Abstract
Domain decomposition methods are widely used as preconditioners for Krylov subspace linear solvers. In the simulation of porous media flow there has recently been a growing interest in nonlinear preconditioning methods for Newton’s method. In this work, we perform a numerical study of a spatial additive Schwarz preconditioned exact Newton (ASPEN) method as a nonlinear preconditioner for Newton’s method applied to both fully implicit or sequential implicit schemes for simulating immiscible and compositional multiphase flow. We first review the ASPEN method and discuss how the resulting linearized global equations can be recast so that one can use standard preconditioners developed for the underlying model equations. We observe that the local fully implicit or sequential implicit updates efficiently handle the local nonlinearities, whereas long-range interactions are resolved by the global ASPEN update. The combination of the two updates leads to a very competitive algorithm. We illustrate the behavior of the algorithm for conceptual one and two-dimensional cases, as well as realistic three dimensional models. A complexity analysis demonstrates that Newton’s method with a fully implicit scheme preconditioned by ASPEN is a very robust and scalable alternative to the well-established Newton’s method for fully implicit schemes.
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Berge, R.L., Klemetsdal, Ø.S., Lie, K.-A.: Unstructured Voronoi grids conforming to lower dimensional objects. Comput. Geosci. 23(1), 169–188 (2019). https://doi.org/10.1007/s10596-018-9790-0
Brooks, R.H., Corey, A.T.: Properties of porous media affecting fluid flow. J Irrigation Drainage Div. 92(2), 61–90 (1966). https://doi.org/10.1061/JRCEA4.0000425
Cai, X. -C., Keyes, D.E.: Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Comput. Sci. 24(1), 183–200 (2002). https://doi.org/10.1137/s106482750037620x
Cai, X.C., Keyes, D.E., Marcinkowski, L.: Non-linear additive schwarz preconditioners and application in computational fluid dynamics. Int. J. Numer. Meth. Fluids 40(12), 1463–1470 (2002). https://doi.org/10.1002/fld.404
Cao, H., Tchelepi, H.A., Wallis, J., Yardumian, H.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (SPE). https://doi.org/10.2118/96809-ms (2005)
Chien, H., Mark, C., Tchelepi, H.A., Yardumian, H.E., Chen, W.H.: Scalable parallel multi-purpose reservoir simulator. In: SPE Symposium on Reservoir Simulation, pp 17–30. https://doi.org/10.2118/37976-MS (1997)
Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: A comparison of upscaling techniques. SPE Reserv Eval Eng 4(04), 308–317 (2001). https://doi.org/10.2118/72469-pa
Demidov, D.: AMGCL: An efficient, flexible, and extensible algebraic multigrid implementation. Lobachevskii J. Math. 40(5), 535–546 (2019). https://doi.org/10.1134/S1995080219050056
Dolean, V., Gander, M.J., Kheriji, W., Kwok, F., Masson, R.: Nonlinear preconditioning: How to use a nonlinear Schwarz method to precondition Newton’s method. SIAM J. Sci. Comput. 38(6), A3357–A3380 (2016)
Gries, S., Stüben, K., Brown, G.L., Chen, D., Collins, D.A.: Preconditioning for efficiently applying algebraic multigrid in fully implicit reservoir simulations. SPE J. 19(4), 726–736 (2014). https://doi.org/10.2118/163608-PA
Hamon, F.P., Mallison, B.T., Tchelepi, H.A.: Implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy. Comput. Methods Appl. Mech. Eng. 311, 599–624 (2016). https://doi.org/10.1016/j.cma.2016.08.009
Henson, V.E., Yang, U.M.: BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002). https://doi.org/10.1016/S0168-9274(01)00115-5
Jenny, P., Lee, S.H., Tchelepi, H.A.: Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media. J. Comput. Phys. 217(2), 627–641 (2006). https://doi.org/10.1016/j.jcp.2006.01.028
