Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids
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Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.
KeywordsCell-centred finite-volume Higher resolution method Two-phase flow Gravity Capillary pressure Vt and Va formulations CVD MPFA
We thank Dr’s S. Lamine and B. Huisman of Shell and M. Pal of Maersk for helpful discussions.
- 1.Barth, T., Jespersen, D.C.: The design and application of upwind schemes on unstructured meshes. AIAA paper 89–0366 (1989)Google Scholar
- 2.Trangenstein, J.A.: Multi-phase Flow in Porous Media: Mechanics, Mathematics and Numerics. Lecture Notes, IBM Scientific Center, Bergen, Norway. UCLR-97053 Lawrence Livermore National Laboratory, USA (1987)Google Scholar
- 4.Karimi-Fard, M., Firoozabadi, A.: Numerical simulaton of water injection in 2-D fractured media using discrete fractured model. SPE RE & E, 117–126 (2003)Google Scholar
- 7.Edwards, M.G.: Global and local central non-upwind finite volume schemes for hyperbolic conservation laws in porous media. Int. J. Numer. Meth. Fluids 64(7), 793–811 (2010)Google Scholar
- 8.Friis, H.A., Evje, S.: Numerical treatment of two-phase flow in capillary heterogeneous porous media by finite-volume approximations. Int. J. Numer. Anal. Mod. 9(3), 505–528 (2012)Google Scholar
- 10.Leveque, R.J.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press (2004)Google Scholar
- 11.Godunov, S.K.: A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb. (1959)Google Scholar
- 12.Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, Amsterdam/New York (1977)Google Scholar
- 13.Aziz, K., Settari, A.: Petroleum Reservoir Simulation. Elsevier (1979)Google Scholar
- 14.Radu, F.A., Nordbotten, J.M., Pop, I.S., Kumar, K.: A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. https://doi.org/10.1016/j.cam.2015.02.051 (2015)
- 15.Bastian, P.: A fully-coupled discontinuous galerkin method for two-phase flow in porous media with discontinuous capillary pressure. arXiv:1309.7555v1 (2013)
- 18.Wheeler, M.F., Xue, G.: Accurate locally conservative discretizations for modeling multiphase flow in porous media on general hexahedra grids. ECMOR XII (2010)Google Scholar
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