Implicit High-Resolution Compositional Simulation With Optimal Ordering of Unknowns and Adaptive Spatial Refinement
- Ø.S. Klemetsdal (Norwegian University of Science and Technology/SINTEF) | O. Møyner (Norwegian University of Science and Technology/SINTEF) | K.-A. Lie (SINTEF)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Simulation Conference, 10-11 April, Galveston, Texas, USA
- Publication Date
- Document Type
- Conference Paper
- 2019. Society of Petroleum Engineers
- 5 Reservoir Desciption & Dynamics, 5.5 Reservoir Simulation
- Reordering, Discontinuous Galerkin, Adaptive refinement, Compositional flow
- 5 in the last 30 days
- 120 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 9.50|
|SPE Non-Member Price:||USD 28.00|
High-resolution discretizations can be advantageous in compositional simulation to reduce excessive numerical diffusion that tends to mask shocks and fingering effects. In this work, we outline a fully implicit, dynamic, multilevel, high-resolution simulator for compositional problems on unstructured polyhedral grids. We rely on four ingredients: (i) sequential splitting of the full problem into a pressure and a transport problem, (ii) ordering of grid cells based on intercell fluxes to localize the nonlinear transport solves, (iii) higher-order discontinuous Galerkin (dG) spatial discretization with order adaptivity for the component transport, and (iv) a dynamic coarsening and refinement procedure. For purely cocurrent flow, and in the absence of capillary forces, the nonlinear transport system can be perturbed to a lower block-triangular form. With counter-current flow caused by gravity or capillary forces, the nonlinear system of discrete transport equations will contain larger blocks of mutually dependent cells on the diagonal. In either case, the transport subproblem can be solved efficiently cell-by-cell or block-by-block because of the natural localization in the dG scheme. In addition, we discuss how adaptive grid and order refinement can effectively improve accuracy. We demonstrate the applicability of the proposed solver through a number of examples, ranging from simple conceptual problems with PEBI grids in two dimensions, to realistic reservoir models in three dimensions. We compare our new solver to the standard upstream-mobility-weighting scheme and to a second-order WENO scheme.
|File Size||1 MB||Number of Pages||22|
Aarnes, J. E.,Hauge, V. L., and Efendiev, Y. 2007. Coarsening of three-dimensional structured and unstructured grids for subsurface flow. Adv. Water Resour., 30(11):2177-2193. doi: 10.1016/j.advwatres.2007.04.007.
Adam, A.,Pavlidis, D.,Percival, J. R.,Salinas, P.,Loubens, R. D.,Pain, C. C.,Muggeridge, A. H., and Jackson, M. D. 2017. Dynamic mesh adaptivity for immiscible viscous fingering. In SPE Reservoir Simulation Conference, 20-22 February, Montgomery, Texas, USA. Society of Petroleum Engineers. doi: 10.2118/182636-MS.
Amooie, M. A. and Moortgat, J. 2018. Higher-order black-oil and compositional modeling of multiphase compressible flow in porous media. International Journal of Multiphase Flow, 105:45-59. doi: 10.1016/j.ijmultiphaseflow.2018.03.016.
Appleyard, J. R. and Cheshire, I. M. 1982. The cascade method for accelerated convergence in implicit simulators. In European Petroleum Conference, pages 113-122, SPE 12804. doi: 10.2118/12804-MS.
Bell, J. B.,Trangenstein, J. A., and Shubin, G. R. 1986. Conservation laws of mixed type describing three-phase flow in porous media. SIAM Journ. Appl. Math., 46(6):1000-1017. doi: 10.1137/0146059.
Berger, M. J. and Oliger, J. 1984. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of computational Physics, 53(3):484-512. doi: 10.1016/0021-9991(84)90073-1.
Brenier, Y. and Jaffré, J. 1991. Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal., 28(3):685-696. doi: 10.1137/0728036.
Christie, M.A. and Blunt,M. J. 2001. Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng., 4:308-317. doi: 10.2118/72469-PA.
Coats, K. H. 2000. A note on IMPES and some IMPES-based simulation models. SPE Journal, 5(03):245-251. doi: 10.2118/65092-PA.
Cusini, M.,Fryer, B.,Van Kruijsdijk, C.P., and Hajibeygi, H. 2017. Algebraicdynamicmultilevelmethodforcompositionalsimulations. In SPE Reservoir Simulation Conference, 20-22 February, Montgomery, Texas, USA. Society of Petroleum Engineers. doi: 10.2118/182644-MS.
