Fully discrete finite element analysis of multiphase flow in groundwater hydrology

SIAM Journal on Numerical Analysis

Volumn 34, Issue 6, 1997, Pages 2228-2253

  Chen, Zhangxin ^a   Ewing, Richard E ^b  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Air water system; Compressible flow; Error estimate; Finite element; Mixed method; Numerical experiments; Porous media; Time discretization

Indexed keywords

EID: 0000761187     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995290063     Document Type: Article

References (41)

1. S. ANTONCEV, On the solvability of boundary value problems for degenerate two-phase porous flow equations, Dinamika Sploshn. Sredy Vyp., 10 (1972), pp. 28-53 (in Russian).
2. D. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 7-32.
3. J. BEAR, Dynamics of Fluids in Porous Media, Dover, New York, 1972.
4. F. BREZZI, J. DOUGLAS, JR., R. DURÁN, AND M. FORTIN, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), pp. 237-250.
5. F. BREZZI, J. DOUGLAS, JR., M. FORTIN, AND L. MARINI, Efficient, rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér, 21 (1987), pp. 581-604.
6. F. BREZZI, J. DOUGLAS, JR., AND L. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217-235.
7. M. CELIA AND P. BINNING, Two-phase unsaturated flow: One dimensional simulation and air phase velocities, Water Resources Research, 28 (1992), pp. 2819-2828.
8. G. CHAVENT AND J. JAFFRÉ, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam, 1978.
9. Z. CHEN, Analysis of mixed methods using conforming and nonconforming finite element methods, RAIRO Modèl. Math. Anal. Numér., 27 (1993), pp. 9-34.
10. Z. CHEN AND J. DOUGLAS, JR., Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems, Mat. Ap. Comput., 10 (1991), pp. 137-160.
11. Z. CHEN AND J. DOUGLAS, JR., Prismatic mixed finite elements for second order elliptic problems, Calcolo, 26 (1989), pp. 135-148.
12. Z. CHEN, M. ESPEDAL, AND R. EWING, Continuous-time finite element analysis of multiphase flow in groundwater hydrology, Appl. Math., 40 (1995), pp. 203-226.
13. Z. CHEN AND R. EWING, Stability and convergence of a finite element method for reactive transport in ground water, SIAM J. Numer. Anal., 34 (1995), pp. 881-904.
14. Z. CHEN. R. EWING, AND M. ESPEDAL, Multiphase flow simulation with various boundary conditions, in Numerical Methods in Water Resources, Vol. 2, A. Peters, et al., eds., Kluwer Academic Publishers, Amsterdam, Netherlands, 1994, pp. 925-932.
15. S. CHOU AND Q. LI, Mixed finite element methods for compressible miscible displacement in porous media, Math. Comp., 57 (1991), pp. 507-527.
16. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
17. J. DOUGLAS, JR., Finite difference methods for two-phase incompressible flow in porous media, SIAM J. Numer. Anal., 20 (1983), pp. 681-696.
18. J. DOUGLAS, R. EWING, AND M. WHEELER, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Modél Math. Anal. Numér., 17 (1983), pp. 17-33.
19. J. DOUGLAS, JR. AND J. ROBERTS, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp., 41 (1983), pp. 441-459.
20. J. DOUGLAS, JR. AND J. ROBERTS, Global estimates for mixed methods for second order elliptic problems. Math. Comp., 45 (1985), pp. 39-52.
21. N. ESPEDAL AND R. EWING, Characteristic Petrov-Galerkin subdomain methods for two phase immiscible flow, Comput. Methods Appl. Mech. Engrg., 64 (1987), pp. 113-135.
22. R. EWING AND T. RUSSELL, Efficient time-stepping methods for miscible displacement problems in porous media, SIAM J. Numer. Anal., 19 (1982), pp. 1-66.
23. R. EWING AND M. WHEELER, Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case, in Mathematical Methods in Energy Research, K. I. Gross, ed., SIAM, Philadelphia, PA, 1984, pp. 40-58.
24. R. EWING, Y. YUAN, AND G. LI, Timestepping along characteristics for a mixed finite element approximation for compressible flow of contamination from nuclear waste in porous media, SIAM J. Numer. Anal., 26 (1989), pp. 1513-1524.
25. K. FADIMBA AND R. SHARPLEY, A priori estimates and regularization for a class of porous medium equations, Nonlinear World, 2 (1995), pp. 13-41.
26. K. FADIMBA AND R. SHARPLEY, Galerkin finite element method for a class of porous medium equations, preprint.
27. D. HILLEL, Fundamentals of Soil Physics, Academic Press, San Diego, CA, 1980.
28. C. JOHNSON AND V. THOMÉE, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Modél. Math. Anal. Numér., 15 (1981), pp. 41-78.
29. H. MOREL-SEYTOUX, Two-phase flows in porous media, Advances in Hydroscience, 9 (1973), pp. 119-202.
30. 3, Numer. Math., 35 (1980), pp. 315-341.
31. ∞-Convergence of finite element approximation, in Proc. 2nd Conference on Finite Elements, Rennes, France, 1975.
32. D. PEACEMAN, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977.
33. O. PIRONNEAU, On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math., 38 (1982), pp. 309-332.
34. T. POTEMPA AND M. WHEELER, An Implicit Diffusive Numerical Procedure for a Slightly Compressible Miscible Displacement Problem in Porous Media, Lecture Notes in Math. 1066, Springer-Verlag, Berlin, 1983, pp. 239-263.
35. P. RAVIART AND J. THOMAS, A Mixed Finite Element Method for Second Order Elliptic Problems, Lecture Notes in Math. 606, Springer-Verlag, Berlin, 1977, pp. 292-315.
36. M. ROSE, Numerical methods for flow through porous media I, Math. Comp., 40 (1983), pp. 437-467.
37. A. SCHATZ, V. THOMÉE, AND L. WAHLBIN, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math., 33 (1980), pp. 265-304.
38. ∞ estimates for the finite element method on irregular meshes, Math. Comp., 30 (1976), pp. 681-697.
39. D. SMYLIE, A Near Optimal Order Approximation to a Class of Two Sided Nonlinear Degenerate Parabolic Partial Differential Equations, Ph.D. thesis, University of Wyoming, Laramie, WY, 1989.
40. V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math. 1054, Springer-Verlag, Berlin, New York, 1984.
41. 2 error estimates for Galerkin approximation to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723-759.