Computational Geosciences

, Volume 20, Issue 3, pp 421–435 | Cite as

Compositional flow modeling using a multi-point flux mixed finite element method

  • Gurpreet Singh
  • Mary F. Wheeler


We present a general compositional formulation using multi-point flux mixed finite element (MFMFE) method on general hexahedral grids. The mixed finite element framework allows for local mass conservation, accurate flux approximation, and a more general treatment of boundary conditions. The multi-point flux inherent in MFMFE scheme allows the usage of a full permeability tensor. The proposed formulation is an extension of single and two-phase flow formulations presented by Wheeler and Yotov, SIAM J. Numer. Anal. 44(5), 2082–2106 (35) with similar convergence properties. Furthermore, the formulation allows for black oil, single-phase and multi-phase incompressible, slightly and fully compressible flow models utilizing the same design for different fluid systems. An accurate treatment of diffusive/dispersive fluxes owing to additional velocity degrees of freedom is also presented. The applications areas of interest include gas flooding, CO 2 sequestration, contaminant removal, and groundwater remediation.


Compositional flow Multipoint flux mixed finite element method General hexahedral grids CO 2 enhanced oil recovery 


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  1. 1.
    Acs, G., Doleschall, S., Farkas, E.: General purpose compositional model. Old SPE J. 25(4), 543–553 (1985)Google Scholar
  2. 2.
    Baker, L.E., Pierce, A.C., Luks, K.D.: Gibbs energy analysis of phase equilibria. SPE J. 22(5), 731–742 (1982)CrossRefGoogle Scholar
  3. 3.
    Brezzi, F., Douglas, J., Duràn, R.Jr., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51(2), 237–250 (1987)CrossRefGoogle Scholar
  4. 4.
    Chang, Y.-B.: Development and application of an equation of state compositional simulator (1990)Google Scholar
  5. 5.
    Chen, C., Wang, Y., Li, G.: Closed-loop reservoir management on the brugge test case. Comput. Geosci. 14, 691–703 (2010)CrossRefGoogle Scholar
  6. 6.
    Coats, K.: An equation of state compositional model. Old SPE J. 20(5), 363–376 (1980)Google Scholar
  7. 7.
    Farkas, E.: Linearization techniques of reservoir-simulation equations: fully implicit cases. SPE J. 3(4) (1998)Google Scholar
  8. 8.
    Firoozabadi, A., Pan, H.: Fast and robust algorithm for compositional modeling: part i-stability analysis testing. SPE J. 7(1), 79–89 (2002)CrossRefGoogle Scholar
  9. 9.
    Fussell, L.T., Fussell, D.D.: An iterative technique for compositional reservoir models. SPE J. 19(4) (1979)Google Scholar
  10. 10.
    Fussell, L.T.: Technique for calculating multiphase equilibria. SPE J. 19(4), 203–210 (1979)CrossRefGoogle Scholar
  11. 11.
    Hajibeygi, H., Tchelepi, H.A.: Compositional multiscale finite-volume formulation. SPE J. 19(2) (2014)Google Scholar
  12. 12.
    Heidemann, R.A., Michelsen, M.L.: Instability of successive substitution. Ind. Eng. Chem. Res. 34(3), 958–966 (1995)Google Scholar
  13. 13.
    Ingram, R., Wheeler, M.F., Yotov, I.: A Multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48(4), 1281–1312 (2010)CrossRefGoogle Scholar
  14. 14.
    Kazemi, H., Vestal, C.R., Shank, D.G.: An efficient multicomponent numerical simulator. SPE J. 18(5) (1978)Google Scholar
  15. 15.
    Lauser, A., Hager, C., Helmig, R., Wohlmuth, B.: A new approach for phase transitions in miscible multi-phase flow in porous media. Adv. Water Resour. 34(8), 957–966 (2011)Google Scholar
  16. 16.
    Martinez, M.J., Stone, C.M.: Considerations for developing models of multiphase flow in deformable porous media. SANDIA REPORT, SAND2008-5887 (2008)Google Scholar
  17. 17.
    Mehra, R.K., Heidemann, R.A., Aziz, K.: Computation of multiphase equilibrium for compositional simulation. SPE J. 22(1), 61–62 (1982)Google Scholar
  18. 18.
    Michelsen, M.L.: The isothermal flash problem. part i. stability. Fluid Phase Equilib. 9(1), 1–19 (1982a)CrossRefGoogle Scholar
