Abstract
A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective.
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References
T. Arbogast, M.F. Wheeler and I. Yotov, Logically rectangular mixed methods for groundwater flow and transport on general geometry, Dept. Comp. Appl. Math. TR94-03, Rice University, Houston, TX (January 1994).
T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal. 34 (1997) (828–852.
O. Axelsson, Iterative Solution Methods(Cambridge University Press, Cambridge, 1994).
K. Aziz and A. Sethari, Petroleum Reservoir Simulation (Applied Science Publisher, 1983).
R. Bank, T. Chan, W. Coughran and K. Smith, The alternate-block-factorization procedure for systems of partial differential equations, BIT 29 (1989) (938–954.
R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (SIAM, Philadelphia, PA, 1994).
J. Bear, Dynamics of Fluids in Porous Media (Elsevier, New York, 1972).
G. Behie and P. Forsyth, Incomplete factorization methods for fully implicit simulation of enhanced oil recovery, SIAM J. Sci. Statist. Comput. 5 (1984) (543–561.
G. Behie and P. Vinsome, Block iterative methods for fully implicit reservoir simulation, Soc. of Petroleum Engineers J. (1982) (658–668.
P. Bjorstad, W.C. Jr. and E. Grosse, Parallel domain decomposition applied to coupled transport equations, in: Seventh International Conference on Domain Decomposition Methods for Scientific Computing, eds. D. Keyes and J. Xu, Como, Italy (American Mathematical Society, 1993).
P.N. Brown, A. Hindmarsh and L. Petzold, Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput. 15 (1994) (1467–1488.
P.N. Brown and Y. Saad, Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Statist. Comput. 11 (1990) (450–481.
P.N. Brown and Y. Saad, Convergence theory of nonlinear Netwon-Krylov algorithms, SIAM J. Optim. 4 (1994) (297–330.
G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation (North-Holland, Amsterdam, 1986).
C.N. Dawson, M.F. Wheeler and C.S. Woodward, A two-grid finite difference scheme for nonlinear parabolic equations, Center for Applied Scientific Computing UCRL-JC-125599, Lawrence Livermore National Laboratory, Livermore, CA (October 1996) to appear in SIAM J. Numer. Anal.
J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).
L.J. Durlovsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Research 27 (1991) (699–708.
S. Eisenstat and H. Walker, Globally convergent inexact Newton methods, SIAM J. Optim. 4 (1994) 393–422.
S. Eisenstat and H. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput. 17 (1996) (16–32.
A. Ern, V. Giovangigli, D. Keyes and M.D. Smooke, Towards polyalgorithmic linear system solvers for nonlinear elliptic problems, SIAM J. Sci. Comput. 15 (1994) (681–703.
R. Ewing, The mathematics of reservoir simulation, in: Frontiers in Applied Mathematics (SIAM, Philadelphia, PA, 1983).
P.A. Forsyth, Y.S. Wu and K. Pruess, Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media, Advances in Water Resources 18 (1995) (25–38.
R.A. Freeze and J.A. Cherry, Groundwater (Prentice-Hall, Englewood Cliffs, NJ, 1979).
G. Golub and C. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, MD, 1989).
S. Gomez and J. Morales, Performance of Chebyshev iterative method, GMRES and ORTHOMIN on a set of oil reservoir simulation problems, in: Mathematics for Large Scale Computing (J.C. Diaz, Basel, 1989) (pp. 265–295.
T. Kerkhoven and Y. Saad, On acceleration methods for coupled nonlinear elliptic problems, Numer. Math. 60 (1992) (525–548.
H. Klíe, Krylov-secant methods for solving large-scale systems of coupled nonlinear parabolic equations, Ph.D. thesis, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX (September 1996).
H. Klíe, M. Ramé and M. Wheeler, Fast Krylov-secant solvers for systems of nonlinear partial differential equations, Tech. Rep. TR96-30, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX (1996).
H. Klíe, M. Ramé and M. Wheeler, Two-stage preconditioners for inexact Newton methods in multi-phase reservoir simulation, Tech. Rep. CRPC-TR96641, Center for Research on Parallel Computation, Rice University, Houston, TX (1996).
V. Kumar, A. Grama, A. Gupta and G. Karypis, Introduction to Parallel Computing: Design and Analysis of Algorithms (Benjamin/Cummings, 1994).
C. Mattax and R. Dalton, Reservoir Simulation, SPE Monograph Series 13 (Richardson, TX, 1990).
J. Nedelec, Mixed finite elements in R3, Numer. Math. 35 (1980) (315–341.
D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation (Elsevier, Amsterdam, 1977).
P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in: Mathematical Aspects of Finite Element Methods, eds. I. Galligani and E. Magenes, Lecture Notes in Mathematics 606 (Springer, Berlin, 1977) pp. 292–315.
M.T. van Genuchten, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Amer. J. 44 (1980) (892–898.
J. Wallis, Two-step preconditioning, private communication (1994).
J. Watts, A method of improving line succesive overrelaxation in anisotropic problems-a theoretical analysis, Soc. of Petroleum Engineers J. (1973) (105–118.
J. Wheeler and R. Smith, Reservoir simulation on a hypercube, in: 64th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, SPE paper No. 19804, San Antonio, TX (1989).
L. Wigton, D. Yu and N. Young, GMRES acceleration of computational fluid dynamics codes, in: Proceedings 1985 AIAA Conference, Denver, CO (1985).
J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996) (1759–1777.
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Dawson, C.N., Klíe, H., Wheeler, M.F. et al. A parallel, implicit, cell‐centered method for two‐phase flow with a preconditioned Newton–Krylov solver. Computational Geosciences 1, 215–249 (1997). https://doi.org/10.1023/A:1011521413158
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DOI: https://doi.org/10.1023/A:1011521413158