×

Upstream weighting and mixed finite elements in the simulation of miscible displacements. (English) Zbl 0568.76096

A finite element method for approximating incompressible miscible displacements in porous media is presented and analysed. A mixed finite element approximation is used for the pressure equation while a discontinuous upstream weighting scheme in conjunction with a mixed finite element method is employed for the concentration equation. Error estimates, which remain valid for vanishing diffusion, are derived.

MSC:

76S05 Flows in porous media; filtration; seepage
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] 1. F. BREZZI, On the existence, uniqueness and approximation of saddle-point problem arising from Lagrangien multipliers. R.A.I.R.O., Anal. Numér. 2 (1974), pp. 129-151 Zbl0338.90047 MR365287 · Zbl 0338.90047
[2] 2. G. CHAVENT, G. COHEN, M. DUPUY, J JAFFRE, I. RIBERA, Simulation of two dimensional water flooding using mixed finite elements, SPEJ. 24 (1984), pp. 382-390.
[3] 3. J. DOUGLAS Jr., R. E. EWING, M. F. WHEELER, The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O., Anal. Numér. 17 (1983), pp. 17-33. Zbl0516.76094 MR695450 · Zbl 0516.76094
[4] 4. J. DOUGLAS Jr., J. E. ROBERTS, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp. 41 (1983), pp. 441-459. Zbl0537.76062 MR717695 · Zbl 0537.76062
[5] 5. R. E. EWING, M. F. WHEELER, Galerkin methods for miscible displacement problem in porous media, SIAM J. Numér. Anal. 17 (1980), pp. 351-365. Zbl0458.76092 MR581482 · Zbl 0458.76092
[6] 6. M. FORTIN, Résolution numérique des équations de Navier Stokes par des éléments finis du type mixte, Rapport INRIA n^\circ 184, INRIA Le Chesnay (1976).
[7] 7. J JAFFRE, Éléments finis mixtes et décentrage pour les équations de diffusion-convection, Calcolo 23 (1984), pp. 171-197. Zbl0562.65077 MR799619 · Zbl 0562.65077
[8] 8. P. JOLY, La méthode des éléments finis mixtes appliquée au problème de diffusion-convection, Thèse de 3e cycle, Université Pierre-et-Marie Curie, Paris (1982).
[9] 9. C. JOHNSON, V. THOMÉE, Error estimates for some mixed finite element methods for parabolic type problems, R.A.I.R.O., Anal. Numér., 15 (1981), pp. 41-78. Zbl0476.65074 MR610597 · Zbl 0476.65074
[10] 10. P LESAINT, P. A. RAVIART, On a finite element method for solving the neutron transport equation. Mathematical Aspect of Finite Elements in Partial Differential Equations, Ed. Carl de Boor, Academic Press (1974), pp. 89-123. Zbl0341.65076 MR658142 · Zbl 0341.65076
[11] 11. P. A. RAVIART, J. M. THOMAS, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method, Eds. I. Galligani and E. Magenes, Lecture Notes in Mathematics 606, Springer Verlag (1977), pp. 292-315. Zbl0362.65089 MR483555 · Zbl 0362.65089
[12] 12. T. F. RUSSELL, Finite elements with characteristics for two-component incompressible miscible displacement, 6th SPE Symposium on Reservoir Simulation, New Orléans, SPE 10500 (1982).
[13] 13. M. F. WHEELER, B. L. DARLOW, Interior penalty Galerkin methods for miscible displacement problems in porous media, Computational Methods in Nonlinear Mechanics, Ed. J. T. Oden, North Holland (1980). Zbl0444.76081 MR576923 · Zbl 0444.76081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.