Journal of Scientific Computing

, Volume 55, Issue 3, pp 688–717 | Cite as

Numerical Modeling of Degenerate Equations in Porous Media Flow

Degenerate Multiphase Flow Equations in Porous Media
  • Eduardo Abreu
  • Duilio Conceição


In this paper is introduced a new numerical formulation for solving degenerate nonlinear coupled convection dominated parabolic systems in problems of flow and transport in porous media by means of a mixed finite element and an operator splitting technique, which, in turn, is capable of simulating the flow of a distinct number of fluid phases in different porous media regions. This situation naturally occurs in practical applications, such as those in petroleum reservoir engineering and groundwater transport. To illustrate the modelling problem at hand, we consider a nonlinear three-phase porous media flow model in one- and two-space dimensions, which may lead to the existence of a simultaneous one-, two- and three-phase flow regions and therefore to a degenerate convection dominated parabolic system. Our numerical formulation can also be extended for the case of three space dimensions. As a consequence of the standard mixed finite element approach for this flow problem the resulting linear algebraic system is singular. By using an operator splitting combined with mixed finite element, and a decomposition of the domain into different flow regions, compatibility conditions are obtained to bypass the degeneracy in order to the degenerate convection dominated parabolic system of equations be numerically tractable without any mathematical trick to remove the singularity, i.e., no use of a parabolic regularization. Thus, by using this procedure, we were able to write the full nonlinear system in an appropriate way in order to obtain a nonsingular system for its numerical solution. The robustness of the proposed method is verified through a large set of high-resolution numerical experiments of nonlinear transport flow problems with degenerating diffusion conditions and by means of a numerical convergence study.


Degenerate convection-diffusion Operator splitting Mixed finite elements Finite volume central scheme Porous media Three-phase flow 



The authors wish to thank Professors Dan Marchesin and Jim Douglas Jr. for enlightening discussions during the preparation of this work. We would like to thank the anonymous reviewers for their thorough evaluation and highly appreciate the constructive recommendations for improving this manuscript.


