Operator-splitting techniques are applied to the nonlinear coupled systems of partial differential equations used to model multiphase or multicomponent flows in porous media. Convection is treated by time-stepping along the characteristics of the associated pure convection problem, and diffusion is modeled via Petrov-Galerkin finite element techniques. Accurate approximations of the fluid velocities needed in the modified method of characteristic time-stepping are obtained by mixed finite element methods. Substructuring techniques are used to develop self-adaptive local grid-refinement enhance ments to the operator-splitting methods described above. Aspects of large-scale computation will be emphasized.
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References
1.
Appleyard, J.R., Cheshire, I.M., and Pollard, R.K.1981. Special techniques for fully-implicit simulators . Presented at the European Symposium on Enhanced Oil Recovery, Bournemouth, U.K.
2.
Barrett, J.W., and Morton, K.W.1984. Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems. Comput. Methods Appl. Mech. Engrg.45:97-122.
3.
Bramble, J.H., Ewing, R.E., Pasciak, J.E., and Schatz, A.H.1988. A preconditioning technique for the efficient solution of problems with local grid refinement. Comput. Methods Appl. Mech. Engrg.67:149-159.
4.
Bramble, J.H., Pasciak, J.E., and Schatz, A.H.1986. An iterative method for elliptic problems on regions partitioned into substructures. Math. Comp.46:361-370.
5.
Chavent, G.1976. A new formulation of diphasic incompressible flows in porous media. In Lecture Notes in Math. 503.
6.
Dahle, H., Espedal, M.S., and Ewing, R.E.1988. Characteristic Petrov-Galerkin subdomain methods for convection diffusion problems. In IMA vol. 11, Numerical simulation in oil recovery, edited by M. F. Wheeler.Berlin: Springer-Verlag , pp. 77-88.
7.
Demkowitz, L., and Oden, J.T.1986. An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables. Comput. Methods Appl. Mech. Engrg.55:63-87.
8.
Douglas, J. Jr., Dupont, T., and Ewing, R.E.1979. Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem. SIAM J. Numer. Anal.16:503-522.
9.
Elinan, H.C.1981. Preconditioned conjugate-gradient methods for non-symmetric systems of linear equations. Research Report 203New Haven, Conn.: Yale University, Department of Computer Science .
10.
Espedal, M., and Ewing, R.E.1987. Characteristic Petrov-Galerkin subdomain methods for two-phase immiscible flow. Comput. Methods Appl. Mech. Engrg.64:113-135.
11.
Ewing, R.E.1978. Timestepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal.15:1125-1150.
12.
Ewing, R.E.1983. Problems arising in the modeling of processes for hydrocarbon recovery. In The mathematics of reservoir simulation , edited by R. E. Ewing, Frontiers in applied mathematics, vol. 1. Philadelphia: SIAM, pp. 3-34.
13.
Ewing, R.E., Lazarov, R.D., and Vassilevski, P.S.1988. Local refinement techniques for elliptic problems on cell-centered grids. Report No. 1988-15. Laramie, Wyo.: Institute for Scientific Computation .
14.
Ewing, R.E., McCormick, S., and Thomas, J.1984. The fast composite grid method for solving differential boundary-value problems. In Proceedings fifth ASCE specialty conference. New York: ASCE, pp. 1453-1456.
15.
Ewing, R.E., Russell, T.F., and Wheeler, M.F.1983. Simulation of miscible displacement using mixed methods and a modified method of characteristics. In SPE reprint series no. 20, numerical simulation II. Dallas: Society of Petroleum Engineers.
16.
Ewing, R.E., Russell, T.F., and Wheeler, M.F.1984. Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics . Comput. Methods Appl. Mech. Engrg.47:73-92.
17.
Mandel, J., and McCormick, S.1988. Iterative solution of elliptic equations with refinement: the two-level case. Presented at the second international symposium on domain decomposition methods, Los Angeles , January 1988.
18.
McCormick, S., and Thomas, J.1986. The fast adaptive composite grid method for elliptic boundary value problems. Math. Camp.46:439-456.
19.
Meijerink, J.A.1983. Iterative methods for the solution of linear equations based on incomplete block factorizations of the matrix. SPE 12262. In Proceedings, seventh SPE symposium on reservoir simulation . Dallas: SPE.
20.
Obeysekare, U., Allen, M., Ewing, R.E., and George, J.H.1987. Application of conjugate gradient-like methods to a hyperbolic problem in porous media flow. Internat. J. Numer. Methods Fluids7:551-566.
21.
Russell, T.F.1985. The timestepping along characteristics with incomplete iteration for Galerkin approximation of miscible displacement in porous media . SIAM J. Numer. Anal.22:970-1013.
22.
Vinsome, P.K.W.1976. Orthomin, an iterative method for solving sparse sets of simultaneous linear equations. SPE 5729. In Proceedings of the fourth SPE symposium on numerical simulation of reservoir performance. Dallas: SPE.
