Abstract
Immiscible two‐phase flow in porous media can be described by the fractional flow model. If capillary forces are neglected, then the saturation equation is a non‐linear hyperbolic conservation law, known as the Buckley–Leverett equation. This equation can be numerically solved by the method of Godunov, in which the saturation is computed from the solution of Riemann problems at cell interfaces. At a discontinuity of permeability this solution has to be constructed from two flux functions. In order to determine a unique solution an entropy inequality is needed. In this article an entropy inequality is derived from a regularisation procedure, where the physical capillary pressure term is added to the Buckley‐Leverett equation. This entropy inequality determines unique solutions of Riemann problems for all initial conditions. It leads to a simple recipe for the computation of interface fluxes for the method of Godunov.
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References
M.B. Allen III, G.A. Behie and J.A. Trangenstein, Multiphase Flow in Porous Media (Springer, Berlin, 1988).
A.K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers, London, 1979).
J. Bear, Hydraulics of Groundwater (McGraw-Hill, New York, 1979).
A.C. Berkenbosch, E.F. Kaasschieter and J.H.M. ten Thije Boonkkamp, Finite-difference methods for one-dimensional hyperbolic conservation laws, Numerical Methods for Partial Differential Equations 10 (1994) 225–269.
G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation (North-Holland, Amsterdam, 1986).
C.J. van Duijn, J. Molenaar and M.J. de Neef, The effects of capillary forces on immiscible two-phse flow in heterogeneous porous media, Transport in Porous Media 21 (1995) 71–93.
M.I.J. van Dijke, S.E.A.T.M. van der Zee and C.J. van Duijn, Multi-phase flow modelling of air sparging, Advances in Water Resources 18 (1995) 319–333.
M.Th. van Genuchten, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society 44 (1980) 892–898.
T. Gimse and N.H. Risebro, Riemann problems with a discontinuous flux function, in: Proceedings of the 3rd International Conference on Hyperbolic Problems, eds. B. Engquist and B. Gustafsson (Chartwell-Bratt, Uppsala, 1991) pp. 488–502.
T. Gimse and N.H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM Journal on Mathematical Analysis 23 (1992) 635–648.
E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws (Ellipses, Paris, 1991).
R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997).
C. Hirsch, Numerical Computation of Internal and External Flows (Wiley, Chichester, 1990).
J. Jaffré, Flux calculation at the interface between two rock types for two-phase flow in porous media, Transport in Porous Media 21 (1995) 195–207.
H.P. Langtangen, A. Tveito and R. Winther, Instability of Buckley-Leverett flow in a heterogeneous medium, Transport in Porous Media 9 (1992) 165–185.
R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1990).
J. Molenaar, Entropy conditions for heterogeneity induced shocks in two-phase flow problems, in: Mathematical Modelling of Flow Through Porous Media, eds. A.P. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelić (World Scientific, Singapore, 1995).
H.J. Morel-Seytoux, Two-phase flows in porous media, Advances in Hydroscience 9 (1973) 119–202.
W. Proskurowski, A note on solving the Buckley-Leverett equation in the presence of gravity, Journal of Computational Physics 41 (1981) 136–141.
E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer, Berlin, 1997).
M. Wangen, Vertical migration of hydrocarbons modelled with fractional flow theory, Geophysical Journal International 115 (1993) 109–131.
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Kaasschieter, E. Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium. Computational Geosciences 3, 23–48 (1999). https://doi.org/10.1023/A:1011574824970
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DOI: https://doi.org/10.1023/A:1011574824970