Jenny, P., Tchelepi, H.A., Lee, S.H.: Unconditionally convergent nonlinear solver for hyperbolic conservation laws with s-shaped flux functions. J Comput. Phys. 228(20), 7497–7512 (2009). https://doi.org/10.1016/j.jcp.2009.06.032
Jiang, J., Tchelepi, H.A.: Nonlinear acceleration of sequential fully implicit (SFI) method for coupled flow and transport in porous media. Comput. Methods Appl. Mech. Eng. 352, 246–275 (2019). https://doi.org/10.1016/j.cma.2019.04.030
Kaarstad, T., Froyen, J., Bjorstad, P., Espedal, M.: A massively parallel reservoir simulator. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (SPE). https://doi.org/10.2118/29139-MS (1995)
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998). https://doi.org/10.1137/S1064827595287997
Killough, J.E., Wheeler, M.F.: Parallel iterative linear equation solvers: An investigation of domain decomposition algorithms for reservoir simulation. In: SPE Symposium on Reservoir Simulation. https://doi.org/10.2118/16021-MS (1987)
Klemetsdal, Ø.S.: Efficient Solvers for Field-Scale Simulation of Flow and Transport in Porous Media. Phd thesis, Norwegian University of Technology and Science (2019)
Klemetsdal, Ø.S., Lie, K.-A.: Dynamic coarsening and local reordered nonlinear solvers for simulating transport in porous media. SPE J., (January), 1–20. https://doi.org/10.2118/201089-PA (2020)
Klemetsdal, Ø.S., Møyner, O., Lie, K.-A.: Robust nonlinear Newton solver with adaptive interface-localized trust regions. SPE J. 24(04), 1576–1594 (2019). https://doi.org/10.2118/195682-PA
Klemetsdal, Ø.S., Moncorgé, A., Nilsen, H.M., Møyner, O., Lie., K.-A.: An adaptive sequential fully implicitdomain-decomposition solver. SPE J. In press (2021a)
Klemetsdal, Ø.S., Møyner, O., Moncorgé, A., Nilsen, H.M., Lie, K.-A.: High-resolution compositional reservoir simulation with dynamic coarsening and local timestepping for unstructured grids. SPE J. In press (2021b)
Lacroix, S., Vassilevski, Y., Wheeler, J., Wheeler, M.F.: Iterative solution methods for modeling multiphase flow in porous media fully implicitly. SIAM J. Sci Comput. 25(3), 905–926 (2003). https://doi.org/10.1137/S106482750240443X
Li, J., Wong, Z.Y., Tomin, P., Tchelepi, H.: Sequential implicit Newton method for coupled multi-segment wells. In: SPE Reservoir Simululation Conference. https://doi.org/10.2118/193833-MS (2019)
Lie, K.-A.: An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User guide for the MATLAB Reservoir Simulation Toolbox (MRST). Cambridge University Press, Cambridge (2019). https://doi.org/10.1017/9781108591416
Lie, K.-A., Møyner, O., Natvig, J.R., Kozlova, A., Bratvedt, K., Watanabe, S., Li, Z.: Successful application of multiscale methods in a real reservoir simulator environment. Comput. Geosci. 21(5-6), 981–998 (2017). https://doi.org/10.1007/s10596-017-9627-2
Linga, G., Møyner, O., Møll, H., Moncorgė, A., Lie, K.-A.: An implicit local time-stepping method based on cell reordering for multiphase flow in porous media. J. Comput. Phys.: X 6, 100051 (2020). https://doi.org/10.1016/j.jcpx.2020.100051
Liu, L., Keyes, D.E.: Field-split preconditioned inexact Newton algorithms. SIAM J. Comput. Sci 37(3), A1388–A1409 (2015). https://doi.org/10.1137/140970379
Liu, L., Keyes, D.E., Sun, S.: Fully implicit two-phase reservoir simulation with the additive Schwarz preconditioned inexact Newton method. In: SPE Reservoir Characterization and Simulation Conference and Exhibition, 16-18 September, Abu Dhabi, UAE. Society of Petroleum Engineers (SPE) (2013)
Luo, L., Cai, X.-C., Keyes, D.E.: Nonlinear preconditioning strategies for two-phase flows in porous media discretized by a fully implicit discontinuous galerkin method. SIAM J. Sci. Comput. (0), S317–S344 (2021)
Manzocchi, T., et al.: Sensitivity of the impact of geological uncertainty on production from faulted and unfaulted shallow-marine oil reservoirs: objectives and methods. Petrol. Geosci. 14(1), 3–15 (2008). https://doi.org/10.1144/1354-079307-790
