Adaptive split-operator methods for modeling transport phenomena in porous medium systems

Advances in Water Resources

Volumn 34, Issue 10, 2011, Pages 1268-1282

  Gasda, Sarah E ^a   Farthing, Matthew W ^b   Kees, Christopher E ^b   Miller, Cass T ^a  

^a UNIVERSITY OF NORTH CAROLINA   (United States)

^b U S ARMY ENGINEER RESEARCH AND DEVELOPMENT CENTER   (United States)

Author keywords

Algorithms; Error control; Error estimation; Reactive transport

Indexed keywords

APPROXIMATION APPROACH; APPROXIMATION ERRORS; COMPUTATIONAL EFFORT; COUPLED PROBLEMS; DIFFERENTIAL OPERATORS; DYNAMIC ADJUSTMENT; ENVIRONMENTAL MODEL; ENVIRONMENTAL MODELING; ERROR CONTROL; ERROR ESTIMATORS; FULLY-COUPLED; MODELING TRANSPORT; NONLINEAR REACTION; ORDERS OF MAGNITUDE; PARALLEL COMPUTATION; POROUS MEDIUM SYSTEMS; REACTIVE TRANSPORT; ROBUST SOLUTIONS; SPLIT OPERATOR METHOD; STANDARD METHOD;

ALGORITHMS; ERROR ANALYSIS; MATHEMATICAL OPERATORS; POROUS MATERIALS;

ESTIMATION;

ALGORITHM; ERROR ANALYSIS; FLUID FLOW; METHODOLOGY; MODELING; NONLINEARITY; POROUS MEDIUM; REACTIVE TRANSPORT;

EID: 80052842321     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2011.06.004     Document Type: Article

References (71)

