle Scholar
  25. 25.
    Maher, K., Steefel, C.I., White, A.F., Stonestrom, D.A.: The role of reaction affinity and secondary minerals in regulating chemical weathering rates at the Santa Cruz Soil Chronosequence, California. Geochim. Cosmochim. Acta 73, 2804–2831 (2009)CrossRefGoogle Scholar
  26. 26.
    Mayer, K.U., MacQuarrie, K.T.B.: Solution of the MoMaS reactive transport benchmark with MIN3P–model formulation and simulation results. Comput. Geosci. (2010, this issue). doi: 10.1007/s10596-009-9158-6
  27. 27.
    Mayer, K.U., Benner, S.G., Frind, E.O., Thornton, S.F., Lerner, D.L.: Reactive transport modeling of processes controlling the distribution and natural attenuation of phenolic compounds in a deep sandstone aquifer. J. Contam. Hydrol. 53, 341–368 (2001)CrossRefGoogle Scholar
  28. 28.
    Mayer, K.U., Frind, E.O., Blowes, D.W.: Multicomponent reactive transport modeling in variably saturated porous media using a generalized formulation for kinetically controlled reactions. Water Resour. Res. 38, 1174 (2002). doi: 10:1029/2001WR000862 CrossRefGoogle Scholar
  29. 29.
    Mayer, K.U., Benner, S.G., Blowes, D.W.: Process-based reactive transport modeling of a permeable reactive barrier for the treatment of mine drainage. J. Contam. Hydrol. 85, 195–211 (2006)CrossRefGoogle Scholar
  30. 30.
    Molinero, J., Samper, J.: Large-scale modeling of reactive solute transport in fracture zones of granitic bedrocks. J. Contam. Hydrol. 82, 293–318 (2006)CrossRefGoogle Scholar
  31. 31.
    Molins, S., Mayer, K.U.: Coupling between geochemical reactions and multicomponent gas diffusion and advection—a reactive transport modeling study. Water Resour. Res. 43, W05435 (2007). doi: 10.1029/2006WR005206 CrossRefGoogle Scholar
  32. 32.
    Nowack, B., Mayer, K.U., Oswald, S.E., Van Beinum, W., Appelo, C.A.J., Jacques, D., Seuntjens, P., Gerard, F., Jaillard, B., Schnepf, A., Roose, T.: Verification and intercomparison of reactive transport codes to describe root-uptake. Plant and Soil 285, 305–321 (2006)CrossRefGoogle Scholar
  33. 33.
    Prommer, H., Aziz, L.H., Bolaño, N., Taubald, H., Schüth, C.: Modelling of geochemical and isotopic changes in a column experiment for degradation of TCE by zero-valent iron. J. Contam. Hydrol. 97, 13–26 (2008)Google Scholar
  34. 34.
    Reeves, H., Kirkner, D.J.: Multicomponent mass transport with homogeneous and heterogeneous chemical reactions: effect of the chemistry on the choice of numerical algorithm. 2. Numerical results. Water Resour. Res. 24, 1730–1739 (1988)CrossRefGoogle Scholar
  35. 35.
    Saaltink, M.W., Carrera, J., Ayora, C.: A comparison of two approaches for reactive transport modelling. J. Geochem. Explor. 6970, 97–101 (2000)CrossRefGoogle Scholar
  36. 36.
    Saaltink, M.W., Carrera, J., Ayora, C.: On the behavior of approaches to simulate reactive transport. J. Contam. Hydrol. 48, 213–235 (2001)CrossRefGoogle Scholar
  37. 37.
    Salvage, K.M., Yeh, G.T.: Development and application of a numerical model of kinetic and equilibrium microbiological and geochemical reactions (BIOKEMOD). J. Hydrol. 209, 27–52 (1998)CrossRefGoogle Scholar
  38. 38.
    Selim, H.M., Mansell, R.S.: Analytical solution of the equation for transport of reactive solutes through soils. Water Resour. Res. 12, 528–532 (1976)CrossRefGoogle Scholar
  39. 39.
    Shen, H., Nikolaidis, N.P.: A direct substitution method for multicomponent solute transport in ground water. Ground Water 35, 67–78 (1997)CrossRefGoogle Scholar
  40. 40.
    Siegel, P., Mosé, R., Ackerer, Ph., Jaffre, J.: Solution of the advection–diffusion equation using a combination of discontinuous and mixed finite elements. Int. J. Num. Methods Fluids 24, 595–613 (1997)MATHCrossRefGoogle Scholar
  41. 41.
    Spiessl, S.M., MacQuarrie, K.T.B., Mayer, K.U.: Identification of key parameters controlling dissolved oxygen migration and attenuation in fractured crystalline rocks. J. Contam. Hydrol. 95, 141–153 (2008)CrossRefGoogle Scholar
  42. 42.
