Energy inequalities in compositional simulation

Main Article Content

Jiří Mikyška Ondřej Polívka


We investigate the single-phase flow of a mixture composed of $n$ components in a porous medium under the influence of pressure gradients, viscosity, and gravity. For the case of the single-phase flow (i.e. assuming that under given conditions the phase splitting will not occur), we derive an energy inequality for the continuous problem. Then, we propose the fully implicit discretization of the transport problem which uses a combination of the mixed-hybrid finite element method for the velocity approximation and the finite volume method for the discretization of the transport equations. We prove that the proposed numerical scheme fulfills a discrete version of the energy inequality. A numerical experiment reveals exponential decay of the Helmholtz free energy when a system approaches the equilibrium state. 

Article Details

How to Cite
MIKYŠKA, Jiří; POLÍVKA, Ondřej. Energy inequalities in compositional simulation. Proceedings of the Conference Algoritmy, [S.l.], p. 224-233, feb. 2016. Available at: <>. Date accessed: 22 sep. 2017.


[1] G. Ács, S. Doleschall, E. ́ Farkas, General Purpose Compositional Model, SPE Journal 25(4) (1985), pp. 543–553 (SPE-10515-PA).

[2] J. Bear, A. Verruijt, Modeling Groundwater Flow and Pollution, D. Reidel Publishing Company, Dordrecht, Holland, 1987.

[3] J.A. Carrillo, A. Ju ̈ngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy Dissipation Methods for Degenerate Parabolic Problems and Generalized Sobolev Inequalities, Monatshefte f ̈ ur Mathematik, 133, 1–82, 2001.

[4] Z. Chen, Y. Ma, and G. Huan, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006.

[5] Z. Chen, R. E. Ewing From Single-Phase To Compositional Flow: Applicability Of Mixed Finite Elements, Transport in Porous Media, (1997), pp. 225–242.

[6] H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmond, Paris, 1856.

[7] A. Firoozabadi, Thermodynamics of Hydrocarbon Reservoirs, McGraw-Hill, NY (1998).

[8] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York Inc. (1991).

[9] T. Jindrová, J. Mikyška, Fast and Robust Algorithm for Calculation of Two-Phase Equilibria at Given Volume, Temperature, and Moles, Fluid Phase Equilibria, Vol. 353 (2013), pp. 101–114.

[10] T. Jindrová, J. Mikyška, General algorithm for multiphase equilibria calculation at givenvolume, temperature, and moles, Fluid Phase Equilibria, 393: 7–25, 2015.

[11] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge (2002).

[12] J. Lohrenz, B. G. Bray, C. R. Clark. Calculating Viscosities of Reservoir Fluids From Their Compositions, Journal of Petroleum Technology, Oct. (1964), pp. 1171–1176.

[13] J. Maryška, M. Rozložník, M. Tuma, Mixed-hybrid finite element approximation of the potential fluid flow problem, Journal of Computational and Applied Mathematics, 63 (1995), pp. 383–392.

[14] M. L. Michelsen, The Isothermal Flash Problem. 1. Stability, Fluid Phase Equilibria, 9(1) (1982), pp. 1–19.

[15] M. L. Michelsen, The Isothermal Flash Problem. 2. Phase Split Calculation, Fluid Phase Equilibria, 9(1) (1982), pp. 21–40.

[16] M.L. Michelsen, J.M. Mollerup, Thermodynamic Models: Fundamentals and Computational Aspects, Tie-Line Publications, 2004.

[17] J. Mikyška, A. Firoozabadi, A New Thermodynamic Function for Phase-Splitting at Constant Temperature, Moles, and Volume, AIChE Journal, 57(7) (2011), pp. 1897–1904.

[18] J. Mikyška, A. Firoozabadi, Investigation of Mixture Stability at Given Volume, Temperature, and Number of Moles, Fluid Phase Equilibria, Vol. 321 (2012), pp. 1–9.

[19] D. Y. Peng, D. B. Robinson, A New Two-Constant Equation of State, Industrial and Engineering Chemistry: Fundamentals 15 (1976), pp. 59–64.

[20] O. Polívka, J. Mikyška, Numerical simulation of multicomponent compressible flow in porous medium, Journal of Math-for-Industry Vol. 3 (2011C-7), (2011) pp. 53–60.

[21] O. Polívka, J. Mikyška, Compositional Modeling in Porous Media using Constant Volume Flash and Flux Computation without the Need for Phase Identification, Journal of Computational Physics, 272: 149–169, 2014.

[22] L. C. Young, R. E. Stephenson, A Generalized Compositional Approach for Reservoir Simulation, SPE Journal, 23(5) (1983), pp. 727–742.