  1. 1.
    Aarnes, J.E., Krogstad, S., Lie, K.-A.: A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. Multiscale Model. Simul. 5(2), 337–363 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aarnes, J.E., Hauge, V.L., Efendiev, Y.: Coarsening of three-dimensional structured and unstructured grids for subsurface flow. Adv. Water Resour. 30(11), 2177–2193 (2007) CrossRefGoogle Scholar
  3. 3.
    Aarnes, J.E., Krogstad, S., Lie, K.-A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12, 297–315 (2008) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Abreu, E., Furtado, F., Pereira, F.: On the numerical simulation of three-phase reservoir transport problem. Transp. Theory Stat. Phys. 33, 1–24 (2004) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abreu, E., Douglas, J., Furtado, F., Marchesin, D., Pereira, F.: Three-phase immiscible displacement in heterogeneous petroleum reservoirs. Math. Comput. Simul. 73, 2–20 (2006) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Abreu, E., Douglas, J., Furtado, F., Pereira, F.: Operator splitting based on physics for flow in porous media. Int. J. Comput. Sci. 2, 315–335 (2008) Google Scholar
  7. 7.
    Azevedo, A., Marchesin, D., Plohr, B.J., Zumbrun, K.: Capillary instability in models for three-phase flow. Z. Angew. Math. Phys. 53, 713–746 (2002) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Azevedo, A., Souza, A., Furtado, F., Marchesin, D., Plohr, B.: The solution by the wave curve method of three-phase flow in virgin reservoirs. Transp. Porous Media 83, 99–125 (2010) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Blunt, M.J., Thiele, M.R., Batycky, R.P.: A 3D field scale streamline-based reservoir simulator. SPE Reserv. Eng. 246–254 (1997) Google Scholar
  10. 10.
    Borges, M.R., Furtado, F., Pereira, F., Amaral Souto, H.P.: Scaling analysis for the tracer flow problem in self-similar permeability fields multiscale model. Multiscale Model. Simul. 7, 1130–1147 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Brooks, R.H., Corey, A.T.: Hydraulic properties of porous media. In: Hydrology Paper No. 3, pp. 1–27. Colorado State University, Fort Collins (1964) Google Scholar
  12. 12.
    Bürger, R., Coronel, A., Sepúlveda, M.: A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modelling sedimentation-consolidation processes. Math. Comput. 75, 91–112 (2006) MATHGoogle Scholar
  13. 13.
    Cavalli, F., Naldi, G., Puppo, G., Semplice, M.: High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Anal. 45, 2098–2119 (2007) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chavent, G., Jaffré, J.: Mathematical models and finite elements for reservoir simulation. In: Studies in Applied Mathematics, vol. 17. North-Holland, Amsterdam (1986) Google Scholar
  15. 15.
    Chen, Z., Ewing, R.E.: Comparison of various formulation of three-phase flow in porous media. J. Comput. Phys. 132, 362–373 (1997) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chertock, A., Doering, C.R., Kashdan, E., Kurganov, A.: A fast explicit operator splitting method for passive scalar advection. J. Sci. Comput. 45, 200–214 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Chrispella, J.C., Ervin, V.J., Jenkinsa, E.W.: A fractional step θ-method for convection-diffusion problems. J. Math. Anal. Appl. 333, 204–218 (2007) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Corey, A., Rathjens, C., Henderson, J., Wyllie, M.: Three-phase relative permeability. Trans. Am. Inst. Min. Metall. Eng. 207, 349–351 (1956) Google Scholar
  19. 19.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  20. 20.
    Dahle, H.K., Ewing, R.E., Russell, T.F.: Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation. Comput. Methods Appl. Mech. Eng. 122, 223–250 (1995) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    di Chiara Roupert, R., Chavent, G., Schafer, G.: Three-phase compressible flow in porous media: total differential compatible interpolation of relative permeabilities. J. Comput. Phys. 229, 4762–4780 (2010) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Doughty, C., Pruess, K.: Modeling supercritical carbon dioxide injection in heterogeneous porous media. Vadose Zone J. 3, 837–847 (2004) CrossRefGoogle Scholar
  23. 23.
    Douglas, J. Jr., Furtado, F., Pereira, F.: On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Comput. Geosci. 1, 155–190 (1997) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Douglas, J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19(5), 871–885 (1982) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Douglas, J., Pereira, F., Yeh, L.-M.: A locally conservative Eulerian-Lagrangian method for flow in a porous medium of a mixture of two components having different densities. In: Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics, vol. 552, pp. 138–155. Springer, Berlin (2000) CrossRefGoogle Scholar
  26. 26.
    Dria, D.E., Pope, G.A., Sepehrnoori, K.: Three-phase gas/oil/brine relative permeabilities measured under CO2 flooding conditions. Soc. Pet. Eng. J. 20184, 143–150 (1993) Google Scholar
