1887

Abstract

Summary

During the solution of fully-implicit reservoir simulation time-steps, it is often observed that the computed Newton updates may be very sparse, considering computer precision. This sparsity can be as high as 95% and can vary largely from one iteration to the next.

In recent work, a mathematically sound framework was developed to predict the sparsity pattern before the full linear system is solved. The theory is restricted to general, scalar nonlinear advection-diffusion-reaction problems in multidimensional and heterogeneous settings. This theory had been applied to reduce the size of the linear systems that were computed during sequential implicit timesteps for two-phase flow. The results confirmed that the linearization computations and the linear solution processes may be localized by as much as 95% while retaining the exact Newton convergence behavior and final solution. Inspired by the great success of that methodology, this work develops algorithmic extensions in order to devise localization algorithms for fully-implicit coupled multicomponent problems.

We propose, apply, and test a novel algorithm to resolve a system of hyperbolic equations obtained from an Equation of State (EOS) based compositional simulator. When applied to various fully-implicit flow and multicomponent transport simulations, involving six thermodynamic species, on the full SPE 10 geological model, the observed reduction in computational effort ranges between six to forty-nine fold depending on the level of locality present in the system. We apply this algorithm to several injection and depletion scenarios with and without gravity and capillarity in order to investigate the adaptivity and robustness of the proposed method to the underlying heterogeneity and complexity. We demonstrate that the algorithm enables efficient and robust full-resolution fully implicit simulation without the errors introduced by adaptive discretization methods or the stability concerns of semi-implicit approaches.

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2018-09-03
2024-03-29
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References

