Fast Simulation of Polymer Injection in Heavy-Oil Reservoirs on the Basis of Topological Sorting and Sequential Splitting
- Knut-Andreas Lie (SINTEF ICT) | Halvor Møll Nilsen (SINTEF ICT) | Atgeirr Flø Rasmussen (SINTEF ICT) | Xavier Raynaud (SINTEF ICT)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2014
- Document Type
- Journal Paper
- 991 - 1,004
- 2014.Society of Petroleum Engineers
- 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex)
- reservoir simulation, reordering, polymer flooding, nonlinear solvers
- 4 in the last 30 days
- 390 since 2007
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We present a set of algorithms for sequential solution of flow and transport that can be used for efficient simulation of polymer injection modeled as a compressible two-phase system. Our formulation gives a set of nonlinear transport equations that can be discretized with standard implicit upwind methods to conserve mass and volume independent of the timestep. In the absence of gravity and capillary forces, the resulting nonlinear system of discrete transport equations can be permuted to lower triangular form with a simple topological-sorting method from graph theory. This gives an optimal nonlinear solver that computes the solution cell by cell with local iteration control. The single-cell systems can be reduced to a nested set of nonlinear scalar equations that can be bracketed and solved with standard gradient or root-bracketing methods. The resulting method gives orders-of-magnitude reduction in run times and increases the feasible timestep sizes. In fact, one can prove that the solver is unconditionally stable and will produce a solution for arbitrarily large timesteps. For cases with gravity, the same method can be applied as part of a nonlinear Gauss–Seidel method. Altogether, our results demonstrate that sequential splitting combined with single-point upwind discretizations can become a viable alternative to streamline methods for speeding up simulation of advection-dominated systems.
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Aarnes, J. E., Krogstad, S., Lie, K.-A., et al. 2006. Fast Sequential Implicit Porous Media Flow Simulations Using Multiscale Finite Elements and Reordering of Cells for Solution of Nonlinear Transport Equation. Oral presentation given at ECMOR X, 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, The Netherlands.
AGMG 2012. Iterative Solution with AGgregation-Based Algebraic MultiGrid, http://homepages.ulb.ac.be/~ynotay/AGMG/.
Alsofi, A. M. and Blunt, M. J. 2010. Streamline-Based Simulation of Non-Newtonian Polymer Flooding. SPE J. 15 (4): 895–905. http://dx.doi.org/10.2118/123971-PA.
Blatt, M. and Bastian, P. 2007. The Iterative Solver Template Library. In Applied Parallel Computing: State of the Art in Scientific Computing, ed. B. Kåström, E. Elmroth, J. Dongarra, J. Wasniewski, 666–675. Berlin, Germany: Springer-Verlag.
Booth, R. 2008. Miscible Flow Through Porous Media. PhD dissertation, University of Oxford, Oxford, UK (2008).
Bratvedt, F., Gimse, T. and Tegnander, C. 1996. Streamline Computations for Porous Media Flow Including Gravity. Transport Porous Med. 25 (1): 63–78. http://dx.doi.org/10.1007/BF00141262.
Clemens, T., Abdev, J. and Thiele, M. R. 2011. Improved Polymer-Flood Management Using Streamlines. SPE J. 14 (2): 171–181. http://dx.doi.org/10.2118/132774-MS.
Datta-Gupta, A. and King, M. J. 2007. Streamline Simulation: Theory and Practice, Vol. 11. Richardson, Texas: Textbook Series, SPE.
Eclipse Technical Description Manual, 2009.2 edition. 2009. Sugar Land, Texas: Schlumberger.
Fletcher, P., Cobos, S., Jaska, C., et al. 2012. Improving Heavy Oil Recovery Using An Enhanced Polymer System. Paper SPE 154045 presented at the SPE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 14–18 April. http://dx.doi.org/10.2118/154045-MS.
Ford, J. A. 1995. Improved Algorithms Of Illinois-Type For The Numerical Solution Of Nonlinear Equations. Technical Report CSM-257, University of Essex, Colchester, UK.
Gao, C. H. 2011. Scientific Research and Field Applications of Polymer Flooding in Heavy Oil Recovery. J. Pet. Explor. Prod. Tech. 1 (2–4): 65–70. http://dx.doi.org/10.1007/s13202-011-0014-6.