Moncorgė, A., Tchelepi, H.A., Jenny. P.: Sequential fully implicit formulation for compositional simulation using natural variables. J. Comput. Phys. 371, 690–711 (2018). https://doi.org/10.1016/j.jcp.2018.05.048
Moncorgé, A., Møyner, O., Tchelepi, H.A., Jenny, P.: Consistent upwinding for sequential fully implicit multiscale compositional simulation. Comput. Geosci. 24, 533–550 (2020). https://doi.org/10.1007/s10596-019-09835-6
Møyner, O: Nonlinear solver for three-phase transport problems based on approximate trust regions. Comput. Geosci. 21(5-6), 999–1021 (2017)
Møyner, O., Lie, K.-A.: A multiscale restriction-smoothed basis method for compressible black-oil models. SPE J. 21(06), 2079–2096 (2016). https://doi.org/10.2118/173265-PA
Møyner, O., Moncorgé A.: Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow. Comput. Geosci., 789–806. https://doi.org/10.1007/s10596-019-09848-1 (2019)
Møyner, O., Tchelepi, H.A.: A mass-conservative sequential implicit multiscale method for isothermal equation of state compositional problems. SPE J. (23), 2376–2393. https://doi.org/10.2118/182679-PA (2018)
Natvig, J.R., Lie, K.-A.: 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 (2008). https://doi.org/10.1016/j.jcp.2008.08.024
Nordbotten, J., Bjørstad, P.: On the relationship between the multiscale finite-volume method and domain decomposition preconditioners. Comput. Geosci. 12(3), 367–376 (2008)
Peng, D.-Y., Robinson, D.B.: A new two-constant equation of state. Ind. Eng. Chem. Res. 15(1), 59–64 (1976). https://doi.org/10.1021/i160057a011
Redlich, O., Kwong, J.N.S.: On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem Rev 44(1), 233–244 (1949). https://doi.org/10.1021/cr60137a013
Skogestad, J.O., Keilegavlen, E., Nordbotten, J.M.: Domain decomposition strategies for nonlinear flow problems in porous media. J Comput. Phys. 234, 439–451 (2013). https://doi.org/10.1016/j.jcp.2012.10.001
Soave, G.: Equilibrium constants from a modified Redlich–Kwong equation of state. Chem Eng Sci 27(6), 1197–1203 (1972). https://doi.org/10.1016/0009-2509(72)80096-4
Stueben, K.: An introduction to algebraic multigrid. Appendix in book “Multigrid” by U. Trottenberg, C.W. Oosterlee and A. Schueller, pp 413–532. Academic Press, Cambridge (2001)
Stueben, K., Clees, T., Klie, H., Lu, B., Wheeler, M.F.: Algebraic multigrid methods (AMG) for the efficient solution of fully implicit formulations in reservoir simulation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (SPE). https://doi.org/10.2118/105832-ms (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). https://doi.org/10.1137/0149044
Wallis, J.R.: Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reservoir Simulation Symposium, 15-18 November, San Francisco, California. Society of Petroleum Engineers. https://doi.org/10.2118/12265-MS (1983)
Wallis, J.R., Kendall, R.P., Little, T.E.: Constrained residual acceleration of conjugate residual methods. In: PE Reservoir Simulation Symposium. https://doi.org/10.2118/13536-MS (1985)
Watts, J.W.: A compositional formulation of the pressure and saturation equations. SPE Reserv. Eng. 1(03), 243–252 (1986). https://doi.org/10.2118/12244-pa
Wong, Z.Y.: Sequential-Implicit Newton’s Method for Geothermal Reservoir Simulation. PhD thesis, Stanford University (2018)
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The authors thank Halvor Møll Nilsen for fruitful discussions and TOTAL E&P for funding and allowing the publication of this work.
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Klemetsdal, Ø., Moncorgé, A., Møyner, O. et al. A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow. Comput Geosci 26, 1045–1063 (2022). https://doi.org/10.1007/s10596-021-10090-x
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DOI: https://doi.org/10.1007/s10596-021-10090-x