1. Abreu E., Douglas J., Furtado F., Marchesin D., Pereira F. Three-phase immiscible displacement in heterogeneous petroleum reservoirs. Math Comput Simul 2006, 73(1-4):2-20. [special issue].
2. Akkerman I., Bazilevs Y., Kees C.E., Farthing M.W. Isogeometric analysis of free-surface flow. J Comput Phys 2011, 230(11):4137-4152.
3. Alexandrov V., Owczarz W., Thomson G., Zlatev Z. Parallel runs of a large air pollution model on a grid of Sun computers. Math Comput Simul 2004, 65(6):557-577.
4. Bader F.G. Analysis of double-substrate limited growth. Biotechnol Bioeng 1978, 20(2):183-202.
5. Barry D.A., Miller C.T., Culligan P.J., Bajracharya K. Split operator methods for reactive chemical transport in groundwater. Proceeding of the international conference on modelling and simulation 1995: MODSIM95 1995, 125-132. Modelling and Simulation Society of Australia, Inc., New South Wales, Australia.
6. Barry D.A., Bajracharya K., Miller C.T. Alternative split-operator approach for solving chemical reaction groundwater transport models. Adv Water Resour 1996, 19(5):261-275.
7. Barry D.A., Miller C.T., Culligan-Hensley P.J. Temporal discretization errors in non-iterative split-operator approaches to solving chemical reaction/groundwater transport models. J Contam Hydrol 1996, 22(1/2):1-17.
8. 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 1997, 43:331-341.
9. Barry D.A., Prommer H., Miller C.T., Engesgaard P.K., Brun A., Zheng C. Modelling the fate of oxidisable organic contaminants in groundwater. Adv Water Resour 2002, 25(8-12):945-983.
10. Bear J. Dynamics of fluids in porous media 1972, Elsevier, New York.
11. Becker R., Rannacher R. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer 2002, 10:1-102.
12. Belfort B., Carrayrou J., Lehmann F. Implementation of Richardson extrapolation in an efficient adaptive time stepping method: Applications to reactive transport and unsaturated flow in porous media. Transp Porous Media 2007, 69(1):123-138.
13. Bell L.S.J., Binning P.J. A split operator approach to reactive transport with the forward particle tracking Eulerian-Lagrangian localized adjoint method. Adv Water Resour 2004, 27:323-334.
14. Blom J.G., Verwer J.G. A comparison of integration methods for atmospheric transport-chemistry problems. J Comput Appl Math 2000, 126(1-2):381-396.
15. Botchev M.A., Verwer J.G. A new approximate matrix factorization for implicit time integration in air pollution modeling. J Comput Appl Mat 2003, 157(2):309-327.
16. Brenan K.E., Campbell S.L., Petzold L.R. The numerical solution of initial value problems in differential-algebraic equations 1996, Society for Industrial and Applied Mathematics, Philadelphia, PA.
17. Carrayrou J., Mose R., Behra P. Operator-splitting procedures for reactive transport and comparison of mass balance errors. J Contam Hydrol 2004, 68(3-4):239-268.
18. Coats K., Thomas L., Pierson R. Compositional and black oil reservoir simulation. SPE Reservoir Eval Eng 1998, 1(4):372-379.
19. Coats K.H. Impes stability: selection of stable timesteps. SPE J 2003, 8(2):181-187.
20. Coats K.H. Impes stability: the CFL limit. SPE J 2003, 8(3):291-297.
21. Coumou D., Matthaei S., Geiger S., Driesner T. A parallel FE-FV scheme to solve fluid flow in complex geologic media. Comput Geosci 2008, 34(12):1697-1707.
22. Dawson C.N., Vanduijn C.J., Wheeler M.F. Characteristic-Galerkin methods for contaminant transport with nonequilibrium adsorption kinetics. SIAM J Numer Anal 1994, 31(4):982-999.
23. Djouad R., Michelangeli D.V. Investigation of splitting gas and aqueous operators in atmospheric multiphase box models. Atmos Res 2004, 71(4):253-263.
24. Engesgaard P., Kipp K.L. A geochemical transport model for redox-controlled movement of mineral fronts in groundwater flow systems - A case of nitrate removal by oxidation of pyrite. Water Resour Res 1992, 28(10):2829-2843.
25. Eriksson K., Johnson C. Adaptive finite element methods for parabolic problems. 1: a linear model problem. SIAM J Numer Anal 1991, 28(1):43-77.
26. Eriksson K., Johnson C. Adaptive finite-element methods for parabolic problems. 5: long-time integration. SIAM J Numer Anal 1995, 32(6):1750-1763.
27. Estep D. Error estimates for multiscale operator decomposition for multiphysics models. Multiscale methods: bridging the scales in science and engineering 2009, Oxford University Press.
28. Estep D., Holst M., Mikulencak D. Accounting for stability: a posterior error estimates based on residual and variational analysis. Commun Numer Methods Eng 2002, 18(1):15-30.
29. Estep D., Ginting V., Ropp D., Shadid J.N., Tavener S. An a posteriori - a priori analysis of multiscale operator splitting. SIAM J Numer Anal 2008, 46(3):1116-1146.
30. Faragó I. A modified iterated operator splitting method. Appl Math Model 2008, 32:1542-1551.
31. Farago I., Havasi A., Zlatev Z. Richardson-extrapolated sequential splitting and its application. J Comput Appl Math 2009, 226(2):218-227. [special issue].
32. Geiser J. Consistency of iterative operator-splitting methods: theory and applications. Numer Methods Partial Differ Eqn 2009, 26:135-158.
33. Geiser J. Iterative operator-splitting methods for nonlinear differential equations and applications. Numer Methods Partial Differ Eqn, in press.
34. Georgiev K. An algorithm for parallel implementations of an Eulerian smog model. Numer Methods Appl 2003, 2542:463-470.
