Multi-stage preconditioners for thermal-compositional-reactive flow in porous media. (English) Zbl 07506171
Summary: We present a family of multi-stage preconditioners for coupled thermal-compositional-reactive reservoir simulation problems. The most common preconditioner used in industrial practice, the Constrained Pressure Residual (CPR) method, was designed for isothermal models and does not offer a specific strategy for the energy equation. For thermal simulations, inadequate treatment of the temperature unknown can cause severe conver-gence degradation. When strong thermal diffusion is present, the energy equation exhibits significant elliptic behavior that cannot be accurately corrected by CPR’s second stage. In this work, we use Schur-complement decompositions to extract a temperature subsystem and apply an Algebraic MultiGrid (AMG) approximation as an additional preconditioning stage to improve the treatment of the energy equation. We present results for several two-dimensional hot air injection problems using an extra heavy oil, including challenging reactive In-Situ Combustion (ISC) cases. We show improved performance and robustness across different thermal regimes, from advection dominated (high Péclet number) to diffusion dominated (low Péclet number). The number of linear iterations is reduced by 40–85% compared to standard CPR for both homogeneous and heterogeneous media, and the new methods exhibit almost no sensitivity to the thermal regime.
Keywords:
multi-stage preconditioning; thermal-compositional reservoir simulation; porous media; iterative methodsReferences:
| [1] | Briggs, P. J.; Baron, P.; Fulleylove, R. J.; Wright, M. S., Development of heavy-oil reservoirs, J. Pet. Technol., 40, 206-214 (1988) |
| [2] | Prats, M., Thermal Recovery, SPE Monograph Series, vol. 7 (1982) |
| [3] | Burger, J. G.; Sahuquet, B., Les méthodes thermiques de production des hydrocarbures - 5. Combustion in-situ, principes et études de laboratoire, Rev. Inst. Fr. Pét., 32, 141-188 (1975) |
| [4] | Lake, L. W., Enhanced Oil Recovery (1989), Prentice Hall Inc.: Prentice Hall Inc. Old Tappan, NJ |
| [5] | Burger, J. G., Spontaneous ignition in oil reservoirs, SPE J., 16, 73-81 (1976) |
| [6] | Crookston, R. B.; Culham, W. E.; Chen, W. H., A numerical simulation model for thermal recovery processes, SPE J., 19, 37-58 (1979) |
| [7] | Coats, K. H., In-situ combustion model, SPE J., 20, 533-554 (1980) |
| [8] | Youngren, G. K., Development and application of an in-situ combustion reservoir simulator, SPE J., 20, 39-51 (1980) |
| [9] | Young, L. C.; Stephenson, R. E., A generalized compositional approach for reservoir simulation, SPE J., 23, 727-742 (1983) |
| [10] | Rubin, B.; Buchanan, L. W., A general purpose thermal model, SPE J., 25, 202-214 (1985) |
| [11] | Michelsen, M. L., The isothermal flash problem. Part I. Stability, Fluid Phase Equilib., 9, 1-19 (1982) |
| [12] | Michelsen, M. L., The isothermal flash problem. Part II. Phase-split calculation, Fluid Phase Equilib., 9, 21-40 (1982) |
| [13] | Whitson, C. H.; Michelsen, M. L., The negative flash, Fluid Phase Equilib., 53, 51-71 (1989) |
| [14] | CMG, STARS Technical Description, 2016. |
| [15] | Schlumberger, ECLIPSE reservoir simulator, Technical Description, 2015. |
| [16] | Cao, H., Development of techniques for general purpose simulators (2002), Stanford University Stanford, CA, Ph.D. thesis |
| [17] | Lapene, A., Etude expérimentale et numérique de la combustion in-situ d’huiles lourdes (2010), Institut National Polytechnique de Toulouse, Ph.D. thesis |
| [18] | Li, X. S., An overview of SuperLU: algorithms, implementation, and user interface, ACM Trans. Math. Softw., 31, 302-325 (2005) · Zbl 1136.65312 |
| [19] | Kourounis, D.; Fuchs, A.; Schenk, O., Toward the next generation of multiperiod optimal power flow solvers, IEEE Trans. Power Syst., 33, 4005-4014 (2018) |
| [20] | Saad, Y., Iterative Methods for Sparse Linear Systems, vol. 82 (2003), SIAM · Zbl 1031.65046 |
