Abstract
The compositional model is often used to describe multicomponent multiphase porous media flows in the petroleum industry. The fully implicit method with strong stability and weak constraints on time-step sizes is commonly used in mainstream commercial reservoir simulators. In this paper, we develop an efficient multistage preconditioner for the fully implicit compositional flow simulation. The method employs an adaptive setup phase to improve the parallel efficiency on GPUs. Furthermore, a multicolor Gauss–Seidel algorithm based on the adjacency matrix is applied in the algebraic multigrid methods for the pressure part. Numerical results demonstrate that the proposed algorithm achieves good parallel speedup while yielding the same convergence behavior as the corresponding sequential version.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002). https://doi.org/10.1023/A:1021291114475
Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24(5), 1329–1360 (2008). https://doi.org/10.1002/num.20320
Al-Shaalan, T.M., Klie, H.M., Dogru, A.H., Wheeler, M.F.: Studies of robust two stage preconditioners for the solution of fully implicit multiphase flow problems. In: SPE Reservoir Simulation Symposium (2009). https://doi.org/10.2118/118722-MS
Aziz, K., Settari, A.: Petroleum reservoir simulation, p. 687. Applied Science Publishers, London (1979)
Brandt, A., Mccormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for sparse matrix equations. In: Sparsity and its applications, pp. 257–284. Cambridge University Press, Cambridge (1984)
Cao, H., Tchelepi, H.A., Wallis, J.R., Yardumian, H.E.: Parallel scalable unstructured CPR-Type linear solver for reservoir simulation. In: SPE Annual Technical Conference and Exhibition, vol. SPE-96809 (2005). https://doi.org/10.2118/96809-MS
Chen, Z.X., Huan, G.R., Ma, Y.L.: Computational methods for multiphase flows in porous media. Society for Industrial and Applied Mathematics, New York (2006). https://doi.org/10.1137/1.9780898718942
Chen, Z., Liu, H., Yu, S., Hsieh, B., Shao, L.: GPU-based parallel reservoir simulators. In: Domain decomposition methods in science and engineering XXI, pp. 199–206. Springer, Cham (2014)
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
Coats, K.H.: An equation of state compositional model. Soc. Petrol. Eng. J. 20(05), 363–376 (1980)
Collins, D.A., Nghiem, L.X., Li, Y.K., Grabonstotter, J.E.: An efficient approach to adaptive-implicit compositional simulation with an equation of state. SPE Reserv. Eng. 7(02), 259–264 (1992). https://doi.org/10.2118/15133-PA
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen differenzengleichungen der mathematischen physik. Mathematische Annalen 100(1), 32–74 (1928). https://doi.org/10.1007/BF01448839
Esler, K., Gandham, R., Patacchini, L., Garipov, T., Samardzic, A., Panfili, P., Caresani, F., Pizzolato, A., Cominelli, A.: A Graphics Processing Unit-based, industrial grade compositional reservoir simulator. SPE J. 27(01), 597–612 (2022). https://doi.org/10.2118/203929-PA
Falgout, R.D.: An introduction to algebraic multigrid computing. Comput. Sci. Eng. 8(6), 24–33 (2006). https://doi.org/10.1109/MCSE.2006.105
Feng, C.S., Shu, S., Xu, J.C., Zhang, C.S.: A multi-stage preconditioner for the black oil model and its OpenMP implementation. In: Lecture Notes in Computational Science and Engineering 98, 141–153 (2014)
Fussell, L.T., Fussell, D.D.: An iterative technique for compositional reservoir models. Soc. Petrol. Eng. J. 19(04), 211–220 (1979)
Hu, X.Z., Xu, J.C., Zhang, C.S.: Application of auxiliary space preconditioning in field-scale reservoir simulation. Sci. China Math. 56(12), 2737–2751 (2013). https://doi.org/10.1007/s11425-013-4737-3
Kang, Z., Deng, Z., Han, W., Zhang, D.: Parallel reservoir simulation with OpenACC and domain decomposition. Algorithms 11(12), 213 (2018)
Killough, J., Kossack, C.: Fifth comparative solution project: evaluation of miscible flood simulators. In: SPE Symposium on Reservoir Simulation (1987). https://doi.org/10.2118/16000-MS
LeVeque, R.J.: Finite volume methods for hyperbolic problems. Meccanica 39, 88–89 (2002)
Li, Z., Wu, S.H., Zhang, C.S., Xu, J.C., Feng, C.S., Hu, X.Z.: Numerical studies of a class of linear solvers for fine-scale petroleum reservoir simulation. Comput. Vis. Sci. 18(2–3), 93–102 (2017). https://doi.org/10.1007/s00791-016-0273-3
