The Multiscale Finite-Volume Method on Stratigraphic Grids
- Olav Møyner (SINTEF) | Knut-Andreas Lie (SINTEF)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- October 2014
- Document Type
- Journal Paper
- 816 - 831
- 2014.Society of Petroleum Engineers
- 4.3.4 Scale, 5.1.5 Geologic Modeling
- unstructured, multiscale, msfvm
- 7 in the last 30 days
- 396 since 2007
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Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute-intensive is to use multiscale methods that solve coarsened flow problems by use of a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, the multiscale finite-volume (MsFV) method, is well-studied for 2D Cartesian grids but has not been implemented for stratigraphic and unstructured grids with faults in three dimensions. We present an open-source implementation of the MsFV method in three dimensions along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using generalized minimal residual (GMRES) method or as a preconditioner in combination with a standard inexpensive smoother. We conduct a series of numerical experiments in which approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeabilities and geometries. On the one hand, the results show that the MsFV method can produce accurate approximations for geological models with pinchouts, faults, and nonneighboring connections, but on the other hand, they also show that the method can fail quite spectacularly for highly heterogeneous and anisotropic systems in a way that cannot efficiently be mitigated by iterative approaches. Thus, the MsFV method is, in our opinion, not yet sufficiently robust to be applied as a black-box solver for models with industry-standard complexity. However, extending the method to realistic grids is an important step on the way toward a fast and accurate multiscale solution of large-scale reservoir models. In particular, our open-source implementation provides an efficient framework suitable for further experimentation with partitioning algorithms and MsFV variants.
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Aarnes, J.E., Krogstad, S., and Lie, K.-A. 2006. A Hierarchical Multiscale Method for Two-Phase Flow Based Upon Mixed Finite Elements and Nonuniform Coarse Grids. Multiscale Model. Simul. 5 (2): 337–363 (electronic).
Aarnes, J.E., Krogstad, S., and Lie, K.-A. 2008. Multiscale Mixed/Mimetic Methods on Corner-Point Grids. Comput. Geosci. 12 (3): 297–315. http://dx.doi.org/10.1007/s10596-007-9072-8.
Alpak, F.O., Pal, M., and Lie, K.-A. 2012. A Multiscale Method for Modeling Flow in Stratigraphically Complex Reservoirs. SPE J. 17 (4): 1056–1070. http://dx.doi.org/10.2118/140403-PA.
Bonfigli, G. and Jenny, P. 2009. Recent Developments in the Multi-Scale-Finite-Volume Procedure. J. Comput. Phys. 228: 5129–5147.
Christie, M.A. and Blunt, M.J. 2001. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Res Eval Eng 4: 308–317. http://dx.doi.org/10.2118/66599-PA or url: http://www.spe.org/csp/.
Efendiev, Y. and Hou, T.Y. 2009. Multiscale Finite Element Methods, Vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer Verlag.
Hajibeygi, H., Bonfigli, G., Hesse, M.A. et al. 2008. Iterative Multiscale Finite-Volume Method. J. Comput. Phys. 227 (19): 8604–8621.
Hajibeygi, H., Deb, R., and Jenny, P. 2011a. Multiscale Finite Volume Method for Non-Conformal Coarse Grids Arising From Faulted Porous Media. Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 21–23 February. http://dx.doi.org/10.2118/142205-MS.
Hajibeygi, H. and Jenny, P. 2009. Multiscale Finite-Volume Method for Parabolic Problems Arising From Compressible Multiphase Flow in Porous Media. J. Comput. Phys. 228 (14): 5129–5147.
Hajibeygi, H. and Jenny, P. 2011. Adaptive Iterative Multiscale Finite Volume Method. J. Comput. Phys. 230 (3): 628–643.
Hajibeygi, H. and Jenny, H.A. 2013. Compositional Multiscale Finite-Volume Formulation. Paper presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. http://dx.doi.org/10.2118/163664-MS.
Hajibeygi, H., Karvounis, D., and Jenny, P. 2011b. A Hierarchical Fracture Model for the Iterative Multiscale Finite Volume Method. J. Comput. Phys. 230 (24): 8729–8743.
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.
Hesse, M.A., Mallison, B.T., and Tchelepi, H.A. 2008. Compact Multiscale Finite Volume Method for Heterogeneous Anisotropic Elliptic Equations. Multiscale Model. Simul. 7 (2): 934–962 (electronic).
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.
Jenny, P. and Lunati, I. 2009. Modeling Complex Wells With the Multi-Scale Finite-Volume Method. J. Comput. Phys. 228 (3): 687–702.
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. http://dx.doi.org/10.1007/s10596-007-9074-6.
Künze, R. and Lunati, I. 2012. An Adaptive Multiscale Method for Density-Driven Instabilities. J. Comput. Phys. 231 (17): 5557–5570. http://dx.doi.org/10.1016/j.jcp.2012.02.025.