35. Hairer E., Wanner G. Solving ordinary differential equations. II: stiff and differential-algebraic problems 1996, Springer-Verlag, Berlin.
36. Hairer E., Norsett S.P., Wanner G. Solving ordinary differential equations. I: nonstiff problems 1993, Springer-Verlag, Berlin.
37. Herzer J., Kinzelbach W. Coupling of transport and chemical processes in numerical transport models. Geoderma 1989, 44:115-127.
38. Holden H., Karlsen K.H., Lie K.A. Operator splitting methods for degenerate convection-diffusion equations. ii: numerical examples with emphasis on reservoir simulation and sedimentation. Comput Geosci 2000, 4(4):287-322.
39. Hundsdorfer W., Portero L. A note on iterated splitting schemes. J Comput Appl Math 2007, 201:146-152.
40. Hundsdorfer W., Verwer J.G. Numerical solution of time-dependent advection-diffusion-reaction equations 2003, Springer Verlag Publishers, Berlin, Germany.
41. Hundsdorfer W., Verwer J.G. A note on splitting errors for advection-reaction equations. Appl Numer Math 1995, 18(1-3):191-199.
42. Jackson K.R., Sacks-Davis R. An alternative implementation of variable step-size multistep formulas for stiff ODEs. ACM Trans Math Softw 1980, 6:295-318.
43. Kanney J.F., Miller C.T., Kelley C.T. Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems. Adv Water Resour 2003, 26(3):247-261.
44. Kees C.E., Miller C.T. C++ implementations of numerical methods for solving differential-algebraic equations: design and optimization considerations. Assoc Comput Mach, Trans Math Softw 1999, 25(4):377-403.
45. Lanser D., Verwer J.G. Analysis of operator splitting for advection-diffusion-reaction problems from air pollution modelling. J Comput Appl Math 1999, 111(1-2):201-216.
46. Lu B., Wheeler M.F. Iterative coupling reservoir simulation on high performance computers. Pet Sci 2009, 6(1):43-50.
47. Mao X., Prommer H., Barry D., Langevin C., Panteleit B., Li L. Three-dimensional model for multi-component reactive transport with variable density groundwater flow. Environ Model Softw 2006, 21(5):615-628.
48. McLachlan R.I., Quispel G.R.W. Splitting methods. Acta Numer 2002, 341-434.
49. Miller C.T., Rabideau A.J. Development of split-operator, Petrov-Galerkin methods to simulate transport and diffusion problems. Water Resour Res 1993, 29(7):2227-2240.
50. Minkoff S.E., Kridler N.M. A comparison of adaptive time stepping methods for coupled flow and deformation modeling. Appl Math Model 2006, 30(9):993-1009.
51. Mossman D., AlMulki N. One-dimensional unsteady flow and unsteady pesticide transport in a reservoir. Ecol Model 1996, 89(1-3):259-267.
52. Osako I., Datta-Gupta A., King M. Timestep selection during streamline simulation through transverse flux correction. SPE J 2004, 9(4):450-464.
53. Palatnik B., Piskarev V. The use of a high-accuracy numerical scheme to model two-phase flow through a porous medium. Comput Math Math Phys 1996, 36(11):1581-1590.
54. Parkhurst DL Appelo CAJ. Users guide to PHREEQC (version 2) - a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations. Technical report, US Geological Survey, Denver, Colorado, 1999.
55. Pope S.B., Ren Z. Efficient implementation of chemistry in computational combustion. Flow Turbul Combust 2009, 82(4):437-453. [special issue].
56. Prommer H., Barry D., Zheng C. MODFLOW/MT3DMS-based reactive multicomponent transport modeling. Ground Water 2003, 41(2):247-257.
57. Rannacher R. Adaptive Galerkin finite element methods for partial differential equations. J Comput Appl Math 2001, 128:205-233.
58. Ren Z., Pope S.B. Second-order splitting schemes for a class of reactive systems. J Comput Phys 2008, 227(17):8165-8176.
59. Richardson L.F. The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philos Trans R Soc London, Ser A 1910, 210:350-357.
60. Shampine L.F. Numerical solution of ordinary differential equations 1994, Chapman and Hall Mathematics, New York.
61. Shampine L.F., Reichelt M.W. The MATLAB ODE suite. SIAM J Sci Comput 1997, 18(1):1-22.
62. Shampine L.F., Gladwell I., Thompson S. Solving ODEs with Matlab 2002, Cambridge University Press, Cambridge, UK.
63. Simpson M.J., Landman K.A. Analysis of split operator methods applied to reactive transport with Monod kinetics. Adv Water Resour 2007, 30(9):2026-2033.
64. Söderlind G. Automatic control and adaptive time-stepping. Numer Algorithms 2002, 31:281-310.
65. Söderlind G., Wang L. Evaluating numerical ODE/DAE methods, algorithms and software. J Comput Appl Math 2006, 185:244-260.
66. Steefel C.I., Macuarrie K.T.B. Approaches to modeling of reactive transport in porous media. Reactive transport in porous media: general principles and applications to geochemical processes 1996, 83-129. Mineralogical Society of America, Washington, DC. P.C. Lichtner, C.I. Steefel, E.H. Oelkers (Eds.).
67. Strang G. On the construction and comparison of difference schemes. SIAM J Numer Anal 1968, 5(3):506-517.
68. Valocchi A.J., Malmstead M. Accuracy of operator splitting for advection-dispersion-reaction problems. Water Resour Res 1992, 28(5):1471-1476.
69. Verwer J.G., Spee E.J., Blom J.G., Hundsdorfer W. A second-order Rosenbrock method applied to photochemical dispersion problems. SIAM J Sci Comput 1999, 20(4):1456-1480.
70. Wessling S., Kessels W., Schmidt M., Krause U. Investigating dynamic underground coal fires by means of numerical simulation. Geophys J Int 2008, 172(1):439-454.
71. Yeh G.T., Tripathi V.S. A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components. Water Resour Res 1989, 25(1):93-108.