Abstract
This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set of system unknowns comprises both element and face pressures, in addition to the cell saturations, resulting in a \(3\times 3\) block-structured Jacobian matrix with a \(2\times 2\) inner pressure problem. The CPR method is one of the most established techniques for reservoir simulations, but most research focused on solutions for Two-Point Flux Approximation (TPFA)-based discretizations that do not readily extend to our problem formulation. Therefore, we designed a dedicated two-stage strategy, inspired by the CPR algorithm, where a block preconditioner is used for the pressure part with the aim at exploiting the inner \(2\times 2\) structure. The proposed preconditioning framework is tested by an extensive experimentation, comprising both synthetic and realistic applications in Cartesian and non-Cartesian domains.
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Magras, J.F., Quandalle, P., Bia, P.: High-performance reservoir simulation with parallel ATHOS. In: SPE Reserv. Simul. Symp. Houston, Texas, USA: Society of Petroleum Engineers, pp. SPE–66342–MS. (2001). Available from: https://onepetro.org/spersc/proceedings/01RSS/All-01RSS/Houston, Texas/133525
Hu, X., Wu, S., Wu, X.H., Xu, J., Zhang, C.S., Zhang, S., et al.: Combined preconditioning with applications in reservoir simulation. Multiscale. Model. Simul. 11(2), 507–521 (2013). https://doi.org/10.1137/120885188
Esler, K., Gandham, R., Patacchini, L., Garipov, T., Samardzic, A., Panfili, P., et al.: A graphics processing unit–based, industrial grade compositional reservoir simulator. SPE J, pp. SPE–203929–PA. (2021). https://doi.org/10.2118/203929-PA
Wallis, JR.: Incomplete Gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reserv. Simul. Symp. San Francisco, California: Society of Petroleum Engineers, pp. 325–334. (1983). Available from: https://doi.org/10.2118/12265-MS
Wallis, J.R., Kendall, R.P., Little, T.E.: Constrained residual acceleration of conjugate residual methods. In: SPE Reserv. Simul. Symp. Dallas, Texas: Society of Petroleum Engineers, pp. SPE–13536–MS. (1985). Available from: https://doi.org/10.2118/13536-MS
Zhou, Y., Jiang, Y., Tchelepi, H.A.: A scalable multistage linear solver for reservoir models with multisegment wells. Comput. Geosci. 17(2), 197–216 (2013). https://doi.org/10.1007/s10596-012-9324-0
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). https://doi.org/10.1007/s10596-018-9737-5
Cremon, M.A., Castelletto, N., White, J.A.: Multi-stage preconditioners for thermal-compositional-reactive flow in porous media. J. Comput. Phys. 418, 109607 (2020). https://doi.org/10.1016/j.jcp.2020.109607
Klemetsdal, Ø.S., Møyner, O., Lie, K.A.: Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions. Comput. Geosci. 24(2), 459–476 (2020). https://doi.org/10.1007/s10596-019-9827-z
Lie, K.A.: An introduction to reservoir simulation using MATLAB/GNU Octave. Cambridge, United Kingdom: Cambridge University Press (2019). Available from: https://www.cambridge.org/core/product/identifier/9781108591416/type/book
Alvestad, J., Baxendale, D., Bao, K., Blatt, M., Hove, J., Lauser, A., et al.: OPM flow: Reference manual. Oslo, Norway: Equinor ASA (2022). Available from: https://opm-project.org/wp-content/uploads/2022/05/OPM_Flow_Reference_Manual_2022-04_Rev-0_Reduced.pdf
Rasmussen, A.F., Sandve, T.H., Bao, K., Lauser, A., Hove, J., Skaflestad, B., et al.: The open porous media flow reservoir simulator. Comput. Math. with Appl. 81, 159–185 (2021). https://doi.org/10.1016/j.camwa.2020.05.014
Schlumberger: Eclipse: Technical description (2020)
Schlumberger: Intersect: Technical description (2020)
Halliburton: Nexus: Technical reference guide (2014)
Lacroix, S., Vassilevski, Y.V., Wheeler, M.F.: Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numer. Linear. Algebr. with Appl. 8(8), 537–549 (2001). https://doi.org/10.1002/nla.264
Singh, G., Pencheva, G., Wheeler, M.F.: An approximate Jacobian nonlinear solver for multiphase flow and transport. J. Comput. Phys. 375, 337–351 (2018). https://doi.org/10.1016/j.jcp.2018.08.043
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). https://doi.org/10.1137/S106482750240443X
Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E.: Parallel scalable unstructured CPR-type linear solver for reservoir simulation. In: SPE Annu. Tech. Conf. Exhib. Dallas, Texas: Society of Petroleum Engineers, pp. SPE–96809–MS. (2005). Available from:https://doi.org/10.2118/96809-MS
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
Nardean, S., Ferronato, M., Abushaikha, A.: Linear solvers for reservoir simulation problems: An overview and recent developments. Arch. Comput. Methods Eng. 29(6), 4341–4378 (2022). https://doi.org/10.1007/s11831-022-09739-2
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. SIAM J. Sci. Comput. 42(4), B1014–B1040 (2020). https://doi.org/10.1137/19M1292023
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 (2019). https://doi.org/10.1016/j.cma.2019.112575
T Camargo, J., White, J.A., Castelletto, N., Borja, R.I.: Preconditioners for multiphase poromechanics with strong capillarity. Int. J. Numer. Anal. Methods Geomech. 45(9), 1141–1168 (2021). https://doi.org/10.1002/nag.3192
Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Vol. 15 of Springer Series in Computational Mathematics. New York, NY: Springer-Verlag New York (1991). Available from: http://link.springer.com/10.1007/978-1-4612-3172-1
Abushaikha, A.S., Voskov, D.V., Tchelepi, H.A.: Fully implicit mixed-hybrid finite-element discretization for general purpose subsurface reservoir simulation. J. Comput. Phys. 346, 514–538 (2017). https://doi.org/10.1016/j.jcp.2017.06.034
Abushaikha, A.S., Terekhov, K.M.: A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability. J. Comput. Phys. 406, 109194 (2020). https://doi.org/10.1016/j.jcp.2019.109194
Li, L., Abushaikha, A.: A fully-implicit parallel framework for complex reservoir simulation with mimetic finite difference discretization and operator-based linearization. Comput. Geosci. (2021). https://doi.org/10.1007/s10596-021-10096-5
Kuznetsov, Y.A.: Spectrally equivalent preconditioners for mixed hybrid discretizations of diffusion equations on distorted meshes. J. Numer. Math. 11(1), 61–74 (2003). https://doi.org/10.1163/156939503322004891
Maryska, J., Rozlozník, M., Tuma, M.: Schur complement systems in the mixed-hybrid finite element approximation of the potential fluid flow problem. SIAM J. Sci. Comput. 22(2), 704–723 (2000). https://doi.org/10.1137/S1064827598339608
Nardean, S., Ferronato, M., Abushaikha, A.S.: A novel and efficient preconditioner for solving Lagrange multipliers-based discretization schemes for reservoir simulations. In: ECMOR XVII - 17th Eur. Conf. Math. Oil Recover. Edinburgh: European Association of Geoscientists & Engineers, pp. 1–12. (2020). Available from: https://www.earthdoc.org/content/papers/10.3997/2214-4609.202035072
Nardean, S., Ferronato, M., Abushaikha, A.S.: A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based Darcy flow simulations. J. Comput. Phys. 110513 (2021). https://doi.org/10.1016/j.jcp.2021.110513
Nardean, S., Ferronato, M., Abushaikha, A.: A blended CPR/block preconditioning approach for mixed discretization schemes in reservoir modeling. In: ECMOR 2022 Eur. Conf. Math. Geol. Reserv. The Hague, The Netherlands: European Association of Geoscientists & Engineers, pp. 1–15. (2022). Available from: https://www.earthdoc.org/content/papers/10.3997/2214-4609.202244067
Coats, K.H.: An equation of state compositional model. SPE J. 20(5), 363–376 (1980). https://doi.org/10.2118/8284-PA
Peaceman, D.W.: Interpretation of well-block pressures in numerical reservoir simulation. SPE J. 18(3), SPE–6893–PA (1978). https://doi.org/10.2118/6893-PA
Chen, Z., Huan, G., Ma, Y.: Computational methods for multiphase flows in porous media. Society for Industrial and Applied Mathematics. Philadelphia, PA, USA (2006). Available from: https://doi.org/10.1137/1.9780898718942
Corey, A.T.: The interrelation between gas and oil relative permeabilities. Prod. Mon. 19(1), (1954)
Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media. Colorado State University, Fort Collins, Colorado, USA (1964)
Aziz, K., Settari, A.: Petroleum reservoir simulation. Applied Science Publishers, London, United Kingdom (1979)
Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Math. Asp. Finite Elem. Methods. Lect. Notes Math, pp. 292–315. Springer, Berlin, Heidelberg (1977). Available from: http://link.springer.com/10.1007/BFb0064470
