Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models
- Authors
- Hui Zhou (ConocoPhillips Subsurface Technology) | Hamdi A. Tchelepi (Stanford University)
- DOI
- https://doi.org/10.2118/141473-PA
- Document ID
- SPE-141473-PA
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Journal
- Volume
- 17
- Issue
- 02
- Publication Date
- June 2012
- Document Type
- Journal Paper
- Pages
- 523 - 539
- Language
- English
- ISSN
- 1086-055X
- Copyright
- 2012. Society of Petroleum Engineers
- Disciplines
- 4.1.2 Separation and Treating, 4.3.4 Scale
- Keywords
- Large-scale simulation, Reservoir Simulation, Multiscale method
- Downloads
- 2 in the last 30 days
- 465 since 2007
- Show more detail
- View rights & permissions
| SPE Member Price: | USD 5.00 |
| SPE Non-Member Price: | USD 35.00 |
Summary
An efficient two-stage algebraic multiscale solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU) or the Additive Schwarz (AS) method, is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.
| File Size | 4 MB | Number of Pages | 17 |
References
Aarnes, J.E. 2004. On the use of a mixed multiscale finite element methodfor greater flexibility and increased speed or improved accuracy in reservoirsimulation. Multiscale Model. Simul. 2 (3): 421-439. http://dx.doi.org/10.1137/030600655.
Aarnes, J.E. and Hou, T.Y. 2002. Multiscale Domain Decomposition Methods forElliptic Problems with High Aspect Ratios. Acta Mathematicae ApplicataeSinica (English Series) 18 (1): 63-76. http://dx.doi.org/10.1007/s102550200004.
Aavatsmark, I. 2002. An Introduction to Multipoint Flux Approximations forQuadrilateral Grids. Comput. Geosci. 6 (3-4): 405-432. http://dx.doi.org/10.1023/A:1021291114475.
Arbogast, T. 2002. Implementation of a Locally Conservative NumericalSubgrid Upscaling Scheme for Two-Phase Darcy Flow. Comput. Geosci. 6 (3): 453-481. http://dx.doi.org/10.1023/a:1021295215383.
Arbogast, T. and Bryant, S.L. 2002. A Two-Scale Numerical Subgrid Techniquefor Waterflood Simulations. SPE J. 7 (4): 446-457.SPE-81909-PA. http://dx.doi.org/10.2118/81909-PA.
Balay, S., Buschelman, K., Eijkhout, V., et al. 2008. PETSc users manual,Revision 3.0.0. Technical Report ANL-95/11, US DOE/Argonne National Laboratory,Argonne, Illinois.
Cai, X.C., Gropp, W.D., and Keyes, D.E. 1992. A comparison of some domaindecomposition algorithms for nonsymmetric elliptic problems. In FifthInternational Symposium on Domain Decomposition Methods for PartialDifferential Equations, ed. D.E. Keyes, T.F. Chan, G.A. Meurant, J.S.Scroggs, and R.G. Voigt, 224-235. Philadelphia, Pennsylvania: Proceedings inApplied Mathematics, SIAM.
Chen, Y., Durlofsky, L.J., Gerritsen, M., and Wen, X.H. 2003. A coupledlocal-global upscaling approach for simulating flow in highly heterogeneousformations. Adv. Water Resour. 26 (10): 1041-1060. http://dx.doi.org/10.1016/S0309-1708(03)00101-5.
Chen, Z. and Hou, T. 2003. A mixed multiscale finite element method forelliptic problems with oscillating coefficients. Mathematic Computation 72: 541-576. http://dx.doi.org/10.1090/S0025-5718-02-01441-2.
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. SPE-72469-PA. http://dx.doi.org/10.2118/72469-PA.
Deutsch, C.V. and Journel, A.G. 1998. GSLIB: Geostatistical SoftwareLibrary and User's Guide, second edition. Oxford, UK: Oxford UniversityPress.
Durlofsky, L.J. 1991. Numerical Calculation of Equivalent Grid BlockPermeability Tensors for Heterogeneous Porous Media. Water Resour. Res. 27 (5): 699-708. http://dx.doi.org/10.1029/91WR00107.
Graham, I., Lechner, P., and Scheichl, R. 2007. Domain decomposition formultiscale PDEs. Numerische Mathematik 106 (4): 589-626. http://dx.doi.org/10.1007/s00211-007-0074-1.
Hajibeygi, H., Bonfigli, G., Hesse, M.A., and Jenny, P. 2008. Iterativemultiscale finite-volume method. J. Comput. Phys. 227 (19):8604-8621. http://dx.doi.org/10.1016/j.jcp.2008.06.013.
Hesse, M.A., Mallison, B.T., and Tchelepi, H.A. 2009. Compact MultiscaleFinite Volume Method for Heterogeneous Anisotropic Elliptic Equations.Multiscale Modeling & Simulation 7 (2): 934-962. http://dx.doi.org/10.1137/070705015.