  19. 19.
    Michelsen, M.L.: The isothermal flash problem. part ii. phase-split calculation. Fluid Phase Equilib. 9(1), 21–40 (1982b)CrossRefGoogle Scholar
  20. 20.
    Michelsen, M.L.: Calculation of multiphase equilibrium. Comput. Chem. Eng. 18(7), 545–550 (1994)CrossRefGoogle Scholar
  21. 21.
    Nghiem, L.X., Fong, D.K., Aziz, K.: Compositional modeling with an equation of state. SPE J. 21(6) (1981)Google Scholar
  22. 22.
    Okuno, R., Johns, R.T., Sepehrnoori, K.: A new algorithm for rachford-rice for multiphase compositional simulation. SPE J. 15(2), 313–325 (June 2010)CrossRefGoogle Scholar
  23. 23.
    Pan, H., Firoozabadi, A.: Fast and robust algorithm for compositional modeling: part ii-two-phase flash computations. SPE J. 8(4), 380–391 (2003)Google Scholar
  24. 24.
    Peng, D.-Y., Robinson, D.B.: A new two-constant equation of state. Indust. Eng. Chem. Fund. 15(1), 59–64 (1976)CrossRefGoogle Scholar
  25. 25.
    Peters, E., Arts, R., Brouwer, G., Geel, C.: Results of the Brugge benchmark study for flooding optimisation and history matching. SPE 119094-MS. SPE Reservoir Simulation Symposium (2009)Google Scholar
  26. 26.
    Rachford, H.H., Rice, J.D.: Procedure for use of electronic digital computers in calculating flash vaporization hydrocarbon equilibrium. Trans. Am. Instit. Min. Metall. Eng. 195, 327–328 (1952)Google Scholar
  27. 27.
    Roebuck, I.F. Jr., Henderson, G.E., Douglas, J. Jr., Ford, W.T.: The compositional reservoir simulator: case I-the linear model. Old SPE J. 9(01), 115–130 (1969)Google Scholar
  28. 28.
    Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. Math. Reserv. Simul. 1, 35–106 (1983)CrossRefGoogle Scholar
  29. 29.
    Singh, G., Pencheva, G., Kumar, K., Wick, T., Ganis, B., Wheeler, M.F.: Impact of Accurate Fractured Reservoir Flow Modeling on Recovery Predictions. SPE Hydraulic Fracturing Technology Conference (2014)Google Scholar
  30. 30.
    Sun, S., Firoozabadi, A.: Compositional Modeling in Three-Phase Flow for CO2 and other Fluid Injections using Higher-Order Finite Element Methods. SPE Annual Technical Conference and Exhibition (2009)Google Scholar
  31. 31.
    Thomas, S.G.: On some problems in the simulation of flow and transport through porous media. PhD. Thesis (2009)Google Scholar
  32. 32.
    Watts, J.W.: A compositional formulation of the pressure and saturation equations. SPE Reserv. Eng. 1(3), 243–252 (1986)CrossRefGoogle Scholar
  33. 33.
    Wheeler, M.F., Xue, G.: Accurate locally conservative discretizations for modeling multiphase flow in porous media on general hexahedra grids. Proceedings of the 12th European Conference on the Mathematics of Oil Recovery-ECMOR XII, publisher EAGE (2011)Google Scholar
  34. 34.
    Wheeler, M., Xue, G., Yotov, I.: A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121(1), 165–204 (2011a)CrossRefGoogle Scholar
  35. 35.
    Wheeler, M.F., Yotov, I.: A Multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44(5), 2082–2106 (2006)CrossRefGoogle Scholar
  36. 36.
    Wheeler, M.F., Xue, G., Yotov, I.: A family of multipoint flux mixed finite element methods for elliptic problems on general grids. Proc. Comput. Sci. 4, 918–927 (2011b)CrossRefGoogle Scholar
  37. 37.
    Wilson, G.: A modified redlich-kwong equation of state applicable to general physical data calculations. Paper No15C, 65th AIChE National meeting (1968)Google Scholar
  38. 38.
    Wong, T.W., Firoozabadi, A., Aziz, K.: Relationship of the volume-balance method of compositional simulation to the Newton-Raphson method. SPE Reserv. Eng. J. 5(3) (1990)Google Scholar
  39. 39.
    Young, L., Stephenson, R: A generalized compositional approach for reservoir simulation. Old SPE J. 23 (5), 727–742 (1983)Google Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, POB 5.324The University of Texas at AustinAustinUSA

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