Cusini, M. and Hajibeygi, H. 2018. Algebraic dynamic multilevel (adm) method for simulations of multiphase flow with an adaptive saturation interpolator. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802254.
Epshteyn, Y. and Riviere, B. 2007. Fully implicit discontinuous finite element methods for two-phase flow. Appl. Numer. Math., 57(4):383-401. doi: 10.1016/j.apnum.2006.04.004.
Gries, S.,Stuben, K.,Brown, G. L.,Chen, D., and Collins, D. A. 2014. Preconditioning for efficiently applying algebraic multigrid in fully implicit reservoir simulations. SPE J., 19(04):726-736. doi: 10.2118/163608-PA.
Hamon, F. P. and Tchelepi, H. A. 2014. Ordering-based nonlinear solver for fully-implicit simulation. In ECMOR XIV - 14th European Conference on the Mathematics of Oil Recovery, Catania, Sicily, Italy, 8-1 1 September 2014. EAGE. doi: 10.3997/2214-4609.20141767.
Hauge, V. L.,Lie, K.-A., and Natvig, J. R. 2012. Flow-based coarsening for multiscale simulation of transport in porous media. Comput. Geosci., 16(2):391-408. doi: 10.1007/s10596-011-9230-x.
Heinemann, Z. E.,Gerken, G., and von Hantelmann, G. 1983. Using local grid refinement in a multiple-application reservoir simulator. In SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. doi: 10.2118/12255-MS.
Hoteit, H. and Chawathe, A. 2016. Making field-scale chemical enhanced-oil-recovery simulations a practical reality with dynamic gridding. SPEJournal, 21(06):2-220. doi: 10.2118/169688-PA.
Hoteit, H. and Firoozabadi, A. 2005. Multicomponent fluid flow by discontinuous Galerkin and mixed methods in unfractured and fractured media. Water Resources Research, 41(11). doi: 10.1029/2005WR004339.
Hoteit, H. and Firoozabadi, A. 2006. Compositional modeling by the combined discontinuous Galerkin and mixed methods. SPE J., 11(01):19-34. doi: 10.2118/90276-PA.
Hoteit, H. and Firoozabadi, A. 2018. Modeling of multicomponent diffusions and natural convection in unfractured and fractured media by discontinuous Galerkin and mixed methods. International Journal for Numerical Methods in Engineering, 114(5):535-556. doi: 10.1002/nme.5753.
Klemetsdal, ø. S.,Møyner, O., and Lie, K.-A. 2018a. Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802251.
Klemetsdal, ø. S.,Rasmussen, A. F.,Møyner, O., and Lie, K.-A. 2018b. Nonlinear gauss-seidel solvers with higher order for black-oil models. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802130.
Krogstad, S.,Hauge, V. L., and Gulbransen, A. F. 2011. Adjoint multiscale mixed finite elements. SPEJ.,16(1):162-171. doi: 10.2118/119112-PA.
Kwok, F. and Tchelepi, H. 2007. Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media. J. Comput. Phys., 227(1):706-727. doi: 10.1016/j.jcp.2007.08.012.
Kwok, F. and Tchelepi, H. A. 2008. Convergence of implicit monotone schemes with applications in multiphase flow in porous media. SIAM J. Numer. Anal., 46(5):2662-2687. doi: 10.1137/070703922.
Lie, K.-A.,Møyner, O.,Natvig, J. R.,Kozlova, A.,Bratvedt, K.,Watanabe, S., and Li, Z. 2017. Successful application of multiscale methods in a real reservoir simulator environment. Comput. Geosci., 21(5):981-998. doi: 10.1007/s10596-017-9627-2.
Lie, K.-A.,Mykkeltvedt, T. S., and Møyner, O. 2018. Fully implicit WENO schemes on stratigraphic and fully unstructured grids. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802269.
Lie, K.-A.,Nilsen,H.M.,Rasmussen,A. F., and Raynaud,X. 2014. Fast simulation ofpolymer injection in heavy-oil reservoirs on the basis of topological sorting and sequential splitting. SPE Journal, 19(06):991-1004. doi: 10.2118/163599-PA.
Moortgat, J. 2017. Adaptive implicit finite element methods for multicomponent compressible flow in heterogeneous and fractured porous media. Water Resources Research, 53(1):73-92. doi: 10.1002/2016WR019644.
