Numerical methods for a model for compressible miscible displacement in porous media

Authors:
Jim Douglas and Jean E. Roberts

Journal:
Math. Comp. **41** (1983), 441-459

MSC:
Primary 65M60; Secondary 76S05

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717695-3

MathSciNet review:
717695

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Abstract: A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The system is consistent with the usual model for incompressible miscible displacement. Two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture. The concentration is treated by a Galerkin method in both procedures, while the pressure is treated by either a Galerkin method or by a parabolic mixed finite element method. Optimal order estimates in and essentially optimal order estimates in are derived for the errors in the approximate solutions for both methods.

**[1]**F. Brezzi,*On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**8**(1974), no. R-2, 129–151 (English, with loose French summary). MR**0365287****[2]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[3]**Jim Douglas Jr.,*The numerical simulation of miscible displacement in porous media*, Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979) North-Holland, Amsterdam-New York, 1980, pp. 225–237. MR**576907****[4]**J. Douglas, Jr., M. F. Wheeler, B. L. Darlow & R. P. Kendall, "Selfadaptive finite element simulation of miscible displacement in porous media,"*SIAM J. Sci. Statist. Comput.*(To appear.)**[5]**Jim Douglas Jr.,*Simulation of miscible displacement in porous media by a modified method of characteristic procedure*, Numerical analysis (Dundee, 1981) Lecture Notes in Math., vol. 912, Springer, Berlin-New York, 1982, pp. 64–70. MR**654343****[6]**Jim Douglas Jr., Richard E. Ewing, and Mary Fanett Wheeler,*The approximation of the pressure by a mixed method in the simulation of miscible displacement*, RAIRO Anal. Numér.**17**(1983), no. 1, 17–33 (English, with French summary). MR**695450****[7]**Richard E. Ewing and Mary Fanett Wheeler,*Galerkin methods for miscible displacement problems in porous media*, SIAM J. Numer. Anal.**17**(1980), no. 3, 351–365. MR**581482**, https://doi.org/10.1137/0717029**[8]**R. E. Ewing & M. F. Wheeler, "Galerkin methods for miscible displacement problems with point sources and sinks--unit mobility ratio case. (To appear.)**[9]**C. I. Goldstein & R. Scott, "Optimal -estimates for some Galerkin methods for the Dirichlet problem,"*SIAM J. Numer. Anal.*(To appear.)**[10]**Claes Johnson and Vidar Thomée,*Error estimates for some mixed finite element methods for parabolic type problems*, RAIRO Anal. Numér.**15**(1981), no. 1, 41–78 (English, with French summary). MR**610597****[11]**J. A. Nitsche,*𝐿_{∞}-convergence of finite element approximation*, Journées “Éléments Finis”}, address=Rennes, date=1975, (1975)**[12]**D. W. Peaceman,*Fundamentals of Numerical Reservoir Simulation*, Elsevier, New York, 1977.**[13]**D. W. Peaceman, "Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement,"*Soc. Pet. Eng. J.*, 1966, pp. 213-216.**[14]**P.-A. Raviart and J. M. Thomas,*A mixed finite element method for 2nd order elliptic problems*, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR**0483555****[15]**Peter H. Sammon,*Numerical approximations for a miscible displacement process in porous media*, SIAM J. Numer. Anal.**23**(1986), no. 3, 508–542. MR**842642**, https://doi.org/10.1137/0723034**[16]**A. H. Schatz, V. C. Thomée, and L. B. Wahlbin,*Maximum norm stability and error estimates in parabolic finite element equations*, Comm. Pure Appl. Math.**33**(1980), no. 3, 265–304. MR**562737**, https://doi.org/10.1002/cpa.3160330305**[17]**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, https://doi.org/10.1090/S0025-5718-1976-0436617-2**[18]**J. M. Thomas,*Sur l'Analyse Numérique des Méthodes d'Éléments Finis Hybrides et Mixtes*, Thèse, Université Pierre et Marie Curie, Paris, 1977.**[19]**Mary Fanett Wheeler,*A priori 𝐿₂ error estimates for Galerkin approximations to parabolic partial differential equations*, SIAM J. Numer. Anal.**10**(1973), 723–759. MR**0351124**, https://doi.org/10.1137/0710062

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0717695-3

Article copyright:
© Copyright 1983
American Mathematical Society