  1. Allgower, E., Böhmer, K., Potra, F. and Rheinboldt, W.
    [1986] A mesh-independence principle for operator equations and their discretizations. Society of Industrial Applied Mathematics Journal on Numerical Analysis, 23(1), 160–169.
    [Google Scholar]
  2. Axelsson, O.
    [1994] Iterative Solution Methods.Cambridge University Press.
    [Google Scholar]
  3. Aziz, K. and Settari, A.
    [1979] Petroleum Reservoir Simulation.Elsevier Applied Science.
    [Google Scholar]
  4. Berger, M. and Colella, P.
    [1989] Local adaptive mesh bibinement for shock hydrodynamics. Journal of computational Physics, 82(1), 64–84.
    [Google Scholar]
  5. Cai, X., Keyes, D. and Marcinkowski, L.
    [2002] Non-linear additive Schwarz preconditioners and application in computational fluid dynamics. International journal for numerical methods in fluids, 40(12), 1463–1470.
    [Google Scholar]
  6. Cao, H., Tchelepi, H., Wallis, J. and Yardumian, H.
    [2005] Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation. In: SPE Annual Technical Conference and Exhibition.
    [Google Scholar]
  7. Chen, K.
    [2005] Matrix Preconditioning Techniques and Applications. Cambridge University Press.
    [Google Scholar]
  8. Chen, Z., Huan, G. and Ma, Y.
    [2006] Computational Methods for Multiphase Flows in Porous Media. Computational Science and Engineering. Society of Industrial Applied Mathematics.
    [Google Scholar]
  9. Coats, K.
    [2000] A note on IMPES and some IMPES-based simulation models.Society of Petroleum Engineers, 5, 245–251.
    [Google Scholar]
  10. Dawson, C., Klíe, H., Wheeler, M. and Woodward, C.
    [1997] A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton–Krylov solver.Computational Geosciences, 1(3–4), 215–249.
    [Google Scholar]
  11. Deuflhard, P.
    [2011] Newton methods for nonlinear problems: affine invariance and adaptive algorithms, 35. Springer.
    [Google Scholar]
  12. Engstler, C. and Lubich, C.
    [1997] Multirate extrapolation methods for differential equations with different time scales. Computing, 58(2), 173–185.
    [Google Scholar]
  13. Gear, C. and Wells, D.
    [1984] Multirate linear multistep methods. BIT Numerical Mathematics, 24(4), 484–502.
    [Google Scholar]
  14. Gerritsen, M. and Lambers, J.
    [2008] Integration of local-global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations. Computational Geosciences, 12(2), 193–208.
    [Google Scholar]
  15. Hornung, R. and Trangenstein, J.
    [1997] Adaptive mesh bibinement and multilevel iteration for flow in porous media. Journal of computational Physics, 136(2), 522–545.
    [Google Scholar]
  16. Hwang, F. and Cai, X.
    [2005] A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations. Journal of Computational Physics, 204(2), 666–691.
    [Google Scholar]
  17. Kelley, C. and Sachs, E.
    [1991] Mesh independence of Newton-like methods for infinite dimensional problems. Journal of Integral Equations and Applications, 3(4), 549–573.
    [Google Scholar]
  18. Killough, J.E. and Kossack, C.A.
    [1987] Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators. In: SPE Symposium on Reservoir Simulation.
    [Google Scholar]
  19. Lacroix, S., Vassilevski, Y. and Wheeler, M.
    [2003] Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numerical Linear Algebra with Applications, 8(8), 537–549.
    [Google Scholar]
  20. Moncorgé, A., Tchelepi, H. and Jenny, P.
    [2018] Sequential Fully Implicit Formulation for Compositional Simulation using Natural Variables.Journal of Computational Physics (To appear).
    [Google Scholar]
  21. Mφyner, O. and Tchelepi, H.
    [2018] A Mass-Conservative Sequential Implicit Multiscale Method for Isothermal Equation of State Compositional Problems.SPE Journal (To appear).
    [Google Scholar]
  22. Müller, S. and Stiriba, Y.
    [2007] Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. Journal of Scientific Computing, 30(3), 493–531.
    [Google Scholar]
  23. Sheth, S. and Younis, R.
    [2016] Localized computation of Newton updates in fully-implicit two-phase flow simulation.Procedia Computer Science, 80, 1392–1403.
    [Google Scholar]
  24. [2017a] Localized Linear Systems in Sequential Implicit Simulation of Two-Phase Flow and Transport. Society of Petroleum Engeers Journal, 22(05), 1542–1569.
    [Google Scholar]
  25. [2017b] Localized solvers for general full-resolution implicit reservoir simulation. In: SPE Reservoir Simulation Conference.
    [Google Scholar]
  26. Skogestad, J., Keilegavlen, E. and Nordbotten, J.
    [2012] Domain decomposition strategies for nonlinear flow problems in porous media.Journal of Computational Physics, 234, 439–451.
    [Google Scholar]
  27. Sun, S., Keyes, D. and Liu, L.
    [2013] Fully Implicit Two-phase Reservoir Simulation With the Additive Schwarz Preconditioned Inexact Newton Method. In: Society of Petroleum Engineers Reservoir Characterisation and Simulation Conference and Exhibition.
    [Google Scholar]
  28. Trangenstein, J.
    [2002] Multi-scale iterative techniques and adaptive mesh bibinement for flow in porous media. Advances in Water Resources, 25(8), 1175–1213.
    [Google Scholar]
  29. Wallis, J., Kendall, R., Little, T. and et al.
    [1985] Constrained Residual Acceleration of Conjugate Residual Methods. In: SPE-13536-MS in Proceedings of the SPE Reservoir Simulation Symposium.
    [Google Scholar]
  30. Younis, R.
    [2013] A Sharp Analytical Bound on the Spatiotemporal Locality in General Two-Phase Flow and Transport Phenomena.Procedia Computer Science, 18, 473–480.
    [Google Scholar]
  31. Younis, R., Tchelepi, H. and Aziz, K.
    [2010] Adaptively Localized Continuation-Newton Method-Nonlinear Solvers That Converge All the Time. SPE Journal, 15(02), 526–544.
    [Google Scholar]
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