Manea, A.M., Almani, T.: A massively parallel algebraic multiscale solver for reservoir simulation on the GPU architecture. In: SPE Reservoir Simulation Conference (2019)
Meyerink, J.A.: Iterative methods for the solution of linear equations based on incomplete block factorization of the matrix. In: SPE Reservoir Simulation Symposium, vol. SPE-12262 (1983). https://doi.org/10.2118/12262-MS
Middya, U., Manea, A., Alhubail, M., Ferguson, T., Byer, T., Dogru, A.: A massively parallel reservoir simulator on the GPU architecture. In: SPE Reservoir Simulation Conference (2021). https://doi.org/10.2118/203918-MS
Napov, A., Notay, Y.: An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34(2), 1079–1109 (2012). https://doi.org/10.1137/100818509
NVIDIA: CUDA C++ programming guide (2022). https://docs.nvidia.com/cuda/pdf/CUDA_C_Programming_Guide.pdf
Odeh, A.S.: Comparison of solutions to a three-dimensional black-oil reservoir simulation problem. J Petrol Technol 33(01), 13–25 (1981). https://doi.org/10.2118/9723-PA
OpenACC: OpenACC programming and best practices guide (2022). http://www.openacc.org/
OpenCAEPoro: mlticomponent multiphase porous media flow simulator (2022). https://faspdevteam.github.io/OpenCAEPoro
Peaceman, D.W.: Fundamentals of numerical reservoir simulation. In: Developments in petroleum science, p. 190. Elsevier, Amsterdam (1977)
Peng, D., Robinson, D.B.: A rigorous method for predicting the critical properties of multicomponent systems from an equation of state. AIChE J. 23(2), 137–144 (1977). https://doi.org/10.1002/aic.690230202
Qiao, C.: General purpose compositional simulation for multiphase reactive flow with a fast linear solver. PhD thesis, The Pennsylvania State University (2015)
Quandalle, P., Savary, D.: An implicit in pressure and saturations approach to fully compositional simulation. In: SPE Symposium on Reservoir Simulation (1989)
Saad, Y.: Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, New York (2003). https://doi.org/10.1137/1.9780898718003
Schlumberger: ECLIPSE Technical Description Version 2017.2 (2017). https://www.software.slb.com/products/eclipse
Stüben, 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 (2007). https://doi.org/10.2118/105832-MS
Sudan, H., Klie, H., Li, R., Saad, Y.: High performance manycore solvers for reservoir simulation. In: European Conference on the Mathematics of Oil Recovery (2010). https://doi.org/10.3997/2214-4609.20144961
Wallis, J.R.: Incomplete gaussian elimination as a preconditioning for generalized conjugate gradient acceleration. In: SPE Reservoir Simulation Symposium (1983). 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 Reservoir Simulation Symposium (1985). https://doi.org/10.2118/13536-MS
Xu, J.C.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)
Xu, J.C.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)
Yang, B., Liu, H., Chen, Z.X.: Accelerating linear solvers for reservoir simulation on GPU workstations. Society for Computer Simulation International 1, 1–8 (2016). https://doi.org/10.22360/SpringSim.2016.HPC.007
Zhang, C.S.: Linear solvers for petroleum reservoir simulation. J. Numer. Methods Comput. Appl. 43(1), 1–26 (2022). https://doi.org/10.12288/szjs.s2021-0813
Zhao, L., Feng, C.S., Zhang, C.S., Shu, S.: Parallel multi-stage preconditioners with adaptive setup for the black oil model. Comput. Geosci. 168, 105230 (2022). https://doi.org/10.1016/j.cageo.2022.105230
Acknowledgements
This work was supported by the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20210607) and Postgraduate Scientific Research Innovation Project of Xiangtan University (No. XDCX2021B110). Li and Zhang were partially supported by the National Science Foundation of China (No. 11971472). Feng was partially supported by the Excellent Youth Foundation of SINOPEC (No. P20009). Shu was partially supported by the National Science Foundation of China (No. 11971414).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhao, L., Li, S., Zhang, CS. et al. An improved multistage preconditioner on GPUs for compositional reservoir simulation. CCF Trans. HPC 5, 144–159 (2023). https://doi.org/10.1007/s42514-023-00136-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42514-023-00136-0
Keywords
- Compositional model
- Fully implicit method
- multistage preconditioner
- multicolor Gauss–Seidel
- GPU
- Compute unified device architecture (CUDA)