| [21] | Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [22] | White, J. A.; Borja, R. I., Block-preconditioned Newton-Krylov solvers for fully coupled flow and geomechanics, Comput. Geosci., 15, 647-659 (2011) · Zbl 1367.76034 |
| [23] | Haga, J. B.; Osmes, H.; Langtangen, H. P., A parallel block preconditioner for large-scale poroelasticity with highly heterogeneous material parameters, Comput. Geosci., 16, 723-734 (2012) |
| [24] | 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, 726-736 (2014) |
| [25] | White, J. A.; Castelletto, N.; Tchelepi, H. A., Block-partitioned solvers for coupled poromechanics: a unified framework, Comput. Methods Appl. Mech. Eng., 303, 55-74 (2016) · Zbl 1425.74497 |
| [26] | Gaspar, F. J.; Rodrigo, C., On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics, Comput. Methods Appl. Mech. Eng., 326, 526-540 (2017) · Zbl 1439.74413 |
| [27] | White, J. A.; Castelletto, N.; Klevtsov, S.; Bui, Q. M.; Osei-Kuffuor, D.; Tchelepi, H. A., A two-stage preconditioner for multiphase poromechanics in reservoir simulation, Comput. Methods Appl. Mech. Eng., 357, 112575:1-112575:24 (2019) · Zbl 1442.76118 |
| [28] | Wallis, J. R., Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration, (Proceedings - SPE Reservoir Simulation Symposium. Proceedings - SPE Reservoir Simulation Symposium, San Francisco, CA, USA (1983)) |
| [29] | Wallis, J. R.; Kendall, R. P.; Little, T. E., Constrained residual acceleration of conjugate residual methods, (Proceedings - SPE Reservoir Simulation Symposium. Proceedings - SPE Reservoir Simulation Symposium, Dallas, TX, USA (1985)) |
| [30] | Lacroix, S.; Vassilevski, Y.; Wheeler, J. A.; Wheeler, M. F., Iterative solution methods for modeling multiphase flow in porous media fully implicitly, SIAM J. Sci. Comput., 25, 905-926 (2003) · Zbl 1163.65310 |
| [31] | Cao, H.; Tchelepi, H. A.; Wallis, J. R.; Yardumian, H. E., Parallel scalable unstructured CPR-type linear solver for reservoir simulation, (Proceedings - SPE Annual Technical Conference and Exhibition. Proceedings - SPE Annual Technical Conference and Exhibition, Dallas, TX, USA (2005)) |
| [32] | Ruge, J. W.; Stüben, K., Algebraic multigrid, (Multigrid Methods (1987), SIAM), 73-130 |
| [33] | Stüben, K., A review of algebraic multigrid, J. Comput. Appl. Math., 128, 281-309 (2001) · Zbl 0979.65111 |
| [34] | Tchelepi, H. A.; Jiang, Y., Scalable multistage linear solver for coupled systems of multisegment wells and unstructured reservoir models, (Proceedings - SPE Reservoir Simulation Symposium. Proceedings - SPE Reservoir Simulation Symposium, The Woodlands, TX, USA (2009)) |
| [35] | Voskov, D. V.; Tchelepi, H. A., Comparison of nonlinear formulations for two-phase multi-component EoS based simulation, J. Pet. Sci. Eng., 82, 101-111 (2012) |
| [36] | Zhou, Y.; Jiang, Y.; Tchelepi, H. A., A scalable multistage linear solver for reservoir models with multisegment wells, Comput. Geosci., 17, 197-216 (2013) · Zbl 1382.86007 |
| [37] | Cusini, M.; Lukyanov, A. A.; Natvig, J.; Hajibeygi, H., Constrained pressure residual multiscale (CPR-MS) method for fully implicit simulation of multiphase flow in porous media, J. Comput. Phys., 299, 472-486 (2015) · Zbl 1351.76055 |
| [38] | Li, G.; Wallis, J. R.; Shaw, G., A parallel linear solver algorithm for solving difficult large scale thermal models, (Proceedings - SPE Reservoir Simulation Symposium (2015)) |
| [39] | Li, G.; Wallis, J. R., Enhanced constrained pressure residual ECPR preconditioning for solving difficult large scale thermal models, (Proceedings - SPE Reservoir Simulation Symposium (2017)) |
| [40] | Roy, T.; Jönsthövel, T. B.; Lemon, C.; Wathen, A. J., A block preconditioner for non-isothermal flow in porous media, J. Comput. Phys., 395, 636-652 (2019) · Zbl 1452.65053 |
| [41] | Roy, T.; Jönsthövel, T. B.; Lemon, C.; Wathen, A. J., A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media (2019) |
| [42] | Voskov, D.; Zhou, Y.; Volkov, O. (2012), AD-GPRS technical description |
| [43] | Garipov, T. T.; Tomin, P.; Rin, R.; Voskov, D. V.; Tchelepi, H. A., Unified thermo-compositional-mechanical framework for reservoir simulation, Comput. Geosci., 22, 1039-1057 (2018) · Zbl 1401.86012 |