    Steefel, C.I., Lasaga, A.C.: A coupled model for transport of multiple chemical species and kinetic precipitation/dissolution reactions with application to reactive flow in single phase hydrothermal systems. Am. J. Sci. 294, 529–592 (1994)Google Scholar
  43. 43.
    Steefel, C.I., MacQuarrie, K.T.B.: Approaches to modelling of reactive transport in porous media. In: Lichtner, P.C., Steefel, C.I., Oelkers, E.H. (eds.) Reactive Transport in Porous Media, vol. 34, pp. 82–129. Reviews in Mineralogy, Mineralogical Society of America, Washington (1996)Google Scholar
  44. 44.
    Steefel, C.I., Carroll, S., Zhao, P.H., Roberts, S.: Cesium migration in Hanford sediment: a multisite cation exchange model based on laboratory transport experiments. J. Contam. Hydrol. 67, 219–246 (2003)CrossRefGoogle Scholar
  45. 45.
    Sun, Y., Petersen, J.N., Clement, T.P.: Analytical solutions for multiple species reactive transport in multiple dimensions. J. Contam. Hydrol. 35, 429–440 (1999)CrossRefGoogle Scholar
  46. 46.
    Toride, N., Leij, F.J., van Genuchten, M.T.: A comprehensive set of analytical solutions for nonequilibrium solute transport with first-order decay and zero-order production. Water Resour. Res. 29, 2167–2182 (1993)CrossRefGoogle Scholar
  47. 47.
    Valocchi, A.J., Malmstead, M.: Accuracy of operator-splitting for advection–dispersion–reaction problems. Water Resour. Res. 28, 1471–1476 (1992)CrossRefGoogle Scholar
  48. 48.
    van Genuchten, M.T.: Analytical solutions for chemical transport with simultaneous adsorption, zero-order production and first-order decay. J. Hydrol. 49, 213–233 (1981)CrossRefGoogle Scholar
  49. 49.
    van Genuchten, M.T., Wierenga, P.J.: Mass transfer studies in sorbing porous media. 1. Analytical solutions. Soil Sci. Soc. Am. J. 40, 473–480 (1976)CrossRefGoogle Scholar
  50. 50.
    van Genuchten, M.T., Wierenga, P.J., O’Connor, G.A.: Mass transfer studies in sorbing porous media. 3. Experimental evaluation with 2,4,5-T1. Soil Sci. Soc. Am. J. 41, 278–285 (1976)Google Scholar
  51. 51.
    van der Lee, J., De Windt, L., Lagneau, V., Goblet, P.: Module-oriented modeling of reactive transport with HYTEC. Comput. Geosci. 29, 265–275 (2003)CrossRefGoogle Scholar
  52. 52.
    van der Lee, J., Langeau, V.: Rigorous methods for reactive transport in unsaturated porous medium coupled with chemistry and variable porosity. In: Miller, C.T., Farthing, M.W., Gray, W.G., Pinder, G.F. (eds.) Computational methods in water resources (CMWR XV), vol. 48(1), pp. 861–868. Elsevier (2004)Google Scholar
  53. 53.
    Walter, A.L., Frind, E.O., Blowes, D.W., Ptacek, C.J., Molson, J.W.: Modelling of multicomponent reactive transport in groundwater, 2. Metal mobility in aquifers impacted by acidic mine tailings discharge. Water Resour. Res. 30, 3149–3158 (1994)CrossRefGoogle Scholar
  54. 54.
    Yeh, G.T., Tripathi, V.S.: A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resour. Res. 25, 93–108 (1989)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Jérôme Carrayrou
    • 1
  • Joachim Hoffmann
    • 2
  • Peter Knabner
    • 2
  • Serge Kräutle
    • 2
  • Caroline de Dieuleveult
    • 3
    • 4
  • Jocelyne Erhel
    • 3
  • Jan Van der Lee
    • 5
  • V. Lagneau
    • 5
  • K. Ulrich Mayer
    • 6
  • Kerry T. B. MacQuarrie
    • 7
  1. 1.Institut de Mécanique des Fluides et des Solides, Laboratoire d’Hydrogéologie et de Géochimie de StrasbourgUniversity of Strasbourg, UMR 7517 UdS-CNRSStrasbourgFrance
  2. 2.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany
  3. 3.INRIA RennesCampus de BeaulieuRennesFrance
  4. 4.ANDRAChâtenay-MalabryFrance
  5. 5.Mines ParisTechFontainebleau CedexFrance
  6. 6.Department of Earth and Ocean SciencesUniversity of British ColumbiaVancouverCanada
  7. 7.Department of Civil EngineeringUniversity of New BrunswickFrederictonCanada

Personalised recommendations