  27. 27.
    Durlofsky, L.J.: A triangle based mixed finite element-finite volume technique for modeling two phase flow through porous media. J. Comput. Phys. 105, 252–266 (1993) MATHCrossRefGoogle Scholar
  28. 28.
    Durlofsky, L.J.: Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resour. Res. 30, 965–973 (1994) CrossRefGoogle Scholar
  29. 29.
    Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. Presented at 8th International Forum on Reservoir Simulation Iles Borromees, Italy, June 20–24 (2005) Google Scholar
  30. 30.
    Durlofsky, L.J., Jones, R.C., Milliken, W.J.: A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media. Adv. Water Resour. 20, 335–347 (1997) CrossRefGoogle Scholar
  31. 31.
    Fortin, M., Brezzi, F.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics. Springer, Berlin (1991) Google Scholar
  32. 32.
    Furtado, F., Pereira, F.: Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows. Comput. Geosci. 7, 115–135 (2003) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Gasda, S.E., Farthing, M.W., Kees, C.E., Miller, C.T.: Adaptive split-operator methods for modeling transport phenomena in porous medium systems. Adv. Water Resour. 34, 1268–1282 (2011) CrossRefGoogle Scholar
  34. 34.
    Gerritsen, M.G., Durlofsky, L.J.: Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid Mech. 37, 211–238 (2006) CrossRefGoogle Scholar
  35. 35.
    Glimm, J., Sharp, D.M.: Stochastic methods for the prediction of complex multiscale phenomena. Q. Appl. Math. 56, 741–765 (1998) MathSciNetMATHGoogle Scholar
  36. 36.
    Glimm, J., Sharp, D.M.: Prediction and the quantification of uncertainty. Physica D 133, 142–170 (1999) CrossRefGoogle Scholar
  37. 37.
    Hauge, V.L., Aarnes, J.E., Lie, K.A.: Operator splitting of advection and diffusion on non-uniformly coarsened grids. In: Proceedings of ECMOR XI-11th European Conference on the Mathematics of Oil Recovery, Bergen, Norway, 8–11 September (2008) Google Scholar
  38. 38.
    Hauge, V.L., Lie, K.-A., Natvig, J.R.: Flow-based grid coarsening for transport simulations. In: Proceedings of ECMOR XII-12th European Conference on the Mathematics of Oil Recovery (EAGE), Oxford, UK, 6–9 September (2010) Google Scholar
  39. 39.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Isaacson, E., Marchesin, D., Plohr, B., Temple, J.B.: Multiphase flow models with singular Riemann problems. Comput. Appl. Math. 11, 147–166 (1992) MathSciNetMATHGoogle Scholar
  42. 42.
    Karlsen, K.H., Risebro, N.H.: Corrected operator splitting for nonlinear parabolic equations. SIAM J. Numer. Anal. 37, 980–1003 (2000) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Karlsen, K.H., Lie, K.-A., Natvig, J.R., Nordhaug, H.F., Dahle, H.K.: Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. J. Comput. Phys. 173, 636–663 (2001) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Kippe, V., Aarnes, J.E., Lie, K.A.: A comparison of multiscale methods for elliptic problems in porous media flow. Comput. Geosci. 12(3), 377–398 (2008) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Kurganov, A., Petrova, G., Popov, B.: Adaptive semi-discrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29, 2381–2401 (2007) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Leverett, M.C.: Capillary behavior in porous solids. Trans. Soc. Pet. Eng. 142, 152–169 (1941) Google Scholar
  47. 47.
    Li, K.: More general capillary pressure and relative permeability models from fractal geometry. J. Contam. Hydrol. 111(1–4), 13–24 (2010) CrossRefGoogle Scholar
  48. 48.
    Marchesin, D., Plohr, B.: Wave structure in WAG recovery. Soc. Pet. Eng. J. 71314, 209–219 (2001) Google Scholar
  49. 49.
    Nessyahu, N., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Oleinik, O.: Discontinuous solutions of non-linear differential equations. Transl. Am. Math. Soc. 26(2), 95–172 (1963) MathSciNetGoogle Scholar
  51. 51.
    Pencheva, G., Thomas, S.G., Wheeler, M.F.: Mortar coupling of multiphase flow and reactive transport on non-matching grids. In: Eymard, R., Herard, J.M. (eds.) Finite Volumes for Complex Applications V (Problems and Perspectives), Aussois-France, October, vol. 5, pp. 135–143. Wiley, New York (2008) Google Scholar
  52. 52.
    Rossen, W.R., van Duijn, C.J.: Gravity segregation in steady-state horizontal flow in homogeneous reservoirs. J. Pet. Sci. Eng. 43, 99–111 (2004) CrossRefGoogle Scholar
  53. 53.
    Shi, J., Hu, C., Shu, C.-W.: A technique for treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002) MATHCrossRefGoogle Scholar
  54. 54.
    Stone, H.L.: Probability model for estimating three-phase relative permeability. J. Pet. Technol. 22, 214–218 (1970) Google Scholar
  55. 55.
    Titareva, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238–260 (2004) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Applied Mathematics, IMECCUniversity of Campinas (UNICAMP)CampinasBrazil
  2. 2.Department of MathematicsFederal University Rural of Rio de JaneiroSeropédicaBrazil

Personalised recommendations