Stabilized finite element methods for miscible displacement in porous media
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 28 (1994) no. 5, p. 611-665
@article{M2AN_1994__28_5_611_0,
     author = {Wei, Yuting},
     title = {Stabilized finite element methods for miscible displacement in porous media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {28},
     number = {5},
     year = {1994},
     pages = {611-665},
     zbl = {0853.76042},
     mrnumber = {1295589},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1994__28_5_611_0}
}
Wei, Yuting. Stabilized finite element methods for miscible displacement in porous media. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 28 (1994) no. 5, pp. 611-665. http://www.numdam.org/item/M2AN_1994__28_5_611_0/

[1] J. Bear, 1988, Dynamics of Fluids in Porous Media, Dover Publication, Inc., New York.

[2] R. B. Bird, W. E. Stewart, E. N. Lighfoot, 1966, Transport Phenomena, John Wiley, New York.

[3] F. Brezzi, M. Fortin, 1991, Mixed and Hybrid Finite Element Methods, Springer-Verlag. | MR 1115205 | Zbl 0788.73002

[4] A. N. Brooks, T. J. R. Hughes, 1982, Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Meths, 32, 199-259. | MR 679322 | Zbl 0497.76041

[5] B. Cockburn, C. W. Shu, 1991, The Runge-Kutta local projection P' discontinuous Galerkin finite element method for scalar conservation laws, RAIRO-Model. Math. Anal. Numer., 25, 337-361. | Numdam | MR 1103092 | Zbl 0732.65094

[6] C. T. Dawson, 1991, Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. Numer. Anal., 28, 1282-1309. | MR 1119271 | Zbl 0741.65068

[7] J. Jr. Douglas, 1982, Simulation of miscible displacement in porous media by a modified method of characteristics procedure, In Numerical Analysis, Dundee 1981, vol. 912 of Lecture Notes in Mathematics, Springer-Verlag, Berlin. | MR 654343 | Zbl 0476.76100

[8] J. Jr. Douglas, 1984, Numerical methods for the flow of miscible fluids in porous media, In Numerical Methods in Coupled Systems, pp. 405-439, John Wiley and Sons Ltd., London, R. W. Lewis, P. Bettess and E. Hinton, Eds. | Zbl 0585.76138

[9] J. Jr. Douglas, R. E. Ewing, M. F. Wheeler, 1983, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R. A. I. R. O., Anal. Numér., 17, 249-265. | Numdam | MR 702137 | Zbl 0526.76094

[10] J. Jr. Douglas, R. E. Ewing, M. F. Wheeler, 1983, The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O., Anal. Numér., 17, 17-33. | Numdam | MR 695450 | Zbl 0516.76094

[11] J. Jr. Douglas, J. L. Hensley, Y. Wei, L. Yeh, J.Jaffré, P. J.Paes Leme, T. S.Ramakrishnam, D. J.Wilkinson, 1992, A derivation for Darcy's law for miscible fluids and a revised model for miscible displacement in porous media, In Mathematical Modeling in Water Resources, vol. 2, pp. 165-178, Computational Mechamcs Publications, Elsevier Applied Science, Southampton, Boston, T. F. RUSSELL, E. R. EWING, C. A. BREBBIA, W. G. GRAY, G. F. PINDER, Eds.

[12] J. Jr. Douglas, J. E. Roberts, 1983, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp., 41, 441-459. | MR 717695 | Zbl 0537.76062

[13] J. Jr. Douglas, T. F. Russell, 1982, Numerical methods for convectiondominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal.,19, 871-885. | MR 672564 | Zbl 0492.65051

[14] R. Durán, 1988, On the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method, SIAM J. Numer. Anal., 25, 989-1001. | MR 960861 | Zbl 0661.76096

[15] R. E. Ewing, 1983, The mathematics of reservoir simulation, Frontiers in Applied Mathematics, SIAM, Philadelphia. | MR 770577 | Zbl 0533.00031

[16] R. E Ewing, T. F. Russell, M. F. Wheeler, 1983, Simulation of miscible displacement using mixed methods and a modified method of characteristics, In Proceedings, Seventh SPE Symposium on Reservoir Simulation, pp. 71-81, Dallas, Texas, Society of Petroleum Engineers, Paper SPE 12241.