Moortgat, J. and Firoozabadi, A. 2016. Mixed-hybrid and vertex-discontinuous-Galerkin finite element modeling of multiphase compositional flow on 3d unstructured grids. Journal of Computational Physics, 315:476-500. doi: 10.1016/j.jcp.2016.03.054.
Mostaghimi, P.,Kamali, F.,Jackson, M. D.,Muggeridge, A.H., and Pain, C. C. 2016. Adaptive mesh optimization for simulation of immiscible viscous fingering. SPEJournal, 21(06):2-250. doi: 10.2118/173281-PA.
Møyner, O. and Moncorge, A. 2018. Nonlinear domain decomposition scheme for sequential fully implicit formulation of compositional multiphase flow. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802128.
Møyner, O. and Tchelepi, H. A. 2018. A mass-conservative sequential implicit multiscale method for isothermal equation of state compositional problems. SPEJ. doi: 10.2118/915-PA.
MRST 2018. The MATLAB Reservoir Simulation Toolbox. www.sintef.no/MRST.
Müller, B.,Kummer, F., and Oberlack, M. 2013. Highly accurate surface and volume integration on implicit domains by means of momentfitting. Int. J. Numer. Meth. Eng., 96(8):512-528. doi: 10.1002/nme.4569.
Mykkeltvedt, T. S.,Raynaud, X., and Lie, K.-A. 2017. Fully implicit higher-order schemes applied to polymer flooding. Comput. Geosci., 21(5):1245-1266. doi: 10.1007/s10596-017-9676-6.
Natvig, J. R.,Lie, K.,Eikemo, B., and Berre, I. 2007. An efficient discontinuous galerkin method for advective transport in porous media. Adv. Water Resour., 30(12):2424-2438. doi: 10.1016/j.advwatres.2007.05.015.
Natvig, J. R. and Lie, K.-A. 2008. 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. doi: 10.1016/j.jcp.2008.08.024.
Nilsson, J.,Gerritsen,M., and Younis,R.2005. A novel adaptive anisotropic grid framework for efficient reservoir simulation. In SPE reservoir simulation symposium. Society of Petroleum Engineers. doi: 10.2118/93243-MS.
Rivière, B. and Wheeler, M. F. 2002. Discontinuous Galerkin methods for flow and transport problems in porous media. Comm. Numer. Meth. Eng., 18(1):63-68. doi: 10.1002/cnm.464.
Salinas, P.,Pavlidis, D.,Xie, Z.,Pain, C. C., and Jackson, M. D. 2017. A double control volume finite element method with dynamic unstructured mesh optimization. In SPE Reservoir Simulation Conference, 20-22 February, Montgomery, Texas, USA. Society of Petroleum Engineers. doi: 10.2118/182647-MS.
Sheth, S.,Moncorge, A., and Younis, R. M. 2018. Localized linear systems for fully-implicit simulation of multiphase multicomponent flow in porous media. In ECMOR XVI-16th European Conference on the Mathematics of Oil Recovery. doi: 10.3997/2214-4609.201802123.
Sheth, S. M. and Younis, R. M. 2017a. Localized linear systems in sequential implicit simulation oftwo-phase flow and transport. SPE Journal, 22(5):1542 - 1569. doi: 10.2118/173320-PA.
Sheth, S. M. and Younis, R. M. 2017b. Localized solvers for general full-resolution implicit reservoir simulation. In SPE Reservoir Simulation Conference, 20-22 February, Montgomery, Texas, USA. Society of Petroleum Engineers. doi: 10.2118/182691-MS.
Trangenstein,J.A. and Bell,J.B.1989. Mathematical structure of the black-oil model for petroleum reservoir simulation. SIAM J. Appl. Math., 49(3):749-783. doi: 10.1137/0149044.
Van Batenburg, D. W.,Bosch, M.,Boerrigter, P. M., De Zwart, A. H., and Vink, J. C. 2011. Application of dynamic gridding techniques to IOR/EOR processes. In SPE Reservoir Simulation Symposium, 21-23 February, The Woodlands, Texas, USA. doi: 10.2118/141711-MS.
Watts, J.1986. A compositional formulation ofthe pressure and saturation equations. SPEReservoirEng., 1(03):243-252. doi: 10.2118/12244-PA.
Xiao, H. and Gimbutas, Z. 2010. A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Comput. Math. Appl., 59(2):663-676. doi: 10.1016/j.camwa.2009.10.027.