Lee, S.H., Wolfsteiner, C., and Tchelepi, H. 2008. Multiscale Finite-Volume Formulation for Multiphase Flow in Porous Media: Black Oil Formulation of Compressible, Three-Phase Flow With Gravity. Comput. Geosci. 12 (3): 351–366. http://dx.doi.org/10.1007/s10596-007-9069-3.
Lie, K., Krogstad, S., Ligaarden, I. et al. 2012. Open-Source MATLAB Implementation of Consistent Discretisations on Complex Grids. Comput. Geosci. 16: 297–322. http://dx.doi.org/10.1007/s109596-011-9244-4.
Lunati, I. and Jenny, P. 2006. Multiscale Finite-Volume Method for Compressible 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 for Density-Driven Flow in Porous Media. Comput. Geosci. 12 (3): 337–350. http://dx.doi.org/10.1007/s10596-007-9071-9.
Lunati, I. and Lee, S.H. 2009. An Operator Formulation of the Multiscale Finite-Volume Method With Correction Function. Multiscale Modeling & Simulation 8 (1): 96–109. http://dx.doi.org/10.1137/080742117.
Lunati, I., Tyagi, M., and Lee, S.H. 2011. An Iterative Multiscale Finite Volume Algorithm Converging to the Exact Solution. J. Comput. Phys. 230 (5): 1849–1864. http://dx.doi.org/10.1016/j.jcp.2010.11.036.
Metis. 2012. Metis—Serial Graph Partitioning and Fill-Reducing Matrix Ordering. url: http://glaros.dtc.umn.edu/gkhome/views/metis.
Møyner, O. 2012. Multiscale Finite-Volume Methods on Unstructured Grids. MS thesis, Norwegian University of Science and Technology.
Møyner, O. and Lie, K.-A. 2013. A Multiscale Two-Point Flux-Approximation Method. submitted.
MRST. 2013. The MATLAB Reservoir Simulation Toolbox, Version 2013b. http://www.sintef.no/MRST/.
Natvig, J.R., Lie, K.-A., Krogstad, S. et al. 2012. Grid Adaption for Upscaling and Multiscale Methods. In Proceedings of the ECMOR XIII–13th European Conference on the Mathematics of Oil Recovery, Biarritz, France: EAGE.
Natvig, J.R., Skaflestad, B., Bratvedt, F. et al. 2011. Multiscale Mimetic Solvers for Efficient Streamline Simulation of Fractured Reservoirs. SPE J. 16 (4). http://dx.doi.org/10.2118/119132-PA.
Nordbotten, J.M. and Bjørstad, P. 2008. On the Relationship Between the Multiscale Finite-Volume Method and Domain Decomposition Preconditioners. Comput. Geosci. 12 (3): 367–376. http://dx.doi.org/10.1007/s10596-007-9066-6.
Nordbotten, J.M., Keilegavlen, E., and Sandvin, A. 2012. Mass Conservative Domain Decomposition for Porous Media Flow. In Finite Volume Method-Powerful Means of Engineering Design, ed. R. Petrova. Rijeka, Croatia: InTech Europe.
Sandve, T.H., Berre, I., Keilegavlen, E. et al. 2013. Multiscale Simulation of Flow and Heat Transport in Fractured Geothermal Reservoirs: Inexact Solvers and Improved Transport Upscaling. In Proceedings of Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, February 11–13, Stanford, California.
Sandvin, A., Keilegavlen, E., and Nordbotten, J.M. 2013. Auxiliary Variables for 3D Multiscale Simulations in Heterogeneous Porous Media. J. Comput. Phys. 238: 141–153. http://dx.doi.org/10.1016/j.jcp.2012.12.016.
Sandvin, A., Nordbotten, J., and Aavatsmark, I. 2011. Multiscale Mass Conservative Domain Decomposition Preconditioners for Elliptic Problems on Irregular Grids. Comput. Geosci.15: 587–602. http://dx.doi.org/10.1007/s10596-011-9226-6.
Smith, B.F., Bjørstad, P.E., and Gropp, W.D. 1996. Parallel Multilevel Methods for Elliptic Partial Differential Equations. In Domain Decomposition. Cambridge: Cambridge University Press.
Wang, Y., Hajibeygi, H., and Tchelepi, H.A. 2012. Algebraic Multiscale Linear Solver for Heterogeneous Elliptic Problems. In ECMOR XIII—13th European Conference on the Mathematics of Oil Recovery, Biarritz, France: EAGE.
Wang, Y., Hajibeygi, H., and Tchelepi, H.A. 2013. Algebraic Multiscale Solver for Flow in Heterogeneous Porous Media. J. Comput. Phys. 259: 284–303. http://dx.doi.org/10.1016/j.jcp.2013.11.024.
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 (electronic). http://dx.doi.org/10.1137/050640771.
Zhou, H. and Tchelepi, H.A. 2008. Operator-Based Multiscale Method for Compressible Flow. SPE J. 13 (2): 267–273. http://dx.doi.org/10.2118/106254-PA.
Zhou, H. and Tchelepi, H.A. 2012. Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models. SPE J. 17 (2): 523–539. http://dx.doi.org/10.2118/141473-PA.