Gmelig Meyling, R. H. J. 1990. A Characteristic Finite Element Method for Solving Non-Linear Convection-Diffusion Equations on Locally Refined Grids. In 2nd European Conference on the Mathematics of Oil Recovery, Arles, France, ed. D. Guerillot, and O. Guillon, 255–262.
Gmelig Meyling, R. H. J. 1991. Numerical methods for solving the nonlinear hyperbolic equations of porous media flow. In Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990), 503–517. Studentlitteratur: Lund, Sweden.
NTNU 2012. The Norne Benchmark Case, http://www.ipt.ntnu.no/~norne/wiki/doku.php).
Kwok, F. and Tchelepi, H. 2007. Potential-Based Reduced Newton Algorithm for Nonlinear Multiphase Flow in Porous Media. J. Comput. Phys. 227 (1): 706–727. http://dx.doi.org/10.1016/j.jcp.2007.08.012.
Lie, K.-A., Natvig, J. R. and Nilsen, H. M. 2012a. Discussion of Dynamics and Operator Splitting Techniques for Two-Phase Flow with Gravity. Int. J. Numer. Anal. Mod. 9 (3): 684–700.
Lie, K.-A., Nilsen, H. M., Rasmussen, A. F., et al. 2012b. An Unconditionally Stable Splitting Method Using Reordering for Simulating Polymer Injection. Oral presentation given at ECMOR XIII, 13th European Conference on the Mathematics of Oil Recovery, Biarritz, France, 10–13 September.
Natvig, J. R. and Lie, K.-A. 2008a. Fast Computation of Multiphase Flow in Porous Media by Implicit Discontinuous Galerkin Schemes with Optimal Ordering of Elements. J. Comput. Phys. 227 (24): 10108–10124. http://dx.doi.org/10.1016/j.jcp.2008.08.024.
Natvig, J. R. and Lie, K.-A. 2008b. On Efficient Implicit Upwind Schemes. Oral presentation given at ECMOR XI, Bergen, Norway, 8–11 September.
Natvig, J. R., Lie, K.-A. and Eikemo, B. 2006. Fast Solvers for Flow in Porous Media based on Discontinuous Galerkin Methods and Optimal Reordering. Oral presentation given at the XVI International Conference on Computational Methods in Water Resources, Copenhagen, Denmark, 18–22 June.
Natvig, J. R., Lie, K.-A., Eikemo, B., et al. 2007. An Efficient Discontinuous Galerkin Method for Advective Transport in Porous Media. Adv. Water Resour. 30 (12): 2424–2438. http://dx.doi.org/10.1016/j.advwatres.2007.05.015.
Notay, Y. 2010. An Aggregation-Based Algebraic Multigrid Method. Electron. Trans. Numer. Anal. 37: 123–140.
OPM 2012. The Open Porous Media (OPM) Initiative, http://www.opm-project.org/.
Shahvali, M., Mallison, B., Wei, K., et al. 2012. An Alternative to Streamlines for Flow Diagnostics on Structured and Unstructured Grids. SPE J. 17 (3): 768–778. http://dx.doi.org/10.2118/146446-PA.
Shahvali, M. and Tchelepi, H. 2013. Efficient Coupling for Nonlinear Multiphase Flow with Strong Gravity. Paper SPE 163659 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–21 February. http://dx.doi.org/10.2118/163659-MS.
Thiele, M. R., Batycky, R. P., Pöllitzer, S., et al. 2010. Polymer-Flood Modeling Using Streamlines. SPE J. 13 (2): 313–322. http://dx.doi.org/10.2118/115545-PA.
Todd, M. R. and Longstaff, W. J. 1972. The Development, Testing, and Application Of a Numerical Simulator for Predicting Miscible Flood Performance. J. Pet. Tech. 24 (7): 874–882. http://dx.doi.org/10.2118/3484-PA.
Wassmuth, F. R., Green, K., Arnold, W., et al. 2009. Polymer Flood Application to Improve Heavy Oil Recovery at East Bodo. J. Cdn. Pet. Tech. 48 (2): 55–61. http://dx.doi.org/10.2118/09-02-55.
Xiaodong, K., Jian, Z., Fujie, S., et al. 2011. A Review of Polymer EOR on Offshore Heavy Oil Field in Bohai Bay, China. Paper SPE 144932 presented at the SPE Enhanced Oil Recovery Conference, Kuala Lumpur, Malaysia, 19–21 July. http://dx.doi.org/10.2118/144932-MS.