Abstract
The paper discusses the numerical algorithm constructing a three-dimensional model for a flow of two-phase incompressible fluid caused by the mass force of gravity in a porous medium. The algorithm is based on a combination of a hybrid upwind method with an explicit scheme for determination of the saturation. The hybrid upwinding allows us to take into account flows of fluid of various nature (in this case, viscous and gravitational flows) separately, which is extremely important in the case of gravitational flow with opposite directions of phase flows. The explicit scheme being extremely simple in implementation provides a small dispersion of solutions on discontinuities. The proposed algorithm is illustrated by the results of numerical experiments demonstrating the monotonicity of the method considered in this paper.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 19–11–00048
Funding statement: The work was supported by the Russian Science Foundation (project No. 19–11–00048).
References
[1] K. Aziz and A. Settari, Petroleum Reservoir Simulation Applied Science Publishers, London, 1979.Search in Google Scholar
[2] G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Fluid Flows Through Natural Rocks Kluwer Academic Publishers, Boston, 1990.10.1007/978-94-015-7899-8Search in Google Scholar
[3] P. G. Bedrikovetskii and V. I. Maron, Gravitational separation of oil and water in beds of limited thickness. Fluid Dynamics 21 (1986), 242–251.10.1007/BF01050176Search in Google Scholar
[4] Y. Brenier and J. Jaffre, Upstream differencing for multiphase flow in reservoir simulation. SIAM J. Numer. Anal. 28 (1991), No. 3, 685–696.10.1137/0728036Search in Google Scholar
[5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods Springer-Verlag, New York, 1991.10.1007/978-1-4612-3172-1Search in Google Scholar
[6] S. E. Buckley and M. C. Leverett, Mechanism of fluid displacement in sands. Trans. of the A.I.M.E. 146 (1942), 107–116.10.2118/942107-GSearch in Google Scholar
[7] I. A. Charny, Underground Hydrodynamics Gostoptehizdat, Moscow, 1963 (in Russian).Search in Google Scholar
[8] G. Chavent and J. Jaffre, Mathematical Models and Finite Elements for Reservoir Simulation Elsevier, Amsterdam, 1986.Search in Google Scholar
[9] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media SIAM, Philadelphia, 2006.10.1137/1.9780898718942Search in Google Scholar
[10] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980), 1–21.10.1090/S0025-5718-1980-0551288-3Search in Google Scholar
[11] L. P. Dake, Fundamentals of Reservoir Engineering Developments in Petroleum Science, Vol. 8. Elsevier, New York, 1978.Search in Google Scholar
[12] G. Enchery, R. Masson, S. Wolf, and R. Eymard, Mathematical and numerical study of an industrial scheme for two-phase flows in porous media under gravity. Comput. Methods Appl. Math. 2 (2002), No. 4, 325–353.10.2478/cmam-2002-0019Search in Google Scholar
[13] R. Eymard, T. Gallouet, and P. Joly, Hybrid finite element techniques for oil recovery simulation. Comput. Methods Appl. Mech. Engrg. 74 (1989), No. 1, 83–98.10.1016/0045-7825(89)90088-1Search in Google Scholar
[14] F. P. Hamon and H. A. Tchelepi, Analysis of hybrid upwinding for fully implicit simulation of three-phase flow with gravity. SIAM J. Numer. Anal. 54 (2016), No. 3, 1682–1712.10.1137/15M1020988Search in Google Scholar
[15] F. P. Hamon, B. T. Mallison, and H. A. Tchelepi, Implicit hybrid upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity. Comput. Methods Appl. Mech. Engrg. 331 (2018), 701–727.10.1016/j.cma.2017.10.008Search in Google Scholar
[16] F. P. Hamon and B. T. Mallison, Fully implicit multidimensional hybrid upwind scheme for coupled flow and transport. Comput. Methods Appl. Mech. Engrg. 358 (2020), DOI: 10.1016/j.cma.2019.112606.10.1016/j.cma.2019.112606Search in Google Scholar