| [44] | Darcy, H., Les fontaines publiques de la ville de Dijon: exposition et application (1856), Victor Dalmont |
| [45] | Muskat, M.; Meres, M. W., The flow of heterogeneous fluids through porous media, J. Appl. Phys., 7, 346-363 (1936) · JFM 62.1565.01 |
| [46] | Incropera, F. P.; Lavine, A. S.; Bergman, T. L.; DeWitt, D. P., Fundamentals of Heat and Mass Transfer (2007), Wiley |
| [47] | Iranshahr, A.; Voskov, D. V.; Tchelepi, H. A., Phase equilibrium computations are no longer the bottleneck in thermal compositional EoS based simulation, (Proceedings - SPE Reservoir Simulation Symposium (2009)) |
| [48] | Lapene, A.; Nichita, D. V.; Debenest, G.; Quintard, M., Three-phase free-water flash calculations using a new modified Rachford-Rice equation, Fluid Phase Equilib., 297, 121-128 (2010) |
| [49] | Arrhenius, S., Über die reaktionsgeschwindigkeit bei der inversion von rohrzucker durch säuren, Z. Phys. Chem., 4, 226-248 (1889) |
| [50] | Aziz, K.; Settari, A., Petroleum Reservoir Simulation (1979), Elsevier Applied Science Publishers: Elsevier Applied Science Publishers London, UK |
| [51] | Voskov, D. V.; Tchelepi, H. A.; Younis, R., General nonlinear solution strategies for multiphase multicomponent EoS based simulation, (Proceedings - SPE Reservoir Simulation Symposium. Proceedings - SPE Reservoir Simulation Symposium, The Woodlands, TX, USA (2009)) |
| [52] | Zaydullin, R.; Voskov, D. V.; Tchelepi, H. A., Comparison of EoS-based and K-values-based methods for three-phase thermal simulation, Transp. Porous Media, 116, 663-686 (2017) |
| [53] | Coats, K. H., A note on IMPES and some IMPES-based simulation models, SPE J., 5, 245-251 (2000) |
| [54] | Gries, S., System-AMG approaches for industrial fully and adaptive implicit oil reservoir simulations (2015), Universität zu Köln, Ph.D. thesis |
| [55] | Baker, A. H.; Falgout, R. D.; Kolev, Tz. V.; Yang, U. M., Multigrid smoothers for ultraparallel computing, SIAM J. Sci. Comput., 33, 2864-2887 (2011) · Zbl 1237.65032 |
| [56] | Falgout, R. D.; Yang, U. M., hypre: a library of high performance preconditioners, (Sloot, P. M.A.; Hoekstra, A. G.; Tan, C. J.K.; Dongarra, J. J., Computational Science — ICCS 2002. Computational Science — ICCS 2002, ICCS 2002. Computational Science — ICCS 2002. Computational Science — ICCS 2002, ICCS 2002, Lecture Notes in Computer Science, vol. 2331 (2002)), 632-641 · Zbl 1056.65046 |
| [57] | Bui, Q. M.; Elman, H. C.; Moulton, J. D., Algebraic multigrid preconditioners for multiphase flow in porous media, SIAM J. Sci. Comput., 39, S662-S680 (2017) · Zbl 1392.65024 |
| [58] | Bui, Q. M.; Wang, L.; Osei-Kuffuor, D., Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions, Adv. Water Resour., 114, 19-28 (2018) |
| [59] | Bui, Q. M.; Osei-Kuffuor, D.; Castelletto, N.; White, J. A., A scalable multigrid reduction framework for multiphase poromechanics of heterogeneous media, SIAM J. Sci. Comput., 42, B379-B396 (2020) · Zbl 1435.65153 |
| [60] | Kozdon, J.; Mallison, B.; Gerritsen, M. G., Robust multi-d transport schemes with reduced grid orientation effects, Transp. Porous Media, 78, 47-75 (2009) |
| [61] | HSL, A collection of FORTRAN codes for large scale scientific computation, 2002. |
| [62] | Christie, M. A.; Blunt, M. J., Tenth SPE comparative solution project: a comparison of upscaling techniques, SPE Reserv. Eval. Eng., 4, 308-317 (2001) |
| [63] | Dechelette, B.; Heugas, O.; Quenault, G.; Bothua, J.; Christensen, J., Air injection-improved determination of the reaction scheme with ramped temperature experiment and numerical simulation, J. Can. Pet. Technol., 45, 41-47 (2006) |
| [64] | Kovscek, A.; Castanier, L. M.; Gerritsen, M., Improved predictability of in-situ-combustion enhanced oil recovery, SPE Reserv. Eval. Eng., 16, 172-182 (2013) |
| [65] | Nissen, A.; Zhu, Z.; Kovscek, A.; Castanier, L.; Gerritsen, M., Upscaling kinetics for field-scale in-situ-combustion simulation, SPE Reserv. Eval. Eng., 18, 158-170 (2015) |
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