[17] R. E. Ewing, T. F. Russell, M. F. Wheeler, 1984, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comp. Meth. Appl. Mech. Eng., 47, 73-92. | MR 777394 | Zbl 0545.76131

[18] L. P. Franca, S. L. Frey, 1992, Stabilized finite element methods: II. The incompresible Navier-Stokes Equations, Comput. Meths. Appl. Mech. Engrg., 99, 209-233. | MR 1186727 | Zbl 0765.76048

[19] L. P. Franca, S. L. Frey, T. J. R. Hughes, 1992, Stabilized finite element methods : I. Application to the advective-diffusive model, Comput. Meths. Appl. Mech. Engrg., 95, 253-276. | MR 1155924 | Zbl 0759.76040

[20] V. Girault, P.- A. Raviart, 1986, Finite Element Methods for Navier-Stokes Equations, Theory and Algonthms, Springer-Verlag, Berlin, Heidelberg, New York. | MR 851383 | Zbl 0585.65077

[21] A. Harten, S. Osher, 1987, Uniformly high-order accurate non-oscilatory schemes I, SIAM J. Numer. Anal., 24 279-309 | MR 881365 | Zbl 0627.65102

[22] T. J. R. Hughes, A. N. Brooks, 1979, A multidimensional upwind scheme with no cross-wind diffusion, In Finite Element Methods for Convection Dominated Flows, pp. 19-35. ASME, New York, 1979 T. J. R. HUGUES, ed. | MR 571679 | Zbl 0423.76067

[23] T. J. R. Hughes, A. N. Brooks, 1982, A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions : application to the streamline upwind procedure, In Finite Element Methods in Fluids, Wiley, Chichester, R. H. GALLAGHER, ed.

[24] T. J. R. Hughes, L. P. Franca, G. M. Hulbert, 1989, A new finite element formulation for computational fluid dynamics : VIII The Galerkin/least-square method for convective-diffusive equations, Comput. Meths..Appl. Engrg., 73, 173-189. | MR 1002621 | Zbl 0697.76100

[25] J. Jaffre, J. E. Roberts, 1985, Upstream weighting and mixed finite elements in the simulation of miscible displacements, Modélisation Mathématique et Analyse Numérique, 19, 443-460. | Numdam | MR 807326 | Zbl 0568.76096

[26] C. Johnson, J. Saranen, 1986, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp., 47, 1-18. | MR 842120 | Zbl 0609.76020

[27] C. Johnson, V. Thomée, 1981, Error estimates for some mixed finite element methods for parabolic type problems, R. A. I. R. O., Anal. Numér., 14, 41-78. | Numdam | MR 610597 | Zbl 0476.65074

[28] S. Osher, 1984, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal., 22, 947-961. | MR 799122 | Zbl 0627.35061

[29] D. W. Peaceman, 1966, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Petroleum Engr. J., 6, 213-216.

[30] D. W. Peaceman, 1977, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York.

[31] T. S. Ramakrishnam, D. J. Wilkinson, pivate communication.

[32] T. F. Russell, 1985, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal., 22, 970-1013. | MR 799124 | Zbl 0594.76087

[33] C. W. Shu, 1987, TVB uniformly high-order schemes for conservation laws, Math. of. Comp., 49, 105-121. | MR 890256 | Zbl 0628.65075

[34] A. Szepessy, 1991, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, R. A. I. R. O. Modél. Math. Anal. Numér., 26, 749-782. | Numdam | MR 1135992 | Zbl 0751.65061

[35] Y. Wei, Discontinuous Galerkin - mixed finite element methods for convection - dominated diffusion problems, to appear.