[17] J. Jaffre, Flux calculation at the interface between two rock types for two phase flow in porous media. Transp. Porous Media 21 (1995), 195–207.10.1007/BF00617405Search in Google Scholar
[18] P. Jenny, H. A. Tchelepi, and S. H. Lee, Unconditionally convergent nonlinear solver for hyperbolic conservation laws with S-shaped flux functions. J. Comp. Phys. 228 (2009), 7497–7512.10.1016/j.jcp.2009.06.032Search in Google Scholar
[19] E. Keilegavlen, J. E. Kozdon, and B. T. Mallison, Multidimensional upstream weighting for multiphase transport on general grids. Comput. Geosci. 16 (2012), No. 4, 1021–1042.10.1007/s10596-012-9301-7Search in Google Scholar
[20] A. N. Konovalov, Problems of Multiphase Fluid Filtration World Scientific, Hong Kong, 1994.10.1142/2330Search in Google Scholar
[21] J. E. Kozdon, B. T. Mallison, and M. G. Gerritsen, Multidimensional upstream weighting for multiphase transport in porous media. Comput. Geosci. 15 (2011), No. 3, 399–419.10.1007/s10596-010-9211-5Search in Google Scholar
[22] N. N. Kuznetsov and S. A. Voloshin, On monotone difference approximations for a first order quasi-linear equation. Soviet Math. Dokl. 17 (1976), 1203–1206.Search in Google Scholar
[23] F. Kwok and H. A.Tchelepi, Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media. SIAM J. Numer. Anal. 46 (2008), No. 5, 2662–2687.10.1016/j.jcp.2007.08.012Search in Google Scholar
[24] Yu. M. Laevsky, P. E. Popov, and A. A. Kalinkin, Simulation of two-phase fluid filtration by mixed finite element method. Matem. Model. 22 (2010), No. 3, 74–90.Search in Google Scholar
[25] L. W. Lake, Enhanced Oil Recovery Englewood Cliffs, Prentice-Hall, NJ, 1989.Search in Google Scholar
[26] V. I. Lebedev, Explicit difference schemes for solving stiff problems with a complex or separable spectrum. Comput. Math. Math. Phys. 40 (2000), No. 12, 1729–1740.Search in Google Scholar
[27] S. H. Lee, Y. Efendiev, and H. A. Tchelepi, Hybrid upwind discretization of nonlinear two-phase flow with gravity. Advances in Water Resources 82 (2015), 27–38.10.1016/j.advwatres.2015.04.007Search in Google Scholar
[28] S. H. Lee and Y. Efendiev, Hybrid discretization of multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces. Comput. Geosci. 22 (2018), 1403–1421.10.1007/s10596-018-9760-6Search in Google Scholar
[29] G. I. Marchuk, Splitting and alternating direction methods. In: Handbook of Numerical Analysis (Eds. P. G. Ciarlet and J.-L. Lions). North-Holland, Amsterdam, 1990, pp. 197–462.10.1016/S1570-8659(05)80035-3Search in Google Scholar
[30] A. Moncorge, O. Moyner, H. A. Tchelepi, and P. Jenny, Consistent upwinding for sequential fully implicit multiscale compositional simulation. Comput. Geosci. 24 (2020), 533–550.10.1007/s10596-019-09835-6Search in Google Scholar
[31] D. W. Peaceman, Fundamentals of Numerical Reservoir Simulation Elsevier, Amsterdam, 1977.10.1016/S0376-7361(08)70233-4Search in Google Scholar
[32] P. A. Raviart and J. M. Thomas, A mixed finite element method for 2-nd order elliptic problems. Mathematical aspects of finite element methods. Lecture Notes in Mathematics 606 (1977), 292–315.10.1007/BFb0064470Search in Google Scholar
[33] P. Sammon, An analysis of upstream differencing. SPE Reservoir Engrg. 3 (1988), No. 3, 1053–1056.10.2118/14045-PASearch in Google Scholar
[34] J. W. Sheldon, B. Zondek, and W. T. Cardwell, One-dimensional, incompressible, noncapillary, two-phase fluid flow in a porous medium. Petroleum Trans., AIME 216 (1959), 290–296.10.2118/978-GSearch in Google Scholar
[35] R. Younis, H. A. Tchelepi, and K. Aziz, Adaptively localized continuation-Newton: nonlinear solvers that converges all the time. SPE J. 15 (2010), No. 2, 526–544.10.2118/119147-PASearch in Google Scholar
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