Zhang, N., Abushaikha, A.S.: An implementation of mimetic finite difference method for fractured reservoirs using a fully implicit approach and discrete fracture models. J. Comput. Phys. 110665 (2021). 10.1016/j.jcp.2021.110665
Maryška, J., Rozložník, M., Tůma, M.: Mixed-hybrid finite element approximation of the potential fluid flow problem. J. Comput. Appl. Math. 63(1–3), 383–392 (1995). https://doi.org/10.1016/0377-0427(95)00066-6
Mosé, R., Siegel, P., Ackerer, P., Chavent, G.: Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? Water Resour. Res. 30(11), 3001–3012 (1994). https://doi.org/10.1029/94WR01786
Younes, A., Ackerer, P., Delay, F.: Mixed finite elements for solving 2-D diffusion-type equations. Rev. Geophys. 48(1), RG1004 (2010). https://doi.org/10.1029/2008RG000277
Younis, R.M.: Modern advances in software and solution algorithms for reservoir simulation [PhD dissertation]. Stanford University (2011). Available from:https://stacks.stanford.edu/file/druid:fb287kz3299/RMY_PHD_THESIS-augmented.pdf
Bui, Q.M., Elman, H.C., Moulton, J.D.: Algebraic multigrid preconditioners for multiphase flow in porous media. SIAM J. Sci. Comput. 39(5), S662–S680 (2017). https://doi.org/10.1137/16M1082652
Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), A1079–A1109 (2012). https://doi.org/10.1137/100818509
Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–146 (2010)
Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012). https://doi.org/10.1137/110835347
Eisenstat, S.C., Elman, H.C., Schultz, M.H.: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983). https://doi.org/10.1137/0720023
Jiránek, P., Rozložník, M., Gutknecht, M.H.: How to make simpler GMRES and GCR more stable. SIAM J. Matrix Anal. Appl. 30(4), 1483–1499 (2009). https://doi.org/10.1137/070707373
Falgout, R.D., Yang, U.M.: HYPRE: A library of high performance preconditioners. In: Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J. (eds.) Comput. Sci. — ICCS 2002, pp. 632–641. Springer, Berlin, Heidelberg (2002). Available from: http://link.springer.com/10.1007/3-540-47789-6_66
Balay, S., Abhyankar, S., Adams, M.F., Benson, S., Brown, J., Brune, P., et al.: PETSc/TAO users manual - ANL-21/39 - Revision 3.17. Argonne National Laboratory (2022)
Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object-oriented numerical software libraries. In: Mod. Softw. tools Sci. Comput. Boston, MA: Birkhäuser Boston, pp. 163–202. (1997). Available from: http://link.springer.com/10.1007/978-1-4612-1986-6_8
Christie, M.A., Blunt, M.J.: Tenth SPE comparative solution project: A comparison of upscaling techniques. In: SPE Reserv. Simul. Symp. Houston, Texas: Society of Petroleum Engineers, pp. 308–317. (2001). Available from: https://doi.org/10.2118/66599-MS
Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986). https://doi.org/10.1137/0907058
Coats, K.H.: IMPES stability: Selection of stable timesteps. SPE J. 8(02), 181–187 (2003). https://doi.org/10.2118/84924-PA
Saad, Y.: Iterative methods for sparse linear systems. Philadelphia, USA: Society for Industrial and Applied Mathematics (2003). Available from: http://epubs.siam.org/doi/book/10.1137/1.9780898718003
Ferronato, M., Franceschini, A., Janna, C., Castelletto, N., Tchelepi, H.A.: A general preconditioning framework for coupled multiphysics problems with application to contact- and poro-mechanics. J Comput Phys. 398, 108887 (2019). https://doi.org/10.1016/j.jcp.2019.108887
Franceschini, A., Castelletto, N., Ferronato, M.: Approximate inverse-based block preconditioners in poroelasticity. Comput. Geosci. 25(2), 701–714 (2021). https://doi.org/10.1007/s10596-020-09981-2
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Nardean, S., Ferronato, M. & Abushaikha, A. Block constrained pressure residual preconditioning for two-phase flow in porous media by mixed hybrid finite elements. Comput Geosci 28, 253–272 (2024). https://doi.org/10.1007/s10596-023-10238-x
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DOI: https://doi.org/10.1007/s10596-023-10238-x
Keywords
- Constrained pressure residual
- Block preconditioning
- Two-phase flow in porous media
- Mixed hybrid finite elements