Use of Multiple Multiscale Operators To Accelerate Simulation of Complex Geomodels
- Knut-Andreas Lie (SINTEF) | Olav Møyner (SINTEF) | Jostein R. Natvig (Schlumberger Information Solutions)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2017
- Document Type
- Journal Paper
- 1,929 - 1,945
- 2017 Norway. Published by the Society of Petroleum Engineers. This is an open access article.
- Multiscale simulation
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Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use both a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features such as faults, fractures, facies, or other geobodies; or a particular computational challenge such as propagating displacement and chemical fronts, wells being turned on or off, and others. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a nonoverlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse gridblock. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators—computed on different coarse grids and possibly also by different basis-function formulations—to be combined into one iterative procedure.
Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators that have different spatial resolution and that each individual operator can be designed to target, among others, challenging grids, including faults, pinchouts, and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.
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Akkutlu, I. Y., Efendiev, Y., Vasilyeva, M. et al. 2017. Multiscale Model Reduction for Shale Gas Transport in a Coupled Discrete Fracture and Dual-Continuum Porous Media. J. Nat. Gas Sci. Eng. (in press) https://doi.org/10.1016/j.jngse.2017.02.040.
Bush, L., Ginting, V., and Presho, M. 2014. Application of a Conservative, Generalized Multiscale Finite Element Method to Flow Models. J. Comput. Appl. Math. 260: 395–409. https://doi.org/10.1016/j.cam.2013.10.006.
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. https://doi.org/10.2118/72469-PA.
Chung, E. T., Efendiev, Y., and Li, G. 2014. An Adaptive GMsFEM for High-Contrast Flow Problems. J. Comput. Phys. 273: 54–76. https://doi.org/10.1016/j.jcp.2014.05.007.
Cortinovis, D. and Jenny, P. 2014. Iterative Galerkin-enriched Multiscale Finite-Volume Method. J. Comput. Phys. 277: 248–267. https://doi.org/10.1016/j.jcp.2014.08.019.
Cortinovis, D. and Jenny, P. 2017. Zonal Multiscale Finite-Volume Framework. J. Comput. Phys. 337: 84–97. https://doi.org/10.1016/j.jcp.2017.01.052.
Cusini, M., van Kruijsdijk, C., and Hajibeygi, H. 2016. Algebraic Dynamic Multilevel (ADM) Method for Fully Implicit Simulations of Multiphase Flow in Porous Media. Journal of Computational Physics 314: 60–79. https://doi.org/10.1016/j.jcp.2016.03.007.
Dolean, V., Jolivet, P., Nataf, F. et al. 2014. Two-Level Domain Decomposition Methods for Highly Heterogeneous Darcy Equations. Connections With Multiscale Methods. Oil & Gas Science and Technology–Rev. IFP Energies Nouvelles 69 (4): 731–752. https://doi.org/10.2516/ogst/2013206.
Efendiev, Y. and Hou, T. Y. 2009. Multiscale Finite Element Methods, Vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences. New York: Springer Verlag.
Efendiev, Y., Galvis, J., and Wu, X.-H. 2011. Multiscale Finite Element Methods for High-Contrast Problems Using Local Spectral Basis Functions. J. Comput. Phys. 230 (4): 937–955. https://doi.org/10.1016/j.jcp.2010.09.026.
Efendiev, Y., Galvis, J., and Hou, T. Y. 2013. Generalized Multiscale Finite Element Methods (GMsFEM). J. Comput. Phys. 251: 116–135. https://doi.org/10.1016/j.jcp.2013.04.045.
Fossen, H. and Hesthammer, J. 1998. Structural Geology of the Gullfaks Field. In Structural Geology in Reservoir Characterization, ed. Coward M. P., Johnson H., and Daltaban T. S., Vol. 127, pp. 231–261. Geological Society Special Publication.
Hajibeygi, H. and Tchelepi, H. A. 2014. Compositional Multiscale Finite-Volume Formulation. SPE J. 19 (2): 316–326. SPE-163664-PA. https://doi.org/10.2118/163664-PA.
Hauge, V. L. 2010. Multiscale Methods and Flow-based Gridding for Flow and Transport in Porous Media. PhD thesis, Norwegian University of Science and Technology.
Hauge, V. L., Lie, K.-A., and Natvig, J. R. 2012. Flow-Based Coarsening for Multiscale Simulation of Transport in Porous Media. Comput. Geosci. 16 (2): 391–408. https://doi.org/10.1007/s10596-011-9230-x.
Hou, T. Y. and Wu, X.-H. 1997. A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media. J. Comput. Phys. 134: 169–189. https://doi.org/10.1006/jcph.1997.5682.
Jenny, P., Lee, S. H., and Tchelepi, H. A. 2003. Multi-scale Finite-Volume Method for Elliptic Problems in Subsurface Flow Simulation. J. Comput. Phys. 187: 47–67. https://doi.org/ 10.1016/S0021-9991(03)00075-5.
Jenny, P. and Lunati, I. 2009. Modeling Complex Wells With the Multi-Scale Finite-Volume Method. J. Comput. Phys. 228 (3): 687–702. https://doi.org/10.1016/j.jcp.2008.09.026.
Karypis, G. and Kumar, V. 1998. A Fast and High-Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comp. 20 (1): 359–392. https://doi.org/10.1137/S1064827595287997.
Kippe, V., Aarnes, J. E., and Lie, K.-A. 2008. A Comparison of Multiscale Methods for Elliptic Problems in Porous Media Flow. Comput. Geosci. 12 (3): 377–398. https://doi.org/10.1007/s10596-007-9074-6.
