Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations
- Sebastian Gries (Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)) | Klaus Stüben (Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)) | Geoffrey L. Brown (Computer Modelling Group Limited) | Dingjun Chen (Computer Modelling Group Limited) | David A. Collins (Computer Modelling Group Limited)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- August 2014
- Document Type
- Journal Paper
- 726 - 736
- 2014.Society of Petroleum Engineers
- 5.5 Reservoir Simulation
- Constrained Pressure residual, preconditioning, decoupling, dynamic rowsumming, Algebraic Multigrid
- 1 in the last 30 days
- 318 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 10.00|
|SPE Non-Member Price:||USD 30.00|
Fully implicit black-oil simulations result in huge, often very ill-conditioned, linear systems of equations for different unknowns (e.g., pressure and saturations). It is well-known that the underlying Jacobian matrices contain both hyperbolic and nearly elliptic subsystems (corresponding to saturations and pressure, respectively). Because a reservoir simulation is typically driven by the behavior of the pressure, contrained-pressure-residual (CPR) -type two-stage preconditioning methods to solve the coupled linear systems are a natural choice and still belong to the most popular approaches. After a suitable extraction and decoupling, the computationally most costly step in such two-stage methods consists in solving the elliptic subsystems accurately enough. Algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. Hence, in recent years, corresponding CPR-AMG approaches have been extensively used in practice. Unfortunately, if applied in a straightforward manner, CPR-AMG does not always work as expected. In this paper, we discuss the reasons for the lack of robustness observed in practice, and present remedies. More precisely, we will propose a preconditioning strategy (based on a suitable combination of left and right preconditioning of the Jacobian matrix) that aims at a compromise between the solvability of the pressure subproblem by AMG and the needs of the outer CPR process. The robustness of this new preconditioning strategy will be demonstrated for several industrial test cases, some of which are very ill-conditioned. Furthermore, we will demonstrate that CPR-AMG can be interpreted in a natural way as a special AMG process applied directly to the coupled Jacobian systems.
|File Size||1 MB||Number of Pages||11|
Axelsson, O. 1994. Iterative Solution Methods, Cambridge University Press.
Aziz, K., and Settari, A. 1979. Petroleum Reservoir Simulation, Applied Sciences Publishers.
Bank, R., Chan, T., Coughran, W. et al. 1989. The Alternate-Block-Factorization Procedure for Systems of Partial Differential Equations, AT & T Bell Laboratories, BIT Computer Science and Numerical Mathematics, Lawrence.
Batycky, R.P., Förster, M., Thiele, M.R. et al. 2009. Parallelization of a Commercial Streamline Simulator and Performance on Practical Models. Paper SPE 118684 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2−4 February. http://dx.doi.org/10.2118/118684-MS.
Boehmer, K., Hemker, P., and Stetter, H. 1984. The Defect Correction Approach. Computing Supplement 5: 1−32.
Bridson, R. and Tang, W. 2001. A Structural Diagnosis of Some IC Orderings. SIAM J. Sci. Comput. 22 (5): 1527–1532.
Cao, H. 2002. Development of Techniques for General Purpose Simulators, PhD thesis, Stanford University.
Cao, H., Tchelepi, H., and Wallis, J. 2005. Parallel Scalable Unstructured CPR-type Linear Solver for Reservoir Simulation. Paper SPE 96809 presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 9−12 October. http://dx.doi.org/10.2118/96809-MS.
Chow, E. and Saad, Y. 1997. Experimental Study of ILU Preconditioners for Indefinite Matrices. J. Comput. Appl. Math. 86 (2): 387−414.
Christie, M.A. and Blunt, M.J. 2001. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Res Eval & Eng 4 (4): 308−317. http://dx.doi.org/10.2118/72469-PA.
Cleary, A., Falgout, R., Henson, V. et al. 1998. Robustness and Scalability of Algebraic Multigrid. SIAM J. Sci. Comput. 21: 1886−1908.
Clees, T. 2005. AMG Strategies for PDE Systems With Applications in Industrial Semiconductor Simulation, Shaker Verlag.
Clees, T. and Ganzer, L. 2010. An Efficient Algebraic Multigrid Solver Strategy for Adaptive Implicit Methods in Oil-Reservoir Simulation. SPE J. 15 (3): 670−681. http://dx.doi.org/10.2118/105789-PA.
Coats, K. 2000. A Note on IMPES and Some IMPES-Based Simulation Models. SPE J. 5 (3): 245−251. http://dx.doi.org/10.2118/65092-PA.
