Numerical studies of a class of linear solvers for fine-scale petroleum reservoir simulation
Tóm tắt
Numerical simulation based on fine-scale reservoir models helps petroleum engineers in understanding fluid flow in porous media and achieving higher recovery ratio. Fine-scale models give rise to large-scale linear systems, and thus require effective solvers for solving these linear systems to finish simulation in reasonable turn-around time. In this paper, we study convergence, robustness, and efficiency of a class of multi-stage preconditioners accelerated by Krylov subspace methods for solving Jacobian systems from a fully implicit discretization. We compare components of these preconditioners, including decoupling and sub-problem solvers, for fine-scale reservoir simulation. Several benchmark and real-world problems, including a ten-million-cell reservoir problem, were simulated on a desktop computer. Numerical tests show that the combination of the alternating block factorization method and multi-stage subspace correction preconditioner gives a robust and memory-efficient solver for fine-scale reservoir simulation.
Tài liệu tham khảo
Al-Shaalan, T.M., Klie, H.M., Dogru, A.H., Wheeler, M.F., et al.: Studies of robust two stage preconditioners for the solution of fully implicit multiphase flow problems. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2009)
Appleyard, J., Cheshire, I., Pollard, R.: Special techniques for fully implicit simulators. In: Preceeding, European Symposium on Enhanced Oil Recovery, Bournemouth, England, pp. 395–408 (1981)
Appleyard, J., et al.: Nested factorization. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1983)
Baker, A.H., Jessup, E.R., Kolev, T.V.: A simple strategy for varying the restart parameter in GMRES (m). J. Comput. Appl. Math. 230(2), 751–761 (2009)
Bank, R.E., Chan, T.F., Coughran Jr., W.M., Smith, R.K.: The Alternate-Block-Factorization procedure for systems of partial differential equations. BIT Numer. Math. 29(4), 938–954 (1989)
Behie, A., et al.: Comparison of nested factorization, constrained pressure residual, and incomplete factorization preconditionings. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. (1985)
Behie, A., Vinsome, P., et al.: Block iterative methods for fully implicit reservoir simulation. Soc. Pet. Eng. J. 22(05), 658–668 (1982)
Behie, G.A., Forsyth Jr., P.: Incomplete factorization methods for fully implicit simulation of enhanced oil recovery. SIAM J. Sci. Stat. Comput. 5(3), 543–561 (1984)
Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E. et al.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2005)
Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, vol. 2. SIAM, Philadelphia (2006)
Christie, M., Blunt, M., et al.: Tenth SPE comparative solution project: a comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2001)
Dogru, A.H., Fung, L.S., Al-Shaalan, T.M., Middya, U., Pita, J.A. et al.: From mega cell to giga cell reservoir simulation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2008)
Dogru, A.H., Fung, L.S.K., Middya, U., Al-Shaalan, T.M., Byer, T., Hoy, H., Hahn, W.A., Al-Zamel, N., Pita, J.A., Hemanthkumar, K., et al.: New frontiers in large scale reservoir simulation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2011)
Eberhard, J., Attinger, S., Wittum, G.: Coarse graining for upscaling of flow in heterogeneous porous media. Multiscale Model. Simul. 2(2), 269–301 (2004)
Eberhard, J., Wittum, G.: A coarsening multigrid method for flow in heterogeneous porous media. In: Multiscale Methods in Science and Engineering. Springer, pp. 111–132 (2005)
Graham, I., Hagger, M.: Unstructured additive Schwarz-conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comput. 20(6), 2041–2066 (1999)
Hu, X., Liu, W., Qi, G., Xu, J., Yan, Y., Zhang, C.-S., Zhang, S.: A fast auxiliary space preconditioners for numerical reservoir simulations. In: SPE Reservoir Characterization and Simulation Conference (2011)
Hu, X., Xu, J., Zhang, C.-S.: Application of auxiliary space preconditioning in field-scale reservoir simulation. Sci. China Math. 56(12), 2737–2751 (2013)
Klie, H.M.: Krylov-Secant Methods for Solving Large-Scale Systems of Coupled Nonlinear Parabolic Equations. PhD thesis, Rice University (1997)
Kwok, W.H.F.: Scalable Linear and Nonlinear Algorithms for Multiphase Flow in Porous Media. PhD thesis, Stanford University (2007)
Lacroix, S., Vassilevski, Y., Wheeler, J., Wheeler, M.: Iterative solution methods for modeling multiphase flow in porous media fully implicitly. SIAM J. Sci. Comput. 25(3), 905–926 (2003)
Lacroix, S., Vassilevski, Y.V., Wheeler, M.F.: Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numer. Linear Algebra Appl. 8(8), 537–549 (2001)
Saad, Y.: A flexible inner-outer preconditioned gmres algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Scheichl, R., Masson, R., Wendebourg, J.: Decoupling and block preconditioning for sedimentary basin simulations. Comput. Geosci. 7(4), 295–318 (2003)
Stüben, K.: Algebraic Multigrid (AMG): An Introduction with Applications. GMD-Forschungszentrum Informationstechnik Sankt Augustin (1999)
Stueben, K., Clees, T., Klie, H., Lu, B., Wheeler, M.F., et al.: Algebraic multigrid methods (AMG) for the efficient solution of fully implicit formulations in reservoir simulation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2007)
Trangenstein, J.A., Bell, J.B.: Mathematical structure of the black-oil model for petroleum reservoir simulation. SIAM J. Appl. Math. 49(3), 749–783 (1989)
Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56(3), 179–196 (1996)
Wallis, J., et al.: Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1983)
Wang, F., Xu, J.: A crosswind block iterative method for convection-dominated problems. SIAM J. Sci. Comput. 21(2), 620–645 (1999)
Watts, J., Shaw, J., et al.: A new method for solving the implicit reservoir simulation matrix equation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2005)
Wittum, G.: On the robustness of ILU smoothing. SIAM J. Sci. Stat. Comput. 10(4), 699–717 (1989)
Wu, S., Feng, C., Zhang, C.-S., Li, Q., et al.: A multilevel preconditioner and its shared memory implementation for new generation reservoir simulator. Pet. Sci. 11, 540–549 (2014)
Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci. 18(01), 77–105 (2008)