Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media. (English) Zbl 1514.76002
Summary: This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approach – and in particular, FSMSN – results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication.
MSC:
| 76-10 | Mathematical modeling or simulation for problems pertaining to fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
Keywords:
nonlinear solver; field-split preconditioning methods; two-phase flow; coupled multi-physics problemsSoftware:
NewtonLibReferences:
| [1] | Aziz, K., Settari, A.: Petroleum reservoir simulation, vol. 476. Applied Science Publishers London (1979) |
| [2] | Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation, Elsevier Scientific Publishing Company (1977) |
| [3] | Deuflhard, P.: Newton methods for nonlinear problems: affine invariance and adaptive algorithms, vol. 35, Springer (2011) · Zbl 1226.65043 |
| [4] | Younis, R.M.: Modern advances in software and solution algorithms for reservoir simulation, Stanford University (2011) |
| [5] | Jenny, P.; Tchelepi, HA; Lee, SH, Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions, J Comput. Phys., 228, 20, 7497-7512 (2009) · Zbl 1391.76553 |
| [6] | Wang X.; Tchelepi H. A., Trust-region based solver for nonlinear transport in heterogeneous porous media, J. Comput. Phys., 253, 114-137 (2013) · Zbl 1349.76406 |
| [7] | Li, B.; Tchelepi H. A., Nonlinear analysis of multiphase transport in porous media in the presence of viscous, buoyancy, and capillary forces, J. Comput. Phys., 297, 104-131 (2015) · Zbl 1349.76820 |
| [8] | Møyner, O., Nonlinear solver for three-phase transport problems based on approximate trust regions, Comput. Geosci., 21, 5, 999-1021 (2017) · Zbl 1396.76061 |
| [9] | Møyner, O.; Tchelepi, HA, A mass-conservative sequential implicit multiscale method for isothermal equation-of-state compositional problems, SPE J., 23, 6, 2376-2393 (2018) |
| [10] | Møyner, O., Tchelepi, H.: A multiscale restriction-smoothed basis method for compositional models. In: SPE Reservoir Simulation Conference (2017). Paper presented at the SPE Reservoir Simulation Conference, Montgomery, Texas, USA, February 2017. Paper Number: SPE-182679-MS doi:10.2118/182679-MS |
| [11] | Moncorgé, A.; Møyner, O.; Tchelepi, HA; Jenny, P., Consistent upwinding for sequential fully implicit multiscale compositional simulation, Comput. Geosci., 24, 2, 533-550 (2020) · Zbl 1434.76128 |
| [12] | 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 environmentr. In: ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery. European Association of Geoscientists & Engineers (2016) |
| [13] | Moncorgé, A.; Tchelepi, HA; Jenny, P., Sequential fully implicit formulation for compositional simulation using natural variables, J. Comput. Phys., 371, 690-711 (2018) · Zbl 1415.76725 |
| [14] | Jiang, J.; Tchelepi, HA, 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) · Zbl 1441.76080 |
| [15] | Franc, J., Møyner, O., Tchelepi, H.A.: Coupling-strength criteria for sequential implicit formulations. In: SPE Reservoir Simulation Conference (2021). Paper presented at the SPE Reservoir Simulation Conference, On-Demand, October 2021. Paper Number: SPE-203909-MS doi:10.2118/203909-MS |
| [16] | Li, J.; Tomin, P.; Tchelepi, HA, Sequential fully implicit Newton method for compositional flow and transport, J. Comput. Phys., 444, 110541 (2021) · Zbl 07515443 |
| [17] | Watanabe, S., Li, Z., Bratvedt, K., Lee, S.H., Natvig, J.R.: A stable multi-phase nonlinear transport solver with hybrid upwind discretization in multiscale reservoir simulator. In: ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery (2016). Publisher: European Association of Geoscientists & Engineers. Source: ECMOR XV - 15th European Conference on the Mathematics of Oil doi:10.3997/2214-4609.201601852 |