Klemetsdal, Ø. S., Berge, R. L., Lie, K.-A. et al. 2017. Unstructured Gridding and Consistent Discretizations for Reservoirs With Faults and Complex Wells. Presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, USA, 20–22 February. SPE-182666-MS. https://doi.org/10.2118/182666-MS.
Kozlova, A., Walsh, D., Chittireddy, S. et al. 2016. A Hybrid Approach to Parallel Multiscale Reservoir Simulator. Presented at the ECMOR XV–15th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands. EAGE. https://doi.org/10.3997/2214-4609.201601889.
Künze, R., Lunati, I., and Lee, S. H. 2013. A Multilevel Multiscale Finite-Volume Method. J. Comput. Phys. 255: 502–520. https://doi.org/10.1016/j.jcp.2013.08.042.
Lie, K., Krogstad, S., Ligaarden, I. S. et al. 2012. Open-source MATLAB Implementation of Consistent Discretisations on Complex Grids. Comput. Geosci. 16: 297–322. https://doi.org/10.1007/s10596-011-9244-4.
Lie, K.-A., Natvig, J. R., Krogstad, S. et al. 2014. Grid Adaptation for the Dirichlet–Neumann Representation Method and the Multiscale Mixed Finite-Element Method. Comput. Geosci. 18 (3): 357–372. https://doi.org/10.1007/s10596-013-9397-4.
Lie, K.-A. 2016. An Introduction to Reservoir Simulation Using MATLAB: User Guide for the Matlab Reservoir Simulation Toolbox (MRST). SINTEF ICT, http://www.sintef.no/Projectweb/MRST/publications, 3rd edition.
Lie, K.-A., Kedia, K., Skaflestad, B. et al. 2017a. A General Non-Uniform Coarsening and Upscaling Framework for Reduced-Order Modelling. Presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, USA, 20–22 February. SPE-182681-MS. https://doi.org/10.2118/182681-MS.
Lie, K.-A., Møyner, O., Natvig, J. R. et al. 2017b. Successful Application of Multiscale Methods in a Real Reservoir Simulator Environment. Comput. Geosci. https://doi.org/10.1007/s10596-017-9627-2.
Lunati, I. and Lee, S. H. 2009. An Operator Formulation of the Multiscale Finite-Volume Method With Correction Function. Multiscale Model. Simul. 8 (1): 96–109. https://doi.org/10.1137/080742117.
Manea, A., Hajibeygi, H., Vassilevski, P. et al. 2016. Enriched Algebraic Multiscale Linear Solver. Presented at the ECMOR XV–15th European Conference on the Mathematics of Oil Recovery. https://doi.org/10.3997/2214-4609.201601894.
Manea, A., Hajibeygi, H., Vassilevski, P. et al. 2017. Parallel Enriched Algebraic Multiscale Solver. Presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, USA, 20–22 February. SPE-182694-MS. https://doi.org/10.2118/182694-MS.
Møyner, O. and Lie, K.-A. 2014. A Multiscale Two-Point Flux-Approximation Method. J. Comput. Phys. 275: 273–293. https://doi.org/10.1016/j.jcp.2014.07.003.
Møyner, O. and Lie, K.-A. 2016a. A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models. SPE J. 21 (6): 2079–2096. SPE-173265-PA. https://doi.org/10.2118/173265-PA.
Møyner, O. and Lie, K.-A. 2016b. A Multiscale Restriction-Smoothed Basis Method for High-Contrast Porous Media Represented on Unstructured Grids. J. Comput. Phys. 304: 46–71. https://doi.org/10.1016/j.jcp.2015.10.010.
Møyner, O. and Tchelepi, H. A. 2017. A Multiscale Restriction-Smoothed Basis Method for Compositional Models. Presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, USA, 20–22 February. SPE 182679-MS. https://doi.org/10.2118/182679-MS.
Nordbotten, J., Aavatsmark, I., and Eigestad, G. 2007. Monotonicity of Control Volume Methods. Numer. Math. 106 (2): 255–288. https://doi.org/10.1007/s00211-006-0060-z.
Skaflestad, B. and Krogstad, S. 2008. Multiscale/Mimetic Pressure Solvers With Near-Well Grid Adaption. In Proc., ECMOR XI–11th European Conference on the Mathematics of Oil Recovery, Number A36, Bergen, Norway. EAGE. https://doi.org/10.3997/2214-4609.20146387.
Vanek, P., Mandel, J., and Brezina, M. 1996. Algebraic Multigrid by Smoothed Aggregation for Second- and Fourth-Order Elliptic Problems. Computing 56 (3): 179–196. https://doi.org/10.1007/BF02238511.
Wang, Y., Hajibeygi, H., and Tchelepi, H. A. 2014. Algebraic Multiscale Solver for Flow in Heterogeneous Porous Media. J. Comput. Phys. 259: 284–303. https://doi.org/10.1016/j.jcp.2013.11.024.
Wang, Y., Hajibeygi, H., and Tchelepi, H. A. 2016. Monotone Multiscale Finite Volume Method. Comput. Geosci. 20 (3): 509–524. https://doi.org/10.1007/s10596-015-9506-7.
Wolfsteiner, C., Lee, S. H., and Tchelepi, H. A. 2006. Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation. Multiscale Model. Simul. 5 (3): 900–917. https://doi.org/10.1137/050640771.