Collins, D.A., Grabenstetter, J.E., and Sammon, P.H. 2003. A Shared-Memory Parallel Black-Oil Simulator With a Parallel ILU Linear Solver. Paper SPE 79713 presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 3−5 February. http://dx.doi.org/10.2118/79713-MS.
Forsyth, P.A. and Sammon, P.H. 1986. Practical Considerations for Adaptive Implicit Methods in Reservoir Simulation. J. Comput. Physics 62 (2): 265−281.
Geiger, S., Huangfu, Q., Reid, F. et al. 2009. Massively Parallel Sector Scale Discrete Fracture and Matrix Simulations. Paper SPE 118924 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2−4 February. http://dx.doi.org/10.2118/118924-MS.
Hendrikson, B. and Rothberg, E. 1999. Improving the Run Time and Quality of Nested Dissection Ordering. SIAM J. Sci. Comput. 20 (2): 468−489.
Jiang, Y. 2007. Techniques for Modeling Complex Reservoirs and Advanced Wells, PhD thesis, Stanford University.
Klie, H. 1996. Krylov-Secant Methods for Solving Large-Scale Systems of Coupled Nonlinear Parabolic Equations, Rice University, Houston, Texas.
Klie, H., Ramé, M., and Wheeler, M. 1996. Two-stage Preconditioners for Inexact Newton Methods in Multi-Phase Reservoir Simulation, CRPC Report, Rice University, Houston, Texas.
Lacroix, S., Vassilevski, Y., and Wheeler, M. 2000. Iterative Solvers of the Implicit Parallel Accurate Reservoir Simulator (IPARS), I: Single Processor Case. Numer. Linear Algebra Appl.
Lacroix, S., Vassilevski, Y., Wheeler, J. et al. 2003. Iterative Solution Methods for Modeling Multiphase Flow in Porous Media Fully Implicitly. SIAM J. on Scientific Computing 25: 905−926.
Lions, P.L. 1988. On the Schwarz Alternating Method. In Domain Decomposition Methods for Partial Differential Equations, ed. Glowinski et al. Philadelphia: SIAM.
Masson, R., Quandalle, P., Requena, S. et al. 2004. Parallel Preconditioning for Sedimentary Basin Simulations. Lecture Notes in Computer Sci. 2907/2004: 59−81.
Naumovich, A., Förster, M., and Dwight, R. 2010. Algebraic Multigrid Within Defect Correction for the Linearized Euler Equations. Numer. Linear Algebra with Appl.17 (2−3): 307−324.
Ruge, J. and Stüben, K. 1986. Algebraic Multigrid (AMG). In Multigrid Methods, ed. S.F. McCormick, Philadelphia: SIAM, Frontiers in Applied Mathematics Vol. 5.
Saad, Y. 1996. Iterative Methods for Sparse Linear Systems, Boston, Massachusettes: PWS Publishing Company.
Scheichl, R., Masson, R., and Wendebourg, J.2003. Decoupling and Block Preconditioning for Sedimentary Basin Simulations. Comput. Geosci. 7: 295−318.
Schwarz, H. 1870. Über Einen Grenzübergang Durch Alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15: 272−286.
Stüben, K. 2001a. A Review of Algebraic Multigrid. J. Comput. Appl. Math. 128: 281−309.
Stüben, K. 2001b. An Introduction to Algebraic Multigrid, pp. 413−532. In Multigrid, Academic Press.
Stüben, K. 2007. Solving Reservoir Simulation Equations. Paper presented at the 9th International Forum on Reservoir Simulation, Abu Dhabi, United Arab Emirates, 9−13 December.
Stüben, K. 2012. SAMG User’s Manual, Fraunhofer SCAI.
Stüben, K., Clees, T., Klie, H. et al. 2007. Algebraic Multigrid Methods (AMG) for the Efficient Solution of Fully Implicit Formulations in Reservoir Simulation. Paper SPE 105832 presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 26−28 February. http://dx.doi.org/10.2118/105832-MS.
Trottenberg, U., Oosterlee, C., Schüller, A. et al. 2001. Multigrid, Academic Press.
Wallis, J. 1983. Incomplete Gaussian Elimination as a Preconditioning for Generalized Conjugate Gradient Acceleration. Paper SPE 12265 presented at the SPE Reservoir Simulation Symposium, San Francisco, California, 15−18 November. http://dx.doi.org/10.2118/12265-MS.
Wallis, J., Kendall, R., Little, T. et al. 1985. Constrained Residual Acceleration of Conjugate Residual Methods. Paper SPE 13536 presented at the SPE Reservoir Simulation Symposium, Dallas, Texas, 10−13 February. http://dx.doi.org/10.2118/13536-MS.