| [18] | Younis, RM; Tchelepi, HA; Aziz, K., Adaptively localized continuation-Newton method—nonlinear solvers that converge all the time, SPE J, 15, 2, 526-544 (2010) |
| [19] | Jiang, J.; Tchelepi, HA, Dissipation-based continuation method for multiphase flow in heterogeneous porous media, J. Comput. Phys., 375, 307-336 (2018) · Zbl 1416.76152 |
| [20] | Natvig, JR; 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) · Zbl 1218.76029 |
| [21] | Kwok, F.; Tchelepi, HA, Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media, J. Comput. Phys., 227, 1, 706-727 (2007) · Zbl 1388.76258 |
| [22] | Hamon, FP; Tchelepi, HA, Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow, Comput. Geosci., 20, 5, 909-927 (2016) · Zbl 1391.76805 |
| [23] | Klemetsdal, ØS; Rasmussen, AF; Møyner, O.; Lie, KA, Efficient reordered nonlinear Gauss-Seidel solvers with higher order for black-oil models, Comput. Geosci., 24, 593-603 (2020) · Zbl 1434.86015 |
| [24] | Klemetsdal, SØ; Møyner, O.; Lie, KA, Robust nonlinear Newton solver with adaptive interface-localized trust regions, SPE J., 24, 4, 1576-1594 (2019) |
| [25] | Toft, R., Lie, K.-A., Møyner, O.: Full approximation scheme for reservoir simulation. In: Norsk IKT-konferanse for forskning og utdanning (2018). Norsk Informatikkonferanse, https://ojs.bibsys.no/index.php/NIK/article/view/503 |
| [26] | Christensen, MLC; Vassilevski, PS; Villa, U., Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties, J. Comput. Appl. Math., 340, 691-708 (2018) · Zbl 1432.76160 |
| [27] | Lee, CS; Hamon, FP; Castelletto, N.; Vassilevski, PS; White, JA, Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems, Comput. Methods Appl. Mech. Eng., 372, 113432 (2020) · Zbl 1506.76113 |
| [28] | Lee, CS; Hamon, FP; Castelletto, N.; Vassilevski, PS; White, JA, An aggregation-based nonlinear multigrid solver for two-phase flow and transport in porous media, Comput. Math. Appl., 113, 282-299 (2022) · Zbl 1504.65186 |
| [29] | Cai, X-C; Keyes, DE; Marcinkowski, L., Non-linear additive Schwarz preconditioners and application in computational fluid dynamics, Int. J. Numer. Methods Fluids, 40, 1463-1470 (2002) · Zbl 1025.76040 |
| [30] | Dolean, V.; Gander, MJ; 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, A3357-A3380 (2016) · Zbl 1352.65326 |
| [31] | Klemetsdal, SØ; Moncorgé, A.; Nilsen, HM; Møyner, O.; Lie, KA, An adaptive sequential fully implicit domain-decomposition solver, SPE J., 27, 1, 566-578 (2022) |
| [32] | Møyner, O.; Moncorgé, A., Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow, Comput. Geosci., 24, 2, 789-806 (2020) · Zbl 1434.86005 |
| [33] | Klemetsdal, SØ; Moncorgé, A.; Møyner, O.; Lie, KA, 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) · Zbl 1496.76133 |
| [34] | Luo, L., Cai, X.-C., Keyes, D.E.: Nonlinear preconditioning for two-phase flows. European Association of Geoscientists and Engineers. Source: Fifth EAGE Workshop on High Performance Computing for Upstream, Sep 2021, Volume 2021, p. 1-5 doi:10.3997/2214-4609.2021612015 (2021) |
| [35] | Luo, L.; Liu, L.; Cai, X-C; Keyes, DE, Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids, J. Comput. Phys., 409, 109312 (2020) · Zbl 1435.76038 |
| [36] | Luo, L.; Cai, X-C; Keyes, DE, Nonlinear preconditioning strategies for two-phase flows in porous media discretized by a fully implicit discontinuous Galerkin method, SIAM J. Sci. Comput., 43, 5, S317-S344 (2021) · Zbl 1490.65203 |
| [37] | Skogestad, JO; Keilegavlen, E.; Nordbotten, JM, Two-scale preconditioning for two-phase nonlinear flows in porous media, Transp. Porous Media, 114, 2, 485-503 (2016) |