Hou, T.Y. and Wu, X.-H. 1997. A Multiscale Finite Element Method forElliptic Problems in Composite Materials and Porous Media. J. Comput.Phys. 134 (1): 169-189. http://dx.doi.org/10.1006/jcph.1997.5682.
Jenny, P., Lee, S.H., and Tchelepi, H. 2004. Adaptive multiscalefinite-volume method for multiphase flow and transport in porous media.Mulitscale Modeling and Simulation 3 (1): 50-64. http://dx.doi.org/10.1137/030600795.
Jenny, P., Lee, S.H., and Tchelepi, H. 2006. Adaptive fully implicitmulti-scale finite-volume method for multi-phase flow and transport inheterogeneous porous media. J. Comput. Phys. 217 (2):627-641. http://dx.doi.org/10.1016/j.jcp.2006.01.028.
Jenny, P., Lee, S.H., and Tchelepi, H. 2003. Multi-scale finite-volumemethod for elliptic problems in subsurface flow simulation. J. Comput.Phys. 187 (1): 47-67. http://dx.doi.org/10.1016/S0021-9991(03)00075-5.
Kippe, V., Aarnes, J.E., and Lie, K.-A. 2008. A comparison of multiscalemethods for elliptic problems in porous media flow. Comput. Geosci. 12 (3): 377-398. http://dx.doi.org/10.1007/s10596-007-9074-6.
Lee, S., Wolfsteiner, C., and Tchelepi, H. 2008. Multiscale finite-volumeformulation for multiphase flow in porous media: black oil formulation ofcompressible, three-phase flow with gravity. Comput. Geosci. 12 (3): 351-366. http://dx.doi.org/10.1007/s10596-007-9069-3.
Lee, S.H., Zhou, H., and Tchelepi, H.A. 2009. Adaptive multiscalefinite-volume method for nonlinear multiphase transport in heterogeneousformations. J. Comput. Phys. 228 (24): 9036-9058. http://dx.doi.org/10.1016/j.jcp.2009.09.009.
Lunati, I. and Jenny, P. 2006. Multiscale finite-volume method forcompressible multiphase flow in porous media. J. Comput. Phys. 216 (2): 616-636. http://dx.doi.org/10.1016/j.jcp.2006.01.001.
Lunati, I. and Jenny, P. 2008. Multiscale finite-volume method fordensity-driven flow in porous media. Comput. Geosci. 12(3): 337-350. http://dx.doi.org/10.1007/s10596-007-9071-9.
Lunati, I. and Jenny., P. 2004. Multi-scale finite-volume method for highlyheterogeneous porous media with shale layers. Paper presented at the 9thEuropean Conference on the Mathematics of Oil Recovery (ECMOR), Cannes, France,30 August-2 September.
Nordbotten, J. and Bjørstad, P. 2008. On the relationship between themultiscale finite-volume method and domain decomposition preconditioners.Comput. Geosci. 12 (3): 367-376. http://dx.doi.org/10.1007/s10596-007-9066-6.
Peaceman, D.W. 1977. Fundamentals of Numerical Reservoir Simulation.Oxford, UK: Elsevier Publishing.
Saad, Y. 1996. Iterative Methods for Sparse Linear Systems. Boston,Massachusetts: PWS Publishing Company.
Smith, B.F., Bjørstand, P.E., and Gropp, W.D. 1996. Domain Decomposition:Parallel Multilevel Methods for Elliptic Partial Differential Equations.Cambridge, UK: Cambridge University Press.
Stüben, K. 2000. An Introduction to Algebraic Multigrid. InMultigrid, ed. U. Trottenberg, C.W. Ooosterlee, and A. Schüller,Appendix, 413-532. London: Academic Press.
Tchelepi, H.A., Jenny, P., Lee, S.H., and Wolfsteiner, C. 2007. AdaptiveMultiscale Finite-Volume Framework for Reservoir Simulation. SPE J.12 (2): 188-195. SPE-93395-PA. http://dx.doi.org/10.2118/93395-PA.
Toselli, A. and Wildlund, O. 2005. Domain DecompositionMethods--Algorithms and Theory. Berlin, Germany: Springer Series inComputational Mathematics, Springer-Verlag.
Wallis, J.R. and Tchelepi, H.A. 2010. Apparatus, method and system forimproved reservoir simulation using an algebraic cascading class linear solver.US Patent No. 7,684,967.
Zhou, H. 2006. Operator based multiscale method for compressibleflow. MS thesis, Stanford University, Stanford, California.
Zhou, H. and Tchelepi, H.A. 2008. Operator-Based Multiscale Method forCompressible Flow. SPE J. 13 (2): 267-272.SPE-106254-PA. http://dx.doi.org/10.2118/106254-PA.