| [38] | Wong, ZY; Kwok, F.; Horne, RN; Tchelepi, HA, Sequential-implicit Newton’s method for multiphysics simulation, J. Comput. Phys., 391, 155-178 (2019) · Zbl 1452.86003 |
| [39] | Liu, L.; Keyes, D., Field-split preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 37, 3, A1388-A1409 (2015) · Zbl 1328.65122 |
| [40] | Sammon, PH, An analysis of upstream differencing, SPE Reserv. Eng., 3, 3, 1053-1056 (1988) |
| [41] | Brenier, Y.; Jaffré, J., Upstream differencing for multiphase flow in reservoir simulation, J. Numer. Anal., 28, 3, 685-696 (1991) · Zbl 0735.76071 |
| [42] | Lee, SH; Efendiev, Y.; Tchelepi, HA, Hybrid upwind discretization of nonlinear two-phase flow with gravity, Adv. Water Resour., 82, 27-38 (2015) |
| [43] | Lee, SH; Efendiev, Y., Hybrid discretization of multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces, Comput. Geosci., 22, 1403-1421 (2018) · Zbl 1406.76059 |
| [44] | Hamon, FP; Mallison, BT; Tchelepi, HA, Implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy, Comput. Methods Appl. Mech. Eng., 311, 599-624 (2016) · Zbl 1439.76109 |
| [45] | Hamon, FP; Tchelepi, HA, Analysis of hybrid upwinding for fully-implicit simulation of three-phase flow with gravity, SIAM J. Numer. Anal., 54, 3, 1682-1712 (2016) · Zbl 1382.65269 |
| [46] | Hamon, FP; Mallison, BT, Fully implicit multidimensional hybrid upwind scheme for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 358, 112606 (2020) · Zbl 1441.76078 |
| [47] | Bosma, S.B.M., Hamon, F.P., Mallison, B.T., Tchelepi, H.A.: Smooth implicit hybrid upwinding for compositional multiphase flow in porous media, vol. 388 (2022) · Zbl 1507.76195 |
| [48] | Brenner, K.; Masson, R.; Quenjel, E., Vertex approximate gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media, J. Comput. Phys., 409, 109357 (2020) · Zbl 1435.76042 |
| [49] | Eisenstat, S.; Walker, H., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 1, 16-32 (1996) · Zbl 0845.65021 |
| [50] | Dembo, R.; Eisenstat, S.; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 2, 400-408 (1982) · Zbl 0478.65030 |
| [51] | Eisenstat, S.; Walker, H., Globally convergent inexact Newton methods, SIAM J. Optim., 4, 2, 393-422 (1994) · Zbl 0814.65049 |
| [52] | Zhou, Y., Jiang, J., Tomin, P.: Inexact methods for black-oil Sequential Fully Implicit (SFI) scheme. In: SPE Reservoir Simulation Conference (2021). Paper presented at the SPE Reservoir Simulation Conference, On-Demand, October 2021. Paper Number: SPE-203900-MS doi:10.2118/203900-MS |
| [53] | Jiang, J.; Tomin, P.; Zhou, Y., 5, Comput. Geosci., 23, 1709-1730 (2021) · Zbl 1473.86008 |
| [54] | Sheth, S., Moncorgé, A.: Inexact Newton method for general purpose reservoir simulation. arXiv preprint arXiv:1912.06568 |
| [55] | Hamon, FP; Mallison, BT; Tchelepi, HA, Implicit hybrid upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity, Comput. Methods Appl. Mech. Eng., 331, 701-727 (2018) · Zbl 1439.76156 |
| [56] | Alali, AH; Hamon, FP; Mallison, BT; Tchelepi H. A., Finite-volume simulation of capillary-dominated flow in matrix-fracture systems using interface conditions, Comput. Geosci, 25, 1, 17-33 (2021) · Zbl 1453.86002 |
| [57] | Saad, Y.; Schultz, MH, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 |
| [58] | Brown, PN; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11, 3, 450-481 (1990) · Zbl 0708.65049 |
| [59] | Christie, MA; Blunt, MJ, Tenth SPE comparative solution project: A comparison of upscaling techniques, Soc. Pet. Eng., 4, 308-317 (2001) |
| [60] | Fokkema, DR; Sleijpen, GLG; der Vorst, HAV, Accelerated inexact Newton schemes for large systems of nonlinear equations, SIAM J. Sci. Comput., 19, 2, 657-674 (1998) · Zbl 0916.65050 |
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